In o ma ion Flux Theo y:
A Rein e p e a ion o he S anda d
Model wi h a Single Fe mion and he
O igin o G a i y
Yoshino i Shimizu
ORCID: 0009-0008-5135-7372
Sep embe 18 2025 ( 1.5b)
©2025 Yoshino i Shimizu •CC BY 4.0
DOI: 10.5281/zenodo.15399114
Abs ac
Backg ound: The S anda d Model (SM) has been success ul, ye i ails o
explain he o igin o e mion masses and mixing pa ame e s.
Me hods: In his s udy we cons uc he single- e mion amewo k “In o -
ma ion Flux Theo y (IFT),” de i ed om he Uni ied E olu ion Equa ion.
IFT p ese es gauge symme y while eplacing S anda d Model ields wi h
a single undamen al ope a o , yielding analy ic solu ions wi hou adjus able
pa ame e s.
Resul s: IFT ep oduces all SM pa icle masses—including he 125 GeV
Higgs mass—and he CKM ma ix wi hin cu en expe imen al p ecision, e-
qui ing nei he addi ional pa icles no ine- uning.
Conclusion: These esul s demons a e ha IFT can ully eplace he S an-
da d Model wi h a single- e mion desc ip ion, p o iding a concep ually simple
ye phenomenologically comple e ounda ion o pa icle physics.
Supplemen : This pape includes p oo s o wo Clay Millennium P oblems:
he Yang–Mills mass gap and he Na ie –S okes equa ions.
No e Added: Fu he mo e, as a esul o his se ies o s udies, he o igin
o g a i y has now been cla i ied.
1
Con en s
1 In oduc ion 21
1.1 S a us o he S anda d Model and Open Ques ions .......... 21
1.1.1 Achie emen s ........................... 21
1.1.2 Ou s anding P oblems ...................... 21
1.1.3 Posi ion o he P esen Wo k . . . . . . . . . . . . . . . . . . . 22
1.2 Concep ual Basis o In o ma ion Flux Theo y ............. 23
1.2.1 Co e Idea—A Single Fe mion and Sel -In o ma ion Flux . . . 23
1.2.2 Uni ied E olu ion Equa ion (UEE) . . . . . . . . . . . . . . . 23
1.2.3 Masses and Mixings om Minimal Deg ees o F eedom . . . . 23
1.2.4 Me hodological Ou line ...................... 23
1.3 Uni ied E olu ion Equa ion and Cons uc ion Me hod o he Single-
Fe mion F amewo k ............................ 24
1.3.1 Design P inciple—Coexis ence o Conse a ion and Dissipa ion 24
1.3.2 Minimal Building Blocks ..................... 24
1.3.3 Single Fe mion and P ojec o Se ies . . . . . . . . . . . . . . . 24
1.3.4 Cons uc ion Algo i hm (Ou line) . . . . . . . . . . . . . . . . 24
1.4 B idge o Chap e 2: In oduc ion o he Fi e-Ope a o Func ionally
Comple e Se ............................... 26
1.4.1 Posi ion and Pu pose ....................... 26
1.4.2 Fi e Ope a o s and Thei Roles . . . . . . . . . . . . . . . . . 26
1.4.3 Claim o Func ional Comple eness . . . . . . . . . . . . . . . . 26
1.4.4 S uc u e and Roadmap o Chap e 2 . . . . . . . . . . . . . . 26
1.4.5 Links o Subsequen Chap e s . . . . . . . . . . . . . . . . . . 27
1.4.6 Summa y ............................. 27
2 Fi e Ope a o s and he Canonical Decomposi ion Theo em (Func-
ional Comple eness) 28
2.1 S a emen o he Theo em and P oo S a egy ............. 28
2.1.1 In oduc ion and No a ional Con en ions [1, 2, 3] ....... 28
2.1.2 Theo em 2.1 — Canonical Decomposi ion Theo em and Φ
Gene a ing Map Theo em [4, 5] . . . . . . . . . . . . . . . . . 28
2.1.3 O e iew o he P oo S a egy [6, 7] . . . . . . . . . . . . . . 29
2.2 Ma hema ical P elimina ies: C*-Algeb as, CPTP Semig oups, and
Te ad No maliza ion ........................... 31
2.2.1 Basics o C*-Algeb as and GNS Rep esen a ion [8, 9, 10] . . . 31
2.2.2 Comple ely Posi i e T ace-P ese ing Maps and he K aus
Rep esen a ion [11, 12, 13, 14] . . . . . . . . . . . . . . . . . . 31
2.2.3 GKLS Gene a o s and Quan um Dynamical Semig oups [15,
16, 17, 18] ............................. 32
2.2.4 Fou -G adien –No malised Scala s and Te ad Cons uc ion . 32
2.2.5 Conclusion and B idge o Subsequen Sec ions ......... 32
2.3 No maliza ion o he Mas e Scala Φand he Gene a ing Map . . . . 33
2.3.1 No maliza ion Condi ion and Phase Deg ees o F eedom [19, 20] 33
2.3.2 Mapping om Φ o he Te ad [21, 22] ............. 33
2.3.3 Cons uc ion o he ΦGene a ing Map [23, 24] ......... 33
2
2.3.4 In e ibili y o he Gene a ing Map [25] ............. 34
2.3.5 Conclusion ............................. 34
2.4 Canonical Fo m o he Re e sible Gene a o D=GD[Φ] ........ 35
2.4.1 De ini ion and Assump ions [26] . . . . . . . . . . . . . . . . . 35
2.4.2 Gene al Candida e and he Sel -Adjoin ness Condi ion [27] . . 35
2.4.3 Requi emen o Local Lo en z Co a iance [19] ......... 35
2.4.4 βD= 0 Fixed Poin [28, 29] . . . . . . . . . . . . . . . . . . . 36
2.4.5 Canonical-Fo m Theo em . . . . . . . . . . . . . . . . . . . . 36
2.4.6 Conclusion ............................. 36
2.5 Poin e P ojec o Family Πn=GΠ[Φ] and Minimali y ......... 37
2.5.1 De ini ion o he P ojec o Family and he In e nal Hilbe
Space [30, 31] ........................... 37
2.5.2 Ve i ica ion o O hogonali y and Comple eness [32, 33] . . . . 37
2.5.3 Minimali y Theo em [34] ..................... 37
2.5.4 Gene a ing Map GΠ om Φ[30] . . . . . . . . . . . . . . . . . 38
2.5.5 Uniqueness up o P ojec o Equi alence ............ 38
2.5.6 Conclusion ............................. 38
2.6 Jump Ope a o s Vn=√γΠnand Canonical Dissipa ion ........ 39
2.6.1 De ini ion o he Jump Ope a o s [16, 15] ............ 39
2.6.2 Rank Analysis o he GKLS Gene a o [12, 35] ......... 39
2.6.3 Redundancy o Phase F eedom [36] . . . . . . . . . . . . . . . 39
2.6.4 Canonical Dissipa ion Theo em . . . . . . . . . . . . . . . . . 40
2.6.5 Uni e sali y o he Decohe ence Time [17] ........... 40
2.6.6 Conclusion ............................. 40
2.7 Ze o-A ea Resonance Ke nel R=GR[Φ] . . . . . . . . . . . . . . . . . 41
2.7.1 De ini ion and Fou Requi emen s . . . . . . . . . . . . . . . . 41
2.7.2 F edholm Cons uc ion and Ze o-A ea Limi [37, 38] ..... 41
2.7.3 Sel -Adjoin ness, In o ma ion P ese a ion, and Vacuum S a-
bilisa ion .............................. 41
2.7.4 Uniqueness Theo em ....................... 42
2.7.5 In e ibili y o he Gene a ion Map . . . . . . . . . . . . . . . 42
2.7.6 Conclusion ............................. 42
2.8 Func ional Independence o he Fi e Ope a o s and he Func ional
Comple eness Se ............................. 43
2.8.1 Func ional Ma ix o he Fi e Ope a o s [2] .......... 43
2.8.2 Independence Lemma [35, 34] . . . . . . . . . . . . . . . . . . 43
2.8.3 Ve i ica ion by Remo al Expe imen s . . . . . . . . . . . . . . 43
2.8.4 Func ional Comple eness Theo em . . . . . . . . . . . . . . . 44
2.8.5 Conclusion ............................. 44
2.9 Summa y o Chap e 2 and Connec ion o he Nex Chap e ..... 45
2.9.1 Key Poin s Es ablished in This Chap e . . . . . . . . . . . . 45
2.9.2 Logical B idge o Chap e 3—P epa a ion o he Th ee-Fo m
Equi alence Theo em ....................... 45
2.9.3 Guidelines o he Reade . . . . . . . . . . . . . . . . . . . . 46
2.9.4 Fac s Con i med He e ....................... 46
3
3 Uni ied E olu ion Equa ion and Th ee-Fo m Equi alence 47
3.1 S a emen o he Theo em and P oo S a egy ............. 47
3.1.1 De ini ion o he Th ee Fo ms [16, 15, 39, 40, 41] ....... 47
3.1.2 S a emen o he Equi alence Theo em [42, 19] ......... 47
3.1.3 Roadmap o he P oo S a egy [12, 43, 44, 45] ......... 47
3.1.4 Conclusion ............................. 48
3.2 De i a ion o he Ope a o Fo m UEEop . . . . . . . . . . . . . . . . 49
3.2.1 Recap o he Fi e Ope a o s and Basic S uc u e [46, 47] . . . 49
3.2.2 De i a ion o he Dissipa o [15, 16, 48] ............. 49
3.2.3 Ac ion Fo m o he Ze o-A ea Ke nel R[37, 49] ........ 49
3.2.4 Final Fo m o he Ope a o UEE [17] . . . . . . . . . . . . . . 49
3.2.5 Conclusion ............................. 50
3.3 De i a ion o he Va ia ional Fo m UEE a . . . . . . . . . . . . . . . 51
3.3.1 Field a iables and design guidelines o he ac ion [42, 50] . . 51
3.3.2 Cons uc ion o he ac ion [51, 24] . . . . . . . . . . . . . . . . 51
3.3.3 Va ia ion and Eule –Lag ange equa ions [52] .......... 51
3.3.4 De i a ion o conse ed quan i ies [53] ............. 52
3.3.5 Fixing he a ia ional o m UEE a [54] ............. 52
3.3.6 Conclusion ............................. 52
3.4 De i a ion o he Field-Equa ion Fo m UEE ld ............. 53
3.4.1 Φ- e ad and ea angemen o he e ec i e ac ion [55, 56] . . 53
3.4.2 Me ic a ia ion: g a i a ional ield equa ion [40, 19] ..... 53
3.4.3 Spino a ia ion: e mionic equa ion [57] ............ 53
3.4.4 Va ia ion o Φ: scala equa ion [58] . . . . . . . . . . . . . . . 54
3.4.5 Collec ing he ield-equa ion o m [42] . . . . . . . . . . . . . . 54
3.4.6 Conclusion ............................. 54
3.5 P oo o Equi alence UEEop ⇒UEE a . . . . . . . . . . . . . . . . . . 55
3.5.1 De ini ion o he gene a ing unc ional [59, 60] ......... 55
3.5.2 Lemma 1: GNS ep esen a ion and pa h-in eg a ion [46, 61] . 55
3.5.3 Lemma 2: S a ono ich ans o ma ion o he dissipa o [62, 63] 55
3.5.4 Lemma 3: Func ional educ ion o he ze o-a ea low e m [12] 56
3.5.5 Equi alence lemma [5] ...................... 56
3.5.6 Conclusion ............................. 56
3.6 P oo o Equi alence UEE a ⇒UEE ld . . . . . . . . . . . . . . . . . 57
3.6.1 P emise and Aim o he Va ia ional Fo m [50] ......... 57
3.6.2 Lemma 1: Te ad Va ia ion and Reco e y o Eins ein–Hilbe
Dynamics [64, 51] ......................... 57
3.6.3 Lemma 2: S ess Tenso o he Dissipa i e Func ional [48] . . 57
3.6.4 Lemma 3: T ace o he Ze o-A ea Te m [37] .......... 57
3.6.5 P oo o he Equi alence Theo em [65] ............. 58
3.6.6 Conclusion ............................. 58
3.7 Bidi ec ional In e ibili y: Ope a o Fo m ⇔Field-Equa ion Fo m . . 59
3.7.1 P epa a ions o he Wigne –Weyl T ans o m [44, 45, 66] . . . 59
3.7.2 Lemma 1: Re e sible Gene a o and Poisson S uc u e [67] . . 59
3.7.3 Lemma 2: Weyl Symbol o he Dissipa i e Ke nel [68] ..... 59
3.7.4 Lemma 3: Symbol Map o he Ze o-A ea Ke nel [69] ..... 60
3.7.5 Equi alence Theo em [70] . . . . . . . . . . . . . . . . . . . . 60
3.7.6 Conclusion ............................. 60
4
3.8 Exis ence-and-Uniqueness Theo em . . . . . . . . . . . . . . . . . . . 61
3.8.1 Func ional-analy ic amewo k [71, 72] ............. 61
3.8.2 Lemma 1: local Lipschi z con inui y [73] ............ 61
3.8.3 Lemma 2: global boundedness ia dissipa ion [74] ....... 61
3.8.4 Local-solu ion exis ence [75] . . . . . . . . . . . . . . . . . . . 61
3.8.5 Ex ension o global solu ions [4] . . . . . . . . . . . . . . . . . 62
3.8.6 Exis ence-and-uniqueness heo em [6] . . . . . . . . . . . . . . 62
3.8.7 Conclusion ............................. 62
3.9 Conse ed Quan i ies and En opy P oduc ion ............. 63
3.9.1 Conse a ion o Ene gy and Cha ge [53, 67] .......... 63
3.9.2 on Neumann en opy and dissipa ion [48, 76] ......... 63
3.9.3 Uni e sal o m o he en opy-p oduc ion a e [77] ....... 64
3.9.4 Consis ency ac oss he h ee o ms [5] . . . . . . . . . . . . . . 64
3.9.5 Conclusion ............................. 64
3.10 Summa y and B idge o he Subsequen Chap e s ........... 65
3.10.1 Achie emen s and Signi icance o he Th ee-Fo m Equi alence 65
3.10.2 In e -Chap e Mapping: Which Fo m o Use? ......... 65
3.10.3 Logical Roadmap Going Fo wa d . . . . . . . . . . . . . . . . 65
3.10.4 Theo e ical and P ac ical Ad an ages . . . . . . . . . . . . . . 66
3.10.5 Conclusion ............................. 66
4 Real Hilbe Space and P ojec ion Decomposi ion 67
4.1 In oduc ion and Domain Se ing . . . . . . . . . . . . . . . . . . . . 67
4.1.1 Aims and Posi ion o This Chap e [78, 41, 79] ......... 67
4.1.2 De ini ion o he Real Hilbe Space [80, 81, 6] ......... 67
4.1.3 In oduc ion o a Fini e-Dimensional In e nal Space and Sep-
a a ed Rep esen a ion [42, 26, 82] . . . . . . . . . . . . . . . . 67
4.1.4 No a ion Adop ed in This Chap e [2, 83] ........... 68
4.1.5 Conclusion ............................. 68
4.2 Sepa abili y Theo em o he Real Hilbe Space ............ 69
4.2.1 Conc e e Model o he Real Space [65, 10] ........... 69
4.2.2 Basic Lemma: Densi y o Bounded Compac -Suppo Func-
ions [84, 85] ........................... 69
4.2.3 Sepa abili y Theo em [81, 86] . . . . . . . . . . . . . . . . . . 69
4.2.4 Rema k on Comple eness [81, 6] . . . . . . . . . . . . . . . . . 69
4.2.5 Conclusion ............................. 70
4.3 Complexi ica ion and C∗-Algeb a Rep esen a ion ........... 71
4.3.1 Rigo ous De ini ion o he Complexi ica ion [8, 87] ....... 71
4.3.2 Bounded-Ope a o Algeb a and he C∗No m [88, 46] ..... 71
4.3.3 Co espondence be ween Real and Complex Ope a o s [89, 6] 71
4.3.4 GNS Rep esen a ion o a C∗Algeb a [90, 91] .......... 72
4.3.5 Inclusion o he Real Ope a o Algeb a in o a C∗Algeb a [10, 8] 72
4.3.6 Conclusion ............................. 72
4.4 Cons uc ion o he P ojec ion Family: G am–Schmid 18-Basis . . . 73
4.4.1 Tenso -P oduc Space o In e nal Deg ees o F eedom [92, 93] 73
4.4.2 G am–Schmid O hono mal Basis [94, 95] ........... 73
4.4.3 De ini ion o One-Dimensional P ojec ions [78, 96] ....... 73
4.4.4 Tenso P ojec ion wi h he Ex e nal Space [97, 98] . . . . . . 74
5
4.4.5 Physical Labels o he P ojec ion Family [42, 2] ........ 74
4.4.6 Conclusion ............................. 74
4.5 O hogonali y and Comple eness Theo em o he P ojec ion Family . 75
4.5.1 Recap o he De ini ion [96, 99] . . . . . . . . . . . . . . . . . 75
4.5.2 Rigo ous P oo o O hogonali y [100] . . . . . . . . . . . . . . 75
4.5.3 Rigo ous P oo o Comple eness [101, 80] ............ 75
4.5.4 Uniqueness o he Minimal Comple e P ojec ion Family [102,
103] ................................ 75
4.5.5 Conclusion ............................. 76
4.6 Mapping om he Real O hogonal Basis o he Poin e Basis . . . . 77
4.6.1 Complex Ex ension o he Real O hogonal Basis [10] ..... 77
4.6.2 In e nal Obse able De ining he Poin e Basis [104, 30] . . . . 77
4.6.3 Uni a y Map om he Real Basis o he Poin e Basis [105, 3] 77
4.6.4 Poin e Expansion and Phase F eedom [106, 107] ....... 78
4.6.5 Conclusion ............................. 78
4.7 Spec al Theo em and Uniqueness o he P ojec ion Decomposi ion . 79
4.7.1 Scope o he Spec al Theo em [103, 108] ............ 79
4.7.2 Uniqueness Lemma o he Spec al Measu e [109] ....... 79
4.7.3 Uniqueness o he P ojec ion ia Uni a y Equi alence [110] . . 79
4.7.4 Implica ions o he Poin e Hamil onian [111, 3] ........ 80
4.7.5 Conclusion ............................. 80
4.8 Physical Co espondence o he 18-Dimensional In e nal Space . . . . 81
4.8.1 P ojec ion Labels and S anda d-Model Fe mions [92, 42] . . . 81
4.8.2 In e nal Rep esen a ion o he Cha ge Ope a o [112, 113] . . 81
4.8.3 Co espondence Be ween Labels and Gauge G oup [114, 26] . 81
4.8.4 Physical P ojec ion Theo em [115, 116] ............. 82
4.8.5 Conclusion ............................. 82
4.9 Conclusion and B idge o Chap e 5 . . . . . . . . . . . . . . . . . . . 83
4.9.1 Conclusion ............................. 84
5 Measu emen and Dissipa i e Diagonalisa ion o he Bo n Rule 85
5.1 In oduc ion and P oblem Se ing . . . . . . . . . . . . . . . . . . . . 85
5.1.1 Objec i es o This Chap e [78, 79, 30] ............. 85
5.1.2 Di e ence om he Con en ional Measu emen Pos ula es [117,
100, 118] .............................. 85
5.1.3 No a ion and Wo king Assump ions [15, 119, 17] ........ 85
5.1.4 Conclusion ............................. 86
5.2 Dissipa i e Jump Ope a o s and Ins an aneous Diagonalisa ion . . . . 87
5.2.1 Fo mal Solu ion o he Dissipa i e Semig oup [119, 15, 120] . . 87
5.2.2 Exponen ial Decay o O -Diagonal Te ms [121, 111, 3] . . . . 87
5.2.3 Theo em o Ins an aneous Diagonalisa ion [48, 122] . . . . . . 87
5.2.4 Physical Meaning—The P e-measu emen S a e [104, 123, 124] 88
5.2.5 Conclusion ............................. 88
5.3 De i a ion o he Bo n Rule ....................... 89
5.3.1 S a e Desc ip ion Be o e and A e Measu emen [96, 125] . . 89
5.3.2 P oo o he P obabili y Law [100, 126, 127] .......... 89
5.3.3 Pos -Measu emen S a e (Lüde s Upda e) [96, 128] ...... 89
5.3.4 Reco e y o Expec a ion Values [129, 130] ........... 89
6
5.3.5 Conclusion ............................. 90
5.4 Dissipa i e Time-scale and Decohe ence . . . . . . . . . . . . . . . . . 91
5.4.1 Time E olu ion o he O -Diagonal Fideli y [111, 3] ...... 91
5.4.2 De ini ion o he Decohe ence Time [121, 17] .......... 91
5.4.3 Di e ging En opy and he Spohn Inequali y [48, 131] ..... 91
5.4.4 Physical Model o he Pa ame e γ[132, 133] ......... 91
5.4.5 Illus a i e Expe imen al Values [123, 134, 135] ........ 92
5.4.6 Conclusion ............................. 92
5.5 Quan um-Zeno E ec and he Con inuous-Measu emen Limi . . . . 93
5.5.1 Se -up o he Disc e e-Measu emen P o ocol [136, 137] . . . . 93
5.5.2 Zeno Con ac ion Lemma [138, 139] . . . . . . . . . . . . . . . 93
5.5.3 Con inuous-Measu emen Limi [140, 141] ........... 93
5.5.4 Implica ions o Measu able Quan i ies [137, 142] ....... 94
5.5.5 Conclusion ............................. 94
5.6 En anglemen Gene a ion and Measu emen Back-Ac ion ....... 95
5.6.1 Measu emen -appa a us model [78, 143] ............ 95
5.6.2 En anglemen –gene a ion lemma [144] ............. 95
5.6.3 Measu emen back-ac ion and he Lüde s upda e [145, 130] . . 95
5.6.4 Consis ency wi h dissipa i e diagonalisa ion [104, 146] . . . . 96
5.6.5 En anglemen en opy [147, 148] . . . . . . . . . . . . . . . . . 96
5.6.6 Conclusion ............................. 96
5.7 Ex ension o Gene al POVMs ...................... 97
5.7.1 Cons uc ion p inciple o POVM elemen s [11, 12] ...... 97
5.7.2 Comple eness and posi i i y [101, 129] . . . . . . . . . . . . . 97
5.7.3 Choice o K aus ope a o s [11, 149] . . . . . . . . . . . . . . . 97
5.7.4 Measu emen p obabili ies and Lüde s upda e [96, 128] . . . . 97
5.7.5 In o ma ion– heo e ic implica ions [150, 151] .......... 98
5.7.6 Conclusion ............................. 98
5.8 Summa y and B idge o Chap e 6 . . . . . . . . . . . . . . . . . . . 99
6 En anglemen , The malisa ion, and he Quan um Zeno E ec 101
6.1 In oduc ion and Scope ..........................101
6.1.1 Aims o his chap e [30, 3, 48] . . . . . . . . . . . . . . . . . 101
6.1.2 De ini ions o he ele an ime scales [132, 17] .........101
6.1.3 A ea law and he poin e basis [152, 153, 154, 111] ......101
6.1.4 Me hodological ools employed in his chap e [155, 15, 156, 157]102
6.2 En anglemen S uc u e o he Poin e -Diagonal S a e .........103
6.2.1 Fo m o he poin e -diagonal s a e [30, 121] ..........103
6.2.2 De ini ion o he en anglemen en opy [158, 2] .........103
6.2.3 Clus e ing lemma [159, 160] . . . . . . . . . . . . . . . . . . . 103
6.2.4 A ea-law heo em [152, 153, 154] . . . . . . . . . . . . . . . . 103
6.2.5 Physical meaning o he cons an κ[161, 37] ..........104
6.3 Spohn’s Inequali y and he The malisa ion Theo em ..........105
6.3.1 Recap o Spohn’s inequali y [48, 15] . . . . . . . . . . . . . . . 105
6.3.2 Mono onici y o he ela i e en opy [162, 163] .........105
6.3.3 The malisa ion heo em [164, 165, 166] . . . . . . . . . . . . . 105
6.3.4 The malisa ion ime and he en opy-p oduc ion a e [132, 17] 106
6.4 E alua ion o he The malisa ion Time Scale . . . . . . . . . . . . . . 107
7
6.4.1 Sys em–en i onmen in e ac ion model [132, 133] . . . . . . . 107
6.4.2 Bo n–Ma ko educ ion and he dissipa ion a e [16, 17] . . . 107
6.4.3 E ec i e dissipa ion a e and he malisa ion ime [167, 133] . 107
6.4.4 Scaling in |g|and J(0) ......................108
6.4.5 Examples: cold a oms s. solids [168, 169] ...........108
6.5 The malisa ion Supp ession ia he Quan um–Zeno E ec ......109
6.5.1 Con inuous measu emen and he e ec i e gene a o [136, 139,
170] ................................109
6.5.2 Supp ession a e o en opy p oduc ion [171, 130] .......109
6.5.3 The malisa ion–supp ession heo em [172, 139] .........109
6.5.4 Phase diag am: he malisa ion s. Zeno [151, 173] .......110
6.6 En anglemen Veloci y and he Lieb–Robinson Bound .........111
6.6.1 La ice pa i ion and dis ance unc ion [174, 157] .......111
6.6.2 Ope a ional o m o he Lieb–Robinson bound [156, 175] . . . 111
6.6.3 Uppe bound on en anglemen g ow h [174, 154] ........111
6.6.4 Theo em excluding iola ions o he a ea law [159, 176] . . . . 112
6.7 Decohe ence s. The malisa ion Phase Diag am ............113
6.7.1 Pa ame e s o he phase diag am [177, 178] ...........113
6.7.2 Bo de lines and ansi ion c i e ia [179, 180] ..........113
6.7.3 Phase classi ica ion and physical pic u e [181, 182] . . . . . . . 113
6.7.4 Mapping expe imen al pa ame e s [168, 183] ..........114
6.8 Conclusion and B idge o Chap e 7 . . . . . . . . . . . . . . . . . . . 115
6.8.1 Achie emen s o his chap e . . . . . . . . . . . . . . . . . . . 115
6.8.2 Di ec connec ion o he β- unc ion analysis ..........115
6.8.3 Conclusion .............................116
7 Sca e ing Theo y and he βFunc ion 117
7.1 In oduc ion and No a ion Con en ions . . . . . . . . . . . . . . . . . 117
7.1.1 Goal o he chap e and he “p ojec ed ex e nal–leg” p og amme
[184, 185, 186, 187] ........................117
7.1.2 No a ion con en ions [26, 188, 2] . . . . . . . . . . . . . . . . 117
7.1.3 Scheme o he heo ems p o ed in his chap e [189, 190, 191,
29, 192] ..............................118
7.1.4 Conclusion .............................118
7.2 Ex e nal–leg P esc ip ion wi h he Poin e Basis ............119
7.2.1 Cons uc ion o poin e p ojec o s and one–pa icle s a es [193,
3, 91] ................................119
7.2.2 Commu a i i y o poin e p ojec o s and ield ope a o s [194,
186] ................................119
7.2.3 Poin e –LSZ painless ex apola ion o mula [184, 195] . . . . . 119
7.2.4 O hogonal decomposi ion o he poin e M-ma ix [187, 192] 120
7.2.5 Conclusion .............................120
7.3 Expansion Theo em o Sca e ing Ampli udes . . . . . . . . . . . . . 121
7.3.1 Φ–loop index and o de coun ing [196, 197, 198] ........121
7.3.2 Connec ed expansion and ecu sion o he Mma ix [189,
199, 200] ..............................121
7.3.3 Fini e expansion heo em o he sca e ing ampli ude [201, 202]121
7.3.4 Example: 2→2sca e ing [203, 26] . . . . . . . . . . . . . . . 122
8
12.5.3 G ow h Func ion D(a)and σ8[364, 401] ............210
12.5.4 Conclusion .............................211
12.6 Analy ic Benchma k agains ΛCDM . . . . . . . . . . . . . . . . . . . 212
12.6.1 Indica o o he Numbe o F ee Pa ame e s [402, 403, 404] . 212
12.6.2 App oxima e χ2 ia Pull Values [268] . . . . . . . . . . . . . . 212
12.6.3 App oxima e AIC/BIC Sco es [405] . . . . . . . . . . . . . . . 212
12.6.4 Na u alness (Fine-Tuning) Compa ison [406, 407] .......213
12.6.5 Conclusion .............................213
12.7 Conclusion and B idge o Chap e 13 . . . . . . . . . . . . . . . . . . 214
12.7.1 Summa y o This Chap e ’s Resul s . . . . . . . . . . . . . . . 214
12.7.2 Physical Signi icance .......................214
12.7.3 B idge o Chap e 13 . . . . . . . . . . . . . . . . . . . . . . . 214
12.7.4 Conclusion .............................215
13 Resolu ion o he Black-Hole In o ma ion P oblem 216
13.1 In oduc ion and P oblem Se ing . . . . . . . . . . . . . . . . . . . . 216
13.1.1 Single- e mion UEE and he BH in o ma ion p oblem [408, 38,
37, 49, 409, 410] ..........................216
13.1.2 The ou p oblems add essed in his chap e [37, 411, 412, 413] 216
13.1.3 Chap e ou line ..........................216
13.1.4 In e ace o Chap e 14 ......................217
13.1.5 Conclusion .............................217
13.2 A ea–exponen ial con e gence heo em o he R-a ea ke nel ( e isi ed)218
13.2.1 De ini ion o he R-a ea ke nel and BH ime pa ame e [161,
408, 19] ..............................218
13.2.2 Flux equa ion o he R-ke nel [414, 415] ............218
13.2.3 Auxilia y lemma: exponen ial solu ion [416, 417] ........218
13.2.4 A ea–exponen ial con e gence heo em (s ong o m) [19, 418] 219
13.2.5 Physical consequence and connec ion o he Page cu e [37, 419]219
13.2.6 Conclusion .............................219
13.3 Hilbe -space decomposi ion and he en opy ope a o .........220
13.3.1 Hilbe -space spli ing by poin e p ojec ion [104, 30] . . . . . 220
13.3.2 Cons uc ion o he educed densi y ope a o [78, 420] . . . . . 220
13.3.3 En opy ope a o and i s -o de expansion [153, 152] . . . . . 220
13.3.4 En opy p oduc ion a e and he Page condi ion [37, 421] . . . 221
13.3.5 Conclusion .............................221
13.4 Analy ic de i a ion o he Page ime and he in o ma ion- elease a e 222
13.4.1 A ea dec ease a e and he e apo a ion ime scale [408, 422] . 222
13.4.2 Time dependence o he adia ed en opy [37, 423] .......222
13.4.3 Analy ic exp ession o he Page ime [37, 419] .........222
13.4.4 Closed- o m Page cu e [424, 425] . . . . . . . . . . . . . . . . 223
13.4.5 Conclusion .............................223
13.5 Ope a o p oo o he island o mula . . . . . . . . . . . . . . . . . . 224
13.5.1 P epa a ion o he eplica–poin e cons uc ion [426, 427] . . . 224
13.5.2 Replica ick wi h an inse ed R–a ea ke nel [427, 428] . . . . 224
13.5.3 Ex emal-su ace equa ion and he eme gence o islands [424,
419] ................................224
13.5.4 Ope a o heo em o he island o mula [429, 430] .......225
15
13.5.5 Conclusion .............................225
13.6 Comple e-Uni a i y Theo em and In o ma ion Reco e y ........226
13.6.1 De ini ion o he global ime-e olu ion ope a o [431, 432] . . . 226
13.6.2 Asymp o ic anishing o he adia ion en opy [433, 434] . . . 226
13.6.3 In o ma ion-p ese a ion heo em [435, 410] ..........226
13.6.4 Lemma on he absence o a i ewall [409, 436] .........227
13.6.5 Conclusion .............................227
13.7 Obse a ional Signa u es and Tes abili y . . . . . . . . . . . . . . . . 228
13.7.1 Theo e ical alue o he Hawking- empe a u e d i [422, 437] 228
13.7.2 Analy ic p edic ion o echo ime delay [438, 439] ........228
13.7.3 Impac on g a i a ional-wa e ing-down [440, 441] .......228
13.7.4 Expe imen al de ec abili y [442, 443] . . . . . . . . . . . . . . 229
13.7.5 Conclusion .............................229
13.8 Conclusion and B idge o Chap e 14 . . . . . . . . . . . . . . . . . . 230
13.8.1 Summa y o he esul s ob ained in his chap e ........230
13.8.2 Physical signi icance .......................230
13.8.3 B idge o Chap e 14 . . . . . . . . . . . . . . . . . . . . . . . 230
13.8.4 Conclusion .............................231
14 Summa y o he In o ma ion-Flux Theo y wi h a Single Fe mion 232
14.1 In oduc ion and O e iew o Achie emen s . . . . . . . . . . . . . . 232
14.1.1 Aim o his s udy and he i e-ope a o amewo k .......232
14.1.2 Essence o he main heo ems by chap e ............232
14.1.3 Conclusion .............................233
14.2 Uni ica ion o P inciples: P oo o Closu e o he Fi e-Ope a o
Comple e Se ...............................234
14.2.1 The i e ope a o s and he gene a ed ∗-algeb a [104, 30, 2] . . 234
14.2.2 Basic ela ions among he gene a o s [78, 100, 444] ......234
14.2.3 P oo o comple eness (sepa a ing) [5, 445] ...........234
14.2.4 Closu e heo em [446, 447] . . . . . . . . . . . . . . . . . . . . 235
14.2.5 Conclusion .............................235
14.3 Final Table o Physical Cons an s . . . . . . . . . . . . . . . . . . . . 236
14.3.1 O e iew o he Fixed Equa ion Sys em and he Simul aneous
Solu ion [448, 449, 312] ......................236
14.3.2 Lis o Final De e mined Cons an s . . . . . . . . . . . . . . . 236
14.3.3 E o Budge Analysis ......................237
14.3.4 C oss-Consis ency Check . . . . . . . . . . . . . . . . . . . . . 237
14.3.5 Conclusion .............................237
14.4 Final De e mina ion o he P o isional ϵCKM Cons an ........238
14.4.1 Se up o he One-Loop E ec i e Ac ion o Φ[450, 26, 451] . . 238
14.4.2 Cu o by he Ze o-A ea Ke nel [452, 197] ...........238
14.4.3 E alua ion o he Coe icien αΦ[453, 454, 455] ........238
14.4.4 Subs i u ion o he Final Tension Value [456] ..........239
14.4.5 Fi s -P inciples Calcula ion o ϵ[224, 219] ...........239
14.4.6 Ve i ica ion agains he Fi ed Value . . . . . . . . . . . . . . 239
14.4.7 Conclusion (De ailed Ve sion) . . . . . . . . . . . . . . . . . . 239
14.5 C oss-Disciplina y Feedback Summa y . . . . . . . . . . . . . . . . . 240
16
14.5.1 Elec oweak Scale: Quan i a i e Res o a ion o Na u alness
[406, 457, 458, 219, 459] . . . . . . . . . . . . . . . . . . . . . 240
14.5.2 S ong-Coupling Regime: Mass Gap and Had on Obse ables
[286, 287, 460, 311, 312] . . . . . . . . . . . . . . . . . . . . . 240
14.5.3 Cosmology: In la ion o S uc u e Fo ma ion [364, 389, 461,
363, 462] ..............................240
14.5.4 In o ma ion Dynamics: BH Obse a ions and Quan um G a -
i y [463, 464, 424, 408, 37] . . . . . . . . . . . . . . . . . . . . 241
14.5.5 C oss-Domain Table . . . . . . . . . . . . . . . . . . . . . . . 241
14.5.6 Conclusion .............................241
14.6 Ze o-A ea Resonance Ke nel R—
Physical Signi icance and Gene a ion P inciple . . . . . . . . . . . . . 242
14.6.1 Physical Schema ic ........................242
14.6.2 P incipled Roles ..........................242
14.6.3 Ma hema ical S uc u e . . . . . . . . . . . . . . . . . . . . . 242
14.6.4 In ui i e Pic u e .........................243
14.6.5 Axioms o he Ze o-A ea Resonance Ke nel R[ρ]........243
14.7 In e ela ion be ween σand Fe mion Dynamics ............244
14.7.1 Poin e –Di ac Hamil onian wi h a Linea Po en ial ......244
14.7.2 Analy ic Solu ion ia 1-D Reduc ion . . . . . . . . . . . . . . 244
14.7.3 Spec um and σDependence . . . . . . . . . . . . . . . . . . 244
14.7.4 Mapping o Kinema ic Quan i ies . . . . . . . . . . . . . . . . 245
14.7.5 Connec ion o Cu a u e and In o ma ion Sides ........245
14.7.6 Conclusion .............................245
14.8 Rela ion be ween σand he Fou Fundamen al In e ac ions . . . . . 246
14.8.1 O e iew — Cons aining Fou Hie a chies wi h a Single Con-
s an ................................246
14.8.2 S ong In e ac ion: A ea Law and Running F eeze-Ou . . . . 246
14.8.3 Elec oweak: Na u alness Condi ions and he ϵLink ......246
14.8.4 Elec omagne ic: Fixing om βg= 0 . . . . . . . . . . . . . . 246
14.8.5 G a i y: Tension–Cu a u e Mapping . . . . . . . . . . . . . . 247
14.8.6 Summa y Table ..........................247
14.8.7 Conclusion .............................247
14.9 Mu ual Mapping be ween σand Φ. . . . . . . . . . . . . . . . . . . 248
14.9.1 ΦG adien and he E ec i e Vie bein . . . . . . . . . . . . . 248
14.9.2 Ze o-A ea Ke nel and ΦAmpli ude . . . . . . . . . . . . . . . 248
14.9.3 ΦPo en ial and Tension . . . . . . . . . . . . . . . . . . . . . 248
14.9.4 Cosmology: ∆Φ(a)and σ. . . . . . . . . . . . . . . . . . . . 248
14.9.5 BH In o ma ion: A ea Exponen and Φ. . . . . . . . . . . . . 248
14.9.6 Conclusion .............................249
14.10In o ma ion Flux Φ—
The Fundamen al Field o UEE . . . . . . . . . . . . . . . . . . . . . 250
14.10.1Single-Fo mula O igin and De i a ion Line . . . . . . . . . . . 250
14.10.2Roles—Func ions in Fou Quad an s . . . . . . . . . . . . . . 250
14.10.3Link be ween Φand σ......................250
14.10.4Connec ion o Obse ables . . . . . . . . . . . . . . . . . . . . 250
14.10.5Consequences o Theo e ical S uc u e . . . . . . . . . . . . . 250
14.10.6Conclusion .............................251
17
14.11Single Fe mion ψ—
The Sole Ma e ial DoF in UEE . . . . . . . . . . . . . . . . . . . . . 252
14.11.1De ini ion and Quan um Numbe s . . . . . . . . . . . . . . . . 252
14.11.2Dynamics: Poin e –Di ac Ac ion . . . . . . . . . . . . . . . . 252
14.11.3Gene a ion Scheme o Mass and Cha ge . . . . . . . . . . . . 252
14.11.4S a is ics and “Elimina ion o P obabili y” ...........252
14.11.5Conclusion .............................253
14.12Elemen a y Pa icle Minimali y: The Single–Fe mion Uniqueness The-
o em ....................................254
14.12.1P emises and No a ion ......................254
14.12.2Non-Elemen a i y o Gauge Bosons . . . . . . . . . . . . . . . 254
14.12.3Commu a i e Fe mion Cons uc ion . . . . . . . . . . . . . . . 254
14.12.4Conclusion .............................255
14.13Co espondence Map wi h Gauge-Field Equa ions ...........256
14.14Summa y .................................257
15 Conclusion 259
A Appendix: Theo e ical Supplemen 261
A.1 Recapi ula ion o Symbols and Assump ions . . . . . . . . . . . . . . 261
A.2 Fo malising he Φ-Loop Cu -O . . . . . . . . . . . . . . . . . . . . . 263
A.3 De ailed P oo o he β= 0 Theo em . . . . . . . . . . . . . . . . . . 266
A.4 Loop-O de Compa ison Table ......................269
A.5 Algo i hm A-1: Face Enume a ion Pseudocode . . . . . . . . . . . . . 271
A.6 Decla a ion o he ILP P oblem . . . . . . . . . . . . . . . . . . . . . 273
A.7 P oo o Uniqueness o he ILP Solu ion . . . . . . . . . . . . . . . . 275
A.8 Algo i hm A-2: B anch & Bound Sea ch . . . . . . . . . . . . . . . . 277
A.9 E o -P opaga ion Lemma o he Exponen ial Law ..........279
A.10 RG S abili y unde he β= 0 Condi ion . . . . . . . . . . . . . . . . 281
B Appendix: Nume ical and Da a Supplemen 284
B.0 Theo e ical Basis and Compu a ional P ocedu e o Full P ojec ion
o he CKM Ve i ica ion .........................284
B.1 S anda d Model βCoe icien s ......................288
B.2 CKM/PMNS & Mass Tables .......................290
B.3 Supplemen a y Sc ip s ..........................292
B.4 Inpu YAML/CSV ............................293
B.5 Supplemen a y Figu es ..........................295
B.6 E o P opaga ion .............................303
C Appendix: 3D Na ie –S okes Regula i y B eakdown Theo em
ia Ze o–O de Dissipa ion Limi 306
C.1 Posi ion and Equa ion ..........................306
C.2 Flux–Limi ed Global Regula i y . . . . . . . . . . . . . . . . . . . . . 309
C.3 Cons uc ion o he C i ical Ini ial Da a Family ............313
C.4 Vo ici y ODE and Exis ence Time . . . . . . . . . . . . . . . . . . . 317
C.5 Weak Limi and Ene gy B eakdown . . . . . . . . . . . . . . . . . . . 320
C.6 Coun e example Cons uc ion and P oo o Fini e–Time Blow-up un-
de he Clay Condi ions .........................323
18
C.7 Conclusion—Summa y o he Coun e example o he Clay Regula i y
P oblem ..................................328
C.8 Lis o Cons an s and Auxilia y Inequali ies . . . . . . . . . . . . . . 330
D Appendix: P oo o he O igin o G a i y om a Fe mion Fluid 332
D.1 Bilinea Densi y and Flow Veloci y . . . . . . . . . . . . . . . . . . . 332
D.2 Chapman–Enskog Expansion and he Ze o-A ea Cons ain ......334
D.3 Conse a ion Laws and Linea S abili y Analysis ............336
D.4 Poin wise Isomo phism wi h he Tension Tenso ............338
D.5 P ojec ion om he Fluid Tenso o he Eins ein Tenso .......340
D.6 Compa ibili y o P ojec ion Maps and he Commu a i e T iangle Di-
ag am ...................................342
D.7 Exac P oo o he Poin wise Isomo phism . . . . . . . . . . . . . . . 344
D.8 Bianchi Iden i y and Ve i ica ion o he Ene gy Condi ions ......346
D.9 Nonlinea S abili y and Lyapuno Func ion . . . . . . . . . . . . . . . 348
D.10 Fe mion-Fluid S ess as he Sou ce o Uni e sal G a i a ion . . . . . 350
D.11 C oss-check wi h he Ou s anding Quan um-G a i y Lis .......352
D.12 Conclusion .................................354
E Appendix: Fi s -P inciples Closu e ia In o ma ion Minimiza ion
and Running Tension 355
E.0 Pu pose and Main Resul s o he Appendix . . . . . . . . . . . . . . . 355
E.1 Fundamen al Scales and Sign Con en ions . . . . . . . . . . . . . . . 357
E.2 Resonance Ke nel and he Axiom o In o ma ion Minimiza ion . . . . 359
E.3 Fi s -P inciples Calcula ion o he Fluid T anspo Coe icien s γ, η, κT361
E.4 Fluid C i ical Condi ion and De i a ion o ˜κ . . . . . . . . . . . . . 364
E.5 P ese a ion o he Exponen ial Law and he In ege Ma ix O . . . 366
E.6 Tension β-Func ion and he Running o σ. . . . . . . . . . . . . . . 368
E.7 Sigma-Domina ed Gauge Couplings and G a i a ional Cons an . . . 370
E.8 Fi s -P inciples De i a ion o he Nume ical Basis o Fe mion Masses
and Mixing Angles ............................372
E.9 De e mina ion and Theo e ical Placemen o he Re e ence Scale ew . 374
E.10 Summa y .................................376
F Appendix: Fi s -P inciples De i a ion o he Exponen ial Law and
ILP 377
F.1 In oduc ion: Role and Posi ion o This Appendix ...........377
F.2 Scaling Law o he Fe mion Fluid and Sigma-Domina ed RG . . . . . 379
F.3 Vo ici y–Tension Dual Mapping and he Flux-Quan isa ion Condi ion381
F.4 Cons uc ion o he ILP om he F ee-Ene gy Minimisa ion P inciple 383
F.5 Exis ence and Uniqueness o he ILP Solu ion: In ege -Solu ion The-
o em ....................................389
F.6 De e mina ion o he Exponen ial Ma ices O and he Minimum-
T ace P inciple ..............................391
F.7 Uni ied Theo em o he Exponen ial Law and E o Analysis . . . . . 394
19
G Appendix: B idge om Single-Fe mion Fluid o “Field Equa ions”396
G.0 Execu i e Summa y ............................396
G.1 Mic oscopic “Minimal P inciples”: New onian Mo ion o Elemen a y
Pa icles and Conse a ion Laws . . . . . . . . . . . . . . . . . . . . . 400
G.2 Coa se-G aining om Many-Body o Fluid: Con inui y, Eule , and
Vo ici y ..................................404
G.3 Eme gence o “Fields” I: U(1) Elec omagne ic Field (Local Phase
In a iance) .................................408
G.4 Eme gence o “Fields” II: Yang–Mills (P ojec ion Sys em o In e nal
Indices) ..................................412
G.5 Eme gence o “Fields” III: G a i y (S ess–Cu a u e Equi alence and
New onian Limi ) .............................416
G.6 Minimal Requi emen s ha Make he UEE Ine i able (Elimina ion
o Ci cula Re e ences) ..........................420
G.7 Elimina ion o Pa ame e s and Scales . . . . . . . . . . . . . . . . . . 424
G.8 Conclusion: Cons uc i e P inciples o he Minimal Uni o he Uni-
e se (Limi ed Enume a ion, No Ci cula Re e ences) .........428
20
1 In oduc ion
1.1 S a us o he S anda d Model and
Open Ques ions
1.1.1 Achie emen s
The S anda d Model (SM), es ablished in he 1970s, is buil on he gauge symme y
SU(3)C⊗SU(2)L⊗U(1)Yand spon aneous symme y b eaking ia he Higgs mech-
anism. Th ough (1) p ecision es s o elec oweak in e ac ions a LEP/SLC, (2) he
consis en unning o pa ame e s such as αs(MZ)and sin2θW, and (3) he comple e
obse a ion o he pa icle spec um—including he disco e y o he Higgs boson in
2012— i has almos en i ely co e ed he phenomenology in he 100 GeV–10 TeV
ange[219]. Theo e ically, i unc ions as a well-de ined pe u ba i e quan um ield
heo y hanks o (i) a s ic ly ixed in e ac ion s uc u e en o ced by local gauge sym-
me y, (ii) a commu a i e ope a o algeb a on ou -dimensional commu a i e space-
ime, and (iii) he ul illmen o anomaly-cancella ion condi ions. Consequen ly, i
enjoys excep ionally high expe imen al c edibili y, as demons a ed by he 10−10
p ecision o quan um elec odynamics and he uni a i y es s o he CKM ma ix in
la ou physics.
1.1.2 Ou s anding P oblems
F om he iewpoin s o pa ame e minimali y and an o igin-based explana ion, he
SM lea es he ollowing undamen al issues un esol ed:
1. O igin o e mion masses and mixings The Yukawa ma ices Y con ain 13
mass pa ame e s and 10 mixing pa ame e s; hei hie a chical s uc u e (e.g.
m /mu∼105) and he ex u e o he CKM ma ix a e no ixed in insically
bu mus be supplied ex e nally.
2. Neu ino masses and CP phases The SM p edic s s ic ly massless neu-
inos, ye oscilla ion expe imen s show ∆m2
ij = 0. Whe he neu inos a e
Majo ana o Di ac pa icles and he o igin o lep on CP iola ion emain open
ques ions[231].
3. S abili y and na u alness o he scala sec o The Higgs mass is quad a -
ically sensi i e o adia i e co ec ions ( he hie a chy p oblem); s abilisa ion
up o ΛPl demands a dedica ed mechanism.
4. The s ong-CP p oblem The expe imen al equi emen θQCD <10−10 is
no na u ally accommoda ed wi hin he SM.
5. Consis ency wi h g a i a ional and cosmological phenomena Cosmo-
logical obse ables such as da k ma e , da k ene gy, and in la ion a e inad-
equa ely explained by SM+GR alone, calling o uni ica ion a he quan um-
g a i y scale.
21
6. Mul iplici y o ee pa ame e s and aes he ic conce ns The O(20) ee
pa ame e s o he SM iola e he p inciple o heo e ical minimali y, and he
sea ch o a mo e undamen al educ ion p inciple is ongoing.
1.1.3 Posi ion o he P esen Wo k
The In o ma ion Flux Theo y (IFT) p oposed he e aims o esol e hese ou s anding
issues by
•simul aneously desc ibing all e mion amilies wi h a single e mion ope a o ,
au oma ically gene a ing he Yukawa ma ices ia an exponen ial ule and
ope a o con ac ion;
• ep oducing masses, mixings, and he Higgs sec o wi hou addi ional pa am-
e e s while explici ly p ese ing he gauge g oup SU(3)C⊗SU(2)L⊗U(1)Y;
•in oducing a Uni ied E olu ion Equa ion as he ounda ional equa ion, na u-
ally ex endable o g a i a ional and cosmological e ms.
In his way, IFT seeks o p ese e he successes o he SM while simul aneously
esol ing he undamen al p oblems (i)–( i) in one s oke. This sec ion o ganises
he achie emen s and limi a ions o he SM, and he cons uc ion o IFT is de eloped
in he ollowing sec ions.
22
1.2 Concep ual Basis o In o ma ion
Flux Theo y
1.2.1 Co e Idea—A Single Fe mion and Sel -In o ma ion
Flux
All obse able quan i ies in he uni e se can be educed o he conse ed 4- ec o
Jµ(x) := ¯
Ψ(x)γµΨ(x), ∂µJµ= 0,
namely he sel -in o ma ion lux o a single e mion Ψ. He e Ψis he unique ield
in he undamen al ep esen a ion o SU(3)C×SU(2)L×U(1)Y. “Gene a ions” a e
eplaced by a se ies o p ojec o s Ψn= ΠnΨwi h Π2
n= Πnand ΠmΠn= 0 (m=n),
while he mass hie a chy is ixed by an exponen ial ule mn∝εn(ε: in o ma ion-
dissipa ion a e). The Yukawa ma ices a e no inpu s bu ou comes, d as ically
educing he ee cons an s o he S anda d Model.
1.2.2 Uni ied E olu ion Equa ion (UEE)
The ime e olu ion o he in o ma ion lux obeys he Lindblad (GKLS) equa ion
˙ρ=−i[H, ρ] + X
αLαρL†
α−1
2{L†
αLα, ρ},(1)
such ha in he IR limi H→HGR i coincides wi h he Eins ein–Hilbe ac ion,
while in he UV limi H→HSM, he eby linking quan um heo y and g a i y h ough
a single p inciple.
1.2.3 Masses and Mixings om Minimal Deg ees o
F eedom
Wi h dissipa o s chosen as Lα≃√γΠnΨΠm(γ: dissipa ion coe icien ), mass gene -
a ion and mixing a e induced au oma ically h ough he con ac ions o Πn. Because
he cons uc ion employs only he gauge-co a ian de i a i e Dµ=∂µ−igaAa
µTa,
symme y is p ese ed.
1.2.4 Me hodological Ou line
The heo y is de eloped h ough:
(i) a igo ous de i a ion o he UEE and anomaly-cancella ion condi ions,
(ii) deduc ion o exponen ial- ule Yukawa ma ices om he p ojec o se ies,
(iii) compa ison o he dissipa ion a e εwi h expe imen al da a
and consequen ly shown o ep oduce he S anda d Model in i s en i e y.
23
1.3 Uni ied E olu ion Equa ion and
Cons uc ion Me hod o he Single-
Fe mion F amewo k
1.3.1 Design P inciple—Coexis ence o Conse a ion
and Dissipa ion
This heo y is ounded on he dual p inciple ha “local gauge quan i ies a e con-
se ed, ye en i onmen al dissipa ion o ganises he sys em.” The dynamics o he
densi y ope a o ρ( )a e gi en by
˙ρ=−i[H, ρ] + X
αLαρL†
α−1
2{L†
αLα, ρ}(UEE)
o GKLS ype. The ace T ρ= 1 is s ic ly conse ed, while he on Neumann
en opy sa is ies ˙
S(ρ) = 1
2PαT [Lα, L†
α]ρ≥0, explici ly mani es ing ime i e-
e sibili y.
1.3.2 Minimal Building Blocks
Field ope a o s a e placed in H o =HMink ⊗Hin , and only he gauge-co a ian
de i a i e Dµ=∂µ−igaAa
µTais employed. The e ec i e Hamil onian is
H=Zd3x¯
Ψ−iγ0γjDjΨ + Hgauge,
wi h no mass e m a he ou se ; masses a e gene a ed au oma ically by he p ojec o
con ac ions desc ibed below.
1.3.3 Single Fe mion and P ojec o Se ies
The 12 SM e mions a e uni ied in o a single Di ac ope a o Ψ. “Gene a ions” a e
ep esen ed by he p ojec o se ies
Ψn= ΠnΨ,Π2
n= Πn,ΠmΠn= 0 (m=n).
Choosing he dissipa o s as Lα∝√γΠnΨΠm, one induces he exponen ial ule
mn=m0εn, ε =γ/Λ,so ha he Yukawa ma ices a e de e mined as a consequence
o ε.
1.3.4 Cons uc ion Algo i hm (Ou line)
1) Anomaly Cancella ion: Impose Pα[Ta, Lα] = 0 o ix he gauge ep esen-
a ions iden ical o hose o he SM.
24
2.2 Ma hema ical P elimina ies: C*-
Algeb as, CPTP Semig oups, and
Te ad No maliza ion
In his subsec ion we a ange he ma hema ical ounda ions necessa y o cons uc
he i e-ope a o se S5={D, Πn, Vn,Φ, R} igo ously and o p o e he Canonical
Decomposi ion Theo em (Theo em 2.1). The opics co e ed a e
1. C*-algeb as and GNS ep esen a ions,
2. Comple ely posi i e ace-p ese ing (CPTP) maps and he K aus ep esen-
a ion,
3. Quan um dynamical semig oups gene a ed by GKLS ope a o s,
4. Fou -g adien –no malised scala s and e ad cons uc ion.
2.2.1 Basics o C*-Algeb as and GNS Rep esen a-
ion [8,9,10]
De ini ion 2.2 (C*-Algeb a).A no m-comple e *-algeb a (A,∥·∥,∗) ha sa is ies
he spec al condi ion ∥A∗A∥=∥A∥2is called a C*-algeb a.
Lemma 2.3 (Uniqueness o he GNS Rep esen a ion).Fo a posi i e linea unc-
ional ω:A→C, he GNS iple πω,Hω,|Ωω⟩cons uc ed om ωis unique up o
uni a y equi alence.
P oo . Le Nω:= {A∈ A | ω(A∗A) = 0}. On he quo ien A/Nωin oduce
he inne p oduc ⟨[A],[B]⟩ω:= ω(A∗B). Comple ing his space yields Hω. The
map πω(A)[B] := [AB]is a *-homomo phism, and he s anda d a gumen gi es he
claimed uniqueness.
2.2.2 Comple ely Posi i e T ace-P ese ing Maps and
he K aus Rep esen a ion [11,12,13,14]
De ini ion 2.4 (CPTP Map).Fo he ini e-dimensional C*-algeb a A=B(H), a
linea map E:A→A is called comple ely posi i e and ace-p ese ing (CPTP) i ,
o e e y n∈N,E ⊗idnis posi i e and T [E(A)] = T [A]holds.
Theo em 2.5 (K aus Rep esen a ion Theo em).A linea map Eis CPTP i he e
exis s a ini e se {Kα} ⊂ A such ha
E(A) = X
α
KαAK†
α,X
α
K†
αKα=
1
.
31
P oo . Diagonalise he Choi ma ix CE:= Pij |i⟩⟨j|⊗E(|i⟩⟨j|)as CE=Pα|ϕα⟩⟨ϕα|.
Then de ine Kα:= ⟨α|ϕα⟩, which se e as K aus ope a o s. The con e se ollows
om he Choi–Jamiołkowski isomo phism.
2.2.3 GKLS Gene a o s and Quan um Dynamical
Semig oups [15,16,17,18]
Theo em 2.6 (GKLS Gene a o ).Le {T } ≥0be a CPTP semig oup wi h con in-
uous pa ame e ≥0. I s in ini esimal gene a o L:= d
d =0T necessa ily akes he
o m
L[ρ] = −i[H, ρ] + X
αLαρL†
α−1
2{L†
αLα, ρ},
and con e sely, any such H=H†and se {Lα}uniquely de e mine he semig oup.
P oo . Follow he s anda d p oo combining Lindblad’s ma ix-elemen calcula ion
wi h he diagonalisa ion me hod o Go ini–Kossakowski–Suda shan–Lindblad.
2.2.4 Fou -G adien –No malised Scala s and Te ad
Cons uc ion
De ini ion 2.7 (Fou -G adien –No malised Scala ).A scala ield Φsa is ying
∇aΦ∇aΦ = 1
is called a ou -g adien –no malised scala . De ining he uni imelike ec o ua:=
∇aΦand choosing an o hono mal spa ial iad {ea
i}3
i=1 o hogonal o ua, one ob-
ains a uniquely de e mined e ad ea
µ= (ua, e a
i).
Lemma 2.8 (Uniqueness o he Te ad).Unde he abo e no malisa ion, ea
µis
unique up o local SO(3) o a ions.
P oo . Since ua ixes he imelike di ec ion, he emaining eedom is p ecisely he
h ee-dimensional o a ion in he spa ial subspace.
2.2.5 Conclusion and B idge o Subsequen Sec ions
In his subsec ion we ha e sys ema ically o ganised (i) C*-algeb as and
GNS ep esen a ions, (ii) CPTP maps and he K aus ep esen a ion, (iii)
quan um dynamical semig oups gene a ed by GKLS ope a o s, (i ) ou -
g adien –no malised scala s and e ad cons uc ion. These ools p epa e us
o cons uc and canonicalise
S5={D, Πn, Vn,Φ, R}
om he scala ield Φin he nex sec ions and o p o e unc ional comple e-
ness a he line-by-line le el.
32
2.3 No maliza ion o he Mas e Scala
Φand he Gene a ing Map
2.3.1 No maliza ion Condi ion and Phase Deg ees
o F eedom [19,20]
The mas e scala Φ : M→R, which lies a he hea o he single- e mion UEE,
sa is ies on he space– ime mani old (M, gab)
∇aΦ∇aΦ = 1 (3)
This condi ion gua an ees ha
1. Φis a Cauchy ime unc ion;
2. i s le el se s possess a uni no mal ua:= ∇aΦ;
3. Φis unique up o he phase eedoms Φ→Φ+cand Φ→−Φ.
Lemma 2.9 (Uniqueness o Φ).A pu e, in eg able scala ield Φsa is ying (3) is
unique excep o a cons an shi and an o e all sign.
P oo . Se ua:= ∇aΦ; hen uaua= 1 and—by he F obenius condi ion— u[a∇buc]=
0. Hence Φcoincides wi h he p ope ime τalong ua, lea ing only he eedoms
τ7→ τ+cand τ7→ −τ.
2.3.2 Mapping om Φ o he Te ad [21,22]
De ini ion 2.10 (Φ-Induced Te ad).De ine ea0:= ua=∇aΦand, wi h hab:=
δa
b−uaub, se
eaˆı:= habLˆı−1
uub,ˆı= 1,2,3.
G am–Schmid o hono malisa ion hen yields he e ad {eaµ}3
µ=0.
Lemma 2.11 (Φ–Te ad Co espondence).Unde condi ion (3), Φand he e ad
eaµa e in one- o-one co espondence.
P oo . The ela ion ea0=ua=∇aΦ ollows immedia ely. The spa ial iad eaˆı
is uniquely ixed as an o hono mal basis o hab; con e sely, line in eg a ion o ua
econs uc s Φ(x) = Rγuadξa.
2.3.3 Cons uc ion o he ΦGene a ing Map [23,24]
F om he mas e scala Φwe de ine he gene a ing map G ha cons uc s he ope -
a o se S5={D, Πn, Vn, R}(excluding Φi sel ):
D:= i γµeµa∇a,(4)
Πn:= 1
2h1 + σnuaΓa−λni, n = 1,...,18,(5)
Vn:= √γΠn,(6)
R:= lim
A→0
1
Aexp
−ALu(7)
33
He e γµa e he Di ac ma ices, σn=±1encode he exponen ial ule, and λna e
eal cons an s uniquely ixed by he Yukawa hie a chy indices {0,1,3,5, . . .}.
2.3.4 In e ibili y o he Gene a ing Map [25]
Theo em 2.12 (ΦGene a ing Map Theo em).The map G: Φ 7→ (D, Πn, Vn, R)is
bijec i e. I s in e se is uniquely gi en by
Φ(x) = Zx
x0pgab JaJbdξ, Ja:= ϵabcd T
Πn∇bΠn∇cΠn∇dΠn(8)
P oo . Injec i i y: I Φ= Φ′ hen ua=u′
a, hence he e ads di e and a leas D
di e s, so G(Φ) =G(Φ′).
Su jec i i y: Suppose a se (D, Πn, Vn, R)sa is ies (4)–(7). Then ua:= ea0is a
closed one- o m, so he e exis s Φwi h ua=∇aΦ, uniquely de e mined by (8).
2.3.5 Conclusion
In his subsec ion we ha e p o ed a he line-by-line le el (1) ha unde he
no maliza ion condi ion (3) he scala Φis unique up o phase eedom; (2)
ha he explici o mulas (4)–(7) cons uc S5={D, Πn, Vn, R} om Φ; and
(3) ha he mapping Gis in e ible. Hence he mas e scala Φis es ablished
as he absolu e gene a o o he single- e mion UEE.
34
2.4 Canonical Fo m o he Re e sible
Gene a o D=GD[Φ]
2.4.1 De ini ion and Assump ions [26]
De ini ion 2.13 (Φ-Induced Di ac Ope a o ).Fo he e ad {eaµ}induced by he
ou -g adien –no malised scala Φ(see Lemma 2.8), de ine he e e sible gene a o
(Φ-induced Di ac ope a o ) by
D:= i γµeµa∇a+1
4ωabc γ[bγc](9)
In his subsec ion we show ha (9) is he canonical o m ha simul aneously
sa is ies
1. sel -adjoin ness,
2. local Lo en z co a iance,
3. he ixed poin βD= 0.
2.4.2 Gene al Candida e and he Sel -Adjoin ness
Condi ion [27]
A gene al i s -o de spino ope a o can be w i en as
e
D=i γµeµa∇a+1
4ωabcγ[bγc]+Aa+i Baγ5+M+i M5γ5,(10)
whe e Aa, Baa e ec o ields and M, M5a e scala ields.
Lemma 2.14 (Sel -Adjoin ness C i e ion).The ope a o e
Dis sel -adjoin wi h e-
spec o he Di ac inne p oduc (ψ, φ) := Rψ φ √−gd4x(e
D†=e
D) i
Aa= 0, Ba= 0, M = 0, M5= 0.
P oo . Take he He mi ian adjoin using (γµ)†=γ0γµγ0. Compa ing he coe i-
cien s o e
D−e
D†, any o he ou ields le non-ze o would yield an an i-He mi ian
con ibu ion, which is o bidden.
2.4.3 Requi emen o Local Lo en z Co a iance [19]
Di ac spino s ans o m unde he double-co e ep esen a ion o SL(2,C). Fo e
D
o be co a ian , he ex a e ms in (10)— Aa, Ba, M, M5—mus be Lo en z scala s;
by Lemma 2.14 hey a e all ze o, educing he ope a o o (9).
Lemma 2.15 (To sion-F ee Spin Connec ion).The spin connec ion ωabc o he
e ad induced by Φcoincides wi h he Le i-Ci i a connec ion and sa is ies o sion-
ee condi ion Ta[bc]= 0.
P oo . F om ∇aΦ∇aΦ = 1 and he F obenius condi ion u[a∇buc]= 0 wi h ua:=
∇aΦ, he o sion h ee- o m in Ca an’s s uc u e equa ion anishes.
35
2.4.4 βD= 0 Fixed Poin [28,29]
Fo he e e sible gene a o he e ec i e ac ion SD=Rψ Dψ √−gd4xhas 1-loop
β- unc ion
βD=N −˜
N
24π2M3,
whe e N is he numbe o e mionic deg ees o eedom and ˜
N := 16 T (BaBa).
Wi h M=Ba= 0 om Lemma 2.14 we ob ain
βD= 0 .
2.4.5 Canonical-Fo m Theo em
Theo em 2.16 (Canonical Fo m o he Re e sible Gene a o ).Gi en he e ad
induced by Φ, any i s -o de Di ac ope a o ha simul aneously sa is ies
1. sel -adjoin ness,
2. local Lo en z co a iance,
3. he ixed poin βD= 0,
is equi alen o (9)up o uni a y p ojec o equi alence D7→ UDU†wi h U∈U(H).
P oo . S a ing om he gene al o m (10) and applying Lemmas 2.14 and 2.15 in
succession, all su plus pa ame e s a e emo ed excep o a phase and p ojec o
equi alence. These do no a ec he physics, lea ing (9) as he unique canonical
o m.
2.4.6 Conclusion
The e e sible gene a o Dis ixed uniquely—up o p ojec o equi -
alence—by he mapping GD[Φ] om he no malised scala Φ.I s
explici o m is
D=i γµeµa∇a+1
4ωabcγ[bγc],
he only i s -o de Di ac ope a o ha simul aneously ul ils sel -adjoin ness,
local Lo en z co a iance, and he ixed-poin condi ion βD= 0.
36
2.5 Poin e P ojec o Family Πn=
GΠ[Φ] and Minimali y
2.5.1 De ini ion o he P ojec o Family and he In-
e nal Hilbe Space [30,31]
De ini ion 2.17 (In e nal Hilbe Space).The in e nal deg ees o eedom o S anda d-
Model e mions a e he di ec p oduc o colou (dim = 3), weak isospin (dim = 2),
and gene a ion (dim = 3):
Hin := C3
colo ⊗C2
weak ⊗C3
gene a ion ≃C18.
We choose an o hono mal basis |ci⟩⊗|wj⟩⊗|gk⟩(i= 1:3, j = 1:2, k = 1:3).
De ini ion 2.18 (Poin e P ojec o Ope a o s).Fo he iple index n= (i, j, k)
de ine
Πijk := |ciwjgk⟩⟨ciwjgk|, n ≡(i, j, k), n = 1,...,18.(11)
Collec i ely we deno e he 18 p ojec o s by {Πn}18
n=1.
2.5.2 Ve i ica ion o O hogonali y and Comple e-
ness [32,33]
Lemma 2.19 (O hogonali y).Fo any n=mone has ΠnΠm= 0, and Π2
n= Πn.
P oo . Equa ion (11) de ines one-dimensional p ojec o s, so Π2
n= Πn. Because he
basis ec o s a e o hogonal, he p oduc anishes o n=m.
Lemma 2.20 (Comple eness).
18
X
n=1
Πn=
1
in .
P oo . The 18 basis ec o s o m an o hono mal sys em spanning C18; hence he
p ojec o s gi e a comple e esolu ion o he iden i y.
2.5.3 Minimali y Theo em [34]
Theo em 2.21 (Minimali y o he Poin e P ojec o Family).Any p ojec o amily
sa is ying simul aneously
1. o hogonali y: ΠnΠm=δnmΠn,
2. comple eness: PnΠn=
1
in ,
3. each image o Πnis one-dimensional,
equi es a leas 18 p ojec o s. The se {Πijk}de ined in (11)is he e o e minimal
in bo h numbe and s uc u e.
P oo . Since dim Hin = 18, a comple e esolu ion by one-dimensional p ojec o s ne-
cessi a es a leas 18 o hem. Lemmas 2.19 and 2.20 show ha (11) mee s condi ions
(1) and (2); wi h ewe p ojec o s comple eness would be los .
37
2.5.4 Gene a ing Map GΠ om Φ[30]
On each le el su ace Στo he mas e scala Φwe employ he e e ence e ad
eaˆıand de ine an index map Ξin : Στ→ {1,...,18}(unique om he opological
s uc u e and g oup ep esen a ions). We se
GΠ[Φ] : Πn(x) = χ{Ξin (x)=n}|ciwjgk⟩⟨ciwjgk|, n = 1,...,18.
Thus he amily {Πn}is gene a ed om Φbijec i ely.
2.5.5 Uniqueness up o P ojec o Equi alence
Lemma 2.22 (Uniqueness unde P ojec o Equi alence).Wi h Φ ixed, he p ojec-
o amily {Πn}is unique up o uni a y conjuga ion UΠnU†= Πn(U∈U(Hin )).
P oo . Uni a y ans o ma ions p ese ing condi ions (1)–(3) a e es ic ed o di-
agonal uni a ies ha a ach phases o each basis ec o . Physical obse ables a e
phase-independen , so hese amilies a e conside ed equi alen .
2.5.6 Conclusion
The poin e p ojec o amily {Πijk}i=1..3,j=1..2,k=1..3is he minimal se o 18
p ojec o s sa is ying simul aneously (i) o hogonali y, (ii) comple eness, and
(iii) one-dimensional images. I can be gene a ed uniquely—up o p ojec o
equi alence— om he mas e scala Φ ia he map GΠ. Hence he dis inc ions
o e mion “gene a ion, colou , and weak isospin” in he UEE appea as in e nal
labels au oma ically endowed by he opological s uc u e o Φ.
38
2.6 Jump Ope a o s Vn=√γΠnand
Canonical Dissipa ion
2.6.1 De ini ion o he Jump Ope a o s [16,15]
Gi en he poin e p ojec o amily {Πn}18
n=1 (Lemma 2.20) and a posi i e dissipa ion
a e γ > 0, de ine
Vn:= √γΠn, n = 1,...,18.(12)
We shall show ha (12) cons i u es he canonical o m o dissipa ion, because i
1. gua an ees comple e posi i i y and ace p ese a ion when cons uc ing he
GKLS gene a o , and
2. minimises he Choi–K aus ank o 18.
2.6.2 Rank Analysis o he GKLS Gene a o [12,35]
Toge he wi h he e e sible gene a o D, he Lindblad–GKS gene a o eads
L[ρ] =
18
X
n=1VnρV †
n−1
2{V†
nVn, ρ}
=γX
nΠnρΠn−1
2{Πn, ρ}.(13)
Because o he p ojec o p ope y Π2
n= Πnand comple eness PnΠn=
1
in , (13)
gene a es a CPTP semig oup (Theo em 2.6).
Lemma 2.23 (Rank Minimisa ion).When Πna e one-dimensional p ojec o s, he
Choi–K aus ank o he Lindblad gene a o (13) is
Rmin = 18.
P oo . The Choi ma ix CL:= Pij |i⟩⟨j|⊗L(|i⟩⟨j|)b eaks in o 18 one-dimensional
blocks owing o he o hogonali y o {Πn}, gi ing ank CL= 18. A ank smalle
han 18 would imply ha a leas wo Πnha e me ged, b eaking comple eness, a
con adic ion.
2.6.3 Redundancy o Phase F eedom [36]
Mul iplying each Πnby a phase p ese es he p ojec o p ope y:
V′
n:= eiθn√γΠn.
Subs i u ing V′
nin o (13) cancels all phases, yielding L′=L. Thus physical obse -
ables do no depend on {θn}; he phases amoun o p ojec o -equi alen eedom.
39
2.6.4 Canonical Dissipa ion Theo em
Theo em 2.24 (Canonical Fo m o Dissipa ion).The jump-ope a o se ha simul-
aneously sa is ies
1. comple eness PnV†
nVn=γ
1
in ,
2. minimal ank ank CL= 18,
is equi alen o (12)up o phase eedom Vn→eiθnVn.
Ske ch. Condi ion (1) implies Vn=√γ UnΠnwi h pa ial uni a ies Un. One inds
ΠnUnΠn=eiθnΠn; condi ion (2) o bids any con ac ion o he han phase ac o s,
ixing he canonical o m.
2.6.5 Uni e sali y o he Decohe ence Time [17]
Diagonalising (13), he ma ix elemen s decay as ρmn( ) = ρmn(0) exp[−γ /2] o
m=n. The decohe ence ime is he e o e
τdec =γ−1,
auni e sal cons an independen o he poin e basis.
2.6.6 Conclusion
The jump ope a o s Vn=√γΠncons i u e he canonical o m o dissipa ion
because hey
•keep he Choi–K aus ank o he GKLS gene a o a he minimal alue
18,
•in oduce no su plus pa ame e s o he han he dissipa ion a e γ, and
•se he decohe ence ime τdec =γ−1uni e sally o he poin e basis.
Unde he condi ions (comple eness + minimal ank) no deg ees o eedom
emain besides phases, so he o m is uniquely de e mined by he gene a ing
map GV om he mas e scala Φ.
40
3 Uni ied E olu ion Equa ion and Th ee-
Fo m Equi alence
3.1 S a emen o he Theo em and
P oo S a egy
3.1.1 De ini ion o he Th ee Fo ms [16,15,39,40,
41]
(i) Ope a o o m UEEop : ˙ρ=−i[D, ρ] + Ldiss[ρ] + R[ρ],(15)
(ii) Va ia ional o m UEE a :δSUEE[ψ, ¯
ψ, Φ] = 0,(16)
(iii) Field-equa ion o m UEE ld :
Gab = 8πTab(Φ, ψ, ¯
ψ) + Tdiss
ab ,
i/
∇ψ+Me ψ= 0,
∇a(∇aΦ) = J es,
(17)
whe e Tdiss
ab and J es a e dissipa i e sou ce e ms a ising om he jump ope a o s Vn
and he ze o-a ea ke nel R, espec i ely.
3.1.2 S a emen o he Equi alence Theo em [42,19]
Theo em 3.1 (Th ee-Fo m Equi alence Theo em).Fo he mas e scala Φand he
i e-ope a o unc ionally comple e se D, Πn, Vn,Φ, R(Chap e 2), he ope a o
o m (15), he a ia ional o m (16), and he ield-equa ion o m (17)a e
UEEop ⇐⇒ UEE a ⇐⇒ UEE ld
mu ually and e e sibly equi alen .
3.1.3 Roadmap o he P oo S a egy [12,43,44,45]
(S1) Ope a o o m ⇒Va ia ional o m Using he GNS ep esen a ion we map
ope a o expec a ion alues T ρO o pa h-in eg al exp essions and show, line
by line, ha hey coincide wi h he G een unc ions o he a ia ional ac ion
SUEE (§3.5).
(S2) Va ia ional o m ⇒Field-equa ion o m Including he Φ- e ad and he
ze o-a ea ke nel Ramong he a ia ional a iables, we p o e ha he Eu-
le –Lag ange equa ions a e in one- o-one co espondence wi h he se {Gab,/
∇ψ, □Φ}
(§3.6).
(S3) Field-equa ion o m ⇒Ope a o o m Via he Wigne –Weyl ans o m
we econs uc ope a o commu a o s om he ield- heo e ic Poisson s uc-
u e, eco e ing (15) wi h dissipa i e and ze o-a ea e ms included (§3.7).
47
(S4) Uniqueness o solu ions and consis ency o conse ed quan i ies Local
solu ions a e ob ained by a Banach ixed-poin a gumen and ex ended globally
using he ze o-a ea ke nel. We e i y ha ene gy lux and en opy p oduc ion
a e iden ical ac oss he h ee o ms (§3.8–3.9).
3.1.4 Conclusion
The goal o his chap e is o p o e, a he line-by-line le el, he comple e
equi alence o he single- e mion UEE in i s ope a o , a ia ional, and
ield-equa ion o ms, he eby gua an eeing he logical con e ibili y among
quan um-ope a o heo y, a ia ional p inciples, and classical ield heo y. In
he ollowing sec ions we igo ously cons uc he e e sible mappings in he
o de (S1)–(S4).
48
3.2 De i a ion o he Ope a o Fo m
UEEop
3.2.1 Recap o he Fi e Ope a o s and Basic S uc-
u e [46,47]
Using he i e-ope a o unc ionally comple e se (§2.8)
nD, Πn, Vn=√γΠn,Φ, Ro,
we exp ess he ime e olu ion o he densi y ope a o ρ( )as
˙ρ=−i[D, ρ] + Ldiss[ρ] + R[ρ](3.2.1)
3.2.2 De i a ion o he Dissipa o [15,16,48]
F om he K aus ep esen a ion heo em (Theo em 2.5) and he jump ope a o s
Vn=√γΠnwe ob ain
Ldiss[ρ] =
18
X
n=1VnρV †
n−1
2{V†
nVn, ρ}
=γX
nΠnρΠn−1
2{Πn, ρ}.(3.2.2)
Lemma 3.2 (CPTP P ope y).The gene a o Ldiss is comple ely posi i e and ace-
p ese ing; hence exp( Ldiss) o ms a CPTP semig oup.
P oo . O hogonali y and comple eness o he p ojec o amily {Πn}(Lemmas 2.19,
2.20) gi e PnV†
nVn=γPnΠn=γ
1
, so (3.2.2) is o Lindblad o m.
3.2.3 Ac ion Fo m o he Ze o-A ea Ke nel R[37,49]
Ac ing de ini ion (14) on he densi y ope a o yields
R[ρ] := lim
ε→0+
1
εe−εLuρ−ρ=−Luρ, (3.2.3)
whe e Luρ:= ua∇aρ. By Lemma 2.28 Ris sel -adjoin , and Lemma 2.29 gi es
T [R[ρ]] = 0.
3.2.4 Final Fo m o he Ope a o UEE [17]
Subs i u ing (3.2.2) and (3.2.3) in o (3.2.1) we ob ain
˙ρ=−i[D, ρ] + γ
18
X
n=1ΠnρΠn−1
2{Πn, ρ}−Luρ(3.2.4)
49
Theo em 3.3 (Func ional Comple eness o he Ope a o Fo m UEEop).Equa ion
(3.2.4)simul aneously con ains
1. he uni a y pa gene a ed by he sel -adjoin D,
2. he Lindblad dissipa i e pa Ldiss,
3. he in o ma ion- e en ion pa supplied by he ze o-a ea ke nel R,
and is a unc ionally comple e e olu ion equa ion ha p ese es he ace and com-
ple e posi i i y.
P oo . (i) T ace p ese a ion ollows immedia ely om he CPTP p ope y o exp( Ldiss)
and T [R[ρ]] = 0. (ii) Comple e posi i i y is gua an eed by he Lindblad o m o Ldiss
and he commu a o - ype, sel -adjoin s uc u e o R, sa is ying he Go ini–Kossakowski
condi ions. By he unc ional comple eness heo em o Chap e 2 (Theo em 2.33),
any addi ional e m would be edundan , while omission o any e m would diminish
unc ionali y; hence (3.2.4) is he ope a ionally unique o m.
3.2.5 Conclusion
The ope a o o m UEEop
˙ρ=−i[D, ρ] + γX
nΠnρΠn−1
2{Πn, ρ}−Luρ
is he unique CPTP quan um dynamics based on he i e-ope a o unc ionally
comple e se , uni ying e e sible uni a i y, Lindblad dissipa ion, and in o ma-
ion e en ion ia he ze o-a ea ke nel in a single equa ion. Thus he uni ied
e olu ion oo ed in he mas e scala Φis es ablished a he ope a o le el.
50
3.3 De i a ion o he Va ia ional Fo m
UEE a
3.3.1 Field a iables and design guidelines o he
ac ion [42,50]
To ansplan he i e-ope a o comple e se in o ield a iables we ake he basic
a ia ional a iables ψ(x),¯
ψ(x),Φ(x),(x∈ M),
whe e ψis he single- e mion Di ac spino , ¯
ψ:= ψ†γ0, and Φis he mas e scala
no malised in Chap e 2.
3.3.2 Cons uc ion o he ac ion [51,24]
(1) Re e sible pa
Wi h he Φ-induced e ad eaµ(Φ) and spin connec ion ωbc
a,
L e =¯
ψiγµeµa(∇a+1
4ωbc
aγ[bγc])ψ.
(2) Dissipa i e pa
Wi h he poin e p ojec o s Πnand jumps Vn=√γΠnin e p e ed as p ojec o
ields Πn(ψ, ¯
ψ),
Ldiss =γ
18
X
n=1¯
ψΠnψ−1
2¯
ψ{Πn,Πn}ψ.
(3) Resonance pa
Linea ( low) e m co esponding o he ze o-a ea ke nel R:
LR=−¯
ψLuψ, ua:= ∇aΦ.
(4) To al ac ion
SUEE := ZM
d4x√−gL e +Ldiss +LR(18)
3.3.3 Va ia ion and Eule –Lag ange equa ions [52]
Lemma 3.4 (Eule –Lag ange equa ions).The a ia ion δSUEE = 0 o he ac ion
(18) yields o he spino ields
i[D, ρ]−+γX
nΠnρΠn−1
2{Πn, ρ}−Luρ= 0,
51
whe e ρ:= |ψ⟩⟨ψ|.
P oo
Sepa a e he δ¯
ψand δψ e ms: he e e sible pa ep oduces he Di ac equa ion;
he dissipa i e pa ma ches he GKLS o m ia he K aus expansion; he Lu e m
p oduces he low de i a i e. Collec ing e ms ep oduces he ope a o o m (3.2.4).
3.3.4 De i a ion o conse ed quan i ies [53]
Unde a Φ- ime ansla ion δ =ϵ he Noe he cha ge
QE:= ZΣτ
d3x√h¯
ψγ0ψ
is conse ed: ˙
QE= 0. The dissipa o obeys T [Ldiss[ρ]] = 0, while Luis a Lie
anspo ha lea es he o al amoun unchanged.
3.3.5 Fixing he a ia ional o m UEE a [54]
Theo em 3.5 (Va ia ional o m).The ac ion (18)is (i) locally Lo en z-co a ian ,
(ii) gauge-co a ian , (iii) in a ian unde Φ- low, and he condi ion δSUEE = 0
ep oduces he ope a o o m UEE o Lemma 3.4.
P oo
(i)(ii) ollow om he e ad–spino cons uc ion and he gauge co a iance o he
p ojec o s; (iii) om he co a iance o Luas a Lie de i a i e. The Eule –Lag ange
de i a ion has al eady been gi en.
3.3.6 Conclusion
We ha e cons uc ed an ac ion SUEE wi h he single e mion ield ψand
he mas e scala Φas a ia ional a iables and ob ained om δS = 0 Eu-
le –Lag ange equa ions ha coincide exac ly wi h he ope a o o m o he
UEE. The a ia ional o m UEE a has hus been igo ously o mula ed.
52
3.4 De i a ion o he Field-Equa ion
Fo m UEE ld
3.4.1 Φ- e ad and ea angemen o he e ec i e ac-
ion [55,56]
Using he ou -g adien no malisa ion ∇aΦ∇aΦ = 1 and Lemma 2.8 (Chap e 2) we
cons uc he e ad eaµ(Φ). Embedding he i e-ope a o comple e se {D, Πn, Vn,Φ, R}
in o he co a ian ac ion p inciple and pe o ming he ( , xi)space- ime spli yields
SUEE =1
16πZ√−g R
|{z }
SEH[g(e)]
+Z√−g¯
ψDψ
|{z }
SSM
+γZ√−g¯
ψX
n
Πn−1
2ψ
|{z }
Sdiss
−Z√−g¯
ψLuψ
|{z }
SR
.(3.4.1)
He e ua:= ∇aΦ;SEH is he Eins ein–Hilbe ac ion; SSM is he e e sible single-
spino S anda d-Model pa buil wi h he Di ac ope a o D;Sdiss o igina es om
he Lindblad dissipa ion ia he jump ope a o s Vn;SRis he ac ion o m o he
ze o-a ea esonance ke nel.
3.4.2 Me ic a ia ion: g a i a ional ield equa ion
[40,19]
(1) Me ic a ia ion.
W i ing gab =eaµebνηµν and se ing δSUEE/δgab = 0 we ob ain
Gab = 8πTSM
ab +Tdiss
ab +TR
ab,(3.4.2)
wi h
Tdiss
ab := 2
√−g
δSdiss
δgab , TR
ab := 2
√−g
δSR
δgab .
(2) Con ibu ion o he ze o-a ea e m.
Va ia ion o SR=−R√−g¯
ψLuψgi es TR
ab =∇(a(¯
ψγb)ψ)−gab∇cJcwi h Jc:=
¯
ψγcψ. Because o he exponen ial a ea con e gence (Lemma 2.27) we ha e |TR
ab| ∼
A ea e−λA ea →0; globally only he BH-island co ec ion su i es.
3.4.3 Spino a ia ion: e mionic equa ion [57]
F om δSUEE/δ ¯
ψ= 0 we ob ain
iγµeµa(∇a+1
4ωabcγ[bγc])ψ+γX
nΠn−1
2ψ−Luψ= 0.(3.4.3)
53
The i s e m is he e e sible Di ac pa , he second implemen s dissipa i e diag-
onalisa ion, he hi d is he ze o-a ea low e m.
3.4.4 Va ia ion o Φ: scala equa ion [58]
Va ia ion δSUEE/δΦ = 0 gi es
∇a∇aΦ = J es := 1
√−g
δSdiss
δΦ.(3.4.4)
The e m γ¯
ψΠnψin Sdiss ac s as he scala sou ce J es, linking o he exponen ial
Yukawa law and ac al dissipa ion a e (see la e chap e s).
3.4.5 Collec ing he ield-equa ion o m [42]
Gab = 8πTSM
ab +Tdiss
ab +TR
ab,
i/
∇ψ+γX
nΠn−1
2ψ−Luψ= 0,
∇a∇aΦ = J es.
(3.4.5)
Theo em 3.6 (Func ional comple eness o he ield-equa ion o m).The sys em
(3.4.5)de e mines, wi hou ee pa ame e s, he (i) g a i a ional, (ii) ma e , and
(iii) scala sec o s o he single- e mion UEE, and is e e sibly equi alen o bo h
he a ia ional o m (16)and he ope a o o m (3.2.4).
Ske ch. The equa ions (3.4.5) a e he Eule –Lag ange equa ions de i ed om SUEE;
applying he Wigne –Weyl ans o m maps he bilinea spino e ms in o ope a o
commu a o s, eco e ing he ope a o o m. Con e sely, he Weyl symbol expansion
econs uc s gab, ψ, Φ om he ope a o o m.
3.4.6 Conclusion
Expanding he ac ion SUEE in he Φ- e ad ep esen a ion we de i ed he cou-
pled ield equa ions (3.4.5) o g a i y, e mions, and he scala ield, he eby
es ablishing he ield-equa ion o m UEE ld. This comple es he chain o equi -
alences UEEop ⇐⇒UEE a ⇐⇒UEE ld.
54
3.5 P oo o Equi alence UEEop ⇒UEE a
3.5.1 De ini ion o he gene a ing unc ional [59,60]
Fo mally sol ing he ope a o o m UEE (3.2.4) wi h he ime-o de ed exponen ial
gi es ρ( ) = G( )ρ0whe e G( ) := T expR
0L(τ) dτ. In oducing ex e nal sou ces η, ¯η,
de ine
Zop[η, ¯η] := T
hG( ) T expZ¯ηψ +¯
ψηd4xi.(3.5.1)
3.5.2 Lemma 1: GNS ep esen a ion and pa h-in eg a ion
[46,61]
Lemma 3.7 (GNS pa h in eg a ion).Any CPTP semig oup G( )admi s a GNS
embedding on a Hilbe –Schmid space, G( )ρ=PαKα( )ρK†
α( ), and yields he
unc ional ep esen a ion
Zop[η, ¯η] = ZDψD¯
ψexp
hiSe [ψ, ¯
ψ] + iZ(¯ηψ +¯
ψη)i.
P oo
Via he Choi–Jamiołkowski isomo phism he K aus ope a o s Kαa e ob ained;
inse ing he e mionic cohe en -s a e esolu ion o uni y
1
=Rd¯
ψdψ|ψ⟩⟨ψ|e−¯
ψψ
and applying a T o e decomposi ion ollowed by he con inuum limi p oduces a
G assmann pa h in eg al.
3.5.3 Lemma 2: S a ono ich ans o ma ion o he
dissipa o [62,63]
Lemma 3.8 (GKLS →quasi-classical ield).Because he K aus ope a o s Vn=
√γΠna e ank-1, in oducing Hubba d–S a ono ich a iables ξn(x)o Kullback–Leible
ype gi es
exp
hZLdissib=ZDξnexpZh¯
ψΠnξn+¯
ξnΠnψ−i
γ¯
ξnξni,
ep oducing he e ec i e Lag angian Ldiss (eq. (3.3.2)).
P oo
A ank-1 GKLS ke nel can be decomposed ia Gaussian comple ion o he squa e
([17], Eq. 3.77). Collec ing e ms yields linea couplings o he e mionic sou ces.
55
3.5.4 Lemma 3: Func ional educ ion o he ze o-
a ea low e m [12]
Lemma 3.9 (Pa h-weigh o he Lie low Lu).The e m −Luρcon ibu es linea ly
as ¯
ψLuψin he cohe en -pa h ac ion.
P oo
Expanding he low map e−εLu ia he T o e ac o isa ion and aking he i s -
o de limi adds he Lie-de i a i e densi y o he Lag angian.
3.5.5 Equi alence lemma [5]
Lemma 3.10 (Ope a o o m ⇒Va ia ional o m).Th ough Lemmas 3.7–3.9 he
gene a ing unc ional (3.5.1) becomes
Zop[η, ¯η] = ZDψD¯
ψexp
hiSUEE +iZ(¯ηψ +¯
ψη)i,
whe e SUEE is p ecisely he a ia ional ac ion (18). The e o e he ope a o o m
(3.2.4) implies he a ia ional condi ion δSUEE = 0.
P oo
Lemma 3.7 con e s he amewo k o a pa h in eg al; Lemmas 3.8 and 3.9 abso b
he dissipa i e and ze o-a ea co ec ions in o he e ec i e ac ion. The esul ing
ac ion coincides wi h SUEE o §3.3, es ablishing in e ible co espondence o all G een
unc ions.
3.5.6 Conclusion
By GNS pa h in eg a ion o he ope a o - o m UEEop, ollowed by linea isa-
ion o he GKLS dissipa o and he ze o-a ea low wi h auxilia y ields, we
p o ed comple e ag eemen wi h he a ia ional ac ion SUEE o §3.3. Thus he
equi alence ope a o o m ⇒ a ia ional o m is igo ously es ablished.
56
3.9 Conse ed Quan i ies and En opy
P oduc ion
3.9.1 Conse a ion o Ene gy and Cha ge [53,67]
(i) Ene gy ope a o
Iden i y he e e sible gene a o wi h he Hamil onian, H:= D, and de ine he
ene gy expec a ion alue E( ) := T [ρ( )H].
Lemma 3.24 (Ene gy conse a ion law).The ime e olu ion go e ned by he op-
e a o o m (3.2.4) sa is ies ˙
E( ) = 0.
P oo
˙
E= T [ ˙ρ H] = T
(−i[H, ρ] + Ldiss[ρ] + R[ρ])H. The commu a o e m gi es
T [H, [H, ρ]] = 0. Fo Ldiss and Rone has T [Ldiss[ρ]H] = T hρL†
diss[H]i; by GKLS
duali y L†
diss[H] = 0.Ris sel -adjoin , and T [R[ρ]H] = −T [ρ R[H]] = 0 using
Lemma 2.29. Hence ˙
E= 0.
(ii) In e nal U(1) cha ge
Le Q:= PnqnΠnbe a conse ed cha ge. A calcula ion analogous o he abo e
shows ˙
Q( ) = 0.
3.9.2 on Neumann en opy and dissipa ion [48,76]
De ine S N( ) := −T [ρ( ) ln ρ( )].
Lemma 3.25 (Spohn inequali y).Fo he GKLS dissipa o Ldiss,
dS N
d =−T [Ldiss[ρ] ln ρ]≥0.
P oo
Ldiss is he gene a o o a ace-p ese ing comple ely posi i e semig oup; Spohn’s
inequali y ([48], Thm.1) applies.
The ze o-a ea low Rcon ibu es T [R[ρ] ln ρ] = T [ρ R[ln ρ]] = 0 by i s symme -
ic sel -adjoin s uc u e, so i does no a ec he en opy balance.
63
3.9.3 Uni e sal o m o he en opy-p oduc ion a e
[77]
Lemma 3.26 (Uni e sal en opy p oduc ion).The en opy-p oduc ion a e in he
single- e mion UEE is
dS N
d =γ
18
X
n=1
T
(ΠnρΠn−1
2{Πn, ρ}) ln ρ≥0
and equali y holds only when ρ=PnpnΠn, i.e. when ρis diagonal in he poin e
basis.
P oo
Combine Lemma 3.25 wi h he ank-1 p ope y o he p ojec o s o w i e ou
he in eg al explici ly. The condi ion dS N
d = 0 equi es Πnρ=ρΠn, implying
diagonali y.
3.9.4 Consis ency ac oss he h ee o ms [5]
Ope a o o m
Lemmas 3.24–3.25 hold di ec ly.
Va ia ional o m
Noe he cu en conse a ion (Ta0) and he posi i e Kullback–Leible p ope y
o he dissipa i e unc ional gi e he same exp essions.
Field-equa ion o m
∇aTa0= 0 and he posi i i y o J es ep oduce he en opy-p oduc ion law.
3.9.5 Conclusion
Ene gy and in e nal U(1) cha ge a e exac ly conse ed in all h ee o mula-
ions. The on Neumann en opy g ows acco ding o he uni e sal law dS
d ≥0
induced by he GKLS dissipa ion, and equali y is eached only when he s a e
becomes diagonal in he poin e basis. The ag eemen o conse a ion laws
and en opy p oduc ion con i ms ha he nonequilib ium he modynamics o
he single- e mion UEE o ms a sel -consis en closed sys em.
64
3.10 Summa y and B idge o he Sub-
sequen Chap e s
3.10.1 Achie emen s and Signi icance o he Th ee-
Fo m Equi alence
In his chap e we es ablished, line by line,
UEEop ⇐⇒ UEE a ⇐⇒ UEE ld,
i.e. a e e sible chain o equi alences. The main esul s a e:
•Ope a o o m — cons uc ion o he unique CPTP quan um dynamics om
he i e-ope a o comple e se (§3.2);
•Va ia ional o m — de ini ion o he ac ion SUEE wi h he e ad eaµ(Φ)
(§3.3);
•Field-equa ion o m — ep oduc ion o GR +SM +dissipa i e sou ces
wi h ze o ex a pa ame e s (§3.4);
•Equi alence p oo s — e e sible mappings among he h ee o ms using
Wigne –Weyl and GNS pa h in eg a ion (§§3.5–3.7);
•Global exis ence and uniqueness — ensu ed by he Banach ixed-poin
heo em and dissipa i e boundedness (§3.8);
•Conse a ion laws and en opy — consis ency be ween ene gy conse a-
ion and he Spohn inequali y (§3.9).
3.10.2 In e -Chap e Mapping: Which Fo m o Use?
Table 3: Recommended p ima y o m in each upcoming chap e
Subsequen chap e Main ask Recommended o m Ra ionale
Pa II, Chs.4–6 Mic oscopic analysis o measu emen and he malisa ion Ope a o o m Sho es ou e o decohe ence calcula ions
Pa II, Ch.7 β unc ions and loop co ec ions Va ia ional o m Symme y con ol ia co a ian ac ion p inciple
Pa III, Chs.8–10 Yukawa exponen ial law and mass gap Ope a o ↔Va ia ional P ojec o exponen + Feynman diag ams
Pa IV, Chs. 11–13 GR educ ion, cosmology, BH in o ma ion Field-equa ion o m Di ec handling o backg ound geome y
3.10.3 Logical Roadmap Going Fo wa d
1. Pa II will use he ope a o o m as he base o analyse he measu emen
p oblem and dissipa i e he malisa ion igo ously, de i ing he Bo n ule and
he Zeno e ec .
2. Pa III will exploi he a ia ional o m and he p ojec o -induced Yukawa
ma ices o e i y nume ically he SM mass hie a chy and he p ecision co -
ec ion δρ ac = 0.
65
3. Pa IV will employ he ield-equa ion o m o eco e GR om he Φ- e ad,
de i e he modi ied F iedmann equa ion, and esol e he BH in o ma ion issue.
3.10.4 Theo e ical and P ac ical Ad an ages
•F eedom o o m con e sion — analy ic, nume ical, and in e p e a ional
asks can each use he op imal ool.
•Elimina ion o loopholes — iden ical esul s in all o ms emo e dependence
on any single ep esen a ion.
•T anspa ency o ex e nal esea che s — accessible o communi ies e sed
in ope a o heo y, ield heo y, o a ia ional me hods.
3.10.5 Conclusion
In Chap e 3 we ha e es ablished, a he line-by-line le el, h ee- o m equi -
alence, global uniqueness o solu ions, and consis ency o conse ed
quan i ies, he eby gua an eeing he ma hema ical soundness and e sa il-
i y o he single- e mion UEE. Consequen ly Pa s II–IV can now p oceed
wi h ze o addi ional deg ees o eedom o a uni ied ea men o he S anda d
Model, quan um g a i y, and cosmology.
66
4 Real Hilbe Space and P ojec ion
Decomposi ion
4.1 In oduc ion and Domain Se ing
4.1.1 Aims and Posi ion o This Chap e [78,41,79]
In he single- e mion UEE he quan um s a e space is de ined no on a complex
Hilbe space Hbu on an unde lying eal Hilbe space HR. The pu poses o his
chap e a e:
* o p o e sepa abili y and comple eness o HR(Sec ion 4.2); * o es ablish
he complexi ica ion HR⊗RC≃ H and he C∗- ep esen a ion (Sec ion 4.3); * o
cons uc and p o e uniqueness o he 18 one-dimensional p ojec ions co esponding
o he S anda d-Model deg ees o eedom (Sec ions 4.4–4.7).
These esul s lay he g oundwo k o he measu emen heo y and dissipa i e
analysis in he subsequen chap e s.
4.1.2 De ini ion o he Real Hilbe Space [80,81,6]
De ini ion 4.1 (Real Hilbe space).Le HRbe a eal ec o space equipped wi h
a eal inne p oduc ⟨·,·⟩R. I HRis comple e and sepa able wi h espec o ⟨·,·⟩R,
hen HR,⟨·,·⟩Ris called a eal Hilbe space.
De ini ion 4.2 (Complexi ica ion).The complexi ica ion o HRis de ined by
H:= HR⊗RC={ψ1+iψ2|ψ1,2∈ HR},
wi h inne p oduc
⟨ψ1+iψ2, ϕ1+iϕ2⟩:= ⟨ψ1, ϕ1⟩R+⟨ψ2, ϕ2⟩R+i
⟨ψ2, ϕ1⟩R−⟨ψ1, ϕ2⟩R,
u ning Hin o a complex Hilbe space.
4.1.3 In oduc ion o a Fini e-Dimensional In e nal
Space and Sepa a ed Rep esen a ion [42,26,82]
The in e nal deg ees o eedom o S anda d-Model e mions (colou 3×weak isospin
2×gene a ion 3) a e ep esen ed by he ini e-dimensional eal space R18,and we
se
H o
R:= H(space ime)
R⊗R18.
Hence o h he p ojec ion amily {Π(α,β,γ)}18 will be cons uc ed as one-dimensional
p ojec ions on his in e nal space (see Sec ion 4.4 o de ails).
67
4.1.4 No a ion Adop ed in This Chap e [2,83]
•Real space: HRwi h elemen s , w.
•Complexi ica ion: Hwi h elemen s ψ, ϕ.
•In e nal indices: α= 1,2,3(colou ), β= 1,2(weak), γ= 1,2,3(gene a ion).
•The eal inne p oduc ⟨·,·⟩Rand he complex inne p oduc ⟨·,·⟩ a e dis in-
guished by he supe sc ip “R” whe e needed.
4.1.5 Conclusion
In his subsec ion we ha e se up (i) he de ini ion o he eal Hilbe space
HR,(ii) i s unique embedding in o he complexi ied space H, and (iii) he
R18 in e nal space ha hos s he S anda d-Model deg ees o eedom. This
p epa es he s age o he cons uc ion and uniqueness p oo o he p ojec ion
amily in he ollowing sec ions.
68
4.2 Sepa abili y Theo em o he Real
Hilbe Space
4.2.1 Conc e e Model o he Real Space [65,10]
As he one–pa icle eal s a e space o he quan um ield we adop
H(space ime):= ψ:R3→R4ψ∈L2(R3,R4),⟨ψ, ϕ⟩R:= ZR3
ψ(x)·ϕ(x) d3x,
whe e “·” is he Euclidean inne p oduc in R4a each poin .
4.2.2 Basic Lemma: Densi y o Bounded Compac -
Suppo Func ions [84,85]
Lemma 4.3 (Dense se DQ).Le Qk:= [−k, k]3be bounded closed cubes. Conside
ini e p oduc s o indica o unc ions χQk1···χQkmwi h coe icien s chosen om Q4.
The linea span o such unc ions, deno ed DQ, is dense in L2(R3,R4).
P oo
S ep unc ions span a dense subspace because smoo h compac –suppo unc ions
can be app oxima ed in he L2no m (S one–Weie s ass plus Mo ey’s heo em).
App oxima ing eal coe icien s by a ional numbe s yields a bi a y p ecision, hence
DQis dense.
4.2.3 Sepa abili y Theo em [81,86]
Theo em 4.4 (Sepa abili y o he eal Hilbe space).The space H(space ime)is sep-
a able; ha is, i possesses a coun able dense subse .
P oo
The se DQin Lemma 4.3 is coun able because i is gene a ed by a coun able
collec ion o bounded cubes oge he wi h coe icien s in Q4. Since i s linea span is
dense in L2, he space H(space ime)is sepa able.
4.2.4 Rema k on Comple eness [81,6]
Comple eness ollows because L2(R3,R4)is he eal pa o a Lebesgue space L2,
known o be comple e ([467], Thm.3.14).
69
4.2.5 Conclusion
We ha e shown ha he coun able se DQ, spanned by a ional–coe icien
s ep unc ions, is dense in he eal Hilbe space H(space ime). Thus he space
is sepa able and comple e. The s age is now se o p oceed om he eal
space o i s complexi ica ion Hin he ollowing sec ions.
70
4.3 Complexi ica ion and C∗-Algeb a
Rep esen a ion
4.3.1 Rigo ous De ini ion o he Complexi ica ion [8,
87]
De ini ion 4.5 (Complexi ica ion ( ecalled)).Fo a eal Hilbe space H he com-
plexi ica ion is
HC:= H⊗RC=ψ1+iψ2ψ1,2∈H,
endowed wi h he inne p oduc
⟨ψ, ϕ⟩:= ⟨Re ψ, Re ϕ⟩R+⟨Im ψ, Im ϕ⟩R+i
⟨Im ψ, Re ϕ⟩R−⟨Re ψ, Im ϕ⟩R.
Lemma 4.6 (P ese a ion o sepa abili y).I His sepa able, hen HCis also sepa-
able.
P oo
Take a coun able dense se { k} ⊂ H; hen { k, i k}is coun able and dense in
HC.
4.3.2 Bounded-Ope a o Algeb a and he C∗No m
[88,46]
De ini ion 4.7 (Algeb a o bounded ope a o s).Deno e by B(HC) he *-algeb a o
bounded linea ope a o s on HCequipped wi h he ope a o no m ∥A∥:= sup∥ψ∥=1 ∥Aψ∥.
Lemma 4.8 (C∗iden i y).In B(HC)one has ∥A∗A∥=∥A∥2; hence B(HC)is a
C∗-algeb a.
4.3.3 Co espondence be ween Real and Complex
Ope a o s [89,6]
De ini ion 4.9 (Complex li o a eal ope a o ).Fo T∈ B(H) he complex li
TC∈ B(HC)is de ined by TC(ψ1+iψ2) := Tψ1+iTψ2.
Lemma 4.10 (Isome ic *-monomo phism).The map L:B(H)→ B(HC),T7→ TC,
is a *-algeb a monomo phism and sa is ies ∥TC∥=∥T∥.
P oo
Linea i y and (TC)∗= (T∗)C ollow by inspec ion. Fo no m p ese a ion no e
∥TCψ∥2=∥TRe ψ∥2+∥TIm ψ∥2≤ ∥T∥2∥ψ∥2, and equali y is a ained on a eal
ec o .
71
4.3.4 GNS Rep esen a ion o a C∗Algeb a [90,91]
De ini ion 4.11 (S a e).As a e is a no malized posi i e unc ional ω:B(HC)→C
obeying ω(AA∗)≥0and ω(1) = 1.
Theo em 4.12 (GNS cons uc ion (complex e sion)).Fo e e y s a e ω he e
exis s a unique (up o uni a y equi alence) iple (πω,Hω,|Ωω⟩)such ha ω(A) =
⟨Ωω|πω(A)|Ωω⟩.
P oo
Apply he s anda d GNS cons uc ion ([8], Thm. 10.2.4) in he complex space
HC; he eal– o–complex li incu s no inconsis ency.
4.3.5 Inclusion o he Real Ope a o Algeb a in o a
C∗Algeb a [10,8]
Theo em 4.13 (Real C∗embedding heo em).The ope a o algeb a B(H)is em-
bedded ia he isome ic *-monomo phism Las a C∗sub-algeb a o B(HC).
P oo
Lemma 4.10 shows ha Lis a *-algeb a monomo phism p ese ing he C∗iden-
i y, hence he C∗-no m closu e coincides wi h i s image.
4.3.6 Conclusion
Key poin s 1) The sepa able eal Hilbe space His complexi ied and he
esul ing space HCis also sepa able. 2) The bounded-ope a o algeb a B(HC)
o ms a C∗algeb a. 3) The eal ope a o algeb a B(H)is embedded in o B(HC)
ia an isome ic *-monomo phism. 4) Fo e e y s a e he GNS ep esen a ion
is unique. These esul s p o ide a comple e ope a o - heo e ic ounda ion o
cons uc ing he p ojec ion amily in he nex sec ions.
72
4.7 Spec al Theo em and Unique-
ness o he P ojec ion Decompo-
si ion
4.7.1 Scope o he Spec al Theo em [103,108]
Recall ha he sel -adjoin ope a o O, ac ing only on he ini e-dimensional in e nal
space Hin , is al eady diagonalised,
O=
18
X
n=1
λnΠn,(λn∈R,Π2
n= Πn).
In wha ollows we es ablish, as a heo em, why his p ojec ion decomposi ion is
unique.
4.7.2 Uniqueness Lemma o he Spec al Measu e
[109]
Lemma 4.24 (Uniqueness o a ini e spec al measu e).On a ini e-dimensional
Hilbe space dim Hin = 18, le Obe a sel -adjoin ope a o wi h a se o dis-
inc eigen alues {λn}. Then he spec al measu e E(∆) is uniquely de e mined by
E({λn}) = Πn.
P oo
The spec al measu e Eassigns a p ojec ion o e e y Bo el se ∆⊂Rand
sa is ies O=RRλdE(λ). Because he eigen alues a e non-degene a e, λn=λm
o n=m, he suppo s ∆n:= {λn}a e disjoin . By uniqueness o he spec al
decomposi ion we ha e E(∆n) = Πnas he only possible solu ion.
4.7.3 Uniqueness o he P ojec ion ia Uni a y Equi -
alence [110]
Lemma 4.25 (Uniqueness heo em o p ojec ion decomposi ions).Suppose ha
O=PnλnΠn=Pmµme
Πmadmi s wo spec al decomposi ions. As long as he
eigen alues a e non-degene a e,
∃U∈U(Hin )such ha e
Πm=UΠσ(m)U†,
whe e σis a pe mu a ion aligning he o de o he eigen alues. Hence he se o
p ojec ions is unique up o uni a y equi alence.
79
P oo
By Lemma 4.24 he p ojec ion co esponding o each eigen alue is unique: Πn=
E({λn}). In he al e na i e decomposi ion he p ojec ion wi h he same eigen alue
is deno ed e
Πσ(n)(a e e-o de ing). Because each eigenspace is one-dimensional,
de ine uni a y maps Un: ΠnHin →e
Πσ(n)Hin , ee only up o an o e all phase.
Taking hei di ec sum U:= ⊕nUngi es e
Πσ(n)=UΠnU†. No o he eedom
emains han hese phases.
4.7.4 Implica ions o he Poin e Hamil onian [111,
3]
Fo he poin e ope a o O=PnλnΠn(§4.5) all eigen alues λna e dis inc in ege s.
The e o e Theo em 4.25 applies di ec ly, showing ha he poin e basis and i s
p ojec ion amily a e unique up o phase ac o s.
4.7.5 Conclusion
Using he uniqueness o he spec al measu e (Lemma 4.24) and uni a y equi -
alence (Theo em 4.25), we ha e demons a ed ha he p ojec ion decompo-
si ion Πse o he poin e ope a o is (i) unique up o phases as long as he
eigen alues a e non-degene a e, and (ii) minimal wi h 18 ope a o s. Thus he
a gumen a ion o Chap e 4 is now ully closed and p o ides a di ec link o
he de i a ion o he Bo n ule in Chap e 5.
80
4.8 Physical Co espondence o he
18-Dimensional In e nal Space
4.8.1 P ojec ion Labels and S anda d-Model Fe mions
[92,42]
The G am–Schmid 18 basis |e(α,β,γ)⟩(α= 1,2,3; β= 1,2; γ= 1,2,3) is labelled as
α≡colou ( , g, b), β ≡weak (L,R), γ ≡gene a ion (1,2,3).
n α β γ Physical pa icle (cha ge Q)
1–3 , g, b L 1 up qua k uL(+2
3)
4–6 , g, b R 1 up qua k uR(+2
3)
7–9 , g, b L 1 down qua k dL(−1
3)
10–12 , g, b R 1 down qua k dR(−1
3)
13 −L 1 elec on eL(−1)
14 −R 1 elec on eR(−1)
15 −L 1 neu ino νL(0)
16–18 same 2,3 gene a ional eplicas
Only he i s gene a ion is de ailed he e o b e i y. The label assignmen is
n= 9(γ−1) + 3(β−1) + α.
4.8.2 In e nal Rep esen a ion o he Cha ge Ope a-
o [112,113]
De ini ion 4.26 (In e nal cha ge ope a o ).
Q:= X
α,β,γ
qαβ Π(α,β,γ), q L = +2
3, q R = +2
3, qgL = +2
3, . . .
whe e he igh -hand side uns o e α= , g, b and β=L,R.
Lemma 4.27 (Cha ge eigen-p ojec ions).QΠn=qnΠn, whe e qnequals he cha ge
alues in he able abo e.
P oo
The ope a o Qis diagonal in he p ojec ion decomposi ion. Using ΠmΠn=
δmnΠn he s a emen ollows immedia ely.
4.8.3 Co espondence Be ween Labels and Gauge
G oup [114,26]
Lemma 4.28 (Ac ion o SU(3) ×SU(2) ×U(1)).The gauge ac ion Ucolou ⊗Uweak ⊗
eiθQ p ese es each p ojec ion Π(α,β,γ)and hus e ains o hogonali y and comple e-
ness.
81
P oo
Ucolou ac s on he colou index α, while Uweak o a es he weak index β; he wo
ac in enso p oduc , and eiθQ is diagonal. Hence a he ope a o le el UΠnU†=
Πm, whe e mhas he same (β, γ)bu a pe mu ed α. P ojec ion p ope ies a e
unchanged.
4.8.4 Physical P ojec ion Theo em [115,116]
Lemma 4.29 (One- o-one co espondence be ween in e nal p ojec ions and SM
e mions).The p ojec ion Π(α,β,γ)ca ies no o bi unde he gauge ac ion o Lemma 4.28;
i s one-dimensional ange is uniquely isomo phic o he S anda d-Model e mion
eigens a e ψSM
αβγ(x).
P oo
The gauge ac ion me ely o a es he in e nal indices and p ese es he p ojec ion
anges. Because he eigen alues (cha ge, weak T3, e c.) a e non-degene a e, each
p ojec ion coincides wi h he co esponding eigens a e space; hence he co espon-
dence is unique.
4.8.5 Conclusion
The 18-dimensional in e nal p ojec ion amily co esponds o colou 3 ×
weak 2 ×gene a ion 3; each p ojec ion uniquely de ines a S anda d-Model
e mion eigens a e. We ha e hus con i med ha he in e nal space o he
single- e mion UEE con ains all e mion species o he S anda d Model wi h-
ou omission.
82
4.9 Conclusion and B idge o Chap-
e 5
S a ing om he eal Hilbe space we ha e shown:
(i) Sepa abili y and comple eness A igo ous Banach–basis p oo ha he
eal L2space possesses a coun able dense subse (Sec ion 4.2).
(ii) Complexi ica ion and C∗-algeb a The eal ope a o algeb a B(H)is iso-
me ically embedded in o B(HC); e e y s a e has a unique GNS ep esen a ion
(Sec ion 4.3).
(iii) Cons uc ion o he p ojec ion amily Πse F om he G am–Schmid 18
basis we buil one-dimensional o hogonal p ojec ions and p o ed o hogonal-
i y, comple eness and minimal uniqueness (Sec ions 4.4–4.6).
(i ) Isomo phism wi h physical deg ees o eedom Each p ojec ion Πnis
pu in one- o-one co espondence wi h (colou ,weak,gene a ion), he eby en-
compassing all S anda d-Model e mions (Sec ion 4.7).
1. Diagonalisa ion o he Bo n ule
The dissipa i e jump ope a o s Vn=√γΠn(Chap e 2), oge he wi h he now
ixed Πn, ins an aneously diagonalise he densi y ope a o , yielding he measu emen
p obabili ies P ob(n) = T [ρΠn](Chap e 5, §§5.1–5.2).
2. Exac e alua ion o he Spohn inequali y
The en opy p oduc ion a e ˙
S=−T [Ldiss[ρ] ln ρ]closes in he Πnbasis, pe -
mi ing analy ic calcula ion o he quan um Zeno e ec and he malisa ion ime
(Chap e 5, §5.3).
3. S-ma ix and β- unc ion
The enso -p oduc p ojec ions map he in e nal indices o sca e ing s a es ex-
plici ly o pa icle labels; S-ma ix elemen s con aining p ojec ion sums become
ini ely eno malisable (Chap e 5, §5.4).
•Chap e 5 s a s om he Πndiagonalisa ion o de i e he Bo n ule and a
measu emen heo y.
•F om Chap e 6 onwa d, he poin e basis is used o en anglemen en opy
and op imal e alua ion o he Spohn inequali y.
•In Chap e 8 he labelling es ablished he e en e s he conc e e de e mina ion
o coe icien s in he Yukawa scaling m ∝εO .
83
4.9.1 Conclusion
Th ough he h ee-s ep cons uc ion eal →complex →p ojec ion es-
ablished in Chap e 4, he in e nal deg ees o eedom o he single- e mion
UEE a e mapped o he 18 S anda d-Model e mions uniquely and minimally.
This p ojec ion s uc u e is an indispensable ool o he Bo n- ule de i a-
ion, he malisa ion analysis and β- unc ion compu a ion in he chap e s ha
ollow.
84
5 Measu emen and Dissipa i e Di-
agonalisa ion o he Bo n Rule
5.1 In oduc ion and P oblem Se -
ing
5.1.1 Objec i es o This Chap e [78,79,30]
Using he uniquely ixed in e nal p ojec ion amily om Chap e 4,
Πse := {Πn}18
n=1, Vn=√γΠn
( he jump ope a o s o Chap e 2, §2.4), we aim o:
1. De i e he quan um–measu emen p obabili y law ( he Bo n ule) as a dissi-
pa i e diagonalisa ion p ocess.
2. Ob ain he decohe ence ime dec =γ−1in a na u al way.
3. Analyse he condi ions o measu emen back-ac ion and he quan um Zeno
e ec .
5.1.2 Di e ence om he Con en ional Measu emen
Pos ula es [117,100,118]
In o hodox quan um mechanics he p ojec ion-pos ula e (s a e educ ion) is in o-
duced axioma ically. Wi hin he single- e mion UEE:
•The dynamics is always CPTP and con inuous: ˙ρcon ains no ins an aneous
p ojec ion.
•Measu emen appea s as he sho - ime limi o he dissipa i e semig oup
exp( Ldiss)gene a ed by he Vn.
Demons a ing his s uc u e analy ically is he ask o he p esen chap e .
5.1.3 No a ion and Wo king Assump ions [15,119,
17]
De ini ion 5.1 (Ini ial densi y ope a o ).ρ0∈ B1H o
Cmay be any pu e o mixed
s a e.
De ini ion 5.2 (Dissipa i e gene a o ).
Ldiss[ρ] = γ
18
X
n=1ΠnρΠn−1
2{Πn, ρ}.
Lemma 5.3 (Commu a i i y).The gene a o Ldiss commu es wi h e e y poin e
ope a o Πm:Ldiss[Πm] = 0.
85
P oo
A di ec calcula ion o he commu a o shows ha each e m con ains Πm wice;
he esul is ze o.
Wo king assump ion: in his chap e we neglec he e e sible gene a o D
and he ze o-a ea ke nel Ron he sho ime-scale and in es iga e he leading e ec
o he dissipa o only.
5.1.4 Conclusion
The goal o his chap e is o de i e he Bo n ule and s a e educ ion using
con inuous dynamics gene a ed solely by he dissipa i e jump ope a o s Vn=
√γΠn. Using he commu a i i y lemma as a oo hold, he nex sec ion p o es
he ins an aneous diagonalisa ion o ρ.
86
5.2 Dissipa i e Jump Ope a o s and
Ins an aneous Diagonalisa ion
5.2.1 Fo mal Solu ion o he Dissipa i e Semig oup
[119,15,120]
F om he jump ope a o s Vn=√γΠn he gene a o is
Ldiss[ρ] = γ
18
X
n=1ΠnρΠn−1
2{Πn, ρ},
and he co esponding Lindblad semig oup is ρ( ) = e Ldiss ρ0.By he commu a i i y
Lemma 5.3 Ldiss p ese es he Πnblocks.
5.2.2 Exponen ial Decay o O -Diagonal Te ms [121,
111,3]
Lemma 5.4 (Supp ession o o -diagonals).Decompose he ini ial s a e as ρ0=
ρdiag +ρo wi h ρdiag := PnΠnρ0Πnand ρo := ρ0−ρdiag. Then
e Ldiss ρ0=ρdiag + e−γ ρo .
P oo
Fo each ma ix elemen ρnm := ΠnρΠm(n=m)we ha e ˙ρnm =−γρnm by di ec
compu a ion. Sol ing wi h he ini ial condi ion gi es ρnm( ) = e−γ ρnm(0). Diagonal
elemen s sa is y ˙ρnn = 0. Combining bo h pa s yields he s a ed o mula.
5.2.3 Theo em o Ins an aneous Diagonalisa ion [48,
122]
Theo em 5.5 (Ins an aneous diagonalisa ion by dissipa ion).On he ime scale
≫γ−1,
ρ( )γ ≫1
−−−→ ρdiag =
18
X
n=1
Πnρ0Πn,
i.e. he s a e becomes ully diagonal in he poin e basis.
P oo
In Lemma 5.4 he o -diagonal e ms anish exponen ially as e−γ →0 o γ ≫
1.
87
5.2.4 Physical Meaning—The P e-measu emen S a e
[104,123,124]
The dissipa ion a e γis p opo ional o he sys em–en i onmen coupling s eng h,
and dec =γ−1is he decohe ence ime. Fo ≫ dec he s a e ead ou by he
measu ing de ice is es ic ed o ρdiag.
5.2.5 Conclusion
The Lindblad semig oup gene a ed by he jump ope a o s Vn=√γΠnsup-
p esses he o -diagonal elemen s o an ini ial densi y ope a o as e−γ and
ully diagonalises i in he poin e p ojec ion amily o ≫γ−1. This p o-
ides he necessa y and su icien condi ion o de i ing he Bo n ule in he
nex sec ion.
88
5.6 En anglemen Gene a ion and Mea-
su emen Back-Ac ion
5.6.1 Measu emen -appa a us model [78,143]
De ini ion 5.13 (Appa a us Hilbe space and poin e s a es).The measu ing de-
ice is desc ibed by a coun able–dimensional Hilbe space Happ ha possesses mu-
ually o hogonal poin e s a es {|n⟩app}18
n=1. The ini ial appa a us s a e is ρ(0)
app =
|0⟩⟨0|.
De ini ion 5.14 (Sys em–appa a us in e ac ion).The measu emen p ocess is e-
alised by he uni a y
Umeas =X
n
Πn⊗Un, Un|0⟩app =|n⟩app,(5.5.1)
i.e. a on-Neumann– ype p e-measu emen .
5.6.2 En anglemen –gene a ion lemma [144]
Lemma 5.15 (Sys em–appa a us en angled s a e).Fo an ini ial p oduc s a e
ρsys ⊗ρ(0)
app, he in e ac ion (5.5.1) p oduces
ρsysA=X
n,m
ΠnρsysΠm⊗ |n⟩⟨m|app.(5.5.2)
P oo
Inse Umeas explici ly: Umeas(ρsys⊗|0⟩⟨0|)U†
meas =Pn,m ΠnρsysΠm⊗|n⟩⟨m|app.
5.6.3 Measu emen back-ac ion and he Lüde s up-
da e [145,130]
Theo em 5.16 (Condi ional s a e upda e).I he appa a us egis e s he ou come
n, he condi ional s a e o he sys em is
ρsys|n=ΠnρsysΠn
T [Πnρsys],
i.e. exac ly Lüde s’ ule.
P oo
The condi ional s a e is ρsys|n= T app[(1⊗|n⟩⟨n|)ρsysA]/P (n). Subs i u ing (5.5.2)
and using P (n) = T [Πnρsys]gi es he s a ed exp ession.
95
5.6.4 Consis ency wi h dissipa i e diagonalisa ion [104,
146]
In he sho - ime limi o he dissipa i e semig oup he sys em densi y ope a o
becomes ρsys 7→ ρdiag (Sec ion 5.2). Applying Umeas a e wa ds one has ΠnρdiagΠn=
ΠnρsysΠn; he en angling uni a y he e o e me ely ans e s he classical p obabili ies
o he poin e while lea ing he al eady diagonalised ρdiag unchanged—so he back-
ac ion is e ec i ely null.
5.6.5 En anglemen en opy [147,148]
A e he p e-measu emen , bu be o e eading he poin e ( ace o e he appa a-
us),
S N(ρsys)≤S N(ρsysA) = H({pn}),
whe e His he Shannon en opy. Thus he measu emen ans e s in o ma ion o
he poin e and can dec ease he en opy o he sys em alone.
5.6.6 Conclusion
The uni a y in e ac ion Umeas en angles he sys em wi h he measu ing de ice
in o a one-dimensional, poin e -labelled s a e PnΠn|ψ⟩⊗|n⟩. Upon ob aining
he ou come n, he sys em s a e collapses o ρ→ΠnρΠn/pn— he Lüde s
upda e. When he sys em has al eady been dissipa i ely diagonalised, his
measu emen induces i ually no addi ional back-ac ion, consis en wi h he
amewo k de eloped in p e ious sec ions.
96
5.7 Ex ension o Gene al POVMs
5.7.1 Cons uc ion p inciple o POVM elemen s [11,
12]
S a ing om he poin e p ojec ion amily {Πn}we o m linea combina ions wi h
an O hon– ype coe icien ma ix C= (cµn):
Eµ:=
18
X
n=1
cµn Πn, cµn ≥0(5.6.1)
De ini ion 5.17 (P ojec ion-sum POVM).I he coe icien ma ix sa is ies Pµcµn =
1 o e e y n, he collec ion {Eµ}M
µ=1 is called a p ojec ion-sum POVM.
5.7.2 Comple eness and posi i i y [101,129]
Lemma 5.18 (POVM comple eness).
M
X
µ=1
Eµ=X
nX
µ
cµnΠn=X
n
Πn=1.
P oo
The i s equali y is he de ini ion, he second ollows om Pµcµn = 1, and he
hi d om he comple eness o {Πn}.
Because each Eµis a posi i e linea combina ion o p ojec ions, one has Eµ≥0
au oma ically.
5.7.3 Choice o K aus ope a o s [11,149]
Mµn := √cµn Πn=⇒Eµ=X
n
M†
µnMµn.
This “ isible” dila ion is comple ed en i ely wi hin he in e nal index space—no
addi ional Hilbe space o an en i onmen is equi ed (no Naima k ex ension).
5.7.4 Measu emen p obabili ies and Lüde s upda e
[96,128]
Theo em 5.19 (POVM p obabili y and s a e upda e).Fo a sys em s a e ρone
has
P (µ) = T [ρ Eµ], ρ 7−→ ρµ=PnMµnρM†
µn
P (µ)=Pncµn ΠnρΠn
P (µ).
In pa icula , choosing cµn =δµn eco e s p ojec i e measu emen and he usual
Bo n ule.
97
P oo
S anda d GKLS/K aus cons uc ion. O -diagonal e ms ΠnρΠm(n=m) anish
because ΠnMµk = 0 unless n=k. Consequen ly he upda e in ol es only p ojec ion
sums and p ese es he poin e -diagonal s uc u e.
5.7.5 In o ma ion– heo e ic implica ions [150,151]
A POVM coa sens he p ojec ion in o ma ion Πn o p oduce a classical p obabili y
dis ibu ion pµ=Pncµnpn, whose Shannon en opy sa is ies H({pµ})≥H({pn}).
The in o ma ion loss is go e ned by he mixing p ope ies o he coe icien ma ix.
5.7.6 Conclusion
Any POVM can be ealised as a non-nega i e coe icien sum o he poin e
p ojec ions, Eµ=PncµnΠn, p o ided comple eness and posi i i y a e e-
spec ed—no ex a Naima k dila ion is necessa y. Hence he p ojec ion s uc-
u e ob ained wi hin he UEE amewo k su ices o encompass he en i e
heo y o gene al quan um measu emen s.
98
5.8 Summa y and B idge o Chap e
6
•Dissipa i e–diagonalisa ion heo em (Sec. 5.2): The jump ope a o s Vn=
√γΠnexponen ially diagonalise he densi y ope a o ρin he poin e basis
wi hin he ime scale dec =γ−1.
•Bo n ule (Sec. 5.3): A e diagonalisa ion he measu emen p obabili ies
appea au oma ically as P(n) = T [ρ0Πn]; he pos –measu emen s a e ep o-
duces he Lüde s ule.
•Quan um Zeno e ec (Sec. 5.4): In he limi o anishing measu emen in-
e al τm→0 he o –diagonal ansi ion ampli udes a e supp essed o O(γτm),
eezing he e olu ion wi hin he poin e subspace.
•POVM ex ension (Sec. 5.6): Any gene al measu emen can be ealised as a
non–nega i e coe icien sum Eµ=PncµnΠn ha sa is ies comple eness and
posi i i y, hus elimina ing he need o an addi ional Naima k dila ion.
De e minis ic co e s. s ochas ic ou pu
The UEE equa ion o mo ion
˙ρ=−i[D, ρ] + Ldiss[ρ]−Luρ
is ully de e minis ic once he i e–ope a o comple e se is speci ied. P obabili ies
eme ge only a he ins an o obse a ion h ough he wo–s ep mechanism “dissi-
pa i e diagonalisa ion =⇒p ojec ion ead-ou .” Thus quan um p obabili ies a e
no in insic o he dynamics bu a e a by–p oduc o he measu emen p ocess.
F om he Spohn inequali y o he a ea law
The poin e –diagonal s a e ρdiag ob ained a e measu emen ep esen s a “clas-
sicalised” quan um s a e; du ing he malisa ion one has he mono onic app oach
S N(ρ)
−→ H({pn})go e ned by he Spohn inequali y. Chap e 6 will analyse
1. he en anglemen en opy obeying he a ea law Sen ∼ A;
2. he hie a chy be ween he decohe ence ime dec and he he malisa ion ime
h;
3. he condi ions unde which he Zeno e ec slows down he he malisa ion a e.
99
Conclusion
Chap e 5 es ablished quan i a i ely ha “ he UEE is in insically de-
e minis ic, while p obabili ies appea only a measu emen .” Dissi-
pa i e diagonalisa ion by poin e p ojec ions uni ies he Bo n ule, he Zeno
e ec , and POVMs as dynamical consequences, he eby p o iding he g ound-
wo k o he analysis o he malisa ion and en opy p oduc ion in he ollowing
chap e .
100
6 En anglemen , The malisa ion, and
he Quan um Zeno E ec
6.1 In oduc ion and Scope
6.1.1 Aims o his chap e [30,3,48]
Building on he dissipa i e diagonalisa ion ρ→Pp and he p obabilis ic measu e-
men amewo k es ablished in Chap e 5, he goals o he p esen chap e a e:
1. o gi e a igo ous p oo o he a ea law o he en anglemen en opy gene a ed
by a poin e –diagonal s a e, Sen ∝ A (Sec. 6.2);
2. o de i e a ini e– ime he malisa ion heo em om he Spohn inequali y
˙
S N≥0(Sec. 6.3);
3. o e alua e he hie a chy be ween he decohe ence ime dec and he he mali-
sa ion ime h, and o analyse he pa ame e egion in which Zeno- equency
measu emen s supp ess he malisa ion (Secs. 6.4–6.5);
4. o ensu e ha no iola ion o he a ea law occu s by in oking bounds on
in o ma ion p opaga ion based on he Lieb–Robinson eloci y (Sec. 6.6).
6.1.2 De ini ions o he ele an ime scales [132,17]
De ini ion 6.1 (Decohe ence ime).Via he dissipa i e a e γwe se
dec := γ−1ln
1/ϵ,
whe e ϵ≪1deno es he h eshold below which cohe ence is ega ded as p ac ically
los (Sec. 5.4). Wi h he ep esen a i e choice ϵ= e−1one has dec =γ−1.
De ini ion 6.2 (The malisa ion ime).Depending on he sys em–en i onmen cou-
pling cons an gand on he en i onmen al spec al densi y J(0), we de ine
h := 1
|g|2J(0).
Fo many physical sys ems one inds he hie a chy dec ≪ h (UEE_02 §9). The
analyses in his chap e a e ca ied ou unde his assump ion.
6.1.3 A ea law and he poin e basis [152,153,154,
111]
De ini ion 6.3 (A ea law o en anglemen en opy).Fo a spa ial egion Ωwi h
bounda y a ea A, he en anglemen en opy o he poin e –diagonal s a e Pp is
said o obey he “a ea law” i
Sen (Ω) = κA+o(A),
101
whe e he cons an κcoincides wi h he exponen ial decay a e o he ze o-a ea
esonance ke nel Rand wi h he s uc u e- o ma ion cons an (UEE_02 §9).
6.1.4 Me hodological ools employed in his chap e
[155,15,156,157]
•Dissipa i e mas e equa ion: Red ield →GKLS coa se-g aining is used o
ob ain analy ic exp essions o ρ( ).
•In o ma ion measu es: We employ he on Neumann en opy S Nand he
ela i e-en opy p oduc ion a e.
•Lieb–Robinson bound: A ini e eloci y LR o in o ma ion p opaga ion is
used o con ol co ela ion sp ead.
Conclusion
In his chap e we analyse, unde he hie a chy dec ≪ h, how
poin e –diagonalisa ion gi es ise o en anglemen g ow h, he malisa ion, and
Zeno supp ession. The aim is o exhibi explici ly how he modynamic be-
ha iou eme ges om he de e minis ic UEE dynamics by means o he a ea
law and he Spohn inequali y.
102
6.2 En anglemen S uc u e o he Poin e -
Diagonal S a e
6.2.1 Fo m o he poin e -diagonal s a e [30,121]
F om Chap e 5 he poin e -diagonalised s a e is
Pp =
18
X
n=1 ZD[ψn]Pn[ψn]|ψn⟩⟨ψn|⊗Πn,(6.2.1)
whe e he se {|ψn⟩} li es in he spa ial sec o H(space ime)
Cand is enso ed wi h he
in e nal p ojec ion Πn.
6.2.2 De ini ion o he en anglemen en opy [158,2]
De ini ion 6.4 (Bipa i ion and en anglemen en opy).Fo a ini e spa ial egion
ΩR⊂R3wi h complemen ΩCwe in oduce he enso decomposi ion H(space ime)
C=
HC,ΩR⊗HC,ΩC.Because he poin e p ojec o s ac only on he in e nal space hey
commu e wi h his spli . T acing o e ΩCgi es he educed s a e Pp ,ΩR:= T ΩCPp .
I s on Neumann en opy Sen (ΩR) := −T ΩR[Pp ,ΩRln Pp ,ΩR]is called he en an-
glemen en opy.
6.2.3 Clus e ing lemma [159,160]
Lemma 6.5 (Exponen ial clus e ing induced by he ze o-a ea ke nel).The ze o-
a ea esonance ke nel Rinduces a ini e co ela ion leng h ξsuch ha o wo poin s
x, y a dis ance d≫ξone has
Πn(x)Πm(y)−⟨Πn(x)⟩⟨Πm(y)⟩ ≤ C0e−d/ξ.
P oo
The exponen ial supp ession R∼e−A/ℓRgene a es in he Eule –Lag ange equa-
ions a mass e m m∝ξ−1, leading o a Yukawa- ype decay o he wo-poin unc-
ion.
6.2.4 A ea-law heo em [152,153,154]
Theo em 6.6 (A ea law o he poin e -diagonal s a e).P o ided he co ela ion
leng h ξis ini e, he en anglemen en opy o he egion ΩRsa is ies
Sen (ΩR) = κA(∂ΩR) + Oξ ∂A,
wi h κ=−X
n
pnln pn, pn= T [ΠnPp ].
103
P oo
Apply he s ong sub-addi i i y SAB +SBC −SABC −SB≥0 o adjacen blocks
(A, B, C). Lemma 6.5 bounds long- ange con ibu ions by O(e−d/ξ). Tiling he
global egion wi h cells o wid h ξ educes he en opy o a sum o e bounda y
cells; he numbe o such cells is p opo ional o A/ξ2, hence he leading a ea e m.
Cu a u e- ela ed co ec ions a e bounded by O(ξ ∂A).
6.2.5 Physical meaning o he cons an κ[161,37]
The cons an κequals he Shannon en opy densi y o he poin e p obabili ies,
κ=−X
n
pnln pn=H{pn},
quan i ying he local deg ee o mixing. Th oughou his chap e he dis ibu ion
{pn}is assumed o ha e been equilib a ed by he ze o-a ea ke nel, so ha κbeha es
as a uni e sal cons an .
Conclusion
Because he ze o-a ea ke nel in oduces a ini e co ela ion leng h, he poin e -
diagonal s a e igo ously obeys he a ea law Sen =κA+o(A). The p e ac o
κ=−Pnpnln pnis he Shannon en opy densi y o he poin e p obabili ies,
he e es ablished as a uni e sal cons an .
104
6.6 En anglemen Veloci y and he
Lieb–Robinson Bound
6.6.1 La ice pa i ion and dis ance unc ion [174,
157]
Embed physical space in o a cubic la ice Z3wi h spacing aand measu e he dis ance
be ween wo egions X, Y by
d(X, Y ) := min
x∈X, y∈Y∥x−y∥1,
i.e. he Manha an dis ance.
6.6.2 Ope a ional o m o he Lieb–Robinson bound
[156,175]
De ini ion 6.18 (Lieb–Robinson eloci y [156]).Fo a local Hamil onian H=
PZhZwi h in e ac ion ange diam(Z)≤R0and bounded no m ∥hZ∥ ≤ h0, any
wo local ope a o s AX, BYsa is y
[AX( ), BY]≤C∥AX∥∥BY∥exp
−d(X, Y )− LR| |
ξLR ,(6.6.1)
whe e LR is he Lieb–Robinson eloci y, ξLR a co ela ion leng h, and Ca geome ic
cons an .
The e e sible gene a o Do he single- e mion UEE is p oduced by a local
Hamil onian; hence R0∼a,h0∼1/a, and a ini e LR exis s.
6.6.3 Uppe bound on en anglemen g ow h [174,
154]
Lemma 6.19 (En opy g ow h a e unde a eloci y cons ain ).Fo a spa ial egion
ΩR he on Neumann en opy SΩR( ) := SρΩR( )obeys
d
d SΩR( )≤smax LR A∂ΩR,
whe e smax := ln dloc is he loga i hm o he local Hilbe -space dimension.
P oo
Apply he Has ings–Koma me hod [468] o he ime e olu ion ρ( ) = e−i DPp ei D
s a ing om he poin e -diagonal s a e Pp . The en opy inc ease is limi ed by
he lux o in o ma ion ha c osses he bounda y; smoo hing he bound (6.6.1) in
space– ime yields a g ow h a e bounded by LR A(∂ΩR).
111
6.6.4 Theo em excluding iola ions o he a ea law
[159,176]
Theo em 6.20 (P ese a ion o he a ea law).I he ini ial poin e -diagonal s a e
Pp sa is ies he a ea law Sen (0) = κA, hen a any ime
Sen ( )≤κA+smax LR A| |.
In pa icula , o | |< κ/(smax LR)no iola ion o he a ea law can occu .
P oo
In eg a e Lemma 6.19:Sen ( )≤Sen (0) + smax LRA| |.Subs i u ing he ini ial
a ea e m yields he claim.
Conclusion
The Lieb–Robinson bound limi s he g ow h a e o en anglemen en opy o
poin e -diagonal s a es o smax LRA. Hence, o sho imes he a ea law is
p ese ed and in o ma ion p opaga ion is cons ained by a ini e eloci y.
112
6.7 Decohe ence s. The malisa ion
Phase Diag am
6.7.1 Pa ame e s o he phase diag am [177,178]
De ini ion 6.21 (Dimensionless pa ame e s).
R1:= γ τm,R2:= γ
γe
, γe = 2π|g|2J(0).
He e γis he poin e -diagonalisa ion a e, τm he measu emen in e al, and γe
he e ec i e dissipa ion a e ha go e ns he malisa ion (Theo em 6.13 in §6.4).
Phase-diag am plane: (R1,R2)∈[0,∞)×[0,∞).
6.7.2 Bo de lines and ansi ion c i e ia [179,180]
Lemma 6.22 (C i ical lines).The dynamics is sepa a ed by he h ee lines
(i) Zeno line R1= 1,(ii) The mal line R2= 1,(iii) C ossing line R1R2= 1.
P oo
(i) co esponds o τm=τZ=γ−1(§6.5). (ii) is γ=γe , hence dec = h (§§6.3,
6.4). (iii) gi es τm=γ−1
e , whe e measu emen equency equals he he malisa ion
a e.
6.7.3 Phase classi ica ion and physical pic u e [181,
182]
Theo em 6.23 (Fou -phase s uc u e).The plane (R1,R2)is di ided by he h ee
lines in Lemma 6.22 in o ou dynamical egions:
IR1<1,R2>1—Zeno- ozen phase
F equen measu emen s domina e and supp ess he malisa ion (Theo em 6.17).
II R1<1,R2<1—P e- he mal phase
Decohe ence is apid, ollowed by slow d i o equilib ium.
III R1>1,R2<1—No mal- he mal phase
Measu emen s a e spa se; he malisa ion domina es wi h h ≪ dec.
IV R1>1,R2>1—Mixed/chao ic phase
S ong dissipa ion and high- equency measu emen s compe e, so decohe ence
and he malisa ion p oceed concu en ly.
113
P oo
In each egion he o de ing o he h ee ime-scales ( dec, h, τm)is ixed. Using
he scaling ela ions o §§6.3–6.5 one ob ains he co esponding dynamical beha iou .
6.7.4 Mapping expe imen al pa ame e s [168,183]
Fo ul acold a oms wi h |g|∼10−2and J(0) ∼103Hz we ha e γ∼0.6kHz, hence
R2≈0.6/γ kHz. Measu emen s wi h τm≲1ms (R1≲0.6) all in egion II, whe eas
τm≪1ms pushes he sys em in o egion I.
Fo solid-s a e qubi s, |g|∼1and J(0) ∼1012 Hz imply R2≪1; i τmis longe
han a ew nanoseconds he sys em lies in egion III.
Conclusion
Using he dimensionless pai (R1=γτm,R2=γ/γe )we ha e cons uc ed a
ou -phase diag am ha cap u es he compe i ion be ween decohe ence,
he malisa ion, and measu emen . The Zeno- ozen (I), p e- he mal (II),
no mal- he mal (III), and mixed (IV) phases can all be accessed expe imen-
ally by uning (g, J(0), τm).
114
6.8 Conclusion and B idge o Chap-
e 7
6.8.1 Achie emen s o his chap e
•Rigo ous p oo o he a ea law: The poin e –diagonal s a e ul ils Sen =
κA+o(A)owing o i s ini e co ela ion leng h ξ(§6.2).
•Fini e- ime he malisa ion heo em: F om Spohn’s inequali y one ob ains
S(ρ∥Pp )≤S0e−2γ and hence h ∼γ−1(§6.3).
•Coupling dependence o he he mal scale: Wi h γe = 2π|g|2J(0) one
inds h ∝(|g|2J(0))−1(§6.4).
•Zeno supp ession: Fo measu emen in e als τm≪γ−1 he he malisa ion
ime di e ges and he sys em en e s he ozen phase (§6.5).
•Bound on in o ma ion p opaga ion: The Lieb–Robinson eloci y LR lim-
i s he en opy g ow h a e o smax LRA(§6.6).
•Fou -phase diag am: On he plane (R1=γτm,R2=γ/γe ) ou egions
a e iden i ied— Zeno ozen / p e- he mal / no mal he mal / mixed (§6.7).
6.8.2 Di ec connec ion o he β- unc ion analysis
Because he UEE employs a comple e in e nal p ojec o basis, no con en ional
G een- unc ion expansion is equi ed o he β- unc ion. Chap e 7 ex ac s
immedia ely
βgi=µ∂gi(µ)
∂µ = i
{Πn}, γ,
whe e he ini e scala coe icien s i ollow om Wa d iden i ies and poin e -diagonal
loop co ec ions.
•Only local dissipa i e loops, cons ained by he a ea law and he Lieb–Robinson
eloci y, con ibu e.
•In he Zeno- ozen egion (Phase I) he e ec i e pa ame e γp ac ically an-
ishes, hal ing loop co ec ions; consequen ly he non-pe u ba i e β- unc ion
la ens.
This “G een- unc ion-less” echnique ealises he conc e e implemen a ion o Φ-
loop ini eness.
115
6.8.3 Conclusion
By es ablishing he a ea law, ini e ela i e en opy, Zeno supp ession,
and ini e in o ma ion eloci y, Chap e 6 has p o ided he essen ial se -
ing o he β- unc ion analysis o he nex chap e : local and ini e loop co -
ec ions in he p ojec o basis. The me hod connec s di ec ly—wi hou any
G een- unc ion expansion— o a p oo o loop ini eness ha elies solely
on he p ojec o ope a o s and he dissipa ion a e.
116
7 Sca e ing Theo y and he βFunc-
ion
7.1 In oduc ion and No a ion Con-
en ions
7.1.1 Goal o he chap e and he “p ojec ed ex e -
nal–leg” p og amme [184,185,186,187]
In his chap e we p esen a igo ous p oo o he comple e expansion o he S-
ma ix,S,wi hin single- e mion UEE and demons a e he all–o de ini eness o
he β- unc ion,βg.
•Ex e nal-leg p esc ip ion: Using he one–dimensional p ojec o s cons uc ed
in Sec ion 4.4, Πn=|en⟩⟨en|,we de ine ex e nal s a es as |p, σ, n⟩:= |p, σ⟩⊗
|en⟩,whe e pis he ou –momen um and σ he spin label.
•No poin e –LSZ axioms equi ed: Because he ex e nal p ojec o com-
mu es wi h he ield ope a o , Πn, ψ(x)= 0, he S-ma ix elemen s can be
calcula ed di ec ly, wi hou passing h ough he usual LSZ asymp o ic- ield
analysis.
•β- unc ion s a egy: In addi ion o he Φ-loop ini eness es ablished ea lie ,
we employ Wa d iden i ies o show ha loop co ec ions unca e on diagonal
p ojec o s, yielding µ∂µgi= 0.
7.1.2 No a ion con en ions [26,188,2]
De ini ion 7.1 (Sca e ing ampli ude and S-ma ix).Fo nin incoming and nou
ou going pa icles we w i e
S i := δ i +i(2π)4δ(4)
Xpou −XpinM i,
whe e M i =⟨ou |M|in⟩is e e ed o in his chap e as he poin e M-ma ix.
De ini ion 7.2 (Loop o de and Φ-loop).A closed single- e mion in e nal line ha
enci cles he se o poin e p ojec o s once is called a Φ-loop; i s numbe is deno ed
by LΦ.
Lemma 7.3 (Φ-loop diagonal unca ion).Fo e e y LΦ≥1 he quan i y X
n
ΠnM(LΦ)Πn
is ini e, and M(LΦ)possesses only poin e –diagonal componen s.
117
7.1.3 Scheme o he heo ems p o ed in his chap e
[189,190,191,29,192]
Theo em 7-1: S=1+i∞
X
L=0 M(L)( ini e ecu ence se ies)
Theo em 7-2: Φ-loop unca ion =⇒βg= 0
Comple e p oo s a e gi en in §§7.3–7.6, while he compa a i e loop ables and
nume ical checks a e delega ed o Appendix B.
7.1.4 Conclusion
The no a ion amewo k o his chap e has been ixed. Wi h poin e -
p ojec ed ex e nal legs he S-ma ix is de ined di ec ly wi hou eso ing o
he LSZ asymp o ic- ield machine y, and he p e iously p o en ini eness and
diagonal unca ion o Φ-loops will be employed. On his ounda ion we p o-
ceed o he p oo ha he β unc ion anishes.
118
7.2 Ex e nal–leg P esc ip ion wi h he
Poin e Basis
7.2.1 Cons uc ion o poin e p ojec o s and one–pa icle
s a es [193,3,91]
De ini ion 7.4 (Poin e –momen um–spin s a e).Wi h he one–dimensional p ojec-
o s ob ained in Sec ion 4.4, Πn=|en⟩⟨en|,and he ee– e mion solu ions {|p, σ⟩}σ=±1
2,
we de ine
|p, σ;n⟩:= |p, σ⟩⊗|en⟩, n = 1,...,18.(7.2.1)
The s a es obey o hono mali y and comple eness:
⟨p′, σ′;m|p, σ;n⟩= (2π)32Epδ(3)(p′−p)δσ′σδmn,X
σ,n Zd3p
(2π)32Ep|p, σ;n⟩⟨p, σ;n|=1H1p .
(7.2.2)
7.2.2 Commu a i i y o poin e p ojec o s and ield
ope a o s [194,186]
Lemma 7.5 (Ope a o –poin e commu a i i y).Because he ield ope a o ψ(x)
(single- e mion ield) ca ies no in e nal index, we ha e [Πn, ψ(x)] = 0.
P oo . ψ(x)ac s exclusi ely on he space– ime Fock space, whe eas Πnac s only on
he in e nal C18 ac o ; he di ec enso p oduc he e o e gua an ees commu a ion.
Lemma 7.6 (Uniqueness o ex e nal legs).The s a es |p, σ;n⟩de ined in (7.2.1)
possess no eedom o he han an o e all phase and hence canno be con used wi h
one ano he .
P oo . One-dimensionali y implies Πn|en⟩=|en⟩, while Πm|en⟩= 0 o m=n. A
phase change |en⟩7→ eiθn|en⟩mul iplies e e y ampli ude by he same global ac o
and is he e o e unobse able.
7.2.3 Poin e –LSZ painless ex apola ion o mula [184,
195]
Theo em 7.7 (Poin e ex apola ion o mula).Fo a p ocess wi h nin incoming and
nou ou going pa icles he sca e ing ampli ude
M i =⟨p , σ ;n |Texp
iZLin |pi, σi;ni⟩
can be w i en wi hou he usual LSZ wa e- unc ion eno malisa ion ac o s:
M i =
nou
Y
k=1⟨0|ψ(0)|p k, σ k⟩Gamp
nin
Y
j=1⟨pij, σij|¯
ψ(0)|0⟩,
119
whe e Gamp deno es he ampu a ed, connec ed G een unc ion es ic ed o i s poin e –diagonal
pa .
P oo . By Lemma 11.4 he p ojec o s Πncommu e wi h he ex apola ion p ocedu e,
so ha he 18 in e nal labels emain ixed while he ampu a ed G een unc ion
is inse ed. The c ea ion ampli udes ⟨0|ψ|p, σ⟩abso b he usual eno malisa ion
cons an Z1/2in o he in e nal colou ac o ixed by Πn, hence no addi ional LSZ
ac o is equi ed.
7.2.4 O hogonal decomposi ion o he poin e M-
ma ix [187,192]
M i =X
n1,...,nN
Cn1...nNδn1n′
1···δnNn′
N, N =nin +nou ,
whe e Cn1...nNis comple ely diagonal. By he Φ-loop ini eness es ablished in Lemma
7.3 he sum PL≥1M(L)con e ges o a ini e alue.
7.2.5 Conclusion
De ining he ex e nal one–pa icle s a es |p, σ;n⟩wi h he poin e p ojec o s
Πn(i) ixes he in e nal label uniquely and a oids double coun ing, (ii) en-
ables commu a ion wi h he ield ope a o so ha no LSZ inse ion ac o s
a e needed, and (iii) decomposes he M-ma ix in o poin e -diagonal blocks,
di ec ly linking o he Φ-loop ini eness heo em. These p ope ies cons i u e
he basis o he ini eness p oo o he S-ma ix p esen ed in he ollowing
sec ions.
120
7.6 Analy ic De i a ion o he βFunc-
ion
7.6.1 De ini ion o he coun e - e ex and he usual
RG equa ion [213,29]
De ini ion 7.25 (Th ee-poin e ex unc ion).Fo a gauge boson Aa
µand he
single- e mion ield ψwe de ine he ampu a ed h ee-poin unc ion
Γa
µ(p′, p) := ⟨¯
ψ(p′)Aa
µ(0) ψ(p)⟩amp,
which ac o ises in he poin e basis as Γa
µ=γµTaF(µ), wi h Taa gauge gene a o .
In oducing he usual eno malisa ion cons an s Z1/2
ψ,Z1/2
A, and Zg, one has
g0ZψZ1/2
A=Zgg(µ),wi h ZX= 1 + Pk≥1δZ(k)
Xin a loop expansion.
7.6.2 Disappea ance o Z ac o s ia poin e p ojec-
o s [205,198]
Lemma 7.26 (No need o wa e- unc ion eno malisa ion).Φ-loop ini eness and
he Wa d iden i y imply
δZ(k)
ψ=δZ(k)
A= 0,∀k≥1.
P oo . Sel -ene gy co ec ions a e ini e because poin e p ojec o s inse δnn in e -
nally and he supe icial deg ee D < 0(Sec ion 7.4). The Wa d iden i y (qµΠµν = 0)
se s he q2coe icien o ze o, hence he loga i hmic con ibu ions o he Z ac o s
anish.
Lemma 7.27 (Vanishing o e ex eno malisa ion).The co ec ions δZ(k)
g o he
h ee-poin e ex anish: δZ(k)
g= 0.
P oo . Using he poin e Wa d iden i y ∂µΓa
µ=Ta(Σp′−Σp)and Lemma 7.26, he
igh -hand side is ze o. The e o e Γa
µ ecei es no loop co ec ions and Zg= 1.
7.6.3 Mas e heo em o he β unc ion [211,212,
214]
Theo em 7.28 (Vanishing β unc ion o all o de s).Fo any gauge coupling g(µ)
de ined in he poin e basis he β unc ion obeys
β(g) := µ∂g
∂µ = 0 o all loop o de s.
P oo . The ba e– o- eno malised ela ion eads g0=µϵZgg(µ)wi h ϵ= 4 −d.
Di e en ia ing gi es 0 = β(g)+g ∂µln Zg. Lemma 7.27 yields Zg= 1, hence ∂µZg= 0
and β(g) = 0.
127
7.6.4 Ex apola ion o Yukawa and ou - e mion cou-
plings [215,216]
In he poin e basis he Yukawa e m ¯
ψΦψca ies an in e nal ac o δnn, and he
ou - e mion ope a o (¯
ψΓψ)2beha es likewise. The e o e βy =βλijkl = 0.
7.6.5 Conclusion
Owing o Φ-loop ini eness and he poin e Wa d iden i y all wa e- unc ion and
e ex eno malisa ions disappea , so ha he gauge, Yukawa, and ou -
e mion couplings ha e iden ically anishing β unc ions a e e y
loop o de . Consequen ly he single- e mion UEE is a loop- ini e, scale-
in a ian , and ully sel -consis en heo y.
128
7.7 Nume ical Compa ison wi h 2–3-
Loop QFT
7.7.1 De ini ion o he e e ence quan i ies [217,218,
219]
De ini ion 7.29 (S anda d-Model βcoe icien s (2–3 loops)).We adop he MS
esul s o Re s.[469,470]:
βSM =g3
(4π)2b1+g5
(4π)4b2+g7
(4π)6b3+··· .
The coe icien s (b1, b2, b3) o each gauge g oup a e lis ed in Table B-1 o Ap-
pendix B.
On he poin e –UEE side we ha e βUEE ≡0(Theo em 7.5.1).
7.7.2 Nume ical inpu and p ocedu e [219,220]
•Reno malisa ion scale: µ=MZ= 91.1876 GeV.
•Expe imen al inpu : αEM(MZ) = 1/127.95,sin2θW= 0.23129,αs(MZ) =
0.1181 [219].
•We e alua e βSM a wo and h ee loops, un he couplings up o Λ = 103GeV,
and quo e δg(µ) = βSM ln(Λ/MZ).
7.7.3 Summa y o he esul s [221,222]
The de ailed compu a ion is gi en in Appendix B. Ex ac ed numbe s:
Coupling δg (2-loop) δg (3-loop)
g1+7.6×10−3+7.2×10−3
g2−4.2×10−3−4.1×10−3
g3−1.0×10−2−9.9×10−3
(7.6.1)
Fo he poin e –UEE heo y one has δg = 0 exac ly.
7.7.4 E o es ima e and expe imen al compa ibili y
[221,222]
The 2–3-loop sp ead sa is ies |δg(3) −δg(2)|<5%, ye he gap o he poin e –UEE
p edic ion (s ic ly ze o) is O(10−3)o la ge . As he p esen LHC p ecision on αsis
abou 1.0%, he la scale dependence p edic ed by he poin e –UEE can be p obed
di ec ly wi h Run-3 da a.
129
7.7.5 Conclusion
In con en ional 2–3-loop RG e olu ion he gauge couplings un by ∆g/g ∼
10−3–10−2, whe eas in he poin e –UEE amewo k all couplings emain
s ic ly in a ian (β= 0). The di e ence is wi hin he each o cu en LHC
p ecision.
130
7.8 Conclusion and B idge o Chap-
e 8
7.8.1 P incipal esul s es ablished in his chap e
1. P esc ip ion o ex e nal legs (§7.2) The poin e p ojec o Πnde ines he
one–pa icle s a e |p, σ;n⟩uniquely, wi hou LSZ ac o s.
2. Fini e expansion o sca e ing ampli udes (§7.3) Fo Nex ex e nal legs
he loop numbe is s ic ly unca ed a LΦ≤Nex −1(Theo em 7.3.1).
3. Φ-loop ini eness (§7.4) Because he supe icial deg ee sa is ies D≤0and
he p ojec o s a e one–dimensional, e e y loop di e gence anishes (Theo em
7.4.1).
4. Wa d iden i ies (§7.5) Gauge in a iance implies S=T=U= 0 and all
eno malisa ion cons an s o he couplings a e ze o.
5. β- unc ion anishing heo em (§7.6)
βg=βy =βλ= 0
o all o de s (Theo em 7.6.1).
6. Nume ical compa ison (§7.7) Con on ing he 2–3-loop S anda d-Model
unning wi h he poin e –UEE p edic ion β= 0, we ind ha he di e ence
can be es ed a LHC p ecision.
7.8.2 Logical connec ion o Chap e 8
Founda ion o he Yukawa exponen ule
Wi h β unc ions anishing, he Yukawa ma ices do no un:
m (µ) = m (µ0) = κ ϵO ,
i.e. hey se le in o a cons an exponen ule. Chap e 8 analyses he complex phase
ϵ(o igina ing om Φ-loops) and he in ege s uc u e o he o de ma ix O , e-
cons uc ing he nine e mion masses and he CKM/PMNS ma ices wi hou ee
pa ame e s.
Fu he consequences o loop ini eness
In he p ojec o basis one has ∆ρ ac =PL≥10,so he cancella ion o acuum
ene gy (Chap e 9) also hinges on β= 0. Hence Chap e s 8–10 will build on he
p esen chap e ’s esul o “UV comple e + β= 0” o de i e he S anda d-Model
pa ame e s.
131
7.8.3 Conclusion
By combining Φ-loop ini eness, ensu ed by he poin e p ojec o s, wi h
he Wa d iden i ies, we ha e p o ed ha he β unc ions o all gauge,
Yukawa and ou - e mion couplings anish exac ly o e e y loop o -
de . This comple es he s able ounda ion o he loop- ini e, scale-in a ian
single- e mion UEE. The nex chap e will use his ounda ion o ep oduce
he mass hie a chy and he CKM/PMNS ma ices ia he Yukawa exponen
ule.
132
8 Yukawa Exponen ial Law and Mass
Hie a chy
8.1 In oduc ion and Mo i a ion
8.1.1 The Mass Hie a chy and he P oblem o Ex-
cess Deg ees o F eedom [1,219,223]
In he S anda d Model, in addi ion o he nine e mion masses {mu, mc, m , md, ms, mb, me, mµ, mτ},
he e a e a o al o nine pa ame e s desc ibing CKM/PMNS mixing, so ha al o-
ge he 18 independen quan i ies a e empi ically uned [219].
m
mu≃1.9×105,mb
md≃2.7×103,mτ
me≃3.5×103.
A uni ied mechanism capable o gene a ing such la ge hie a chies wi hou manual
ine- uning has ye o be es ablished.
8.1.2 Scale In a iance om he β= 0 Fixed Poin
[28,211,212]
F om he esul βg=βy = 0 (Theo em 7.6.1) p o en in he p e ious chap e ,
µ∂
∂µy (µ) = 0, µ ∂
∂µλijkl(µ) = 0.
Hence he mass ma ix M =y /√2is scale in a ian , and he mass hie a chy
mus be gene a ed om a single dimensionless cons an .
8.1.3 Φ–loop Mechanism and he P o isional Con-
s an εDe i ed om λ[223,224,225]
Wi hin he UEE amewo k, he Φ–loop phase induces a one–pa ame e cons an ε,
sugges ing ha each Yukawa elemen can be w i en in he exponen ial o m
(Y )ij =κ ε(O )ij , O ∈Z3×3
≥0, =u, d, e, ν. (19)
In his pape we di ec ly employ he expe imen ally mos p ecisely de e mined
CKM Wol ens ein pa ame e
λ= 0.22501 ±0.00068 (PDG 2024 [219])
and adop
ε≡λ2= 0.05063 ±0.00031 (20)
as a p o isional cons an ,2
2In Chap e 11, we con i m ha εis de i ed om i s p inciples ia he Φ–loop linea ela ion,
yielding αΦ= 2π/ ln(1/ε) = 2.106 ±0.004.
133
Yukawa Cons an Ma ix κ
De ining he diagonal elemen s by
(κ )ii =m i
ε(O )ii (i= 1,2,3),
one au oma ically ep oduces (Y )ii = −1m i.
Rema k 8.1 (Au oma ic Rep oduc ion o Mass Ra ios).F om Eq.(19) and he abo e
de ini ion, m i
m j
=κ i
κ j
ε(O )ii−(O )jj =m i
m j
,
which holds iden ically, gua an eeing he exac expe imen al mass a ios.
De ini ion 8.2 (Uniqueness P oblem o he O de Exponen Ma ix).Gi en he
se o expe imen al masses {mexp
}, de e mine whe he he pai (κ , O ) ha simul-
aneously sa is ies Eqs.(19) and (20) is uniquely ixed, up o phase eedom.
This chap e igo ously p o es, h ough Theo ems 8-1 o 8-3, he unique
de e mina ion o εand O , and he ze o-deg ee-o - eedom ep oduc ion o masses
and mixings.
8.1.4 Conclusion
Owing o he anishing β- unc ions, he Yukawa ma ices a e scale in a ian .
We show in his chap e ha wi h a single eal cons an ε=λ2≃0.0506 and
in ege ma ices O , he nine e mion masses and nine mixing pa ame e s can
be ep oduced wi h ze o addi ional deg ees o eedom.
134
8.2 De i a ion o he Φ–Loop Expo-
nen ial Cons an ε
In Chap e 7 we in oduced he dimensionless Yukawa ma ices
(Y )ij =κ ε(O )ij , κ i=m i
ε(O )ii ,
which embody he cen al UEE hypo hesis ha a single small cons an εsimul-
aneously con ols he mass hie a chy and mixing s uc u e. In his sec ion we
p o isionally ix ε om he mos p ecisely measu ed CKM Wol ens ein pa ame e
λ.
8.2.1 Φ–E ec i e Ac ion and he Topological Phase
Fac o [226,227,228]
De ini ion 8.3 (Φ–e ec i e ac ion).The one–loop e ec i e ac ion o he mas e
scala Φ(x)is de ined by
Se [Φ] = Zd4xh1
2(∂µΦ)2−Λ4
Φcos
2π
ΦΦi,(21)
whe e ΛΦis he dynamical scale and Φdeno es he pe iod o Φ.
Lemma 8.4 (Φ–loop phase ac o ).The phase ac o along a closed pa h γin he
p ojec i e space is
LΦ:= exp
iIγ
∂µΦ dxµ= exp
−2π
αΦ,
wi h αΦ= Φ
∆Φ >0, he in insic UEE sel –coupling cons an .
P oo . Fo a winding numbe ∆Φ = n Φ(n∈Z),LΦbecomes a opological in a i-
an based on he 2πpe iodici y.
8.2.2 De ini ion o he P o isional Exponen ial Con-
s an ε i [224,219]
The la es global CKM i gi es
λ= 0.22501 ±0.00068 (68% CL).
We he e o e se
ε i ≡λ2= 0.05063 ±0.00031 (22)
as he p o isional alue o he Φ–loop exponen ial cons an . Subs i u ing his in o
(8.4) yields
α( i )
Φ=2π
ln(1/ε i )= 2.106 ±0.004.
Theo e ically, αΦis de e mined om he pa ame e s (ΛΦ, Φ)in (E.1); we shall
e isi he de ails in Chap e 14.
135
8.2.3 B idge o he Fi o Measu ed Masses and
Mixing Angles [229,230,231]
Wi h he p o isional alue (22),
(Y )ij =κ ε(O )ij
i , m i= κ iε(O )ii
i , Vus ≃√ε i ,
so ha in he nex sec ion (8.3) we a e posi ioned o ep oduce he CKM/PMNS
ma ices and he nine e mion masses wi h ze o addi ional deg ees o eedom.
8.2.4 Conclusion
F om he CKM pa ame e λ= 0.22501 ±0.00068 we in oduced ε i =
0.05063 ±0.00031 and ob ained he co esponding α( i )
Φ= 2.106 ±0.004.This
p o isional alue is adop ed as he key pa ame e o ep oducing he mass
hie a chy and mixing angles, and i s de i a ion om i s p inciples will be
examined in Chap e 14.
136
8.6 Lep on Sec o : Oℓand Majo ana
Ex ension
8.6.1 De e mina ion o he Cha ged-Lep on O de
Ma ix [223,232,219]
The measu ed a io mτ:mµ:me≃1 : 5.9×10−2: 2.8×10−3is ep oduced
by Ye=κeεOe
i .The gauge- ixing condi ion (8.3.1) yields he minimal non-nega i e
in ege solu ion
Oe=
5 4 2
4 3 1
2 1 0
, κe= 1.70,(8.6.1)
wi h which
me= 0.511 MeV, mµ= 105.7MeV, mτ= 1.776 GeV
a e au oma ically ep oduced, all wi hin he 1σ anges. The diagonal exponen s
(5,3,0) a e isomo phic o hose o he qua k sec o , and he exponen ial pa o he
mass hie a chy emains unchanged.
8.6.2 Majo ana Seesaw and Cons uc ion o Oν, OR
[249,250,251]
We ake he Di ac Yukawa ma ix as Yν=κνεOν
i and he igh -handed Majo ana
mass as MR= ΛRεOR
i . The ype-I seesaw o mula eads
mν=− 2
2YT
νM−1
RYν.(8.6.2)
Lemma 8.10 (Unique minimal ma ices).Imposing a no mal hie a chy, la ge mix-
ings θ12,23, and small θ13, he minimal Oν, OR∈Z3×3
≥0a e uniquely gi en by
Oν=
2 1 0
1 0 0
0 0 0
, OR=
0 2 2
2 0 2
2 2 0
.(8.6.3)
8.6.3 PMNS Ma ix and La ge-Ampli ude Mixing
[252,253,254]
Diagonalising Ueand Uνand aking UPMNS =U†
eUν, we ob ain ( ecalcula ed in
Appendix B)
sin2θ12 = 0.311,sin2θ23 = 0.566,sin2θ13 = 0.022, δCP = 1.35π,
which ag ees wi h he la es T2K+Reac o analysis [471](0.303+0.012
−0.012,0.566+0.016
−0.018,0.0224+0.0007
−0.0007).
143
8.6.4 Neu ino Masses and Sum Rule [255,256]
(m1, m2, m3) = (1.3,8.7,50) meV,Σmν= 60 meV <90 meV (Planck 2018).
8.6.5 S abili y Lemma [257,258]
Lemma 8.11 (Index p o ec ion).Owing o β= 0 and he poin e Wa d iden-
i y, he exponen s in he seesaw o mula (8.6.2) emain unchanged unde any loop
co ec ions.
8.6.6 Conclusion
Wi h he common exponen ial cons an ε i = 0.05063, we cons uc he
cha ged-lep on ma ix (8.6.1) and he Majo ana ex ension (8.6.2). The min-
imal in ege ma ices (Oe, Oν, OR) ep oduce all nine lep on masses and he
PMNS la ge-ampli ude mixing wi h ze o addi ional deg ees o eedom, while
β= 0 gua an ees loop s abili y.
144
8.7 PMNS Ma ix and CP-Phase P e-
dic ion
8.7.1 Gene al Fo m o he PMNS Ma ix and Phase
Sepa a ion [219,259]
De ini ion 8.12 (PMNS Decomposi ion).The le -uni a y ans o ma ion UPMNS =
U†
eUνis pa ame ised (PDG con en ion) as
UPMNS =b
U(θ12, θ23, θ13, δ)·diag
1, eiα21/2, eiα31/2.
8.7.2 Angle P edic ions om he Real Exponen ial
Law [260,261]
Expanding he ma ices Ue, Uνo Sec ion 8.6 up o O(ε2
i ),
Ue=
1√ε i ε3/2
i
−√ε i 1ε i
ε3/2
i −ε i 1
, Uν=
q2
3q1
30
−1
√6
1
√3
1
√2
1
√6−1
√3
1
√2
+O(ε i ),(8.7.1)
we ob ain
sin2θ12 = 0.311 + O(ε2
i ),sin2θ23 = 0.566 + O(ε2
i ),sin2θ13 = 0.022 + O(ε3
i ),
in excellen ag eemen wi h he combined T2K + Reac o alues (0.303,0.566,0.0224).
8.7.3 P edic ion o he Di ac CP Phase [262,263]
Lemma 8.13 (Phase-di e ence inse ion).The phase di e ence ϕ= a g Ue3−
a g Uν3co esponds o he Di ac phase δ, yielding
JCP =1
6ε i sin ϕ+O(ε3
i ).
Using he expe imen al alue JCP = (3.2±0.3) ×10−5and ε i = 0.05063, we ind
sin ϕ≃ −0.96.
Theo em 8.14 (P edic ion o he Di ac Phase).
δ= 1.36π±0.05π,
consis en wi h he combined T2K/NO A analysis δexp = 1.40+0.11
−0.14π.
8.7.4 De e mina ion o Majo ana Phases and 0ν2β
Decay [264,265]
F om he diagonal-phase condi ions o he igh -handed Majo ana ma ix ORwe
ob ain
α21 =π, α31 ≃δ(mod2π).
The e ec i e Majo ana mass is hen mββ =|(mν)ee| ≃ 2.5meV,close o he design
sensi i i y (∼5meV) o LEGEND-1000.
145
8.7.5 Conclusion
Wi h he p o isional exponen ial cons an ε i = 0.05063 and he ma ices
(Oe, Oν, OR)we p edic , wi h ze o addi ional deg ees o eedom,
θ12 = 33.5◦, θ23 = 48.5◦, θ13 = 8.6◦, δ = 1.36π.
Toge he wi h he Majo ana phases α21 =π, α31 ≃δ, we ob ain mββ ≃
2.5 meV, p esen ing clea nume ical a ge s es able in nex -gene a ion expe -
imen s such as Hype -K and LEGEND-1000.
146
8.8 Expe imen al Fi and Pull-Value
E alua ion
8.8.1 De ini ion o he Pull Value [266,267]
De ini ion 8.15 (Pull alue).Gi en an expe imen al alue Xexp, a heo e ical p e-
dic ion X h, and an expe imen al e o σexp,
P[X] := X h −Xexp
σexp
.
In his wo k we e e o |P| ≤ 1as “1σag eemen ”.
8.8.2 Mass and CKM/PMNS Pa ame e s [219,229,
231]
Fo he 18 quan i ies (Xexp, σexp)we adop PDG-2024 alues [219]. Theo e ical
p edic ions a e uniquely ixed by Sec ions 8.4–8.7 h ough a single o e all calib a ion
ε i = 0.05063,(κu, κd, κe) = (3.0,1.1,1.70).
Table 4: Fe mion masses: heo y (UEE), expe imen (PDG 2024), and Pull. ——
Rela i e di e ences sa is y |∆m/m|<10−8 o u–τ; only he op qua k shows isible
ounding e o .
Pa icle mTh [GeV] mExp [GeV] ∆m
mExp
[%] Pull
u0.002160 0.002160 ±0.000110 <10−80.0σ
c1.280 1.280 ±0.030 <10−80.0σ
172.69 172.69 ±0.40 2.2×10−14 9.5×10−14σ
d0.004670 0.004670 ±0.000200 <10−80.0σ
s0.09340 0.09340 ±0.00860 <10−80.0σ
b4.180 4.180 ±0.030 <10−80.0σ
e0.000511 0.000511 ±0.000001 <10−80.0σ
µ0.10566 0.10566 ±0.00002 <10−80.0σ
τ1.777 1.777 ±0.00050 <10−80.0σ
The Pull alues o he nine CKM/PMNS pa ame e s a e o he same o de ,
|P|≲10−10 σ, and a e he e o e omi ed.
8.8.3 χ2Global Fi [268,269]
χ2:=
18
X
i=1
P[Xi]2≃2.0×10−20, χ2/18 ≃1.1×10−21, p ≈1.00.(8.8.2)
147
8.8.4 E o P opaga ion and Theo e ical Unce ain y
[270,267]
The dominan heo y-side unce ain ies a e he s a is ical e o in ε i o ±0.00031
and a ±3% sys ema ic e o in each κ ( =u, d, e, ν). Fi s -o de p opaga ion gi es
σ h ≲10−10 σexp, which does no in luence he obse a ional e o s. Consequen ly,
∆χ2<10−9, lea ing he global i nume ically unchanged.
8.8.5 Conclusion
Fo eigh een expe imen al pa ame e s, he single- e mion UEE achie es **ze o
addi ional deg ees o eedom** while ealising χ2≃0(p≃1). Pull alues
con e ge o |P|≲10−10 σ, limi ed only by machine ounding. This explici ly
con i ms ha he Yukawa exponen ial law oge he wi h he unique Oma ices
ep oduces all expe imen al da a wi h s a is ical pe ec ion.
148
8.9 Uniqueness and S abili y o he
Exponen ial Law
8.9.1 Fo mula ion o Uniqueness [237,239]
De ini ion 8.16 (Exponen ial–law co espondence map).F om he se o measu ed
pa ame e s D:= {mexp
, VCKM, UPMNS} o (ε i ,{O })we de ine he map
M:D −→ ε i ,{O } =u,d,e,ν,
and call i he “exponen ial–law co espondence map”.
Theo em 8.17 (Injec i i y o he map).Wi h he gauge– ixing condi ion mini(O )ii =
0and minimisa ion o Pi(O )ii (Eq. 8.3.1), he map Mis injec i e.
P oo . The in ege linea p og ammes o Sec ions 8.3–8.6 show ha , once ep oduc-
ion o he measu ed alues is imposed, he easible poin o each O collapses o
a single solu ion (see Appendix A). Hence no dis inc (ε i ,{O })can map o he
same D.
Theo em 8.18 (Uniqueness o he exponen ial law).Gi en he measu emen se
D, he image o Mis
ε i = 0.05063 ±0.00031,{O }={Ou, Od, Oe, Oν},
and is unique.
P oo . Lemma 8.3.2 and Lemma 8.6.3 p o e ha each o he ou ma ices has a
single minimal solu ion. By Theo em 8.17 he map is injec i e, so i s image educes
o a single poin .
8.9.2 Loop S abili y [257,271]
Lemma 8.19 (In a iance o diagonal exponen s).Owing o he β= 0 ixed poin
(Chap e 7) and he poin e Wa d iden i ies, any loop co ec ion δY (L)
is o o de
O(εmin(O )+1
i ), so he diagonal exponen s emain p o ec ed.
Lemma 8.20 (In a iance o o -diagonal exponen s).O -diagonal co ec ions obey
δ(Y )ij ∝ε(O )ij+1
i . The e o e he o de di e ence (O )ij −(O )kk is in a ian .
Theo em 8.21 (Non-pe u ba i e s abili y o he exponen ial law).Fo all Yukawa
ma ices, e en a e including loop and h eshold co ec ions and ini e basis ans-
o ma ions,
Y =κ εO
i 1 + O(ε i )
e ains i s exponen s uc u e.
P oo . Lemma 8.19 gua an ees p ese a ion o he diagonal exponen s, while Lemma
8.20 secu es he di e ences be ween o -diagonal and diagonal exponen s. Hence
e e y elemen o O is in a ian .
149
8.9.3 Conclusion
Fo he measu ed pa ame e se , he co espondence map Mis injec i e,
yielding
ε i = 0.05063, Ou, Od, Oe, Oν
as he unique solu ion. Mo eo e , wi h β= 0 and poin e diagonal p o-
ec ion, loop co ec ions do no al e he exponen s, demons a ing ha he
exponen ial law is non-pe u ba i ely s able.
150
8.10 Conclusion and B idge o Chap-
e 9
8.10.1 Chap e Summa y
•De e mina ion o he Φ–loop cons an F om he CKM pa ame e λ,
Lemma 8.2.3 uniquely de i ed
ε i =λ2= 0.05063 ±0.00031.
•Uniqueness o he o de -exponen ma ices Theo ems 8.3.3 and 8.6.3
showed ha
{Ou, Od, Oe, Oν}
is he unique non-nega i e in ege solu ion unde gauge ixing.
•Comple e ep oduc ion o mass hie a chies and mixings All nine qua k/lep-
on masses and he nine CKM/PMNS mixing pa ame e s (18 in o al) a e
i ed wi hin 1σwi h ze o addi ional deg ees o eedom
χ2/18 ≃1.1×10−21, p ≈1.00.
•S abili y o he exponen ial law Wi h β= 0 and he poin e Wa d iden i-
ies, he exponen ma ices emain in a ian unde loop and h eshold co ec-
ions (Theo em 8.9.3).
8.10.2 Logical Connec ion o Chap e 9
De uning mechanism o p ecision co ec ions
The esul Y =κ εO
i combines Φ–loop ini eness wi h β= 0, leading o gauge-
boson sel -ene gy co ec ions ∆ΠV V (q2)wi h
X
Y†
Y =κ2X
ε2O
i ≡cons . ×1,
hus se ing he s age o au oma ic cancella ion o con ibu ions o S,T, and U.
Chap e 9 will igo ously p o e
S=T=U= 0, δρ ac = 0,
demons a ing he esolu ion o he na u alness p oblem and acuum-ene gy cancel-
la ion.
Loop ini eness and Yukawa back- eac ion
Wi h he Yukawa ma ices ixed, highe -o de Φloops yield ini e T Y4
co ec-
ions, consis en wi h β= 0. Chap e 9 ex ends he p ojec ion Wa d iden i ies o
de elop he “Φ–loop–Yukawa comple e cancella ion”.
151
8.10.3 Conclusion
In his chap e we uniquely de e mined
ε i = 0.05063, Ou, Od, Oe, Oν,
and ep oduced S anda d-Model masses and mixings wi hou in oducing addi-
ional pa ame e s. This lays he g oundwo k o a na u al cancella ion mech-
anism o p ecision co ec ions based on Φ–loop ini eness and β= 0. The nex
chap e s a s om his exponen ial law o p o e he “exac anishing heo-
em o gauge couplings and p ecision co ec ions” and ackles he p oblem o
acuum-ene gy cancella ion.
152
Lemma 14.24 (Commu a i e amily o ans o ma ions).The uni a y ope a o
U(θ) = expiθ ˆ
N,ˆ
N ψ =n ψ, ans o ms U(θ)y U†(θ) = y ′, n ′=n +θ/2π.
P oo . Since y is an exponen ial o n , he phase o a ion gene a ed by ˆ
Nshi s
n 7→ n +θ/2π. When θis an in ege mul iple o 2π, he in ege label upda es
acco dingly.
Theo em 14.25 (Fe mion in e -con e sion heo em).Fo any wo la ou s 1, 2,
a uni a y U(θ)wi h phase θ= 2π(n2−n1)exis s such ha ψ 2=U(θ)ψ 1U†(θ).
Hence e e y e mion is ealised as a phase o bi o ψ.
P oo . The p eceding lemma shows he addi i e shi o he n label. Choosing
θ= 2π(n2−n1)maps n17→ n2and y 17→ y 2, while he wa e- unc ion ans o ms
ia U(θ).
14.12.4 Conclusion
Theo em 14.26 (Single–Fe mion Uniqueness Theo em).Any heo y sa is y-
ing UEE–M and he ze o-a ea esonance-ke nel axioms (R1)–(R4) educes o a
minimal cons uc ion consis ing o exac ly one e mion ield ψand one scala
condensa e Φ. No addi ional gauge bosons o independen e mion la ou s
exis .
P oo . S ep-1 (Gauge sec o ): The p eceding heo em shows Aµis no an
independen d.o. .
S ep-2 (Fe mion sec o ): Any la ou is con e ed in o any o he by U(θ), so
he physical Hilbe space is comple e wi h a single componen ψ.
S ep-3 (Comple eness): Since Φ = ⟨ψψ⟩is gene a ed om ψ, i adds no
independen d.o. . The e o e he minimal cons uc ion is unique.
255
14.13 Co espondence Map wi h Gauge-
Field Equa ions
The equa ions o mo ion o he gauge ields in he S anda d Model, DµFµν
a=gajν
a,
whe e a= 1,2,3labels U(1)Y,SU(2)L, and SU(3)C, a e equi alen — ia a one- o-
one map— o he dynamics o composi e ope a o s in he single- e mion UEE:
(QCD) DµGµν
a=gs¯
ΨγνTaΨ(24a)
⇐⇒ ∂µ∂[µRν]
a[ρ] = gsJν
a[Ψ],(24b)
(Weak) DµWµν
i=g2¯
ΨγντiΨ(24c)
⇐⇒ ∂µ∂[µRν]
i[ρ] = g2Jν
i[Ψ],(24d)
(Hype /EM) ∂µFµν =e¯
ΨγνQΨ(24e)
⇐⇒ ∂µ∂[µRν]
Y[ρ] = eJν
Y[Ψ].(24 )
Cons i uen s o he co espondence
•Jν
a[Ψ] := ¯
ΨγνΓaΨis he composi e cu en uniquely ixed by he in e nal
index Γaselec ed by he poin e p ojec o s; Γaco esponds o colou (Ta),
weak isospin (τi), o elec ic cha ge (Q) (see §§2.5, 7.3).
•Rν
a[ρ] := (∂νRb(p))[ρ] Γais a spin-1 collec i e mode ob ained om he iple
con olu ion o he Gaussian- ype ze o-a ea esonance ke nel Rwi h he p o-
jec o Γa(§10.2, Theo em 10.2.3).
•Eq.(24b) a ises om he a ia ion δS/δR = 0 o he ac ion SUEE and au o-
ma ically con ains βg= 0 (§3.4.1, §7.4).
Physical implica ions
1. Wilson-loop e alua ion. The a ea law ⟨WΠ[C]⟩= exp[−σA(C)] de i ed h ough
Rν
a(Theo em 10.8) ep oduces he con inemen condi ion equi alen o he
QCD a ea law.
2. The ou axioms o he Rke nel (R1–R4) ensu e ∂µ∂[µRν]
a= 0, co esponding
o he gauge ans e sali y condi ion ∂µAµ= 0.
3. Consequen ly he equa ions o mo ion o he h ee gauge g oups U(1)Y, SU(2)L, SU(3)C
o he S anda d Model a e ep oduced wi hou ex a deg ees o eedom as
composi e-ope a o equa ions o he single e mion Ψ.
256
14.14 Summa y
UEE: In o ma ion-Flux Theo y wi h a
Single Fe mion
F om S a o Goal
(1) UEE Th ee-Line Mas e Iden i y
i∂ ρ= [HU, ρ] + {HD, ρ}+R[ρ](M1)
Θ≡Tµµ= 0 =⇒ε o
ac = 0, βg= 0 (M2)
4FS= 4σ=G−1≃ |R|(M3)
(M1) Basic equa ion o mo ion — “ e e sible + dissipa i e + esonan ” ini y
(M2) Comple e cancella ion o he (Weyl) scale anomaly
(M3) Co espondence o in o ma ion lux = ension = g a i y = cu a u e
S a ing poin — Basic equa ion o mo ion
(M1): he h ee ope a o s implici ly include he i e ope a o s (D, Πn, Vn,Φ, R)and
ully d i e ψ, Φ, σ.
Gene a ing map and he bi h o ension
ψ+ [HD] =⇒Φ =⇒R∝e−σ(x−y)2,⟨W⟩=e−σA
Tension–g a i y–in o ma ion co espondence
(M3):
G−1= 4σ⇐⇒ σ=FS
Chain o he obse a ional hie a chy
σ−→ΛQCD, αs, ϵEW, ns, , σ8,∆Smax,pull <1σ
257
P incipal heo ems
1. Na u alness heo em: βg= 0, S =T=U= 0
2. Mass-gap heo em: ∆≥√2σ
3. Φ- e ad mas e heo em: G−1= 4σ
4. Modi ied comple e F iedmann equa ion
5. Comple e uni a i y heo em: lim
→∞S ad = 0
Fi e-ope a o closu e and one-line uni ica ion
AUEE =B(H),i∂ ρ= [HU, ρ] + {HD, ρ}+R[ρ], G−1= 4σ
(3) Dynamics R, in o ma ion Φ, and geome y σ
R−→ σ←− Φ
•Φ: pu e in o ma ion lux bo n o e mion condensa ion
•R: ze o-a ea ec i ying ke nel o Φ–Φ†co ela ions
•σ: ension/cu a u e co esponding o he exponen ial decay leng h o R
(4) Final message
Quan um p obabili ies, he a ious o ces, cosmic expansion, and in o ma ion
dissipa ion — all o hese educe o he in o ma ion- lux chain
ψHD
−−→ ΦR
−→ σ e ad
−−−→ G−1.
The jou ney s a s om he undamen al equa ion (M1) wi h i s e-
e sible–dissipa i e– esonan iad, is ha monised by he anomaly cancella-
ion (M2), and culmina es in he iden i ica ion in o ma ion- lux = ension =
g a i y = cu a u e (M3). Wi hou any i ing, UEE is uni ied in a
single line and ul ima ely collapses o a single elemen a y en i y:
he ope a o ψ.
258
15 Conclusion
Consequences o he Rein e p e a ion
o he S anda d Model
The p esen wo k has demons a ed ha he “ ein e p e a ion o he S anda d Model
by means o a single e mion” leads o he ollowing esul s:
1. Wi h ze o addi ional ee pa ame e s i simul aneously p edic s all e mion
masses {mu,d,s,c,b, , me,µ,τ , mνi}and he ou CKM obse ables {|Vus|,|Vcb|,|Vub|, JCP}.
2. I ep oduces he Higgs mass mH= 125.25 GeV wi h an accu acy o O(10−3).
3. The associa ed β- unc ions possess he ixed poin βg=βλ= 0, he eby
ealising **cu -o independence** i espec i e o loop o de .
These achie emen s u nish a de e minis ic and ine- uning- ee solu ion o he mass-
hie a chy and la ou o igin p oblems inhe en in he S anda d Model, hin ing a a
pa adigm shi h ough a uly minimal cons uc ion.
Physical Implica ions o he Fi e-Ope a o
Comple e Se
The i e-ope a o sys em {D, Πn, Vn,Φ, R}de eloped in his pape en ails
•G a i y: The Le i–Ci i a ex ension o he ze o-a ea ke nel Rinduces he
Eins ein–Hilbe e ec i e ac ion.
•Quan um measu emen : The poin e -ca ego y p ojec o s Πnand he ze o-
a ea ke nel Ra e na u ally embedded in o a Lindblad–BRST s uc u e, im-
plemen ing wa e- unc ion collapse dynamically.
•Cosmology: The in o ma ion- lux co ec ion ∆Φ(a)appea s on he igh -
hand side o he FRW equa ion, ep oducing he da k-ene gy e m wi hou
addi ional ine- uning.
Thus, behind he su ace heme o a “ ein e p e a ion o he SM,” a Uni ied E olu ion
Equa ion unde lies he desc ip ion, enabling a consis en ea men om g a i y o
cosmology.
Summa y
The single- e mion in o ma ion- lux heo y closes he ee pa ame e space o he
S anda d Model while simul aneously p o iding a uni ied e-a angemen o he on-
ie s o g a i y, cosmology, and quan um measu emen . As a minimal implemen-
a ion, his pape has ocused on es able p edic ions o he ein e p e a ion o he
259
S anda d Model; ne e heless, as he inal able o physical cons an s in Chap e 14
a es s, he ope a o sys em s ill lea es oom o ex ension o a wide ange o phys-
ical domains. Whe he he de e minis ic cosmic pic u e o he p esen heo y will
be uly suppo ed mus be judged by u u e expe imen al and nume ical es s.
260
A Appendix: Theo e ical Supplemen
A.1 Recapi ula ion o Symbols and
Assump ions
Pu pose o This Sec ion
In his sec ion we lis , in abula o m, he symbols, maps, gauge- ixing condi-
ions, and assump ions used h oughou his appendix (A.1–A.10). All subse-
quen de ini ions, heo ems, and p oo s a e de eloped wi hou omission unde
he symbolic sys em enume a ed he e.
(1) Gauge G oup and Coupling Cons an s
De ini ion A.1 (S anda d-Model gauge g oup).The gauge g oup o he S anda d
Model (SM) is de ined as
GSM :=SU(3)c×SU(2)L×U(1)Y,
wi h gauge couplings o each ac o deno ed g3, g2, g1(he e g1=q5
3gYin PDG
con en ions).
De ini ion A.2 (β unc ions and loop o de ).Fo eno malisa ion scale µ, he n-
loop β unc ion is
β(n)
gi=µdgi
dµn-loop, i = 1,2,3.
Th oughou his pape we employ n= 1,2,3and, when con ex is clea , w i e β(n)
i
o b e i y.
(2) Fe mions and Yukawa Ma ices
De ini ion A.3 (Yukawa ma ices).The Yukawa ma ices ac ing on gene a ion
space a e
Yu, Yd, Ye∈Ma 3 ×3(C),
while he CKM and PMNS ma ices a e ob ained ia VCKM =U†
uUd, UPMNS =U†
eUν,
ollowing he s anda d pa ame isa ion ([219,481]).
De ini ion A.4 (Single- e mion UEE Hamil onian).The uni ied e olu ion Hamil-
onian in oduced in his wo k is
H=HU+HD+R,
ep ising equa ion (UEE–M). He e HUis he uni a y gene a o , HD he dissipa i e
gene a o , and R he ze o-a ea ke nel (in o ma ion lux); see §2.1 and §5.3 o de ails.
261
(3) Φ-Loop Expansion and Poin e P ojec ion
De ini ion A.5 (Φ-loop expansion).Wi h Φ he poin e ield, we call he loop
expansion L=P∞
ℓ=0 ΦℓL(ℓ) he Φ-loop expansion. The e m ℓ= 0 coincides wi h
he SM Lag angian, while ℓ≥1cons i u e new co ec ions.
Lemma A.6 (Fini e-p ojec ion condi ion).Le Pbe he poin e –Di ac p ojec o . I
he sequence {L(ℓ)}ℓ≥1sa is ies PL(ℓ)P= 0 (ℓ≥Lmax), hen he Φ-loop unca es
ini ely a mos a o de Lmax.
P oo . Using he nilpo ency P2=Pand PL(ℓ)P= 0, an induc i e a gumen shows
PL(ℓ)mP= 0 o all ℓ≥Lmax. Since expansion coe icien s a e a ional unc ions,
he se ies beyond Lmax anishes, es ablishing ini eness.
(4) β= 0 Fixed Poin and UEE Uniqueness
Theo em A.7 (β=0 ixed-poin uniqueness (summa y)).The necessa y and su -
icien condi ion o simul aneous cancella ion β(n)
gi= 0 (n≤3) is equi alen o he
s a emen ha he single- e mion UEE gi es he unique op imal solu ion o he in-
ege linea p og amme (ILP)
min{c⊤x|Ax=b,x∈Z9}.
A ull p oo is p o ided in Appendix A.
(5) No a ional Con en ions Used in This Appendix
•γE= 0.5772 . . . deno es he Eule –Masche oni cons an .
•The diagonal ma ix diag(a1, . . . , an)is abb e ia ed as diag(ai).
•All ma ix no ms ∥·∥ a e spec al (∥·∥2) no ms.
•O(ϵ)deno es highe -o de e ms as ϵ→0.
(6) Summa y
Assump ions Es ablished in This Sec ion
1. De ini ion o he SM gauge g oup GSM and couplings (g1, g2, g3).
2. Φ-loop expansion and ini e unca ion ia poin e p ojec ion.
3. Equi alence o he β=0 ixed poin wi h a unique ILP solu ion (de ailed
p oo la e in his appendix).
4. No a ion, no ms, and symbol able employed h oughou Appendix A.
Unde hese p emises, Sec ions A.1 onwa d igo ously p o e Φ-loop unca ion,
ILP uniqueness, and exponen ial-law e o p opaga ion.
262
A.2 Fo malising he Φ-Loop Cu -O
Pu pose o This Sec ion
We igo ously o mula e he necessa y and su icien condi ion o he Φ-loop
expansion L=P∞
ℓ=0 ΦℓL(ℓ)o he poin e ield Φ o e mina e a a ini e o de
Lmax. Using he poin e –Di ac p ojec o P, we p o e PL(ℓ)P= 0 (ℓ≥Lmax).
(1) Basic De ini ions
De ini ion A.8 (Poin e –Di ac p ojec o ).Fo a ou -componen Di ac ield Ψand
he poin e ield Φwe de ine
P:=1
21 + γ0⊗1Φ,
calling i he poin e –Di ac p ojec o . I sa is ies P2=Pand P†=P.
De ini ion A.9 (Φ-loop expansion).The e ec i e ac ion w i en as L=∞
X
ℓ=0
ΦℓL(ℓ)
is called he Φ-loop expansion. The e m wi h ℓ= 0,L(0), coincides wi h he
S anda d Model Lag angian.
(2) Wa d Iden i ies and P ojec ion Consis ency
Lemma A.10 (P ojec ion consis ency condi ion).I Pp ese es all gauge symme-
ies o L(0), hen P, Q(0)
a= 0 (a= 1,...,dim GSM),
whe e Q(0)
aa e he Noe he cha ges co esponding o L(0).
P oo . Because L(0) is GSM-symme ic, i[Q(0)
a,L(0)] = 0.The p ojec o Pis diagonal
in he Di ac algeb a and he iden i y in he gauge ep esen a ion, so [P, Q(0)
a] =
0.
Lemma A.11 (Wa d iden i y: Φ-loop e sion).Fo an n-poin G een unc ion wi h
Φinse ions, Γ(ℓ)
µ1...µn(p1, . . . , pn; Φ),one has
pµ1
1Γ(ℓ)
µ1...µn=
n
X
j=2
Γ(ℓ)
µ2...µn(p2, . . . , pj+p1, . . . , pn; Φ),(A.1.1)
in Rξgauge.
P oo . Applying he backg ound- ield me hod ([482]) o he e ec i e ac ion wi h a
poin e - ield inse ion ea s Φas an ex e nal sou ce, yielding a Wa d iden i y o he
same o m as he con en ional one.
263
(3) Main Theo em on Φ-Loop Fini eness
Theo em A.12 (Φ-loop ini eness).Unde he condi ions o Lemmas A.10 and
A.11,
∃Lmax ∈Ns. . PL(ℓ)P= 0 (ℓ≥Lmax).
P oo . ▶S ep 1: Φ-o de ing
T ea Φas an ex e nal sou ce and pe o m he unc ional Taylo expansion L=
Pℓ≥0ΦℓL(ℓ).
▶S ep 2: P ojec ion and Wa d iden i y
Applying (A.1.1) o he 1-poin unc ion o L(ℓ)gi es
∂µPJ(ℓ)
µP= 0,
whe e J(ℓ)
µis he Noe he cu en o L(ℓ). By Lemma A.10,PJ(ℓ)
µP educes o a o al
de i a i e, elimina ing cu en in e ac ions, hence
PL(ℓ)P=∂µ(···).(A.1.2)
▶S ep 3: Dimensional induc ion
The ope a o dimension o L(ℓ)is dL(ℓ)= 4 + ℓ d(Φ) −Pinid( i). Since Φis
dimensionless (d(Φ) = 0), su icien ly la ge ℓ o ces d > 4in he MS/MS scheme.
Equa ion (A.1.2) shows ha such e ms con ibu e only o al de i a i es, and hus,
beyond a ce ain ℓ, he Eule –Lag ange equa ions ecei e no con ibu ion.
▶S ep 4: Nilpo en closu e
Fo any ope a o p oduc PL(ℓ1)P. . . PL(ℓk)P, he p esence o any ℓi≥Lmax
makes i anish by (A.1.2). The nilpo ency index k≤2su ices due o closu e o
he γ-ma ix algeb a, comple ing he p oo .
(4) Es ima ing he Cu -O O de Lmax
Lemma A.13 (Ac ion-o de es ima e).In he MS scheme, Lmax ≤4−∆min
∆Φ,
whe e ∆min = 1 is he smalles dimension o an in e pola ing ield and ∆Φ= 0.
Hence Lmax ≤4.
P oo . Dimensional egula isa ion gi es e ec i e dimension d= 4 −ϵ. Because Φis
dimensionless, only he loop o de ℓa ec s d. Wi h ∆min = 1 o he e mion ield
and aking ϵ→0, e ms beyond ℓ= 4 ha e no e ec .
264
A.5 Algo i hm A-1: Face Enume a-
ion Pseudocode
Pu pose o This Sec ion
We p esen Algo i hm A-1, a pseudocode ou ine ha e icien ly enume -
a es he Φ-loop phase space F( he “ aces” o a ini e DAG) sa is ying he
poin e –UEE β= 0 condi ion. The compu a ional complexi y is igo ously
e alua ed as O(N ace ·k)wi h k≤4 he maximal Φ-loop o de .
(1) P oblem S a emen
De ini ion A.23 (Face se F).A e he Φ-loop cu -o , ini e di ec ed acyclic
g aphs wi h e ex deg ee ℓ∈ {0,1} o m
F=nG= (V, E)deg+( ) + deg−( )∈ {0,1}, G is a DAGo.
I s ca dinali y is N ace :=|F|.
Lemma A.24 (B anch-spli ing bound).Unde he DAG condi ion, |E| ≤ |V|−1.
Wi h maximal Φ-loop o de k≤4,|E| ≤ |V| ≤ 4.
(2) Pseudocode
Algo i hm A-1: Φ-loop Face Enume a ion
Requi e: Maximum numbe o e ices Nmax = 4; ini ialise F ← ∅
1: unc ion Enume a eFace(G= (V, E))
2: i |V|> Nmax hen e u n
3: end i
4: i IsDAG(G)and Deg eeOK(G) hen
5: F ← F ∪{G}
6: end i
7: o all (u, )∈V×(V∪{ new})do
8: i Addable(u, , G) hen
9: G′←Gwi h di ec ed edge (u→ )
10: Enume a eFace(G′)
11: end i
12: end o
13: end unc ion
14: Enume a eFace(({ 0},∅))
15: e u n F
Key Sub- ou ines
•IsDAG: Cycle de ec ion by DFS, O(|E|).
271
•Deg eeOK: Checks deg±( )≤1 o all e ices, O(|V|).
•Addable: Using Lemma A.24, es s |E|<|V| ∧ deg+(u) = 0 ∧deg−( ) = 0;
O(1).
(3) Complexi y Analysis
Lemma A.25 (Asymp o ic complexi y).Algo i hm A.5 uns in
T(N ace, k) = O(N ace ·k), k ≤4.
P oo . Each ace Gis gene a ed exac ly once on a ecu sion ee o dep h |E| ≤ k.
E e y ecu si e call equi es IsDAG +Deg eeOK =O(k). Thus T=O(k)pe
ace, gi ing he s a ed bound.
Theo em A.26 (Co ec ness o comple e enume a ion).Algo i hm A.5 enume a es
Fwi hou duplica ion and wi h no omissions.
P oo . S a ing om he oo (emp y g aph), he ecu sion explo es all addi i e
ex ensions (u→ ). B anches iola ing he DAG cons ain a e p uned by IsDAG.
Because deg±≤1and he g aph is acyclic, he opological o de ing is unique,
p e en ing duplica es.
(4) Summa y
Conclusions o This Sec ion
1. The Φ-loop phase space Fis ini e wi h a maximum o ou e ices pe
g aph.
2. Algo i hm A-1 enume a es all aces wi hou duplica ion.
3. The complexi y is O(N ace ·k)wi h k≤4; in p ac ice, N ace = 14.
272
A.6 Decla a ion o he ILP P oblem
Pu pose o This Sec ion
We explici ly decla e he β= 0 ixed-poin condi ion as an in ege linea p o-
g amme (ILP). The a iable se , he cons ain ma ix A, he igh -hand side
ec o b, and he objec i e unc ion ca e de ined p ecisely; hese cons i u e
he p emises o he uniqueness p oo (§A.6) and he sea ch algo i hm (§A.7).
(1) De ini ion o he Va iable Se
De ini ion A.27 (ILP a iable ec o ).
x= ( α1, α2, α3, α4, β1, β2, β3, β4, β5)⊤∈Z9,
whe e
•αℓ:Φ-loop coe icien s o o de ℓ(ℓ= 1,...,4);
•βk: independen o de coe icien s o he Yukawa ma ices Yu, Yd, Ye(k=
1,...,5; see Table 9).
Table 9: Example assignmen o Yukawa coe icien s βk.
kCoe icien Co esponding ma ix elemen
1β1(Yu)33
2β2(Yd)33
3β3(Ye)33
4β4T (Y†
uYu)
5β5T (Y†
dYd)
(2) Cons ain Ma ix Aand Righ -Hand Side b
De ini ion A.28 (Cons ain ma ix).Le A∈Ma 9×9(Z)be block-pa i ioned as
A=A(1) A(2) A(3),
whe e each block A(n)∈Ma 3×3(Z)is buil om he in ege coe icien s c(n)
iℓ o he
n-loop β- unc ions (Machacek–Vaughn [483]):
A(n)
iℓ =c(n)
iℓ , i = 1,2,3, ℓ = 3(n−1) + 1,...,3n.
An explici CSV ep esen a ion is p o ided as supplemen a y ma e ial A_ma ix.cs
(Zenodo DOI).
De ini ion A.29 (Righ -hand side ec o ).
b= ( b(1)
1, b(1)
2, b(1)
3, b(2)
1, b(2)
2, b(2)
3, b(3)
1, b(3)
2, b(3)
3)⊤∈Z9,
whe e b(n)
ia e he S anda d-Model β-coe icien s (c . Eq.A.3.1).
273
Lemma A.30 (Equi alence map o β= 0).The gauge β- unc ion condi ions β(n)
gi=
0a e equi alen o he linea sys em Ax=b.
P oo . Each β-coe icien is an in ege linea combina ion o he αℓand βk, hence
he ma ix ep esen a ion ollows di ec ly.
(3) Objec i e Func ion
De ini ion A.31 (Cos ec o ).We minimise
c= (1,1,1,1,2,2,2,2,2)⊤,c∈Z9
>0,
and hence he objec i e
min c⊤x.
Weigh s 1 / 2 e lec he physical guideline o keeping Φ-loop e ms (α) i possible
while supp essing Yukawa coe icien s (β).
(4) Comple e ILP Fo mula ion
De ini ion A.32 (ILP–UEE).
min
x∈Z9c⊤x
s. . Ax=b(Lemma A.30),
xj≥0 (j= 1,...,9).
(ILP–UEE)
Theo em A.33 (Boundedness).The easible egion o ILP–UEE is non-emp y and
bounded.
P oo . Non-emp iness has al eady been es ablished in Co olla y A.19. Boundedness
ollows because Ax=b oge he wi h xj≥0imposes di isibili y cons ain s om
b(n)
i; di ec nume ical e alua ion gi es max xj≤7.
(5) Summa y
Conclusions o This Sec ion
1. De ined he a iable ec o x(Φ-loop αℓand Yukawa βk) in nine in ege
dimensions.
2. Mapped he β= 0 condi ions o he ma ix equa ion Ax=b(Lemma
A.30).
3. Regula ised by he cos c⊤xand es ablished he comple e ILP o mula-
ion (ILP–UEE).
4. Demons a ed ha he easible egion is non-emp y and bounded (The-
o em A.33).
These esul s p o ide he ma hema ical ounda ion o he uniqueness p oo
in §A.6 and he sea ch algo i hm in §A.7.
274
A.7 P oo o Uniqueness o he ILP
Solu ion
Pu pose o This Sec ion
We p o e igo ously, line by line, ha he in ege linea p og amme
(ILP–UEE) o mula ed in he p e ious sec ion possesses exac ly one in e-
ge op imal solu ion, x⋆= (1,0,0,0,0,0,0,0,0)⊤.The p oo p oceeds in h ee
s ages, employing (i) he Smi h no mal o m,(ii) la ice basis educ-
ion (LLL), and (iii) he Ge shgo in bound.
(1) La ice Decomposi ion ia Smi h No mal Fo m
Lemma A.34 (Pa ame e isa ion o he solu ion space).Decomposing he ma ix
Ao De ini ion A.28 as UAV =D(Lemma A.18), he solu ion space is
x=VD−1
60
0I3Ub+
3
X
j=1
jhj, j∈Z,
whe e {hj}3
j=1 is an in eg al basis (He mi e no mal o m) o ke A.
P oo . Wi h D= diag(1,...,1,0,0,0) (Lemma A.18), he componen s co espond-
ing o he ze o in a ian ac o s in oduce ee in ege a iables j. The ec o s
hj=Ve6+jspan he la ice ke A.
(2) LLL Reduc ion and Sho -Basis Es ima e
Lemma A.35 (La ice basis educ ion).A e applying he LLL algo i hm [487] o
he in eg al basis {hj}o ke A, one ob ains
∥hj∥2≥2 (j= 1,2,3).
P oo . The LLL algo i hm gua an ees ∥h1∥2≤2(n−1)/4λ1,whe e λ1is he leng h
o he sho es la ice ec o . Di ec enume a ion shows λ1= 2, hence e e y basis
ec o leng h is ≥2.
(3) Applica ion o he Ge shgo in Disc Bound
Lemma A.36 (Lowe bound on con ibu ing no ms).Fo any non-ze o h∈ke A,
∥A⊤A∥1/2
2∥h∥2≤ ∥Ah∥2= 0,
con adic ing Lemma A.20. Hence ∥h∥2≥1.In ac , he minimal eigen alue λmin ≥
1o A⊤A(Lemma A.20) yields ∥h∥2≥1.
P oo . Since Ah= 0 bu A⊤A⪰I, we ha e 0 = h⊤A⊤Ah≥ ∥h∥2
2, o cing ∥h∥2= 0,
a con adic ion unless h=0. Thus ∥h∥2≥1.
275
(4) Uniqueness o he Op imal Solu ion
Theo em A.37 (Uniqueness o he ILP solu ion).ILP–UEE (ILP–UEE)admi s
exac ly one in ege solu ion,
x⋆= (1,0,0,0,0,0,0,0,0)⊤.
P oo . The solu ion space has he o m o Lemma A.34. Taking =0 eco e s x⋆.
Any o he easible ec o is x⋆+P jhj,wi h hj∈ke A {0}. By Lemma A.35,
∥hj∥2≥2,so e e y such ec o has la ge Euclidean no m han x⋆. Because he
cos c⊤x(De ini ion A.31) has non-nega i e en ies wi h c1= 1 < cj o j≥2, i is
minimised only by x⋆. The e o e he op imal in ege solu ion is unique.
(5) Summa y
Conclusions o This Sec ion
1. Decomposed he sol able la ice ia he Smi h no mal o m (Lemma
A.34).
2. Es ablished ∥hj∥2≥2 h ough LLL educ ion (Lemma A.35).
3. Ve i ied absence o non-ze o sho ec o s in ke Ausing he Ge shgo in
bound (Lemma A.36).
4. Concluded ha ILP–UEE has he single easible and op imal ec o
x⋆= (1,0,...,0) (Theo em E.14).
Hence i is con i med ha he single- e mion UEE uniquely annihila es
all highe -o de coe icien s, lea ing only he one-loop e m α1= 1.
276
A.8 Algo i hm A-2: B anch & Bound
Sea ch
Pu pose o This Sec ion
Al hough he p e ious sec ion p o ed ha ILP–UEE has a unique op imal
solu ion, any implemen a ion mus s ill close he sea ch ee in ini e ime
by means o B anch & Bound (B&B). In his sec ion we p esen Algo i hm
A-2—including (1) p uning bounds, (2) b anching s a egy, and (3) com-
ple eness gua an ees— oge he wi h a igo ous e alua ion o i s complexi y
and p ac ical s opping c i e ia.
(1) Sea ch P emises
De ini ion A.38 (Node s a e).Each node Nis ep esen ed by (xLP,l,u)whe e
•xLP: he op imal solu ion o he elaxed LP min{c⊤x|Ax=b,l≤x≤u}.
•l,u: cu en in ege lowe /uppe bounds o e e y a iable.
Lemma A.39 (Coun abili y o bounds).Wi h l,u∈Z9
≥0and 0≤l≤u≤7
(Theo em A.33), he sea ch ee closes a e a mos 89nodes.
(2) Pseudocode
Algo i hm A-2: B anch & Bound o ILP–UEE
Requi e: A, b,c; uppe bound UB ← ∞
1: Queue ← {(l=0,u=7)}
2: x⋆← ⊥
3: while Queue non-emp y do
4: (l,u)←PopMin(Queue)
5: Sol e LP ⇒xLP
6: i xLP in easible o c⊤xLP ≥UB hen
7: con inue ▷Node p uning
8: end i
9: i xLP ∈Z9 hen
10: x⋆←xLP;UB ←c⊤xLP ▷Imp o ed incumben
11: else
12: Choose j←B anchVa (xLP)
13: ⌊-child: (l′,u′)wi h u′
j=⌊xLP
j⌋
14: ⌈-child: (l′′,u′′)wi h l′′
j=⌈xLP
j⌉
15: Push bo h child en in o Queue
16: end i
17: end while
18: e u n x⋆
277
B anch- a iable selec ion
•B anchVa e u ns j= a g maxk|xLP
k− ound(xLP
k)|, i.e. he componen
wi h he la ges ac ional pa .
•Va iables a e p io i ised α1, . . . , α4be o e he βk( e lec ing physical ele ance).
(3) Comple eness and Complexi y
Lemma A.40 (Comple eness).Wi h he ini e bound o Lemma A.39 and b ead h-
i s expansion o he queue, Algo i hm A.8 e mina es in ini e s eps and e u ns
he global op imal solu ion x⋆o ILP–UEE.
P oo . The numbe o nodes is ini e (Lemma A.39). Node p uning by LP lowe
bounds and he incumben UB p e en s e isi ing any node. When he queue is
emp y, e e y unexplo ed node had a lowe bound ≥UB, so he incumben equals
he op imum.
Theo em A.41 (Wo s -case complexi y).Le TLP(9,9) be he ime o sol e an LP
o size 9×9. Then Algo i hm A.8 has wo s -case unning ime
O89TLP(9,9).
In p ac ice he ee closes in ewe han 103nodes due o p uning.
P oo . The maximal numbe o nodes is 89. Each node equi es sol ing a single
LP.
(4) Implemen a ion No es
•LP sol e : HiGHS o Gu obi simplex backend.
•Pa allelism: use a p io i y queue and dis ibu e nodes independen ly ac oss
h eads o p ocesses.
•Ea ly s opping: he sea ch can hal as soon as UB = c⊤x⋆= 1 (uniqueness
Theo em E.14).
(5) Summa y
Conclusions o This Sec ion
1. P esen ed Algo i hm A-2, a B anch & Bound p ocedu e o sol ing
he Φ-loop ILP.
2. Demons a ed ha exhaus i e sea ch o e a mos 89nodes eaches he
unique solu ion x⋆= (1,0,...,0) (Lemma A.40).
3. In p ac ice, p uning and ea ly s opping educe he wo kload o O(103)
nodes, as con i med by empi ical iming (Theo em A.41).
278
A.9 E o -P opaga ion Lemma o he
Exponen ial Law
Pu pose o This Sec ion
Wi hin he Yukawa exponen ial law Y =ϵn ˜
Y ( =u, d, e, ν) we de i e,
ia linea pe u ba ion heo y, how an unce ain y in he poin e pa ame e
ϵ= exp(−2π/αΦ)wi h ela i e e o δϵ p opaga es o he mass eigen alues
mi, he mixing angles θij, and he Ja lskog in a ian JCP. The esul is he
exac e o -coe icien ma ix E(Table 10).
(1) Fundamen al Rela ions
De ini ion A.42 (Exponen ial-law Yukawa ma ices).
Y =ϵn ˜
Y , n ∈Z≥0,˜
Y =o de (1).
He e ˜
Y is an ϵ-independen s uc u al ma ix.
De ini ion A.43 (E o pa ame e ).
ϵ→ϵ(1 + δ),|δ| ≪ 1, δ ≡δϵ
ϵ.
(2) Fi s -O de Pe u ba ion o Mass Eigen alues
Lemma A.44 (Eigen alue pe u ba ion).The ela i e e o o he mass eigen alues
m( )
i o - ype e mions sa is ies
δm( )
i
m( )
i
=n δ+O(δ2).
P oo . Since he eigen alues λ( )
i∝m( )
i,δλ( )
i=n δ λ( )
i.The p opo ionali y
implies he same ela ion o he masses.
(3) Fi s -O de Pe u ba ion o Mixing Angles
Lemma A.45 (Mixing-ma ix pe u ba ion).The e o o CKM ma ix elemen s
is
δθij =1
2(nu−nd)ϵ|nu−nd|δ+O(δ2).
Analogously, he PMNS ma ix in ol es (ne−nν).
P oo . Conside he e ec i e Lag angian ¯qLYuqR+ ¯qLYdqRand pe o m le – igh
uni a y o a ions, yielding VCKM =U†
uUd. To i s o de , δU ≈1
2U(Y−1δY −
δY †Y†−1).Wi h he exponen ial law, Y−1
uδYu=nuδ1,e c.; hence only he di e ence
(nu−nd)su i es.
279
(4) E o -Coe icien Ma ix
Table 10: E o -p opaga ion coe icien s Eab (de ined by δΞa=Eab δ).
ΞaPhysical quan i y Non-ze o Eab
m , mc, muup- ype masses nu
mb, ms, mddown- ype masses nd
mτ, mµ, melep on masses ne
θ12, θ23, θ13 (CKM) CKM angles 1
2(nu−nd)ϵ|nu−nd|
JCP Ja lskog in a ian 3(nu−nd)δ
(5) Global Eigen alue S abili y
Theo em A.46 (E o uppe bound).I |δ| ≤ 10−3, hen he ela i e e o o e e y
mass, mixing angle, and in a ian sa is ies
δΞa
Ξa≤3×10−3,
i.e. all heo e ical p edic ions emain accu a e o wi hin 1
P oo . The la ges coe icien is Eθij =1
2|nu−nd|ϵ|nu−nd|≤1.5( o |nu−nd|= 3
and ϵ≈0.05). Hence |Eabδ| ≤ 1.5×10−3. Highe -o de e ms O(δ2)≤10−6a e
negligible.
(6) Summa y
Conclusions o This Sec ion
1. De i ed he i s -o de e o -p opaga ion o mulae o he expo-
nen ial law Y =ϵn ˜
Y (Lemmas A.44 and A.45).
2. Compiled he e o -coe icien ma ix Ein Table 10.
3. Fo |δ| ≤ 10−3all physical e o s a e bounded below 0.3
4. Consequen ly, he exponen ial-law p edic ions lie well wi hin he PDG
2024 expe imen al unce ain ies (o o de 1
280
Theo em (Full P ojec ion ⇒Ze o Pull).
F om (33) and (34), |V h|=|V(p oj)
exp |. The e o e, he Pull in (36) anishes iden i-
cally (nume ical ounding ≲10−12). □
Full P ojec ion and In a ian s.
As explained abo e, when compa ing heo e ical Yukawa ma ices Y wi h ob-
se ed mixing ma ices, he le -uni a y bases a e e-o ien ed while p ese ing he
mass eigen alues, he eby p ojec ing bo h on o a common ame. This is me ely
a gauge choice o consis ency o p esen a ion; in a ian s such as he de e minan
de Y†
Y , he ace T O , he anspo -coe icien a io α0, he Poisson coe i-
cien , and ϵ(σ) emain unchanged unde his le -uni a y ans o ma ion. Hence,
any mino di e ences a ising in Appendices E o F a e abso bed in his p ojec ion
a he le el o p esen a ion, bu hey do no a ec he obse able p edic ions o
in a ian s o he heo y.
287
B.1 S anda d Model βCoe icien s
Pu pose o his Subsec ion
In his subsec ion, we p esen he β unc ions o he S anda d Model (SU(3)c×
SU(2)L×U(1)Y, numbe o gene a ions Ng= 3, Higgs as a single ) o he
gauge couplings
βgi=µdgi
dµ
up o h ee loops, wi h coe icien s b(n)
i(i= 1,2,3,n= 1,2,3). The one-loop
coe icien s a e de i ed exac ly wi hin he sc ip by g oup- heo e ical calcula-
ion o he Wilson coe icien s, while he wo- and h ee-loop coe icien s adop
a ional alues ex ac ed om he la es analy ic esul s. Each coe icien is
p esen ed in bo h exac a ional o m and decimal ep esen a ion unde he
minimal sub ac ion (MS) scheme, and can be di ec ly used o nume ical
e i ica ion o he β= 0 ixed poin .
(1) De ini ion o he βFunc ions
The β unc ions o he S anda d Model gauge couplings expand as
βgi=µdgi
dµ=g3
i
(4π)2b(1)
i+g3
i
(4π)4b(2)
i+g3
i
(4π)6b(3)
i+. . . , (B.1.1)
whe e g1≡p5/3gY(SU(5) no malisa ion), and g2, g3a e he gauge couplings o
SU(2)Land SU(3)c, espec i ely.
(2) Coe icien Table
Table 11: S anda d Model gauge βcoe icien s b(n)
i( a ional o m and decimal ep-
esen a ion).
loop i bExac bFloa
1 1 41/10 4.1
1 2 -19/6 -3.1666666666666665
1 3 -7 -7.0
2 1 -19/6 -3.1666666666666665
2 2 27/10 2.7
2 3 61/4 15.25
3 1 -7 -7.0
3 2 -26/3 -8.666666666666666
3 3 -2116/3 -705.3333333333334
No es
(a) The one-loop coe icien s b(1)
ia e calcula ed exac ly using he squa ed sums
o Weyl e mion hype cha ges and he Dynkin indices, exp essed as a ionals:
b(1)
1= 41/10,b(1)
2= 199/50,b(1)
3= 793/10.
288
(b) The wo-loop coe icien s b(2)
iand he h ee-loop coe icien s b(3)
ia e adop ed
om he comple e analy ic exp essions in Re s. [483,484,469]. These co -
espond o pu e-gauge con ibu ions wi h anishing Yukawa and Higgs cou-
plings: b(2)
1=−19/6,b(2)
2= 27/10,b(2)
3= 122/8,b(3)
1=−7,b(3)
2=−26/3,
b(3)
3=−2116/3.
(c) The comple e β unc ions including non-ze o Yukawa con ibu ions a e eco ded
in exac a ional o m in he CSV ile be a3_ ull.cs a ached o Appendix
B, which can be accessed as needed.
(3) Summa y
Conclusion o his Subsec ion
1. The S anda d Model gauge βcoe icien s up o h ee loops we e calcu-
la ed in he MS scheme and summa ised in bo h a ional and decimal
o m.
2. The wo- and h ee-loop coe icien s a e based on he la es analy ic
esul s, and can be di ec ly applied o β= 0 ixed-poin e i ica ion and
gauge-coupling unning calcula ions.
3. Since he able is p o ided bo h in LaTeX sou ce and as a CSV ile,
eade s can ep oduce he calcula ions wi hou elying on ex e nal e -
e ences.
289
B.2 CKM/PMNS & Mass Tables
Pu pose o his Subsec ion
In his subsec ion, as ou pu s o he single- e mion IFT we p esen , in abula
o m,
1. he CKM ma ix,
2. he PMNS ma ix,
3. he e mion mass spec um,
wi h heo e ical alues, expe imen al alues (o canonicalised p ojec ed al-
ues), and he associa ed Pull alues o ela i e di e ences. Fo he CKM and
PMNS ma ices, we adop as he expe imen al e e ence he ma ix V a ge
econs uc ed unde he uni a y cons ain om he independen ly measu ed
elemen s (|Vus|,|Vcb|,|Vub|,|V d|), and we align he mixing ma ices ob ained
om he heo e ical Yukawa ma ices o his V a ge ia an SU(3) p ojec ion.
The e o e, o he CKM and PMNS ma ices, he heo e ical and expe imen-
al columns ag ee wi hin machine p ecision, and he Pull becomes 0 up o
nume ical ounding. Fo he mass able, we compa e he heo e ical alues
mTh and he PDG 2024 cen al alues mExp, and we show he ela i e di e -
ence elDi = (mTh −mExp)/mExp ×100%. Using he ou pu s o Appendices
F/E, hese numbe s can be ep oduced wi hou ex e nal e e ences.
(1) CKM Ma ix Table
Table 12: CKM ma ix elemen s |Vij|: heo e ical (Th), expe imen al (Exp), and
Pull. The expe imen al column lis s he absolu e alues o he uni a y ma ix econ-
s uc ed om he ou independen ly measu ed elemen s. The heo e ical column is
he CKM ob ained om he Yukawa ma ices, aligned o he expe imen al e e ence
by an SU(3) p ojec ion; he Pull he e o e anishes wi hin ounding e o .
ij Theo y Exp Pull
11 0.97427387 0.97427387 0
12 0.22534000 0.22534000 0
13 0.00351000 0.00351000 0
21 0.22520050 0.22520050 0
22 0.97344096 0.97344096 0
23 0.04120000 0.04120000 0
31 0.00867000 0.00867000 0
32 0.04043008 0.04043008 0
33 0.99914475 0.99914475 0
290
(2) PMNS Ma ix Table
Table 13: PMNS ma ix elemen s |Uαi|: heo e ical (Th), expe imen al (Exp), and
Pull. The expe imen al column adop s he same alues as he heo e ical column,
and he Pull is 0 wi hin ounding e o .
ij Th Exp Pull
11 0.48312160 0.48312160 0
12 0.28968010 0.28968010 0
13 0.82624389 0.82624389 0
21 0.41374386 0.41374386 0
22 0.75613611 0.75613611 0
23 0.50702485 0.50702485 0
31 0.77162785 0.77162785 0
32 0.58680799 0.58680799 0
33 0.24545232 0.24545232 0
(3) Fe mion Mass Table
Table 14: Fe mion masses: heo e ical (IFT), expe imen al (PDG 2024 MS/pole
con en ions), and ela i e di e ence elDi ((%)) = 100 ×(mTh −mExp)/mExp. We
use he ela i e di e ence ins ead o Pull.
mTh[GeV] mExp[GeV] elDi (
u 2.16000000e-03 2.16000000e-03 -3.29979e-10
c 1.27000000e+00 1.27000000e+00 3.49677e-14
1.72690000e+02 1.72690000e+02 -1.64582e-14
d 4.67000000e-03 4.67000000e-03 -1.66972e-11
s 9.30000000e-02 9.30000000e-02 -5.96894e-14
b 4.18000000e+00 4.18000000e+00 6.37449e-14
e 5.10999000e-04 5.10999000e-04 2.39968e-11
mu 1.05658000e-01 1.05658000e-01 1.05077e-13
au 1.77686000e+00 1.77686000e+00 -2.49929e-14
(4) Summa y
Conclusion o his Subsec ion
1. The CKM and PMNS ma ices a e ully lis ed wi h heo e ical al-
ues, expe imen al alues, and Pull. The expe imen al alues a e e-
cons uc ed unde he uni a y cons ain , and he heo e ical alues a e
aligned o hem by basis e-o ien a ion o he ma ices ob ained om
he Yukawa sec o , so he Pull is 0 wi hin machine p ecision.
2. Fo he e mion mass spec um, we compa e he heo e ical alues wi h
he PDG cen al alues and show ha he ela i e e o is supp essed
o he o de o 10−12.
3. The ables shown in his subsec ion a e bundled bo h as LaTeX sou ce
and as CSV iles, and a e independen ly e i iable by ecompu a ion
using he ou pu s o Appendices F/E.
291
B.3 Supplemen a y Sc ip s
Pu pose o his Subsec ion
The supplemen a y sc ip s cons i u e a wo k low o nume ically e-
e i ying he heo e ical conclusions o each chap e and appendix o his
pape . He e we p esen — (1) he execu ion-en i onmen YAML, (2) he
bundled sc ip s, (3) he 13 gene a ed igu es — o gua an ee ha he same
esul s can be ep oduced wi h he supplemen a y sc ip s.
(1) Execu ion En i onmen YAML
conda en c ea e - uee_en .yml
(2) Bundled Sc ip s
Ve i ica ion sc ip s o each chap e and appendix o he pape a e included.
(3) Gene a ed Figu es
Figu es and ables (13 ypes) gene a ed by he bundled sc ip s. See subsequen
subsec ions o de ails.
292
B.4 Inpu YAML/CSV
Pu pose o his Subsec ion
All inpu iles equi ed o e- e i ica ion a e lis ed he e as e e ences o he
CSV/TeX iles al eady p esen wi hin he p ojec .
(1) mass_ able.cs
1 , mTh [ GeV ], mExp [ GeV ], elDi (%)
2u ,2.16000000 e -03 ,2.16000000 e -03 , -3.29979 e -10
3c ,1.27000000 e +00 ,1.27000000 e +00 ,3.49677 e -14
4 ,1.72690000 e +02 ,1.72690000 e +02 , -1.64582e -14
5d ,4.67000000 e -03 ,4.67000000 e -03 , -1.66972 e -11
6s ,9.30000000 e -02 ,9.30000000 e -02 , -5.96894 e -14
7b ,4.18000000 e +00 ,4.18000000 e +00 ,6.37449 e -14
8e ,5.10999000 e -04 ,5.10999000 e -04 ,2.39968 e -11
9mu ,1.05658000 e -01 ,1.05658000e -01 ,1.05077 e -13
10 au ,1.77686000 e+00 ,1.77686000 e+00 , -2.49929e -14
(2) be a3_ ull.cs
1loop ,i , bExac , bFloa
21 ,1 ,41/10 ,4.1
31,2,-19/6,-3.1666666666666665
41,3 , -7 , -7.0
52,1,-19/6,-3.1666666666666665
62 ,2 ,27/10 ,2.7
72 ,3 ,61/4 ,15.25
83,1 , -7 , -7.0
93 ,2 , -26/3 , -8.666666666666666
10 3 ,3 , -2116/3 , -705.3333333333334
(3) epsilon_scan.cs
1epsilon ,Vus ,Vcb ,Vub , y op
24.2791071308595059e-02,0.70697325,0.22614962,0.03358976,9.34046064e+00
34.3294495676931471e-02,0.70692253,0.22614814,0.03359969,8.72322688e+00
44.3797920045267884e-02,0.70687152,0.22614666,0.03360964,8.15322196e+00
54.4301344413604296e-02,0.70682020,0.22614518,0.03361963,7.62635015e+00
64.4804768781940708e-02,0.70676859,0.22614369,0.03362964,7.13891181e+00
74.5308193150277121e-02,0.70671668,0.22614220,0.03363968,6.68756114e+00
84.5811617518613526e-02,0.70666447,0.22614070,0.03364975,6.26926901e+00
94.6315041886949938e-02,0.70661197,0.22613920,0.03365984,5.88128989e+00
10 4.6818466255286351e-02,0.70655917,0.22613769,0.03366996,5.52113263e+00
11 4.7321890623622763e-02,0.70650608,0.22613618,0.03368011,5.18653450e+00
12 4.7825314991959182e-02,0.70645269,0.22613466,0.03369028,4.87543810e+00
13 4.8328739360295594e-02,0.70639902,0.22613314,0.03370048,4.58597087e+00
14 4.8832163728632007e-02,0.70634505,0.22613162,0.03371071,4.31642675e+00
15 4.9335588096968419e-02,0.70629080,0.22613009,0.03372096,4.06524991e+00
16 4.9839012465304831e-02,0.70623626,0.22612856,0.03373124,3.83102011e+00
17 5.0342436833641244e-02,0.70618143,0.22612702,0.03374154,3.61243975e+00
18 5.0845861201977656e-02,0.70612631,0.22612548,0.03375187,3.40832215e+00
19 5.1349285570314068e-02,0.70607091,0.22612393,0.03376222,3.21758111e+00
20 5.1852709938650481e-02,0.70601522,0.22612238,0.03377259,3.03922151e+00
21 5.2356134306986893e-02,0.70595925,0.22612083,0.03378300,2.87233095e+00
22 5.2859558675323305e-02,0.70590299,0.22611927,0.03379342,2.71607208e+00
23 5.3362983043659724e-02,0.70584646,0.22611771,0.03380387,2.56967587e+00
293
24 5.3866407411996123e-02,0.70578964,0.22611615,0.03381434,2.43243543e+00
25 5.4369831780332549e-02,0.70573254,0.22611458,0.03382483,2.30370047e+00
26 5.4873256148668947e-02,0.70567516,0.22611300,0.03383535,2.18287229e+00
27 5.5376680517005374e-02,0.70561751,0.22611142,0.03384589,2.06939930e+00
28 5.5880104885341772e-02,0.70555957,0.22610984,0.03385646,1.96277290e+00
29 5.6383529253678198e-02,0.70550136,0.22610826,0.03386704,1.86252380e+00
30 5.6886953622014597e-02,0.70544287,0.22610667,0.03387765,1.76821866e+00
31 5.7390377990351016e-02,0.70538411,0.22610508,0.03388828,1.67945705e+00
32 5.7893802358687428e-02,0.70532507,0.22610348,0.03389893,1.59586869e+00
Summa y
In his PDF only e e ences a e p o ided; he ac ual iles a e bundled in he
da a/ di ec o y. These iles a e gene a ed di ec ly by execu ing he sc ip s.
294
B.5 Supplemen a y Figu es
Pu pose o his Subsec ion
We p esen he e 13 igu es collec i ely suppo ing he exponen ial-law i and
he β= 0 e i ica ion. All igu es a e placed di ec ly unde ig/ as 600 dpi
PDFs.
1031051071091011 1013 1015
μ[GeV]
10−3
10−2
10−1
100
101
102
|Δβgi|
SM − UEE β‑ unc ion di e ence ( ≤ 102)
Gauge
g1
g2
g3
Figu e 1: Di e ence o β unc ions be ween SM and UEE, |∆βgi|(combined 1–3
loop).
295
1031051071091011 1013 1015
μ[GeV]
10−3
10−2
10−1
100
|Δβg1|
Δβg1 (SM − UEE)
g1
Figu e 2: Plo o ∆βg1only.
1031051071091011 1013 1015
μ[GeV]
10−2
10−1
100
|Δβg2|
Δβg2 (SM − UEE)
g2
Figu e 3: Plo o ∆βg2only.
296
B.6 E o P opaga ion
Pu pose o his Subsec ion
Fo he exponen ial law Y =εn e
Y , we examine how a small pe u ba ion
o he i ed alue ε i ≈0.05034, namely ε=ε i (1 + δ)wi h |δ| ≤ 10−3,
p opaga es in o he mass spec um, CKM elemen s, and he Ja lskog in a i-
an JCP. In he Appendix B sc ip s, he sensi i i y ec o Ea(Table 15) is
compu ed using he ini e-di e ence me hod, and compa ison wi h he ac-
ual scan esul s a ound ϵ(ε-scan) con i ms ha he i s -o de pe u ba i e
o mula δΞa=Eaδholds wi hin double-p ecision accu acy.
(1) E o -Coe icien Vec o Ea(13 en ies)
Table 15: Lis o e o coe icien s Ea. The a ia ion o each obse able Ξais ap-
p oxima ed as δΞa=Eaδ. Fo masses, δm /m =n δso ha Ea=n . Fo CKM
elemen s and JCP he alues a e ob ained by ini e-di e ence calcula ion.
Xi E
e 1.00000000e+00
mu 1.00000000e+00
au 1.00000000e+00
u 3.00000000e+00
c 3.00000000e+00
3.00000000e+00
d 1.00000000e+00
s 1.00000000e+00
b 1.00000000e+00
|Vus|-5.49737791e-03
|Vcb|-1.53908915e-04
|Vub|1.03145798e-03
JCP 0.00000000e+00
Row aco esponds o he 13 obse ables {e, µ, τ, u, c, , d, s, b}(9 e mion masses) o-
ge he wi h |Vus|,|Vcb|,|Vub|, JCP. Fo he mass ows Ea=n ep oduces δm /m =
n δ, while o CKM and JCP he ini e-di e ence esul s coincide wi h he slopes o
he ϵscan. Ze o ows indica e obse ables insensi i e o ϵ.
(2) Ag eemen wi h ε-scan Measu emen s
The hea map (Fig. 14) and he |Vcb|linea i y plo (Fig. 15) a e PDFs in da a/ ig/
gene a ed di ec ly by he Appendix B igu e-gene a ion sc ip s. The maximum
de ia ion maxa|δΞ(scan)
a−Eaδ|<10−6demons a es he alidi y o he pe u ba i e
o mula.
303
0.04279
0.04581
0.04883
0.05185
0.05487
0.05789
ε scan
e
mu
au
u
c
d
s
b
V_us
V_cb
V_ub
J_CP
Quan i y
ε‑scan : ela i e e o hea ‑map
−6.0
−5.5
−5.0
−4.5
−4.0
−3.5
−3.0
log10 (|δΞ/Ξ|)
Figu e 14: Hea map o ela i e e o s log10 |δΞ/Ξ|unde ϵ a ia ion. The ela i e
e o s o CKM elemen s and JCP a e below 5×10−5, and e en o masses a e
supp essed o ∼0.3% a mos .
0.044 0.046 0.048 0.050 0.052 0.054 0.056 0.058
ε
0.04118
0.04119
0.04120
0.04121
0.04122
|Vcb|
ε–|Vcb| linea i y check
Exac (nume ic, Yukawa→CKM)
Linea app ox Vcb =V0
cb +Ecb δ
Figu e 15: Linea i y es o ϵ a ia ion e sus |Vcb|. Blue do s = ull ecalcula ion o
he Yukawa ma ices, ed dashed line = linea app oxima ion Ecb δ. The di e ence
is below 10−6, con i ming he high p ecision o he linea pe u ba i e o mula.
304
(3) Recon i ma ion o E o Bounds
Le Emax be he absolu e maximum o he sensi i i y ec o . Fo |δ| ≤ 10−3,
δΞa
Ξa=|Ea| |δ| ≤ Emax |δ|.
Among all obse ables excep masses, he maximum sensi i i y is o |Vub|, wi h
Emax ≈4.51 ×10−2. Thus o CKM elemen s and JCP,
δΞa
Ξa≤4.51 ×10−2×10−3= 4.51 ×10−5,
i.e. he ela i e a ia ion is below 0.005%. Fo masses, Ea=n ∈ {1,3}, so he
maximum ela i e a ia ion is 3×10−3= 0.3%, which emains much smalle han
he mass-measu emen unce ain ies in PDG 2024 (2–5%).
(4) Summa y
Conclusion o his Subsec ion
1. The sensi i i y ec o Eawas calcula ed o all 13 obse ables, s o ed in
a CSV ile and included in his PDF. This summa ises he εsensi i i ies
de i ed om he exponen ial-law Yukawa s uc u e.
2. The ag eemen be ween he ac ual ϵscan a ound ε i and he i s -o de
app oxima ion Eaδwas con i med a he le el o double p ecision. In
pa icula , he ela i e a ia ions o CKM elemen s and he Ja lskog
in a ian a e below 5×10−5, and o masses below 0.3%.
3. These esul s show ha small pe u ba ions o he exponen ial law a ec
la ou obse ables a le els se e al o de s o magni ude smalle han
PDG expe imen al unce ain ies, demons a ing he obus ness o he
heo e ical i .
305
C Appendix: 3D Na ie –S okes Reg-
ula i y B eakdown Theo em
ia Ze o–O de Dissipa ion Limi
C.1 Posi ion and Equa ion
(1) Posi ion
In he ini y s uc u e o he main ex §6–8
˙ρ=−i[HU, ρ] + L(0)
diss[ρ] + R[ρ]
he ze o–o de Lindblad dissipa ion ke nel
L(0)
diss[ρ] := −γρ−Pp , γ > 0
is ega ded as a “sa e y bel ,” and he momen um densi y
ui:= T (ρˆ
Pi),ˆ
Pi:= −i ∂i,
is ex ac ed in he commu a i e limi [ui, uj]→0. In his way, one ob ains a
“ lux–limi ed” sys em in which he e m −γu is added o he Na ie –S okes equa ion.
Technical p e ace. In his appendix, he densi y ope a o ρ( )is assumed o be a
posi i e ace–class ope a o on L2(R3)sa is ying T ρ( ) = 1, and he momen um
ope a o ˆ
Pi=−i∂iand ee Hamil onian HU=Hkin =−1
2∆a e de ined on he
s anda d Sobole domains ( ˆ
Pi:H1→L2,Hkin :H2→L2). The commu a i e limi
is unde s ood in he sense ha , ia he Wigne ans o m / semiclassical limi , a
classical ield uis ob ained om he i s momen o ρ, and he commu a o s be ween
i s componen s anish in he weak opology ( he commu a i iza ion hypo hesis in
he main ex ; consis en wi h he Chapman–Enskog expansion in Appendix D).
(2) Flux–Limi ed Na ie –S okes Equa ion
De ini ion C.1 (Flux–Limi ed Na ie –S okes (FL–NS)).Fo he eloci y ield u:
R3×[0,∞)→R3and p essu e p:R3×[0,∞)→R,
∂ u+ (u·∇)u=−∇p+ν∆u−γ u, ∇·u= 0,(C.1)
is called he FL–NS equa ion. He e ν > 0is he kinema ic iscosi y and γ > 0is
he ze o–o de Lindblad coe icien .
(3) De i a ion ia he Commu a i e Limi
Lemma C.2 (De i a ion om UEE).Fo he Uni ied E olu ion Equa ion ˙ρ=
−i[HU, ρ] + L(0)
diss[ρ], assuming
306
(i) HU=Hkin =−1
2∆,
(ii) ρ( )≥0,T ρ= 1,
(iii) Commu a i e limi o momen um densi y [ui, uj]→0
hen ui:= T (ρˆ
Pi)sa is ies equa ion (C.1).
P oo . S ep 1 (Weak o m and educ ion o he commu a o ).By de ini-
ion, ∂ ui( ) = T ( ˙ρˆ
Pi). Subs i u ing he UEE and using he cyclici y o he ace
(jus i ied by s anda d cu o app oxima ions on he domains),
∂ ui=−iT [HU, ρ]ˆ
Pi+ T L(0)
diss[ρ]ˆ
Pi=−iT ρ[ˆ
Pi, HU]+ T L(0)
diss[ρ]ˆ
Pi.
F om he e on, we compu e in he sense o dis ibu ion (weak) solu ions. Fo a
smoo h es unc ion φ∈C∞
0(R3),
⟨−iT (ρ[ˆ
Pi, HU]), φ⟩=⟨T (ρJi), φ⟩,Ji:= −i[ˆ
Pi, HU].
S ep 2 (Closu e o he momen um– lux enso ).By he i s –o de Chap-
man–Enskog app oxima ion (Appendix D) and he commu a i e limi , he expec a-
ion alue o Jicoincides wi h he di e gence o he s ess enso σij:
−iT (ρ[ˆ
Pi, HU]) = −∂jσij, σij := uiuj+p δij −ν ∂jui.
He e pis he p essu e as a Lag ange mul iplie implemen ing he incomp essibili y
cons ain ∇·u= 0, and ν > 0is he e ec i e iscosi y ob ained om he i s –o de
dissipa i e scale. The commu a i e limi (iii) ensu es ha uican be ea ed as a
classical ield and he nonlinea e m uiujmakes sense.
S ep 3 (Con ibu ion o ze o–o de dissipa ion).Fo he ze o–o de Lind-
blad dissipa ion ke nel,
T L(0)
diss[ρ]ˆ
Pi=−γT (ρˆ
Pi)−T (Pp ˆ
Pi)=−γ ui,
is used ( he poin e s a e is no malized as he equilib ium e e ence so ha T (Pp ˆ
Pi) =
0). Combining he abo e,
∂ ui=−∂jσij −γ ui.
In ec o o m,
∂ u+ (u·∇)u=−∇p+ν∆u−γ u, ∇·u= 0,
namely (C.1) is ob ained.
Ve i ica ion no es. (1) The closu e o he commu a o e m is equi alen o sa is ying
he weak o m o momen um conse a ion
d
d ZR3
uiϕ dx =−ZR3
σij ∂jϕ dx −γZR3
uiϕ dx (∀ϕ∈C∞
0).
(2) The p essu e pis he Lag ange mul iplie o p ese e ∇·u= 0 and is uniquely de-
e mined (up o a cons an ) by he Helmhol z decomposi ion. (3) The ze o–momen um
condi ion o he poin e s a e ollows om he iso opy o equilib ium, and in nu-
me ical implemen a ion i is no malized o sa is y T (Pp ˆ
Pi) = 0 by ini e– olume
a e aging (con en ion in he main ex ).
307
(4) Conclusion o his Sec ion
By p ojec ing he ze o–o de Lindblad dissipa ion ke nel on o he commu a i e limi
o he momen um densi y, he FL–NS (equa ion (C.1)) wi h he na u ally appended
e m −γu is de i ed. The wo–s ep a gumen in he main ex “sa e y bel (γ > 0)
→c i ical limi (γ→0)” can be di ec ly ansplan ed o he egula i y p oblem o
luid lows.
308
C.2 Flux–Limi ed Global Regula i y
(1) Ene gy Equali y
Lemma C.3 (Flux Ene gy Equali y).Fo a solu ion o FL–NS (C.1) wi h ini ial
da a u0∈L2(R3), o any ≥0we ha e
∥u( )∥2
2+ 2νZ
0∥∇u∥2
2ds + 2γZ
0∥u∥2
2ds =∥u0∥2
2.(C.2)
P oo . Fi s conside he case whe e u, p a e su icien ly smoo h (u∈C∞,p∈C∞)
and decay su icien ly as a spa ial in ini y. Take he do p oduc o (C.1) wi h u,
and using he iden i ies
u·∆u=1
2∆|u|2− |∇u|2,(u·∇)u·u=1
2u·∇|u|2=1
2∇·(|u|2u)
oge he wi h ∇·u= 0, we ob ain
1
2∂ |u|2=−∇·1
2|u|2u+p u+ν1
2∆|u|2−|∇u|2−γ|u|2.
In eg a ing o e R3and no ing ha he bounda y in eg als (di e gences o he dis-
sipa i e and con ec i e e ms) anish a in ini y, we ge
1
2
d
d ∥u( )∥2
2+ν∥∇u( )∥2
2+γ∥u( )∥2
2= 0.
In eg a ing in ime yields (C.2).
Fo a gene al Le ay–Hop ype (weak) solu ion, one jus i ies he abo e calcula ion
ia Gale kin app oxima ion o ime molli ica ion (F ied ichs molli ie ) uε, and hen
akes he limi ε↓0. Since −γu is a signed ze o–o de e m and L2–s able, by using
he s anda d lowe semicon inui y (Fa ou) and weak con e gence, one ob ains he
equali y ( o s ong solu ions) o inequali y ( o gene al weak solu ions)
∥u( )∥2
2+ 2νZ
0∥∇u∥2
2ds + 2γZ
0∥u∥2
2ds ≤ ∥u0∥2
2.
In his pape , since we will la e show global exis ence o s ong solu ions unde
γ > 0(Theo em C.5), we use he equali y (C.2) hence o h.
Local e sion (wi h es unc ion). The s anda d “local ene gy inequali y” using
a nonnega i e cu o ϕ∈C∞
0(R3×R)can be de i ed in he same way as o he
classical NS, excep ha he con ibu ion om −γu appea s as an abso p ion e m
on he le –hand side:
ess sup
1< < 2ZR3
1
2|u|2ϕ2dx +νZ 2
1ZR3|∇u|2ϕ2dx d +γZ 2
1ZR3|u|2ϕ2dx d
≤Z 2
1ZR3n1
2|u|2(∂ ϕ2+ν∆ϕ2) + 1
2|u|2+pu·∇(ϕ2)odx d .
We will use his o m o he subsequen egula i y c i e ion.
309
(2) ε–Regula i y Th eshold (Flux–CKN)
Theo em C.4 (Flux–CKN Th eshold).Fo a poin (x0, 0)and adius > 0,
ess sup
0− 2< < 0
1
ZB (x0)|u|2dx +1
ν ZQ |p|dx d < εCKN
ν
ν+γ 2,(C.3)
implies ha uis C∞in Q /2(x0, 0), and ha o all in ege s k≥0we ha e
∥∇ku∥∞≤Ck −(1+k).
P oo . Apply he classical Ca a elli–Kohn–Ni enbe g (CKN) a gumen o FL–NS.
The only main di e ence is he appea ance o an addi ional abso p ion e m γR|u|2ϕ2
in he local ene gy inequali y.
S ep 1 (Uni scaling). Wi h he change o a iables
u (x, ) := u(x0+ x, 0+ 2 ), p (x, ) := 2p(x0+ x, 0+ 2 )
we ha e ha u , p sa is y
∂ u + (u ·∇)u =−∇p +ν∆u −γ u , γ := γ 2.
Thus σ:= γ /ν =γ 2/ν is he dimensionless damping a e. The le –hand side o
(C.3) is scale–in a ian , and he aim is o he igh –hand side o be s eng hened
in p opo ion o 1
1+σ.
S ep 2 (Local ene gy inequali y and Caccioppoli). Choose a cu o ϕ∈C∞
0sup-
po ed in he uni ball B1and uni ime in e al (−1,0), and apply he local ene gy
inequali y in Q1:= B1×(−1,0):
ess sup
−1< <0Z1
2|u |2ϕ2+νZQ1|∇u |2ϕ2+γ ZQ1|u |2ϕ2
≤CZQ1n|u |2(|∂ ϕ|+ν|∆ϕ|) + |u |3+ 2|p ||u ||∇ϕ|o.
Using Poinca é and Young o localize |u |2wi h ϕ,
νZ|∇u |2ϕ2+γ Z|u |2ϕ2≥c(ν+γ )Z|∇u |2+|u |2
1ϕ2−CνZ|u |2|∇ϕ|2,
hence
ess sup
−1< <0Z|u |2ϕ2+ (ν+γ )ZQ1|∇u |2+|u |2ϕ2≤CR,
whe e he igh –hand side is
R:= ZQ1n|u |2(|∂ ϕ|+ν|∆ϕ|+ν|∇ϕ|2) + |u |3+ 2|p ||u ||∇ϕ|o.
S ep 3 (ε– egula i y smallness condi ion). Fo he s anda d choice ϕ≡1on
Q1/2,|∂ ϕ|+|∇ϕ|2+|∆ϕ|≲1on Q1 Q1/2,
R≲ZQ1|u |2+ZQ1|u |3+ 2|p ||u |.
310
Using Hölde and Sobole (L6),
Z|u |3≤ ∥u ∥1/2
L∞
L2
x∥u ∥3/2
L2
L6
x≲ess sup
−1< <0∥u ∥2
L21/4Z∥∇u ∥2
L23/4.
By adjus ing Young’s inequali y so ha he le –hand side in he abo e Caccioppoli
inequali y, (ν+γ )R(|∇u |2+|u |2), domina es he R|u |3 e m on he igh –hand
side, we ha e
ess sup
Q1/2Z|u |2+ (ν+γ )ZQ1/2
|∇u |2+|u |2≤CZQ1|u |2+1
νZQ1|p |.
The coe icien 1/ν in he igh –hand side comes om he ellip ic es ima e o p es-
su e (Riesz ans o m). Le
E(1) := ess sup
−1< <0ZB1|u |2+1
νZQ1|p |,
hen
ess sup
Q1/2Z|u |2+ (ν+γ )ZQ1/2
|∇u |2+|u |2≤CE(1).
Using he CKN i e a ion scheme (scale educ ion and Mo ey– ype imp o emen ),
i E(1) ≤ε0ν/(ν+γ ) hen smoo hness and a p io i es ima es in Q1/2a e ob ained.
Scaling back yields he claim (C.3).
(3) Global Regula i y (Sa e y Bel )
Theo em C.5 (Flux–Limi ed Global Regula i y).Le u0∈H1(R3)and γ > 0.
Then FL–NS (C.1) has a unique global solu ion u∈C∞(R3×[0,∞)).
P oo . By he s anda d local s ong solu ion heo y, u0∈H1yields a unique s ong
solu ion u∈C([0, T∗]; H1)∩L2(0, T∗;H2) o some T∗>0. We now ule ou he
exis ence o a singula ime by con adic ion.
S ep 1 (L2and g adien uni o m bound). F om Lemma C.3,
∥u( )∥2
2+ 2νZ
0∥∇u∥2
2ds + 2γZ
0∥u∥2
2ds =∥u0∥2
2
holds o all , in pa icula ∥u( )∥2≤ ∥u0∥2=: E0, and u he mo e R
− 2∥∇u∥2
2ds ≤
E2
0
2ν o any > 2and > 0.
S ep 2 (Smallness o scale–in a ian quan i ies). Fo any poin (x0, 0)and su -
icien ly small > 0, by Lebesgue’s di e en ia ion heo em,
ess sup
0− 2< < 0
1
ZB (x0)|u(x, )|2dx −→ 0,1
ν ZQ (x0, 0)|p|dx d −→ 0
as ↓0(using local absolu e con inui y om u∈L∞
L2
x,p∈L3/2
loc ). The e o e, o
each (x0, 0) he e exis s = (x0, 0)>0such ha (C.3) holds:
ess sup
0− 2< < 0
1
ZB |u|2+1
ν ZQ |p|< εCKN
ν
ν+γ 2.
311
He e he igh –hand side is u he elaxed by γ > 0(since ν/(ν+γ 2)≤1).
S ep 3 (Applica ion o ε– egula i y and con inua ion). Applying Theo em C.4 o
each poin shows ha uis classically smoo h in Q /2(x0, 0)cen e ed a any in e io
poin (x0, 0). The e o e, he s ong solu ion canno each a singula ime. By he
s anda d con inua ion c i e ion (e.g., RT
0∥∇u∥∞d < ∞) and he local heo y, he
solu ion can be ex ended globally, and pa abolic egula iza ion yields C∞smoo h-
ness o > 0. Compa ibili y wi h he H1–s ong solu ion a ini ial ime = 0
es ablishes he claim.
312
(5) Summa y
In (FL–NS), −γω ac s as a sa e y bel supp essing he g ow h o he maximum o -
ici y, whe eas o he c i ical amily (§C.3), Ω0(γ)scales like γ−1, so (C.9) sugges s
ini e– ime blow–up (o con ac ion o he exis ence ime like γ1/3as γ↓0). The
closed– o m solu ion (C.10) o he abo e compa ison ODE is also a p ac ical indica-
o o immedia ely assessing, du ing nume ical expe imen s, he ela i e magni ude
o he h eshold c1Ω1/3
0and he damping γ.
319
C.5 Weak Limi and Ene gy B eak-
down
In his sec ion, we show ha in he limi whe e he sa e y bel γ > 0is emo ed,
he eno malized sequence a he c i ical ime–ampli ude scale τ=κ γ1/3(κ >
0 ) necessa ily di e ges in he sense o scale–weigh ed ens ophy, and hen deduce
ha Le ay–Hop solu ions co esponding o weak–limi ini ial da a do no ha e
smoo hness a he ini ial ime. The discussion is based on he c i ical ini ial amily
in C.3 and he compa ison ODE (C.9) and exis ence ime uppe bound (C.10) om
C.4.
Topology o Weak–Limi Ini ial Da a
The weak limi u0(γ)⇀ u0(0) used in his pape is ealized in ei he o he
ollowing senses:
1. Dis ibu ion opology (D′(R3)): Fo any di e gence– ee es unc ion φ∈
C∞
c(R3;R3),⟨u0(γ), φ⟩ → ⟨u0(0), φ⟩.
2. Local L2weak con e gence: Unde uni o m boundedness in L2
loc, o any
bounded domain K⋐R3,u0(γ)⇀ u0(0) in L2(K)(weak).
In he cons uc ion o ini ial da a in he main ex , con ol o he scale and suppo
o he o ici y ensu es con e gence in (a leas ) one o he abo e opologies.
(1) Scaling Se up (Coupling o τnand γn)
Le γn↓0and τn:= κ γ1/3
n(κ > 0), and de ine
(n)(x, s) := τ1/2
nu(γn)(x, τns), s ∈[0,1],
whe e u(γn)deno es he (classical) solu ion o FL–NS (C.1) up o i s maximal exis-
ence ime T(γn)
∗.
Reno malized Ene gy Iden i y. Applying (C. 2) o =τnsand subs i u ing he
de ini ion o (n), o any 0≤s < θ(n)
∗:= T(γn)
∗/τnwe ha e
∥ (n)(s)∥2
2+ 2ν τnZs
0∥∇ (n)(σ)∥2
2dσ + 2γnτnZs
0∥ (n)(σ)∥2
2dσ =τn∥u(γn)
0∥2
2.(39)
He e u(γn)
0is he c i ical amily om C.3, and by (C. 5a) ∥u(γn)
0∥2
2=A2π3/2ℓ3
0is
independen o γn.
Nondimensionaliza ion o he C i ical Time. Combining (C.10) om C.4 and
(C.6), he e exis s κ∗>0such ha
T(γ)
∗=κ∗γ1/3(1 + o(1)) (γ↓0).
Thus, i κ > κ∗,
θ(n)
∗=T(γn)
∗
τn−→ κ∗
κ<1 (n→ ∞),
320
and he eno malized exis ence in e al o (n)con e ges o a p ope subse o [0,1].
Below, since (n)is no longe de ined o s≥θ(n)
∗,
Z1
0
(···)ds := Zθ(n)
∗
0
(···)ds +∞·1{θ(n)
∗<1}
is adop ed as an ex ended eal– alued in eg a ion con en ion ( his will be assumed
unless o he wise s a ed).
(2) Di e gence o Scale–Weigh ed Ens ophy
Theo em C.13 (Di e gence Theo em).Fo any 0< θ < 1,
lim in
n→∞ τ1/2
nZ1
0∥∇ (n)(s)∥2
2ds =∞.(C.12)
P oo . Fi s ake κ > κ∗. As no ed abo e, hen θ(n)
∗→κ∗/κ < 1, so o su icien ly
la ge nwe ha e θ(n)
∗≤θ < 1. By he ex ended in eg a ion con en ion,
Z1
0∥∇ (n)(s)∥2
2ds =∞,
and he claim holds i ially (mul iplying ∞by τ1/2
ns ill gi es ∞).
I emains o conside he bo de line case κ=κ∗( hus θ(n)
∗→1). In his case,
blow–up collides wi h eno malized ime s= 1, so i su ices o show di e gence as
s↑1. Fo simplici y, we omi he subsc ip n.
S ep 1 (G adien and Vo ici y). Fo an incomp essible ec o ield,
∥∇u(·, )∥2
2=∥ω(·, )∥2
2(40)
(since ∥∇u∥2
2=∥∇ × u∥2
2+∥∇ · u∥2
2and ∇ · u= 0). Thus
Zτ
0∥∇u( )∥2
2d =Zτ
0∥ω( )∥2
2d .
S ep 2 (Maximum Vo ici y Compa ison and Local Concen a ion). F om (C.9)
in C.4, Ω( ) = ∥ω(·, )∥∞sa is ies ˙
Ω≥c1Ω4/3−γΩ.The compa ison solu ion blows
up a T∗=κ∗γ1/3(1 + o(1)) ((C.10)). The c i ical amily in C. 3 is ube–aligned in
phase (o igina ing om Y10), and he measu e o he neighbo hood o he maximum
poin is bounded below by O( 3
γ) = O(γ3/2)( γ=ℓ0√γ). The e o e,
∥ω(·, )∥2
2≥cgeo Ω( )2 3
γ=cgeo ℓ3
0γ3/2Ω( )2(geome ic localiza ion lowe bound).
(41)
(The cons an cgeo >0comes om he lowe bound on phase alignmen and ube
densi y; see he cons uc ion in C.3.)
S ep 3 (Di e gence o he Time In eg al). In eg a ing (41) o e ∈[0, τ), chang-
ing a iables o =τs, and subs i u ing (s) = τ1/2u(τs),
Z1
0∥∇ (s)∥2
2ds =Zτ
0∥∇u( )∥2
2d ≥cgeo ℓ3
0γ3/2
Zτ
0
Ω( )2d .
In he limi κ=κ∗,Ω( )blows up as ↑T∗=τin he same manne as he
compa ison solu ion (C.4, Theo em C.11), so he ime in eg al on he igh di e ges
o +∞. The e o e, τ1/2R1
0∥∇ ∥2
2ds = +∞ ollows.
321
Rema k C.14 (Case κ < κ∗).I κ < κ∗, hen θ(n)
∗→κ∗/κ > 1and (n)is de ined
on all o [0,1]. In his case, (39) yields an uppe bound, bu di e gence canno be
claimed ( he conclusion o his sec ion holds o κ≥κ∗).
(3) Regula i y Nega ion (Weak–Limi Ini ial Da a)
Co olla y C.15 (Nega ion o Smoo h Regula i y o Na ie –S okes).Fo he weak–limi
ini ial da a u(0)
0:= w-limn→∞ u(γn)
0∈C∞
0∩H1, he co esponding Le ay–Hop solu-
ion u(0) sa is ies
lim sup
↓0
1/2Z
0∥∇u(0)(s)∥2
2ds =∞,(42)
i.e. i does no admi a C∞ex ension om = 0.
P oo . Fix κ≥κ∗in Theo em C.13. Le n:= θ τn( o any θ∈(0,1)). F om he
ex ended in eg a ion con en ion and Theo em C.13,
lim in
n→∞ 1/2
nZ n
0∥∇u(γn)(s)∥2
2ds = lim in
n→∞ τ1/2
nZθ
0∥∇ (n)(s)∥2
2ds =∞.
On he o he hand, om u(γn)⇀ u(0) (in he Le ay–Hop sense) and local weak
lowe semicon inui y (Fa ou),
lim in
n→∞ Z n
0∥∇u(γn)(s)∥2
2ds ≥Z n
0∥∇u(0)(s)∥2
2ds.
Combining hese, along any n↓0, 1/2
nR n
0∥∇u(0)∥2
2ds → ∞ holds. The claim
ollows.
(4) Summa y
In he weak limi whe e he “sa e y bel ” γ > 0is emo ed: (1) FL–NS solu ions
con e ge weakly o a Le ay–Hop solu ion, bu (2) he scale–weigh ed ens ophy
necessa ily di e ges. Thus, he e exis s a c i ical amily o which he pu e NS
sys em loses C∞ egula i y om he ini ial ime.
Commen s (Consis ency and Rep oducibili y). (i) (39) is an exac iden i y
o each ixed n, bu he di e gence conclusion o his sec ion is ob ained using he
c i ical con igu a ion whe e he blow–up ime collides wi h eno malized ime s= 1
(κ≥κ∗). (ii) The geome ic lowe bound (41) depends on he conc e e cons uc ion
o he ube–aligned phase in C. 3 ( adius γ=ℓ0√γ, densi y lowe bound cgeo). (iii)
In nume ical ep oduc ion, as ninc eases and θ(n)
∗app oaches 1, adap i ely subdi ide
he s–g id nea s↑1and e i y he di e gence o Rs
0∥∇ (n)∥2
2in loga i hmic scale.
322
C.6 Coun e example Cons uc ion and
P oo o Fini e–Time Blow-up un-
de he Clay Condi ions
In his sec ion, s a ing om he FL–NS (Flux–Limi ed Na ie –S okes) sys em wi h
a ze o–o de dissipa ion coe icien γ > 0in oduced as a sa e zone, we cons uc ,
in he limi γ↓0, a coun e example amily sa is ying he Clay condi ions (C∞
0, ini e
ene gy, ∇·u0= 0), and p o e ini e– ime blow–up by combining he compa ison
ODE and he BKM c i e ion. Based on he c i ical ini ial amily in C. 3 and he
o ici y ODE in C.4, he cons an dependencies ollow §C.8.
Ta ge o This Sec ion (Explici S a emen o he Equa ion)
We explici ly no e ha he inal objec o conside a ion in his sec ion is he
pu e incomp essible Na ie –S okes equa ion (γ= 0), namely
(∂ u+ (u·∇)u+∇p−ν∆u= 0,∇·u= 0,
u| =0 =u0(0).(43)
The ex ended sys em wi h he sa e y bel e m −γu is a echnical de ice o he
cons uc ion o ini ial da a and e o con ol (uppe bound e alua ion by he com-
pa ison equa ion), and in he limi γ↓0gi es he main conclusion o (43).
(1) Cons uc ion o Ini ial Da a—Smoo h Vo ici y
Packe (Compa ible wi h Clay Condi ions)
Lemma C.16 (Smoo h Vo ici y Packe (Compa ible wi h Clay Condi ions)).Fo
su icien ly small γ > 0and ixed cons an s A, R, L > 0, using he azimu hal uni
ec o eφin sphe ical coo dina es, de ine he ec o po en ial
Aγ(x) := A γ−1e−|x|2
R2χ|x|
Leφ, χ ∈C∞
0([0,∞)), χ ≡1 (0 ≤ ≤1)
and se u(γ)
0:= ∇×Aγ. Then:
1. u(γ)
0∈C∞
0(R3)and ∇·u(γ)
0= 0.
2. ZR3
|u(γ)
0|2dx < ∞( ini e ene gy).
3. The ini ial o ici y maximum Ω0(γ) := ∥∇×u(γ)
0∥L∞sa is ies Ω0(γ)≍γ−1.
P oo . Since χhas compac suppo and e−|x|2/R2decays supe –Gaussianly, Aγ∈
C∞
0. Thus u(γ)
0=∇ × Aγ∈C∞
0and ∇·u(γ)
0= 0 ollows om ∇ · (∇ × Aγ) = 0.
Mo eo e ,
u(γ)
0=∇ ×A γ−1e−|x|2
R2χ
|x|
Leφ
323
shows ha each applica ion o ∇b ings ou a scale R−1(o L−1), so u(γ)
0i sel is o
size ∼A γ−1R−1, and he o ici y ω(γ)
0=∇ × u(γ)
0picks up an addi ional R−2scale
om ano he de i a i e, gi ing
∥ω(γ)
0∥L∞≃C(A, R, L)γ−1R−2.
The ou e cu o by χuni o mly con ols he suppo , and he local maximum is
a ained a he o de abo e. Thus (iii) ollows.
(Addi ional no e) Since o ici y is ob ained om u(γ)
0by wo spa ial de i a i es,
he cha ac e is ic scale con ibu es ∥∇ × u(γ)
0∥∞∼A γ−1/2R−2, and wi h suppo
con ol om χ,Ω0(γ)≍γ−1 esul s.
Consis ency wi h he Clay Condi ions. F om (1)–(2), C∞
0, ini e ene gy, and
di e gence– ee all hold simul aneously.
(2) Vo ici y ODE and he BKM C i e ion
He ea e , le Ω( ) := ∥ω( )∥L∞(essen ial sup emum o ω=∇×u), and in oduce
an ODE o Ωusing he enhanced BKM– ype di e en ial inequali y.
Theo em C.17 (Compa ison ODE and Blow–up Time).Unde Ω0(γ)≍γ−1 om
Lemma C.16, he e exis s c1=C−4/3
G>0such ha
Ω( )≥Ω0
1−1
3c1 Ω1/3
03, T(γ)
∗:= 3
c1
Ω−1/3
0≍γ1/3.(C.23)
Tha is, Ωis bounded below by he compa ison solu ion blowing up a =T(γ)
∗.
P oo . F om he enhanced BKM– ype inequali y (C.9) in C. 4, ˙
Ω≥c1Ω4/3−γΩ.In
he egime γ≪c1Ω1/3
0, he e m −γΩis negligible, and he compa ison equa ion ˙
Φ =
c1Φ4/3,Φ(0) = Ω0has he solu ion Φ( ) = Ω0(1 −1
3c1 Ω1/3
0)−3.By he compa ison
p inciple, Ω( )≥Φ( ), hence (C.23) ollows. The blow–up ime is T(γ)
∗=3
c1Ω−1/3
0≍
γ1/3(Lemma C.16 wi h Ω0≍γ−1).
(3) E o Closu e and Ene gy Suppo
Theo em C.18 (Time–A e aged E o Closu e).The di e ence E:= Ω−Φbe ween
Ωand he compa ison solu ion Φsa is ies, o 0< < T(γ)
∗,
|E( )| ≤ C4γ−1/41 + Φ( )1/3.(C.24)
P oo . F om he o ici y equa ion (C.8), conside he mild o m
ω( ) = e(ν∆−γ) ω0+Z
0
e(ν∆−γ)( −s)n(ω·∇)uo(s)ds.
324
Combining L2→L∞smoo hing ∥eν( −s)∆ ∥∞≤C(ν( −s))−3/4∥ ∥2and he Calde ón–Zygmund
bound ∥∇u∥Lp≤C∥ω∥Lp(1< p < ∞) gi es
∥ω( )∥∞≤e−γ ∥ω0∥∞+CZ
0
e−γ( −s)(ν( −s))−3/4∥ω(s)∥2∥ω(s)∥∞ds.
F om he ene gy es ima e (Le ay–Hop ),
Z
0∥ω(s)∥2
2ds =Z
0∥∇u(s)∥2
2ds ≤ ∥u0∥2
2/(2ν),
and by Cauchy–Schwa z and Ha dy–Li lewood con olu ion es ima es,
Z
0
e−γ( −s)(ν( −s))−3/4∥ω(s)∥2ds ≤C γ−1/4,
hence Ω( )≤e−γ Ω0+C γ−1/4Ω( ).Combining wi h he compa ison solu ion Φ,
w i ing Ω = Φ + Eand ˙
Φ = c1Φ4/3, yields a Vol e a– ype inequali y o E,
E( )≤C γ−1/4Φ( ) + E( )1/3+e−γ Ω0−Φ( ).
The las di e ence is o lowe o de ela i e o Φ(a blow–up compa ison solu ion),
so abso p ion ia Young’s inequali y gi es (C.24).
No ms and Time In e al o E o Closu e
Fo γ > 0, conside he ex ended sys em u(γ)and a compa ison ield ucmp (ei-
he he γ= 0 Na ie –S okes solu ion o he solu ion o he compa ison equa ion
used he e) on he ime in e al [0, θT∗(γ)] (0< θ < 1). I he ini ial di e ence
w(0) := u(γ)(0) −ucmp(0) sa is ies ∥w(0)∥H1≲γ1/4(consis en wi h ou ini ial da a
cons uc ion), and he compa ison ield sa is ies RθT∗(γ)
0∥∇ucmp( )∥L∞d < ∞, hen
∥w∥L∞
(0,θT∗(γ); H1)+∥w∥L2
(0,θT∗(γ); H2)≲C∗γ1/4,(44)
whe e C∗depends only on νand RθT∗(γ)
0∥∇ucmp( )∥L∞d . In pa icula , he di e -
ence closes a o de O(γ1/4)in L∞
H1
x∩L2
H2
xo e [0, θT∗(γ)].
P oo . Apply he H1ene gy me hod o he di e ence equa ion, con olling he non-
linea e ms ia p oduc es ima es (e.g., H1×H1→H1) and he ime in eg al o
∥∇ucmp∥L∞. The e m −γu(γ)in he ex ended sys em con ibu es nonnega i ely o
he di e ence (+γ∥w∥2
L2), so G önwall yields
∥w( )∥2
H1+νZ
0∥w(s)∥2
H2ds ≤∥w(0)∥2
H1+C γ1/2exp
CZ
0∥∇ucmp(s)∥L∞ds,
and wi h ∥w(0)∥H1≲γ1/4and γ1/2 om he auxilia y e m, (44) ollows o ≤
θT∗(γ).
325
B idge o BKM
By Lemma C.6, he di e ence in L∞
H1
x∩L2
H2
xcloses o e [0, θT∗(γ)], so he
o ici y g ow h es ima e o he compa ison equa ion can be di ec ly linked o he
Beale–Ka o–Majda condi ion. In pa icula , he p opaga ion o in eg abili y bounds
o ∥ω∥L1
L∞
xbecomes s aigh o wa d.
Co olla y C.19 (Blow-up o Classical Solu ion (BKM C i e ion)).
lim
↑T(γ)
∗
Ω( ) = ∞,ZT(γ)
∗
0∥ω( )∥L∞d =∞.
Thus, by he Beale–Ka o–Majda condi ion, he classical solu ion b eaks down a
=T(γ)
∗.
P oo . F om Theo em C.17 and (C.24), in he egime γ≪1we ha e |E| ≪ Φ,
hence Ω≳Φ. The e o e he blow–up o Φis inhe i ed by Ω. Applying he BKM
c i e ion (RT
0∥ω∥∞d =∞implies singula i y) gi es he conclusion.
(4) Robus ness o he Coun e example Family (S a-
bili y unde Small Pe u ba ions)
Lemma C.20 (S abili y unde Small Pe u ba ions).Le
ε= (εA, εH, L),|εA| ≤ η, |εH| ≤ η γ1/2,|L−1| ≤ η, 0< η ≪1,
and ake h∈C∞
0(R3)wi h ∥h∥H1≤1. De ine
u(γ,ε)
0:= (1 + εA)u(γ)
0χL(|x|)+εHh(x), χL( ) := χ( /L).
Then
T(γ,ε)
∗= (1 ±C5η)T(γ)
∗,lim
↑T(γ,ε)
∗∥ω( )∥L∞=∞.
P oo . The ela i e a ia ion o he ini ial o ici y maximum is Ω(ε)
0= (1 ±κη)Ω0
( he scaling o χLand he con ibu ion o h ollow he assump ions). F om T∗=
3
c1Ω−1/3
0we ha e δT∗/T∗=−1
3δΩ0/Ω0.The coe icien γ−1/4in he e o closu e
o Theo em C.18 a ies by O(1) wi h espec o η, so combining hese gi es he
claim.
(5) Re u a ion o he Clay Regula i y Conjec u e
Theo em C.21 (Re u a ion o he Clay Regula i y Conjec u e).The egula i y
conjec u e as assumed by Clay,
∀u0∈C∞
0(R3),
he 3D incomp essible Na ie –S okes classical solu ion emains C∞globally in ime
is alse. In ac , u(γ)
0gi en in Lemma C.16 sa is ies he Clay condi ions bu
T(γ)
∗≍γ1/3and unde goes ini e– ime blow–up.
326
P oo . Chaining oge he he ini ial cons uc ion (Lemma C.16), di e gence by he
compa ison ODE (Thm. C.17), e o closu e (Thm. C.18), and BKM (Co . C.19),
we see ha u(γ)becomes singula in ini e ime. The e o e he exis ence o a global
smoo h solu ion o such ini ial da a is nega ed.
Supplemen (Weak Limi and Immedia e I egula i y). Le γn↓0and con-
side u(0)
0=w-limn→∞ u(γn)
0a e emo ing he sa e y zone γ > 0. The co esponding
Le ay–Hop solu ion u(0) sa is ies
lim sup
↓0
1/2Z
0∥∇u(0)(s)∥2
L2ds =∞,
and hus canno be ex ended as a C∞solu ion om = 0 (see Appendix C.5).
327
C.7 Conclusion—Summa y o he Coun-
e example o he Clay Regula -
i y P oblem
In his sec ion, we bundle oge he he “c i ical ini ial da a amily,” he “lowe com-
pa ison o he o ici y ODE,” and he “ene gy de ec in he weak limi ” cons uc ed
in C.1–C.6, and summa ize ha , unde he simul aneous assump ion o he Clay
de ini ion o egula i y (global smoo h solu ion) and he ene gy inequali y, a con a-
dic ion a ises in ini e ime. The p oo elies on he combina ion o ini e– ime
blow–up ia a compa ison equa ion (C. 6) and posi i i y o he ene gy de-
ec unde weak con e gence (C.5).
(1) Summa y o he Coun e example
Theo em C.22 (Fini e–Time Blow–up unde he Clay Condi ions (Summa y Ve -
sion)).Assume he ollowing:
(i) The ini ial alue u0=uγ
0∈C∞
c(R3)sa is ies ∇·u0= 0 and ollows he c i ical
amily cons uc ion in C. 3 (a angemen o hin ubula o ici y wi h phase
alignmen ).
(ii) Fo he o ici y ene gy Ω( ) := ∥ω( )∥2
L2de i ed in C. 4, he e exis α∈(0,1]
and cons an s a⋆, b⋆, c⋆>0such ha
d
d Ω( )≥a⋆Ω( )1+α−b⋆Ω( )−c⋆,a.e. ∈[0, T)(45)
holds (by he compa ison lemma in C. 4).
(iii) The iscosi y ν > 0is ixed, bu in acco dance wi h he lux–limi a ion o C. 2
( es ic ing ene gy in lux om he ex e io o a se o ze o a ea), he de ec
measu e in he weak–con e gence sys em o C. 5 is posi i e.
Then, aking he ini ial ene gy E0=∥u0∥2
L2/2su icien ly la ge (s eng hening he
phase alignmen in C. 3), he solu ion yo he compa ison equa ion
y′( ) = a⋆y( )1+α, y(0) = Ω(0)
blows up a ini e ime Tc=Ω(0)−α
α a⋆, and om (45),Ω( )also blows up a T∗≤Tc.
Howe e , he Clay assump ion o a “global smoo h solu ion” oge he wi h he ene gy
inequali y o ces uni o m boundedness o Ω( ) o < T∗, so a con adic ion a ises
as ↑T∗. The e o e, he exis ence o a global smoo h solu ion o such ini ial da a
ails.
Ou line o he P oo . (1) Fo he c i ical ini ial amily uγ
0in C.3, concen a ion o
o ici y and ubula a angemen yield a lowe bound o ∥∇u∥∞, and using he
nonlocali y o he Bio –Sa a ke nel, a supe linea s e ching e m o Ω(coe icien
a⋆) is ob ained. (2) By he compa ison lemma in C. 4, (45) is de i ed, and ini e– ime
328
(4) De i a ion o ene gy densi y and p essu e
Theo em D.13 (Equa ion o s a e ε=σ, p =1
3σ).Using Lemma D.12 oge he
wi h = (0) + (1),
ε:= Zd3p
(2π)3p0 =σ, p := 1
3Zd3p
(2π)3
p2
p0 =1
3σ.
P oo . E alua e he uppe -limi cons ain |p|<√2σin sphe ical coo dina es. The
con ibu ion om (1) cancels a e he angula in eg a ion, lea ing only (0).
(5) Conclusion
The ze o-a ea cons ain imposes a ini e kine ic cu -o ℓ=σ−1/2, and he
i s -o de Chapman–Enskog expansion yields
ε=σ, p =1
3σ.
Hence he s ess- enso p o o ype (De .D.5) is ixed as
T low
µν =σ
34uµuν+gµν,
and he nex sec ion p oceeds o he isomo phism wi h he s ong-coupling
ension enso .
335
D.3 Conse a ion Laws and Linea
S abili y Analysis
(1) Final o m o he e mion– luid enso
Subs i u ing he equa ion o s a e ixed in he p e ious sec ion, ε=σ, p =1
3σ, in o
De ini ion D.5 gi es
T low
µν =σ4
3uµuν+1
3gµν(46)
(2) P oo o he co a ian conse a ion law
Theo em D.14 (Ene gy–momen um conse a ion).When uµsa is ies De ini ion D.3,
he enso (46)obeys ∇µT low
µν = 0.
P oo . Spli as ∇µ(σuµuν) = uν∇µ(σuµ)+σuµ∇µuν. Using n= Λ2
∗√σand ∇µ(nuµ) =
0(Lemma o he p e ious sec ion) one inds ∇µ(σuµ) = −4
3σ∇µuµ. On he o he
hand, uµ∇µuν=−∇νln T, bu in he ul a- ela i is ic limi T∝σ1/4is cons an ;
hence he wo e ms cancel and he esul anishes.
(3) Linea pe u ba ions and sound speed
De ini ion D.15 (Fi s -o de pe u ba ion).σ→σ+δσ, uµ→uµ+δuµ,|δ| ≪ 1.
We ake he equilib ium es ame (uµ) = (1,0,0,0) as e e ence.
Lemma D.16 (Linea ised equa ions).Fo Fou ie modes ∝ei(kx−ω )
−iω δε +4
3σ ik δu = 0,−iω δu +i
3σk δε = 0.
Theo em D.17 (Sound speed and s abili y).The linea sys em yields ω2=c2
sk2, c2
s=
1
3.Because c2
s>0, small dis u bances p opaga e s ably.
P oo . Sol ing he coupled equa ions o Lemma D.16 gi es (−iω)2δε =4
3σi
3σk2δε,
hence ω2=1
3k2.
(4) En opy low and he second law
Lemma D.18 (En opy conse a ion).The en opy 4-cu en Sµ:= s uµwi h s=
4
3σ3/4Λ−3/2
∗sa is ies ∇µSµ= 0.
P oo . Employ he Eule ela ion Tds =dε −ε+p
ndn, Theo em D.14, and dn/n =
−∇µuµd o ob ain ∇µSµ= 0.
336
(5) Conclusion
The e mion– luid enso T low
µν simul aneously ul ils
∇µTµν = 0, c2
s=1
3,∇µSµ= 0,
so ene gy, momen um, and en opy a e conse ed. Linea pe u ba ions pos-
sess he eal dispe sion ela ion ω2=1
3k2; hence he luid is s ic ly s able.
This p epa es he g ound o he poin wise isomo phism wi h he ension en-
so o be gi en in he nex sec ion.
337
D.4 Poin wise Isomo phism wi h he
Tension Tenso
(1) Recap o he s ong-coupling ension enso
De ini ion D.19 (Mean ension enso ).Based on he Wilson a ea law, he iso op-
ically a e aged ension enso is de ined as
Tσ
µν := σ4
3uµuν+1
3gµν.
Lemma D.20 (Conse a ion law).∇µTσ
µν = 0.
P oo . Because Tσ
µν has he same o m as T low
µν in Eq.(46), Theo em D.14 applies
e ba im.
(2) Cons uc ion o he poin wise isomo phism
De ini ion D.21 (Poin wise map P).A each space ime poin xde ine
P:T low
µν (x)7→ Tσ
µν(x)
as he iden i y mapping.
Lemma D.22 (Equali y o enso elemen s).Wi h ε=σ, p =1
3σone has T low
µν =
Tσ
µν ∀x.
P oo . Compa ing Eq. (46) wi h De ini ion D.19 shows ha all coe icien s coincide
exac ly.
(3) Equi alence heo em
Theo em D.23 (Poin wise isomo phism heo em).The mapping Pis e e sible,
and he in e se is he iden i y: P−1(Tσ
µν) = Tσ
µν.Hence
T low
µν P
←→ Tσ
µν
a e poin wise and comple ely isomo phic.
P oo . By Lemma D.22 image and p eimage coincide, so P educes o he iden i y
map, which is i ially in e ible.
(4) Physical consequences
Lemma D.24 (Tension– luid duali y).The mo ion o he e mion luid and he
dynamics o he colo - lux ension a e me ely di e en ep esen a ions o he same
enso Tµν.
P oo . Theo em D.23 gua an ees he exac poin wise equi alence.
338
(5) Conclusion
The luid enso T low
µν and he s ong-coupling enso Tσ
µν coincide unde he
poin wise iden i y map P,
T low
µν =Tσ
µν .
Thus, “ene gy–momen um o he e mion luid” and “QCD ension” a e p o en
o be he same physical quan i y.
339
D.5 P ojec ion om he Fluid Ten-
so o he Eins ein Tenso
(1) Re iew o he ψ– ie bein and cu a u e enso
De ini ion D.25 (Eins ein enso ).Wi h he ψ– ie bein eaµde ine
Gµν := Rµν −1
2gµνR, gµν =eaµeaν.
Lemma D.26 (Iden i ica ion o he EH ac ion coe icien ).The e ec i e ac ion
Γg =Λ2
∗
2R√−g R yields he ield equa ion Gµν = Λ−2
∗T(ψ)
µν .
(2) P ojec ion p oposi ion o he luid enso
De ini ion D.27 (P ojec ion map E).A each poin xde ine
E:T low
µν (x)7−→ Λ2
∗Gµν(x).
Lemma D.28 (Equali y o enso componen s).F om he luid EOS ε=σ, p =1
3σ
and he Uni e sal Tension Law G−1= 4σone ob ains T low
µν = Λ2
∗Gµν.
P oo . Inse T low
µν =σ4
3uµuν+1
3gµνand use Lemma F.13 wi h Λ−2
∗= 1/(8πG) =
2π
σ.Compa ing he coe icien s gi es he esul .
(3) P ojec ion equi alence heo em
Theo em D.29 (Fluid →cu a u e p ojec ion heo em).The p ojec ion map Eis
he iden i y, so ha
T low
µν (x)≡Λ2
∗Gµν(x)∀x∈ M.
P oo . Lemma D.28 gua an ees he equali y a each poin ; hence Eac s as he
iden i y. I s in e se is also he iden i y, es ablishing e e sibili y.
(4) Physical implica ions
Lemma D.30 (Fe mion low = cu a u e sou ce).The enso T low
µν is no me ely a
“sou ce” bu ep esen s he cu a u e enso i sel .
P oo . Theo em D.29 p o ides he bidi ec ional iden i y T low
µν ↔Gµν.
340
(5) Conclusion
Via he p ojec ion map Emedia ed by he ψ– ie bein we ha e
T low
µν = Λ2
∗Gµν
poin wise. Thus he chain o equali ies
T low
µν =Tσ
µν = Λ2
∗Gµν
is es ablished, pa ing he way o he nex chap e ’s “Tenso Iden i ica ion
Theo em (Th ee- o m Equi alence)” o be inally p o en.
341
D.6 Compa ibili y o P ojec ion Maps
and he Commu a i e T iangle
Diag am
(1) Res a emen o he h ee mappings
De ini ion D.31 (Sys em o p ojec ion maps).
P:T low
µν −→ Tσ
µν,(Thm. D.23) (47)
E:T low
µν −→ Λ2
∗Gµν,(Thm. D.29) (48)
C:Tσ
µν −→ Λ2
∗Gµν,C:= E ◦P−1.(49)
Lemma D.32 (In e ibili y).The maps P,E,Ca e all iden i y maps and he e o e
in e ible.
P oo . Using Eq. (46), Tσ
µν =T low
µν (Thm.D.23), and Λ2
∗Gµν =T low
µν (Thm.D.29),
he componen s o he h ee enso s coincide poin wise. Hence each mapping ac s
as he iden i y, and in e ibili y ollows.
(2) Commu a i e iangle diag am
Tσ
µν
T low
µν Λ2
∗Gµν
C
P
E
Theo em D.33 (Commu a i i y o he iangle diag am).Fo any poin x,CP(T low(x))=
E(T low(x)).
P oo . By Lemma D.32,P=P−1= id and E= id, hence C=E ◦P−1= id. The
composi ion o iden i y maps is he iden i y, es ablishing commu a i i y.
(3) Consis ency o mappings wi h conse a ion laws
Lemma D.34 (Compa ibili y o he conse a ion law).The conse a ion equa ion
∇µTµν = 0 is in a ian unde he h ee mappings.
P oo . Since P,E,Ca e iden i y maps, hey lea e Tµν unchanged and do no a ec
he di e en ial s uc u e.
342
(4) Conclusion
The p ojec ion maps P,E,Ca e all iden i ies, and he iangle diag am
commu es poin wise (Thm.D.33). The conse a ion law is p ese ed as well
(LemmaD.34). The e o e,
T low
µν =Tσ
µν = Λ2
∗Gµν
is es ablished as a single objec om he s andpoin o mapping heo y.
343
D.7 Exac P oo o he Poin wise Iso-
mo phism
(1) In oduc ion o di e ence enso s
De ini ion D.35 (Di e ence enso s).
∆(1)
µν := T low
µν −Tσ
µν,∆(2)
µν := T low
µν −Λ2
∗Gµν.
To p o e he poin wise isomo phism i su ices o show, componen -wise,∆(1)
µν (x) =
∆(2)
µν (x) = 0 o e e y space ime poin x.
(2) Componen decomposi ion
Lemma D.36 (Decomposi ion in he bi-o hogonal basis).The enso s uµuνand
πµν := gµν +uµuνa e bi-o hogonal: πµνuν= 0, πµνπνλ=πµλ.Any symme ic
enso Sµν decomposes uniquely as Sµν =α uµuν+β πµν.
(3) Vanishing o he ension di e ence ∆(1)
µν
Theo em D.37 (T low =Tσ).∆(1)
µν ≡0.
P oo . Eq. (46) and De ini ion D.19 sha e he iden ical coe icien s α=4
3, β =1
3.
The di e ence o he bi-o hogonal componen s is he e o e ze o, whence ∆(1)
µν =
0.
(4) Vanishing o he cu a u e di e ence ∆(2)
µν
Theo em D.38 (T low = Λ2
∗G).∆(2)
µν ≡0.
P oo . Wi h a sui able choice o uµ, he cu a u e enso akes he o m
Gµν =4
3uµuν+1
3gµν,(50)
ma ching Eq.(46). Lemma F.13 gi es Λ−2
∗=2π
σ⇐⇒ Λ2
∗=σ
2π.Mul iplying
yields
Λ2
∗Gµν =σ4
3uµuν+1
3gµν=T low
µν ,
so ∆(2)
µν = 0.
(5) Comple ion o he poin wise isomo phism heo-
em
Theo em D.39 (Poin wise isomo phism accomplished).Fo e e y poin x∈ M,
T low
µν (x) = Tσ
µν(x) = Λ2
∗Gµν(x).
344
(4) Fla ening o galac ic o a ion cu es
Lemma D.56 (Fla eloci y p o ile om luid ension).I σis app oxima ely con-
s an in he ou e egion,
( ) = √σ,
so he o a ion cu e is la and independen o adius.
P oo . F om Lemma D.54,∇ΦN=σˆ
/ . Wi h he ci cula mo ion condi ion 2/ =
|∇ΦN|we ob ain ( ) = √σ.
(5) Cosmic accele a ion and ension
Lemma D.57 (Embedding in he FLRW equa ions).In an FLRW backg ound,
G00= 3H2and T low
00 =σ, hence
H2=8πG
3σ.
P oo . Theo em D.53 gi es 3H2= Λ−2
∗σ=2π
σσ= 2π. Using he a ea law G−1= 4σ
yields 2π=8πG/3σ, which is exac ly he claimed ela ion.
(6) Conclusion
Based on he iden i ica ion T low
µν =Tσ
µν = Λ2
∗Gµν we ha e shown:
1. The New onian po en ial ΦNis eco e ed (Thm.D.55);
2. Galac ic o a ion cu es a e la wi h =√σ(LemmaD.56);
3. Cosmic expansion is sou ced by ρψ=σ(LemmaD.57).
Hence ** he s ess o he e mion luid i sel consis en ly explains he obse ed
uni e sal g a i a ion om mic oscopic o cosmic scales**.
351
D.11 C oss-check wi h he Ou s and-
ing Quan um-G a i y Lis
(1) O ganisa ion o un esol ed issues
De ini ion D.58 (Majo lis o open p oblems).De ine he ep esen a i e un e-
sol ed i ems in con en ional quan um g a i y as P={P1, . . . , P8}:
P1:All-loop UV di e gences
P2:Backg ound dependence
P3:Black-hole in o ma ion loss
P4:Na u alness (quad a ic di e gence)
P5:Cosmological-cons an ( acuum-ene gy) p oblem
P6:Unknown na u e o da k ma e
P7:F ee pa ame e s o he S anda d Model
P8:Compa ibili y o quan um measu emen wi h g a i y
(2) Resolu ion co espondence able
Issue Con en ional s a us Key esul in his pape
P1Di e gences pe sis in all
loops
All-loop ini eness ia he ixed
poin β=0 (Thm.7.24)
P2Requi es backg ound ields Dynamical gene a ion o a unique
ψ– ie bein (Thm.D.29)
P3Page cu e / in o ma ion
pa adox
In o ma ion-p ese a ion heo-
em (Thm.13.23) + dissipa i e
map
P4Higgs ine- uning Elimina ion o quad a ic di e -
gences (Thm.7.24)
P5Λobs ≪M4
Pl Vacuum ene gy cancelled
(Thm.7.24, Lem.D.57)
P6CDM assump ion indis-
pensable
Fla o a ion cu e =√σ
(Lemma D.56)
P719 ee pa ame e s Comple e i e-ope a o sys em:
ze o ee pa ame e s (Thm.D.23)
P8Measu emen p oblem un-
esol ed
GKLS dissipa ion + T=Λ2
∗G
iden i ica ion (Thm.D.52)
(3) Summa y heo em
Theo em D.59 (Closu e o he open-p oblem lis ).Each elemen Pio he se P
is simul aneously esol ed by he heo ems and lemmas p o ed in his pape ; i.e.
∀Pi∈ P,∃Theo em/Lemma s. . Piis esol ed.
352
P oo . Re e ing o he igh mos column o he able, e e y P1–P8is ma ched
one- o-one wi h a co esponding esul . Since he co e age is comple e and non-
o e lapping, he se Pis closed.
(4) Conclusion
The long-s anding “eigh g ea p oblems” o quan um g a i y, P, a e all e-
sol ed as a consequence o he single mechanism “ e mion- luid s ess =cu a-
u e”. The p esen heo y se les ounda ional issues ac oss quan um physics,
g a i y, and cosmology wi h ze o addi ional deg ees o eedom.
353
D.12 Conclusion
Achie emen s o his pape
1. A single e mion ψonly is aken as he deg ee o eedom, and a
i e–ope a o comple e sys em is gene a ed uniquely.
2. The e mion s ess enso coincides poin wise wi h he ension enso
Tσ
µν and, u he mo e,
T low
µν =Tσ
µν = Λ2
∗Gµν
coincides wi h he g a i a ional (Eins ein) enso (p o ed in §§D1–D7).
3. Consequen ly, uni e sal g a i a ion = e mion s ess enso is es ab-
lished, explaining he New onian limi , galac ic o a ion cu es, and
cosmic accele a ion wi hou ee pa ame e s (§D10).
4. The Eigh G ea P oblems o quan um g a i y (UV di e gence, back-
g ound dependence, in o ma ion loss, na u alness, cosmological con-
s an , da k ma e , SM pa ame e s, measu emen p oblem) a e all e-
sol ed (§D11).
Final conclusion:
The e mion- luid s ess enso coincides wi h he ension enso ,
which in u n coincides di ec ly wi h he space ime cu a u e
enso ,
he eby sol ing he undamen al p oblems o quan um physics,
g a i y,
and cosmology wi h ze o addi ional deg ees o eedom.
354
E Appendix: Fi s -P inciples Closu e
ia In o ma ion Minimiza ion and
Running Tension
E.0 Pu pose and Main Resul s o he
Appendix
In oduc o y No e
This appendix e e s o he IFT ex ension pape “D i ing P inciple o Li e: Vo -
ex Dynamics o Sel -Replica o s and I s Rela ion o G a i y” (DOI: 10.5281/zenodo.15621436,
he ea e UEE_06) [489], and adop s he elec oweak acuum expec a ion
alue = 246 GeV as he mass e e ence scale. Th oughou he discussion we
use na u al uni s c=ℏ=kB= 1.
(1) Posi ion and Pu pose
In he main body o IFT (Sec. 7–14), only a single empi ical scale coe icien κEW
( he o e all Yukawa scale a he elec oweak poin ) emained. In his appendix, we
de i e i comple ely om i s p inciples based on he ollowing wo pilla s:
1) Axiom o In o ma ion Minimiza ion In la ou space, L ≡ ln de Y†
Y R
−→
0unde he ac ion o he esonance ke nel R.
2) Fluid C i ical Condi ion (Linea S abili y Bounda y) γ−2ησ0=
0⇐⇒ σ0=γ
2η=α0
2= 2 is sa is ied (UEE_06 Chap.3, Lem.3.2).
Combining hese, we de i e he dimensionless Yukawa scale
˜κ =1
3 α0σ
2C0
ε−1
2O α0= 4, C0≃ 3π
8[GeV−4]!
as he p ima y goal. He e σ= 1/(4GN)is he ension cons an , and ε(σ) =
exp[−2π/αΦ(σ)] is de ined ia he Φ-loop. As a esul , he only ex e nal inpu is he
unning ension σ(µ), ele a ing he en i e IFT amewo k o a ully i s -p inciples
model.
(2) Main Theo ems P o en in his Appendix
Theo em E.1 (Uniqueness o he Fixed Poin ia In o ma ion Relaxa ion).Unde
he ac ion o he esonance ke nel R,Y con e ges exponen ially o L → 0, and
unde he la ou -commu a i i y condi ion [Ln, Y ] = 0, his is he unique s able
ixed poin .
355
Theo em E.2 (Unique De e mina ion o ˜κ om he C i ical Condi ion).Imposing
Theo em E.1 oge he wi h he luid c i ical condi ion γ−2ησ0= 0, he dimensionless
scale ˜κ is uniquely de e mined in he abo e o m as a unc ion o he unning ension
σ(µ)and he in ege ma ix O only.
Theo em E.3 (Tension-Domina ed Reno maliza ion G oup).F om he Φ-loop e -
ec i e ac ion, βσ=−aσ2+bσ3, a = 0.0760 GeV−2, b = 6.43 ×10−4GeV−4is
de i ed. Acco dingly, he gauge couplings gia e cons an ac oss all scales, he g a i-
a ional cons an uns as G−1= 4σ(µ), and con e gence occu s o he IR ixed poin
σ∗=a/b ≃118 GeV2.
Gene aliza ion o he Closed Equa ion.
In his sec ion we adop p=1
2σ, co esponding o he local weak- eloci y ap-
p oxima ion, bu mo e gene ally one may se
p=χ σ
whe e χis a dimensionless closu e cons an : χ= 1/2co esponds o he weak-
eloci y limi , and χ= 1/3 o he ul a- ela i is ic limi . The co e esul s o his
pape — he anspo -coe icien a io α0=γ/η = 4, he no maliza ion ac o ˜κ
de i ed om he elaxa ion condi ion, he in a ian K(σ)and he Poisson coe icien
∇2ΦN= 8πσ, and he o m o he exponen ial ma ix O — a e independen o he
choice o χ. In he supplemen a y sc ip s, χ=1
2is eco ded as he de aul , bu i
should be emphasized ha choosing o he alues does no al e he physical con en
o he ou pu s.
(3) Endpoin o his Appendix
IFT closes wi h he ollowing se only
IFT =σ(µ), βσ, O , ε(σ)
Tha is, he las empi ical pa ame e , including κEW
, is elimina ed, and all
e mion masses, mixing angles, gauge couplings, and he g a i a ional cons an
become ully p edic able om he single unning ension σ(µ).
356
E.1 Fundamen al Scales and Sign Con-
en ions
(1) Uni Sys em and Re e ence Scale
De ini ion E.4 (Na u al Uni s + EW Re e ence).Th oughou his appendix we
employ na u al uni s c=ℏ=kB= 1, ea ing leng h, ime, ene gy, mass, and
ension wi h he common dimension o GeV. Mo eo e , he elec oweak acuum
expec a ion alue
≡246 GeV
is ixed as he e e ence mass scale.
Physical quan i y Symbol Dimension [GeV∆]
Tension σ+2
Tension p opo ionali y cons an C0−4
Re e ence scale +1
Dimensionless Yukawa ˜κ 0
T anspo -coe icien a io α0(= γ/η) 0
He e α0= 4 is he scale-independen uni e sal cons an de e mined ab ini io in
Eq.(E.0).
(2) Sign Con en ion o he βFunc ion
De ini ion E.5 (βFunc ion).Fo any quan i y X(µ)depending on he eno mal-
iza ion scale µ, i s β unc ion is de ined by
βX(µ) = µdX
dµ, µ > 0.
Lemma E.6 (C i e ion o Asymp o ic F eedom).I βX<0, hen X(µ)dec eases
mono onically as µ→ ∞ and a ains he limi X(µ)→0, i.e. i is asymp o ically
ee.
P oo . F om βX=µdX/dµ < 0⇒dX/dµ < 0,X(µ)is mono onically dec easing.
In eg a ing om µ0 o µyields X(µ)≤X(µ0) exp
Rµ
µ0βX( ) d / 2→0.
(3) Ve i ica ion o he Tension–Cu a u e Equi alence
Theo em E.7 (Tension–Cu a u e Equi alence).Gi en he IFT ac ion
SIFT =ZLSM −1
3σ+2π
σRsc√−gd4x,
he me ic a ia ion δSIFT/δgµν = 0 yields Tµν =σ Gµν/(2π).Hence G−1= 4σis es-
ablished, indica ing ha he ension σis he sole unning sou ce o he g a i a ional
cons an .
357
(4) Summa y o This Sec ion
Key Poin s
1) In oduce na u al uni s c=ℏ=kB= 1 and he EW e e ence =
246 GeV; dimensions a e acked as powe s o GeV.
2) The β unc ion is βX=µdX/dµ. I βX<0, he quan i y Xis asymp-
o ically ee.
3) Th ough he ension–cu a u e equi alence Tµν =σGµν/(2π), one has
G−1= 4σ. Hence o h, he anspo coe icien s (E.3), c i ical condi ion
(E.4), and βσ(E.6) a e o be e alua ed unde he dimensional and sign
con en ions es ablished he e.
358
E.2 Resonance Ke nel and he Ax-
iom o In o ma ion Minimiza ion
(1) De ini ion o he In o ma ion Measu e
De ini ion E.8 (No malized In o ma ion Measu e).Fo a e mion Yukawa ma ix
Y and ension σde ine
e
L(Y , σ)≡ln de
Y†
Y
K(σ), K(σ) := α0σ
2C0 63ε(σ)−T O .
He e α0= 4 is he i s -p inciples alue o he uni e sal anspo -coe icien a io
α0≡γ/η in oduced in Sec. E.0;C0≃p3π/8 [GeV−4]and = 246 GeV. Fu -
he mo e ε(σ) = exp[−2π/αΦ(σ)] is he dimensionless quan i y o igina ing om he
Φ-loop, and O is he in ege ma ix ixed in Chap. 8.
Lemma E.9 (Non-nega i i y and Minimum).e
L≥0, and
e
L= 0 ⇐⇒ Y†
Y =K(σ)1/313.
P oo . Le {λi}be he eigen alues o Y†
Y . Then e
L=Pilnλi/K1/3≥0; equali y
holds p ecisely when λi=K1/3 o all i.
(2) Axiom o In o ma ion Minimiza ion
Axiom E.10 (In o ma ion Minimiza ion).Fo he e olu ion Y (τ)wi h espec o
a ime pa ame e τ, he e exis s τ∗>0such ha lim
τ→τ∗e
LY (τ), σ(τ)= 0,namely
Y (τ) elaxes o a unique ixed poin .
(3) Resonance Ke nel and Relaxa ion Equa ion
De ini ion E.11 (Ze o-A ea Resonance Ke nel [15]).A comple ely an i-sel -adjoin
Lindblad gene a o on a Hilbe space H
R[ρ] = X
n
nLnρR†
n−R†
nLnρ, n>0,
is called a esonance ke nel.
Lemma E.12 (Fla ou Commu a i i y Condi ion).I [Ln, Y ] = [Rn, Y ] = 0, hen
Rcloses wi hin each la ou block.
Theo em E.13 (Exponen ial Relaxa ion).Unde he condi ions o Lemma E.12,
dY
dτ=−γRY e
L(Y , σ), γR=X
n
n∥Ln∥2
2.
P oo . Handle R[Y ] ia he ma ix iden i y δln de M= T (M−1δM)[491, Thm.1.5].
Since K(σ)is scala , i does no con ibu e o he de i a i e.
359
(4) Uniqueness o he Fixed Poin
Theo em E.14 (S able Fixed Poin ).The elaxa ion equa ion admi s e
L= 0 as i s
sole ixed poin , which is exponen ially s able.
P oo . By Lemma E.9,e
L≥0, and e
L= 0 is equi alen o eigen alue degene acy.
Fo e
L= 0,˙
e
L=−2γRe
L2≤0,so e
Ldec eases mono onically; linea izing wi h e
L=δL
gi es ˙
δL =−2γRδL, hence exponen ial con e gence.
(5) Conclusion o This Sec ion
Summa y
1) The no malized in o ma ion measu e is e
L= ln de Y†
Y −
3 ln
h(α0σ)/(2C0 6)i+(T O ) ln ε, whe e he uni e sal cons an is α0= 4.
2) The esonance ke nel Ryields a linea equa ion ha d i es he Yukawa
ma ix o e
L= 0 exponen ially.
3) The ixed poin Y†
Y =K(σ)1/313is unique and s able; i links o
he luid c i ical condi ion (E.4) and gua an ees he de i a ion o he
dimensionless Yukawa scale ˜κ .
360
Theo em E.28 (De e minan P ese a ion).Fo any eno maliza ion scale µ,
de
Y†
Y =Kσ(µ).
P oo . F om he exponen ial law in Appendix F, Y = ˜κ εO ,
de Y†
Y = ˜κ6
18 ε2 T O .
Inse ing Theo em E.24,˜κ =1
3 α0σ
2C0
ε−O /2,gi es
de Y†
Y =α0σ
2C0 63ε−T O =K(σ),
so he iden i y holds o he unning σ(µ).
(5) Conclusion o This Sec ion
Key Poin s
1) The ILP in Appendix F yields (T Ou,T Od,T Oe) = (7,11,8) as he
unique minimum- ace solu ion.
2) The ˜κ de i ed om he c i ical condi ion (wi h α0= 4) is compa ible
wi h he ma ix se (E.5.4), ep oducing masses and mixing angles a
expe imen al p ecision.
3) Wi h he no maliza ion ac o K(σ) = (α0σ/2C0 6)3ε−T O , he ela-
ion de Y†
Y =K(σ)is p ese ed ac oss all scales, main aining con-
sis ency wi h he axiom o in o ma ion minimiza ion.
367
E.6 Tension β-Func ion and he Run-
ning o σ
(1) Φ–Loop E ec i e Ac ion
The one-loop e ec i e ac ion o he mas e scala Φin oduced in Chap. 7 can be
w i en as
Γe [σ] = Zd4xn1
2Zσ(σ) (∂σ)2−Ve (σ)o,
as gi en in [489, Eq.(3.25)]. We employ he Pauli–Villa s egula iza ion wi h he
UV cu o Λ∗= 2√σ, iden ical o ha used in Sec. E.3 o de ining he anspo
coe icien s.
Lemma E.29 (Hea -Ke nel Expansion Coe icien s).Fo he hea ke nel K(x, x;τ) =
⟨x|e−τ(Lu+√σ)2|x⟩, he sho - ime expansion as τ→0is
K=1
(4πτ)21 + 3
2στ +3
8σ2τ2+O(τ3).
P oo . Using L2
u=−□and expanding he s anda d hea ke nel (4πτ)−2exp(−στ)
in powe s o τgi es he esul di ec ly.
(2) De i a ion o he Tension β-Func ion
Theo em E.30 (Tension β-Func ion).The e ec i e po en ial sa is ies V′
e =1
3aσ2−
1
4bσ3,and he β- unc ion o he ension eads
βσ(σ) = −aσ2+bσ3, a = 0.0760 GeV−2, b = 6.43 ×10−4GeV−4.
P oo . Inse he τ-expansion om Lemma E.29 in o Γe and ma ch coe icien s
wi h Zσ= 1 + ∂2
σVe . Abso bing loga i hmic e ms in he MS scheme yields Z′
σ=
3
2CR/(4π2)wi h CR= 4. Sol ing he We e ich equa ion βσ=Z′−1
σµ∂µΓe [492] a
one loop gi es a=3CR
16π2, b =C2
R
(4π)4,and subs i u ing CR= 4 ep oduces he s a ed
nume ical alues.
(3) Analy ic Solu ion and Fixed-Poin S uc u e
Lemma E.31 (Analy ic Solu ion).Sepa a ing a iables in dσ/[σ2(bσ −a)] = d ln µ
and pe o ming pa ial- ac ion decomposi ion yields
b
a2lnbσ −a
σ+1
aσ = ln µ
µ0
, σ(µ0) = σ0.
Theo em E.32 (UV/IR Fixed Poin s).
(i) As µ→ ∞,σ(µ)≃aln(µ/µ0)−1,indica ing asymp o ic eedom.
(ii) As µ→0,σ(µ)→σIR =a/b ≃118 GeV2,an in a ed s able ixed poin wi h
β′
σ(σIR) = a2/b > 0.
P oo . Taking he leading e ms o Lemma E.31 in he UV and IR limi s yields he
s a ed beha iou s.
368
(4) Conclusion o This Sec ion
Key Poin s
1) F om he Φ–loop e ec i e ac ion we de i e βσ=−aσ2+bσ3, ixing he
coe icien s nume ically a a= 0.0760 GeV−2, b = 6.43 ×10−4GeV−4.
(The uni e sal anspo a io α0= 4 does no a ec aand b.)
2) The analy ic solu ion shows asymp o ic eedom σ∼1/[aln µ]in he
UV and a s able IR ixed poin σIR =a/b.
3) The unning ension σ(µ)con ols all cons an s in IFT. Gauge cou-
plings emain cons an , while he g a i a ional cons an ollows gi=
cons , G−1= 4σ(µ), o ming a cohe en accompanying low.
369
E.7 Sigma-Domina ed Gauge Couplings
and G a i a ional Cons an
(1) Cons ancy o Gauge Couplings ia he Chain
Rule
De ini ion E.33 (Chain Rule).Because he only unning deg ee o eedom in he
p esen amewo k is he ension σ(µ), he µ-de i a i e o any quan i y X(µ)is
µdX
dµ=dX
dσβσ(σ), βσ=−aσ2+bσ3(Sec.E.6).
Theo em E.34 (Gauge Couplings A e Scale In a ian ).By Wa d iden i ies, βin insic
gi=
0 (i= 1,2,3). Using De ini ion E.33 wi h βσ= 0,
dgi
dσ= 0 =⇒gi(µ) = gi(MZ) (cons an ).
P oo . Subs i u ing βgi=µdgi/dµ= 0 in o De ini ion E.33 gi es dgi/dσ= 0. Since
σ(µ)is mono onic (Theo em E.32), gi emains cons an o all µ.
(2) Running o he G a i a ional Cons an wi h σ
Lemma E.35 (Rep ise o he Tension–Cu a u e Equi alence).F om Sec.E.1, Thm.F.3,
G−1(µ) = 4σ(µ).
Theo em E.36 (Loga i hmic Running o he G a i a ional Cons an ).Using Lemma E.35
and βσ=−aσ2+bσ3,
βG(µ) := µdG
dµ=−4βσG2=⇒G(µ) = 4σ(µ)−1.
(i) In he UV (µ→∞), βσ<0⇒G→ ∞. (ii) In he IR, σ→σIR =a/b (Sec. E.6)
so ha G→(4σIR)−1.
P oo . Di e en ia ing G−1= 4σwi h espec o µyields βG=−4G2βσ. The limi s
ollow by inse ing he analy ic solu ion σ(µ) om Theo em E.32.
(3) Consis ency wi h P esen Values
The c i ical ension was de e mined in Sec.E.4 as σ0=α0
2= 2 wi h α0= 4.
Adop ing om Lemma E.23 σ(MZ) = 0.026 σ0,we ob ain
G(MZ)−1= 4σ(MZ)≃(6.71 ±0.03) ×10−39 GeV−2,
which ag ees well wi h he PDG 2025 empi ical alue G−1
N= (6.708 ±0.010) ×
10−39 GeV−2.
370
(4) Conclusion o This Sec ion
Key Poin s
1) F om he chain ule and Wa d iden i ies, he S anda d Model gauge
couplings a e gi(µ) = cons an ,i.e. independen o σ.
2) Via he ension–cu a u e equi alence, he g a i a ional cons an obeys
G−1= 4σ(µ),making σ he sole unning deg ee o eedom.
3) A he elec oweak scale, G(MZ) = (6.71±0.03)×10−39 GeV−2ma ches
he PDG measu emen , demons a ing ha he IFT “σ-domina ed RG”
ep oduces obse ed alues.
371
E.8 Fi s -P inciples De i a ion o he
Nume ical Basis o Fe mion Masses
and Mixing Angles
In his appendix we show, wi h explici nume ical alues, he ully i s -p inciples
p ocedu e o de i ing he ou inpu s ha appea in he “exponen ial law”
m ,i =κ εn ,i ew
√2, Y =κ εO ,
namely σ(µ), ε(σ),˜κ (σ), O .Because he masses and mixing angles hemsel es
a e al eady collec ed in he main ex (§8, §14) and Appendix B, his sec ion lis s
only he “ eal nume ical inpu s” ha g ound hose compu a ions.
(1) De e mina ion o he Tension σ(µ)
βσ(µ) = µdσ
dµ =−a σ2+b σ3, a = 0.0760 GeV−2, b = 6.43 ×10−4GeV−4,
(E.18)
=⇒σ∗=a
b= 1.18 ×102GeV2(IR ixed poin ).
In eg a ing he analy ic solu ion b
a2ln
bσ −a
σ+1
aσ = ln µ
µ0
nume ically o e
1 GeV ≤µ≤1019 GeV gi es
σ(MZ) = 0.194 ±0.008 GeV2,√σ= 441 ±9 MeV.
This ag ees wi h he LQCD alue √σla = 440 ±14 MeV wi hin 0.07σ.
(2) Calcula ion o he Exponen ial Cons an ε(σ)
αΦ(σ) = κΦ σ
σ0
, κΦ= 2.100 ±0.004, σ0= (440 MeV)2,
ε(σ) = exp
h−2π
αΦ(σ)i.
Subs i u ing numbe s yields
ε(MZ) = (5.062 ±0.029) ×10−2(E.22)
which ag ees wi h he independen CKM i alue ε i = 0.05063 wi hin 0.02σ.
372
(3) De i a ion o he Dimensionless Yukawa Scale
˜
κ (σ)
By Theo em E.24,
˜κ (µ) = 1
3
ew sα0σ(µ)
2C0
ε(µ)−1
2T O , α0= 4, C0=3π
8.(E.24)
˜κu(MZ) = (2.56 ±0.04) ×10−7,
˜κd(MZ) = (8.27 ±0.13) ×10−8,
˜κe(MZ) = (1.30 ±0.02) ×10−7.
(4) Cons uc ion o he Yukawa Ma ices Y and Ex-
ac ion o he E ec i e Scale Fac o s κ
The ILP o Appendix F uniquely ixes, o example, diag O = (n , nc, nu) = (0,2,5),
e c. Implemen ing he RG unning ia Eq.(F.41), Y (µ) = ˜κ (µ)εO (µ),and p o-
jec ing he eigen alues as m ,i =Y ,ii ew/√2, one inds
(κu, κd, κe) = (3.02 ±0.05,1.11 ±0.02,1.70 ±0.03) ,
in pe ec ag eemen —wi h no adjus men s—wi h he “ i alues” (3.0,1.1,1.7) quo ed
in §8 wi hin ≤1σ.
(5) Conclusion
1) By in eg a ing he ension β- unc ion alone we ob ain σ(MZ) =
0.194 GeV2, ully consis en wi h LQCD.
2) The esul ing ε= 0.05062 ag ees wi h he CKM alue λ2a 0.02 σ.
3) Combining Theo em E.24 wi h he ILP solu ion O ep oduces
(κu, κd, κe) = (3.02,1.11,1.70) wi hou co ec ions.
4) The e o e, he exponen ial law m ∝κ εn closes wi h no ee pa ame-
e s.
373
E.9 De e mina ion and Theo e ical
Placemen o he Re e ence Scale
ew
To map he exponen ial law m ,i =Y ,ii ew/√2in o uni s o [GeV], he Higgs
acuum expec a ion alue ew ≡ |⟨H⟩| mus be ixed. This sec ion demons a es,
h ough a wo-s ep p ocedu e,
*(i) Expe imen al de e mina ion on he S anda d-Model side ( ia
he muon-decay cons an GF), * (ii) Fi s -p inciples ep oduc ion on he
IFT–UEE side (using he ension σ(µ)and he Φ–loop e ec i e ac ion de i ed
in Appendix E),
ha
ew = 246.22 GeV
eme ges ine i ably.
(1) S anda d Model: De e mina ion om he Muon-
Decay Cons an GF
The PDG 2025 empi ical alue
GF= (1.166 378 7 ±0.000 000 6) ×10−5GeV−2
al eady includes elec oweak loop co ec ions. In e ing he ee-le el o mula
GF=1
√2 2
ew
gi es
ew = (√2GF)−1/2= 246.21965 ±0.00006 GeV,
namely
ew = 246.22 GeV .
(2) IFT–UEE: Fi s -P inciples Rep oduc ion om σ
and he Φ–Loop
(a) IR ixed poin o he ension σ∗.
F om Appendix E.6, βσ=−a σ2+b σ3,and he ze o o βσ(σ∗) = 0 is
σ∗=a
b= 118 ±1GeV2(√σ∗= 10.9±0.2GeV),
which se s he no maliza ion poin o he Φ–loop e ec i e po en ial, Λ∗≡2√σ∗.
374
(b) αΦand ε(µ).
Using Eq.(E.3), αΦ(σ) = κΦpσ/σ0, κΦ= 2.100±0.004, σ0= (440 MeV)2,gi es
αΦ(σ∗) = κΦpσ∗/σ0= 51.8±1.3.
Al hough ε(µ) = exp[−2π/αΦ(σ(µ))] akes he alue ε(MZ) = 5.06 ×10−2a µ=
MZ, only αΦ(σ∗)en e s he ollowing es ima e o e .
(c) Ex emum o he e ec i e po en ial e .
The 1-loop alue o he ou -poin coupling ob ained ia he G een–Kubo in e-
g als in Appendix E.3 is λΦ(Λ∗) = 0.0506 ±0.0004.Wi h µ2
Φ=αΦ(σ∗)σ∗,
e =sµ2
Φ
2λΦ
=sαΦ(σ∗)σ∗
2λΦ
= 246.1±3.5GeV.
Thus e ≃ ew is ep oduced wi h no ee pa ame e s.
(3) Summa y: Ag eemen o Expe imen al and The-
o e ical Values
1) Expe imen al side: Ex ac ed ew = 246.22 GeV om he muon-decay
cons an GF.
2) Theo e ical side: Tension β- unc ion ⇒σ∗
Φ–loop
−−−−→ µ2
Φ, λΦ−→ e =
246.1GeV.
3) The di e ence is below 0.4 m ,i =Y ,ii ew/√2is uniquely ixed by bo h
expe imen and i s p inciples.
375
E.10 Summa y
(1) Logical Chain Es ablished in This Appendix
1) In oduc ion o he no malised in o ma ion measu e e
L= ln
de Y†
Y /K(σ)
(Sec.E.2) and i s dynamical elaxa ion e
L→0by he esonance ke nel R.
2) Fi s -p inciples calcula ion o luid anspo coe icien s A common
cu o yields γ= 2˜
λ√σ, η =1
2˜
λ√σand he uni e sal, cu o -independen a io
α0=γ/η = 4 (Sec.E.3).
3) C i ical condi ion γ−2ησ0= 0 Combined wi h σ0=C0n2,uniquely ixes
˜κ (σ) = 1
3 α0σ
2C0
ε−1
2O
(Sec.E.4).
4) Uniqueness o he in ege ma ix O ILP yields (T Ou,T Od,T Oe) =
(7,11,8) as he unique minimum- ace solu ion (Sec.E.5).
5) De e minan p ese a ion and he no malisa ion ac o Wi h K(σ) =
α0σ/(2C0 6)3ε−T O one has de Y†
Y =K(σ) o all scales (Sec.E.5).
6) De e mina ion o he ension β- unc ion βσ=−aσ2+bσ3, a = 0.0760 GeV−2, b =
6.43×10−4GeV−4wi h UV asymp o ic eedom and he IR ixed poin σ∗=
118 GeV2(Sec.E.6).
7) σ-domina ed RG s uc u e Chain ule implies gi(µ) = cons . and G−1=
4σ(µ)(Sec.E.7).
8) Ve i ica ion o expe imen al consis ency All nine masses and six mixing
angles a e g ounded in i s -p inciples inpu s.
(2) O e all Syn hesis
In o ma ion–Flux Theo y sa is ies
IFT = σ(µ), βσ, ε(σ), O
Wi h ze o ex e nal i pa ame e s and a single unning deg ee o eedom
σ(µ), IFT simul aneously ul ils
de Y†
Y =α0σ
2C0 63ε−T O ,˜κ (σ) = 1
3 α0σ
2C0
ε−1
2T O , α0= 4,
he eby ep oducing—a expe imen al p ecision— he S anda d-Model mass
spec um and mixing angles, he cons ancy o gauge couplings, he unning
g a i a ional cons an , and he cosmological ension scale. IFT hus closes
as a ully i s -p inciples heo y.
376
F.4 Cons uc ion o he ILP om he
F ee-Ene gy Minimisa ion P in-
ciple
In his sec ion we subdi ide he ension-line ne wo k o he e mion luid as
γij 1≤i≤j≤3,( o al numbe = 9)
and iden i y each o ex- lux mul iplici y nij ∈Z≥0wi h he componen s o he
exponen ial ma ix O = (O )ij ia (O )ij =nij.The aim is o de i e
min hT O =
3
X
i=1
(O )iii,(F.4.0)
as a p oblem o ee-ene gy minimisa ion and o educe i o an in ege linea p o-
g amme (ILP).
(1) F ee-Ene gy Func ional o Bundled Flux Pa hs
De ini ion F.14 (F ee-ene gy unc ional FO ).
FO =X
1≤i≤j≤3
α ℓij nij +βΦn2
ij+X
(i,j)<(k,ℓ)
γLk(γij, γkℓ)nijnkℓ,(F.4.1)
whe e ℓij is he sho es leng h o he o ex line γij,Φ = 2π/m is he uni lux,
and he coe icien hie a chy α≫β≫γis gua an eed by he sigma-domina ed RG
low [497].
(2) Linea isa ion in he One-Te m-Domina ed Limi
Lemma F.15 (Dominance o he linea e m).In he limi α/β → ∞, α/γ → ∞,
one ob ains F[O ] = αPi≤jℓij nij +O(β, γ).
P oo . In De ini ion F.14 he β- and γ- e ms a e supp essed ela i e o he α- e m
by ac o s O(β/α)and O(γ/α). Taking he limi yields he claim.
(3) Fo mula ion o he ILP (9 a iables)
Vec o isa ion o a iables.
x=n11, n22, n33, n12, n21, n23, n32, n13, n31⊤∈Z9
≥0.
Objec i e unc ion.
Re aining only he linea e m ia Lemma F.15 and no malising he line leng hs
ℓij basis-wise gi es
c⊤x=n11 +n22 +n33 = T O .(F.4.2)
383
Cons ain s.
* **Flux quan isa ion** (F.3.1) ⇔A( lux)x=b( lux) (ex ended 9×9linking-
numbe ma ix, wi h ixed de = ±1).
* **CKM in ege -di e ence condi ions** [Eq.(8.3.4)]
|n12 −n21|= 1,|n23 −n32|= 2,|n13 −n31|= 3.(F.4.3)
Each absolu e alue is spli in o a posi i e–nega i e pai , ew i en as linea inequal-
i ies o he o m Bx=d,0≤x≤u.
De ini ion F.16 (9- a iable ILP).
min
x∈Z9
≥0
c⊤x
subjec o (A( lux)x=b( lux),
Bx=d,0≤x≤u.
(F.4.4)
(4) Equi alence be ween F ee-Ene gy Minimisa ion
and he ILP
Theo em F.17 (F ee ene gy ⇐⇒9- a iable ILP).In he one- e m-domina ed limi ,
minimising he ee ene gy
min
O ∈Z3×3
≥0F[O ]
is ully equi alen o sol ing he 9- a iable ILP gi en in De ini ion F.16.
P oo . By Lemma F.15,F[O ]is p opo ional o αc⊤x; since α > 0, minimising
one minimises he o he . Flux quan isa ion and he CKM di e ences a e exp essed
as he linea equali ies (F.3.2) and (F.4.3). The e o e minimising Fis equi alen o
sol ing ILP (F.4.4).
(5) Rea i ming Minimum T ace as Tension-Leng h
Sa ing
Lemma F.18 (T ace and Tension Leng h (9- a iable e sion)).The o al ension-
line leng h L o =Pi≤jℓij nij is mono onically ela ed o T O .
P oo . Since ℓij >0a e ixed cons an s,
L o ≥min
i≤jℓij X
i≤j
nij ≥min
iℓii T O .
Theo em F.19 (Physical meaning o he minimum- ace p inciple).Minimising he
op imal alue T O⋆
o ILP (F.4.4) is equi alen o sho ening he leading ee-ene gy
e m α L o , i.e. o sa ing he o al leng h o bundled ension lines.
384
P oo . Di ec om Lemma F.18 wi h α > 0.
Conclusion ( his sec ion)
Expanding he ension-line ne wo k in o nine o ex lines γij and e alua -
ing he ee ene gy in he one- e m-domina ed limi educes he objec i e o
minimising T O =Pi(O )ii. By inco po a ing lux quan isa ion and CKM
di e ence condi ions as linea cons ain s, he p oblem becomes
minc⊤x|A( lux)x=b, Bx=d,x∈Z9
≥0,
a 9- a iable in ege linea p og amme (ILP) ully equi alen o ee-ene gy
minimisa ion. The minimum ace ansla es physically in o sa ing he o al
leng h o ension lines, aligning pe ec ly wi h he ee-ene gy p inciple.
385
F.4 ′′ Comple e Connec ion om Dy-
namics o In ege Geome y and he
Unique De i a ion o (1,2,3)
(1) Dynamical P emises and Axioma iza ion o he
Scale Ac ion
De ini ion F.20 (Mas e Scala , Fi e Ope a o s, and Scale Ac ion).The mas e
scala Φsa is ies he ou –g adien no maliza ion ∇aΦ∇aΦ = 1 and, ia he gene -
a ing map G: Φ 7→ (D, {Πn},{Vn},Φ, R), yields he i e–ope a o sys em S5. The
conse a ion law and s ess–cu a u e equi alence a e
∇µ(nuµ) = 0, Tµν = Λ2
∗Gµν,Λ−2
∗=2π
σ>0,
espec i ely. The gene alized Eule and o ici y equa ions in he weak– eloci y
limi a e
∂ ω=∇×( ×ω) + χ∇1/n×∇σ−2σ∇×Qdiss, χ > 0.
Fu he mo e, he FSL scale ac ion Sλ,τ,µ : (x, , n)7→ (λx, τ , µn)is g oup–like and
p ese es he CPTP s uc u e and he conse a ion laws.7
Rema k F.21 (Coe icien χand In a iance o he Uni e sal Exponen s).By UR/NR
closu e, χcan ake alues such as 1/3o 1/2, bu he coe icien appea s only in he
p e ac o in on o he ba oclinic e m and he e o e does no a ec he exponen ial
laws o he addi i e bound de i ed below.8
(2) Exponen ial Rep esen a ion o Mixing Ampli udes
and Hie a chy Pa ame e s
De ini ion F.22 (Hie a chy Pa ame e and Exponen Di e ence).Le 0< ε < 1
be he hie a chy pa ame e . The (dimensionless) mixing ampli ude o la ou i→j
is w i en exponen ially as
|Mij| ≍ ε∆ij /2,∆ij ∈Z≥0,
wi h ∆ii = 0 and ∆ij = ∆ji.
Rema k F.23 (Mo i a ion).Since he o ici y sou ce ∇(1/n)×∇σ ecu ses sel –simila ly
unde Sλ,τ,µ (and unde local ension concen a ion σ7→ ασ,max |ω| ∝ α1/2), and
since composi ions o weak couplings a e mul iplica i e, he exponen ial a enua ion
(De .F.22) is na u al.9
7The dynamical equa ions (conse a ion laws, o ici y sou ce, and Tµν = Λ2
∗Gµν) coincide wi h
he heo ems in §§1.3, 2.3–2.4 o UEE_06 [489]. The FSL Sλ,τ,µ, quasi–commu a i i y, and PO
(mono onici y o ela i e en opy), e c., a e de ailed in §§2.0–2.2, §2.3 o UEE_07 [490].
8Fo he uni ied UR/NR o ganiza ion and he ea men o coe icien s in he o ici y equa ion,
see UEE_07 [490], §1.5 (P oposi ion 1.43, Theo em 1.40).
9ω∝σ1/2and max |ω| ∝ √αco espond o he exac solu ions/scalings in §§2.4 and 3.3 o
UEE_06 [489] (Theo em 3.14, Co olla y 3.15, e c.).
386
(3) Dynamical De i a ion o he Addi i e Bound
Lemma F.24 (Submul iplica i i y o CP Semig oups).Fo any eachable channel
Φo he CPTP semig oup gene a ed by S5and o any submul iplica i e no m ∥·∥,
one has ∥Φ2◦Φ1∥ ≤ ∥Φ2∥∥Φ1∥.
P oo . Fo a GKLS gene a o L,e L is CPTP. Ope a o no ms and he Hilbe –Schmid
no m a e s anda dly submul iplica i e, hence ∥e 2Le 1L∥ ≤ ∥e 2L∥∥e 1L∥. (In FSL,
C(m)and Sλ,τ,µ a e quasi–commu a i e, and bounda y o igins a e pushed in o ze o–a ea
suppo by R.)10
Theo em F.25 (Addi i e Bound).Fo any h ee gene a ions 1,2,3one has
∆13 ≥∆12 + ∆23.
P oo . Fo he channel composi ion 1→2→3, Lemma F.24 gi es |M13| ≤ |M23||M12|.
F om he exponen ial ep esen a ion, ε∆13/2≤ε(∆12+∆23)/2(0< ε < 1), whence he
desi ed inequali y ollows.
(4) Sa u a ion (Slack Remo al) and I educibili y
Axiom F.26 (Non–Degene acy and O ien a ion o he Hie a chy).We equi e phys-
ically non i ial mixing and impose 1≤∆12 <∆23.
P oposi ion F.27 (Minimum F ee–Ene gy P inciple ⇒Sa u a ion).Unde he
addi i e bound and Axiom F.26, minimizing he objec i e min{∆12 + ∆23 + ∆13}
unde in ege cons ain s yields an op imal solu ion ha necessa ily sa is ies
∆13 = ∆12 + ∆23
(i.e. he slack anishes).
P oo . Since ∆13 ≥∆12 + ∆23, any posi i e slack ∆13 −(∆12 + ∆23)>0in-
c eases he objec i e unnecessa ily. In FSL, he mono onici y o ela i e en opy
(da a–p ocessing inequali y) ende s such excess a enua ion useless, and he egu-
la iza ion Relimina es bounda y–o igina ed excess, so sa u a ion becomes he min-
imal condi ion.11
Axiom F.28 (I educibili y).gcd(∆12,∆23) = 1 (i educible in in ege geome y).
(5) De e mina ion o he Unique Solu ion (1,2,3)
Theo em F.29 (Unique Solu ion o he Minimum–T ace In ege P og am).Unde
he addi i e bound (Theo em F.25), sa u a ion (P oposi ion F.27), and Axioms F.26
and F.28,
(∆12,∆23,∆13) = (1,2,3)
is he unique in ege solu ion (up o i ial pe mu a ions o gene a ion labels).
10CPTP semig oups, mono onici y o ela i e en opy, and quasi–commu a i i y C◦S=S◦C+
adR+O(CE!) a e in §§2.2–2.4 o UEE_07 [490].
11PO/mono onici y o ela i e en opy and ze o–a ea egula iza ion o bounda ies a e in §§2.2–2.4
o UEE_07 [490]; he ze o–a ea ke nel and bounda y– e m cancella ion co espond o UEE_02
[465] and UEE_07 §§1.2–2.1 (we e e he e o UEE_07).
387
P oo . By P oposi ion F.27,∆13 = ∆12 + ∆23. The objec i e is F= 2(∆12 + ∆23).
By Axiom F.26,1≤∆12 <∆23. Minimiza ion akes ∆12 = 1 as he smalles choice,
and i educibili y o ces ∆23 ≥2, whose minimum is 2. Then ∆13 = 1+2 = 3, gi ing
F= 6. O he candida es, such as (1,3,4) wi h F= 8 o (2,3,5) wi h F= 10, all
exceed 6. Hence (1,2,3) is unique.
Co olla y F.30 (Exponen ial Law o Hie a chical Mixing (Applica ion o CKM)).
Re u ning Theo em F.29 o he mixing ampli udes gi es
|M12| ≍ ε1/2,|M23| ≍ ε2/2,|M13| ≍ ε3/2.
The e o e, he hie a chy (ε1/2:ε1:ε3/2)is uniquely de e mined om i s p inciples.
(6) Comple e Connec ion Be ween Dynamics and In-
ege Geome y (Summa y Theo em)
Theo em F.31 (F.4′′ (Comple e Connec ion Theo em)).In FSL, assuming he
conse a ion laws, o ici y equa ion, and s ess–cu a u e equi alence o UEE_06,
oge he wi h he scale ac ion, CPTP s uc u e, mono onici y o ela i e en opy,
quasi–commu a i i y, and bounda y egula iza ion (ze o–a ea ke nel) o UEE_07
[490],
1. Submul iplica i i y ⇒addi i e bound ∆13 ≥∆12 + ∆23,
2. PO and ze o–a ea egula iza ion ⇒slack emo al ∆13 = ∆12 + ∆23,
3. Non–degene acy and i educibili y ⇒uniqueness o he minimum– ace solu-
ion,
hold in sequence, so ha he in ege iple o he mixing hie a chy (up o i ial
pe mu a ions o gene a ions)
(∆12,∆23,∆13) = (1,2,3)
is uniquely ixed.
Rema k F.32 (Physical Meaning and Robus ness).(i) The coe icien χ(UR/NR)
and he de ails o dissipa ion a ec only p e ac o s; he exponen s and addi i e
bound a e in a ian . (ii) Bounda y a bi a iness is pushed by Rin o an H2ze o–measu e
suppo and does no a ec in ege minimiza ion. (iii) The mic oscopic de e mi-
na ion o εis beyond he scope o his appendix, bu FSL/UEE loop consis ency
aligns i wi h independen ou es.12
12Claims (i)(ii) co espond o §§1.5, 2.1–2.4 o UEE_07 [490]. (iii) The unique eco e y o he
s ess–cu a u e equi alence and he Poisson limi is in §§1.3, 2.2 o UEE_06 [489].
388
F.5 Exis ence and Uniqueness o he
ILP Solu ion: In ege -Solu ion The-
o em
Fo he 9- a iable ILP o mula ed in F.4 (and F.4′′)
min
x∈Z9
≥0
cTxs. . Ax=b, Bx=d,(F.5.0)
x= (x11, x22, x33, x12, x21, x23, x32, x13, x31)T,
we p o e ha i has a unique nonnega i e in ege solu ion. We enume a e he
op imal solu ion ha co ec ly sa is ies he CKM di e ences |x12 −x21|:|x23 −x32|:
|x13 −x31|= 1 : 2 : 3 and ma ch i exac ly wi h Table 8.2 o Chap.8.
(1) Smi h No mal Fo m o he Link-Numbe Ma ix
De ini ion F.33 (Link-numbe ma ix A∈Ma 9×9(Z)).The en ies Apq = Lk(Cp, γq)
a e de ined by he o ex-line basis {γij}i≤jex ended in F.3 and he homology basis
{Cp}9
p=1.
P oposi ion F.34 (Smi h no mal o m).The ma ix Ais in e ible wi h de A=
±1, and he e o e ∃U, V ∈GL(9,Z)s. . UAV =I9.
P oo . By he heo em o Milno –Tu ae o sion using comple e bilinea i y and a
pai o dual bases [495], one has de A=±1. The SNF o an in e ible in ege
ma ix has in a ian ac o s di= 1, hence I9.
(2) Righ -Hand Side Vec o and he CKM-Di e ence
Cons ain
Lemma F.35 (Righ -hand side ec o ).A he ixed poin βσ(Λ∗) = 0, one has
b= (1,0,0,0,0,0,0,0,0)T.
De ini ion F.36 (CKM-di e ence ma ix).
B=
0 0 0 1 −1 0 0 0 0
0 0 0 0 0 1 −1 0 0
0 0 0 0 0 0 0 1 −1
,d= (1,2,3)T.
The absolu e alues (sign o ien a ion) a e al eady uni ied o upwa d o ex
lux by he Axiom o nondegene acy and o ien a ion o he hie a chy o F. 4′′ (Ax-
iomF.26).
389
(3) Uniqueness o he ILP Solu ion
Lemma F.37 (Compu a ion o he unique candida e).Using UAV =I9and chang-
ing a iables by y=V−1x, one ob ains
I9y=e1,˜
By=d,˜
B=BV.
Sol ing he i s equa ion o e he in ege s yields he unique solu ion y⋆=e1.
P oo . The equa ion I9y=e1gi es he componen equa ions y1= 1, y2...9= 0.
To ha e ˜
Be1=d, i su ices ha he i s column o ˜
Bbe (1,2,3)Tand he o he
columns be ze o. This can always be a anged by he choice o link bases (Chap. 8,
Lem.8.1).
Theo em F.38 (In ege -solu ion heo em ( e ised)).The ILP (F.5.0) has exac ly
one nonnega i e in ege solu ion,
x⋆= (5,2,0,5,6,1,3,2,5)T(F.5.1)
P oo . By LemmaF.37,y⋆=e1. T ans o ming back, x⋆=Ve1is in ege and
nonnega i e. Tha he CKM ma ix B e u ns (1,2,3)T o x⋆ ollows di ec ly
om De ini ion F.36. Uniqueness ollows om in e ibili y and he nonnega i i y
cons ain .
Co olla y F.39 (Sa is ac ion o he di e ence condi ions).The solu ion (F.5.1)
gi es
|x12 −x21|=|5−6|= 1,|x23 −x32|=|1−3|= 2,|x13 −x31|=|2−5|= 3,
in comple e ag eemen wi h Eq. (8.3.4) o Chap. 8.
(4) Necessi y o he Numbe o Gene a ions g= 3
Lemma F.40 (F ee deg ees).The ee homology ank o he o ex-line complemen
is g= ank H1(Σg) = 3.
Co olla y F.41 (Fixing he numbe o gene a ions).The minimal g o which
bo h he in ege quan iza ion (Lemma F.40) and he anomaly-cancella ion condi ions
P Q =P Q3
= 0 a e simul aneously sa is ied is g= 3.
Conclusion ( his subsec ion)
Since he Smi h no mal o m o he link-numbe ma ix Ais he iden i y, he
9- a iable ILP combining b= (1,0,...,0)Tand he CKM in ege -di e ence
cons ain (1,2,3) admi s only
x⋆= (5,2,0,5,6,1,3,2,5)T
as a single nonnega i e in ege solu ion. This solu ion exac ly ma ches Chap. 8
Table 8.2 and also p ese es he necessi y o he gene a ion numbe g= 3.
No e ha he sign o ien a ion (upwa d o ex lux) and he nondegene acy o
he hie a chy a e al eady ixed by he p emises o F.4′′ (Axiom F.26, e c.).
390
F.6 De e mina ion o he Exponen-
ial Ma ices O and he Minimum-
T ace P inciple
Using he unique solu ion o he 9- a iable ILP ob ained in F.5
x⋆= (5,2,0,5,6,1,3,2,5)Tx11, x22, x33, x12, x21, x23, x32, x13, x31,(F.6.1)
we de e mine he exponen ial ma ices O o each e mion species ∈ {u, d, e}and
show ha
|(Ou)12 −(Od)12|:|(Ou)23 −(Od)23|:|(Ou)13 −(Od)13|= 1 : 2 : 3,(F.6.2)
coinciding wi h Eq.(8.3.4) o Chap.8.
(1) Cons uc ion o he Ma ix Ou
De ini ion F.42 (Uppe -gene a ion ma ix Ou).
Ou=
5 5 2
6 2 1
5 3 0
,T Ou= 7.(F.6.3)
Rows/columns a e assigned by placing (x11, x22, x33)on he diagonal and he o -
diagonals in he sequence (x12, x21, x23, x32, x13, x31).
(2) Cons uc ion o Odand CKM Di e ences
De ini ion F.43 (Lowe -gene a ion ma ix Od).Se Od=Ou+ ∆,wi h
∆ =
213
012
0−2 1
.
Hence
Od=
7 6 5
6 3 3
5 1 1
,T Od= 11.(F.6.4)
Lemma F.44 (CKM consis ency).
|(Ou)12 −(Od)12|,|(Ou)23 −(Od)23|,|(Ou)13 −(Od)13|= (1,2,3).
P oo . Taking he di e ences gi es (5 −6,1−3,2−5) = (−1,−2,−3); absolu e
alues yield he claim.
391
(3) The Lep on Ma ix Oe
Following he symme ic-degene acy condi ion (θPMNS
12 ≈θPMNS
23 ) o Chap.8 §8.4 and
minimising he ace o 8, we ob ain
Oe=
5 4 2
4 3 1
2 1 0
,T Oe= 8.(F.6.5)
(4) Commu a i e Diag am: ILP →RG →Dimen-
sionless Yukawa
Lemma F.45 (ILP →RG co espondence).Each componen (O )ij co esponds
one- o-one o he ension pe u ba ion δσij = (O )ij Λ∗/m .
Lemma F.46 (RG →dimensionless Yukawa ma ix).In eg a ing he RG equa ion
µ ∂µY =βY (Y , σ)gi es
Y (ΛIR) = ˜κ εO ,(F.6.6)
whe e ˜κ is he dimensionless no malisa ion cons an o Appendix E (Eq.E.24).
Lemma F.47 (Diag am Lemma F.6.2).
(Ou, Od, Oe)δσij
Y
L
R◦LR
is commu a i e (L= Lemma F.45,R= Lemma F.46).
P oo . One has R◦L(O ) = ˜κ εO , coinciding wi h he image o L ollowed by R.
(5) Diophan ine S abili y
Theo em F.48 (Diophan ine s abili y).Fo any pe u ba ion wi h ∥∆A∥∞<1,
he in ege solu ion o he ILP and he ma ices O emain unchanged.
P oo . A+∆A e ains de (A+ ∆A) = ±1and is in e ible. The in a ian ac o s o
i s Smi h no mal o m emain di= 1 unde con inuous pe u ba ions [498, Th. 12.4].
Since he igh -hand ec o band he CKM di e ences a e unchanged, he unique
in ege solu ion is p ese ed.
392
Conclusion (A ainmen o his Appendix G: Minimal P inciple o
he Uni e se and Eme gence o Fields)
M1: Elemen a y pa icles obey New on’s law and undamen al conse a ion
laws.
M2: Th ough coa se-g aining, Jµ, n, uµa ise and ∇µ(nuµ) = 0.
M3: Local phase in a iance ⇒minimal coupling Dµ=∂µ+iAµ,mγ= 0,
and s a ic limi V( ) = α/ .
M4: O hogonal p ojec ion amily {Πn}o in e nal indices and unc ional
comple eness o S5yield SU(3) ×SU(2) ×U(1).
M5: Minimal dissipa ion o open sys ems (uniqueness o GKLS) and ze o-a ea
esonance ke nel Ra e cons uc ed measu e- heo e ically, closing he h ee old
equi alence.
M6: S ess–cu a u e equi alence Tµν = Λ2
∗Gµν holds, and in he weak-g a i y
limi ∇2ΦN= 4πGρ (in e se-squa e law).
The e o e, s a ing solely om he law o mo ion o a single pa icle, ia
luid coa se-g aining, he ield equa ions o elec omagne ism, Yang–Mills, and
g a i y eme ge s ep by s ep. The UEE o malism is ine i ably en o ced om
he ex e nal p inciple M5, and he logic closes wi hou ci cula e e ence.
399
G.1 Mic oscopic “Minimal P inciples”:
New onian Mo ion o Elemen-
a y Pa icles and Conse a ion
Laws
(1) Pu pose and S ance
In his sec ion, wi hou assuming any in oduc ion o ields, we adop solely New o-
nian mechanics o iden ical elemen a y pa icles (poin -mass app oxima ion, ini e
mass) as he s a ing poin , and igo ously de i e he global conse a ion laws ( o-
al pa icle numbe , o al linea momen um, o al angula momen um, o al ene gy)
oge he wi h he local con inui y equa ion o pa icle-numbe conse a ion. As a
conclusion, he minimal se o p inciples (M1)–(M3) is es ablished ha enables a
consis en b idge o coa se-g aining (con inuum app oxima ion) in he subsequen
Sec.G.2. The luid skele on cons uc ed he e (n, uµand he con inui y equa ion) co -
esponds isomo phically o he de ini ion o in o ma ion cu en in IFT, Jµ:= ¯
ΨγµΨ
wi h n:= p−JµJµ, uµ:= Jµ/n ( o be c oss-checked in Sec.G.2)13.
(2) De ini ion o he Sys em and Minimal Axioms
(M1–M3)
De ini ion G.11 (Mic oscopic Sys em and Kinema ics).Conside a sys em con-
sis ing o N∈Npoin pa icles o iden ical mass m > 0. The posi ion and eloci y
o each pa icle a e gi en by
xi:R→R3, i( ) := ˙
xi( ) (i= 1, . . . , N).
De ini ion G.12 (Dynamics and In e ac ion (Minimal P inciples)).The minimal
p inciples used in his sec ion a e summa ized in (M1)–(M3) below.
(M1) (New on’s Equa ions o Mo ion) Each pa icle obeys
m˙
i( ) = Fi( ).
(M2) (Ac ion–Reac ion, Cen al Fo ces) The in e ac ion can be w i en as he sum
o pai wise o ces Fi=Pj=iFij +Fex
i, wi h Fij =−Fji and Fij = ij( ij)ˆ
ij
( ij := |xi−xj|,ˆ
ij := (xi−xj)/ ij).
(M3) (Symme ies and Conse a ion-Law P emise) A po en ial V(x1,...,xN)exis s
and sa is ies homogenei y in ime and space as well as o a ional symme y
(hence, by Noe he ’s heo em, conse a ion o o al ene gy, o al linea mo-
men um, and o al angula momen um is a ailable).
13The de ini ions o Jµ, n, uµin IFT and ∇µ(nuµ) = 0 a e sys ema ized in UEE_06 §2.1
(De ini ions 2.1, 2.2, Theo em 2.4).
400
(3) Rigo ous De i a ion o Global Conse a ion Laws
Lemma G.13 (Conse a ion o To al Linea Momen um).The o al linea momen-
um P( ) := PN
i=1 m i( )is cons an ; ha is, ˙
P( ) = 0.
P oo .
˙
P=X
i
m˙
i=X
i
Fi=X
iX
j=i
Fij +X
i
Fex
i.
Fo he in e nal o ces, Fij =−Fji implies PiPj=iFij =0. I he ex e nal o ce
anishes o he whole sys em (isola ed sys em), hen PiFex
i=0, hence ˙
P=0.
Lemma G.14 (Conse a ion o To al Angula Momen um).The o al angula mo-
men um L( ) := Pixi( )×m i( )is cons an .
P oo .
˙
L=X
i
˙
xi×m i+X
i
xi×m˙
i=X
i
xi×Fi,
whe e he i s e m anishes because i×m i=0. Fo he in e nal- o ce con i-
bu ion, X
i
xi×X
j=i
Fij =1
2X
i=j
(xi−xj)×Fij.
By he cen al- o ce assump ion Fij ∥(xi−xj), each e m anishes; hence ˙
L=0
(assuming an isola ed sys em).
Lemma G.15 (Conse a ion o To al Ene gy).When Vhas no explici ime de-
pendence, he o al ene gy E:= Pi
1
2m| i|2+V(x1,...,xN)is cons an .
P oo .
˙
E=X
i
m i·˙
i+X
i∇xiV·˙
xi=X
i
Fi· i−X
i
F(po )
i· i= 0,
whe e F(po )
i:= −∇xiV, we se Fi=F(po )
i, and used ime in a iance.
(4) Local Rep esen a ion: Pa icle-Numbe Densi y
and Con inui y Equa ion
As he s a ing poin o local conse a ion laws, de ine he mic oscopic pa icle-
numbe densi y and mic oscopic pa icle lux by
ρmic(x, ) :=
N
X
i=1
δx−xi( ),jmic(x, ) :=
N
X
i=1
i( )δx−xi( ).(53)
Theo em G.16 (Con inui y Equa ion o Pa icle-Numbe Conse a ion).In he
dis ibu ional sense,
∂ ρmic(x, ) + ∇·jmic(x, ) = 0 (54)
holds iden ically.
401
P oo . Le φ∈C∞
c(R3)be a bi a y. F om De ini ion (53),
⟨∂ ρmic, φ⟩=∂ X
i
φ(xi( )) = X
i∇φ(xi( )) ·˙
xi( ) = ZR3∇φ(x)·jmic(x, )d3x.
By in eg a ion by pa s, ⟨∂ ρmic, φ⟩=−⟨∇·jmic, φ⟩. Hence (54) holds as a dis ibu-
ion.
Lemma G.17 (Coa se-G aining and Mapping o he Fluid Skele on (P epa a ion)).
Wi h a spa ial smoo hing ke nel Wℓ(x)≥0(RWℓ= 1,ℓa mesoscale), de ine he
con olu ions
n(x, ) := (ρmic ∗Wℓ)(x, ),u(x, ) := (jmic ∗Wℓ)(x, )
n(x, ).
Then Theo em G.16 isomo phically becomes ∂ n+∇·(nu) = 0 (no limi is equi ed
wi h ixed ℓ).
P oo . Con ol e (54) wi h Wℓand use he commu a ion o ∂ and ∇wi h con olu ion
and linea i y; he claim ollows immedia ely.
Rema k (C oss-Check wi h IFT)
The abo e n, uco espond o he classical limi o he hype bolic no maliza ion
o Jµin IFT, n:= p−JµJµand uµ:= Jµ/n, and he con inui y equa ion coincides
wi h he heo em (pa icle-numbe conse a ion) in UEE_0614. This co espondence
will be li ed o enso ial o m and used in he subsequen Sec. G.2.
(5) Summa y o he “Minimal P inciples” Es ablished
in This Sec ion
F om he abo e, he ollowing non-ci cula , minimal assump ions and conclusions
ha e been es ablished:
(M1) New onian mo ion (De ini ion G.12).
(M2) Ac ion– eac ion and cen al o ces (De ini ion G.12).
(M3) Homogenei y o ime and space and o a ional symme y (De ini ion G.12).
F om hese, (i) conse a ion o o al linea momen um, o al angula momen um,
and o al ene gy (Lemmas G.13,G.14,G.15), and (ii) local pa icle-numbe conse -
a ion (Theo em G.16) ha e been de i ed line-by-line. Up o his poin , he discus-
sion has no assumed he UEE o malism a all (UEE will be in oduced ine i ably
in G.6 om he minimal equi emen s o coa se-g aining =open sys ems15).
14See UEE_06 §2.1, De ini ions 2.1, 2.2 and Theo em 2.4. I is gi en in he o m ∇µ(nuµ) = 0.
15The p ocedu e ha makes UEE ine i able om ex e nal p inciples (CPTP, e lec ion posi i i y,
co a iance) is o ganized in he a ached oadmap G.6.
402
Execu i e Conclusion o This Sec ion (G.1):
1. F om he h ee minimal condi ions—New on’s law (M1), ac ion– eac ion
and cen al o ces (M2), and homogenei y o space ime wi h o a ional
symme y (M3)—we igo ously de i ed he global conse a ion laws o
he sys em (P,L, E) and he local pa icle-numbe conse a ion equa-
ion ∂ n+∇·(nu) = 0.
2. The (n, u)ob ained by coa se-g aining ag ee wi h he no maliza ion o
he in o ma ion cu en Jµin IFT (classical limi ) and se e as a non-
ci cula s a ing poin o he subsequen luid equa ions (G.2)—and
u he o he U(1), Yang–Mills, and g a i a ional ield equa ions
(G.3–G.5).a
aFo he de ini ion o Jµand he con inui y equa ion in IFT, see UEE_06 §2.1; o
he connec ion o g a i y (s ess–cu a u e equi alence), see Appendix D o UEE_05 and
UEE_06 §1.3/§2.2.
403
G.2 Coa se-G aining om Many-Body
o Fluid: Con inui y, Eule , and
Vo ici y
(1) Aim and Posi ion
In his sec ion, s a ing om he mic oscopic deg ees o eedom o he single- e mion
luid (IFT), we om i s p inciples de i e, ia coa se-g aining, he s anda d luid
equa ions (con inui y equa ion, Eule equa ion, o ici y equa ion), and embed, in
a o m consis en wi h conse a ion laws, he e ec s o dissipa ion in oduced in
he UEE-IFT o malism (GKLS ype) and o he ze o-a ea esonance ke nel R.
The amewo k is p esen ed along wo ou es: (A) a coa se-g ained de i a ion om
he New onian mo ion o a mic oscopic many-body sys em, and (B) a coo dina ed
de i a ion om he IFT conse a ion equa ions ∇µ(nuµ) = 0,∇µTµν = 0, and hei
equi alence is p o ed in a heo em gi en below.
(2) De ini ion o he Coa se-G aining Ope a o and
Field Va iables
Coa se-g aining is de ined using a local a e aging ke nel Wℓ(x)(posi i e, C∞,RWℓ=
1, e en unc ion, momen s o o de O(ℓ2)).
De ini ion G.18 (Empi ical Measu e and Coa se-G ained Fields).Conside a sys-
em o poin pa icles wi h mic oscopic iden i ie s α= 1, . . . , N (mass mα, posi ion
qα( ), eloci y α( )). Fo he empi ical measu e
µN( , x) :=
N
X
α=1
δ(x−qα( )),
de ine he coa se-g ained densi y, momen um densi y, and eloci y by
ρℓ( , x) := X
α
mαWℓ(x−qα( )),(55)
jℓ( , x) := X
α
mα α( )Wℓ(x−qα( )),(56)
uℓ:= jℓ/ρℓ.(57)
In he non ela i is ic app oxima ion, he ela ion o he na u al IFT a iables (n, uµ)
is ρℓ≃mn, and uℓcoincides wi h he spa ial eloci y in IFT.
Lemma G.19 (Con inui y Equa ion (Exac Fo m o Fini e ℓunde Coa se-G ain-
ing)).Assuming only he pa icle equa ions o mo ion ˙
qα= α, one has
∂ ρℓ+∇·(ρℓuℓ) = 0 exac ly. (58)
404
P oo . I Wℓhas no ime dependence, hen
∂ ρℓ=X
α
mα∂ Wℓ(x−qα) = −X
α
mα˙
qα·∇Wℓ(x−qα).
Also,
∇·(ρℓuℓ) = ∇· X
α
mα αWℓ(x−qα)!=X
α
mα α·∇Wℓ(x−qα).
Adding he wo exp essions yields ze o, gi ing (58). In IFT, he equi alen conse -
a ion equa ion ∇µ(nuµ) = 0 holds.
(3) De i a ion o he Eule Equa ion: New onian
Limi and IFT Limi
Le Fαbe he o ce ac ing on pa icle α. The ime e olu ion o he momen um
densi y is
∂ jℓ=X
α
mα˙
αWℓ−X
α
mα( α⊗ α)·∇Wℓ.
The i s e m on he igh -hand side equals he coa se-g ained o ce densi y ℓ:=
PαFαWℓ. The second e m can be ew i en in he dissipa i e o m o Reynolds
s ess.
De ini ion G.20 (Decomposi ion o he S ess Tenso ).
τℓ:= X
α
mα( α−uℓ)⊗( α−uℓ)Wℓ,Πℓ:= ρℓuℓ⊗uℓ+τℓ.
Unde he iso opic app oxima ion τℓ≈pℓI,pℓbecomes he coa se-g ained p essu e.
Theo em G.21 (Eule Equa ion (In iscid, wi h Ex e nal Fo ce)).When he ex e -
nal o ce densi y ℓis conse a i e, ℓ=−ρℓ∇Φℓ, one has
∂ (ρℓuℓ) + ∇·(ρℓuℓ⊗uℓ) + ∇pℓ=−ρℓ∇Φℓ.(59)
Fu he , in he gene aliza ion including iscosi y and damping (UEE-NS ex ension),
∂ uℓ+ (uℓ·∇)uℓ=−1
ρℓ∇pℓ−∇Φℓ+ν∆uℓ−γuℓ,(60)
holds (wi h ν≥0 he e ec i e iscosi y and γ≥0 he damping).
P oo . F om De ini ion G.20,
∂ jℓ+∇·Πℓ= ℓ.
Subs i u ing he iso opized Reynolds s ess τℓ≈pℓIand Πℓ=ρℓuℓ⊗uℓ+pℓI,
and supplemen ing (GKLS-o igin) e ec i e iscosi y νand damping γas co ec-
ion e ms, yield (59)–(60). The Na ie –S okes ex ension wi h −γuℓis in oduced
igo ously in he UEE appendix (English edi ion).
405
Re-de i a ion om IFT and O igin o he Po en ial.
F om IFT’s s ess–cu a u e equi alence Tµν = Λ2
∗Gµν and he ull o m o he
spino - luid s ess Tµν, in he non ela i is ic limi one ob ains
(ε+p)uµ∇µuν+ (δνµ+uνuµ)∇µp=(dissipa ion),
( he ime componen is he con inui y equa ion). He e pis de e mined ia IFT’s
ension scala σ, and Φℓis in e p e ed as an e ec i e po en ial induced om σ(see
he IFT main ex o p ecise de ini ions and a ia ional compu a ions).
(4) Vo ici y Equa ion and Ba oclinic Te m
De ine he o ici y ωℓ:= ∇×uℓ. Taking he cu l o (60) and using ∇×∇Φℓ=0,
we ha e
Lemma G.22 (Vo ici y T anspo Equa ion (UEE-NS Fo m)).
∂ ωℓ=∇×(uℓ×ωℓ) + ∇ρℓ×∇pℓ
ρ2
ℓ
+ν∆ωℓ−γωℓ.(61)
The second e m on he igh -hand side is he ba oclinic e m.
P oo . Use he ec o iden i ies ∇×[(u·∇)u] = (ω·∇)u−(u·∇)ω+ω(∇·u),
∇ × (ρ−1∇p) = ∇ρ−1× ∇p= (∇ρ× ∇p)/ρ2, and ea ange by he con inui y
equa ion (58) (accoun ing o comp essibili y). No e ha ∇×(−γuℓ) = −γωℓand
∇×(ν∆uℓ) = ν∆ωℓ.
IFT Tension σand Vo ici y Sou ce.
When he p essu e pℓis a local unc ion o σ,pℓ=p(σ, ρℓ), one has ∇pℓ=
∂p
∂σ ∇σ+∂p
∂ρ ∇ρℓ, hence
∇ρℓ×∇pℓ
ρ2
ℓ
=1
ρ2
ℓ
∂p
∂σ (∇ρℓ×∇σ),
i.e., spa ial misalignmen o ∇σ(no pa allel o he densi y g adien ) p oduces
a sou ce o o ici y. This mechanism ag ees wi h he de i a ion o he o ici y
equa ion in IFT.
(5) Equi alence o he Two Rou es (Many-Body Coa se-
G aining ⇔IFT)
Theo em G.23 (Equi alence Theo em).(i) The equa ions (58),(60),(61)ob ained
om a many-body New onian sys em wi h he coa se-g aining and iso opiza ion o
De ini ion G.18 (assump ions on Wℓ, bounded ℓ), and (ii) he co esponding equa-
ions ob ained om IFT conse a ion laws ∇µ(nuµ) = 0,∇µTµν = 0 and he non-
ela i is ic limi o he IFT s ess, gi e he same dynamics when ν, γ a e chosen
as e ec i e coe icien s consis en wi h UEE dissipa ion (GKLS) and he ze o-a ea
na u e o R.
406
P oo . IFT’s space ime conse a ion ∇µTµν = 0 yields, in he non ela i is ic limi ,
he s uc u e o (59) (p essu e, po en ial, dissipa ion). The GKLS- ype dissipa ion
o UEE sa is ies CPTP and OS e lec ion posi i i y, so i is possible o in oduce
iscosi y and damping (ν, γ) compa ible wi h he de ini ions o conse ed quan i ies
(cha ge, ene gy–momen um). The ke nel Rhas ze o-a ea (measu e-ze o) suppo
and ze o ace, and does no des oy he s uc u e o conse a ion equa ions ( o
de ailed ope a o ela i e-boundedness, see heo ems o UEE). The e o e, by ex-
p essing he e ec i e coe icien s appea ing in (i) by he dissipa i e pa ame e s in
(ii), he equa ion o ms coincide.
(6) Conclusion o This Sec ion (Key Poin s)
Conclusion (G.2):
1. By coa se-g aining a many-body New onian sys em (De ini ion G.18),
he con inui y equa ion (58) is de i ed exac ly om densi y and momen-
um.
2. The Eule equa ion (59) ollows om iso opic s ess and conse a i e
o ces, and in oducing he e ec i e iscosi y νand damping γbased on
UEE dissipa ion yields (60).
3. The o ici y equa ion (61) con ains he ba oclinic e m (∇ρ×∇p)/ρ2,
and ∇σ om IFT’s ension σac s as a o ici y sou ce (non-conse a i e
o ces do no gene a e i ; Φis i o a ional).
4. The non ela i is ic limi om IFT conse a ion laws and
s ess–cu a u e equi alence is equi alen o he de i a ion om
many-body coa se-g aining (Theo em G.23), and UEE’s R(ze o-a ea)
and GKLS dissipa ion con ibu e o he luid equa ions consis en ly
wi h conse a ion and e lec ion posi i i y.
407
G.3 Eme gence o “Fields” I: U(1) Elec-
omagne ic Field (Local Phase
In a iance)
(1) Aim and Posi ion
In his sec ion, s a ing om he undamen al a iables o he single- e mion luid
ob ained in G.1 (New onian mo ion and conse a ion laws o elemen a y pa icles)
and G.2 (coa se-g aining om many-body o luid),
Jµ:= ¯
ΨγµΨ, n := p−JµJµ, uµ:= Jµ
n
(wi h uµuµ=−1,∇µ(nuµ) = 0), we igo ously de i e, line by line and om bo h
he ac ion p inciple and he luid ep esen a ion, ha he U(1) gauge ield Aµand
Maxwell’s equa ions a ise ine i ably om local phase in a iance. The equi alence
o he h ee o ms (ac ion, ope a o , and ield equa ions) o he Uni ied E olu ion
Equa ion (UEE) adop ed he e, and he U(1) de i a ion wi hin he single- e mion
amewo k (IFT), a e sys ema ically p o ided in p e ious wo ks. These cons i u e
he basis o each p oposi ion in his sec ion (equi alence o he h ee o ms o UEE
and a ia ional de i a ion o he gauge ield, as well as he ine i abili y o U(1) om
local phase in a iance in IFT).
(2) Ine i abili y o he Connec ion om Local Phase
In a iance
De ini ion G.24 (Local U(1) T ans o ma ion and Co a iance o he Densi y Op-
e a o ).Fo a smoo h eal unc ion θ(x),
Ψ(x)7→ Ψ′(x) := eiθ(x)Ψ(x),¯
Ψ(x)7→ ¯
Ψ′(x) := ¯
Ψ(x)e−iθ(x).
The necessa y and su icien condi ion o he UEE ime e olu ion ˙ρ=−i[D, ρ] +
L∆[ρ] o be co a ian unde his ans o ma ion as ρ7→ eiθρ e−iθ is ha he di e en ial-
ope a o pa be eplaced by ∂µ7→ Dµ:= ∂µ+iAµ, wi h he in oduc ion o a
connec ion Aµ ans o ming as Aµ7→ Aµ−∂µθ. Mo eo e , each jump ope a o Vj
o he Lindblad pa L∆mus be a gauge scala .
Lemma G.25 (Uniqueness o Minimal Coupling).Unde De . G.24, a e e sible
gene a o sa is ying co a iance unde U(1) ans o ma ions is es ic ed o he o m
D=iγµ(∂µ+iAµ) + ···
(whe e “···” deno es gauge-scala couplings). Adding a mass e m 1
2m2
γAµAµb eaks
gauge in a iance, hence mγ= 0.
P oo . Fo −i[D, ρ] o de ine an isomo phism unde ρ7→ eiθρe−iθ, he co ec ion
D7→ eiθDe−iθ +ieiθ∂ (e−iθ)mus be canceled by he con ibu ion om spa ial
de i a i es o θ. Since ∂µdoes no commu e wi h θ, he in oduc ion o Dµis
necessa y and su icien . The mass e m is no in a ian unde Aµ→Aµ−∂µθ, and
is he e o e o bidden.
408
