Generalised complex geometry in thermodynamical fluctuation theory

En opy 2015,17, 5888-5902; doi:10.3390/e17085888 OPEN ACCESS en opy ISSN 1099-4300 www.mdpi.com/jou nal/en opy A icle Gene alised Complex Geome y in The modynamical Fluc ua ion Theo y P. Fe nández de Có doba and J. M. Isid o * Ins i u o Uni e si a io de Ma emá ica Pu a y Aplicada, Uni e sidad Poli écnica de Valencia, Valencia 46022, Spain; E-Mail: [email p o ec ed].es *Au ho o whom co espondence should be add essed; E-Mail: [email p o ec ed].es. Academic Edi o : Geo ge Ruppeine Recei ed: 29 May 2015 / Accep ed: 19 Augus 2015 / Published: 20 Augus 2015 Abs ac : We p esen a b ie o e iew o some key concep s in he heo y o gene alized complex mani olds. This new geome y in e pola es, so o speak, be ween symplec ic geome y and complex geome y. As such i p o ides an ideal amewo k o analyze he modynamical luc ua ion heo y in he p esence o g a i a ional ields. To illus a e he use ulness o gene alized complex geome y, we examine a simpli ied e sion o he Un uh e ec : he he malising e ec o g a i a ional ields on he Sch oedinge wa e unc ion. Keywo ds: di e en ial-geome ic echniques; luc ua ion heo y; Un uh e ec 1. In oduc ion The heo y o he modynamical luc ua ions p o ides a solid link be ween mac oscopic and mic oscopic physics. Classical luc ua ion heo y [1] o en sheds ligh on coun e in ui i e quan um-mechanical phenomena, hus helping o b idge he gap be ween he classical wo ld and he quan um wo ld. Fo example, Heisenbe g’s unce ain y p inciple can be nicely illus a ed eso ing o he heo y o Gaussian luc ua ions a ound he mal equilib ium [2]. On he o he hand, he heo y o he modynamical luc ua ions can be ecas using he geome ic language o di e en ial mani olds [3–9]. This eexp ession o a physical discipline in mo e abs ac ma hema ical language goes a long way beyond a me e ew i ing o he concep s in ol ed. I ende s he heo y mo e e sa ile, enla ging i s scope. Mo eo e , since he ad en o Eins ein’s gene al ela i i y En opy 2015,17 5889 a cen u y ago, (pseudo) Riemannian geome y belongs o he echnical skills ha any physicis has o mas e (a leas a a wo king le el). This places (pseudo) Riemannian geome y a a an age poin . In he opposi e di ec ion (i.e., he modynamics as applied o geome y) one should men ion a leas wo de elopmen s. The i s one is a whole body o knowledge on he he modynamics o black holes [10,11]. Mo e ecen ly, he eexp ession o Eins ein’s ela i i y as a he modynamics [12,13] has had a - eaching consequences o ou unde s anding o space ime. He e we would like o epo on ano he ecen de elopmen in geome y wi h implica ions on he he modynamics o luc ua ions: he heo y o gene alized complex mani olds [14,15]. In ying o unde s and he ho ny ela ionship be ween g a i y and he quan um [16–19] i has been a gued ha g a i y ac s dissipa i ely on quan um sys ems [20]. Speci ically, in he p esence o a g a i a ional ield, he mal luc ua ions become indis inguishable om quan um luc ua ions [21–23]. This aises he undamen al ques ion: How is one o ea he mal and quan um luc ua ions on he same oo ing? Is i al oge he possible? We will see he e ha gene alized complex mani olds p o ide one iable answe o his ques ion, one ha appea s no o ha e been explo ed ye in he geome ical app oach o he modynamics. 2. Geome y and Fluc ua ions 2.1. Riemannian Geome y As a e y elemen a y example, conside a he modynamical sys em in an equilib ium s a e desc ibed by he ollowing a iables: empe a u e T, p essu e Pand olume V. In he Gaussian app oxima ion, choosing Tand Vas independen a iables, he p obabili y Wo a luc ua ion ∆T,∆Va ound equilib ium is gi en by [24] W=W0exp −CV 2kBT2∆T2+1 2kBT∂P ∂V T ∆V2(1) The he modynamic inequali ies CV>0and (∂P/∂V )T<0ensu e ha he a gumen o he abo e exponen ial is nega i e de ini e. This sugges s conside ing he ollowing (posi i e de ini e) Riemannian me ic on he 2-dimensional mani old coo dina ised by T, V : ds2:= CV 2kBT2dT2−1 2kBT∂P ∂V T dV2=: gijdxidxj(2) The me ic coe icien s gij a e o cou se (T, V )-dependen unc ions. This Riemannian s uc u e encodes all he ele an in o ma ion. Fo example, he a e age alue h (T, V )io an a bi a y unc ion = (T, V ), h (T, V )i=Z−1Z (T, V ) exp −gT T T2−gV V V2√gdTdV(3) whe e Z:= R√gexp (−gT T T2−gV V V2) dTdV, na u ally in ol es he me ic. The ole o Riemannian geome y in luc ua ion heo y is well known and has been e iewed a leng h in [8]. En opy 2015,17 5890 2.2. Symplec ic Geome y As ou s a ing poin he e we will conside a ce ain he modynamical sys em in equilib ium, in o de o a i e a a co esponding symplec ic s uc u e. Again in he Gaussian app oxima ion, he p obabili y Wo a luc ua ion ∆P,∆V,∆T,∆Sis gi en by [24] W=W0exp −1 2kBT(−∆P∆V+ ∆T∆S)(4) Assume an equa ion o s a e F(P, V, T)=0 ha can be sol ed o he empe a u e o ob ain T=g(P, V ). Fo simplici y le us conside an ideal gas, PV =S0T: W=W0exp −1 2kB−S0 ∆P∆V PV +∆T∆S T (5) I is con enien o de ine he dimensionless a iables p1:= −ln P P0, q1:= ln V V0, p2:= ln T T0, q2:= S S0 (6) whe e P0,V0and T0a e e e ence alues. Then Equa ion (5) becomes W=W0exp −S0 2kB (∆p1∆q1+ ∆p2∆q2)(7) We can ega d q1and q2as coo dina es on a he modynamical con igu a ion space S, wi h p1and p2as hei conjuga e momen a. Thus he q1,p1,q2,p2a e Da boux coo dina es o he symplec ic o m ω= dp1∧dq1+ dp2∧dq2(8) In his way we iden i y ∆p1∆q1+ ∆p2∆q2in Equa ion (7) as he symplec ic a ea o a 2-dimensional su ace Finduced by he luc ua ion: ∆p1∆q1+ ∆p2∆q2=ZF (dp1∧dq1+ dp2∧dq2)(9) Finally subs i u ing Equa ion (9) in o Equa ion (7) we ind W=W0exp −S0 2kBZF ω(10) i.e., he p obabili y o his he mal luc ua ion is p opo ional o he exponen ial o he symplec ic a ea o he luc ua ion su ace F. The impo ance o symplec ic s uc u es in classical mechanics is widely ecognized and need ha dly be ecalled [25]. In ac no jus Riemannian geome y, bu also symplec ic geome y, pe ains o he ealm o he mal luc ua ions: he i s law o he modynamics endows he he modynamic phase space wi h a con ac s uc u e, which includes symplec ic geome y as a sub-case [3,4,6,7]. A eal 2n-dimensional mani old Mis symplec ic i he e exis s a closed, non-degene a e, ank 2 an isymme ic enso ield ωij de ined e e ywhe e on M. Le xibe local coo dina es a ound x∈M, so ω=1 2ωijdxi∧dxjwi h ωji =−ωij. Since he ma ix ωij is nonsingula , an in e se πjk exis s such ha En opy 2015,17 5891 ωijπjk =δk i. The Poisson b acke s o wo unc ions , g a e de ined as { , g}:= πjk∂j ∂kg, and he in eg abili y condi ion dω= 0 u ns ou o be equi alen o he Jacobi iden i y o hese Poisson b acke s. In his way he ollowing symplec ic analogue o Equa ion (3) allows one o compu e he a e age alue h io he unc ion on M: h i=Z−1ZM exp (−ω)(11) Abo e, he exponen ial e−ωis de ined by Taylo expansion, powe s being aken wi h espec o he wedge p oduc . Then he 2n-dimensionali y o he symplec ic mani old picks ou jus one di e en ial o m ha can be in eg a ed agains M, namely he 2n- o m (−1)nωn/n!; all o he e ms in he Taylo expansion gi e a anishing con ibu ion when in eg a ed. The ac o (−1)n/n!has been included in he no maliza ion Z. As had o be he case, his a e age in ol es he da a conce ning he symplec ic s uc u e on M. One can also ega d a symplec ic s uc u e as p o iding an isomo phism om he angen space TxMin o he co angen space T∗ xMa each x∈M. Speci ically, he angen ec o X=Xi∂iis mapped in o he 1- o m ω(X) = ξ=ξidxi, wi h ξi=ωij Xj. This iewpoin mo i a es he ollowing de ini ion (equi alen o he abo e, bu mo e use ul o la e applica ions): a symplec ic s uc u e o e a2n-dimensional mani old Mis an isomo phism ωxbe ween he angen and he co angen ibe s o e each poin x∈M, ωx:TxM−→ T∗ xM(12) such ha , unde he ope a ion o aking he linea dual (deno ed by an as e isk), ω∗ x=−ωx,∀x∈M(13) Mo eo e , he in eg abili y condi ion dω= 0 mus be sa is ied. 2.3. Complex Geome y and Kähle Geome y In o mally one could say ha he imagina y uni is he hallma k o quan um mechanics. Tha i = √−1pe ains o he quan um wo ld has been e y in e es ingly a gued ecen ly in [26,27]. Mo e s anda d a gumen s ha e been known o long; such a e he hea equa ion in imagina y ime i , o he ac ha quan um commu a o s [·,·] o mally equal √−1 imes classical Poisson b acke s {·,·}. He e we will b ie ly ecall he ole played by complex s uc u es in he heo y o cohe en s a es [28,29]. Le Mbe a eal 2n-dimensional phase space endowed wi h he symplec ic o m ω. Fo simplici y le us also assume ha Madmi s a holomo phic a las compa ible wi h he symplec ic s uc u e ( his compa ibili y condi ion is called he Kähle p ope y). In plain wo ds, he eal and imagina y pa s o he holomo phic coo dina es zja e Da boux coo dina es o ω(he e assumed dimensionless o simplici y): zj=1 √2qj+ ipj, j = 1, . . . , n (14) Upon quan isa ion, he Da boux coo dina es qjand pjbecome ope a o s Qjand Pjon Hilbe space sa is ying he Heisenbe g algeb a [Qj, Pk] = iδj k. C ea ion and annihila ion ope a o s a e de ined in he s anda d ashion: A† j:= (Qj−iPj)/√2,Aj:= (Qj+ iPj)/√2, and quan um exci a ions a e measu ed En opy 2015,17 5892 wi h espec o a acuum s a e |0isa is ying Aj|0i= 0, o all j= 1, . . . , n. Cohe en s a es |zjia e eigen ec o s o Aj, he eigen alues being he holomo phic coo dina es (14): Aj|zji=zj|zji, j = 1, . . . , n (15) (No sum o e jimplied). In o de o illus a e ou poin le us conside a 1-dimensional ha monic oscilla o . The expec a ion alue o he Hamil onian ope a o H=A†A+ 1/2in he s a e |ziequals hz|H|zi=|z|2+ 1/2. Since he ene gy luc ua ion in he s a e |ziequals (∆H)z=|z|, z ∈C(16) he ela i e luc ua ion goes, o la ge enough |z|, like (∆H)z hz|H|zi≃1 |z|,|z| → ∞ (17) Bu 1/|z|is he in e se o he squa e oo o he Kähle po en ial K(z, ¯z) := |z|2 o he Euclidean me ic on he complex plane C. This simple example illus a es he impo an ole played by complex mani olds in he quan um heo y. E e y complex mani old Madmi s a (posi i e de ini e) He mi ian me ic hijd¯zidzj ha is compa ible wi h he complex s uc u e [30]. Then an analogue o Equa ions (3) and (11) gi es us he a e age alue h io a unc ion on M: h i=Z−1ZM exp −hij ¯zizj√h n Y k=1 d¯zk∧dzk(18) The no maliza ion Zincludes all ac o s o i = √−1coming om he olume elemen , and h:= |de hij|. As had o be he case, his a e age in ol es he da a conce ning he complex s uc u e on M. Fo mally, a complex s uc u e Jo e a eal 2n-dimensional mani old Mis an endomo phism o he angen ib e o e each poin x∈M Jx:TxM−→ TxM(19) sa is ying J2 x=−1,∀x∈M(20) as well as he in eg abili y condi ion ha he Nijenhuis enso N anish iden ically. (We will no w i e down he Nijenhuis enso explici ly; see e e ence [30] o de ails). Roughly speaking, Equa ion (20) exp esses he exis ence o he imagina y uni i = √−1locally a ound he poin x∈M. The in eg abili y condi ion N= 0 ensu es ha he complex coo dina es hus cons uc ed locally uly ans o m holomo phically ac oss di e en coo dina e pa ches on he mani old M. (The Kähle p ope y assumed in Equa ion (14) abo e is an addi ional hypo hesis, ha an a bi a y complex mani old may, bu need no , sa is y in gene al). En opy 2015,17 5893 2.4. Gene alized Complex Geome y Ou o iginal mo i a ion was he s a emen [21–23] ha , in he p esence o a g a i a ional ield, quan um luc ua ions become indis inguishable om he mal luc ua ions. We ha e a gued ha he mal luc ua ions a e associa ed wi h symplec ic s uc u es, while quan um luc ua ions come along wi h complex s uc u es. How, hen, is one o ea he mal and quan um luc ua ions on he same oo ing? This is i ially achie ed by hose phase spaces M ha quali y as Kähle mani olds. Howe e , he Kähle condi ion is e y es ic i e: no only does Mha e o be simul aneously complex and symplec ic; hese wo independen s uc u es also ha e o be compa ible. In e e ences. [3,4] he geome y o he he modynamic phase space (including luc ua ions) esul s in a pa a-Sasakian mani old, which is he con ac -geome y equi alen o a Kähle mani old in symplec ic geome y. This means ha i one es ic s o a p ope e en-dimensional subspace, he geome y is indeed ha o a Kähle mani old. This geome y achie es he goal o ea ing he mal and quan um luc ua ions on he same oo ing. Gene alized complex s uc u es (GCS) also achie e he goal o p o iding a uni ied amewo k o he mal and quan um luc ua ions. The ollowing is a b ie summa y o GCS ex ac ed om [14], duly ailo ed o mee ou needs. Fo simplici y we p e e o wo k locally a ound a poin x∈M. Global issues can be aken ca e o by he co esponding in eg abili y condi ions, o be men ioned along he way whene e necessa y. Fo ou pu poses he 2n-dimensional mani old Mis assumed o be a phase space, ha is, M=T∗S, o a ce ain n-dimensional con igu a ion space S. Ra he han conside ing he ib es TxMo T∗ xMsepa a ely, in gene alized complex geome y one conside s hei di ec sum: o e each poin x∈Mone e ec s he ib e TxM⊕T∗ xM. The o al space o he bundle so cons uc ed is 6n-dimensional: 2ndimensions o he base M,4ndimensions o he ib e. An inne p oduc is de ined on he ib e TxM⊕T∗ xM: hX+ξ, Y +ηi:= 1 2(ξ(Y) + η(X)) (21) Abo e, X, Y ∈TxMa e angen ec o s, while ξ, η ∈T∗ xMa e 1- o ms, all e alua ed a x∈M. I u ns ou ha his inne p oduc is pseudo-Riemann wi h signa u e (2n, 2n). Hence he Lie g oup SO(2n, 2n) ac s on TxM⊕T∗ xMby isome ies. I is con enien o block-decompose he Lie algeb a so(2n, 2n)as ollows: A β B−A∗!(22) The diagonal blocks Aand A∗a e endomo phisms o hei espec i e (sub) ibe s, A∈End(TxM)and A∗∈End(T∗ xM), while he o diagonal blocks Band βconnec hese wo (sub) ibe s as pe B:TxM−→ T∗ xM, β :T∗ xM−→ TxM(23) Mo eo e , upon aking he dual we ha e B∗=−B,β∗=−β. This an isymme y allows us o ega d he block Bas a 2- o m in Λ2T∗ xMi we se B(X) = iXB(24) Fo illus a i e pu poses le us exp ess Equa ion (24) in local coo dina es xia ound a poin x∈M, so Bbecomes he ma ix Bij. Gi en he ec o X=Xj∂j∈TxM, he objec iXBis de ined o be he En opy 2015,17 5894 co ec o whose componen s a e BijXj∈T∗ xM. We see ha his is exac ly he way a symplec ic o m ω beha es. Since ωcan be ega ded as an elemen o Λ2T∗ xM, so can B. (Con a y o ω, howe e , Bneed nei he be closed no non-degene a e). The pa icula isome ies o he ib e TxM⊕T∗ xMob ained by se ing A= 0 = βin Equa ion (22) and exponen ia ing, exp 0 0 B0!= 10 B1!(25) a e he pseudo-o hogonal ans o ma ions X+ξ−→ X+ξ+iXB(26) The isome ies (26), called B- ans o ma ions, will play an impo an ole. A gene alized complex s uc u e o e M, deno ed J, is an endomo phism o he ib e o e each x∈M, Jx:TxM⊕T∗ xM−→ TxM⊕T∗ xM(27) such ha he ollowing wo condi ions hold. Fi s , J2 x=−1,∀x∈M(28) Second, J∗ x=−Jx,∀x∈M(29) The abo e wo condi ions a e o mula ed locally a ound any x∈M; as usual hey need no be compa ible wi h changes o coo dina e cha s on M. The Cou an in eg abili y condi ion, whose alidi y we will hence o h assume wi hou s a ing i s con en s explici ly, ensu es his compa ibili y; see [14,15] o de ails. Compa ing now Equa ions (29) and (13), we a e led o he pa icula case when Ja x∈Mis gi en by Jωx= 0−ω−1 x ωx0!(30) whe e ωis a symplec ic o m. One says ha his Jωde ines a GCS o symplec ic ype. Simila ly, he compa ison o Equa ions (28) and (20) sugges s he pa icula case o a GCS gi en by JJx= −Jx0 0J∗ x!(31) whe e Jis a complex s uc u e. We say ha he abo e JJde ines a GCS o complex ype. Fu he mo e, GCS succeed a in e pola ing be ween he abo e opposi e ypes, he symplec ic ype and he complex ype; le us explain his mo e ca e ully. A poin x∈Mis said o be egula i i possesses a neighbo hood Nxon which he e exis s a Poisson s uc u e ω−1wi h cons an ank. In a neighbo hood Nxo any egula poin x∈Mone can de ine a di eomo phism and a B- ans o ma ion, he combined ac ion o which maps Nxin o he p oduc Cx×Rx⊂Ck×R2n−2k. He e Cxis an open se wi hin he s anda d complex mani old Ck, and Rxis an open se wi hin he s anda d symplec ic mani old R2n−2k. En opy 2015,17 5895 The nonnega i e in ege kis called he ype o he GCS J, he limi ing cases o Equa ions (30) and (31) espec i ely co esponding o k= 0 and k=n. As desc ibed in [14,15], he ype kneed no be cons an ac oss M: i may a y om one poin o ano he in M. In plain wo ds, any gene alized complex mani old ac o izes locally as he p oduc o a complex mani old imes a symplec ic mani old. Finally assume ha Mis a linea space. Then any gene alized complex s uc u e o ype k= 0 is he B- ans o m o a symplec ic s uc u e. This means ha any gene alized complex s uc u e o ype k= 0 can be w i en as e−BJωeB= −ω−1B−ω−1 ω+Bω−1B Bω−1!(32) o a ce ain 2- o m B; use has been made o Equa ions (25) and (30). Simila ly any gene alized complex s uc u e o ype k=no e a linea mani old Mis he B- ans o m o a complex s uc u e, e−BJJeB= −J0 BJ +J∗B J∗!(33) a e using Equa ions (25) and (31). When Mis an a bi a y smoo h mani old, no necessa ily a linea space, s a emen s (32) and (33) emain basically ue, wi h some mino modi ica ions equi ed; see e e ences [14,15] o de ails. The consequences o he abo e become immedia ely appa en . Le us o simplici y assume ha he ype kis cons an ac oss M. Then any GCS wi h an ex emal alue o k,i.e., ei he k= 0 o k=n, can always be educed o he co esponding canonical o m (30) o (31) by means o a B- ans o ma ion. Thus k= 0 co esponds o a he mal desc ip ion o phenomena, while k=nco esponds o a quan um desc ip ion o phenomena, no in e pola ion exis ing be ween he wo desc ip ions. Nonex emal alues o he ype, i.e., such ha 06=k6=n, con ain bo h he mal and quan um desc ip ions simul aneously. A e age alues h io unc ions on gene alized complex mani olds a e de ined by an ob ious modi ica ion o he p oduc o he igh -hand sides o Equa ions (11) and (18). 3. When “Quan um” Becomes “The mal” Any g a i a ional ield is locally equi alen o an accele a ed ame. In an accele a ed ame, quan um becomes he mal; his is basically he con en o he Un uh e ec [31] (in an admi edly lax o mula ion ha is howe e p ecise enough o ou pu poses). Wi hou using he ull appa a us o ela i is ic quan um ield heo y, le us see how quan um can become he mal in he simpli ied se up o he quan um mechanics o a non ela i is ic pa icle. This unde s ood, we will analyse he ole played by he GCS on phase space unde he passage om an ine ial ame o an accele a ed ame. We will conclude ha he ans o ma ion law o he Sch oedinge wa e unc ion unde he passage o a nonine ial ame (as in he Un uh e ec ) is go e ned by a B- ans o ma ion o he GCS on phase space. A ema k is in o de . The g a i a ional ield conside ed he e mus be weak in o de o ule ou e ec s such as, e.g., ela i is ic speeds, o he likely b eakdown o s anda d quan um mechanics in he p esence o e y s ong g a i a ional ields [20]. Such phenomena lie beyond ou scope. En opy 2015,17 5896 3.1. Inclusion o a G a i a ional Field In la Euclidean space R3, le Kdeno e an ine ial ame wi h o igin Oand axes Ox,Oy and Oz. Le K0deno e a uni o mly accele a ed ame, wi h o igin O0and axes O0x0,O0y0and O0z0 espec i ely pa allel o Ox,Oy and Oz. Fo simplici y we will assume ha , a = 0, he wo o igins Oand O0 coincide, hei ela i e eloci y also anishing a = 0. Le he accele a ion ~α o K0wi h espec o Kbe (α, 0,0), wi h αa cons an . Coo dina es (x, y, z)wi h espec o Ka e ela ed o coo dina es (x0, y0, z0)wi h espec o K0as pe x=x0+1 2α 2, y =y0, z =z0, = 0(34) We conside a poin pa icle o mass m ixed o he o igin O0, hus a es wi h espec o K0. I Hdeno es he Hamil onian o he pa icle as seen om he ine ial ame K, hen he Hamil onian H0in K0 eads H0=H−pxα +m 2α2 2(35) wi h he momen a pxand p0 x ela ed as pe p0 x=px−mα . In he ine ial ame Kwe ha e a Sch oedinge equa ion i¯h∂ψ/∂ =Hψ. Ou aim is o de i e a ans o ma ion law o he wa e unc ion ψsuch ha , in he accele a ed ame K0, he Sch oedinge equa ion will ead i¯h∂ψ0/∂ =H0ψ0. Fo his pu pose le us make he Ansa z ψ0=ψexp [ ( )] (36) ( )being an unde e mined unc ion o he ime a iable. In his way we a i e a he ollowing di e en ial equa ion o he unknown unc ion : i¯hd d =−pxα +1 2mα2 2(37) D opping an i ele an in eg a ion cons an and subs i u ing he esul in o Equa ion (36) leads o ψ0= exp −i ¯h1 6mα2 3−1 2pxα 2ψ(38) Clasically, he pa icle is a es in he ame K0, so p0 x= 0 implies px=mα . Quan um-mechanically we can only s a e ha he cen e o mass emains a es a x0= 0, he wa epacke sp eading a ound his a e age posi ion. Wi h his unde s anding we can also se hpxi=px=mα in Equa ion (38). We conclude ha , aking he wa e unc ion in he accele a ed ame o be ψ0= exp i ¯h 1 3mα2 3ψ(39) ensu es he o m in a iance o he Sch oedinge equa ion unde he ans o ma ion om an ine ial ame o an accele a ed ame. Fo ime lapses ha a e sho enough, and/o o accele a ions ha a e weak enough, he speeds a ained will ne e become ela i is ic. Wi hin his limi ed ange, New onian mechanics (and i s quan um coun e pa , he Sch oedinge equa ion) can be us ed.