Equilibrium and nonequilibrium many-body perturbation theory: a unified framework based on the Martin-Schwinger hierarchy

This is an elec onic ep in o he o iginal a icle. This ep in may di e om he o iginal in pagina ion and ypog aphic de ail. Au ho (s): Ti le: Yea : Ve sion: Please ci e he o iginal e sion: All ma e ial supplied ia JYX is p o ec ed by copy igh and o he in ellec ual p ope y igh s, and duplica ion o sale o all o pa o any o he eposi o y collec ions is no pe mi ed, excep ha ma e ial may be duplica ed by you o you esea ch use o educa ional pu poses in elec onic o p in o m. You mus ob ain pe mission o any o he use. Elec onic o p in copies may no be o e ed, whe he o sale o o he wise o anyone who is no an au ho ised use . Equilib ium and nonequilib ium many-body pe u ba ion heo y: a uni ied amewo k based on he Ma in-Schwinge hie a chy an Leeuwen, Robe ; S e anucci, Gianluca an Leeuwen, R., & S e anucci, G. (2013). Equilib ium and nonequilib ium many-body pe u ba ion heo y: a uni ied amewo k based on he Ma in-Schwinge hie a chy. In P og ess in Nonequilib ium G een's Func ions V (PNGF V), 27–31 Augus 2012, Jy äskylä, Finland (A icle 012001). Ins i u e o Physics. Jou nal o Physics: Con e ence Se ies, 427. h ps://doi.o g/10.1088/1742-6596/427/1/012001 2013 This con en has been downloaded om IOPscience. Please sc oll down o see he ull ex . Download de ails: IP Add ess: 130.234.75.141 This con en was downloaded on 15/01/2016 a 11:16 Please no e ha e ms and condi ions apply. Equilib ium and nonequilib ium many-body pe u ba ion heo y: a uni ied amewo k based on he Ma in-Schwinge hie a chy View he able o con en s o his issue, o go o he jou nal homepage o mo e 2013 J. Phys.: Con . Se . 427 012001 (h p://iopscience.iop.o g/1742-6596/427/1/012001) Home Sea ch Collec ions Jou nals Abou Con ac us My IOPscience Equilib ium and nonequilib ium many-body pe u ba ion heo y: a uni ied amewo k based on he Ma in-Schwinge hie a chy Robe an Leeuwen1and Gianluca S e anucci2 1Depa men o Physics, Nanoscience Cen e , FIN 40014, Uni e si y o Jy ¨askyl¨a, Jy ¨askyl¨a, Finland 2Dipa imen o di Fisica, Uni e si `a di Roma To Ve ga a, Via della Rice ca Scien i ica 1, 00133 Rome, I aly Abs ac . We p esen a uni ied amewo k o equilib ium and nonequilib ium many-body pe u ba ion heo y. The mos gene al nonequilib ium many-body heo y alid o gene al ini ial s a es is based on a ime-con ou o iginally in oduced by Kons an ino and Pe el’. The a ious o he well-known o malisms o Keldysh, Ma suba a and he ze o- empe a u e o malism a e hen de i ed as special cases ha a ise unde di e en assump ions. We u he p esen a single simple p oo o Wick’s heo em ha is a he same ime alid in all hese la o s o many- body heo y. I a ises simply as a solu ion o he equa ions o he Ma in-Schwinge hie a chy o he nonin e ac ing many-pa icle G een’s unc ion wi h app op ia e bounda y condi ions. We u he discuss a gene alized Wick heo em o gene al ini ial s a es on he Keldysh con ou and de i e how he o malisms based on he Keldysh and Kons an ino -Pe el’-con ou s a e ela ed o he case o gene al ini ial s a es. 1. In oduc ion In many physical si ua ions we a e in e es ed in knowing he expec a ion alue o some obse able quan i y o a sys em in o ou o equilib ium. Fo quan um sys ems o many iden ical and in e ac ing pa icles a e y con enien ma hema ical objec o ex ac his in o ma ion is he G een’s unc ion. Le ˆρbe he densi y ma ix which desc ibes he sys em a ime, say, 0and ˆ H( ) be he Hamil onian o he sys em o imes > 0. The n-pa icle G een’s unc ion Gnis de ined acco ding o Gn(1 . . . n; 1′. . . n′) = 1 inT hˆρ T nˆ ψH(1) ... ˆ ψH(n)ˆ ψ† H(n′)... ˆ ψH(1′)oi.(1) In his o mula 1 = (x1, 1), 2 = (x2, 2), e c. a e collec i e indices o he posi ion-spin coo dina es x= , σ and ime , he symbol T deno es a ace o e he Fock space, Tis he ime-o de ing ope a o and ˆ ψH(j) = ˆ U( 0, j)ˆ ψ(j)ˆ U( j, 0) a e ield ope a o s in he Heisenbe g pic u e wi h espec o he Hamil onian ˆ H(hence ˆ Uis he e olu ion ope a o ). The quan um a e age o a n-body ope a o can be calcula ed om he equal- ime G een’s unc ion Gn. The di ec e alua ion o Gn om Eq. (1) is, in gene al, an impossible ask. The i s di icul y is b ough by he Hamil onian ˆ H=ˆ H0+ˆ Hin which is ypically he sum o a one-body ope a o ˆ H0and a m-body ope a o ˆ Hin wi h m≥2. Fo ˆ Hin 6= 0 he ield ope a o ˆ ψHin he P og ess in Nonequilib ium G een’s Func ions V (PNGF V) IOP Publishing Jou nal o Physics: Con e ence Se ies 427 (2013) 012001 doi:10.1088/1742-6596/427/1/012001 Published unde licence by IOP Publishing L d 1 Heisenbe g pic u e is a complica ed objec and mus be app oxima ed in some cle e way. The second di icul y consis s in aking he ace o e he Fock space wi h a densi y ma ix ˆρ. The densi y ma ix is a sel -adjoin , posi i e semi-de ini e ope a o wi h uni ace and, he e o e, i can be w i en as ˆρ=e−ˆ X/T [ ˆ X] whe e ˆ Xis a sel -adjoin ope a o . Fo ins ance o sys ems in equilib ium ˆ X=βˆ HMwi h β he in e se empe a u e and ˆ HM=ˆ H−µˆ N he g and-canonical Hamil onian. To make con ac wi h his equilib ium si ua ion we de ine ˆ HM=ˆ X/β so ha ˆρ=e−βˆ HM Z(2) wi h Z= T [e−βˆ HM]. In equilib ium Zis he pa i ion unc ion. In o de o speci y he ini ial p epa a ion o he sys em we can assign ei he ˆρo ˆ HMsince he e is a one- o-one co espondence be ween he wo. I we now sepa a e ˆ HM=ˆ HM 0+ˆ HM in in o he sum o a one-body ope a o ˆ HM 0 and a m-body ope a o ˆ HM in wi h m≥2 hen he ace in Eq. (1) can easily be wo ked ou o ˆ HM in = 0 whe eas we ha e o use sui able app oxima ion schemes o ˆ HM in 6= 0. Di e en Many-Body Pe u ba ion Theo ies (MBPT) ha e been pu o wa d o o e come hese di icul ies. The mos popula MBPT’s a e p obably he ze o- empe a u e ( eal- ime) G een’s Func ion Fo malism (GFF) and he ini e- empe a u e (imagina y- ime) Ma suba a GFF [1]. These wo o malisms a e limi ed o equilib ium si ua ions. Sys ems d i en ou o equilib ium by an ex e nal ield a e usually s udied wi hin he (adiaba ic eal- ime) Keldysh GFF [2, 3]. The Keldysh GFF, howe e , neglec s he e ec o ini ial co ela ions which a e ele an in he sho - ime dynamics o gene al quan um sys ems, such as in ansien dynamics in quan um anspo o in he s udy o a oms and molecules in ex e nal lase ields. The e exis wo al e na i e GFF’s o include ini ial co ela ions. The i s is based on he idea o Kons an ino and Pe el’ [4] and consis s in a aching he imagina y- ime Ma suba a ack o he o iginal Keldysh con ou , see Re s. [3, 5, 6]. The second GFF does ins ead accoun o ini ial co ela ions h ough ex a Feynman diag ams, he e alua ion o which equi es he knowledge o he educed n-pa icle densi y ma ices Γn(x1. . . xn;x′ 1. . . x′ n) = T [ˆρˆ ψ†(x′ 1)... ˆ ψ†(x′ n)ˆ ψ(xn)... ˆ ψ(x1)] (3) whe e he symbol T signi ies a ace o e he Fock space, see Re s. [7, 8, 9, 10, 11, 12]. These las wo o malisms a e bo h exac and hence equi alen . In all he a o emen ioned GFF’s he d essed (in e ac ing) Gnis expanded in powe s o he in e ac ion Hamil onian ( ˆ Hin and/o ˆ HM in ), leading o an expansion o Gnin e ms o he ba e (nonin e ac ing) G een’s unc ions G0,n. The appealing ea u e o any GFF is he possibili y o educing he G0,n o an (an i)symme ized p oduc o G0≡G0,1by means o Wick’s heo em [13]. E en hough he ma hema ical s uc u e o all GFF’s is iden ical, hese o malisms a e usually ea ed as independen p obably due o he ac ha he exis ing p oo s o Wick’s heo em a e e y much o malism-dependen . In his pape we show ha Wick’s heo em is he solu ion o a bounda y p oblem o he Ma in-Schwinge Hie a chy (MSH) [14] and ha di e en GFF’s co espond o di e en domains and pa ame e s o he MSH [15]. In his way we can easily explain he common ma hema ical s uc u e o e e y GFF and see how, e.g., he Keldysh GFF educes o he ze o- empe a u e GFF in equilib ium o he Kons an ino -Pe el’ GFF educes o he Keldysh GFF unde he adiaba ic assump ion. Ou e o mula ion also allows us o p o e a gene alized Wick’s heo em o in e ac ing densi y ma ices ˆρ. This na u ally leads o he diag amma ic expansion wi h ex a Feynman diag ams p e iously men ioned. The gene alized Wick expansion has a o m iden ical o ha o a Laplace expansion o pe manen s/de e minan s ( o bosons/ e mions). Consequen ly, he calcula ion o he a ious p e ac o s is bo h explici and g ea ly simpli ied. In his con ibu ion we only s a e P og ess in Nonequilib ium G een’s Func ions V (PNGF V) IOP Publishing Jou nal o Physics: Con e ence Se ies 427 (2013) 012001 doi:10.1088/1742-6596/427/1/012001 2 γ . . 0- 0+ 2- 1- .. Figu e 1. Con ou γo Eq. (4). The con ou consis s o a o wa d b anch going om 0 o ∞(on his b anch he poin s a e deno ed by z= −) and a backwa d b anch going om ∞ o 0(on his b anch he poin s a e deno ed by z= +). he gene alized Wick’s heo em and e e he eade o Re s. [12, 15] o he p oo . We will, howe e , discuss he equi alence be ween he GFF based on he gene alized Wick’s heo em and he Kons an ino -Pe el’ GFF. 2. Gene al o mula o he G een’s unc ion The n-pa icle G een’s unc ion in Eq. (1) can also be w i en as Gn(1 . . . n; 1′. . . n′) = 1 inT ˆρTe−iRγdz ˆ H(z)ˆ ψ(1) ... ˆ ψ(n)ˆ ψ†(n′)... ˆ ψ(1′).(4) Le us explain his o mula and discuss he equi alence wi h Eq. (1). In Eq. (4) he in eg al is o e he con ou γo Fig. 1 which goes om 0 o ∞and back o 0whe eas Tis he con ou o de ing ope a o which ea anges ope a o s wi h la e con ou a gumen s o he le . We deno e by z= ± he poin s on γlying on he lowe /uppe b anch a a dis ance om he o igin and de ine he ield ope a o s wi h a gumen s on he con ou as ˆ ψ(x, z) = ˆ ψ(x).(5) Mo e gene ally e e y ope a o ˆ O( ) wi h a eal- ime a gumen can be con e ed in o an ope a o ˆ O(z) wi h a con ou - ime a gumen acco ding o he ule ˆ O( +) = ˆ O( −) = ˆ O( ). In pa icula ˆ H( −) = ˆ H( +) = ˆ H( ). The eason o keep he con ou a gumen in Eq. (4) e en o ope a o s ha do no ha e an explici ime dependence (like he ield ope a o s) s ems om he need o speci ying hei posi ion along he con ou , hus ende ing unambiguous he ac ion o T. Once he ope a o s a e o de ed we can omi he ime a gumen s i he e is no ime dependence. Fo ins ance i 1< 2 hen Te−iRγdz ˆ H(z)ˆ ψ(x1, 1−)ˆ ψ†(x2, 2−)=±ˆ U( 0,∞)ˆ U(∞, 2)ˆ ψ†(x2)ˆ U( 2, 1)ˆ ψ(x1)ˆ U( 1, 0) =Tnˆ ψH(x1, 1)ˆ ψ† H(x2, 2)o,(6) whe e he ±sign in he i s equali y is o bosons/ e mions. One can e i y ha Eq. (6) is alid also o 1> 2. This example can easily be gene alized o many ield ope a o s. We conclude ha Eq. (4) is equi alen o Eq. (1) o con ou a gumen s on he uppe b anch o γ. The Gnin Eq. (4) is, howe e , mo e gene al since he con ou a gumen s can lie ei he on he uppe o lowe b anch o γ. Quan i ies like pho oemission cu en s, hype -pola izabili ies and mo e gene ally high-o de esponse p ope ies equi e he knowledge o his mo e gene al G een’s unc ion. The densi y ma ix in Eq. (4) can be inco po a ed in o he con ou o de ing ope a o i we ex end γas illus a ed in Fig. 2 and de ine he Hamil onian wi h imagina y- ime a gumen s as ˆ H( 0−iτ) = ˆ HM. Since e−βˆ HM=e−iR 0−iβ 0 ˆ H(z)=T(e−iR 0−iβ 0 ˆ H(z))(7) P og ess in Nonequilib ium G een’s Func ions V (PNGF V) IOP Publishing Jou nal o Physics: Con e ence Se ies 427 (2013) 012001 doi:10.1088/1742-6596/427/1/012001 3 γ . . . - iβ 0 0 H( ) ^ H( ) ^ H ^ M Figu e 2. Con ou γand Hamil onian along he con ou γ o ge he exac (Kons an ino -Pe el’) G een’s unc ion om Eq. (8). we ha e Gn(1 . . . n; 1′. . . n′) = 1 in T Te−iRγdz ˆ H(z)ˆ ψ(1) ... ˆ ψ(n)ˆ ψ†(n′)... ˆ ψ(1′) T Te−iRγdz ˆ H(z) (8) whe e in he denomina o we ook in o accoun ha T {e−iR 0+ 0− dz ˆ H(z)}=ˆ U( 0,∞)ˆ U(∞, 0) = 1. Equa ion (8) is, by cons uc ion, equi alen o Eq. (4). I gi es he exac G een’s unc ion p o ided ha he in eg al is done along he con ou γo Fig. 2 and p o ided ha he Hamil onian changes along he con ou as illus a ed in he same igu e. We now show ha he G een’s unc ion o e e y GFF can be w i en as in Eq. (8), he only di e ence being he shape o γand he Hamil onian along γ. We men ioned in he in oduc ion ha he calcula ion o he ace simpli ies i he densi y ma ix is o he o m ˆρ0=e−βˆ HM 0/Z0, wi h ˆ HM 0a one-body ope a o and Z0= T [e−βˆ HM 0]. I is possible o u n a ace wi h ˆρin o a ace wi h ˆρ0i he adiaba ic assump ion is ul illed. Acco ding o he adiaba ic assump ion one can gene a e he densi y ma ix ˆρwi h Hamil onian ˆ HM=ˆ HM 0+ˆ HM in s a ing om he densi y ma ix ˆρ0wi h Hamil onian ˆ HM 0and hen swi ching on ˆ HM in adiaba ically, i.e., ˆρ=e−βˆ HM Z=ˆ Uη( 0,−∞)e−βˆ HM 0 Z0 ˆ Uη(−∞, 0) = ˆ Uη( 0,−∞) ˆρ0ˆ Uη(−∞, 0),(9) whe e ˆ Uηis he eal- ime e olu ion ope a o wi h Hamil onian ˆ Hη( ) = ˆ HM 0+e−η| − 0|ˆ HM in ,(10) and ηis an in ini esimally small posi i e cons an . This Hamil onian is equal o ˆ HM 0when → −∞ and is equal o he ull in e ac ing ˆ HMwhen = 0. In gene al he alidi y o he adiaba ic assump ion should be checked case by case. Unde he adiaba ic assump ion we can ew i e Eq. (4) as (omi ing he a gumen s o Gn) Gn=1 inT ˆρ0ˆ Uη(−∞, 0)Te−iRγdz ˆ H(z)ˆ ψ(1) ... ˆ ψ(n)ˆ ψ†(n′)... ˆ ψ(1′)ˆ Uη( 0,−∞).(11) We now see ha i we cons uc he con ou γo Fig. 3 and le he Hamil onian change along he con ou as ˆ H( ±) =      ˆ Hη( ) = ˆ HM 0+e−η| − 0|ˆ HM in o < 0 ˆ H( ) = ˆ H0( ) + ˆ Hin ( ) o > 0 ˆ H(z∈γM) = ˆ HM 0=ˆ H0−µˆ N, P og ess in Nonequilib ium G een’s Func ions V (PNGF V) IOP Publishing Jou nal o Physics: Con e ence Se ies 427 (2013) 012001 doi:10.1088/1742-6596/427/1/012001 4 γ . . . - - iβ 0H( ) ^ H( ) ^ H ^ M 0 H ( ) ^ η H ( ) ^ η Figu e 3. Con ou γand Hamil onian along he con ou γ o ge he adiaba ic (Keldysh) G een’s unc ion om Eq. (8). hen Eq. (11) akes he same o m as Eq. (8). We will e e o his way o calcula ing Gnas he adiaba ic o mula. This is exac ly he o mula used by Keldysh in his o iginal pape [2]. The adiaba ic o mula is co ec only p o ided ha he adiaba ic assump ion is ul illed. We can de i e ye ano he exp ession o Gn o sys ems in equilib ium a ze o empe a u e. In equilib ium ˆ HM=ˆ H−µˆ Nand he e o e ˆ HM 0=ˆ H0−µˆ Nand ˆ HM in =ˆ Hin . Assuming ha ˆ H0and ˆ Hin commu e wi h ˆ N he e olu ion ope a o ˆ Uηin Eq. (9) can be calcula ed wi h Hamil onian ˆ Hη( ) = ˆ H0+e−η| − 0|ˆ Hin (12) since he addi ion o −µˆ Nco esponds o mul iplying ˆ Uηby a phase ac o . In Eq. (9) his phase ac o cancels ou since ˆ Uη(−∞, 0) = [ ˆ Uη( 0,−∞)]†. Fu he mo e, o any ini e con ou - imes in Gnwe can app oxima e he e olu ion ope a o ˆ Uin he ield ope a o s ˆ ψHwi h he e olu ion ope a o Uηsince we can always choose η≪1/| − 0|and hence ˆ Hη∼ˆ H. Thus Eq. (11) becomes Gn=1 inT ˆρ0Te−iRγdz ˆ H(z)ˆ ψ(1) ... ˆ ψ(n)ˆ ψ†(n′)... ˆ ψ(1′) (13) whe e γis a con ou ha goes om −∞ o ∞and back o −∞ and ˆ H( ±) = ˆ Hη( ) is he Hamil onian o Eq. (12). Nex we obse e ha he in e ac ing ˆρcan also be gene a ed s a ing om ˆρ0and hen p opaga ing backwa d in ime om ∞ o 0using he same e olu ion ope a o ˆ Uηsince ˆ Hη( 0− ) = ˆ Hη( 0+ ). In o he wo ds ˆρ=ˆ Uη( 0,∞) ˆρ0ˆ Uη(∞, 0).(14) Compa ing his equa ion wi h Eq. (9) we conclude ha ˆρ0=ˆ Uη(−∞,∞) ˆρ0ˆ Uη(∞,−∞).(15) I he g ound s a e |Φ0io ˆ H0−µˆ Nis nondegene a e hen he ze o- empe a u e ˆρ0=|Φ0ihΦ0| is a pu e s a e and Eq. (15) implies ha hΦ0|ˆ Uη(∞,−∞) = eiα0hΦ0|.(16) We will e e o he adiaba ic assump ion in combina ion wi h equilib ium a ze o empe a u e and wi h he condi ion o no g ound-s a e degene acy as he ze o- empe a u e assump ion. The ze o- empe a u e assump ion can be used o manipula e Eq. (13) a bi mo e. We ha e ˆρ0=|Φ0ihΦ0|=|Φ0ihΦ0|ˆ Uη(∞,−∞) hΦ0|ˆ Uη(∞,−∞)|Φ0i=lim β→∞ e−βˆ HM 0ˆ Uη(∞,−∞) T he−βˆ HM 0ˆ Uη(∞,−∞)i.(17) P og ess in Nonequilib ium G een’s Func ions V (PNGF V) IOP Publishing Jou nal o Physics: Con e ence Se ies 427 (2013) 012001 doi:10.1088/1742-6596/427/1/012001 5 γ . - iβ H ^ M 0 8 H ( ) ^ η Figu e 4. Con ou γand Hamil onian along he con ou γ o ge he ze o- empe a u e G een’s unc ion om Eq. (8). Inse ing his esul in o Eq. (13) we ind ha he ze o- empe a u e G een’s unc ion can again be w i en as in Eq. (8) wi h he con ou γ ha s a s a −∞, goes all he way o ∞and hen down o ∞ − iβ, see Fig. 4, and wi h he Hamil onian ˆ H(z) ha a ies along he con ou as illus a ed in he same igu e. I is wo h no icing ha he con ou γhas he special p ope y o ha ing only a o wa d b anch and ha o he ze o- empe a u e assump ion o make sense he Hamil onian o he sys em mus be ime independen . The e is indeed no eason o expec ha by swi ching on and o he in e ac ion he sys em goes back o he same s a e in he p esence o ex e nal d i ing ields. To summa ize he exac (Kons an ino -Pe el’), adiaba ic (Keldysh) and ze o- empe a u e G een’s unc ions ha e he same ma hema ical s uc u e, gi en by Eq. (8). Wha changes is he con ou and he Hamil onian along he con ou . 3. Wick’s heo em and Many-Body Pe u ba ion Theo y To be conc e e we specialize he discussion o in e ac ion Hamil onians ˆ Hin and ˆ HM in which a e wo-body ope a o s. Highe o de n-body ope a o s lead o mo e oluminous equa ions bu do no ise concep ual complica ions. Thus we w i e ˆ Hin (z) = 1 2Zdx1dx2 (x1, x2;z)ˆ ψ†(x1)ˆ ψ†(x2)ˆ ψ(x2)ˆ ψ(x1).(18) In he exac (Kons an ino -Pe el’) o mula he in e ac ion (x1, x2;z) = (x1, x2) is he in e pa icle in e ac ion o zon he ho izon al b anches whe eas (x1, x2;z) depends on he ini ial p epa a ion o zon he e ical ack. Fo ins ance in equilib ium (x1, x2; 0−iτ) = (x1, x2). On he o he hand in he adiaba ic (Keldysh) and ze o- empe a u e o mula (x1, x2;z) = e−η| − 0| (x1, x2) o zon he ho izon al b anches whe eas (x1, x2;z) = 0 o zon he e ical ack. Le us conside Eq. (8) and w i e he exponen ial o ˆ Has he p oduc o he exponen ials o ˆ H0and ˆ Hin : Gn(1 . . . n; 1′. . . n′) = 1 in T Te−iRγdz ˆ H0(z)e−iRγdz ˆ Hin (z)ˆ ψ(1) ... ˆ ψ(n)ˆ ψ†(n′)... ˆ ψ(1′) T Te−iRγdz ˆ H0(z)e−iRγdz ˆ Hin (z) .(19) The expansion in powe s o ˆ Hin leads o an expansion o Gnin e ms o nonin e ac ing G een’s unc ions G0,n. The G0,n a e ob ained om Eq. (19) by se ing ˆ Hin (z) = 0 o all z∈γ. Fo ins ance o n= 1 we ge G(a;b) = ∞ P k=0 1 k!i 2kR (1; 1′). . . (k;k′)G0,2k+1(a, 1,1′,...;b, 1+,1′+,...) ∞ P k=0 1 k!i 2kR (1; 1′). . . (k;k′)G0,2k(1,1′,...; 1+,1′+,...) ,(20) P og ess in Nonequilib ium G een’s Func ions V (PNGF V) IOP Publishing Jou nal o Physics: Con e ence Se ies 427 (2013) 012001 doi:10.1088/1742-6596/427/1/012001 6 o n= 2 we ge G2(a, b;c, d) = ∞ P k=0 1 k!i 2kR (1; 1′). . . (k;k′)G0,2k+2(a, b, 1,1′...;c, d, 1+,1′+,...) ∞ P k=0 1 k!i 2kR (1; 1′). . . (k;k′)G0,2k(1,1′,...; 1+,1′+,...) ,(21) e c. In hese equa ions a= (xa, a), b= (xb, b) a e collec i e indices like 1,2,..., he in e ac ion (j;j′)≡δ(zj, z′ j) (xj, x′ j;zj) and he in eg als a e o e 1,1′,...,k,k′. The appealing ea u e o any GFF is he possibili y o educing he nonin e ac ing G0,n o a (an i)symme ized p oduc o ( e mions) bosons o one-pa icle G een’s unc ions G0. This educ ion is called Wick’s heo em. The exis ing p oo s o Wick’s heo em a e a he labo ious and di e depending on whe he one is wo king wi h he ze o- empe a u e o Ma suba a o Keldysh G een’s unc ions. Below we gi e a simple and gene al p oo o Wick’s heo em which applies o all cases. We conside a one-body Hamil onian o he o m ˆ H0(z) = Zdx ˆ ψ†(x)h(x, z)ˆ ψ(x).(22) The mo e gene al case o a nondiagonal h(x, x′, z) can be ea ed in a simila manne . The G een’s unc ions G0,n sa is y he nonin e ac ing MSH id dzk −h(k)G0,n(1 . . . n; 1′. . . n′) = n X j=1 (±)k+jδ(k;j′)G0,n−1(1 ... ⊓ k. . . n; 1′... ⊓ j′. . . n′) (23) G0,n(1 . . . n; 1′. . . n′)"−i ←− d dz′ k −h(k′)#= n X j=1 (±)k+jδ(j;k′)G0,n−1(1 ... ⊓ j. . . n; 1′... ⊓ k′. . . n′) (24) whe e he hook o e he a gumen s in G0,n−1means ha hose a iables a e missing. The MSH is a se o coupled di e en ial equa ions o be sol ed on he con ou γo he G een’s unc ion o in e es (exac , adiaba ic, o ze o- empe a u e). In all cases om he de ini ion Eq. (8) i ollows ha he G0,n sa is y he Kubo-Ma in-Schwinge (KMS) ela ions, i.e., he G0,n a e (an i)pe iodic along he con ou γwi h espec o all hei con ou a gumen s. The e o e we can calcula e he G0,n by sol ing he MSH wi h KMS ela ions. We now show ha he solu ion is gi en by he Wick heo em G0,n(1,...,n; 1′,...,n′) =  G0(1; 1′). . . G0(1; n′) . . .. . . G0(n; 1′). . . G0(n;n′) ± (25) whe e he symbol |...|±signi ies he pe manen /de e minan o he case o bosons/ e mions and G0is he solu ion o Eqs. (23) and (24) wi h n= 1, i.e., id dz1 −h(1)G0(1; 1′) = δ(1; 1′), G0(1; 1′)"−i ←− d dz′ 1 −h(1′)#=δ(1; 1′) (26) wi h KMS bounda y condi ions. Expanding he pe manen /de e minan along ow, say, kwe ge G0,n(1,...,n; 1′,...,n′) = n X j=1 (±)k+jG0(k, j′)G0,n−1(1 ... ⊓ k. . . n; 1′... ⊓ j′. . . n′) (27) P og ess in Nonequilib ium G een’s Func ions V (PNGF V) IOP Publishing Jou nal o Physics: Con e ence Se ies 427 (2013) 012001 doi:10.1088/1742-6596/427/1/012001 7 When spli in o a ious componen s hese a e simply he Kadano -Baym equa ions on he ex ended con ou . Le us now w i e he con ou γas γ= ¯γ⊕γM, whe e γMdeno es he e ical (o Ma suba a) ack o he con ou and ¯γ he emaining piece, which is iden ical o he con ou o Fig. 1. The las e m on he .h.s o Eq.(58) can hen be w i en as Zγ d3 Σ(1; 3)G(3; 2) = Z¯γ d3 Σ(1; 3)G(3; 2) −iZdx3Zβ 0 dτ Σ⌉(1; x3τ)G⌈(x3τ; 2) (60) whe e we in oduced he pa ame iza ion z= 0−iτ on he e ical ack γM. Fo a gene al unc ion A(z, z′) on he con ou (spa ial coo dina es supp essed) we u he de ined A⌉(z= ±, τ) = A(z= ±, 0−iτ) (61) A⌈(τ, z = ±) = A( 0−iτ, z = ±).(62) F om he Dyson equa ion (57) and he Lang e h ules on he con ou γwe can u he de i e ha [15, 16] G⌈(1; 2) = −iZd¯x GM(1; ¯x 0)GA(¯x 0; 2) + [GM⋆Σ⌈·GA](1; 2) (63) whe e ⋆deno es a con olu ion be ween 0and 0−iβ on he e ical ack and ·deno es a con olu ion be ween 0and ∞. We u he de ined he ad anced and Ma suba a G een’s unc ions as GA(1; 2) = −θ( 2− 1)[G>(1; 2) −G<(1; 2)] (64) GM(1; 2) = G(x1 0−iτ1;x2 0−iτ2) (65) I we u he use ha Z¯γ dz δ−(z)G(¯xz;x ±) = GA(¯x 0;x ) (66) we ind by inse ing Eq.(63) in o he las e m o Eq.(60) ha −iZdx3Zβ 0 dτ Σ⌉(1; x3τ)G⌈(x3τ; 2) = Z¯γ d3 [Σ⌉⋆ GM⋆Σ⌈](1; 3)G(3; 2) −iZ¯γ d3 [Σ⌉⋆ GM](1; x3 0)δ−(z3)G(3; 2) (67) I we he e o e de ine ΣLby ΣL(1; 2) = −i[Σ⌉⋆ GM](1; x2 0)δ−(z2),(68) we can ew i e he equa ion o mo ion (58) o he G een’s unc ion as (i∂z1−h(1))G(1; 2) = δ(1; 2) + Z¯γ d3 [Σ + Σ⌉⋆ GM⋆Σ⌈+ ΣL](1; 3)G(3; 2).(69) A simila p ocedu e can be ca ied ou o he adjoin equa ion (59). We ind G(1; 2)(−i←− ∂z2−h(2)) = δ(1; 2) + Z¯γ d3G(1; 3)[Σ + Σ⌉⋆ GM⋆Σ⌈+ ΣR](3; 2) (70) whe e we de ined ΣR(1; 2) = −iδ−(z1)[GM⋆Σ⌈](x1 0; 2).(71) P og ess in Nonequilib ium G een’s Func ions V (PNGF V) IOP Publishing Jou nal o Physics: Con e ence Se ies 427 (2013) 012001 doi:10.1088/1742-6596/427/1/012001 14 Gi en he sel -ene gy and he G een’s unc ion wi h a gumen s on he imagina y ack γM, we can ega d Eqs. (69) and (70) as equa ions o mo ion o he G een’s unc ion Gon he con ou ¯γ. These equa ions can be in eg a ed using he nonin e ac ing G een’s unc ion go Eq.(35) since i sa is ies (±i) lim z1,z′ 1→ 0− g(1; 1′) = (±i) lim z1,z′ 1→ 0− G(1; 1′) = Γ(x1;x′ 1).(72) I we, he e o e, de ine he o al sel -ene gy as Σ o = Σ + Σ⌉⋆ GM⋆Σ⌈+ ΣL+ ΣR(73) we can w i e Gin e ms o wo equi alen Dyson equa ions G(1; 2) = g(1; 2) + Z¯γ d3d4g(1; 3)Σ o (3; 4)G(4; 2) (74) G(1; 2) = g(1; 2) + Z¯γ d3d4G(1; 3)Σ o (3; 4)g(4; 2) (75) To check ha hese equa ions a e equi alen o he Eqs. (69) and (70) we need o be ca e ul. The s anda d app oach is o ac wi h he ope a o o he o m i∂z−hand i s adjoin on bo h Dyson equa ions and use he equa ion o mo ion o g (i∂z1−h(1))g(1; 2) = δ(1; 2) (76) g(1; 2)(−i←− ∂z2−h(2)) = δ(1; 2) (77) We need o be ca e ul, howe e , since we canno change in eg a ion and di e en ia ion in he p esence o del a- unc ions unde he in eg al sign. The ele an in eg als o e he del a- unc ions need o be done i s be o e we use Eqs.(76) and (77). In Eq.(74) we ha e an in eg al o he o m Z¯γ d3g(1; 3)ΣR(3; 4) = −iZdx3[g>(1; x3 0)−g<(1; x3 0)][GM⋆Σ⌈](x3 0; 4) (78) On he igh hand side o his equa ion we ecognize he con ou spec al unc ion o Eq.(37) ha sa is ies Eq. (38). We he e o e see ha (i∂z1−h(1)) Z¯γ d3g(1; 3)ΣR(3; 4) = 0 (79) Simila ly we ha e Z¯γ d3 ΣL(3; 4)g(4; 2)(−i←− ∂z2−h(2)) = 0 (80) Then by ac ing wi h i∂z1−h(1) on Eq.(74) we see ha we eco e Eq.(69). Simila ly by ac ing wi h −i←− ∂z2−h(2) om he le on Eq.(75) we eco e Eq.(70). I only emains o check ha he Dyson Eqs.(74) and (75) sa is y he co ec bounda y condi ions. Since in he limi z1, z2→ 0− he con ibu ion o he in eg als on he .h.s. o he equa ions anish we see ha he condi ion (72) is indeed sa is ied. Now we a e eady o discuss he connec ion be ween he o mula ion based on he ini ial co ela ion blocks and he o malism based on in eg a ions along he imagina y ack. By compa ing Eq.(74) o Eq.(52) we see ha σ = Σ o + Σ o gΣ o + Σ o gΣ o gΣ o +...= Σ o 1 1−gΣ o (81) P og ess in Nonequilib ium G een’s Func ions V (PNGF V) IOP Publishing Jou nal o Physics: Con e ence Se ies 427 (2013) 012001 doi:10.1088/1742-6596/427/1/012001 15 and hence Σ o =σ (1 −gΣ o ) = σ −σ gσ +σ gσ gσ −...=σ 1 1 + gσ (82) This yields he expansion o he i educible sel -ene gy Σ o in e ms o he G een’s unc ions gand he co ela ion blocks Cm. We would like o men ion ha Σ o could in p inciple be calcula ed om he app op ia e ex en ion o he Hedin equa ions o include ini ial co ela ions [8]. This would lead o an expansion o Σ o in e ms o he d essed G een’s unc ion Gand co ela ion blocks. Howe e , he i e a i e solu ion o hese equa ions depend on he s a ing poin . In pa icula i we s a wi h a sel -ene gy which con ains only a C2-block hen he i e a i e p ocedu e canno gene a e diag ams wi h C-blocks o highe o de . 7. Conclusions We p esen ed a uni ied amewo k o equilib ium and nonequilib ium many-body pe u ba ion heo y. The mos gene al o malism o nonequilib ium many-body heo y o gene al ini ial s a es is based on he Keldysh con ou o which we a ach a e ical ack desc ibing a gene al ini ial s a e. This idea goes back o he wo ks o Kons an ino and Pe el’ (who conside ed equilib ium ini ial s a es), Danielewicz and Wagne . On his con ou we can s aigh o wa dly p o e a Wick heo em by sol ing he nonin e ac ing Ma in-Schwinge hie a chy o he nonin e ac ing many-body G een’s unc ions wi h KMS bounda y condi ions. This sho p oo o Wick’s heo em does no need any o he usually in oduced heo e ical concep s such as no mal o de ing and con ac ions. The s a emen is simply ha he nonin e ac ing m-pa icle G een’s unc ion is a de e minan o pe manen o one-pa icle G een’s unc ions. We showed how he a ious o he well-known o malisms o Keldysh, Ma suba a and he ze o- empe a u e o malism can be de i ed as special cases ha a ise unde di e en assump ions. We u he discussed a gene alized Wick heo em o gene al ini ial s a es on he Keldysh con ou . I again a ises as a solu ion o he nonin e ac ing Ma in-Schwinge hie a chy o he nonin e ac ing many-body G een’s unc ions bu his ime wi h ini ial condi ions speci ied by ini ial m-body densi y ma ices. The inal esul o Eq.(51) is an elegan al e na i e o he Wick heo em o Eq.(25) o KMS bounda y condi ions. We inally showed how he o malisms based on he Keldysh and Kons an ino -Pe el’-con ou s a e ela ed o he case o gene al ini ial s a es. Re e ences [1] A. L. Fe e and J. D. Walecka, Quan um Theo y o Many-Pa icle Sys ems (McG aw-Hill, New Yo k, 1971). [2] L. V. Keldysh, JETP 20, 1018 (1965). [3] P. Danielewicz, Ann. Phys. (N.Y.) 152, 239 (1984). [4] O. V. Kons an ino and V. I. Pe el’, So . Phys. JETP 12, 142 (1961). [5] M. Wagne , Phys. Re . B 44, 6104 (1991). [6] V. G. Mozo o and G. R¨opke, Ann. Phys. (N.Y.) 278 , 127 (1998) [7] A. G. Hall, J. Phys. A: Ma h. Gen. 8, 214 (1975). [8] D. Semka , D.K emp and M.Boni z, Phys. Re . E 59, 1557 (1999) [9] D. Semka , D.K emp and M.Boni z, J. Ma h. Phys. 41, 7458 (2000) [10] M. Boni z, Quan um Kine ic Theo y (Teubne , 1998). [11] M. Ga ny and M. M. M¨ulle , Phys. Re . D80, 085011 (2009) [12] R. an Leeuwen and G. S e anucci, Phys. Re . B 85, 115119 (2012). [13] G. C. Wick, Phys. Re . 80, 268 (1950). [14] P. C. Ma in and J. Schwinge , Phys. Re . 115, 1342 (1959). [15] G. S e anucci and R. an Leeuwen, Nonequilib ium Many-Body Theo y o Quan um Sys ems: A Mode n In oduc ion (Camb idge Uni e si y P ess, 2013). [16] G. S e anucci and C.-O. Almbladh, Phys. Re . B 69, 195318 (2004) P og ess in Nonequilib ium G een’s Func ions V (PNGF V) IOP Publishing Jou nal o Physics: Con e ence Se ies 427 (2013) 012001 doi:10.1088/1742-6596/427/1/012001 16