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