An Introductory Lecture to the Keldysh Techniquefor Non-equilibrium Green's Functions

Submitted by Inon Sharony

October 23, 2008

Towards credit in the course

The Green's Function Method for Many-Body Theory

Given in the spring semester, 2008 by

Professor K. Kikoin

1

1 MotivationPhysical systems out of equilibrium usually represent one or more of the following behaviors:

• Relaxation: Towards the ground state.

• Dephasing: From a coherent state to a non-coherent state.

• Steady-state: A time-stationary state which is not the ground state. e.g., a system with a source& drain, in which exists a constant current.

And so on...In the case of relaxation, we deal with a system initially prepared (at time t = −∞) in some excited

state, and we explore its return to the ground state. Alternatively, the system begins in some groundstate, and an external �eld begins to be applied. If the external �eld is weak enough, we should recoverfrom our theory of non-equilibrium Green's functions (NEGF), the theory of linear response.

The NEGF method proposes a quantum mechanical formulation of the classical Boltzmann equation.

1.1 Non-equilibrium Dynamics [7]We begin with a short de�nition of non-equilibrium. This is not a rigorous proof, but a mathematicalintroduction to the term �non-equilibrium� with respect to physical problems.

1.1.1 Master Equation Solutions for Equilibrium and Non-equilibrium CasesLooking at a single point in con�guration space, C ≡ ({xi} , {pi}), we �nd that writing a master equationfor a non-equilibrium process will consist of summing on all the processes that increase the probabilityto �nd the system with such a con�guration at time t, and subtracting all the processes that decreasethat probability.

∂tP (C, t) =∑

C′[R (C← C′)P (C′, t)−R (C′ ← C)P (C, t)]

∂tP = −LP

Where R (C← C′) represent the �incoming� and R (C′ ← C) the �outgoing� rates (assumed constant).This master equation seems similar to the Hamilton equation, but di�ers from it in some respects

(e.g., all P 's are non-negative Reals and not Complex as in the case of Ψ).One class of solutions consists of all the time independent solutions, P (C). The di�erence between

the equilibrium and non-equilibrium solutions lies in that the rates R related to the equilibrium solutionsobey detailed balance, whereas in the non-equilibrium case they do not.

1.1.2 Detailed BalanceTake a series of con�gurations:

C1, C2, . . . , Cn−1, Cn.

Consider the products of the rates around the cycle

Π+ ≡ R (C1 ← Cn)R (Cn ← Cn−1) · · ·R (C3 ← C2)R (C2 ← C1)Π− ≡ R (C1 ← C2)R (C2 ← C3) · · ·R (Cn−1 ← Cn) R (Cn ← C1)

Detailed balance entails that Π+ = Π−for all cycles.Writing

R (C← C′)R (C′ ← C)

≡ eA × . . .

Detailed balance becomes

∇×A = 0

A can be written as a gradient of some scalar (e.g. −H/kBT ) so that

P (C) ∝ e−β·H(C)

2

and for all pairs C and C′ we have

R (C← C′) P (C′)−R (C′ ← C)P (C) = 0 (1)i.e., zero current everywhere (e.g., electrostatics).By contrast, if the rates violate detailed balance, we will still have time-independent solutions P (C),

but this time the left-hand-side (LHS) of 1 will not equal zero, and we will get non-trivial current loopsin our system (e.g., magnetostatics). These are called non-equilibrium steady states. Examples for whenwe can get these could be when coupling to two energy reservoirs or two electrodes. In general, we canexpect these if F 6= −∇V.

1.2 Technical Di�erencesAs opposed to the Matsubara GF method, where an analytical continuation was applied (to relatediscrete Imaginary frequencies with temperature), the Keldysh technique we will discuss here uses onlycontinuous Real frequencies. This makes the Keldysh technique more appealing even in equilibriumcases, where the Matsubara method can be more complex. An alternative approach, by Kadanoffand Baym [10] may be more exhaustive, and will not be discussed here.

In most cases we will deal with, we are interested in a cannonic or grand-cannonic ensemble (i.e., theenergy isn't known exactly, but the temperature is). Therefore the averaging now takes on a di�erentmeaning from the quantum expectation value (in equilibrium with respect to some ground state), butrather a quantum ensemble average. Therefore

⟨O

⟩≡ 1

ZTr

[ρO

]

Z ≡ 1Z

Tr [ρ]

This carries some importance as pertains to the initial state of the system, as de�ned by ρ (t = 0), butwe will assume that the system was initially prepared in a pure state. For a more generalized approach,see [6].

2 Non-equilibrium Green's FunctionsWe remember that the Green's function method reveals to be a pertubative technique through thescattering matrix, S ≡ S (−∞,∞) ≡ T exp

[−i

∫∞−∞ V (t) dt

]1, where V is the time-dependent part of

the Hamiltonian and where the time-ordering operator, T , to the right as follows:

TA (1)B (2) =

{A (1)B (2) t1 > t2

A (1)B (2) t1 < t2

where we employed the shorthand notation (1) ≡ (r1, t1). We likewise de�ne the anti-time-orderingoperator, ˆ

T , which operates to the right in the opposite way.Now, as in the equilibrium case

G (1, 2) ≡ −i⟨S−1TΨ(1) Ψ† (2) S

⟩

S−1 ≡ ˆTe+i

R∞−∞ V(t)dt

Since no �adiabatic switching-on� can be assumed in a non-equilibrium situation, we cannot assumethat the state of the system at time t = −∞ di�ers from its state at time t = ∞ by a phase factoronly (Gell-Man & Low theorem) . Therefore, as in the Matsubara method, we will need to concernourselves with a situation where we cannot factorize out the term

⟨S−1

⟩.

The second quantization of the interaction (for a system under an external �eld) can be expressed as:

V (t) ≡∫

Ψ(r, t) U (r, t)Ψ† (r, t) dr

1Throughout this work we will take ~ = 1.

3

We can expand the GF as a pertubative series by taking the S-matrix term-by-term in accordancewith the series expansion of the exponent.

S = S(0) + S(1) + . . .

S(n) ≡ (−i)n

n!T

∫ ∞

−∞

∫ ∞

−∞. . .

∫ ∞

−∞V (t)V (t′) . . .V

(t(n)

)dtdt′ . . . dt(n)

G (1, 2) = G0 (1, 2) + G1 (1, 2) + G2 (1, 2) + . . .

G0 (1, 2) = −i⟨S−1

(0)TΨ(1) Ψ† (2) S(0)

⟩= −i

⟨TΨ(1) Ψ† (2)

⟩

G1 (1, 2) = −i⟨S−1

(0)TΨ(1) Ψ† (2) S(1)

⟩− i

⟨S−1

(1)TΨ(1) Ψ† (2) S(0)

⟩

= −i⟨TΨ(1)Ψ† (2)S(1)

⟩− i

⟨S−1

(1)TΨ(1)Ψ† (2)⟩

= −i

⟨T

[Ψ(1)Ψ† (2)

∫ ∞

−∞Ψ(3) U (3)Ψ† (3) dr3

]⟩− i

⟨ˆT

[∫ ∞

−∞Ψ (3) U (3) Ψ† (3) dr3

]× T

[Ψ(1)Ψ† (2)

]⟩(2)

Obviously, the second and third terms on the RHS are not equivalent due to the di�erent timeordering of the component �eld operators. This leads us to the concept of contour integration.

2.1 Contour IntegrationThe Kadanoff-Baym approach �rst introduced the concept of contour integration in relation to GFby switching the time-ordering operator by a �closed-time-path� integration contour.

This can be extended to thermodynamic systems through an �interaction contour� by including thenegative imaginary time axis.

In the case where the initial state of the system is described at t0 = −∞ as some pure state (allinitial correlations are zero), we can ignore the �hook� part of the contour (t0 − iβ ≤ t ≤ t0). This iscorrect assuming the GF falls o� fast enough with respect to the di�erence of its two time arguments.Thus, the �closed time-path contour� and the �interaction contour� are identical.

2.2 �The Keldysh Contour� [2]2Keldysh showed that the contour can be extended beyond the largest time argument, so that it extendsfrom −∞ to +∞ (this is called the C1 contour) then crosses the real time axis in�nitesimally and returnsfrom +∞ to −∞ (C2). This is named �theKeldysh contour�. Clearly, the C1 part of the contour is time-ordered, and the C2 part is anti-time-ordered. Thus, we can change our perspective from time-orderingto contour-ordering, which we will discuss in the next sections.3

2For a review see [5]3We will take this time to point out that a generalisation of the Green's functions technique to include also thermody-

namic information and correlations has been proposed in [6].

4

We now need to de�ne Green's functions in accordance to the relative location of their two timearguments along the Keldysh contour.

2.3 Introducing: The Non-equilibrium Green's Functions (NEGF)The �eld operators in each correction term to the GF are connected in pairs (as in equilibrium methods).Each pair is averaged over separately as a zero-order GF, G0, where each of the two �eld operators canbe either part of the time-ordered part or anti-time-ordered part.

2.3.1 (Massive) Particle NEGFThe �greater� and �lesser� GF The �greater� (and �lesser�) GF correspond to the cases where the�rst time argument lies on the lower (upper) part of the contour, and the second time argument lies onthe opposite part.4

G> (1, 1′) ≡ −i⟨ΨH (1) Ψ†H (1′)

⟩

G< (1, 1′) ≡ ±i⟨Ψ†H (1′)ΨH (1)

⟩

where in the last line, the positive sign corresponds to a Fermion GF, and the negative sign to aBoson GF.

The relation to the density matrix is evident, when taking coinciding time arguments

G< (r1, t ; r1′ , t) =N

Vρ (r1, t ; r1′ , t) (3)

Due to the commutation relations of the �eld operators, we have another identity for GF withcoinciding time arguments:

G< (r1, t ; r1′ , t)−G> (r1, t ; r1′ , t) = iδ (r1 − r1′)

The �time-ordered� and �anti-time-ordered� GF The �causal� equilibrium GF is now renamedthe �time-ordered� NEGF, and corresponds to cases where both time arguments lie on the upper part ofthe contour (the part which is directed in the positive time direction):

Gc (1, 1′) ≡ −i⟨TΨH (1) Ψ†H (1′)

⟩

≡{

G> (1, 1′) t1 > t1′

G< (1, 1′) t1 < t1′

We similarly de�ne an �anti-time-ordered� NEGF, Gec, using the anti-time-ordering operator, andcorresponding to cases where both time arguments lie on the lower part of the contour. Since the timeordering operator (or anti time ordering operator) is invoked, the type of NEGF chosen does not dependon the relative order of the two time arguments, but only that they are on the same part of the contour(as opposed to the �greater� and �lesser� NEGF).

Thus, it is clear that the �greater� GF is applied when the �rst time argument is greater than thesecond, and vice-versa for the �lesser� GF.

The �advanced� and �retarded� GF The �advanced� and �retarded� GF are de�ned as in equilib-rium GF techniques

Gr (1, 1′) ≡−i

⟨[ΨH (1) , Ψ†H (1′)

]±

⟩t1 > t1′

0 t1 < t1′

Ga (1, 1′) ≡

0 t1 > t1′

+i

⟨[ΨH (1) , Ψ†H (1′)

]±

⟩t1 < t1′

4The �greater� GF is sometimes termed �the particle propagator� and the �lesser� GF �the hole propagator�.

5

These last two GF, as in the equilibrium case, have no cuts and are analytic in the upper (�retarded�)or lower (�advanced�) half-planes of the complex time coordinate. Physically, they may represent thesystem's magnetic susceptibility or dielectric constant.

The �contour-ordered� GF The �contour-ordered� NEGF is now de�ned as:

G (1, 1′) ≡ −i⟨TCΨH (1) Ψ†H (1′)

⟩

≡

Gc (1, 1′) t1, t1′ ∈ C1

G> (1, 1′) t1 ∈ C2, t1′ ∈ C1

G< (1, 1′) t1 ∈ C1, t1′ ∈ C2

Gec (1, 1′) t1, t1′ ∈ C2

this can be written in a matrix notation:

G ≡(

Gc G<

G> Gec

)=

(G−− G−+

G+− G++

)(4)

Where the Gαβ notation signi�es on which of the contours each of the time arguments lie. Confusingly,the right sign corresponds to the �rst time argument and the left sign corresponds to the second timeargument. If a time argument lies on the top contour (C1) it is marked with a negative sign, andvice-versa.

2.3.2 (Massless) Interaction Quanta NEGFIn order to calculate the interaction of particles via quanta such as phonons we need to de�ne NEGF forsuch Bosons.

We begin by de�ning the gauge transformation of the Complex Boson �eld operator, χ, into the Realscalar Boson �eld operator, Φ:

Φ ≡ χ + χ†

Since Φ is Real, we de�ne

D> (1, 1′) ≡ −i 〈ΦH (1)ΦH (1′)〉D< (1, 1′) ≡ −i 〈ΦH (1′)ΦH (1)〉

we note thatD> (1, 1′) = D< (1′, 1)

and that they are both anti-Hermitian.Once again,

Dc (1, 1′) ≡ −i⟨TΦH (1)ΦH (1′)

⟩

and likewise forDec (1, 1′) ≡ −i

⟨ ˆTΦH (1)ΦH (1′)

⟩

For further details see [3].

2.4 Mathematical Relations of the NEGF2.4.1 Conjugation Relations

Ga (1, 1′) = Gr (1′, 1)∗ (5)

Gc (1, 1′) = −Gec (1′, 1)∗

G< (1, 1′) = −G< (1′, 1)∗

G> (1, 1′) = −G> (1′, 1)∗

6

2.4.2 Linear Co-dependence

Ga = Gc −G>

= Gec −G<

Gr = Gc −G<

= Gec −G>

Gc + Gec = G> + G<

This last relation can be used to remove an excess GF equation, using the �Keldysh� GF

Gk ≡ Gec −Gc = G> −G< (6)

2.5 The Fourier Transformed NEGF (FT-NEGF)In the case of homogeneous space and time, we expand the �eld operators in plane-waves (Fouriertransformation)

Ψ (r1, t1) ≡ 1√V

∑p

apei(p·r1−ξpt1)

ξp ≡ εp − µ

Using the correlations of the creation and annihilation operators in momentum space⟨a†pap′

⟩= npδ (p− p′)

⟨apa†p′

⟩=

{[1− np] δ (p− p′) Fermions

[1 + np] δ (p− p′) Bosons

where only at equilibrium is np equal to the Fermi-Dirac (or Bose-Einstein) distribution function,and otherwise is just the non-equilibrium momentum distribution function.

G< (r1, t1 ; r1′ , t1′) = G< (r, t ; 0, 0)

= ± i

V

∫npei(p·r−ξpt) V d3p

(2π)3

inserting a Dirac delta �function� in frequency space allows a further Fourier transformation infrequency:

G< (r, t ; 0, 0) = ±2πi

∫npei(p·r−ωt) d3p

(2π)3δ (ω − ξp)

dω

2π

≡∫

ei(p·r−ωt)G< (p, ω)d4p

(2π)4

and we identify

G< (p, ω) ≡ ±2πinpδ (ω − ξp)G> (p, ω) ≡ ±2πi [1− np] δ (ω − ξp)

and we can use previous identities together with the Sokhatsky-Weierstrass theorem to get

Gc (p, ω) ≡ PP∫

1ω − ξp

+ iπ [±2np − 1] δ (ω − ξp)

7

2.5.1 Spectral Representation [9]We de�ne the spectral function A (p, ω) as follows:

Gr,a (p, ω) ≡ limη→0+

∫A (p, ω′)

ω − ω′ ± iη

dω′

2π

and we note the following resultant identities

A = 2Im [Ga] = −2Im [Gr]

making use of the Lehmann expansion for the equilibrium case we �nd

G< (p, ω) = inpA (p, ω)G> (p, ω) = −inpA (p,−ω)Gk (p, ω) = − [±2np − 1]A (p, ω)

2.5.2 Correspondence to Equilibrium Green's FunctionsAt T=0, we take the averaging over the ground state,

〈. . .〉 → 0 〈. . .〉0 ≡ 〈0 |. . .| 0〉 .

Thus, the Fermi-Dirac distribution function is replaced by a Heaviside function,

np → Θ(|p| − pF ) ≡

1 |p| > pF

12 |p| = pF

0 |p| < pF

.

This will cause one of the �rst order terms to drop, and likewise for higher orders, reverting from theNEGF to the equilibrium GF.

G>T=0 (p, ω) = 0 ⇐ |p| < pF

G<T=0 (p, ω) = 0 ⇐ |p| > pF

3 (Feynman) DiagrammaticsWe can form a diagrammatic technique for the NEGF which is almost identical to the equilibriumFeynman technique used at T=0. Again, free particle propagators (G0) will be represented by solidarrows, and interactions (U) through dashed lines. The relevant form of �Wick's Theorem� for the non-equilibrium case can be proposed via a generalisation of the equilibrium �Wick's Theorem� however wewill not elaborate this here. We just assume that the average over any number of NEGF with theirdi�erent orderings can be simpli�ed to a product of averages over connected pairs of NEGF, with somecoe�cients. More information on these points may be found elsewhere (For instance, [6]).

We return to the Gαβ notation and reiterate that each propagator in the diagram will be assignedtwo signs (±) according to the time-ordering with which its operators are related (time-ordered operatorsget a negative index). 5 In accordance to the matrix representation of the NEGF, we will have foursuch terms, G±±. For each such element we will draw the diagrams complying with each term in thepertubative expansion. Each element di�ers from the others through the �rst and last of its sign indeces.Let us observe an example.

3.1 First OrderIn accordance with the equation for the �rst order correction to the GF in 2 we have two diagrammaticterms for G−−1 .

5The external �eld has two operators, but the dashed line associated with it is only connected to one vertex. Thereforeit is only associated with one sign index.

8

Note that the �rst and last signs are negative, as should be for a term in the expansion of G−−, andthe vertex 3 is integrated over as in the equilibrium Feynman technique (integration over space andtime, and summation over spin indeces).

G−−1 (1, 2) =∫ {

G−−0 (1, 3) G−−0 (3, 2) [−U (3)] + G−+0 (1, 3) G+−

0 (3, 2) [U (3)]}

The negative sign of the interaction in the �rst term is due to the sign with which it is associated, asa term in S.

The same equation may be written for the other elements, Gαβ1 .

3.2 Second OrderWe can extend the de�nitions to second order diagrams for the four terms of the G−− element, G−−2 .(Note that we dropped the vertex numbering, or space & time arguments)

3.3 First Order Two-particle InteractionThese diagrams are formed, as in the equilibrium case, by di�erent combinations of two external-�elddiagrams, where the dashed line now represents an interaction quantum. Notice that the propagator ofthe interaction quanta begins and ends with the same sign.

4 Self-energies and the Dyson EquationWe write the Dyson equation as for G−−, similarly to the equilibrium case

G−− (1, 2) = G−−0 (1, 2) +∫

G−−0 (1, 4) Σ−− (4, 3)G−− (3, 2) +G−0 (1, 4) Σ−+ (4, 3)G+− (3, 2)+G−+

0 (1, 4) Σ+− (4, 3)G−− (3, 2) +G−+0 (1, 4)Σ++ (4, 3) G+− (3, 2)

We can write this same equation with a more general self-energy (note that the coordinate indeceshave also been changed)

We deal with a 2× 2 matrix of the indeces corresponding to the di�erent NEGF (±± ≡ {>, <, c, c})and we can write the Dyson equation in matrix notation:

G = G0 [1 + ΣG]

We can draw diagrams for the di�erent elements of di�erent order terms of the self energy

9

where the top line is comprised of terms of the Σ−− element, and the bottom line of the Σ−+ element.

5 RAK RepresentationUsing a unitary rotation matrix and 6 we can write 4 as

G ≡(

0 Ga

Gr Gk

)

Σ ≡(

Σk Σr

Σa 0

)

With

Σa ≡ Σ−− + Σ+−

Σr ≡ Σ++ + Σ−+

Σk ≡ Σ++ + Σ−−

Three kinetic equations are formed:

Ga (1, 2) = Ga0 (1, 2) +

∫Gc

0 (1, 4)Σa (4, 3) Ga (3, 2) d4X4

And likewise we have second the equation for Gr (1, 2), however it is usually simpler just to use therelations between the �advanced� and �retarded� GFs 5.

The third equation is:

Gk (1, 2) = −ρ (1, 2) +∫ [

Gr0 (1, 4)

{Σk (4, 3)Ga (3, 2) + Σr (4, 3)Gk (3, 2)

}+ Gk

0 (1, 4)Σa (4, 3)Ga (3, 2)]d4X3d

4X4

We can apply the operator [G (0, 1)]−1 from the left of both sides of the equation, where

G−1 (0, 1) = i∂

∂t1+∇2

r1

2m+ µ

G−1 (0, 1) G (1, 2) = δ (r1 − r2)

G−1 (0, 1) Gk0 (1, 2) != 0

because G (1, 2) has poles at r1 − r2, but Gk (1, 2) has no poles!We get

G−1 (0, 1)Gk (1, 2) =∫ {

Σk (1, 3)Ga (3, 2) + Σr (1, 3)Gk (3, 2)}

d4X3 (7)

An alternative derivation may be formulated via the Kadanoff-Baym technique and using theLangreth theorem.6

6See [4, 8].

10

6 Appendix: The Semi-classical NEGF Approximation � TheTransport Equation

6.1 Equivalence to the Boltzmann EquationFor a free gas of classical hard spheres with some distribution function, n (r,p)

dn

dt=

∂n

∂t+

∂n

∂r∂r∂t

=∂n

∂t+

∂n

∂rpm

the LHS of the equation is equal to all the contributions from scattering in di�erent collisions. Thismay be expressed in integral form as the collision integral (Stosszahl), C (n).

If we add an external �eld via the force it a�ects, F

∂n

∂t+

pm

∂n

∂r+ F

∂n

∂p= C (n)

[∂

∂t+

pm

∂

∂r+ F

∂

∂p

]n = C (n)

which is quite similar7 to the Dyson equation 7[i

∂

∂t1+∇2

r1

2m+ µ

]Gk (1, 2) = C (n)

G−1 (0, 1)Gk (1, 2) = C (n)

where we take the quantum equivalent of the classical distribution function n (r,p)→ ρ (r,p), whichsatis�es 3.

6.2 The �Mixed Representation� � the Wigner DistributionWe combine two G−1operators to form a 2-particle operator

R ≡ [G−1 (0, 1)

]† −G−1 (0, 2) = −i

(∂

∂t1+

∂

∂t2

)− 1

2m

(∇2r1 −∇2

r2

)

we separate X1 and X2 into a center-of-mass coordinate and a di�erence coordinate

R ≡ r1 + r22

r ≡ r1 − r2

t ≡ t1 + t22

τ ≡ t1 − t2

and write the Wigner distribution function (a �mixed representation�)

ρ

(R +

12r, t ; R− 1

2r, t

)=

V

N

∫eip·rn (R,p, t)

d3p

(2π)3

in reference to the NEGF, this representation gives (in 4-dimensional notation: X ≡ (R, t) , Ξ ≡ (r, τ))

G< (X, P ) =∫

eiPΞG<

(X +

12Ξ, X − 1

2Ξ

)d4Ξ

(2π)4

which, for simultaneous time arguments in the GF gives the identity

n (R,p, t) = −i

∫G< (X, P )

dω

2π

7For a more rigorous comparison, see [4, 8].

11

Notice, also that R in the new coordinates (at simultaneous times) is

R (R, r, t) = −i

(∂

∂t− i

m∇R∇r

)

The RHS is exactly the RHS of the Boltzmann equation with no external �eld, up to a factor of i.We can generalize this result to a system under an external �eld through the introduction of the vectorpotential up to �rst order in |A|

i∇r −→ i∇r − e

cA

The result will be the Boltzmann equation with an external force F = qE.

6.3 The Collision IntegralWe operate using R on G and use the mixed representation to disregard all short range spatial variations(neglect dependence on r) �nally arriving at a NEGF formulation for the collision integral:

C (n) ∼∫ {−Σ−+ (X, P ) G+− (X, P ) + Σ+− (X,P )G−+ (X,P )

} dω

2π

= iΣ−+ (ξp,p ; r, t) · [1− n (r,p, t)] + iΣ+− (ξp,p ; r, t) · n (r,p, t)

Where the �rst term on the RHS of the last line corresponds to a �gain� of particles, and the secondterm to a �loss�.

Some more speci�c approximations are needed to get results for speci�c models, however we cannote that this result is independent of the same-sign-index self-energies, Σ++ and Σ−−. Therefore, the�rst order correction terms to the participating self energy elements are gotten from the second orderdiagrams of type (here, for Σ−+)

where (from conservation of 4-momentum) we have P ′1 = P + P1 − P ′, and the analytic expressionfor this self energy element is

Σ−+ (P ) = −i

∫G−+ (P ′) G+− (P1)G−+ (P ′1)U2 (P1 − P ′) δ (P + P1 − P ′ − P ′1)

d4P1d4P ′

(2π)8

References[1] Mills, R., Propagators for Many-Particle Systems, Gordon and Breach, New York (1969).

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