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Physics Letters B 802 (2020) 135197

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Feynman rules in terms of the Wigner transformed Green functions

C.X. Zhang

, M.A. Zubkov

a,b,∗

Physics Department, Ariel University, Ariel 40700, Israel

NRC “Kurchatov Institute” -ITEP, B. Cheremushkinskaya 25, Moscow 117259, Russia

a r t i c l e i n f o a b s t r a c t

Article history:

Received

28 November 2019

Received

in revised form 21 December 2019

Accepted

1 January 2020

Available

online 8 January 2020

Editor:

N. Lambert

In the models deﬁned on the inhomogeneous background the propagators depend on the two space–time

momenta rather than on one momentum as in the homogeneous systems. Therefore, the conventional

Feynman diagrams contain extra integrations over momenta, which complicate calculations. We propose

to express all amplitudes through the Wigner transformed propagators. This approach allows us to reduce

the number of integrations. As a price for this the ordinary products of functions are replaced by the

Moyal products. The corresponding rules of the diagram technique are formulated using an example of

the model with the fermions interacting via an exchange by scalar bosons. The extension of these rules to

the other models is straightforward. This approach may simplify calculations in certain particular cases.

The most evident one is the calculation of various non-dissipative currents.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

External ﬁelds provide the nontrivial inhomogeneous background to the ﬁeld theoretical models that describe various physical systems

both in the high energy physics and in the condensed matter physics. Strong magnetic ﬁelds appear in the description of the early universe

[1], of the neutron stars [2] and can be created in heavy-ion collision experiments [3]. In condensed matter physics magnetic ﬁelds cause

a lot of interesting phenomena. Possibly, the most popular one is the quantum Hall effect (QHE), which includes the mysterious fractional

quantum Hall effect (FQHE). In quantum Hall systems, the Hall resistivity R

as a function of magnetic ﬁeld B possesses the plateaus in

the presence of interactions between the electrons, and impurities [4,5]. Elastic deformations in materials is another kind of external ﬁeld

experienced by the charged carriers, and may also change the behavior of the electric transport [6–9].

the presence of external ﬁelds, the translational invariance is broken. Therefore, the two-point Green functions G(x

, x

) can not be

expressed in the form of the function of (x

− x

), and after the Fourier transformation the Green function depends on the initial and

the ﬁnal momenta, that are not equal in general case. An alternative to the Fourier transformation is Wigner transformation [10–12]. The

Wigner-transformed Green function G

(R, p) has certain advantages compared to the ordinary Fourier transform

G(p

, p

). Expressions

in terms of G

(R, p) are more concise. We will see below, that the corresponding Feynmann diagram technique contains the same

amount of integrations over momenta as in the homogeneous theory. The price for this is the appearance of the Moyal products instead

of the ordinary multiplications. Sometimes the resulting expressions for the physical quantities are more useful than those that are

obtained using the conventional Feynmann diagrams. An example is given by the Hall conductivity, which is expressed through the

Wigner transformed Green functions using the Moyal products. The corresponding expression is the topological invariant in phase space,

i.e. its value is not changed under the smooth modiﬁcation of the system (see [15]). The similar (but simpler) constructions were used

earlier to consider the intrinsic anomalous Hall effect and chiral magnetic effect [13]. It has been shown that the corresponding currents

are proportional to the topological invariants in momentum space. This method allows to reproduce the conventional expressions for Hall

conductivity [14], and to prove the absence of the equilibrium chiral magnetic effect. Recently, the electron-electron interactions have

been taken into account, and it has been shown to all orders in perturbation theory that the Hall conductivity (averaged over the system

area) is proportional to the same topological invariant, as in the presence of interactions [16,17]. This proof works equally well for the

Corresponding author.

E-mail

addresses: zhang12345s@sina.com (C.X. Zhang), zubkov@itep.ru (M.A. Zubkov).

https://doi.org/10.1016/j.physletb.2020.135197

0370-2693/

SCOAP

2 C. X. Zhang, M.A. Zubkov / Physics Letters B 802 (2020) 135197

homogeneous systems with intrinsic anomalous Hall effect, and the non-homogeneous systems with quantum Hall effect in the presence

of varying magnetic ﬁeld, and varying electric potential of impurities.

the theories with interactions, Feynman diagrams and Feynman rules are necessary to evaluate various Green functions [18,19], and

various physical amplitudes. Each particular Feynman diagram F (X

( f )

(i)

) with 2N external fermion lines depends on the set of input

coordinates X

(i)

={x

(i)

|a = 1, ..., N} and output coordinates X

( f )

={x

( f )

|a = 1, ..., N}. The fermion lines pass through the whole diagram,

and connect the input/output coordinates into pairs X

= (x

(i)

, x

( f )

) with a = 1, ..., N. If we denote the Wigner-transformed function of

F(X

( f )

(i)

) by F (R|P), where R ={r

= (x

( f )

+ x

(i)

)/2|a = 1, ..., N} and P ={p

|a = 1, ..., N}, while p

are the conjugate momenta with

respect to x

( f )

− x

(i)

, then the relation between F (X

( f )

(i)

) and F (R|P) is

F(X

( f )

(i)

) =



(2π)



( f )

−x

(i)

)

(R|P ),

where D is the dimension of space–time. The physical Green function G(x

( f )

(i)

) corresponding to N incoming fermions, and N outgoing

fermions may be expressed through the Feynman diagrams as the sum over permutations  of the sequence (a = 1, ..., N):

G( X

( f )

(i)

) =





(−1)

P()

F(X

( f )



(i)

) =





(−1)

P()



(2π)

−i



( f )

(a)

−x

(i)

)



(

( f )



+ X

(i)

)/2|P



Here (−1)

P()

is the parity of the permutation . X

( f )



is the set of permutated output coordinates {x

( f )

(a)

|a = 1, ..., N} according to

the permutation . In the present paper we construct the diagram technique for the calculation of the Wigner transformed Feynmann

diagrams F

(R|P). This technique expresses these quantities through the Wigner transformed bare propagators. For deﬁniteness we con-

sider

the model of one Dirac fermion interacting minimally with the scalar ﬁeld. The inhomogeneity is introduced to the system through

the external Abelian gauge ﬁelds that interact with the scalar ﬁeld and the fermion. Those ﬁelds may depend arbitrarily on coordinates.

Correspondingly, each propagator depends on two momenta rather than on one momentum (as in the case of the homogeneous system).

The generalization of our construction to the more general case is straightforward.

The

paper is organized as follows: In Sect. 2 the lagrangian of the model under consideration is introduced and certain basic expressions

for the Wigner transformed Green functions are presented. In Sect. 3, the Feynman rules are obtained for the diagrams, which contain

zero or two external fermion lines. In Sect. 4, the Feynman rules are obtained for the diagrams, which contain two external fermion lines

and several internal fermion loops. In Sect. 5, we discuss the extension of our construction to the diagrams with more than two external

fermion lines. In Sect. 6 we end with the conclusions.

2. Wigner transform and the model under consideration

Wigner transform is also known as Weyl transform [20]. Originally, it was closely intertwined with the development of phase-space

formulation of quantum mechanics (also called “deformation quantization”) in 1930’s (see [21,22] and references therein). Compared

with the operator formulation of quantum mechanics, the phase-space formulation deals with the ordinary functions of coordinates and

momenta: the so-called Wigner function W (q, p), which is considered as a quantum counterpart of classical distribution in phase space

(q, p) [23]. Wigner distribution W (q, p) is not a positive real function. Therefore, strictly speaking, it cannot be considered as a phase space

analogue of the probability distribution. For example, the superposition of two Gaussian wavepackets gives W (q, p) with varying signs

[24]. Despite this drawback, a ﬂuid analog of the quantum entropy ﬂux in phase space can be formulated using Weyl-Wigner formalism,

i.e. in terms of the Wigner function W (q, p) [31]. For positive deﬁned Wigner distribution the analogue of von Neuman entropy may be

deﬁned as [25–27] S

=−



dqdpW (q, p) ln W (q, p). Besides, Wigner distribution provides an expression for the quantum purity [31]

P = 2π



dqdpW (q, p)

. For pure states, P = 1, while for mixed states, P < 1. Therefore 1 − P gives measure of the distance to the pure

state.

addition to its applications in quantum mechanics [28–31], the Wigner transform has been extended to quantum ﬁeld theory and

has been adopted to the investigation of the high energy physics and the many-body effects, including various anomalous transport

phenomena such as the quantum Hall effect, the chiral separation effect, and the chiral vortical effect [32–37]. In this section we will go

along this line to introduce the Wigner transformation of the Green function. Our starting point is the following lagrangian for the Dirac

fermion interacting with the scalar ﬁeld (in the presence of the inhomogeneous background given by the external hauge ﬁelds):

L =

ψ((i∂

− A

)γ

− m)ψ + (i∂

− B

)φ(i∂

− B

)φ − m

− g

ψψφ

(1)

where A

and B

are the vector potentials of the external ﬁelds, which can be different one from another.

For

the fermions the two – point Green function G satisﬁes equation

Q (x

)G(x

, x

) = δ(x

− x

), where

Q (x) = (i∂

− A

(x))γ

− m.

The Wigner transformation of G is deﬁned as

(R, p) =



drG(R + r/2, R − r/2)e

−ipr

. (2)

It satisﬁes the Groenewold equation Q

(R, p)  G

(R, p) = 1[13,38], where Q

is the Weyl symbol of operator

Q , while  =

←−

∂

−→

∂

−

←−

∂

−→

∂

)/2

is the Moyal product.

Similarly

for the bosonic ﬁeld φ operator

U (x) = (i∂

− B

(x))(i∂

− B

(x)) − m

is the inverse bare propagator, and the Wigner –

transformed Green function D

satisﬁes U

(R, p)  D

(R, p) = 1. An important result of Wigner – Weyl calculus is [38–40]

C(x

, x

) =



A(x

, y)B(y, x

)dy ⇒ C

(R, p) = A

(R, p)B

(R, p). (3)

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