DSpace at VNU: Basics of quantum field theory of electromagnetic interaction processes in single-layer graphene

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Basics of quantum field theory of electromagnetic interaction processes in single-layer graphene

View the table of contents for this issue, or go to the journal homepage for more

2016 Adv Nat Sci: Nanosci Nanotechnol 7 035001

(http://iopscience.iop.org/2043-6262/7/3/035001)

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Basics of quantum field theory of

electromagnetic interaction processes in

single-layer graphene

Van Hieu Nguyen

Advanced Center of Physics and Institute of Materials Science, Vietnam Academy of Science and

Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam

University of Engineering and Technology, Vietnam National University, 144 Xuan Thuy, Cau Giay,

Hanoi, Vietnam

E-mail:nvhieu@iop.vast.vn

Received 2 May 2016

Accepted for publication 31 May 2016

Published 5 July 2016

Abstract

The content of this work is the study of electromagnetic interaction in single-layer graphene by

means of the perturbation theory The interaction of electromagneticfield with Dirac fermions

in single-layer graphene has a peculiarity: Dirac fermions in graphene interact not only with

the electromagnetic wave propagating within the graphene sheet, but also with electromagnetic

field propagating from a location outside the graphene sheet and illuminating this sheet The

interaction Hamiltonian of the system comprising electromagneticfield and Dirac fermions

fields contains the limits at graphene plane of electromagnetic field vector and scalar potentials

which can be shortly called boundary electromagneticfield The study of S-matrix requires

knowing the limits at graphene plane of 2-point Green functions of electromagneticfield which

also can be shortly called boundary 2-point Green functions of electromagneticfield As the

first example of the application of perturbation theory, the second order terms in the

perturbative expansions of boundary 2-point Green functions of electromagneticfield as well

as of 2-point Green functions of Dirac fermionfields are explicitly derived Further extension

of the application of perturbation theory is also discussed

Keywords: electromagnetic, graphene, Dirac fermion, perturbation theory, Green function

Classification numbers: 3.00, 5.15

1 Introduction

Soon after the discovery of graphene by Geim and Novoselov

[1–4], the research on graphene rapidly developed and became

a wide interdisciplinary area of science and technology It was

shown[5] that even in the case when the electron spin plays no

role, its quantum states are still described by two-component

wave functions satisfying differential wave equations similar to

relativistic Dirac equation for a massless particles in(2 +

1)-dimensional Minkowski space–time Therefore the charge carriers in graphene are called Dirac fermions

Denote K and K′ two nearest corners of the first Brillouin zone in the reciprocal lattice of the hexagonal crystalline structure of a graphene monolayer They are called Dirac points In the framework of the quantum field theory the spinless fermions in graphene are described by two-component quantum fields y K(r,t) and y K¢(r,t),

= r r = x y

r 1, 2 , Each of them can be considered as a spinor field of a new SU(2) symmetry group similar to the isospinors in theory of elementary particles [6–9] Thus the two-component fieldsy K(r,t) and y K¢(r,t) can be called, for example, quasi-spinors or pseudo-spinors Three Pauli matrices acting on these spinors of a new type will be denoted

|Vietnam Academy of Science and Technology Advances in Natural Sciences: Nanoscience and Nanotechnology Adv Nat Sci.: Nanosci Nanotechnol 7 (2016) 035001 (10pp) doi:10.1088 /2043-6262/7/3/035001

Original content from this work may be used under the terms

of the Creative Commons Attribution 3.0 licence Any

further distribution of this work must maintain attribution to the author (s) and

the title of the work, journal citation and DOI.

2043-6262 /16/035001+10$33.00 1 © 2016 Vietnam Academy of Science & Technology

τi, i = 1, 2, 3 It was shown [5] that for the system of free

Dirac fermions with wave functions having wave vectors in

the neighbors of Dirac points we can use following

approx-imate expression of the Hamiltonian

( )

⁎

t





-+

1

F

where vF is the speed of Dirac fermions

In the study of the interaction between Dirac fermions

and electromagnetic field we must consider electromagnetic

field in the physical three-dimensional space Let us chose to

use the Cartesian coordinate system as follows: the plane of

graphene monolayer is the xOy coordinate plane and,

there-fore, the Oz axis is perpendicular to this plane Then the

coordinate of a point in the physical three-dimensional space

is denoted {r, z} = {x, y, z} The electromagnetic field is

described by the vector potentialA(r, z, t) and scalar potential

field j(r, z, t) From formula (1) it follows that the interaction

Hamiltonian of the system of Dirac fermion fields and

electromagneticfield has the expression

⁎

ò ò

t t

=

+

+

+

+

+

The functionA(r, o, t) and j(r, o, t) of variable r and t are

vectorfield A(r, t) and scalar field j(r, t) on the graphene plane:

=

=

, def. , , ,

Since they are the limits of the vector potentialA(r, z, t) and

the scalar potentialj(r, z, t) of the electromagnetic field when

the point{r, z} tends to the limit {r, o} in the xOy coordinate

plane, which is the boundary of the upper or lower half-space

above or under the single-layer graphene plane, we shortly call

them vector potential and scalar potential of the boundary

electromagneticfield on the graphene plane In terms of A(r, t)

andj(r, t) the interaction Hamiltonian (4) has the expression

⁎

ò ò

t t

=

+ + +

+

+

The charge current densityJ(r, t) and charge density ρ(r, t)

are expressed in terms of Hint(t) by the definition

d d

=

t

J r

A r

,

int

and

dj

t

r

r

,

int

From formula(4) of Hint(t) we have

t t

= +

+

F

and

( )

8

Using Dirac equations derived from Hamiltonian (1) we can demonstrate that charge densityρ(r, t) and charge current densityJ(r, t) satisfy well-known continuity equation

( )

¶

t

r

J r

,

Recently there arose a significant attention to the study of electromagnetic interaction processes in graphene such as non-linear optical processes[10] and plasmon resonance [11–17] The purpose of present work is to elaborate the basics of quantum field theory of electromagnetic interaction processes in the single-layer graphene starting from the interaction Hamiltonian(4)

It was well-known that the most popular approach for the theoretical study of dynamical processes in any quantum system with a given interaction Hamiltonian Hint(t) is to work

in the interaction picture in which thefield operators satisfy the Heisenberg quantum equation of motion of the freefields and, therefore, have the same expressions in terms of the destruction and creation operators of their quanta as those of the corresponding free fields This special feature of the interaction picture permits to establish exact and clearly for-mulated mathematical rules in the calculation of physical quantities of the quantum system These quantities are expressed in terms of interaction Hamiltonian(4)

Since we have chosen to work in the interaction picture, for the study of electromagnetic interaction processes in single-layer graphene we must use the expressions determining the boundary free electromagneticfield on graphene as well as the boundary limits on the graphene plane of Green functions of the free electromagnetic field, which also briefly called the boundary Green functions of free electromagnetic field on graphene plane The boundary electromagnetic field and the 2-point boundary Green function of free electromagneticfield are investigated in the subsequent section 2 In section 3 the 2-point Green functions of boundary electromagneticfield on the graphene plane and Dirac fermionfields of the interacting system of electromagneticfield and Dirac fermions in graphene are studied by means of the perturbation theory The conclu-sion and discusconclu-sion are presented in section4

2 Boundary free electromagneticfield and boundary 2-point Green functions of free electromagneticfield

In order to study electromagnetic interaction processes in graphene it is necessary to use explicit expressions of vector and scalar potentials A(r, t) and j(r, t) of the boundary free electromagneticfield as well as the boundary Green functions

of free electromagneticfield on the graphene plane, and also Green functions of free Dirac fermionfields The laters were

studied in[18] and we shall use the results of this work In the

present section we study vector and scalar potentialsA(r, t)

and j(r, t), and boundary 2-point Green function of free

electromagneticfield The study of boundary 2n-point Green

functions of electromagneticfield with n > 1 by means of the

perturbation theory will be carried out in section3

Vectorfield A(r, t) and complex scalar field ij(r, t) are

the components of the limit at z→ 0 of a 4-vector field Aμ(x)

in the(3 + 1)-dimensional Minkowski space–time: μ = 1, 2,

3, 4; x= {r, z, it}; Aμ(x) = {A(x), ij(x)}

For simplifying equations and calculations in classical

electrodynamics [19] one frequently imposes on Aμ(x) the

Lorentz condition

( )

( )

¶

m m

However, in quantum electrodynamics (QED) this

con-dition cannot hold for the quantum vectorfield Aμ(x) Instead

of condition (10) it was reasonably proposed to assume

another similar but weaker condition imposed on the state

vectors |F ñ1 and |F ñ2 of all physical states of the system:

áF ¶

¶ F ñ = ⋅

m m

In the fundamental research works on QED [20,21] it

was demonstrated that due to the weak Lorentz condition(11)

the electromagnetic waves in the states with longitudinal and

scalar polarizations play no role in all physical processes

Therefore in the Hilbert space of state vectors of all physical

states of the system the vector potentialfield A(x) = A(r, z, t)

has following effective Fourier expansion formula

{

}

[ ( ) ]

ò ò å

x x

p

=

W

´ +

s

=

+ -W

l c

c

k

1

e

l

lz l t

l

k kr k k k

kr k

k

3 2

1

2 2 whereξσklwithσ = ±1 are two 3-component complex unit

vectors characterizing two transversely polarized states of the

electromagnetic plane waves with the wave vector{k, l} The

vectorfield A(r, t) of boundary electromagnetic field is

( )

( )

ò ò å

p

=

W

s

=

l

k

1

13

k kr k k k kr k k

3 2

1

It looks like a linear combination of an un-numerable set

of vector quantumfields A(r, t)l, each of them being labeled

by a value of the continuous index l:

( )

( ) { }

( )

[ ( ) ]

⁎ [ ( ) ]

x

p

=

W

s

=

-W

t

c

k

1

e

14

l

l t l

k kr k k k

kr k

k

1

The vectorfield A(r, t)lwith an index l≠ 0 looks like a massive free vector field with the mass | |l in the (2 +

1)-dimensional Minkowski space–time

Note that matrix elements of scalar field j(r, t) of boundary electromagneticfield between two state vectors |F ñ1 and |F ñ2 of any pairs of two physical states of the system always vanish Therefore there is no necessity to write the explicit expression ofj(r, t)

The 2-point Green function of free electromagnetic field

at T= 0 is defined as follows:

D r, ,z t i T A r, ,z t A 0,0,0 , 15 where the symbol á denote the average of insertedñ expression (containing field operators) in the ground state

| ñ0 of the Dirac fermion gas

This ground state can be considered as the vacuum of electromagneticfield

In QED it was shown that due to the gauge invariance of the theory one always can chose to work in such a gauge that the boundary limit



z 0

of the 2-point Green function (16) of free electromagnetic field has following simple formula [20,21]

( )

( )

( )

ò ò ò

p

=

18

k t

kr

i

0

0

with

d

=

k

k

0

Formula(18) shows thatD mn(r,t)is a linear combination

of an un-numerable set of functionsD mn(r,t)l labeled by the continuous index l

( )

( )

ò ò

d p

=

´

k

1

2 2

0

0

Thus formulae(14) and (15) together with formulae (20) and(21) clearly demonstrated that the boundary vector field A (r, t) and the boundary 2-point Green function D mn(r,t)are effectively represented as the linear combinations (13) and (20) of the quantum vector field A(r, t)land the 2-point Green functions D mn(r,t)l labeled by a continuous index l having real values in the whole infinite interval from −∞ to +∞ It

is interesting to note that quantum boundary vectorfields A(r,

t)lwith l≠ 0 look like quantum massive vector fields with the

continuous mass/l/, and boundary 2-point Green functions

( )

mn

D r,t l also look like those of these quantum massive

vectorfields

3 Perturbation theory

In the present section we develop perturbation theory for

studying electromagnetic interaction processes in single-layer

graphene The S-matrix is expressed in terms of interaction

Hamiltonian(4) as follows

{ ò ( ) } ( )

where the integration with respect to the time variable t is

performed over the whole real axis from −∞ to +∞ By

expanding the exponential function in rhs of formula(22) into

power series, we express S-matrix in the form of a series

( ) ( )

å

= +

=

¥

n n

1 where the nth order term S( n)is

( )

!

( )

ò ò ò

S

i

d d d

n n

1 2 int 1 int 2 int

As the first example of the application of perturbation

theory let us study boundary 2-point Green functions of the

interacting system comprising electromagnetic field in the

whole three-dimensional physical space and the Dirac

fer-mions moving only in the graphene plane They are expressed

in terms of the boundary limits at z→ 0 of the vector potential

field A(r, z, t) and scalar potential field j(r, z, t)

=

=





⎧

⎨

⎪

⎩⎪

z z

0 0

Dirac fermionfieldsy K K, ¢(r,t)and S-matrix as follows:

( - ¢ - ¢ = -) á { ( ) ( ¢ ¢ ñ)} ( )

á ñ

S

ij

j

i

with i, j= 1, 2, 3,

( - ¢ - ¢ = -) á { j( )j( ¢ ¢ ñ)} ( )

á ñ

S

00

and

( )

ab

¢

S

,

K K

,

Using expansion formula(23) of S-matrix, we write each

of 2-point Green functions(26)–(28) in the form of the series: ( - ¢ - ¢ =) å ( - ¢ - ¢)( ) ( )

=

¥

n

0

2

( - ¢ - ¢ =) å ( - ¢ - ¢)( ) ( )

=

¥

n

n

00

0

Dab ¢ - ¢ - ¢ = Dab - ¢ - ¢

=

¥

¢

K K

n

,

0

n running all non-negative integers n = 0, 1, 2 K In the present work we consider the simple case of the Dirac fermion gas at T = 0 with the Fermi level EF = 0 The extension to other more general cases will be done in subsequent works

The first term in the series (29) is a special case of functionD mn(r- ¢r,t- ¢t)determined by formulae(18) and (19):

( )

( )

( )

ò ò ò

p

´ - ¢ - - ¢

k

32

ij

ij

k r

0

i

0 0

0

with

( )=d

k

k

0

Thefirst term in the series (30) can be directly obtained also from formulae(18) and (19):

( )

( )

( )

ò ò ò

p

´ - ¢ - - ¢

k

34

k t t

k r r

i

00 0 0

0

with

( )=

k

k

0 2

The expressions of 2-point Green functions of free Dirac fermion fields can be easily obtained from results demon-strated in[18]:

( )

( )

[ ( ) ( )]

( )

ò ò

p

´

´ D

ab

ab

¢

- ¢ - - ¢

¢

k

k

e

K K

k t t

K K

k r r

i ,

0 0

0

with

( )

*

,

i

,

0 0

0

0

( )= =| |= + ( )

2 2 2

( )

( )

( )

( ) ( ) ( ) ( )

h h

=

=

-q q q q

-⎧

⎨

⎪⎪

⎩

⎪

⎪

⎛

⎝

⎜ ⎞⎠⎟

⎛

⎝

⎜ ⎞⎠⎟

u

v

k

k

1 2

e

1 2

e

39

K

K

k k k k

i 2

i 2

i 2

i 2

/ / / /

( )

( )

( )

( ) ( ) ( ) ( )

h h

=

q q q q

¢

-¢

-⎧

⎨

⎪⎪

⎩

⎪

⎪

⎛

⎝

⎠

⎟

⎛

⎝

⎠

⎟

u

v

k

k

1 2

e

1 2

e

40

K

K

k k k k

i 2

i 2

i 2

i 2

/ / / /

k

1

η and η′ being two arbitrary phase factors | |h = ¢ = 1.| |h

In order to calculate D ij(r- ¢r,t- ¢t) let us consider

matrix element

( )

!

( )

ò ò

=

i

j

j

2

i

2

1 2 int 1 int 2 i

Using formula (4) of the interaction Hamiltonian Hint(t), we

rewrite relation (42) in the form explicitly containing all

quantum field operators of electromagnetic field and Dirac

fermionfields:

!

( )

( )

⁎

⁎

⁎

ò ò ò ò

å å

å

j

´

+

´

´

+

´

´

+

´

= =

+

+

=

+

+

+

+

⎛

⎝

⎜

43

K

K

n K K

K

m K

n

K

K

n K

2

F2

1

2

1

2

1 1 2 2

2

F 1 2

2

1 1

According to formulae (26), in order to find

( - ¢ - ¢)( )

D ij r r,t t 2 it is necessary to calculate also the

average of 2nd order terms S(2)of the S-matrix We have

( )

áS ñ = -i t t áT H t H t ñ⋅

2

2

1 2 int 1 int 2 Using formula (4) of the interaction Hamiltonian Hint(t) we rewrite this matrix element in the form explicitly containing field operators

( )

( )

!

( )

⁎

⁎

⁎

ò ò ò ò

å å

å

j

-á

´ +

´

´ +

´

´ +

´

= =

+

+

= +

+

+

+

⎛

⎝

⎜

45

e v T A t A t

i

n m

K

n K K

n K K

m K K

n n K

n K K

n K

2 F 1 2 1

2

1 1 2 2

2 F 1

2

1 1 2 2

2 2 2 2

2

1 1 2 2

1 1 1 1

1 1 1 1

2 2 2 2

In order to demonstrate the calculation method let us consider in detail the first term in rhs of equation (43) According to the well-known Wick theorem for the average

of any product of quantum freefields in the ground state of the system we have

( )}

( )

( ) ( )

t y

´

t

r

,

46

K

1 1 2 2

2 2

0

1 2 1 2 0

1 2 1 2 0

2 1 2 1 0

a similar formula with the replacement K → K′, τn→ *t , n

τm→ *t , m and

⁎

⁎

t y

t y

´

´

t

r

K

m K

K

n

1 1 2 2

1 1 2 2

2 2

It is obvious that the second term in rhs of equation(43)

vanishes The third term comprises following expression

( )

-+ +

48

ij

1 1 2 2 1 1

0

00 1 2 1 2 0

1 2 1 2 0 2 1 2 1 0

and a similar one with the replacement K→ K′

For calculating D ij(r- ¢r,t- ¢t)( )2 it is necessary to

calculate also matrix elementáS( ) 2ñ.The first term in rhs of

formula(45) contains following expression

( )

( )

t t

á

-+ +

,

49

K

2 1 2 1 0

and a similar one with the replacement K → K′, τn→ *t , n

τm→ *t m The second term vanishes The third term contains

following expressions

( )

( )

y

á

-+ +

50

K

00 1 2 1 2 0

a similar one with the replacement K→ K′, and

y

á

= á

+

+

K

According to the definition (26) we have

( )

( )

52

2

On the basis of relations(43)–(51) it can be demonstrated

that function(52) has following expression

( )

( )

( )

( ) ( ) ( )

ò ò ò ò

å å

t t

- ¢ - ¢ =

= =

¢

¢

, ,

,

,

53

ij

in

jm

K

n K

m K

F 2

1 2

1

2

1 1 0

1 2 1 2 0

2 1 2 1 0

1 2 1 2 0

2 1 2 1 0

In order to calculate D00(r- ¢r,t- ¢t)( )2 we must consider matrix element

( )

!

( )

⁎

⁎

⁎

ò ò ò ò

å å

å

´ +

´

´ +

´

´

= =

+

+

= +

+

+

⎛

⎝

⎜

54

n m

K

n K K

n K K

m K K

m K

n

n K

n K K

n K

2 F 1 2 1

2

1 1 2 2

2 F 1

2

2 2 2 2

2

1 1 2 2

2 2 2 2

According to the definition (27) we have

( )

( )

2

Let us calculate the first matrix element in rhs of relation (55)

( )

!

( )

⁎

⁎

⁎

ò ò ò ò

å å

å

´ +

´

´ +

´

´ +

´

= =

+

+

=

+

+

+

+

⎛

⎝

⎜

56

n m

K

n K K

n K K

K

m K

n

n K

n K K

n K

2 F 1 2 1

2

1 1 2 2

2 F 1

2

1 1

2 2 2 2

2

1 1 2 2

1 1 1 1

1 1 1 1

2 2 2 2

Consider first term in rhs of equation (56) It contains

expression

( )

⁎

⁎

( ) ( )

t

t

t

t

´

+

´

+

+

¢

¢

,

,

57

K

K

n K K

K

m K

nm

n

K

m

K

1 1 2 2

1 2 1 2 0

2 1 2 1 0

1 2 1 2 0

2 1 2 1 0 Second term vanishes and third term contains expression

( )

( ) ( ) ( ) ( )

´

+

´

+

+

¢

¢

,

,

58

K

K

K

K

1 1 2 2

1 2 1 2 0

2 1 2 1 0

1 2 1 2 0

2 1 2 1 0

Using formulae(45), (49) and (50) and similar ones for

determiningáS( ) 2ñtogether with relations(56)–(58), we obtain

( )

( )

( ) ( ) ( ) ( ) ( ) ( )

ò ò ò ò

- ¢ - ¢ =

-¢

¢

, ,

,

59

K K K K

1 2 1 2 0

2 1 2 1 0

1 2 1 2 0

2 1 2 1 0

Consider now Green functions (28) of Dirac fermions

These Green functions are expanded into the series of the

form(31) The second order term in each series is determined

by formula

( ) ( )

( )

ab

¢

,

K K

For the fields y K(r,t) and y K(r¢ ¢,t)+ the first matrix element in rhs formula(59) has the form

( )

!

( )

⁎

⁎

⁎

ò ò ò ò

å å

å

´ +

´

+

´

´ ¢ ¢ +

´

a b

+

= =

+ +

+

=

+

+

⎛

⎝

⎜

61

n m

K

n K K

n K K

K

m K

n

n K K

n K

2 F 1 2 1

2

1 1 2 2

2 F 1

2

1 1 2 2

2 2 2 2

2

1 1 2 2

1 1 1 1

1 1 1 1

Thefirst matrix element in rhs equation (61)

( )

⁎

⁎

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

t t

t t

t t t t

´ +

´

g b b b

g g

g b b b

ab

+ +

+

¢

¢

, ,

, , , ,

,

,

62

K

K

n K K

K

m K K

nm K

m K

nm K

nm

n K

m K

1 1 2 2

1 2 1 2 0

2 1 2 1 0

2 1 2 1 0

0

1 2 1 2 0

1 2 1 2 0

2 1 2 1 0

1 2 1 2 0

2 1 2 1 0

2 2 2

2 1

1 1 1

The second matrix element in rhs equation(61) vanishes,

and the third one is

( )

´

a b

a b

ab

+

+

63

K

K

K

1 1 2 2

0

00 1 2 1 2 0

2

1

Using relations(61)–(63) and similar ones together with

formulae determiningáS( ) 2ñand formula(60), we obtain

( )

( )

( )

( ) ( ) ( )

( ) ( )

ò ò ò ò

ò ò ò ò

t

-ab

g g

g b b b

aa

a a

a b

= =

, , , ,

,

64

K

K

n K

nm

K K

K

F 2

1

2

1

2

1 1 0

1 2 1 2 0

1 2 1 2 0

2

1 2 1 2 0

1 2

2 2 2

1

1 2

2

and

( )

( )

( ) ⁎

( )

( )

( )

( )

ò ò ò ò

ò ò ò ò

t

-ab

g g

g b b b

aa

a a

a b

¢

= =

¢

¢

¢

¢

¢

¢

, , , ,

,

65

K

K

n K

m

K

nm

K

K

K

1

2

1

2

1 1 0

1 2 1 2 0

1 2 1 2 0

2

1 2 1 2 0

1 2

2 2 2

1

1 2

2

Thus the second order terms D ij(r- ¢r,t- ¢t)( )2 and

( - ¢ - ¢)( )

D00 r r,t t 2 of the boundary 2-point Green

func-tions of electromagneticfield are determined by formulae (53)

and(59) They can be represented by the Feynman diagram in figure1(a) The second order termsDab K (r- ¢r,t- ¢t)( ) 2 and

Dab K¢ r- ¢r,t - ¢t 2 of the Dirac fermion fields are deter-mined by formulae(64) and (65) They can be represented by the Feynman diagram infigure1(b)

For shortening expressions(53) and (59) of second order terms of boundary 2-point Green functions of electromagnetic field we introduce the self-energy parts of boundary electro-magneticfield

( )

( ) ( )

t t

t t

-¢

¢

r r

,

,

,

66

nm

n K

n K

m K

2

2 1 2 1 0

1 2 1 2 0

2 1 2 1 0 and

( )

( ) ( )

( ) ( )

¢

¢

,

,

,

67

K K

K K

00 1 2 1 2

2

1 2 1 2 0

2 1 2 1 0

1 2 1 2 0

2 1 2 1 0 Then formulae(53) and (59) become

( )

( )

( )

( )

ò ò ò ò å å

= =

, , ,

68

ij

in

nm mj

2

1 2

1 2

1 1 0

and

( )

( )

( )

ò ò ò ò

, ,

00 1 2 1 2

Similarly, for shortening expressions (64) and (65) of second order terms of 2-point Green functions of Dirac fer-mion fields we introduce the self-energy parts of Dirac fer-mion fields

( )

( ) ( )

,

70

nm

nm

,

2 ,

1 2 1 2 0

1 2 1 2 0

and

( )

( ) ( )

71

00 1 2 1 2 0

Then formulae(64) and (65) can be rewritten in the new

forms

( )

( )

( )

( )

( )

( )

( )

ò ò ò ò

ò ò ò ò

å å t

t

a g g g

g b

b b

aa

a a

a b

= =

72

t t

t t

t t

t t

t t

t t

r r

r r

r r

r r

r r

r r

,

, ,

, , ,

n m

m K

K K K

2

1 1 0 1

2 1

2

,

1 2 1 2

1 1 0

1 2 1 2

1

1 1 1 2

2 2 2

1

1 2 2

and

( )

( )

( )

⁎

⁎ ( )

( )

( )

ò ò ò ò

ò ò ò ò

å å t

t

+

ab

aa

a g g g

g b

b b

aa

a a

a b

¢

¢

= =

¢

¢

¢

¢

¢

,

, ,

, ,

K

K

K

m K

K K K

2

1 1 0

1 2

1

2

,

1 1 0

1

1 1 1 2

2 2

2

1

1 2

2

The self-energy parts (66) and (67) of boundary

electromagneticfield are represented by Feynman diagram in

figure2(a), and the self-energy parts (70) and (71) of Dirac

fermion fields are represented by Feynman diagram in

figure2(b)

Performing the Fourier transformation of 2-point Green

functions

( )

( )

( )

( )

( )

( )

ò ò

ò ò

ò ò

=

=

w w

74

t

t

kr kr kr

0,2

4

, 0,2

4

and of self-energy parts

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

ò ò

ò ò

ò ò

ò ò

w w

t t t t

75

t

nm

nm

K K

kr kr kr kr

4

, ,

4

i

, ,

,

4

we rewrite relations(68), (69) and (72), (73) in the compact forms of algebraic equations

( )

å å

w

´

= =

D

k

ij

mj

2 1 2

1

2

0

0

˜ ( w)( )= ˜ ( w)( )P˜ ( w) ˜( )( w) ( )

D00 k, 2 D00 k, 0 00 k, D00 k, , 77

0

and

( )

( )

å å

= =

78

K

K

n

nm K

m K

2 1 2

1

2

0

Figure 1.Representation of second order terms of(a) boundary 2-point Green functions of electromagnetic field and (b) 2-point Green functions of Dirac fermionfields Continuous lines represent Dirac fermion fields and wavy lines represent electromagnetic field

Figure 2.Representation of self-energy of(a) boundary electro-magneticfield and (b) Dirac fermion fields Continuous lines represent Dirac fermionfields and wavy lines represent electro-magneticfield