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

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Home Search Collections Journals About Contact us My IOPscience Basics of quantum field theory of electromagnetic interaction processes in single-layer graphene This content has been downloaded from IOPscience Please scroll down to see the full text 2016 Adv Nat Sci: Nanosci Nanotechnol 035001 (http://iopscience.iop.org/2043-6262/7/3/035001) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 80.82.78.170 This content was downloaded on 10/01/2017 at 09:20 Please note that terms and conditions apply You may also be interested in: Advanced Solid State Theory: Elements of many-particle physics T Pruschke Quantum field theory of photon–Dirac fermion interacting system in graphene monolayer Bich Ha Nguyen and Van Hieu Nguyen Two-point Green functions of free Dirac fermions in single-layer graphene ribbons with zigzag and armchair edges Van Hieu Nguyen, Bich Ha Nguyen, Ngoc Dung Dinh et al On the relation between reduced quantisation and quantum reduction for spherical symmetry in loop quantum gravity N Bodendorfer and A Zipfel Lectures on Yangian symmetry Florian Loebbert An analytic regularisation scheme on curved space–times with applications to cosmological space–times Antoine Géré, Thomas-Paul Hack and Nicola Pinamonti Scattering on two Aharonov–Bohm vortices E Bogomolny Photon transport in a one-dimensional nanophotonic waveguide QED system Zeyang Liao, Xiaodong Zeng, Hyunchul Nha et al | Vietnam Academy of Science and Technology Advances in Natural Sciences: Nanoscience and Nanotechnology Adv Nat Sci.: Nanosci Nanotechnol (2016) 035001 (10pp) doi:10.1088/2043-6262/7/3/035001 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 May 2016 Accepted for publication 31 May 2016 Published 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 electromagnetic field 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 electromagnetic field and Dirac fermions fields contains the limits at graphene plane of electromagnetic field vector and scalar potentials which can be shortly called boundary electromagnetic field The study of S-matrix requires knowing the limits at graphene plane of 2-point Green functions of electromagnetic field which also can be shortly called boundary 2-point Green functions of electromagnetic field 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 electromagnetic field as well as of 2-point Green functions of Dirac fermion fields 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 Introduction 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 twocomponent quantum fields y K (r, t ) and y K ¢ (r, t ), r = {r1, r2} = {x, y} 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 fields y 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 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)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 © 2016 Vietnam Academy of Science & Technology Adv Nat Sci.: Nanosci Nanotechnol (2016) 035001 V H Nguyen τi, i = 1, 2, 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 approximate expression of the Hamiltonian HG = v F ò dr {y K (r, t )+t (-i) y K (r, t ) + y K ¢ (r , t )+t ⁎ ( - i) y K ¢ (r , t )}, From formula (4) of Hint(t) we have J (r , t ) = ev F {y K (r , t )+ty K (r , t ) + y K ¢ (r , t )+t ⁎y K ¢ (r , t )} and (1 ) r (r , t ) = e {y K (r , t )+y K (r , t ) + y K ¢ (r , t )+y K ¢ (r , t )}⋅ (8 ) 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, therefore, 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 potential A(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 electromagnetic field has the expression Hint (t ) = ev F Using Dirac equations derived from Hamiltonian (1) we can demonstrate that charge density ρ(r, t) and charge current density J(r, t) satisfy well-known continuity equation ¶r (r , t ) (9 ) + J ( r , t ) = ⋅ ¶t Recently there arose a significant attention to the study of electromagnetic interaction processes in graphene such as nonlinear 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 singlelayer 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 the field operators satisfy the Heisenberg quantum equation of motion of the free fields 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 formulated 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 singlelayer graphene we must use the expressions determining the boundary free electromagnetic field 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 electromagnetic field are investigated in the subsequent section In section the 2-point Green functions of boundary electromagnetic field on the graphene plane and Dirac fermion fields of the interacting system of electromagnetic field and Dirac fermions in graphene are studied by means of the perturbation theory The conclusion and discussion are presented in section ò dr {y K (r, t )+t y K (r, t ) + y K ¢ (r , t )+t ⁎ y K ¢ (r , t )} A (r , o , t ) ò + e dr {y K (r , t )+ y K (r , t ) + y K ¢ (r , t )+y K ¢ (r , t )} j (r , o , t ) (2 ) The function A(r, o, t) and j(r, o, t) of variable r and t are vector field A(r, t) and scalar field j(r, t) on the graphene plane: A (r , t ) def = A (r , o , t ) , def j (r , t ) = j (r , o , t ) (3 ) Since they are the limits of the vector potential A(r, z, t) and the scalar potential j(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 electromagnetic field on the graphene plane In terms of A(r, t) and j(r, t) the interaction Hamiltonian (4) has the expression Hint (t ) = ev F ò dr {y K (r, t )+t y K (r, t ) + y K ¢ (r , t )+t ⁎ y K ¢ (r , t )} A (r , t ) ò + e dr {y K (r , t )+y K (r , t ) + y K ¢ (r , t )+y K ¢ (r , t )} j (r , t ) (7 ) (4 ) Boundary free electromagnetic field and boundary 2-point Green functions of free electromagnetic field The charge current density J(r, t) and charge density ρ(r, t) are expressed in terms of Hint(t) by the definition J (r , t ) = dHint (t ) d A (r , t ) (5 ) r (r , t ) = dHint (t ) ⋅ dj (r , t ) (6 ) 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 electromagnetic field as well as the boundary Green functions of free electromagnetic field on the graphene plane, and also Green functions of free Dirac fermion fields The laters were and Adv Nat Sci.: Nanosci Nanotechnol (2016) 035001 V H Nguyen studied in [18] and we shall use the results of this work In the present section we study vector and scalar potentials A(r, t) and j(r, t), and boundary 2-point Green function of free electromagnetic field The study of boundary 2n-point Green functions of electromagnetic field with n > by means of the perturbation theory will be carried out in section Vector field A(r, t) and complex scalar field ij(r, t) are the components of the limit at z → 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 ¶Am (x ) ¶xm = 0⋅ The vector field A(r, t)l with an index l ≠ 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 electromagnetic field between two state vectors |F1ñ and |F2 ñ of any pairs of two physical states of the system always vanish Therefore there is no necessity to write the explicit expression of j(r, t) The 2-point Green function of free electromagnetic field at T = is defined as follows: Dmn (r , z , t ) = - i T {Am (r , z , t ) An (0,0,0)} ñ , where the symbol áñ denote the average of inserted expression (containing field operators) in the ground state | 0ñ of the Dirac fermion gas (10) áñ = |  | 0ñ⋅ However, in quantum electrodynamics (QED) this condition cannot hold for the quantum vector field Aμ(x) Instead of condition (10) it was reasonably proposed to assume another similar but weaker condition imposed on the state vectors |F1ñ and |F2 ñ of all physical states of the system: áF1| ¶Am (x ) ¶xm | F2 đ = 0⋅ (2 p ) ò ò så =1 dk dl (11) Dmn (r , t ) = lim Dmn (r , z , t ) of the 2-point Green function (16) of free electromagnetic field has following simple formula [20, 21] Dmn (r , t ) = (2 p )4 ò dkò dlò dk ei (kr-k t ) D˜mn (k, l, k 0) (18) with W (k , l ) D˜mn (k, l , k 0) = dmn + x ⁎s kl e-i [kr+ lz -W (k, l ) t ] cs+kl }, k2 + l , ò ò dk dl å s =1 i (k + l2 ⋅ - k 02 - io) (19) Formula (18) shows that Dmn (r, t ) is a linear combination (12) dl Dmn (r , t )l (20) 2p of an un-numerable set of functions Dmn (r, t )l labeled by the continuous index l ò Dmn (r , t ) = where ξσkl with σ = ±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 vector field A(r, t) of boundary electromagnetic field is A (r , t ) = (2 p ) (17) z0 ´ {xs kl ei [kr+ lz -W (k, l ) t ] cs kl W (k , l ) = (16) This ground state can be considered as the vacuum of electromagnetic field 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 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 potential field A(x) = A(r, z, t) has following effective Fourier expansion formula A (r , z , t ) = (15) Dmn (r , t )l = dmn W (k , l ) ´ ´ {xs kl ei [kr-W (k, l ) t ] cs kl + x ⁎s kl e-i [kr-W (k, l ) t ] cs+kl }⋅ (2 p ) i (k + ò dkò dk ei (kr-k t ) l2 ⋅ - k 02 - io) (21) (13) 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 Dmn (r, t ) are effectively represented as the linear combinations (13) and (20) of the quantum vector field A(r, t)l and the 2-point Green functions Dmn (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 vector fields A(r, t)l with l ≠ look like quantum massive vector fields with the It looks like a linear combination of an un-numerable set of vector quantum fields A(r, t)l, each of them being labeled by a value of the continuous index l: A (r , t )l = 2p ò dkså =1 {xs kl ei [kr-W (k, l ) t ] cs kl W (k , l ) + x ⁎s kl e-i [kr-W (k, l ) t ] cs+kl }⋅ (14) Adv Nat Sci.: Nanosci Nanotechnol (2016) 035001 V H Nguyen continuous mass /l/, and boundary 2-point Green functions Dmn (r, t )l also look like those of these quantum massive vector fields Using expansion formula (23) of S-matrix, we write each of 2-point Green functions (26)–(28) in the form of the series: Ơ Dij (r - rÂ, t - t ¢)(2n) , Dij (r - r¢ , t - t ¢) = (29) n=0 Perturbation theory ⎡ ⎤ exp ⎢⎣ - i dt Hint (t)⎥⎦ , { } ò ( - i) n S (n) = ò ò is (32) ò with D˜ ij (k, l , k 0)(0) = dij (24) 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 fermions moving only in the graphene plane They are expressed in terms of the boundary limits at z → of the vector potential field A(r, z, t) and scalar potential field j(r, z, t) ⎧ A (r , t ) = lim A (r , z , t ) , ⎪ z0 ⎨ j ( ) = j (r , z , t ) , r , t lim ⎪ ⎩ z0 D 00 (r - r¢ , t - t ¢)(0) = (2 p )4 (26) (27) Sñ ⋅ - k 02 - io) (35) (2 p ) ò dkò dk (36) with K ,K ¢ ˜ ab D (k, k 0)(0) = ⋅ + l2 ´ ei [k (r- r¢) - k (t - t ¢)] K ,K ¢ ˜ ab ´D (k, k 0)(0) , - r¢ , t - t ¢) =- i i (k The expressions of 2-point Green functions of free Dirac fermion fields can be easily obtained from results demonstrated in [18]: and T {S y aK , K ¢ (r , t ) y bK , K ¢ (r¢ , t ¢)+} đ ò dkò dlò dk K ,K ¢ Dab (r - r¢ , t - t ¢)(0) = T {S j (r , t ) j (r¢ , t ¢)} đ Sñ (33) with D˜ 00 (k, l , k 0)(0) = - áS ñ ⋅ - k 02 - io) (34) (25) áT {S A i (r , t ) Aj (r¢ , t ¢)} đ + l2 ´ ei [k (r- r¢) - k (t - t ¢)] D˜ 00 (k, l , k 0)(0) with i, j = 1, 2, 3, K ,K ¢ Dab (r i (k The first term in the series (30) can be directly obtained also from formulae (18) and (19): Dirac fermion fields y K , K ¢ (r, t ) and S-matrix as follows: D 00 (r - r¢ , t - t ¢) = - i ò dkò dlò dk ´ ei [k (r - r¢) - k (t - t ¢)] D˜ ij (k, l , k 0)(0) dt1 dt2 dtn n! ´ T {Hint (t1) Hint (t2) Hint (tn)}⋅ Dij (r - r¢ , t - t ¢) = - i (2 p )4 Dij (r - r¢ , t - t ¢)(0) = (23) n=1 where the nth order term S (31) n running all non-negative integers n = 0, 1, K In the present work we consider the simple case of the Dirac fermion gas at T = with the Fermi level EF = 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 function Dmn (r - r¢ , t - t Â) determined by formulae (18) and (19): Ơ ( n) K ,K ¢ (r - r¢ , t - t ¢)(2n) , å Dab n=0 (22) å S (n), ¥ K ,K ¢ Dab (r - r¢ , t - t ¢) = 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 S=1+ (30) n=0 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 S=T ¥ å D00 (r - r¢, t - t ¢)(2n) , D 00 (r - r¢ , t - t ¢) = uaK , K ¢ (k) ubK , K ¢ (k)* k - E ( k ) + io + (28) vaK , K ¢ ( - k) vbK , K ¢ ( - k)* k + E ( k ) - io , (37) Adv Nat Sci.: Nanosci Nanotechnol (2016) 035001 E (k ) = vF k , k = | k| = k12 + k 22 , ⎧ ⎛ e-iq (k) / ⎞ ⎪ u K (k) = ⎜ ⎟ h, ⎪ ⎝ eiq (k) / ⎠ ⎨ ⎛ e-iq (k) / ⎞ ⎪ K v (k) = ⎜ ⎟ h, ⎪ ⎝- eiq (k) / ⎠ ⎩ V H Nguyen average of 2nd order terms S(2) of the S-matrix We have (38) S (2)ñ = (40) k q (k) = arctg , k1 (41) áS (2) ñ = ´ [y K (r 2, t 2)+tm y K (r 2, t 2) + y K ¢ (r 2, t 2)+t ⁎m y K ¢ (r 2, t 2)]} ñ + 2e 2v F å áT {An (r1, t1) j (r 2, t 2) n=1 áT + y K ¢ (r1, t1)+t ⁎n y K ¢ (r1, t1)] ´ [y K (r 2, t 2)+y K (r 2, t 2) (42) + y K ¢ (r 2, t 2)+y K ¢ (r 2, t 2)]} ñ + e áT {j (r1, t1) j (r 2, t 2) ´ [y K (r1, t1)+y K (r1, t1) + y K ¢ (r1, t1)+y K ¢ (r1, t1)] ´ [y K (r 2, t 2)+y K (r 2, t 2) + y K ¢ (r 2, t 2)+y K ¢ (r 2, t 2)]} đ)⋅ ò dt1ò dr1ò dt2 ò dr2 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 free fields in the ground state of the system we have 2 ⎛ ⎜e2v F2 å å áT {Ai (r , t ) Aj (r¢ , t ¢) An (r1, t1) Am (r2 , t2) ⎝ n = 1m = ´ [y K (r1, t1)+tn y K (r1, t1) + y K ¢ (r1, t1)+t ⁎n y K ¢ (r1, t1)] ´ [y K (r2 , t2)+tm y K (r2 , t2) áT {Ai (r , t ) Aj (r¢ , t ¢) An (r1, t1) Am (r2 , t2) ´ y K (r1, t1)+tn y K (r1, t1) y K (r2 , t2)+ ´ tm y K (r2 , t2)} đ + y K ¢ (r2 , t2)+t ⁎m y K ¢ (r2 , t2)]} đ + 2e2v F å áT {Ai (r , t ) Aj (r¢ , t ¢) An = - {Dij (r - r¢ , t - t ¢)(0) Dnm (r1 - r2 , t1 - t2)(0) n=1 ´ (r1, t1) j (r2 , t2)[y K (r1, t1)+tn y K (r1, t1) + Din (r - r1, t - t1)(0) Djm (r¢ - r2 , t ¢ - t2)(0) + y K ¢ (r1, t1)+t ⁎n y K ¢ (r1, t1)] ´ [y K (r2 , t2)+y K (r2 , t2) + Dim (r - r2 , t - t2)(0) Djn (r¢ - r1, t ¢ - t1)(0)} + e2 áT {Ai (r , t ) Aj (r¢ , t ¢) j (r1, t1) + ´ + (46) ´ Tr [tn DK (r1 - r2 , t1 - t2)(0) tm DK ´ (r2 - r1, t2 - t1)(0)] , + y K ¢ (r2 , t2)+y K ¢ (r2 , t2)]} đ ´ (45) ´ [y K (r1, t1)+tn y K (r1, t1) 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 fermion fields: ( - i) t ) Aj (r¢ , t ¢)} đ = 2! ò dt1ò dr1ò dt2 ò dr2 + y K ¢ (r1, t1)+t ⁎n y K ¢ (r1, t1)] ò ò {S (2) Ai (r , (-i) 2! ´ [y K (r1, t1)+tn y K (r1, t1) T {S (2) A i (r , t ) Aj (r¢ , t ¢)} đ ( - i) dt1 dt2 2! ´ T {Hint (t1) Hint (t2) A i (r , t ) Aj (r¢ , t ¢)} ñ⋅ (44) 2 ⎛ ⎜e 2v F2 å å áT {An (r1, t1) Am (r 2, t 2) ⎝ n = 1m = η and η′ being two arbitrary phase factors | h | = | h ¢ | = In order to calculate Dij (r - r¢ , t - t ¢) let us consider matrix element = ò dt1ò dt2 áT {Hint (t1) Hint (t2)} ñ⋅ Using formula (4) of the interaction Hamiltonian Hint(t) we rewrite this matrix element in the form explicitly containing field operators (39) ⎧ ⎛ eiq (k) / ⎞ ⎪ u K ¢ (k) = ⎜ ⎟ h¢, ⎪ ⎝ e-iq (k) / ⎠ ⎨ ⎛ eiq (k) / ⎞ ⎪ K¢ ⎜ v (k) = ⎟ h¢, ⎪ ⎝- e-iq (k) / ⎠ ⎩ ( - i) 2! a similar formula with the replacement K → K′, τn → tn*, τm → tm*, and j (r2 , t2)[y K (r1, t1)+y K (r1, t1) y K ¢ (r1, t1)+y K ¢ (r1, t1)] [y K (r2 , t2)+y K (r2 , t2) y K ¢ (r2 , t2)+y K ¢ (r2 , t2)]} ñ)⋅ T {Ai (r , t ) Aj (r¢ , t ¢) An (r1, t1) Am (r2 , t2) ´ y K (r1, t1)+tn y K (r1, t1) y K ¢ ´ (r2 , t2)+t ⁎m y K ¢ (r2 , t2)} đ = T {Ai (r , t ) Aj (r¢ , t ¢) An (r1, t1) Am (r2 , t2) (43) ´ y K ¢ (r1, t1)+t ⁎n y K ¢ (r1, t1) y K (r2 , t2)+ ´ tm y K (r2 , t2)} ñ = According to formulae (26), in order to find Dij (r - r¢ , t - t ¢)(2) it is necessary to calculate also the (47) Adv Nat Sci.: Nanosci Nanotechnol (2016) 035001 V H Nguyen It is obvious that the second term in rhs of equation (43) vanishes The third term comprises following expression In order to calculate D00 (r - r¢ , t - t ¢)(2) we must consider matrix element áT {Ai (r, t ) Aj (r¢, t ¢) j (r1, t1) j (r 2, t 2) y K (r1, t1)+ + áT {S (2) j (r, t ) j (r¢, t ¢)} ñ = ´ y (r1, t1) y (r 2, t 2) y (r 2, t 2)} ñ K K K (48) = - Dij (r - r¢, t - t ¢)(0) D00 (r1 - r 2, t1 - t 2)(0) K ò dt1ò dr1ò dt2 ò dr2 2 ⎛ ⎜e 2v F2 å å áT {j (r, t ) j (r¢, t ¢) An (r1, t1) Am (r 2, t 2) ⎝ n = 1m = ´Tr [D (r1 - r 2, t1 - t 2) D (r - r1, t - t1) ] (0) K (-i) 2! (0) ´ [y K (r1, t1)+tn y K (r1, t1) and a similar one with the replacement K → K′ For calculating Dij (r - r¢ , t - t ¢)(2) it is necessary to calculate also matrix element S (2)ñ The first term in rhs of formula (45) contains following expression + y K ¢ (r1, t1)+t ⁎n y K ¢ (r1, t1)] ´ [y K (r 2, t 2)+tm y K (r 2, t 2) + y K ¢ (r 2, t 2)+t ⁎m y K ¢ (r 2, t 2)]} đ áT {An (r1, t1) Am (r2 , t2) y K (r1, t1)+tn y K + 2e 2v F å áT {j (r, t ) j (r¢, t ¢) An (r1, t ) j (r 2, t ) ´ (r1, t1) y K (r2 , t2)+tm y K (r2 , t2)} ñ (54) n=1 ´ [y K (r1, t1)+tn y K (r1, t1) = iDnm (r1 - r2 , t1 - t2)(0) Tr [tn DK (r1 - r2 , t1 - t2)(0) ´ tm DK (r2 - r1, t2 - t1)(0)] + y K ¢ (r1, t1)+t ⁎n y K ¢ (r1, t1)] ´ [y K (r 2, t 2)+y K (r 2, t 2) (49) + y K ¢ (r 2, t 2)+y K ¢ (r 2, t 2)]} đ and a similar one with the replacement K → K′, τn → tn*, τm → tm* The second term vanishes The third term contains following expressions + e áT {j (r, t ) j (r¢, t ¢) j (r1, t1) j (r 2, t 2) ´ [y K (r1, t1)+y K (r1, t1)+y K ¢ (r1, t1)+y K ¢ (r1, t1)] ´ [y K (r 2, t 2)+y K (r 2, t 2) áT {j (r1, t1) j (r2 , t2) y K (r1, t1)+y K (r1, t1) y K + y K ¢ (r 2, t 2)+y K ¢ (r 2, t 2)]} ñ)⋅ ´ (r2 , t2)+y K (r2 , t2)} ñ = iD 00 (r1 - r2 , t1 - t2)(0) ´ Tr [DK (r1 - r2 , t1 - t2)(0) DK (r2 - r1, t2 - t1)(0)] According to the definition (27) we have (50) D00 (r - r¢, t - t ¢)(2) = -i [ T {S (2) j (r, t ) j (r¢, t ¢)} đ -á T {j (r, t ) j (r¢, t ¢)} đá S (2)đ ]⋅ a similar one with the replacement K → K′, and (55) T {j (r1, t1) j (r2 , t2) y K (r1, t1)+y K (r1, t1) ´ y K ¢ (r2 , t2)+y K ¢ (r2 , t2)} đ (55) = T {j (r1, t1) j (r2 , t2) y K ¢ (r1, t1)+y K ¢ (r1, t1) y K ´ (r2 , t2)+y K (r2 , t2)} ñ = (51) Let us calculate the first matrix element in rhs of relation áT {S (2) j (r, t ) j (r¢, t ¢)} đ = (-i) 2! ò dt1ò dr1ò dt2 ò dr2 2 ⎛ ⎜e 2v F2 å å áT {j (r, t ) j (r¢, t ¢) An (r1, t1) Am (r 2, t 2) ⎝ n = 1m = According to the definition (26) we have Dij (r - r¢ , t - t ¢)(2) = - i [ áT {S (2) Ai (r , t ) Aj (r¢ , t ¢) đ ´ [y K (r1, t1)+tn y K (r1, t1) - áT {Ai (r , t ) Aj (r¢ , t ¢) ñáS (2) ñ]⋅ + y K ¢ (r1, t1)+t ⁎n y K ¢ (r1, t1)] ´ [y K (r 2, t 2)+tm y K (r 2, t 2) (52) + y K ¢ (r 2, t 2)+t ⁎m y K ¢ (r 2, t 2)]} đ On the basis of relations (43)–(51) it can be demonstrated that function (52) has following expression Dij (r - r¢ , t - t ¢)(2) = - ie2v F2 + 2e 2v F å áT {j (r, t ) j (r¢, t ¢) An (r1, t1) n=1 ´ j (r 2, t 2)[y K (r1, t1)+tn y K (r1, t1) ò dt1ò dr1ò dt2 ò dr2 + y K ¢ (r1, t1)+t ⁎n y K ¢ (r1, t1)] ´ [y K (r 2, t 2)+y K (r 2, t 2) ´ å å Din (r - r1, t - t1)(0) + y K ¢ (r 2, t 2)+y K ¢ (r 2, t 2)]} ñ n = 1m = ´ Djm (r2 - r¢ , t2 - t ¢)(0) ´ {Tr [tn DK (r1 - r2 , t1 - t2)(0) tm ´DK (r2 - r1, t2 - t1)(0)] + e áT {j (r, t ) j (r¢, t ¢) j (r1, t1) j (r 2, t 2) (53) ´ [y K (r1, t1)+y K (r1, t1) + y K ¢ (r1, t1)+y K ¢ (r1, t1)] ´ [y K (r 2, t 2)+y K (r 2, t 2) + Tr [t ⁎n DK ¢ (r1 - r2 , t1 - t2)(0) + y K ¢ (r 2, t 2)+y K ¢ (r 2, t 2)]} đ)⋅ ´ t ⁎m DK ¢ (r2 - r1, t2 - t1)(0)]}⋅ (56) Adv Nat Sci.: Nanosci Nanotechnol (2016) 035001 V H Nguyen For the fields y K (r, t ) and y K (r¢ , t ¢)+ the first matrix element in rhs formula (59) has the form Consider first term in rhs of equation (56) It contains expression áT {j (r , t ) j (r¢ , t ¢) An (r1, t1) Am (r2 , t2) ´ [y K (r1, t1)+tn y K (r1, t1) = + y (r1, t1)+t ⁎n y K ¢ (r1, t1)] ´ [y K (r2 , t2)+tm y K (r2 , t2) + y K ¢ (r2 , t2)+t ⁎m y K ¢ (r2 , t2)]} đ - D 00 (r - r¢ , t - t ¢)(0) Dnm (r1 ´ {Tr [tn DK (r1 - r2 , t1 - t2)(0) ´ tm DK (r2 - r1, t2 - t1)(0)] + Tr [t ⁎n DK ¢ (r1 - r2 , t1 - t2)(0) ´ t ⁎m DK ¢ (r2 - r1, t2 - t1)(0)]}⋅ áT {S (2) y aK (r, t ) y bK (r¢, t ¢)+} đ = K¢ ò dt1ò dr1ò dt2 ò dr2 2 ⎛ ⎜e 2v F2 å å áT {An (r1, t1) Am (r 2, t 2) y aK (r, t ) y bK (r¢, t ¢)+ ⎝ n = 1m = r2 , t1 - t2)(0) (57) ´ [y K (r1, t1)+tn y K (r1, t1) + y K ¢ (r1, t1)+t ⁎n y K ¢ (r1, t1)] ´ [y K (r 2, t 2)+tm y K (r 2, t 2) + y K ¢ (r 2, t 2)+t ⁎m y K ¢ (r 2, t 2)]} ñ + 2e 2v F å áT {An (r1, t1) j (r 2, t 2) y aK n=1 ´ (r, t ) y bK (r¢, t ¢)+ [y K (r1, t1)+tn y K (r1, t1) Second term vanishes and third term contains expression áT {j (r , t ) j (r¢ , t ¢) j (r1, t1) j (r2 , t2) = (-i) 2! ´ [y K (r1, t1)+y K (r1, t1) + y K ¢ (r1, t1)+y K ¢ (r1, t1)] ´ [y K (r2 , t2)+y K (r2 , t2) + y K ¢ (r2 , t2)+y K ¢ (r2 , t2)]} đ - {D 00 (r - r¢ , t - t ¢)(0) D 00 (r1 - r2 , t1 - t2)(0) + D 00 (r - r1, t - t1)(0) D 00 (r¢ - r2 , t ¢ - t2)(0) + D 00 (r - r2 , t - t2)(0) D 00 (r¢ - r1, t ¢ - t1)(0)} ´ {Tr [DK (r1 - r2 , t1 - t2)(0) ´ DK (r2 - r1, t2 - t1)(0)] + Tr [DK ¢ (r1 - r2 , t1 - t2)(0) ´ DK ¢ (r2 - r1, t2 - t1)(0)]}⋅ + y K ¢ (r1, t1)+t ⁎n y K ¢ (r1, t1)] ´ [y K (r 2, t 2)+y K (r 2, t 2) + y K ¢ (r 2, t 2)+y K ¢ (r 2, t 2)]} ñ + e áT {j (r1, t1) j (r 2, t 2) y aK (r, t ) y bK ´ (r¢, t ¢)+ [y K (r1, t1)+y K (r1, t1) + y K ¢ (r1, t1)+ y K ¢ (r1, t1)] ´ [y K (r 2, t 2)+ y K (r 2, t 2) (58) + y K ¢ (r 2, t 2)+ y K ¢ (r 2, t 2)]} ñ)⋅ (61) The first matrix element in rhs equation (61) áT {An (r1, t1) Am (r2 , t2) y aK (r , t ) y bK (r¢ , t ¢)+ Using formulae (45), (49) and (50) and similar ones for determining S (2)ñ together with relations (56)–(58), we obtain ò ò ò ò D 00 (r - r¢ , t - t ¢)(2) = - ie2 dt1 dr1 dt2 ´ [y K (r1, t1)+tn y K (r1, t1) + y K ¢ (r1, t1)+t ⁎n y K ¢ (r1, t1)] ´ [y K (r2 , t2)+tm y K (r2 , t2) dr2 ´ D 00 (r - r1, t - t1)(0) ´ D 00 (r2 - r¢ , t2 - t ¢)(0) ´ {Tr [DK (r1 - r2 , t1 - t2)(0) ´ DK (r2 - r1, t2 - t1)(0)] + y K ¢ (r2 , t2)+t ⁎m y K ¢ (r2 , t2)]} đ K = Daa (r - r1, t - t1)(0) (tn )a1g1 D gK1g2 (r1 - r2 , t1 - t2)(0) ´ (tm )g2 b D bK2 b (r2 - r¢ , t2 - t ¢)(0) ´ Dnm (r1 - r2 , t1 - t2)(0) + Tr [DK ¢ (r1 - r2 , t1 - t2)(0) K (r - r2 , t - t2)(0) (tm )a2 g2 + Daa ´ DK ¢ (r2 - r1, t2 - t1)(0)]} ´ D gK2 g1 (r2 - r1, t2 - t1)(0) (59) ´ (tn )g1b1 D bK1b (r1 - r¢ , t1 - t ¢)(0) ´ Dnm (r2 - r1, t2 - t1)(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 K - Dab (r - r¢ , t - t ¢)(0) Dnm (r1 - r2 , t1 - t2)(0) ´ {Tr [tn DK (r1 - r2 , t1 - t2)(0) ´ tm DK (r2 - r1, t2 - t1)(0)] K ,K ¢ Dab (r - r¢ , t - t ¢)(2) = - i [ áT -á T {S (2) y aK , K ¢ (r , {y aK , K ¢ (r , t ) y bK , K ¢ (r¢ , t ) y bK , K ¢ (r¢ , + Tr [t ⁎n DK ¢ (r1 - r2 , t1 - t2)(0) t ¢)+} đ t ¢)+} đá S (2)đ ]⋅ ´ t ⁎m DK ¢ (r2 - r1, t2 - t1)(0)]}⋅ (60) (62) Adv Nat Sci.: Nanosci Nanotechnol (2016) 035001 V H Nguyen The second matrix element in rhs equation (61) vanishes, and the third one is and (59) They can be represented by the Feynman diagram in K figure 1(a) The second order terms Dab (r - r¢ , t - t ¢)(2) and K¢ Dab (r - r¢ , t - t ¢)(2) of the Dirac fermion fields are determined by formulae (64) and (65) They can be represented by the Feynman diagram in figure 1(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 electromagnetic field áTr {j (r1, t1) j (r2 , t2) y aK (r , t ) y bK (r¢ , t ¢)+ ´ [y K (r1, t1)+ y K (r1, t1) + y K ¢ (r1, t1)+ y K ¢ (r1, t1)] ´ [y K (r2 , t2)+ y K (r2 , t2) + y K ¢ (r2 , t2)+ y K ¢ (r2 , t2)]} đ K = Daa (r - r1, t - t1)(0) DaK1a2 (r1 - r2 , t1 - t2)(0) ´ DaK2 b (r2 - r¢ , t2 - t ¢)(0) D 00 (r1 - r2 , t1 - t2)(0) Pnm (r1 - r2 , t1 - t2) K + Daa (r - r2 , t - t2)(0) DaK2 a1 (r2 - r1, t2 - t1)(0) = - ie2v F2 {Tr [tn DK (r1 - r2 , t1 - t2)(0) ´ tm DK (r2-r1, t2 - t1)(0)] ´ DaK1b (r1 - r¢ , t1 - t ¢)(0) D 00 (r2 - r1, t2 - t1)(0) ´ {Tr [DK (r1 - r2 , t1 - t2)(0) DK (r2 - r1, t2 - t1)(0)] Tr [DK ¢ (r1 - r2 , t1 - t2)(0) DK ¢ (r2 - r1, t2 - t1)(0)]}⋅ + - r¢ , t - t ¢)(0) D 00 (r1 - r2 , t1 - t2 + - K Dab (r )(0) ´ and (63) P00 (r1-r2, t1 - t2) = - ie2 {Tr [DK (r1 - r2, t1 - t2)(0) ´ DK (r2 - r1, t2 - t1)(0)] Using relations (61)–(63) and similar ones together with formulae determining S (2)ñ and formula (60), we obtain K Dab (r - r¢ , t - t ¢)(2) = ie2v F2 ´ ´ ò dt1ò dr1ò dt2 ò dr2 ´ DK ¢ (r2 - r1, t2 - t1)(0)]}⋅ K (r - r1, t - t1)(0) (tn )a g å å Daa Then formulae (53) and (59) become 1 - r2 , t1 - t2)(0) Dij (r - r¢ , t - t ¢)(2) = ´ (tm )g2 b D bK2 b (r2 - r¢ , t2 - t ¢)(0) ò ò ò ò dr2Daa (r - r1, t - t1)(0) ´ DaK1a2 (r1 ´ DaK2 b (r2 K - r ¢ , t2 - )(0) (68) t ¢)(0) D 00 (r1 - r2 , t1 - t2 )(0) and (64) D 00 (r - r¢ , t - t ¢)(2) = and K¢ Dab (r - r¢ , t - t ¢)(2) = ie2v F2 ´ ´ ò dt1ò dr1ò dt2 ò dr2 ´ D 00 (r - r1, t - t1)(0) ´P00 (r1 - r2 , t1 - t2) ´ D 00 (r2 - r¢ , t2 - t ¢)(0) ò dt1ò dr1ò dt2 ò dr2 K¢ (r - r1, t - t1)(0) (t ⁎n )a g å å Daa n = 1m = D gK1¢g2 (r1 ´ (r2 - r¢ , t2 - t ¢)(0) - r2 , t1 - t2 ò dt1ò dr1ò dt2 ò dr2nå=1må=1Din ´ (r - r1, t - t1)(0) Pnm ´ (r1 - r2 , t1 - t2) Dmj ´ Dnm (r1 - r2 , t1 - t2)(0) + ie2 dt1 dr1 dt2 (67) + Tr [DK ¢ (r1 - r2, t1 - t2)(0) n = 1m = D gK1g2 (r1 (66) Tr [t ⁎n DK ¢ (r1 - r2 , t1 - t2)(0) t ⁎m DK ¢ (r2 - r1, t2 - t1)(0)]} (69) 1 - r2 , t1 - t2)(0) Similarly, for shortening expressions (64) and (65) of second order terms of 2-point Green functions of Dirac fermion fields we introduce the self-energy parts of Dirac fermion fields ´ (t ⁎m )g2 b D bK2¢ b (r2 - r¢ , t2 - t ¢)(0) ´ Dnm (r1 - r2 , t1 - t2)(0) ò ò ò ò dr2Daa (r - r1, t - t1)(0) K¢ + ie2 dt1 dr1 dt2 SgK1,gK2,¢nm (r1 - r2 , t1 - t2) = ie2v F2 D gK1,gK2¢ (r1 - r2 , t1 - t2)(0) ´ DaK1¢a2 (r1 - r2 , t1 - t2 ´ DaK2¢ b (r2 - r¢ , t2 - t ¢)(0) D 00 (r1 - r2 , t1 - t2)(0) )(0) ´ Dnm (r1 - r2 , t1 - t2)(0) (70) and (65) SaK1,aK2¢ (r1 - r2 , t1 - t2) = ie2DaK1,aK2¢ (r1 - r2 , t1 - t2)(0) t ¢)(2) Thus the second order terms Dij (r - r¢ , t and D00 (r - r¢ , t - t ¢)(2) of the boundary 2-point Green functions of electromagnetic field are determined by formulae (53) ´ D 00 (r1 - r2 , t1 - t2)(0) (71) Adv Nat Sci.: Nanosci Nanotechnol (2016) 035001 V H Nguyen Figure Representation of second order terms of (a) boundary 2-point Green functions of electromagnetic field and (b) 2-point Green functions of Dirac fermion fields Continuous lines represent Dirac fermion fields and wavy lines represent electromagnetic field Then formulae (64) and (65) can be rewritten in the new forms K Dab (r - r¢, t - t ¢)(2) = K ò dt1ò dr1ò dt2 ò dr2 Daa ´ (r - r1, t t1)(0) å å (tn )a1g1 SgK1g2, nm ´ n = 1m = ´ (r1 - r 2, t1 - t 2)(tm )g2 b ´ D bK2 b (r - r¢, t - t ¢)(0) + ò dt1ò dr1ò dt2 ò Figure Representation of self-energy of (a) boundary electro- magnetic field and (b) Dirac fermion fields Continuous lines represent Dirac fermion fields and wavy lines represent electromagnetic field K dr 2Daa ´ (r - r1, t - t1)(0) SaK1 a2 ´ (r1 - r 2, t1 - t 2) D aK2 b ´ (r - r¢, t - t ¢)(0) and of self-energy parts (72) and K¢ (r Dab - r¢ , t - t ¢)(2) = (2 p )4 P00 (r , t ) = (2 p )4 SgK1,gK2,¢nm (r , t ) = (2 p )4 SaK1,aK2¢ (r , t ) = (2 p )4 Pij (r , t ) = ò dt1ò dr1ò dt2 ò dr2 K¢ (r - r1, t - t1)(0) ´Daa 2 ´ å å (t ⁎n )a1g1 SgK1¢g2, nm n = 1m = ´ (r1 - r2 , t1 - t2)(t ⁎m )g2 b ´D bK2¢ b (r2 - r¢ , t2 - t ¢)(0) K ,K ¢ ò dkò dw ei (kr-wt ) S˜ g g ,nm (k, w), K ,K ¢ ò dkò dw ei (kr-wt ) S˜ a a (k, w), we rewrite relations (68), (69) and (72), (73) in the compact forms of algebraic equations K¢ (r - r1, t - t1)(0) ´Daa ´SaK1¢a2 (r1 - r2 , t1 - t2) D˜ ij (k, w )(2) = (73) 2 å å D˜ in (k, w)(0) P˜ nm (k, w) n = 1m = The self-energy parts (66) and (67) of boundary electromagnetic field are represented by Feynman diagram in figure 2(a), and the self-energy parts (70) and (71) of Dirac fermion fields are represented by Feynman diagram in figure 2(b) Performing the Fourier transformation of 2-point Green functions ´ D˜ mj (k, w )(0) , (76) (0) ˜ 00 (k, w ) D˜ 00 (k , w ) , D˜ 00 (k, w )(2) = D˜ 00 (k, w )(0) P (77) and K ˜ ab (k, w )(2) = D dk dw e i (kr - wt ) D˜ ij (k, w )(0,2) , Dij (r, = (2p )4 dk dw e i (kr - wt ) D˜ 00 (k, w )(0,2) , (74) D00 (r, t )(0,2) = (2p )4 K,K ¢ K,K ¢ ˜ ab (r, t )(0,2) = dk dw e i (kr - wt ) D (k, w )(0,2) Dab (2p )4 t )(0,2) ò dkò dw ei (kr-wt ) P˜ 00 (k, w), (75) ò ò ò ò dr2 + dt1 dr1 dt2 ´DaK2¢ b (r2 - r¢ , t2 - t ¢)(0) ò dkò dw ei (kr-wt ) P˜ ij (k, w), ò ò 2 K å å D˜ aa (k, w)(0) (tn )a g n = 1m = ˜ gK g , nm (k, ´S ò ò 1 K ˜ b b (k, w )(0) w )(tm )g2 b D K ˜ aa ˜ aK b (k, w )(0) , ˜ aK a (k, w ) D (k, w )(0) S +D 1 2 ò ò (78) Adv Nat Sci.: Nanosci Nanotechnol (2016) 035001 K¢ ˜ ab D (k, w )(2) = 2 V H Nguyen K¢ K¢ ˜ ab ˜ ab D (k , w ) = D (k, w )(0) K¢ å å D˜ aa (k, w)(0) (t ⁎n )a g 1 n = 1m = ˜ gK ¢g , nm (k, ´S K¢ ˜ aa +D (k , ˜ bK ¢ b (k, w )(t ⁎m )g2 b D ˜ aK ¢a (k, w )(0) S 2 w )(0) ˜ aK ¢ b (k, w) D n = 1m = K¢ w )(0) Acknowledgments The author would like to express the deep gratitude to Vietnam Academy of Science and Technology for the support I thank Advanced Center of Physics and Institute of Materials Science for the encouragement and Nguyen Bich Ha for the cooperation References [1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V and Firsov A A 2005 Nature 438 197 [2] Katsnelson M I, Novoselov K S and Geim A K 2006 Nat Phys 520 [3] Geim A K and Novoselov K S 2007 Nat Mater 183 [4] Schedin F, Geim A K, Morozov S V, Hill E W, Blake P, Katsnelson M I and Novoselov K S 2007 Nat Mater 652 [5] Castro Neto A H, Guinea F, Peres NMR, Novoselov K S and Geim A K 2009 Rev Mod Phys 81 109 [6] Weinberg S 1995 The Quantum Theory of Fields vol Foundation (Cambridge: Cambridge University Press) [7] Sterman G 1993 An Introduction to Quantum Field Theory (Cambridge: Cambridge University Press) [8] Brown L S 1995 Quantum Field Theory (Cambridge: Cambridge University Press) [9] Gross F 1993 Quantum Mechanics and Field Theory (New York: Wiley) [10] Mikhailov S A 2016 Phys Rev B 93 085403 [11] Page A F, Ballout F, Hess O and Hamm J M 2015 Phys Rev B 91 075404 [12] Christensen T, Jauho A-P, Wubs M and Mort N A 2015 Phys Rev B 91 125414 [13] Novko D, Despoja V and Šunjič M 2015 Phys Rev B 91 195407 [14] Agarwal A and Vignale G 2015 Phys Rev B 91 245 407 [15] Iurov A, Gumb G, Huang D and Silkin V M 2016 Phys Rev B 93 035404 [16] Pyatkosky P K and Chakraborty T 2016 Phys Rev B 93 085145 [17] Jablan M and Chang D E 2015 Phys Rev Lett 114 236801 [18] Nguyen V H, Nguyen B H and Dinh N D 2016 Adv Nat Sci.: Nanosci Nanotechnol 015013 [19] Jackson W D 1975 Classical Electrodynamics (New York: Wiley) [20] Akhiezer A I and Beresteskii V B 1969 Quantum Electrodynamics (Moscow: Nauka) (in Russian) [21] Bogolubov N N and Shirkov D V 1976 Introduction to Theory of Quantized Fields (Moscow: Nauka) (in Russian) å å D˜ in (k, w )(0) n = 1m = (80) D˜ 00 (k, w ) = D˜ 00 (k, w )(0) + D˜ 00 (k, w )(0) ´ P00 (k, w ) D˜ (k, w ) , (81) K,K ¢ ˜ ab and the Fourier transforms D (k, w ) of total 2-point Dirac fermion Green functions satisfy Dyson equations K K ˜ ab ˜ ab D (k , w ) = D (k, w )(0) 2 K ˜ aa + å åD (k, w )(0) (tn )a1 g1 n = 1m = ˜ bK b (k, w ) ˜ gK g , nm (k, w )(tm )g b D ´S 2 2 K (0) S ˜ aa ˜ aK b (k, ˜ aK a (k, w ) D +D ( k , w ) 1 2 w ), (83) Finally note that by using S-matrix (22) with interaction Hamiltonian (4) it is straightforward to find matrix elements of all electromagnetic interaction processes in graphene singlelayer Thus in the present work we have constructed the basics of theory of electromagnetic interaction processes and phenomena This area of research can be briefly called Graphene quantum electrodynamics Using basic formulae for the Hamiltonian of the interacting system comprising the electromagnetic field in the threedimensional physical space and the Dirac fermion fields in a single-layer graphene sheet we have an efficient method to the theoretical study of electromagnetic interaction processes in single-layer graphene, working in the interaction picture For this purpose we have determined the boundary free electromagnetic field on graphene as well as boundary limits on the graphene plane of 2-point Green functions of the free electromagnetic field, briefly called boundary Green functions of the free electromagnetic field Then we have explicitly constructed the S-matrix and applied the perturbation theory to establish expressions of the boundary 2-point Green functions of electromagnetic field and 2-point Green functions of Dirac fermion fields in the second order of the perturbation theory We have shown that the Fourier transforms D˜ij (k, w )(2) and D˜ 00 (k, w )(2) of second order terms of boundary Green functions of electromagnetic field are determined by equations (76) K,K ¢ ˜ ab and (77), and the Fourier transforms D (k, w )(2) of second order terms of Dirac fermion Green functions are determined by equations (78) and (79) The total boundary 2-point Green functions (26) and (27) of electromagnetic field and total 2-point Dirac fermion Green functions (28) will be calculated in a subsequent work We expect that in the ladder approximation the Fourier transforms D˜ij (k, w ) and D˜ 00 (k, w ) of boundary total 2-point Green functions of electromagnetic field satisfy Dyson equations ´ Pnm (k, w ) D˜ mj (k, w ), K¢ K¢ ˜ aa ˜ aK ¢ b (k, w ) ˜ aK ¢ a (k, w ) D +D (k, w )(0) S 1 2 Conclusion and discussion K¢ ˜ b b (k , w ) ˜ g g , nm (k, w )(t ⁎m )g b D ´S 2 2 (79) D˜ ij (k, w ) = D˜ ij (k, w )(0) + ˜ aa (k, w )(0) (t ⁎n )a g + å åD 1 (82) 10 ... content of this work is the study of electromagnetic interaction in single-layer graphene by means of the perturbation theory The interaction of electromagnetic field with Dirac fermions in single-layer. .. study of electromagnetic interaction processes in single-layer graphene, working in the interaction picture For this purpose we have determined the boundary free electromagnetic field on graphene. .. nonlinear 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 singlelayer