Home Search Collections Journals About Contact us My IOPscience Quantum field theory of photon–Dirac fermion interacting system in graphene monolayer This content has been downloaded from IOPscience Please scroll down to see the full text 2016 Adv Nat Sci: Nanosci Nanotechnol 025003 (http://iopscience.iop.org/2043-6262/7/2/025003) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 113.178.31.26 This content was downloaded on 26/06/2016 at 10:38 Please note that terms and conditions apply | Vietnam Academy of Science and Technology Advances in Natural Sciences: Nanoscience and Nanotechnology Adv Nat Sci.: Nanosci Nanotechnol (2016) 025003 (7pp) doi:10.1088/2043-6262/7/2/025003 Quantum field theory of photon–Dirac fermion interacting system in graphene monolayer Bich Ha Nguyen1,2 and Van Hieu Nguyen1,2 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.ac.vn Received 20 January 2016 Accepted for publication 22 February 2016 Published 30 March 2016 Abstract The purpose of the present work is to elaborate quantum field theory of interacting systems comprising Dirac fermion fields in a graphene monolayer and the electromagnetic field Since the Dirac fermions are confined in a two-dimensional plane, the interaction Hamiltonian of this system contains the projection of the electromagnetic field operator onto the plane of a graphene monolayer Following the quantization procedure in traditional quantum electrodynamics we chose to work in the gauge determined by the weak Lorentz condition imposed on the state vectors of all physical states of the system The explicit expression of the two-point Green function of the projection onto a graphene monolayer of a free electromagnetic field is derived This two-point Green function and the expression of the interaction Hamiltonian together with the two-point Green functions of free Dirac fermion fields established in our previous work form the basics of the perturbation theory of the above-mentioned interacting field system As an example, the perturbation theory is applied to the study of two-point Green functions of this interacting system of quantum fields Keywords: quantum field, Dirac fermion, electromagnetic field, Green function, perturbation theory Classification numbers: 2.01, 3.00, 5.15 graphene is essentially governed by Dirac’s (relativistic) equations [4] in the (2+1)-dimensional Minkowski space-time It is known that in the terminology of quantum field theory the spinless Dirac fermions in graphene monolayers are described by two spinor quantum fields y K (r, t ) and y K ¢ (r, t ), r = {r1, r2} = {x, y} [5] The points K and K′ are the two nearest corners of the first Brillouin zone in the reciprocal lattice of the hexagonal graphene structure They are called Dirac points Since the Dirac fermions are considered as the spinless fermions, the quantum fields y K (r, t ) and y K ¢ (r, t ) are the two-component spinors realizing the fundamental representation of the SU(2) group of rotations in some fictive threedimensional Euclidean space Let us call them the quasi- Introduction After the discovery of graphene by Novoselov et al [1, 2], a new extremely promising interdisciplinary scientific area— the physics, chemistry and technology of graphene and similar two-dimensional hexagonal semiconductors—has emerged and strongly developed as ‘a rapidly rising star on the horizon of materials science and condensed-matter physics, having already revealed a cornucopia of new physics and potential applications’, as Geim et al stated [3] The quantum motion of electrons as spinless point particles in 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 2043-6262/16/025003+07$33.00 © 2016 Vietnam Academy of Science & Technology Adv Nat Sci.: Nanosci Nanotechnol (2016) 025003 B H Nguyen and V H Nguyen spinors or pseudospinors in the analogy with the notion of isospinor used in the theory of elementary particles [6–9] Let us denote τi, i=1, 2, three generators of the SU(2) group of rotations in the fictive three-dimensional Euclidean space We call them the quasi-spin or pseudospin operators acting on the quantum fields of Dirac fermions as twocomponent spinors They are similar in the matrix form but have a quite different physical meaning compared to the Pauli matrices σi, i=1, 2, 3, representing conventional spin operators of spin 1/2 fermions and being generators of the SU(2) group of rotations in the physical three-dimensional space In the unit system with  = c = (c being the light speed in the vacuum) and the approximation assuming the linear dispersion law for the Dirac fermions, the Hamiltonian of the system of free Dirac fermions in the graphene monolayer has the following expression [5] HG = ò dr {y K (r, t )+t (-i ) y K (r, t ) K¢ + y (r , t )+t⁎ ( - i ) y K ¢ (r , Its explicit expression contains only the projected vector potential field with two components def Ai (r , t ) = Ai (r , o , t ) , i = 1, and projected scalar potential field def f (r , t ) = f (r , o , t ) ( ) t0 = ⋅ Then the set of three formulae (2)–(4) can be compactly rewritten as follows Hint (t ) = e å (1 ) + y K ¢ (r , t )+t ⁎ y K ¢ (r , t )} A (r , o , t ) + y (r , ò K¢ + y (r , t )+y K ¢ (r , t )+t ⁎m y K ¢ (r , (7 ) t )}⋅ The study of the interaction of the electromagnetic field with the Dirac fermion field in a graphene monolayer requires the use of explicit formulae determining the projection Am(r, t), m=0, 1, 2, of the electromagnetic field as well as the projection Dmn (r - r¢, t - t ¢) m, n=0, 1, of the twopoint Green function Dmn (r - r¢, z - z ¢ , t - t ¢) of the electromagnetic field onto the graphene plane These formulae are established in section Section is devoted to the study of the interacting system comprising the Dirac fermion fields and electromagnetic field An application of the perturbation theory is presented in section Section contains the conclusion and discussions For simplifying formulae we shall use the unit system with  = c = Projection of free electromagnetic field and its two-point Green function onto graphene monolayer The content of this section is a short presentation of the free electromagnetic field Aμ(x) and its two-point Green function onto the plane xOy of a graphene monolayer In the conventional relativistic quantum field theory [6–11] the electromagnetic field is described by a vector field Aμ(x), μ=1, 2, 3, 4, in the (3+1)-dimensional Minkowski spacetime The coordinate vector x of each point in this space-time has four components xμ, μ=1, 2, 3, 4, x={x1, x2, x3, x4}={x, y, z, it} The two-point Green function of the electromagnetic field Aμ(x) is defined as follows: (2 ) Dirac fermions interact also with the scalar potential field f(r, z, t) The corresponding part of the interaction Hamiltonian is S Hint = e dr {y K (r , t )+y K (r , t ) ò dr {y K (r, t )+tm y K (r, t ) m=0 K¢ Let us chose the Cartesian coordinate system as follows: the plane of a graphene monolayer is the coordinate plane xOy and, therefore, the Oz-axis is perpendicular to this plane The coordinate of a point in the three-dimensional physical space is denoted {r, z}={x, y, z} In conventional quantum electrodynamics it is known [6–11] that three components Ai(r, z, t), i=1, 2, 3, of the vector potential field A(r, z, t) together with the scalar potential field f(r, z, t)=A0(r, z, t) form a four-component vector field Aμ(r, z, t), μ=1, 2, 3, 4, A4(r, z, t)=iA0(r, z, t), in the (3+1)-dimensional Minkowski space-time In order to take into account the interaction between Dirac fermion fields y K (r, t ) and y K ¢ (r, t ) with the vector potential field A(r, z, t), we must perform the substitution -i   -i  + e A (r, o, t ) in the Hamiltonian (1), e being the absolute value of the electron charge [6–12] Then we obtain the following expression of the Hamiltonian of the interaction between the vector potential field A(r, z, t) and Dirac fermion field y K (r, t ) and y K ¢ (r, t ) ò (6 ) Let us denote f(r, t) as A0(r, t) and introduce the matrix t )}Am (r , t )⋅ V Hint = e dr {y K (r , t )+t y K (r , t ) (5 ) (3 ) t )} f (r , o , t ) Dmn (x - x ¢) = -i T {Am (x ) An (x ¢)} đ = -i {q (t - t ¢) Am (x ) An (x ¢)đ The interaction between the electromagnetic field and Dirac fermion fields is completely described by the following total interaction Hamiltonian V S Hint = Hint + Hint + q (t ¢ - t ) An (x ¢) Am (x )ñ }, (8 ) where the symbol á⋅⋅⋅ñ denotes the average of the inserted expression (containing field operators) in the ground state | Gñ (4 ) Adv Nat Sci.: Nanosci Nanotechnol (2016) 025003 B H Nguyen and V H Nguyen of the Dirac fermion gas á⋅⋅⋅ñ = G | ⋅ ⋅⋅| G ñ⋅ For two plane waves propagating along the direction of the Oz-axis we have (9 ) This ground state | Gñ can be considered as the vacuum state of the free electromagnetic field Since the theory of the electromagnetic field is invariant under a class of gauge transformations Am (x )  Am (x ) + ¶c (x ) , ¶xm ¶xm =0 (10) Ai (r , t ) = ¶Am (x ) ¶xm | F2 đ = (2 p ) ´ å ⎛ ⎞ ⎜- i ⎟⋅ ⎜ ⎟ ⎝ ⎠ (15) ò dkò dl {(xs kl )i ei [kr-W (k, l ) t ] cs kl W (k , l ) å (16) (11) It looks like a linear combination of an innumerable set of quantum fields Ai(r, t)l, each of them being labeled by a value of the index l Ai (r , t ) = Ai (r , t )l = (12) 2p ò dk ´ å s =1 2p ò dlAi (r, t )l , (17) {(xs kl )i ei [kr-W (k, l ) t ] cs kl W (k , l ) -i [kr -W (k, l ) t ] c + }⋅ + (xs kl )+ i e s kl x s kl W (k , l ) {ei [kr+ lz -W (k, l ) t ] cs kl s =1 + e-i [kr+ lz -W (k, l ) t ] cs+kl }, (18) The field Ai(r,t)l with an index l≠0 looks like the conventional free vector field with transverse polarizations of a massive particle with the mass |l| and the helicities σ=±1 in the (2+1)-dimensional Minkowski space-time Note that the electromagnetic waves with the scalar polarization play no role in any physical processes Therefore the free scalar field f(r, z, t) effectively does not have the non-vanishing projection onto the graphene plane Now we consider the projection of the two-point Green function k2 + (13) (0) (0) Dmn (x ) = Dmn (k, z , t ) of the free electromagnetic field onto the graphene monolayer In the relativistic quantum electrodynamics [10, 11] it was (0) shown that Dmn (x ) has following general expression l2 , where ξσkl with σ=±1 are two three-component complex unit vectors characterizing two transversely polarized states of the electromagnetic plane waves with the wave vector {k, l} Let us represent each vector ξσkl as a column with three elements x s kl s =1 ò dkò dl W (k , l ) = + (xs kl )i+e-i [kr-W (k, l ) t ] cs+kl }⋅ In the fundamental research works on quantum electrodynamics [10, 11] it was demonstrated that due to condition (12) the electromagnetic waves in the states with longitudinal and scalar polarizations play no role in any physical processes Therefore in the Hilbert space of state vectors of all physical states of the electromagnetic field the vector potential field A(x)=A(r, z, t) has the following effective Fourier expansion formula A (r , z , t ) = (2 p ) ´ was frequently used However, in quantum electrodynamics this condition cannot hold for the quantum vector field Am (x ) Instead of condition (11) it was reasonably proposed to assume another similar but weaker condition imposed on the state vector of all physical states of the electromagnetic field: áF1| x-1 ol  It is straightforward to project the vector field (13) onto the graphene plane to obtain the vector field A||(r, t) with two components the vector field Aμ(x) is not uniquely determined In classical electrodynamics [12] to simplify equations and calculations the vector field Aμ(x) satisfying the following Lorentz condition ¶Am (x ) ⎛1⎞ ⎜ i ⎟, ⎜ ⎟ ⎝0⎠ x+1 ol  ⎛ (xs kl )1 ⎞ ⎜ ⎟  ⎜ ( x s k l ) ⎟⋅ ⎜ ⎟ ⎝ (x s kl )3 ⎠ (0) d4k eikx D˜mn (k ) , (2 p ) km kn ⎤ km kn , - + d (k ) ⎥ k ⎦ i (k - io) k (0) Dmn (x ) = ⎡ (0) D˜mn (k ) = ⎢dmn ⎣ ò (19) (20) where k denotes a four-momentum vector with the components kμ, μ=1, 2, 3, 4, k4=ik0, in the Minkowski space-time, k = (k, l , ik 0) , (14) k2 = k2 + l - k 02, Adv Nat Sci.: Nanosci Nanotechnol (2016) 025003 B H Nguyen and V H Nguyen kx = kr + lz - k t , ò d4k = (0) then Dmn (x ) must satisfy the transversality condition (0) ¶Dmn (x ) ò ò ò dk dl dk ¶xm and d(k2) is a scalar function depending on the choice of the gauge for the free electromagnetic field Since the theory is invariant under gauge transformations of the whole system of all interacting quantum fields, for simplifying the calculations in certain cases one often chose to work in such a gauge that d (k 2) = i (k d (k 2) = 0, meaning that the tensor components: i (k + l2 ⋅ - k 02 - io) ⎛ ⎞ l2 (0) D˜ 33 (k, l , k 0) = ⎜1 - ⎟ k + l - k 02 ⎠ ⎝ ´ , 2 i (k + l - k 02 - io) ( 0) The projection Dmn (r, t ) of the Green function z, t ) onto a graphene monolayer is determined by the following definition def (0) t ) = Dmn (r , o , t ) , (0) (0) D˜ 30 (k, l , k 0) = D˜ 03 (k, l , k 0) lk = - ⋅ , k + l - k 02 i (k2 + l - k 02 - io) (24) where m, n=0, 1, From the above presented formulae it is easy to show that (0) Dmn (r , t) = (2 p )4 ò dkò dlò (0) dk ei (kr- k t ) D˜mn (k, ⎛ ⎞ k 02 (0) D˜ 00 (k, l , k 0) = - ⎜1 + ⎟ 2 k + l - k0 ⎠ ⎝ ´ ⋅ 2 i (k + l - k 02 - io) l , k 0) with i (k ⋅ + l - k 02) (0) dl Dmn (r , t )l (27) 2p ( 0) of an innumerable set of functions Dmn (r, t )l labeled by the index l running all integer values from −∞ to +∞: ´ ò i (2 p ) (34) (35) In order to apply perturbation theory to the study of interacting system comprising the Dirac fermion fields and the electromagnetic field it is necessary to use explicit expressions of following physical quantities: ò dkò dk ei (kr-k t ) ⋅ 2 k + l - k 02 - io (33) Interacting Dirac fermion fields and electromagnetic field tion (0) Dmn (r , t )l = dmn (32) (26) (0) Formula (25) shows that Dmn (r, t ) is a linear combina- (0) Dmn (r , t ) = (31) and (25) (0) D˜mn (k, l , k 0) = dmn l, k ) has the following with i, j=1, 2, (23) (0) Dmn (r, (0) Dmn (r , (30) (0) (0) D˜ i (k, l , k 0) = D˜ 0i (k, l , k 0) kk = - i2 ⋅ , 2 k + l - k i (k + l - k 02 - io) (22) i.e (0) D˜mn (k, l , k 0) = dmn ( 0) D˜mn (k, ⎛ ⎞ ki k j (0) D˜ ij (k, l , k 0) = ⎜dij - ⎟ k + l - k 02 ⎠ ⎝ , ´ 2 i (k + l - k 02 - io) (21) - io) (29) and instead of equation (21) we have the relation In this case formula (20) becomes (0) D˜mn (k ) = dmn = 0, • Dirac fermion fields y K (r, t ) and y K  (r, t ), K K Two-point Green functions Dab (r, t )(0) (r, t )(0) and Dab of free Dirac fermion fields, • Projection A||(r, t) with the two-component Ai(r,t), i=1, 2, of the electromagnetic field onto the graphene monolayer, • Projection Dmn (r, t )(0) , m, n=0, 1, 2, of the two-point Green function of the free electromagnetic field onto the graphene monolayer, and • Interaction Hamiltonian Hint(t) of the system (28) ( 0) Each function Dmn (r, t )l is the two-point Green function of a massive relativistic particle with the mass | l | (in two dimensions) However, if we impose on the state vectors of all physical states of the system the weak Lorentz condition (12), Adv Nat Sci.: Nanosci Nanotechnol (2016) 025003 B H Nguyen and V H Nguyen The interaction Hamiltonian Hint(t) was determined by formula (7) The projection A||(r, t) of the electromagnetic field and the projection Dmn (r, t )(0) , m, n=0, 1, 2, of the two-point Green function of the free electromagnetic field were investigated in the preceding section It remains to establish the explicit expressions of Dirac fermion fields y K (r, t ), y K ¢ (r, t ) and two-point Green functions K¢ K (r, t )(0) of free Dirac fermions Dab (r, t )(0) , Dab In our previous work [13] we derived explicit expresK¢ K sions of two-point Green functions Dab (r, t )(0) (r, t )(0) , Dab of free Dirac fermions in a free Dirac fermion gas at T=0 They depend on the value EF of the Dirac fermion gas For simplicity let us consider the case with EF=0 The extension to other cases is straightforward In the simple case with EF=0 the Dirac fermion fields y K (r, t ) and y K ¢ (r, t ) have the following Fourier expansion formula y K , K ¢ (r , t ) = dk {ei [kr- Ee (k ) t ] u K , K ¢ (k) akK , K ¢ 2p + e-i [kr- Eh (k ) t ] v K , K ¢ (k) b kK , K ¢+}, ò Two-point Green functions of Dirac fermions in free Dirac fermion gas at T=0 have the following definition K ,K ¢ Dab (r - r¢ , t - t ¢)(0) = - i T {Y aK , K ¢ (r , t ) Y bK , K ¢ (r¢ , t ¢)} đ = - i {q (t - t ¢) áY aK , K ¢ (r , t ) Y bK , K ¢ (r , t )+đ - q (t ¢ - t ) áY bK , K ¢ (r , t )+Y aK , K ¢ (r , t )ñ }⋅ (41) Introducing their Fourier transformations K ,K ¢ Dab (r , t )(0) = v K (k) = ⎛ e-iq (k) ⎞ ⎜ iq (k) ⎟ h , ⎝e ⎠ ⎛ e-iq (k) ⎞ ⎜ iq (k) ⎟ h ⎝- e ⎠ (42) w )(0) , we have K ,K ¢ ˜ ab D (k, w )(0) = uaK , K ¢ (k) ubK , K ¢ (k)⁎ w - Ee (k ) + io + (36) vaK , K ¢ ( - k) vbK , K ¢ ( - k)⁎ w + Eh (k ) - io ⋅ (43) Thus the basics for elaborating the perturbation theory of an interacting system comprising Dirac fermion fields and an electromagnetic field were established Perturbation theory The most efficient tool for the theoretical study of interaction processes between quanta of any interacting system of quantum fields is the scattering matrix S, briefly called the S-matrix In the perturbation theory the S-matrix is expressed in terms of the interaction Hamiltonian Hint(t) of the system as follows (37) vF is the speed of the relativistic Dirac fermion in the unit system with c=1, u K (k) = K ,K ¢ ˜ ab (k , ei [kr- wt ] D ò Ee, h (k ) = vF k , k12 + k 22 , ò dk 2p ´ dw where akK , K ¢ and b kK , K ¢ are the destruction operators of the Dirac fermion and Dirac hole, respectively, with wave functions being plane waves, k is the wave vector to be considered also as the momentum of the Dirac fermion or Dirac hole, akK , K ¢+ and b kK , K ¢+ are corresponding creation operators, Ee(k) and Eh(k) are energies of the Dirac fermion and Dirac hole, respectively, with momentum k, k = | k| = (2 p )2 S=T ⎡ ⎤ exp ⎣⎢ - i dt Hint (t )⎦⎥ , { ò } (44) where the integration with respect to the time variable t is performed over the whole real axis from −∞ to +∞ By expanding the exponential function on the right-hand side of formula (44) into power series, we write the S-matrix in the form of a series (38) S=1+ and ¥ å S (n), (45) n=1 u (k) = v K ¢ (k) = K¢ the term S( n) of nth order is ⎛ eiq (k) ⎞ ⎜ -iq (k) ⎟ h ¢ , ⎝e ⎠ ⎛ eiq (k) ⎞ ⎜ -iq (k) ⎟ h ¢ , ⎝- e ⎠ (39) k1 , k2 (40) q (k) = arctg S (n) = ( - i)n n! ´ ò dt1ò dt2 ò dtn T {Hint (t1) Hint (t2) Hint (tn)}⋅ (46) As an example of the application of perturbation theory let us study two-point Green functions of an interacting system comprising Dirac fermion fields and the projection of the electromagnetic field onto the graphene monolayer at T=0 They are expressed in terms of free Dirac fermion fields η and η′ being two arbitrary phase factors | h | = | h ¢ | = Adv Nat Sci.: Nanosci Nanotechnol (2016) 025003 B H Nguyen and V H Nguyen y K (r, t ) and y K ¢ (r, t ), components Ai(r,t) of the projection A||(r,t) of a free electromagnetic field onto the graphene monolayer and S-matrix as follows: Dij (r - r¢ , t - t ¢) = - i T {S Ai (r , t ) Aj (r¢ , t ¢)} đ Sđ The matrix elements on the right-hand side of equations (51) and (52) can be calculated by applying the Wick theorem in quantum field theory They are expressed in terms of K,K ¢ the two-point Green functions Dab (r - r¢, t - t ¢)(0) of free K,K ¢ Dirac fermion fields yab (r, t ) and the projection Dij(0) (r, t ) of two-point Green functions of the free electromagnetic field onto a graphene monolayer By using derived expressions of the above-mentioned matrix elements it is straightforward to calculate second-order terms in the series (49) and (50) We obtain the following result: (47) and K ,K ¢ Dab (r - r¢ , t - t ¢) = -i T {S y aK , K ¢ (r , t ) y bK , K ¢ (r¢ , t ¢)+} đ Sñ ⋅ (48) Using expansion formula (45) of the S-matrix, we write each of the Green functions (47) and (48) in the form of a series: ¥ å Dij (r Dij (r - r¢ , t - t ¢) = K ,K ¢ Dab (r - r¢ , t - t ¢) = - r¢ , t - t ¢)(2n) , K Dab (r - r¢ , t - t ¢)(2) = ´ (49) - r¢ , t - t ¢)(2n) , where å aa (r1 - r2 , t1 - t2) = i ååDnm (r1 - r2 , t1 - t2)(0) K ´ - r , t1 - t2)(0) (tm)b a2 Dij (r - r¢ , t - t ¢)(2) = å ò dt1ò dr1ò dt2 ò dr2nå =0 m=0 ò dt1ò dr1ò dt2 ò dr2 ´ ååDin (r - r1, t - t1)(0) n (51) ´ áT {[y K (r1, t1)+ tn y K (r1, t1) ´ Pnm (r1 - r2 , t1 - t2) Dmj (r2 - r¢ , t2 - t ¢)(0) , where + y K ¢ (r2 , t2)+ t ⁎m y K ¢ (r2 , t2)] ´ An (r1, t1) Am (r2 , t2) Ai (r , t ) Aj (r¢ , t ¢)} đ⋅ Pnm (r1 - r2 , t1 - t2) = - i Tr [tm DK (r1 - r2 , t1 - t2)(0) ´ tn DK (r2 - r1, t2 - t1)(0) Similarly, in order to calculate example, we consider matrix element áT {S (2) y aK (r , t ) y bK (r¢ , t ¢)} ñ = ´ + t ⁎m DK ¢ (r1 - r2 , t1 - t2) t ⁎n DK ¢ (r2 - r1, t2 - t1)] - r¢, t - t ¢), for (56) ( - i )2 2! can be considered as the self-energy part of the projection of the electromagnetic field onto a graphene plane, DK (r, t )(0) and DK  (r, t )(0) being 2ì2 matrices with elements K¢ K (r, t )(0) Dab (r, t )(0) and Dab All higher-order terms in the series (49) and (50) can be calculated analogously Summing them up, we obtain the Dyson equations for the whole Green functions (49) and (50) of interacting quantum fields in the ladder approximation: ò dt1ò dt2 áT {Hint (t1) Hint (t2) yaK (r, t ) y bK (r¢, t ¢)} đ ( - i )2 e dt1 dr1 dt2 2! (55) m + y K ¢ (r1, t1)+ t ⁎n y K ¢ (r1, t1)] ´ [y K (r2 , t2)+ tm y K (r2 , t2) K Dab (r = m is the self-energy part of the Dirac fermion field y K (r, t ), and ò dt1ò dt2 áT {Hint (t1) Hint (t2) Ai (r, t ) Aj (r¢, t ¢)} đ 2! n (tn)a1b1D bK1b (r1 (54) ( - i )2 áT {S (2) Ai (r , t ) Aj (r¢ , t ¢)} đ = 2! = (53) K ,K ¢ (r å Dab n running all non-negative integers n=0, 1, K We have calculated Dij (r - r¢, t - t ¢)(0) and K,K ¢ In order to calculate Dab (r - r¢, t - t ¢)(0) Dij (r - r¢, t - t ¢)(2) let us consider matrix element e2 K (50) ( - i )2 K (r - r1, t - t1)(0) ò dr2Daa ´ å a a (r1 - r2 , t1 - t2) DaK2 b (r2 - r , t2 - t Â)(0) , n=0 Ơ n=0 ´ ò dt1ò dr1ò dt2 å ò ò ò ò dr2nå =0 m=0 ´ áT {[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) Dij (r - r¢ , t - t ¢) = + y K ¢ (r2 , t2)+ t ⁎m y K ¢ (r2 , t2)] ò dt1ò dr1ò dt2 ò dr2 ´ ååDin (r - r1, t - t1)(0) ´ An (r1, t1) Am (r2 , t2) y aK (r , t ) y bK (r¢ , t ¢)+} ñ⋅ n m ´ Pnm (r1 - r2 , t1 - t2) Dmj (r2 - r¢ , t2 - t ¢) (52) (57) Adv Nat Sci.: Nanosci Nanotechnol (2016) 025003 B H Nguyen and V H Nguyen Acknowledgment and K ,K Dab (r - r¢ , t - t ¢) = ´ ò dr2Daa K ,K ¢ ò dt1ò dr1ò dt2 The authors would like to express their deep gratitude to the Advanced Center of Physics and Institute of Materials Science, Vietnam Academy of Science and Technology for its support (r - r1, t - t1)(0) K ,K ¢ ´ å a a (r1 - r2 , t1 - t2) DaK2, Kb ¢ (r2 - r¢ , t2 - t ¢)(0) (58) References [1] Novoselov K S, Geim A K, Mirosov S V, Jiang D, Dubonos S V, Grigorieva I V and Firsov A A 2004 Science 306 666 [2] Novoselov K S, Jiang D, Schedin F, Booth T J, Khotkevich V V, Marozov S V and Geim A K 2005 Proc Natl Acad Sci USA 102 10451 [3] Geim A K and Novoselov K S 2007 Nature Mater 183 [4] Novoselov K S, Geim A K, Mirosov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V and Firsov A A 2005 Nature 438 197 [5] Castro Neto A H, Guinea F, Peres N M R, Novoselov K S and Geim A K 2009 Rev Mod Phys 81 109 [6] Gross F 1993 Relativistic Quantum Mechanics and Field Theory (New York: John Wiley & Sons, Inc.) 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[13] Hieu N V, Ha N B and Dung D N 2016 Adv Nat Sci.: Nanosci Nanotechnol 015013 Conclusion and discussions In the present work we have developed the quantum theory of an interacting system comprising Dirac fermion fields and the projection onto a graphene monolayer of an electromagnetic field The explicit expressions of these fields, the interaction Hamiltonian of the system and the two-point Green functions of free fields as well as the integral equation determining the twopoint Green functions of interacting fields in the ladder approximation were established We have not yet investigated the electromagnetic scattering processes taking place in the graphene monolayer In our subsequent works the presented expressions and equations will be applied to the study of various interaction processes with the participation of photon and Dirac fermions In particular, the application of the whole theoretical tool elaborated in the present work is necessary and also sufficient for the study of physical processes taking place completely inside the graphene monolayer This would be also useful for the study of electromagnetic properties of graphene-based optoelectronic and photonic nanostructures and nanocomposites ... purpose of the present work is to elaborate quantum field theory of interacting systems comprising Dirac fermion fields in a graphene monolayer and the electromagnetic field Since the Dirac fermions... theory of photon Dirac fermion interacting system in graphene monolayer Bich Ha Nguyen1,2 and Van Hieu Nguyen1,2 Advanced Center of Physics and Institute of Materials Science, Vietnam Academy of. .. dtn T {Hint (t1) Hint (t2) Hint (tn)}⋅ (46) As an example of the application of perturbation theory let us study two-point Green functions of an interacting system comprising Dirac fermion fields