Quantum field theory of interacting plasmon–photon–phonon system

Home Search Collections Journals About Contact us My IOPscience Quantum field theory of interacting plasmon–photon–phonon system This content has been downloaded from IOPscience Please scroll down to see the full text 2015 Adv Nat Sci: Nanosci Nanotechnol 035003 (http://iopscience.iop.org/2043-6262/6/3/035003) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 14.163.90.183 This content was downloaded on 27/06/2016 at 08:47 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 (2015) 035003 (11pp) doi:10.1088/2043-6262/6/3/035003 Quantum field theory of interacting plasmon–photon–phonon system Van Hieu Nguyen1,2 and Bich Ha Nguyen1,2 Advanced Center of Physics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay District, Hanoi, Vietnam University of Engineering and Technology, Vietnam National University, 144 Xuan Thuy, Cau Giay District, Hanoi, Vietnam E-mail: nvhieu@iop.vast.ac.vn and bichha@iop.vast.ac.vn Received March 2015 Accepted for publication 18 March 2015 Published 14 April 2015 Abstract This work is devoted to the construction of the quantum field theory of the interacting system of plasmons, photons and phonons on the basis of general fundamental principles of electrodynamics and quantum field theory of many-body systems Since a plasmon is a quasiparticle appearing as a resonance in the collective oscillation of the interacting electron gas in solids, the starting point is the total action functional of the interacting system comprising electron gas, electromagnetic field and phonon fields By means of the powerful functional integral technique, this original total action is transformed into that of the system of the quantum fields describing plasmons, transverse photons, acoustic as well as optic longitudinal and transverse phonons The collective oscillations of the electron gas is characterized by a real scalar field φ(x) called the collective oscillation field This field is split into the static background field φ0(x) and the fluctuation field ζ(x) The longitudinal phonon fields Qal (x ), Qol (x ) are also al ol split into the background fields Q0al (x ), Qol (x ) and dynamical fields q (x ), q (x ) while the transverse phonon fields Qat (x ), Qot (x ) themselves are dynamical fields qat (x ), qot (x ) without background fields After the canonical quantization procedure, the background fields φ0(x), Q0al (x ), Qol (x ) remain the classical fields, while the fluctuation fields ζ(x) and dynamical phonon al fields q (x ), qat (x ), qol (x ), qot (x ) become quantum fields In quantum theory, a plasmon is the quantum of Hermitian scalar field σ(x) called the plasmon field, longitudinal phonons as complex spinless quasiparticles are the quanta of the effective longitudinal phonon Hermitian scalar fields θ a (x ), θ (x ), while transverse phonons are the quanta of the original Hermitian transverse phonon vector fields qat (x ), qot (x ) By means of the functional integral technique the original action functional of the interacting system comprising electron gas, electromagnetic field and phonon fields is transformed into the total action functional of the resultant system comprising plasmon scalar quantum field σ(x), longitudinal phonon effective scalar quantum fields θ a (x ), θ (x ) and transverse phonon vector quantum fields qat (x ), qot (x ) Keywords: functional integral, collective oscillations, fluctuation, plasmon, action functional Classification numbers: 2.09, 3.00 Introduction appearance of an elementary excitation—a complex quasiparticle called a plasmon—and the plasma frequency was also called plasmon frequency (the references on early works on plasmons can be found in the literature [1–3]) In the physical processes with the presence of plasmon the plasmon–photon interaction plays the main role Moreover, in the electron gas of solids there always exists the electron–phonon interaction Since the early works on the collective motion of charged particles in plasma, including the interacting electron gas in solids, it was shown that there exists a resonance of the collective oscillations at some frequency called the plasma frequency This resonance phenomenon was interpreted as the 2043-6262/15/035003+11$33.00 © 2015 Vietnam Academy of Science & Technology 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 Adv Nat Sci.: Nanosci Nanotechnol (2015) 035003 V H Nguyen and B H Nguyen leading to the effective plasmon–phonon interaction Therefore the knowledge on the mutual interaction of plasmon, photon and phonons is necessary for both theoretical and experimental studies on the physical processes and phenomena involving plasmon The present work is devoted to the elaboration of the quantum field theory of the plasmon– photon–phonon interacting system by applying the functional integral technique [4–7] The assumptions comprise only the fundamental principles of electrodynamics and quantum theory of many-body system For the application of mathematical tools of functional integral technique, the physical content of the theory of phonons in solids must be presented in the languages of the quantum field theory This will be done in section Here there is a distinction between longitudinal and transverse phonons While the transverse phonons are described by the transverse phonon vector fields as other transverse vector fields in the theory of the elementary particles, for simplifying the presentation of the formulae related to longitudinal phonons we propose to describe them by some effective scalar fields similar to the quantum fields of spinless particles Moreover, because the interaction of longitudinal phonons with electron is much stronger than that of transverse ones, in the study of physical phenomena and processes with the dominant competition of longitudinal phonons we can neglect the contribution of transverse phonons Thus the transverse phonon fields will be retained only in the particular cases when they play the essential role Section is devoted to the establishment of the expression of total action functional of the interacting plasmon– photon–phonons system It contains all three types of fields: (i) collective oscillation field φ(x); (ii) transverse electromagnetic vector field A(x) and (iii) all phonon fields, both acoustic and optic phonon fields Q aμ (x ), Qoμ (x ), index μ labeling the phonon branches By grouping suitable terms from the formula of total action functional of the whole system it is possible to derive expressions of action functional of different subsystems of related fields The fundamental subsystem is the collective oscillation field φ(x) A short review of the results of previous works related to this field in the harmonic approximation is presented The construction of quantum fields of interacting plasmon–phonon system is the content of section In the harmonic approximation with respect to the collective oscillation field as well as to the fields of both acoustic and optic phonons, the action functional of the subsystem comprising interacting collective oscillation field φ(x) as well as both acoustic and optic phonon fields Q a μ (x ) and Qoμ (x ) is derived Each of longitudinal phonon fields Qal (x ) and Qol (x ) is split into two parts, background field Q0al (x ) or Qol (x ) and dynamical field qat (x ) or qot (x ), while transverse phonon fields Qat (x ), Qot (x ) themselves are dynamical ones qat (x ) and qot (x ) The dynamical phonon fields generate the physical phonons playing the role of dynamical quasiparticles in physical phenomena and processes The construction of the quantum fields of the whole interacting plasmon–photon–phonon system is the content of the section The expression of total action functional of this whole system, described by background fields φ0(x), Q0al (x ) and Qol (x ), fluctuation field ζ(x), electromagnetic field A(x) and dynamical phonon fields qal (x ), qat (x ) and qol (x ), qot (x ), is derived in the harmonic approximation with respect to each of three types of fields: (i) fluctuation field, (ii) electromagnetic field and (iii) all dynamical phonon fields The characterizing features of different subsystems of the whole system are briefly investigated From the obtained expression of total action functional of the whole system it is possible to derive the expressions of the action functional of different interaction vertices The conclusion and discussions are presented in section Phonon quantum fields For using in the study of the interaction of phonons with other quasiparticles in solids by means of the functional integral technique let us construct the quantum fields of phonons There exist many types of phonons with various characteristics in different materials [8] In the present work we limit to the frequently investigated solids: elastic media [3] and crystalline lattices [3, 9] The quantum fields of acoustic and optic phonons will be constructed separately For simplifying formulae we use the notations proposed in our previous works [4–6] and the unit system with ℏ = c = Consider first the acoustic phonons In both abovementioned types of solids there exist one longitudinal and two transverse acoustic phonon branches Denote Q aμ (x ) their quantum fields, where μ = 1, for transverse phonons and μ = for longitudinal one For a definite μth branch between angular frequency ω and wave vector k at small values of k = k there exists a linear relation ω = υμ k We assume that this formula is the dispersion law of the acoustic phonon in general It looks like that of a massless relativistic particle, except for the scaling of spatial coordinates x x → x′ = , k → k′ = υμ k υμ On the basis of the analogy with the free field of relativistic massless particles we have following Lagrange function and action functional of the acoustic phonon in μth branch ⎛ ∂Q a μ ∂Q a μ ⎞ ℒ 0aμ ⎜ Q aμ , , ⎟ ∂t ∂x i ⎠ ⎝ = 2⎤ ⎛ ⎡ ∂Q a μ ⎞ ⎥ ⎢ ⎛ ∂Q a μ ⎞ ⎜ ⎟ − υμ2 ∑ ⎜ ⎟ ⎢⎣ ⎝ ∂t ⎠ ⎝ ∂xi ⎠ ⎥⎦ i=1 (1) and I0aμ ( Q aμ) = ⎛ aμ ∫ dx ℒ a0μ ⎜⎝ Qaμ, ∂Q∂t , ∂Q a μ ⎞ ⎟⋅ ∂x i ⎠ (2) Now we consider the optic phonons In a crystalline lattice with s non-equivalent ions per a primitive cell, s ≠ 1, Adv Nat Sci.: Nanosci Nanotechnol (2015) 035003 V H Nguyen and B H Nguyen wave function Qot (x ) Since the physical origin of the appearance of phonons is the oscillation of ions in solids and the coupling of phonons with electron is caused by the photon exchange between ion and electron, it is natural to believe that the Hamiltonian of the interaction between transverse phonons with electron have the expressions similar to the electron-photon interaction Hamiltonian in the transverse gauge Therefore we assume following expressions of the transverse phonon–electron interaction Hamiltonians: beside three acoustic phonon branches there exist 3(s-1) branches of optic phonons with non-vanishing limiting angular frequency Ωμ at k = Denote Qoμ (x ) the optic phonon field in the μth branch Since k-dependent terms in the dispersion law of optic phonon are very small in comparison with the constant terms Ωμ, let us neglect them Then the optic phonon field Qoμ (x ) has following Lagrange function and action functional ⎡ ⎤ ⎛ ∂Q o μ ∂Q o μ ⎞ ⎢ ⎛ ∂Q o μ ⎞ , ℒ 0oμ ⎜ Q oμ , = − Ω μ2 ( Q oμ) ⎥ ⎜ ⎟ ⎟ ⎥⎦ ∂t ∂xi ⎠ ⎢⎣ ⎝ ∂t ⎠ ⎝ at Hint = −ifa (3) ⎡ ⎛ ∫ d x ⎢⎢⎣ ψ¯ (x) ⎜⎝ ∂∂x − ⎤ ∂⃖ ⎞ ⎟ ψ (x ) ⎥ Q at (x ) ⎥⎦ ∂x ⎠ (9) ⎤ ∂⃖ ⎞ ⎟ ψ (x ) ⎥ Q ot (x ) ⎥⎦ ∂x ⎠ (10) for acoustic transverse phonon and and I0oμ ( Q ) = ∫ dx oμ ⎛ ℒ o0μ ⎜ Q oμ , ⎝ ∂Q o μ ∂Q o μ ⎞ , ⎟⋅ ∂t ∂x i ⎠ ot Hint = −ifo (4) ⎛ (5) γa Hint = gaγ and Ω l2 Ω t2 = ε0 , ε∞ ∫ dx ψ¯ (x ) ψ (x )  Q al (x ) γ0 Hint = goγ = go ∫ dx ψ¯ (x ) ψ (x )  Q (x ), ol Q at (x ) A (x ) (11) ∫ dx Q ot (x ) A (x ) (12) for optic phonon Total functional integral As the extension of total functional integral of the interacting plasmon–photon system studied in the previous work [14] we have following total functional integral of the plasmon–photon–phonon system Z tot = ⎛ ⎞ ∫ [Dψ ] [ Dψ¯ ] ∫ [DA] δ ⎜⎝ ∂∂Ax ⎟⎠ ∫ [ DQaμ] × ∫ [ DQoμ] exp { iItot [ ψ , ψ¯ ; A; Qaμ, Qoμ] }, (13) where Itot ⎡⎣ ψ , ψ¯ ; A; Q aμ , Qoμ⎤⎦ is the total action functional of this system: (7) and ol Hint ∫ dx for acoustic phonon and (6) where ε0 is the static dielectric constant of the medium and ε∞ is the square of the refractive index of the medium at optical frequencies In solids there always exists the electron-phonon interaction In most cases the interaction of longitudinal acoustic or optic phonons with electron is much stronger than that of transverse acoustic or optic phonon, respectively In these cases the longitudinal phonons play a much more important role than the corresponding transverse phonons do, so that the interaction between longitudinal phonons and electron has been intensively studied during a long time It was shown that for various solids the Hamiltonians of the interaction between electron and longitudinal acoustic and optic phonons have following expression [3, 9] al Hint = ga − for optic transverse phonon The interaction of ions in the lattice with the electromagnetic wave, in principle, can also generate the direct coupling of electron with transverse acoustic and optic phonons In the transverse gauge the effective interaction Hamiltonians have the expressions In the special case of isotropic crystals with s = nonequivalent ions per a primitive cell, there exist one longitudinal and two degenerate transverse optic phonon branches with limiting angular frequencies Ωl and Ωt at k → Between Ωl and Ωt there exists following relation Ωl > Ωt ⎡ ∫ d x ⎢⎢⎣ ψ¯ (x) ⎜⎝ ∂∂x Itot [ ψ , ψ¯ ; A ; Q aμ , Q oμ] = I0γ [A] + I0a [ Q aμ] (8) eγ + I0o [ Q oμ] + I e [ ψ , ψ¯ ] + Iint [ ψ , ψ¯ ; A] where ψ (x ) is the electron field operator, ψ¯ (x ) is its Hermitian conjugate The coupling constants ga and go depend on the crystalline and electronic structures of solids Meanwhile, the interaction between electron and transverse phonons was much less known Let us consider the simple case of the lattice with non-equivalent ions per a primitive cell, s = Then beside the two degenerate acoustic transverse phonon branches with wave function Qat (x ) there exist also only two degenerate optic phonon branches with ea + Iint [ ψ , ψ¯ ; Qaμ] + Iinteo [ ψ , ψ¯ ; Qoμ] γa ⎡ γo + Iint ⎣ A ; Q at ⎤⎦ + Iint ⎡⎣ A ; Q ot ⎤⎦ , (14) I0γ [A] is the action functional of the transverse free electromagnetic field in the transverse gauge I0γ [A] = ⎛ ⎡ ⎤2 ∫ dx ⎜⎜ ε0 ⎢⎣ ∂A∂x(0x) ⎥⎦ ⎝ ⎞ ⎡ ∂ ⎤2 −⎢ ∧ A (x ) ⎥ ⎟⎟ , ⎣ ∂x ⎦ ⎠ (15) Adv Nat Sci.: Nanosci Nanotechnol (2015) 035003 V H Nguyen and B H Nguyen ε0 being the static dielectric constant of the medium, I0a ⎡⎣ Q aμ⎤⎦ and I0o ⎡⎣ Qoμ⎤⎦ are the action functional of two systems of all acoustic phonon fields and all optic phonon fields, respectively eo Iint [ ψ , ψ¯ ; Qoμ] = −go ⎡ + ifo ∫ dxψ¯ (x) ψ (x)  Qol (x) ⎛ ∫ dx ⎢⎢⎣ ψ¯ (x) ⎜⎝ ∂∂x − ⎤ ∂⃖ ⎞ ⎟ ψ (x ) ⎥ Q ot (x ) ⎥⎦ ∂x ⎠ (25) I0a [ Q aμ] = ∑ I0aμ [ Qaμ], From expression (11) and (12) of the Hamiltonians describing the coupling of transverse phonons with photon it follows that μ=1 I0o [ Q oμ] = ∑ I0oμ [ Qoμ], (16) μ=1 γa ⎡ γ Iint ⎣ A ; Q at ⎤⎦ = −ga I0aμ ⎡⎣ Q aμ⎤⎦ and I0oμ ⎡⎣ Qoμ⎤⎦ being determined by formulae (1)– (4), I e ⎡⎣ ψ , ψ¯ ⎤⎦ is the action functional of the system of electrons mutually interacting through the Coulomb repulsion I e ⎡⎣ ψ , ψ¯ ⎤⎦ consists of two parts e I [ ψ , ψ¯ ] = I0e [ ψ , ψ¯ ] + e Iint [ψ, ψ¯ ], ( ∂ − i ∂x , ⎡ ⎛ γo ⎡ γ Iint ⎣ A ; Q at ⎤⎦ = −go ⎞⎤ ⎟ (18) exp ) ⎛ ∂ ⎞ ⎛ ∂ ⎞2 ⎜ −i ⎟ + U (x), H ⎜ − i , x⎟ = ⎝ ∂x ⎠ m ⎝ ∂x ⎠ dx dy ψ¯ (x ) ψ (x ) u (x − y) × ψ¯ (y) ψ (y), e2 , ε0 x − y e [ ψ , ψ¯ ; A] = i 2m − ∫ e2 2m Zφ = ⎡ ⎛ ⎤ ∂⃖ ⎞ ⎟ ψ (x ) ⎥ Q at (x ), ⎥⎦ ∂x ⎠ { i } ∫ dx∫ dyφ (x) u (x − y) φ (y) } } ∫ ∫ ∫ [Dφ] exp { i ∫ dx∫ dyφ (x) u (x − y) φ (y) } , (29) as this was proposed in references [4, 5] The bosonic real integration variable φ(x) describing collective oscillations of electron gas was called the collective oscillation field Using formulae (14), (17) and (28), we rewrite the total functional integral (13) in the new form (21) (22) Z tot = Z 0e Zφ × ∫ [Dφ] exp { ⎛ i ∫ dx∫ dyφ (x) u (x − y) φ (y) } ⎞ ∫ [DA] δ ⎜⎝ ∂∂Ax ⎟⎠ exp { iI0γ [A] } ∫ [ DQaμ] { × exp iI0a [ Q aμ] } ∫ [ DQoμ] exp { iI0o [ Qoμ] } γa ⎡ × exp iIint ⎣ A ; Q at ⎤⎦ } exp {iI × F [ φ; A; Q , Q ], { aμ oμ γo ⎡ int ⎣ A ; Q ot ⎤⎦ } (30) ∫ dxψ¯ (x) ψ (x)  Qal (x) − ∫ [Dφ] exp (20) ψ¯ (x ) ψ (x ) A (x )2 ∫ dx ⎢⎢⎣ ψ¯ (x) ⎜⎝ ∂∂x ∫ dx∫ dy ψ¯ (x) ψ (x) u (x − y) ψ¯ (y) ψ (y) where According to formulae (7)–(10) for the electron–phonon interaction Hamiltonians we have + ifa (27) (28) (23) ea Iint [ ψ , ψ¯ ; Qaμ] = −ga i { ⎤ ⎡ ⎛ ∂ ∂⃖ ⎞ ⎟ ψ ( x ) ⎥A ( x ) dx ⎢ ψ¯ (x ) ⎜ − ⎥⎦ ⎢⎣ ∂x ⎠ ⎝ ∂x ∫ dx − Zφ −e is the electron charge It is straightforward to show that eγ Iint Q ot (x ) A (x ) × exp −i dx dyψ¯ (x ) ψ (x ) u (x − y) φ (y) , (19) ∫ ∫ u (x − y) = δ ( x − y0 ) u (x − y), { = e ⎡ m is the effective mass of electron, Iint ⎣ ψ , ψ¯ ⎤⎦ is the action functional of electron-electron Coulomb interaction u (x − y ) = ∫ dx The Coulomb interaction functional (20) is bilinear with respect to the electron density ψ¯ (x ) ψ (x ) This expression can be linearized by means of the Hubbard-Stratonovich transformation H x is the quantum mechanical Hamiltonian of single electron e Iint [ ψ , ψ¯ ] = − (26) (17) ∫ dx ψ¯ (x) ⎢⎣ i ∂∂x0 − H ⎝ −i ∂∂x , x⎠ ⎥⎦ ψ (x), ⎜ Q at (x ) A (x ) and where I0e ⎡⎣ ψ , ψ¯ ⎤⎦ is the action functional of free electron moving in the electrostatic field of ions in the crystalline lattice I0e [ ψ , ψ¯ ] = ∫ dx where Z 0e = (24) ∫ [Dψ ] [ Dψ¯ ] exp { iI0e [ ψ , ψ¯ ] } (31) Adv Nat Sci.: Nanosci Nanotechnol (2015) 035003 V H Nguyen and B H Nguyen and F ⎡⎣ φ; A; Q aμ , Q oμ⎤⎦ = { Z 0e ∫ [Dψ ] [ Dψ¯ ] exp { iI0e [ ψ , ψ¯ ] } ∫ ∫ × exp − i dx dyφ (x ) u (x − y) ψ¯ (y) ψ (x ) { [ ψ , ψ¯ ; A] } exp { iI × exp { iI eo ⎡⎣ ψ , ψ¯ ; Q oμ⎤⎦ }⋅ × exp iI functional of any subsystem of above-mentioned interacting system of four fields φ(x), Aμ(x), Qaμ(x) and Qoμ(x) The first subsystem is the collective oscillation field φ(x) In references [4, 5] it was shown that this field is split into two parts eγ ea ⎡ } aμ ⎤ ⎣ ψ , ψ¯ ; Q ⎦ } φ (x ) = φ0 (x ) + ζ (x ), (32) where φ0(x) is the static background field, φ0(x) = φ0(x, t) = φ0(x) corresponding to the extreme value of the action functional I0[φ] of this field in the harmonic approximation Expanding four last exponential functions in rhs of relation (32) into power series, neglecting the very small terms proportional to m−2 and performing the functional integration over the Grassmann variables, after lengthy but standard calculations we obtain following expression of the functional (32) { } F [ φ ; A ; Q aμ , Q oμ] = exp iW [ φ ; A ; Q aμ , Q oμ] , dx dyφ (x ) u (x − y) φ (y) + W (1,0,0,0) × [ φ ; A ; Q aμ , Q oμ] + W (2,0,0,0) [ φ ; A ; Q aμ , Q oμ], ∫ ∫ I0 [φ] = (33) oμ I0 ⎡⎣ φ0 + ζ ⎤⎦ = I0 [ φ0 ] + Ieff [ζ ] , W [ φ ; A ; Q aμ , Q oμ] ∞ ∞ ∞ ∞ ∑ ∑ ∑ ∑ W (m,n,p,q) [ φ; A; Qaμ , Qoμ], = (34) Ieff [ζ ] = the term W (m, n, p, q) [φ; A; Q aμ, Qoμ] being a homogeneous functional polynome of mth, nth, pth, qth orders with respect to the functions φ(x), A(x), Qaμ(x), Qoμ(x), respectively Substituting expression (33) of functional F [φ; A; Q aμ, Qoμ] into rhs of formula (30), we transform the total functional integral Ztot of the system of four interacting fields φ(x), Aμ(x), Qaμ(x), Qoμ(x) into the form Z 0e Zφ × ⎛ ∫ dx∫ dyζ (x) K (x − y) ζ (y), ∫ ∫ + dx′ dy′u (x − x′) Π (x′ − y′) u (y′ − y) , Π (x − y) = −iG (x − y) G (y − x ), (35) πe K˜ (k , ω) = ω2 − ω 02 − γ 2k , 2 ε0 k ω ( ∫ ∫ ω 02 = γo ⎡ + Iint ⎣ A ; Q ot ⎤⎦ + W ⎡⎣ φ ; A ; Q aμ , Q oμ⎤⎦ γ2 = oμ Since the functional W [φ; A; Q , Q ] is a series of the form (34), the total action functional of the interacting system of four fields φ(x), Aμ(x), Qaμ(x) and Qoμ(x) has following expression ∞ ∫ ∫ ∞ ∞ ) (44) πe n , ε0 m (45) pF , m2 (46) pF is the electron momentum at the Fermi surface In terms of the Fourier transforms ζ˜[k, ω] and K˜ [k, ω] formula (73) becomes dx dyφ (x ) u (x − y) φ (y) γa ⎡ at + I0γ [A] + I0a ⎡⎣ Q aμ⎤⎦ + I0o ⎡⎣ Q oμ⎤⎦ + Iint ⎣ A ; Q ⎤⎦ γo ⎡ ot + Iint ⎣ A ; Q ⎤⎦ + (43) n0 is the electron density and (36) Itot ⎡⎣ φ ; A ; Q aμ , Q oμ⎤⎦ = (42) where ω0 is the plasma frequency of the electron gas dx dyφ (x ) u (x − y ) φ (y ) γa ⎡ + I0γ [A] + I0a ⎡⎣ Q aμ⎤⎦ + I0o ⎡⎣ Q oμ⎤⎦ + Iint ⎣ A ; Q at ⎤⎦ aμ (41) G(x − y) is the two-point Green function of free electron Denote ζ˜[k, ω] and K˜ [k, ω] the Fourier transforms of the field ζ(x) and the kernel K(x − y) It was known that in the case of a homogeneous electron gas ⎞ where the total action functional Itot [φ; A; Q aμ, Qoμ] of this system has following expression Itot ⎡⎣ φ ; A ; Q aμ , Q oμ⎤⎦ = K (x − y ) = u (x − y ) ∫ [Dφ] ∫ [DA] δ ⎜⎝ ∂∂Ax ⎟⎠ ∫ [ DQaμ] ∫ [ DQoμ] exp { iItot [ φ; A; Qaμ, Qoμ]}, (40) where m =0 n =0 p =0 q =0 Z tot = (39) and ζ(x) is the field of small fluctuations around background field φ0(x) We call ζ(x) the fluctuation field In terms of φ0(x) and ζ(x) the action functional I0[φ] has the expression where W [φ; A; Q , Q ] is a functional power series of φ(x), Aμ(x), Qaμ(x), Qoμ(x) as the functional variables: aμ (38) Ieff [ζ ] = ∞ ∑ ∑ ∑ ∑ W (m, n, p, q) ⎡⎣ φ; A; Q aμ , Q oμ⎤⎦ 1 dk dω ζ˜(k, ω)* 2 π (2π ) × K˜ (k, ω) ζ˜(k, ω) ∫ ∫ (47) Setting m=0 n =0 p =0 q =0 (37) σ˜(k, ω) = By grouping suitable terms from the expression in rhs of formula (37), we can derive the expression of total action πe ˜ ζ (k, ω) ε0 ω2k (48) Adv Nat Sci.: Nanosci Nanotechnol (2015) 035003 V H Nguyen and B H Nguyen and introducing the scalar field σ (x , t )with the Fourier transforms σ˜(k, ω), we rewrite Ieff [ζ ] in the new form Ieff [ζ ] = I0pl [σ ] = × where I (m, p, q) [ φ ; Q aμ , Q oμ] = W (m, o, p, q) [ φ ; A ; Q aμ , Q oμ] ∫ d x∫ dt Because the plasmon is the quasiparticle generated by the fluctuation ζ(x) around the background field φ0(x), the state φ0(x) must be considered as the physical vacuum of the plasmon field Therefore the term Iint [φ0 ; Q aμ , Qoμ] must be included into the total action functional of the subsystem comprising only phonon fields Q aμ and Qoμ: ⎫ ⎧ ⎡ ∂σ (x , t ) ⎤2 ⎨⎢ − γ [σ (x , t ) ]2 − ω 02 σ (x , t )2 ⎬⋅ ⎥ ⎩ ⎣ ∂t ⎦ ⎭ ⎪ ⎪ ⎪ ⎪ (49) Functional (49) is the action functional of free plasmon field It has the form similar to the action functional of the Klein–Gordon real scalar field in relativistic quantum field theory [10–13], except for a scaling factor γ at the spatial coordinates After the canonical quantization procedure, real scalar field σ (x , t ) becomes a Hermitian quantum field, whose quantum is plasmon: the quantum plasmon field The expression of σ (x , t ) in terms of the destruction and creation operators of plasmon was known Thus the quantum plasmon field based on the study of collective oscillation field φ(x) as the fundamental subsystem has been constructed Another important subsystem is that of phonon fields developed in the preceding section The quantum fields of interacting plasmon–photon subsystem were also constructed in reference [14] The quantum field theory of plasmon–phonon subsystem is the subject of the next section ph Itot [ Qaμ , Qoμ] = I0a [ Qaμ] + I0o [ Qoμ] + Iint ⎡⎣ φ0 ; Qaμ , Qoμ⎤⎦ (52) In order to avoid the lengthy and complicated formulae and as the simple example, let us limit to the first order approximation (m = 1) with respect to the field φ0(x) in the expression (50) of Iint [φ0 ; Q aμ , Qoμ] Then in the harmonic approximation with respect to the phonon fields (p + q ⩽ 2) we have following action functional of the subsystem of phonon fields Q aμ and Qoμ interacting with the background field φ0(x) of the collective oscillations of the interacting electron gas Ioph [ Q aμ , Q oμ] = + ( ) ⎤ ⎡ oμ − υμ2 ∑ ( ∂ i Q aμ (x ) ) ⎥ + ∑ dx ⎢ Q̇ (x ) ⎣ ⎥⎦ μ i=1 2⎤ − Ω μ2 ( Q oμ (x ) ) ⎦ − dx n (x) ⎡⎣ ga  Q al (x ) ( ∫ + ) ∫ ( ) go (  Q ol (x )) ⎤⎦ + ∫ dx∫ dx′∫ dy φ0 (x′) × u (x′ − x ) Π (x − y) ⎡⎣ ga  Q al (x ) + go  Q ol (x ) ⎤⎦ dx dy ⎡⎣ ga  Q al (x ) + go  Q ol (x ) ⎤⎦ + × Π (x − y) ⎡⎣ ga  Q al (y) + go  Q ol (y) ⎤⎦ dx dx′ dy dz φ0 (x′) u (x′ − x ) + × Π (x − y , x − z) ⎡⎣ ga  Q al (y) + go  Q ol (y) ⎤⎦ × ⎡⎣ ga  Q al (z) + go  Q ol (z) ⎤⎦ , ( ( ∫ ∫ ) ( ) ( ) ( ) ) ( ) ∫ ∫ ∫ ∫ ( ) ( ( ) ) ( ) (53) where ∞ ∑ ∑ I (m,p,o) [ φ; Qaμ , Qoμ] Π (x − y)(x − z) = −[G (x − y) G (y − z) G (z − x ) + G (x − z ) G (z − y ) G (y − x )] m =1 p =1 ∞ ⎡ ̇ aμ ⎢ Q (x ) 2⎣ Now we study in detail the interacting plasmon–phonon system In order to avoid lengthy expression we limit to the harmonic approximation with respect to two types of fields: (i) the collective oscillation field and (ii) both acoustic and optic phonon fields Moreover, since the interaction of longitudinal phonons with electron is much stronger than that of transverse phonons, we can neglect the contribution of transverse phonons in the phenomena and processes in which there exists the competition of longitudinal phonons, and retain the electron–transverse phonon interaction only when the transverse phonons play the essential role In particular, the contribution of transverse phonons must be taken into account when we consider the phenomena and processes with the participation of the transverse electromagnetic field, as this will be performed in the next section First we note that the electron–phonon interaction leads to the interaction of phonons with the collective oscillation field The action functional of the interaction between the fields φ(x), Qaμ(x), Qoμ(x) has the expression of the form Iint [ φ ; Q aμ , Q oμ] = ∑∫ dx μ Quantum fields of interacting plasmon–phonon system ∞ (51) ∞ ∑ ∑I (m,o,q) [ φ; Qaμ , Qoμ] It consists of two parts m =1 q =1 ∞ ∞ ∞ + ∑ ∑∑I (m, p, q)[ φ ; Q aμ , Q oμ], (54) I0ph [ Q aμ , Q oμ] = I0t ⎡⎣ Q at , Q ot ⎤⎦ + I0l ⎡⎣ Q al , Q ol ⎤⎦ , (50) m =1 p =1 q =1 (55) Adv Nat Sci.: Nanosci Nanotechnol (2015) 035003 V H Nguyen and B H Nguyen where the first part I0t ⎡⎣ Q at , Q ot ⎤⎦ = + ∂ 2Q ol (x ) ⎡ ⎤ − υt2 ∑ ∂ i Q at (x ) ⎥ ⎥⎦ i=1 2⎤ ∂ i Q ot (x ) ⎥ ⎦ ∫ dx 12 ⎢⎢ ( Q̇ at (x ) ) ⎣ ⎡ ∫ dx 12 ⎢⎣ ( Q̇ ot (x ) ) − Ω t2 ( ( ) ) ∫ dy∫ dy′x Π (x − y) u (y − y′) φ0 (y′) − go2∫ dy x Π (x − y) (  Q 0ol (y)) − ga go∫ dy x Π (x − y) (  Q 0al (y)) − go2 ∫ dy∫ dz∫ dz′x Π (x − z , y − z) − go (56) is the action functional of the free transverse phonons, and ⎤ − υ l2 ∑ ∂ i Q al (x ) ⎥ ⎥⎦ ⎣ i=1 2⎤ 1⎡ ol dx ⎢ Q̇ (x ) − Ω l2 Q ol (x ) ⎥ 2⎣ ⎦ dxn (x) ⎡⎣ ga  Q al (x ) + go  Q ol (x ) ⎤⎦ Iol ⎡⎣ Q al , Q ol ⎤⎦ = + ∫ ⎡ ∫ dx 12 ⎢⎢ ( Q̇ al (x ) ) ( ) ( ( ) ( − go ga dy dz dz′ x Π (x − z , y − z) ∫ ∫ ∫ ) × ⎡⎣ ga  Q al (x ) + go  Q ol (x ) ⎤⎦ + dx dy ⎡⎣ ga  Q al (x ) + go  Q ol (x ) ⎤⎦ × Π (x − y) ⎡⎣ ga  Q al (y) + go  Q ol (y) ⎤⎦ + dx dx′ dy dz φ0 (x′) u (x′ − x ) × Π (x − y, x − z ) ⎡⎣ ga  Q al (y) + go  Q ol (y) ⎤⎦ × ⎡⎣ ga  Q al (z ) + go  Q ol (z ) ⎤⎦ ( ) ( ( ) ( ) ( al Q al (x ) = Q al (x ) + q (x ), ) ol Q ol (x ) = Q ol (x ) + q (x ) ∫ ∫ ∫ ∫ ( ) ( ( ) ) ( al δ Q (x ) = δIol ⎡⎣ Q al , Q ol ⎤⎦ ) ol δ Q (x ) =0 (57) ol ⎤ l ⎡ al ol⎤ I0l ⎡⎣ Q al , Q ol ⎤⎦ = I0l ⎡⎣ Q al , Q ⎦ + Ieff ⎣ q , q ⎦ ∂t l ⎡ al ol⎤ Ieff ⎣q , q ⎦ = ∫ 1⎡ dx ⎢ q̇ al (x ) ⎢⎣ ( ⎡ ∫ dx 12 ⎣⎢ ( q̇ol (x) ) ⎤ − υ l2 ∑ ∂ i qal (x ) ⎥ ⎥⎦ i=1 2⎤ qol (x ) ⎥ ⎦ ) ( ) ( ) + ∫ dx∫ dy ⎡⎣ g (  q (x )) + g (  q (x )) ⎤⎦ × Π (x − y) ⎡⎣ g (  q (y)) + g (  q (y)) ⎤⎦ + (58) − Ω l2 al a a + ( ) − υ l2  Q al (x ) = ga n (x) al ol o ol o ∫ dx∫ dx′∫ dy∫ dz φ0 (x′) u (x′ − x) × Π (x − y , x − z) ⎡⎣ ga  qal (y) + go  qol (y) ⎤⎦ × ⎡⎣ ga φqal (z) + go  qol (z) ⎤⎦ ( ∫ dy∫ dy′x Π (x − y) u (y − y′) φ0 (y′) − ga2∫ dy x Π (x − y) (  Q al ( y )) − ga go∫ dy x Π (x − y) (  Q 0ol (y)) − ga2 ∫ dy∫ dz × ∫ dz′x Π (x − z , y − z ) (  Q 0al (y) ) u (z − z′) φ0 (z′) − ga go ∫ dy∫ dz∫ dz′ x Π (x − z , y − z ) (  Q ol ( y )) − ga × u (z − z′) φ0 (z′), (62) l where Ieff [qal , qol] is the effective action functional of the fluctuation longitudinal phonon fields qal (x ) and qol (x ) It has following quadratic from it follows the system of differential-integral equations for the background phonon fields Qoal (x ) and Qol o (x ) corresponding to the extreme value of the action functional (57): ∂ 2Q al (x ) (61) It can be shown that in terms of Q0al (x ), Qol (x ) and al ol q (x ), q (x ) the action functional (57) has expression is the total action functional of the acoustic as well as optic longitudinal phonons interacting with the background field φ0(x) of collective oscillation field φ(x) The interaction action functional in this expression leads to the mixing between acoustic and optic phonons From the extreme action principle δIol ⎡⎣ Q al , Q ol ⎤⎦ (60) Denote qal (x ) and qol (x ) the small fluctuations of the longitudinal phonon fields Qal (x ) and Qol (x ) around the background characterized by the functions Q0al (x ), Qol (x ) and considered as the physical vacuum of the system ) ( ) ×  Q 0al (y) u (z − z′) φ0 (z′) ) ( ∫ ∫ ) ×  Q 0ol (y) u (z − z′) φ0 (z′) ∫ ( ) ( ) + ∫ dx∫ dx′∫ dy φ0 (x′) u (x′ − x ) Π (x − y) − − Ω l2 Q 0ol (x ) = go n (x) ∂t ( ) ( ) ( ) ) (63) Since the fluctuation of longitudinal phonon fields generates the quasiparticles participating in various dynamical processes, the fluctuating longitudinal phonon fields qal (x ) and qol (x ) will be called dynamical longitudinal phonon fields (59) Adv Nat Sci.: Nanosci Nanotechnol (2015) 035003 V H Nguyen and B H Nguyen In order to exhibit the property of these fields to be longitudinal let us use following modified Fourier expansion qal, ol (x ) = qal, ol (x , t ) = − = − i (2π )3 i (2π )4 ∫ dk eikx kk ∫ d k 21π ∫ dω ei (kx−ωt ) kk Thus the total action functional of longitudinal phonon fields in the harmonic approximation consists of two parts l ⎡ al l ⎡ al l ⎡ al ol⎤ ol⎤ ol⎤ Ieff ⎣ q , q ⎦ = I0 ⎣ q , q ⎦ + Iint ⎣ q , q ⎦ , a, o θ˜ (k ) a, o θ˜ (k, ω) where (64) I0l ⎡⎣ qal , qol⎤⎦ = so that q al, ol (x ) = (2π )4 ∫ dk e ikx a, o k θ˜ (k ) (65) + In terms of the modified Fourier transforms θ˜ a, o (k ) the action functional (63) becomes l ⎡ al ol ⎤ Ieff ⎣q , q ⎦ = (2π )4 ∫ dk 12 { θ˜ a (k )* ( k 02 − υl2 k2) θ˜ a (k ) } 1 + dk dl φ˜ (l )*u˜(l ) (2π )4 (2π )4 a o ⎡ × ⎣ ga θ˜ (k − l ) + go θ˜ (k − l ) ⎤⎦ * a o × k − l Π˜ (k − l , k ) ⎡⎣ ga θ˜ (k ) + go θ˜ (k )⎤⎦, ∫ ∫ dk Π (x − y , x − z ) = (2π )4 × (2π )4 e Π˜ (k ), (67) ( + θ (k )⁎ k 02 − Ω l2 × ) eikx θ˜ a, o (k ), (73) (74) (2π )4 ⎡ p2 ⎤ ∫ dk e−ik (x−y) ⎢⎢ Π˜ (k ) + 16 mF Π˜ (k, k )⎥⎥ ⎣ ⎦ (75) (68) is the interacting functional describing the elastic scattering of phonons in the effective potential field V(x − y) as well as the mixing between longitudinal acoustic and optic phonons Note that the effective potential V(x − y) is both non-local and non-instantaneous Since plasmons are generated by the fluctuation ζ(x) of the collective oscillation field φ(x) around its static background φ0(x), in order to study the plasmon–phonon interaction first we derive the expression of the action functional Iintfl [ζ; Q aμ , Qoμ] of the interaction between the fluctuation field and phonon fields We have pF , m (69) Iintfl [ ζ ; Q aμ , Q oμ] = Iint ⎡⎣ φ0 + ζ ; Q aμ , Q oμ⎤⎦ , ∫ dk 12 { θ˜a (k )⁎ ( k 02 − υl2 k2)θ˜a (k ) (76) where Iint [φ; Q aμ, Qoμ] is determined by formulae (50) and (51) Because the terms containing transverse phonon fields are very small, we discard them and retain only the terms containing longitudinal fields According to formula (61) each of them consists of two parts: the background longitudinal field Qoal (x ) or Qol o (x ) and the dynamical longitudinal phonon field qal (x ) or qol (x ) Consider again the case of homogeneous electron gas Then in the harmonic approximation with respect to both types of fields (fluctuation field and phonon fields) the action functional can be represented in the θ (k ) + (2π )4 ˜o ∫ dk with ∫ dl e−il (x−y) ∫ dk eik (x−z ) Π˜ (l, k ) (2π )4 (2π )4 ∫ ∫ where pF is the electron momentum at the Fermi level, and formula (66) of the total action functional becomes l ⎡ al ol⎤ Ieff ⎣q , q ⎦ = ) ⎤ ⎥ ⎥⎦ l ⎡ al ol⎤ Iint dx dy ⎡⎣ ga θ a (x ) + go θ o (x ) ⎤⎦ ⎣q , q ⎦ = × V (x − y) ⎡⎣ ga θ a (y) + go θ o (y) ⎤⎦ (66) V (x − y ) = φ˜ (l) u˜(l) = (2π )4 δ (l) and For evidently demonstrating main features of the total action functional (66) let us consider the case of the homogeneous electron gas with a constant electron density n(x) = n0 = const It can be shown that in this case ˜o ( θ a , o (x ) = where Π˜ (k ) and Π˜ (l, k ) are the Fourier transforms of functions Π (x − y ) and Π (x − y, x − z ): ik (x − y ) 2⎤ ⎛ ∂θ a ( x ) ⎞ ⎥ − υ l2 ∑ ⎜ ⎟ ⎝ ∂xi ⎠ ⎥⎦ i=1 ⎣ ⎡ ⎛ o ⎞2 ∂θ ( x ) dx ⎢ ⎜ ⎟ − Ω l2 θ (x ) ⎢⎣ ⎝ ∂t ⎠ ∫ Π (x − y ) = (2π )4 ⎞2 a can be considered as the action functional of a system of two scalar fields θ a (x ) = θ a (x, t ) and θ o (x ) = θ o (x , t ) describing longitudinal acoustic and optic phonons, whose Fourier transforms are the functions θ a (k ) and θ o (k ) in formula (70), ) a o × k Π˜ (k ) ⎡⎣ ga θ˜ (k ) + go θ˜ (k )⎤⎦ ∫ ⎡⎛ ∫ dx 12 ⎢⎢ ⎜⎝ ∂θ∂t(x) ⎟⎠ (72) o o a o + θ˜ (k )* k 02 − Ω l2 θ˜ (k ) + ⎡⎣ ga θ˜ (k ) + go θ˜ (k ) ⎤⎦ * ( (71) } ∫ dk 12 ⎡⎣ ga θ˜a (k ) + go θ˜o (k ) ⎤⎦ ⁎ ⎡ ⎤ pF ˜ a o × ⎢ Π˜ (k ) + Π (k , k ) ⎥ ⎡⎣ ga θ˜ (k ) + go θ˜ (k )⎤⎦ m ⎢⎣ ⎥⎦ (70) Adv Nat Sci.: Nanosci Nanotechnol (2015) 035003 V H Nguyen and B H Nguyen general form as follows: Iintfl ⎡⎣ ζ ; Q al , Q ol ⎤⎦ = It is straightforward to derive the expression of the action functional of the interaction of plasmon with longitudinal phonons from formulae (61), (77), (79) and (81) ∫ dx∫ dy ζ (x) × ∑ ⎡⎣ Fa(i ) (x , y) Qial (y) + Fo(i ) (x , y) Qiol (y)⎤⎦ Quantum fields of interacting plasmon–photon– phonon system i=1 + ∫ dx1∫ dx2 ∫ dy ζ ( x1) ζ ( x2 ) On the basis of the results obtained in preceding sections we consider now the whole system of interacting plasmon, photon and phonons The total action functional of this system has the expression (37) The collective oscillation scalar filed φ(x) is split into two parts: the static background field φ0(x) and the fluctuation field ζ(x) Each longitudinal phonon field Qal (x ), Qol (x ) is also split into two parts: the static background field Qoal (x ), Qol o (x ) and the dynamical longitudinal phonon fields qal (x ), qol (x ), while the transverse phonon fields Qat (x ), Qot (x ) themselves are the dynamical ones without the background fields: Qat (x ) = qat (x ) and Qot (x ) = qot (x ) Plasmon is the quantum of a Hermitian scalar field σ(x) called plasmon field The longitudinal phonons can be considered as the spinless quasiparticles The scalar fields θ a (x ) and θ (x ) having these spinless quasiparticles as their quanta are called effective longitudinal phonon fields In order to shorten lengthy formulae, they are introduced and used instead of the original longitudinal vector fields qal (x ) and qol (x ) Meanwhile the transverse acoustic and optic phonon are the quanta of the original transverse vector fields qat (x ) and qot (x ) Above-mentioned quantized fields have following expansions in terms of the destruction and creation operators of the corresponding quasiparticles: × ∑ ⎡⎣ Fa(i ) ( x1, x2 , y) Qial (y) + Fo(i ) ( x1, x2 , y) Qiol (y)⎤⎦ i=1 + ∫ dx∫ dy1∫ dy2 ζ (x) 3 (i,j ) × ∑∑ ⎡⎣ Faa ( x , y1, y2 ) Qial ( y1) Q jal ( y2 ) i=1 j=1 (i,j ) + Fao ( x , y1, y2 ) Qial ( y1) Q jol ( y2 ) (i,j ) + Foo ( x , y1, y2 ) Qiol ( y1) Q jol ( y2 )⎤⎦ + ∫ dx1∫ dx2 ∫ dy1∫ dy2 ζ ( x1) ζ ( x2 ) 3 (i,j ) × ∑∑ ⎡⎣ Faa ( x1, x2 , y1, y2 ) Qial ( y1) Q jal ( y2 ) i=1 j=1 (i,j ) + Fao ( x1, x2 , y1, y2 ) Qial ( y1) Q jol ( y2 ) (i,j ) + Foo ( x1, x2 , y1, y2 ) Qiol ( y1) Q jol ( y2 )⎤⎦ (77) ˜ The Fourier transform ζ (k, ω) of the fluctuation field ζ (x ) is expressed in terms of the Fourier transform σ˜(k, ω) of the fluctuation field σ (x ) according to formula (48), i.e ε0 ω2k ζ˜(k, ω) = πe σ˜(k, ω) (78) σ (x , t ) = Therefore between ζ (x )and σ (x ) there exists following linear functional relation ∫ dyT (x − y) σ (y), ζ (x ) = × (2π )4 ∫ d k∫ dω ei ⎡⎣ k (x−y)−ω ( x −y ) ⎤⎦ ε0 ω2k πe θ a (x , t ) = ⋅ ∫ dy R (x − y) θ a,o (y) , i (2π )4 θ o (x , t ) = (81) ∫ d k∫ dω ei ⎡⎣ k (x−y)−ω ( x −y ) kk ⋅ 0 (83) ∫ dk 2ωal (k ) (84) and (2π )3 ∫ dk 2ωol (k ) l l ⎡ ⎡ ⎡ ⎤ × ⎣⎢ bl (k) ei ⎣ kx − ωo (k ) t + bl (k)+ e−i ⎣ kx − ωo (k ) t ⎦⎥ , where R (x − y ) = − (2π )3 l l ⎡ ⎡ ⎡ ⎤ × ⎢⎣ a l (k) ei ⎣ kx − ωa (k ) t + a l (k)+ e−i ⎣ kx − ωa (k ) t ⎥⎦ , (80) Similarly, according to formula (64), between the dynamical longitudinal phonon fields qal, ol (x ) and the scalar fields θ a, o (x ) there exists following linear functional relation qal, ol (x ) = ∫ where c(k) and c(k)+ are the destruction and creation operators of the plasmon with momentum k, and ω(k) is its energy, (79) where T (x − y ) = 1 dk π (k ) (2π ) × ⎡⎣ c (k) ei [kx − ω (k ) t + c (k)+ e−i [kx − ω (k ) t ⎤⎦ , (85) where al(k) or bl(k) and al(k)+ or bl(k)+ are the destruction and creation operators, respectively, of the longitudinal acoustic or optic phonon with momentum k and energy (82) Adv Nat Sci.: Nanosci Nanotechnol (2015) 035003 V H Nguyen and B H Nguyen ωal (k ) ≈ υl k or ωol (k ) ≈ Ωl , qat (x , t ) = (2π )3 ∫ dk The total action functional of the whole system can be represented in the form ∑ξ (s) (k) Itot ⎡⎣ ζ ; A ; qal , qat , qol , qot ⎤⎦ = I0 ⎡⎣ ζ ; A ; qal , qat , qol , qot ⎤⎦ + Iint ⎡⎣ ζ ; A ; qal , qat , qol , qot ⎤⎦ , 2ωat (k ) s = t t ⎡ ⎡ ⎡ ⎤ × ⎣ a st (k) ei ⎣ kx − ωa (k ) t + a st (k)+ e−i ⎣ kx − ωa (k ) t ⎦ (86) (90) and qot (x , t ) = (2π )3 ∫ dk 2ωot (k ) where the first term in rhs of equation (90) is the action function of free fields and second term is the interaction action functional Action functional of free fields is the sum of action functional of all six free dynamical fields ∑ξ (s) (k) s=1 t t ⎡ ⎡ ⎡ ⎤ × ⎣ bst (k) ei ⎣ kx − ωo (k ) t + bst (k)+ e−i ⎣ kx − ωo (k ) t ⎦ , (87) ⎤ I0 ⎡⎣ ζ ; A ; qal , qat , qol , qot ⎤⎦ = I0 [ φ0 ] + I0l ⎡⎣ Q 0al , Q ol 0⎦ + Ieff [ζ ] + I0γ [A] + I0l ⎡⎣ q 0al , q 0ol ⎤⎦ + I0t ⎡⎣ q 0at , q 0ot ⎤⎦ where a st (k) or bst (k) and a st (k)+ or bst (k)+ are the destruction and creation operators, respectively, of the transverse acoustic or optic phonon with momentum k, energy ωat (k ) ≈ υt k or ωot (k ) ≈ Ωt and polarization vector ξ(s)(k), s being the index indicating the polarization state of the transverse phonons with momentum k kξ(s) (k) = All six terms in rhs of equation (91) were given in preceding sections The expression of interaction action functional has the general form Iint ⎡⎣ ζ ; A ; qal , qat , qol , qot ⎤⎦ (88) The total action functional of the system of interacting plasmon, photon and phonons has the expression of the form ∞ ∞ ∞ ∞ ∑ ∑ ∑ ∑ W (m,n,p,q) ⎡⎣ φ0 + ζ; A; Qa0μ m =0 n =0 p =0 q =0 + qaμ, Q 0oμ + qoμ] ∞ ( i, j, k1, k 2, l1, l2 ) ∑ ∑ ∑I (92) The term I (i, j, k1, k 2, l1, l2 ) [ζ; A; qal , qat , qol , qot] is a homogeneous functional polynome of ith, jth, k1-th, k2-th, l1-th and l2-th orders with respect to the functional variables ζ(x), A(x), qal(x), qat(x), qol(x) and qot(x), respectively According to formulae (79) and (81), the fluctuation field ζ(x) is a linear functional of plasmon field σ(x), and longitudinal phonon fields qal, ol (x ) are the linear functional of two effective scalar phonons as two spinless quasiparticles Substituting rhs of formulae (79) and (81) into the expression (92), we obtain the interaction action functional in the form of a functional of the scalar fields σ(x), θ a (x ) and θ (x ) of plasmon and longitudinal phonons, and transverse vector fields qat (x ) and qot (x ) of transverse phonons In the diagrammatic representation of the finally derived expression of the interaction action functional, each term is represented by a corresponding vertex with definite external lines The number of external lines of each type indicate the number of corresponding particles or quasiparticles participating in the represented interaction fact γa ⎡ γo + I0o [ Q o0μ + qoμ] + Iint ⎣ A ; qat ⎤⎦ + Iint ⎡⎣ A ; qot ⎤⎦ ∞ ∞ i = j = k1= k = l1= l2 = ∫ ∫ ∞ ∞ × ⎡⎣ ζ ; A ; qal , qat , qol , qot ⎤⎦ dx dy ⎡⎣ φ0 (x ) + ζ (x ) ⎤⎦ u (x − y) × ⎡⎣ φ0 (y) + ζ (y)⎤⎦ + I0γ [A] + I0a [ Q 0aμ + qaμ] + ∞ = ∑∑ ∑ Itot ⎡⎣ φ0 + ζ ; A ; Q a0μ + qaμ, Q o0μ + qoμ⎤⎦ = (91) (89) with Q0at (x ) = Q0ot (x ) = After the quantization procedure the fluctuation field ζ(x), the fluctuating longitudinal phonon fields qal (x ) and qol (x ), the transverse phonon fields qat (x ) and qot (x ) become the field operators, while the background fields φ0(x), Q0al (x )and Qol (x ) remain to be classical fields Quantum fluctuation field is expressed in terms of the plasmon quantum field σ(x) through the linear functional relation (79), longitudinal phonon fields are expressed in terms of the longitudinal phonon effective scalar fields θ a (x ), θ (x ) by means of the linear functional relation (81) The quantum of σ(x) is plasmon, the quanta of θ a, o (x ) are longitudinal phonons, and the quanta of qat, ot (x ) are transverse phonons Thus in the whole system of interacting plasmon, photon and phonons there are three types of dynamical fields: Conclusion and discussions In the present work, by means of the powerful functional integral technique, the quantum fields of the interacting system of plasmons, photons and phonons in electron gas of solids were constructed The starting assumptions are the fundamental principles of electrodynamics and quantum (1) Fluctuation field ζ(x), (2) Transverse electromagnetic field A(x), (3) Longitudinal as well as transverse acoustic and optic phonon fields qal (x ), qat (x ) and qol (x ), qot (x ) 10 Adv Nat Sci.: Nanosci Nanotechnol (2015) 035003 V H Nguyen and B H Nguyen Acknowledgement theory of many-body systems The general form of the formula of total action functional was established The whole system is described by a set of six fields: the scalar plasmon field σ(x), transverse electromagnetic field A(x), the effective scalar fields θ a (x ) and θ (x ) of longitudinal phonons as spinless quasiparticles, the transverse phonon vector fields qat (x ) and qot (x ), The authors would like to express their sincere gratitude to Vietnam Academy of Science and Technology for the support References  qat (x ) =  qot (x ) = [1] Jackson J D 1975 Classical Electrodynamics (New York: John Wiley & Sons, Inc.) [2] Kittel C 2005 Introduction to Solid State Physics (New York: John Wiley & Sons, Inc.) [3] Kittel C 2005 Quantum Theory of Solid (New York: John Wiley & Sons, Inc.) [4] Nguyen V H and Nguyen B H 2012 Adv Nat Sci.: Nanosci Nanotechnol 035009 [5] Nguyen V H and Nguyen B H 2014 Adv Nat Sci.: Nanosci Nanotechnol 035004 [6] Nguyen V H, Nguyen B H and Ngo V T 2015 Adv Nat Sci.: Nanosci Nanotechnol 015014 [7] Nguyen V H and Nguyen B H 2015 Adv Nat Sci.: Nanosci Nanotechnol 023001 [8] Kollár J, Koroó N, Menghhard N and Siklós T 1985 Phonon Physics (Singapore: World Scientific) [9] Frölich H 1954 Adv Phys 325 [10] Weinberg S 1995 The Quantum Theory of Fields: Foundation vol (New York: Cambridge University Press) [11] Gross F 1993 Relativistic Quantum Mechanics and Field Theory (New York: John Wiley & Sons, Inc.) [12] Brown L S 1992 Quantum Field Theory (New York: Cambridge University Press) [13] Sterman G 1993 An Introduction to Quantum Field Theory (New York: Cambridge University Press) [14] Nguyen V H and Nguyen B H 2015 Adv Nat Sci.: Nanosci Nanotechnol 025010 The total interaction action functional of the whole system is a series in which each term represents a definite interaction fact The derived interaction action functional has following particular feature: all terms in its expression represent the non-local and non-instantaneous interaction between the involved quantum fields The Lagrangian and Hamiltonian similar to those in traditional quantum field theory not exist Although each term in the expression of the interaction action functional is the matrix element of a physical process in the first order approximation, the matrix elements of physical processes in higher order approximations cannot be calculated by means of the traditional perturbation theory Therefore it is necessary to elaborate the new method for calculating the matrix elements of physical processes in higher orders from the formula of total action functional derived in the present work Thus a lot of theoretical works should be performed in order to fulfill the construction of a complete quantum theory of physical phenomena and processes with the participation of plasmon 11 ... construction of the quantum field theory of the interacting system of plasmons, photons and phonons on the basis of general fundamental principles of electrodynamics and quantum field theory of many-body systems... of electrodynamics and quantum theory of many-body system For the application of mathematical tools of functional integral technique, the physical content of the theory of phonons in solids must... from the formula of total action functional of the whole system it is possible to derive expressions of action functional of different subsystems of related fields The fundamental subsystem is the