Quantum field theory of interacting plasmon–photon–phonon system

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 Q a l ( x ) , ?> Q o l ( x ) ?> are also split into the background fields Q 0 a l ( x ) , ?> Q 0 o l ( x ) ?> and dynamical fields q a l ( x ) , ?> q o l ( x ) ?> while the transverse phonon fields Q a t ( x ) , ?> Q o t ( x ) ?> themselves are dynamical fields q a t ( x ) , ?> q o t ( x ) ?> without background fields. After the canonical quantization procedure, the background fields φ0(x), Q 0 a l ( x ) , ?> Q 0 o l ( x ) ?> remain the classical fields, while the fluctuation fields ζ(x) and dynamical phonon fields q a l ( x ) , ?> q a t ( x ) , ?> q o l ( x ) , ?> q o t ( 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 0 ( x ) , ?> while transverse phonons are the quanta of the original Hermitian transverse phonon vector fields q a t ( x ) , ?> q o t ( 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 0 ( x ) ?> and transverse phonon vector quantum fields q a t ( x ) , ?> q o t ( x ) ?> .


Introduction
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 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][2][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 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 plasmonphoton-phonon interacting system by applying the functional integral technique [4][5][6][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 2. 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 3 is devoted to the establishment of the expression of total action functional of the interacting plasmonphoton-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 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 4. 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 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 5. The expression of total action functional of this whole system, described by background fields φ 0 (x), x Q ( ) al 0 and x Q ( ), ol 0 fluctuation field ζ(x), electromagnetic field A(x) and dynamical phonon fields 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 6.

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][5][6] and the unit system with ℏ = = c 1. Consider first the acoustic phonons. In both abovementioned types of solids there exist one longitudinal and two transverse acoustic phonon branches. Denote μ x Q ( ) a their quantum fields, where μ = 1, 2 for transverse phonons and μ = 3 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 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 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 In the special case of isotropic crystals with s = 2 nonequivalent ions per a primitive cell, there exist one longitudinal and two degenerate transverse optic phonon branches with limiting angular frequencies Ω l and Ω t at k → 0. Between Ω l and Ω t there exists following relation where ε 0 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 a al int and   where ψ x ( ) is the electron field operator, ψ x ( ) is its Hermitian conjugate. The coupling constants g a and g o 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 2 non-equivalent ions per a primitive cell, s = 2. Then beside the two degenerate acoustic transverse phonon branches with wave function x Q ( ) at there exist also only two degenerate optic phonon branches with wave function x Q ( ). ot 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: for acoustic transverse phonon and 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 a a at int for acoustic phonon and o ot int 0 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 is the action functional of the transverse free electromagnetic field in the transverse gauge ⎤ ⎦ is the action functional of the system of electrons mutually interacting through the Coulomb repulsion.
e e e 0 i n t where ψ ψ I ,ē 0 ⎡ ⎣ ⎤ ⎦ is the action functional of free electron moving in the electrostatic field of ions in the crystalline lattice According to formulae (7)-(10) for the electron-phonon interaction Hamiltonians we have (11) and (12) of the Hamiltonians describing the coupling of transverse phonons with photon it follows that The Coulomb interaction functional (20) is bilinear with respect to the electron density ψ ψ x x ( ) ( ). This expression can be linearized by means of the Hubbard-Stratonovich transformation 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 Expanding four last exponential functions in rhs of relation (32) into power series, neglecting the very small terms proportional to 1 m −2 and performing the functional integration over the Grassmann variables, after lengthy but standard calculations we obtain following expression of the functional (32) , ] a o into rhs of formula (30), we transform the total functional integral Z tot of the system of four interacting fields φ(x), , ] a o is a series of the form (34), the total action functional of the interacting system of four fields φ(x), A μ (x), Q aμ (x) and Q oμ (x) has following expression By grouping suitable terms from the expression in rhs of formula (37), we can derive the expression of total action functional of any subsystem of above-mentioned interacting system of four fields φ(x), A μ (x), Q aμ (x) and Q oμ (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 and ζ(x) is the field of small fluctuations around background field φ 0 (x). We call ζ(x) the fluctuation field. In terms of φ 0 (x) and ζ(x) the action functional I 0 [φ] has the expression G(x − y) is the two-point Green function of free electron. Denote ζ ω k [ , ] and ω K k [ , ] 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 ω 0 is the plasma frequency of the electron gas  Setting  (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][11][12][13], except for a scaling factor γ at the spatial coordinates. After the canonical quantization procedure, real scalar field σ t x ( , ) becomes a Hermitian quantum field, whose quantum is plasmon: the quantum plasmon field. The expression of σ t x ( , ) 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 2. 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.

Quantum fields of interacting plasmon-phonon system
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), Q aμ (x), Q oμ (x) has the expression of the form   Because the plasmon is the quasiparticle generated by the fluctuation ζ(x) around the background field φ 0 (x), the state φ 0 (x) must be considered as the physical vacuum of the plasmon field. Therefore the term φ

⎡ ⎣ ⎤ ⎦
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 φ 0 (x) in the expression (50)  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 μ Q a and μ Q o interacting with the background field φ 0 (x) of the collective oscillations of the interacting electron gas x y g y g y dx dx dy dz x u x x x y x z g y g y It consists of two parts is the action functional of the free transverse phonons, and x y g y g y dx dx dy dz x u x x x y x z g y g y is the total action functional of the acoustic as well as optic longitudinal phonons interacting with the background field φ 0 (x) of collective oscillation field φ(x). The interaction action functional in this expression leads to the mixing between acoustic and optic phonons.
x t x g n g dy dy x y u y y y g dy x y y g g dy x y y g dy dz dz x z y z y u z z z g g dy dz dz x z y z y x t x g n g dy dy x y u y y y g dy x y y g g dy x y y g dy dz dz x z y z y u z z z g g dy dz dz x z y z y u z z z x y g y g y dx dx dy dz x u x x x y x z g y g y g z g z q q q q q q q q q q q q q q , 1 2 ( ) ( ) Since the fluctuation of longitudinal phonon fields generates the quasiparticles participating in various dynamical processes, the fluctuating longitudinal phonon fields x q ( ) al and x q ( ) ol will be called dynamical longitudinal phonon fields.
In order to exhibit the property of these fields to be longitudinal let us use following modified Fourier expansion   where p F is the electron momentum at the Fermi level, and formula (66) of the total action functional becomes can be considered as the action functional of a system of two scalar fields θ θ 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 φ 0 (x), in order to study the plasmon-phonon interaction first we derive the expression of the action functional ζ μ μ of the interaction between the fluctuation field and phonon fields. We have

Quantum fields of interacting plasmon-photonphonon system
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 φ 0 (x) and the fluctuation field ζ(x). Each longitudinal phonon field x Q ( ), 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 is a homogeneous functional polynome of ith, jth, k 1 -th, k 2 -th, l 1 -th and l 2 -th orders with respect to the functional variables ζ(x), A(x), q al (x), q at (x), q ol (x) and q ot (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 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.

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 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 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 do 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.