Quantum field theory of interacting plasmon–photon system

In the framework of functional integral approach, quantum theory of interacting plasmon–photon system was constructed on the basis of general postulates (axioms) called also first principles of electrodynamics and quantum theory of many-body systems. Since plasmons are complex quasiparticles appearing as the resonances in plasma oscillations of the electron gas in solids, we start from the general expression of total action functional of interacting system consisting of electron gas and electromagnetic field. The collective oscillations of electron gas are characterized by a real scalar field φ(x) called the collective oscillation field. In the harmonic approximation the collective oscillations behave like the small fluctuations around a background field φ0(x). The difference between φ(x) and φ0(x) is called the fluctuation field ζ(x). In the case of a homogeneous and isotropic electron gas the fluctuation field ζ(x) is a linear functional of another real scalar field σ(x) satisfying the wave equation similar to the Klein–Gordon equation in relativistic quantum field theory. The quanta of corresponding Hermitian scalar field &sgr; ˆ ( x ) ?> are called plasmons. The real scalar field σ(x) is called plasmonic field. The total action functional of the interacting system of plasmonic and electromagnetic field was derived.


Introduction
During the last two decades research on the interaction of plasmon with photon and other quasiparticles in matter has stimulated the rapid development of a new scientific area: plasmonics [1,2]. However, the authors of both experimental and theoretical works on plasmonic processes and phenomena often accepted corresponding interaction Hamiltonians. In our recent article [3] we have proposed to derive the interaction Hamiltonians or action functional determining matrix elements of all plasmonic processes from basic postulates (axioms), also called first principles, of electrodynamics and quantum theory. For this purpose the effective mathematical tool is functional integral technique [4][5][6]. The ideas proposed in [3] will be implemented in the present work. The concrete expression of the action functional determining matrix elements of many plasmon-photon interaction processes will be exactly derived from three fundamental postulates of quantum theory: (i) assumption on the explicit form of total functional integral of interacting electron-photon system, (ii) extreme action principle, and (iii) canonical quantization procedure.
The original physical system is that of itinerant electrons in solid with the quantum mechanical single electron Hamiltonian where U(x) is potential energy of electron in electrostatic field of ions in the crystal lattice and m is effective mass of electron. We use the unit system with ℏ = = c 1. The abovementioned electron system will also be called the electron gas.
Since we study only physical processes and phenomena in which the electron spin plays no role, we can consider electron as a spinless fermion with anticommuting wave function ψ t x ( , ). Its Hermitian conjugate is denoted ψ t x ( , ). They are often called dynamical Grassmann variables. Being the photon wave function in transverse gauge, the electromagnetic field is a commuting vector field A(x, t) with vanishing divergence For shortening formulae we shall use the following brief notations: In section 2 the functional integral technique will be applied to the study of interacting electron-photon system. The expression of total functional integral of this system will be presented.
In section 3 a real scalar function φ(x) playing the role of the order parameter of collective oscillations in electron gas will be introduced. It will be called the collective oscillation field. The expression of interacting system of collective oscillation field φ(x) and electromagnetic field A(x) will be established.
In the harmonic approximation with respect to scalar field φ(x) this field can be split into two parts, the background static field φ 0 (x) and the fluctuation field ζ(x). In section 4 the total action functional of interacting system of two fields ζ(x) and A(x) will be derived.
In section 5 it will be shown that in the harmonic approximation with respect to collective oscillation field φ(x) and at vanishing absolute temperature T = 0 K, the fluctuation field ζ(x) is expressed in terms of a real scalar field σ(x) such that the quanta of the corresponding quantized scalar field σ x ( ) are plasmons. From the expression of total effective action functional of the system of two fields ζ(x) and A(x) it is straightforward to derive the action functional of interacting system of two fields σ(x) and A(x) in the form of a functional power series with respect to three fields: φ 0 (x), σ(x) and A(x).
The conclusion and discussions will be presented in section 6.

Functional integral of interacting electron-photon system
We start from following expression of the functional integral of the interacting electron-photon system is the action of free electromagnetic field, ψ ψ I ,ē ⎡ ⎣ ⎤ ⎦ is that of electron system in the presence of electron-electron Coulomb interaction and ψ ψ I A ,¯; int ⎡ ⎣ ⎤ ⎦ is the action functional of electron-photon interaction. It was known [7] that in the transverse gauge 2 −e being the electron charge. The action functional of electron-photon interaction has the following expression is the quantum mechanical Hamiltonian of electron-photon interaction In formula (13) we used the following notation i i The total functional integral of interacting electronphoton system has the expression When both electron-electron Coulomb interaction as well as electron-photon interaction are neglected, Z tot becomes the product and that of free electron system If only electron-photon interaction is neglected, Z tot is the product of the functional integral γ Z 0 of free electromagnetic field and that of electron system with only electron-electron Coulomb interaction The system with functional integral (18) was studied in [4,5].

Functional integral of interacting system of collective oscillation field and electromagnetic field
The Coulomb interaction action functional (8) is bilinear with respect to the electron density ψ ψ x x ( ) ( ). In order to linearize this expression we apply celebrated Hubbard-Stratonovich transformation [8,9], as was proposed in [4,5]: The bosonic real integration variable φ(x) in functional integrals (19) and (20) is a real scalar field which can be considered as the order parameter characterizing collective oscillations of electron gas and can be called collective oscillation field. Substituting expression (19) into the rhs of formula (15) and setting determined by formula (21). Expanding exponential functions into the power series, neglecting small terms proportional to 1/m 2 and performing functional integration with respect to Grassmann variables, we obtain (1,0) n(x) being the mean value of electron density ψ ψ Using formulae (23)-(32) we obtain the expression where [, ] with the following terms (2,0) where (1,1) where (2,1) where and so on. can be interpreted as total action functional of the system of two fields φ(x) and A(x). It is a functional power series of the form where the term is a homogeneous functional of mth order with respect to the field φ(x) and nth order with respect to the field A(x). Note that (0,0)

Effective action functional of interacting fluctuation field and electromagnetic field
Before the study of two interacting fields we review some related known results concerning scalar field φ(x) without its interaction with electromagnetic field and limit to the second order (harmonic) approximation for the simplicity of subsequent applications. In [3,4] it was shown that scalar field φ (x) is split into two parts where the physical meaning of functions − u x y ( ), − u x y ( ), n x ( ) and the definition of function Π − x y ( ) were presented in preceding sections. In terms of background field φ 0 (x) and fluctuation field ζ(x) the action functional I 0 [φ] of collective oscillation field φ(x) in the harmonic approximation has the expression φ ζ φ ζ eff Functional (58) can be considered as the effective action functional of fluctuation field ζ(x). The dynamical equation for this field is is a homogeneous functional power of lth, mth and nth orders with respect to the fields φ 0 (x), ζ(x) and A(x) as three functional variables, respectively. In particular in the harmonic approximation with respect to collective oscillation field φ(x), we obtain the following result: where Λ − − x z y z ( , )was defined in formula (48) and so on. Thus, the expression of total effective action functional of interacting system of fluctuation field ζ(x) and electromagnetic field A(x) in the harmonic approximation with respect to collective oscillation field φ(x) was established.

Effective action functional of interacting plasmonic and electromagnetic fields
In order to establish the expression of effective action functional of the interacting system of plasmonic and electromagnetic fields let us start from effective action functional ζ I A [ , ] tot derived in the preceding section and express ζ(x) in terms of the plasmonic field. As a simple example let us consider the case of a homogeneous and isotropic electron gas with a constant electron density n(x) = n at vanishing absolute temperature T = 0 K and denote p F the electron momentum at Fermi surface In [3][4][5] it was shown that in the first-order approximation with respect to the ratio ω k , ω K k ( , ) has the following expression Setting introducing the real scalar field σ σ and so on.

Conclusion and discussions
In the framework of functional integral approach we have established the quantum theory of plasmon and plasmonphoton interaction on the basis of general postulates (axioms), called also first principles, of electrodynamics and quantum theory of many-body systems. Since the plasmons are complex quasiparticles appearing as resonances in plasma oscillations of electron gas, the starting point of our study is the total functional integral of interacting system consisting of electron gas and electromagnetic field. The electron-electron Coulomb interaction was taken into account. The action functional of the above-mentioned interacting system contains a terms bilinear with respect to electron density ψ ψ x x ( ) ( ). By means of the Hubbard-Stratonovich transformation, this bilinear term was linearized and rewritten in a new form containing a real scalar field φ(x) playing the role of the order parameter of collective oscillations of electron gas. The total action functional of the new interacting system consisting of collective oscillation field φ(x) and electromagnetic field A(x) was derived.
Then it was demonstrated that in harmonic approximation the collective oscillation field φ(x) behaves like small fluctuations around its background φ 0 (x), The new real scalar field ζ(x) was called fluctuation field. Total action functional of the interacting system consisting of fluctuation field ζ(x) and electromagnetic field A(x) was established.
Subsequently we have demonstrated that in the case of homogeneous and isotropic electron gas there exists a real scalar field σ(x) satisfying the wave equation similar to Klein-Gordon equation in relativistic quantum field theory such that fluctuation field ζ(x) is a linear functional of σ(x). The quanta of the corresponding quantized Hermitian scalar field σ x ( ) are plasmons, and this scalar field was called quantum plasmonic field. By substituting the linear expression of quantized fluctuation field ζ x ( ) in terms of quantum plasmonic field σ x ( ) into the expression of total action functional ζ I Â [ˆ, ] tot of the interacting system of two fields ζ x ( ) and x Â ( ), we have obtained the expression of total action functional σ J Â [ˆ, ] tot of interacting system of plasmonic and electromagnetic fields. This total action functional determines matrix elements of plasmon-photon interaction processes.
The expression of total action functional σ J Â [ˆ, ] tot shows that plasmon-photon interaction processes are nonlocal. The physical origin of this nonlocality is the spatial extension of each plasmon around its center. Moreover, interaction processes involving plasmons are also not instantaneous: each matrix element contains plasmonic and other fields at different times.
Thus, we have demonstrated that interaction processes involving plasmons are neither local nor instantaneous. This remark suggests that the assumption on the local and instantaneous phenomenological interaction Hamiltonians of various plasmonic processes should be revised.
Finally, let us note that the electron-phonon plays a certain important role in plasmonic processes, but until now there was almost no theoretical research on related subjects. It is worth studying the plasmon-phonon interaction processes, and functional integral technique would be again the effective tool for this purpose.