Basics of quantum plasmonics

The resonance phenomenon in collective oscillations of electron gas was demonstrated by early works on plasma oscillations in solids [1–4]. It was caused by the elementary collective excitations behaving like quasiparticles of a special type called plasmons. In recent years there has been significant progress in the study of the interaction processes in which plasmons play the essential role, and a new very promising area of quantum physics called plasmonics has emerged and was rapidly developed. The variety of plasmonic processes and phenomena is quite broad: the formation of hybrid systems consisting of semiconductor quantum dots and metallic nanoparticles [5, 6], interaction between a metallic nanoparticle and a dipole emitter [7], exciton–plasmon coupling (plexciton) [8–10], exciton–plasmon resonance energy transfer [11, 12], plasmon-enhanced light absorption [13, 14] and fluorescence [15–20], plasmonic-molecular resonance [21–28]. The results of the research on plasmonic processes have led to the creation of plasmonic nanoantennae for various efficient applications [29].

In our previous works [30–32] attempts were made to elaborate the quantum theory of plasmon enabling one to exactly derive the effective action functional, or the effective Hamiltonians in the special cases, of the interactions of plasmons with other quasiparticles in the matter as well as with the electromagnetic field, starting from first principles of electrodynamics and quantum physics. The present article is a topical review of the above-mentioned theoretical works with the extension to include also the derivation of the effective action functional of the interacting photon–plasmon system.

In section 2, applying the canonical quantization procedure of quantum mechanics, we present the theory of canonical quantization of the plasmonic field [31]. The construction of the quantum plasmonic field by means of the functional integral technique is performed in section 3 [30, 32]. In section 4 the effective action functional of the interacting photon–plasmon system is derived by means of the canonical quantization method in quantum mechanics. The same subject as that of section 4 is studied in section 5 in the framework of the functional integral approach. The conclusion and discussions are presented in section 6.

Consider the system of itinerant electrons moving in the electrostatic field generated by the positive charge of the ions in some crystal. Denote n(x, t) the electron density (number of electrons per unit volume) and n₀ its mean value averaged over both space and time. The average charge density –en₀ of electrons, –e being the negative electron charge, compensates the average positive charge of ions in the crystal, and the fluctuating charge density in the crystal is determined by the expression

$$\begin{eqnarray}&&\rho ({\bf x},t)=-e[n({\bf x},t)-{{n}_{0}}].\end{eqnarray} \tag{ 1 }$$

According to the Coulomb law, the charge distribution with the density (1) in a special region V generates a time-dependent electrical field with the potential

$$\begin{eqnarray}&&\varphi ({\bf x},t)=\mathop \int \limits_{V} {\rm d}{\bf x}^{\prime} \frac{\rho ({\bf x}^{\prime} ,t)}{\left| {\bf x}-{\bf x}^{\prime} \right|}\cdot \end{eqnarray} \tag{ 2 }$$

From formula (2) it follows the Poisson equation

$$\begin{eqnarray}&&{{\nabla }^{2}}\varphi ({\bf x},t)=-4\pi \rho ({\bf x},t).\end{eqnarray} \tag{ 3 }$$

The mutual interaction between electrons of the electron gas in the region V gives rise to the potential energy of the electron gas

$$\begin{eqnarray}&&U(t)=\frac{1}{2}\mathop \int \limits_{V} {\rm d}{\bf x}\mathop \int \limits_{V} {\rm d}{\bf x}^{\prime} \rho ({\bf x},t)\frac{1}{\left| {\bf x}-{\bf x}^{\prime} \right|}\rho ({\bf x}^{\prime} ,t)\end{eqnarray} \tag{ 4 }$$

which can be also represented in the form

$$\begin{eqnarray}&&U(t)=\frac{1}{2}\mathop \int \limits_{V} {\rm d}{\bf x}\rho ({\bf x},t)\varphi ({\bf x},t).\end{eqnarray} \tag{ 5 }$$

On the other side, as the consequence of the oscillating displacements of electrons, the fluctuation of the electron density n(r, t) generates the total kinetic energy of the electron gas. Denote $\delta {\bf r}({\bf x},t)$ the displacement vector of the electron having the coordinate x at the time moment t, and m the electron mass. Since the electron has the velocity

$$\begin{eqnarray}&&\delta {\bf \dot{r}}({\bf x},t)=\frac{\partial \delta {\bf r}({\bf x},t)}{\partial t},\end{eqnarray} \tag{ 6 }$$

the whole electron gas has following total kinetic energy

$$\begin{eqnarray}&&T(t)=\frac{m}{2}\mathop \int \limits_{V} {\rm d}{\bf x}\;n({\bf x},t)\delta {\bf \dot{r}}{{({\bf x},t)}^{2}}.\end{eqnarray} \tag{ 7 }$$

Consider now the conservation of the total electron number. Denote dN_S the number of electrons going through the boundary S of a region V from the inside of V into its outside during the time interval from t to t + dt

$$\begin{eqnarray}&&{\rm d}{{N}_{S}}=\oint {\rm d}{\bf S}dt\;n({\bf x},t)\delta {\bf r}({\bf x},t).\end{eqnarray} \tag{ 8 }$$

Using the Ostrogradski–Gauss formula, we rewrite dN_S as follows

$$\begin{eqnarray}&&{\rm d}{{N}_{S}}=\mathop \int \limits_{V} {\rm d}{\bf x}\nabla [n({\bf x},t)\delta {\bf r}({\bf x},t)]{\rm d}t.\end{eqnarray} \tag{ 9 }$$

The decrease of the electron number inside V during that time interval equals

$$\begin{eqnarray}&&{\rm d}{{N}_{V}}=\mathop \int \limits_{V} {\rm d}{\bf x}\;\left[ {{n}_{0}}-n({\bf x},t) \right]{\rm d}t.\end{eqnarray} \tag{ 10 }$$

Because of the conservation of the total electron number, there must be the equality

$$\begin{eqnarray}&&{\rm d}{{N}_{S}}={\rm d}{{N}_{V}}.\end{eqnarray} \tag{ 11 }$$

From equations (1) and (9)–(11) it follows that

$$\begin{eqnarray}&&\rho ({\bf x},t)=e\nabla \left[ n({\bf x},t)\delta {\bf r}({\bf x},t) \right].\end{eqnarray} \tag{ 12 }$$

For the convenience in performing the canonical quantization procedure we decompose the functions $\rho ({\bf x},t),$ $\varphi ({\bf x},t)$and $\delta {\bf r}({\bf x},t)$ into the Fourier series, using a basis consisting of the plane waves satisfying the periodic boundary conditions at the opposite surfaces of a cube with the volume V and normalized in this cube. We have the following formulae

$$\begin{eqnarray}&&\rho ({\bf x},t)=\frac{1}{\sqrt{V}}\mathop \sum \limits_{{\bf k}} {{{\rm e}}^{{\rm i}{\bf kx}}}{{\rho }_{{\bf k}}}(t),\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&\varphi ({\bf x},t)=\frac{1}{\sqrt{V}}\mathop \sum \limits_{{\bf k}} {{{\rm e}}^{{\rm i}{\bf kx}}}{{\varphi }_{{\bf k}}}(t),\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\delta {\bf r}({\bf x},t)=\frac{1}{\sqrt{V}}\mathop \sum \limits_{{\bf k}} {{{\rm e}}^{{\rm i}{\bf kx}}}\sum \limits_{i=1}^{3} {\bf e}_{{\bf k}}^{(i)}q_{{\bf k}}^{(i)}(t),\end{eqnarray} \tag{ 15 }$$

where ${\bf e}_{{\bf k}}^{(i)}$ for each k are three unit vectors satisfying the conditions

$$\begin{eqnarray}&&{\bf ke}_{{\bf k}}^{(i)}=\left\{ \begin{array}{lllllllllllllll} 0,\quad i=1,\;2 \\ k,\quad i=3 \\ \end{array} \right.\end{eqnarray} \tag{ 16 }$$

$k=\left| {\bf k} \right|.$ Two terms with i = 1, 2 are the transverse displacements, while that with i = 3 is the longitudinal displacement along the direction of the wave vector k. From equation (3) and formulae (13) and (14) we obtain

$$\begin{eqnarray}&&{{\varphi }_{{\bf k}}}(t)=\frac{4\pi }{{{{\bf k}}^{2}}}{{\rho }_{{\bf k}}}(t).\end{eqnarray} \tag{ 17 }$$

Consider the approximation in which the fluctuating electron density n(x, t) in the expression of ρ(x, t) is replaced by its average value n₀

$$\begin{eqnarray}&&\rho ({\bf x},t)\approx e{{n}_{0}}\nabla \delta {\bf r}({\bf x},t).\end{eqnarray} \tag{ 18 }$$

In this case from the expressions (13) and (15) and equation (18) we obtain

$$\begin{eqnarray}&&{{\rho }_{{\bf k}}}(t)\approx {\rm i}e{{n}_{0}}kq_{{\bf k}}^{(3)}(t).\end{eqnarray} \tag{ 19 }$$

Therefore the potential energy (4) and the kinetic energy (7) are expressed in terms of the generalized coordinates of the system as follows

$$\begin{eqnarray}&&U=2\pi {{e}^{2}}n_{0}^{2}\mathop \sum \limits_{{\bf k}} q_{{\bf k}}^{(3)}(t)*q_{{\bf k}}^{(3)}(t),\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&T=\frac{1}{2}{{n}_{0}}m\mathop \sum \limits_{{\bf k}} \sum \limits_{i=1}^{3} \dot{q}_{{\bf k}}^{(i)}(t)*\dot{q}_{{\bf k}}^{(i)}(t).\end{eqnarray} \tag{ 21 }$$

Lagrangian of the electron gas as a classical mechanical system has the expression

$$\begin{eqnarray}\begin{array}{lllllllllllllll} L=T-U & = & \frac{1}{2}\mathop \sum \limits_{{\bf k}} \left\{ {{n}_{0}}m\sum \limits_{i=1}^{3} \dot{q}_{{\bf k}}^{(i)}(t)*\dot{q}_{{\bf k}}^{(i)}(t) \right. \\ {} & {} & -\left. 4\pi {{e}^{2}}n_{0}^{2}q_{{\bf k}}^{(3)}(t)*q_{{\bf k}}^{(3)}(t) \right\}. \\ \end{array}\end{eqnarray} \tag{ 22 }$$

The equations of motion are

$$\begin{eqnarray}&&\frac{{{{\rm d}}^{2}}q_{{\bf k}}^{(i)}(t)}{{\rm d}{{t}^{2}}}+4\pi {{e}^{2}}\frac{{{n}_{0}}}{m}{{\delta }_{i3}}q_{{\bf k}}^{(i)}(t)=0.\end{eqnarray} \tag{ 23 }$$

It follows that the transverse coordinates $q_{{\bf k}}^{(1)}(t)$ and $q_{{\bf k}}^{(2)}(t)$ do not oscillate, while the longitudinal one $q_{{\bf k}}^{(3)}(t)$ periodically oscillates with the angular frequency

$$\begin{eqnarray}&&{{\omega }_{0}}=\sqrt{\frac{4\pi {{e}^{2}}{{n}_{0}}}{m}},\end{eqnarray} \tag{ 24 }$$

which is the know plasma frequency of a free electron gas. Since the transverse coordinates do not oscillate, we discard them and replace the notation $q_{{\bf k}}^{(3)}(t)$ by the simpler one ${{q}_{{\bf k}}}(t).$ Lagrangian of the system becomes

$$\begin{eqnarray}\begin{array}{lllllllllllllll} L & = & \frac{1}{2}\mathop \sum \limits_{{\bf k}} \left\{ {{n}_{0}}m{{{\dot{q}}}_{{\bf k}}}(t)*{{{\dot{q}}}_{{\bf k}}}(t) \right. \\ {} & {} & -\left. 4\pi {{e}^{2}}n_{0}^{2}{{q}_{{\bf k}}}(t)*{{q}_{{\bf k}}}(t) \right\}. \\ \end{array}\end{eqnarray} \tag{ 25 }$$

Now we demonstrate that the quantum mechanical system with the Lagrangian (25) can be considered as a classical field $\sigma ({\bf x},t).$ Consider the Fourier transformation of this classical field and its time derivative

$$\begin{eqnarray}&&\sigma ({\bf x},t)=\frac{1}{\sqrt{V}}\mathop \sum \limits_{{\bf k}} {{{\rm e}}^{{\rm i}{\bf kx}}}{{\tilde{\sigma }}_{{\bf k}}}(t),\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&\dot{\sigma }({\bf x},t)=\frac{1}{\sqrt{V}}\mathop \sum \limits_{{\bf k}} {{{\rm e}}^{{\rm i}{\bf kx}}}{{\mathop{\tilde{\sigma }}\limits^{.}}_{{\bf k}}}(t).\end{eqnarray} \tag{ 27 }$$

We have

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \mathop \int \limits_{V} {\rm d}{\bf x}\sigma {{({\bf x},t)}^{2}}=\mathop \sum \limits_{{\bf k}} {{\tilde{\sigma }}_{{\bf k}}}(t)*{{\tilde{\sigma }}_{{\bf k}}}(t), \\ \mathop \int \limits_{V} {\rm d}{\bf x}\dot{\sigma }{{({\bf x},t)}^{2}}=\mathop \sum \limits_{{\bf k}} {{\mathop{\tilde{\sigma }}\limits^{.}}_{{\bf k}}}(t)*{{\mathop{\tilde{\sigma }}\limits^{.}}_{{\bf k}}}(t). \\ \end{array}\end{eqnarray} \tag{ 28 }$$

Setting

$$\begin{eqnarray}&&\sqrt{{{n}_{0}}m}\;{{q}_{{\bf k}}}(t)={{\tilde{\sigma }}_{{\bf k}}}(t),\end{eqnarray} \tag{ 29 }$$

we can represent Lagrangian (25) as that of a real scalar field $\sigma ({\bf x},t)$

$$\begin{eqnarray}&&L(t)=\frac{1}{2}\int {\rm d}{\bf x}\left\{ {{\left[ \frac{\partial \sigma ({\bf x},t)}{\partial t} \right]}^{2}}-\omega _{0}^{2}\sigma {{({\bf x},t)}^{2}} \right\}\cdot \end{eqnarray} \tag{ 30 }$$

This expression is similar to the Lagrangian of a Klein–Gordon real scalar field in the relativistic field theory except for the absence of the term containing the spatial derivative $\nabla \sigma ({\bf x},t)$ of the scalar field $\sigma ({\bf x},t)$ [33–36]. The action functional of this field is

$$\begin{eqnarray}&&I[\sigma ]=\int \limits_{-\infty }^{\infty } {\rm d}t\;L(t).\end{eqnarray} \tag{ 31 }$$

Now we apply the canonical quantization procedure to the real scalar field with the Lagrangian (30). Denote $\delta L(t)$ and $\delta I(\sigma )$ the variations of L(t) and I[σ] when the scalar field $\sigma ({\bf x},t)$ and is time derivative $\dot{\sigma }({\bf x},t)$ are subjected by the infinitely small variations $\delta \sigma ({\bf x},t)$ and $\delta \dot{\sigma }({\bf x},\;t)$,

$$\begin{eqnarray}&&\delta \dot{\sigma }({\bf x},t)=\frac{\partial }{\partial t}\delta \sigma ({\bf x},t).\end{eqnarray} \tag{ 32 }$$

We have

$$\begin{eqnarray}\begin{array}{lllllllllllllll} \delta L(t) & = & \mathop \int \limits_{V} {\rm d}{\bf x}\left[ \frac{\delta L(t)}{\delta \sigma ({\bf x},t)}\delta \sigma ({\bf x},t) \right. \\ {} & {} & +\left. \frac{\delta L(t)}{\delta \dot{\sigma }({\bf x},t)}\frac{\partial }{\partial t}\delta \sigma ({\bf x},t) \right]=\frac{\partial }{\partial t} \\ {} & {} & \times \mathop \int \limits_{V} {\rm d}{\bf x}\frac{\delta L(t)}{\delta \dot{\sigma }({\bf x},t)}\delta \sigma ({\bf x},t)+\mathop \int \limits_{V} {\rm d}{\bf x}\left[ \frac{\delta L(t)}{\delta \sigma ({\bf x},t)} \right. \\ {} & {} & -\left. \frac{\partial }{\partial t}\left( \frac{\delta L(t)}{\delta \dot{\sigma }({\bf x},t)} \right) \right]\delta \sigma ({\bf x},t), \\ \end{array}\end{eqnarray} \tag{ 33 }$$

and therefore

$$\begin{eqnarray}\begin{array}{lllllllllllllll} \delta I[\sigma ] & = & \int \limits_{-\infty }^{\infty } {\rm d}t\delta L(t) \\ {} & = & \mathop \int \limits_{V} {\rm d}{\bf x}\left. \frac{\delta L(t)}{\delta \dot{\sigma }({\bf x},t)}\delta \sigma ({\bf x},t) \right|_{t=-\infty }^{t=+\infty } \\ {} & {} & +\int \limits_{-\infty }^{\infty } {\rm d}t\mathop \int \limits_{V} {\rm d}{\bf x}\left[ \frac{\delta L(t)}{\delta \sigma ({\bf x},t)} \right. \\ {} & {} & -\left. \frac{\partial }{\partial t}\left( \frac{\delta L(t)}{\delta \dot{\sigma }({\bf x},t)} \right) \right]\delta \sigma ({\bf x},t). \\ \end{array}\end{eqnarray} \tag{ 34 }$$

Because of the boundary condition

$$\begin{eqnarray}&&\delta \sigma ({\bf x},t)\to 0\;\;{\rm at}\;\;\;\;t\to \pm \infty ,\end{eqnarray} \tag{ 35 }$$

the first term in the right-hand side of equation (34) vanishes, and we obtain

$$\begin{eqnarray}\begin{array}{lllllllllllllll} \delta I[\sigma ] & = & \int \limits_{-\infty }^{\infty } {\rm d}t\mathop \int \limits_{V} {\rm d}{\bf x}\left[ \frac{\delta L(t)}{\delta \sigma ({\bf x},t)}-\frac{\partial }{\partial t} \right. \\ {} & {} & \left. \times \left( \frac{\delta L(t)}{\delta \dot{\sigma }({\bf x},t)} \right) \right]\delta \sigma ({\bf x},t). \\ \end{array}\end{eqnarray} \tag{ 36 }$$

On the other hand, by definition

$$\begin{eqnarray}&&\delta I[\sigma ]=\int \limits_{-\infty }^{\infty } {\rm d}t\mathop \int \limits_{V} {\rm d}{\bf x}\frac{\delta I(\sigma )}{\delta \sigma ({\bf x},t)}\delta \sigma ({\bf x},t).\end{eqnarray} \tag{ 37 }$$

The comparison of formulae (36) and (37) gives

$$\begin{eqnarray}&&\frac{\delta I[\sigma ]}{\delta \sigma ({\bf x},t)}=\frac{\delta L(t)}{\delta \sigma ({\bf x},t)}-\frac{\partial }{\partial t}\left( \frac{\delta L(t)}{\delta \dot{\sigma }({\bf x},t)} \right)\cdot \end{eqnarray} \tag{ 38 }$$

From the extreme action principle

$$\begin{eqnarray}&&\frac{\delta I[\sigma ]}{\delta \sigma ({\bf x},t)}=0\end{eqnarray} \tag{ 39 }$$

we obtain Lagrange functional equation

$$\begin{eqnarray}&&\frac{\delta L(t)}{\delta \sigma ({\bf x},t)}-\frac{\partial }{\partial t}\left( \frac{\delta L(t)}{\delta \dot{\sigma }({\bf x},t)} \right)=0\cdot \end{eqnarray} \tag{ 40 }$$

Considering $\sigma ({\bf x},t)$ as the canonical coordinate of the field, we have following canonical momentum

$$\begin{eqnarray}&&\pi ({\bf x},t)=\frac{\delta L(t)}{\delta \dot{\sigma }({\bf x},t)}=\dot{\sigma }({\bf x},t)\end{eqnarray} \tag{ 41 }$$

and Hamiltonian functional

$$\begin{eqnarray}\begin{array}{lllllllllllllll} H & = & \mathop \int \limits_{V} {\rm d}{\bf x}\dot{\sigma }({\bf x},t)\pi ({\bf x},t)-L \\ {} & = & \frac{1}{2}\mathop \int \limits_{V} {\rm d}{\bf x}\left\{ {{\left[ \frac{\partial \sigma ({\bf x},t)}{\partial t} \right]}^{2}}+\omega _{0}^{2}\sigma {{({\bf x},t)}^{2}} \right\}. \\ \end{array}\end{eqnarray} \tag{ 42 }$$

After the canonical quantization procedure, the canonical coordinate $\sigma ({\bf x},t)$ and momentum $\pi ({\bf x},t)$ become the operators $\hat{\sigma }({\bf x},t)$ and $\hat{\pi }({\bf x},t),$ and we have following Hamiltonian operator of the quantized field

$$\begin{eqnarray}&&\hat{H}=\frac{1}{2}\mathop \int \limits_{V} {\rm d}{\bf x}\left\{ {{\left[ \frac{\partial \hat{\sigma }({\bf x},t)}{\partial t} \right]}^{2}}+\omega _{0}^{2}\hat{\sigma }{{({\bf x},t)}^{2}} \right\}.\end{eqnarray} \tag{ 43 }$$

Now we expand the canonical field operator $\hat{\sigma }({\bf x},t)$ into the Fourier series of the orthogonal and normalized plane waves

$$\begin{eqnarray}\begin{array}{lllllllllllllll} \hat{\sigma }({\bf x},t) & = & \frac{1}{\sqrt{V}}\frac{1}{\sqrt{2{{\omega }_{0}}}}\mathop \sum \limits_{{\bf k}} \left[ {{{\rm e}}^{{\rm i}({\bf kx}-{{\omega }_{0}}t)}}{{{\hat{a}}}_{{\bf k}}} \right. \\ {} & {} & \left. +{{{\rm e}}^{-{\rm i}({\bf kx}-{{\omega }_{0}}t)}}\hat{a}_{{\bf k}}^{+} \right]\cdot \\ \end{array}\end{eqnarray} \tag{ 44 }$$

Substituting expression (44) of $\hat{\sigma }({\bf x},t)$ into the right hand side of formula (43), after lengthy standard calculations we obtain

$$\begin{eqnarray}&&\hat{H}=\frac{1}{2}\mathop \sum \limits_{{\bf k}} {{\omega }_{0}}\left( \hat{a}_{{\bf k}}^{+}{{{\hat{a}}}_{{\bf k}}}+{{{\hat{a}}}_{{\bf k}}}\hat{a}_{{\bf k}}^{+} \right)\ \;\cdot \end{eqnarray} \tag{ 45 }$$

According to the equation (41), the canonical momentum $\hat{\pi }({\bf x},t)$ equals the time derivative of the canonical field $\hat{\sigma }({\bf x},t).$ Therefore $\hat{\pi }({\bf x},t)$ has following expansion

$$\begin{eqnarray}\begin{array}{lllllllllllllll} \hat{\pi }({\bf x},t) & = & -\frac{{\rm i}}{\sqrt{V}}\sqrt{\frac{{{\omega }_{0}}}{2}}\mathop \sum \limits_{{\bf k}} \left[ {{{\rm e}}^{{\rm i}({\bf kx}-{{\omega }_{0}}t)}}{{{\hat{a}}}_{{\bf k}}} \right. \\ {} & {} & -\left. {{{\rm e}}^{-{\rm i}({\bf kx}-{{\omega }_{0}}t)}}\hat{a}_{{\bf k}}^{+} \right]\cdot \\ \end{array}\end{eqnarray} \tag{ 46 }$$

Inverting the expansions (44) and (46), we obtain the expressions of ${{\hat{a}}_{{\bf k}}}$ and $\hat{a}_{{\bf k}}^{+}$ in terms of the canonical coordinate $\hat{\sigma }({\bf x},0)$ and momentum $\hat{\pi }({\bf x},0)$:

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{{\hat{a}}}_{{\bf k}}} & = & \frac{1}{\sqrt{V}}\mathop \int \limits_{V} {\rm d}{\bf x}{{{\rm e}}^{-{\rm i}{\bf kx}}}\left\{ \sqrt{\frac{{{\omega }_{0}}}{2}}\hat{\sigma }({\bf x},0) \right. \\ {} & {} & +\left. \frac{{\rm i}}{\sqrt{2{{\omega }_{0}}}}\hat{\pi }({\bf x},0) \right\} \\ \end{array}\end{eqnarray} \tag{ 47 }$$

and

$$\begin{eqnarray}\begin{array}{lllllllllllllll} \hat{a}_{{\bf k}}^{+} & = & \frac{1}{\sqrt{V}}\mathop \int \limits_{V} {\rm d}{\bf x}{{{\rm e}}^{-{\rm i}{\bf kx}}}\left\{ \sqrt{\frac{{{\omega }_{0}}}{2}}\hat{\sigma }({\bf x},0) \right. \\ {} & {} & -\left. \frac{{\rm i}}{\sqrt{2{{\omega }_{0}}}}\hat{\pi }({\bf x},0) \right\}. \\ \end{array}\end{eqnarray} \tag{ 48 }$$

According to the canonical quantization rules, between the operators $\hat{\sigma }({\bf x},t)$ and $\hat{\pi }({\bf x},t)$ there exist the following equal-time canonical commutation relations

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \left[ \hat{\sigma }({\bf x},t),\hat{\sigma }({\bf x}^{\prime} ,t) \right]=\left[ \hat{\pi }({\bf x},t),\hat{\pi }({\bf x}^{\prime} ,t) \right]=0, \\ \left[ \hat{\pi }({\bf x},t),\hat{\sigma }({\bf x}^{\prime} ,t) \right]=-{\rm i}\delta (x-{\bf x}^{\prime} ). \\ \end{array}\end{eqnarray} \tag{ 49 }$$

Using two expressions (47) and (48) and commutation relations (49), we derive the following canonical commutation relations

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} [{{{\hat{a}}}_{{\bf k}}},{{{\hat{a}}}_{{\bf l}}}]=[\hat{a}_{{\bf k}}^{+},\hat{a}_{{\bf l}}^{+}]=0, \\ [{{{\hat{a}}}_{{\bf k}}},\hat{a}_{{\bf l}}^{+}]={{\delta }_{{\bf kl}}}. \\ \end{array}\end{eqnarray} \tag{ 50 }$$

These relations together with formula (45) for Hamiltonian $\hat{H}$ show that ${{\hat{a}}_{{\bf k}}}$ and $\hat{a}_{{\bf k}}^{+}$ are the destruction and creation operators of the quasiparticles with the energy ω₀—the plasmons. Thus, we have constructed the quantum field $\hat{\sigma }({\bf x},t)$ whose quanta are the plasmons—the quantum plasmonic field. In the harmonic approximation the plasmons are dispersionless. In order to establish the dispersion law of the plasmons by means of the canonical quantization procedure it is necessary to go beyond the harmonic approximation.

3.1. Basic notions in functional integral method

Consider a physical structure in three-dimensional space, for any vector x in this space we introduce the vector

$$\begin{eqnarray*}&&x=({\bf x},{{x}_{0}})=({\bf x},t)\end{eqnarray*}$$

in the corresponding four-dimensional space–time and denote

$$\begin{eqnarray*}&&\int {\rm d}x=\int {\rm d}{\bf x}\int {\rm d}t.\end{eqnarray*}$$

The integration domain in the three-dimensional space is the whole volume Ω of the structure, while the integral over t is calculated from $-\infty$ to $+\infty$.

Let ${{\varphi }_{x}}=\varphi (x)=\varphi ({\bf x},t)$ be a real scalar function of x. Considering each value ${{\varphi }_{x}}$ as a component of a vector with the continuous vector index x, we extend the notion of multiple integral and introduce the notion of functional integral as follows

$$\begin{eqnarray}&&\int [D\varphi ]\ldots =\int \ldots \mathop{\Pi }\limits_{x}d{{\varphi }_{x}}.\end{eqnarray} \tag{ 51 }$$

Similarly, let ${{\psi }_{x}}=\psi (x)=\psi ({\bf x},t)$ and ${{\bar{\psi }}_{x}}=\bar{\psi }(x)=\bar{\psi }({\bf x},t)$ be two other hermitian conjugate spinor functions. Considering ${{\psi }_{x}}$ and ${{\bar{\psi }}_{x}}$ as the spinor components of two vectors with the continuous vector index x, we introduce the notion of functional integral over $\psi (x)$ and $\bar{\psi }(x)$ as the extension of the definition of the multiple integral to the case of two uncountable sets of integration variables ${{\psi }_{x}}$ and ${{\bar{\psi }}_{x}}$ with the continuous index x

$$\begin{eqnarray}&&\int [D\psi ]\int [D\bar{\psi }]\ldots =\int \ldots \mathop{\Pi }\limits_{x}d{{\psi }_{x}}d{{\bar{\psi }}_{x}}.\end{eqnarray} \tag{ 52 }$$

The functional integral method was a powerful method for the study of relativistic quantum field theory [33–36].

The subject of our study is the electron gas inside the volume Ω of a physical structure. Because the electrons are confined inside this volume, we impose on the functional integration variables the following boundary condition: they must vanish outside volume Ω as well as on its surface.

The Bosonic functional integration variables $\varphi (x)$ are commuting

$$\begin{eqnarray}&&\varphi (x)\varphi (y)=\varphi (y)\varphi (x),\end{eqnarray} \tag{ 53 }$$

while the fermionic ones $\psi (x)$ and $\bar{\psi }(x)$ anticommute each other

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \psi (x)\psi (y)+\psi (y)\psi (x)=0, \\ \psi (x)\bar{\psi }(y)+\bar{\psi }(y)\psi (x)=0, \\ \bar{\psi }(x)\bar{\psi }(y)+\bar{\psi }(y)\bar{\psi }(x)=0. \\ \end{array}\end{eqnarray} \tag{ 54 }$$

Functional integration variables $\psi (x)$ and $\bar{\psi }(x)$ are called the Grassmann variables. Beside of these dynamical variables one often uses also Grassmann parameter $\eta (x)$ and its hermitian conjugate $\bar{\eta }(x).$ They anticommute each other

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \eta (x)\eta (y)+\eta (y)\eta (x)=0, \\ \eta (x)\bar{\eta }(y)+\bar{\eta }(y)\eta (x)=0, \\ \bar{\eta }(x)\bar{\eta }(y)+\bar{\eta }(y)\bar{\eta }(x)=0, \\ \end{array}\end{eqnarray} \tag{ 55 }$$

and also anticommute with the dynamical Grassmann variables

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \eta (x)\psi (y)+\psi (y)\eta (x)=0, \\ \eta (x)\bar{\psi }(y)+\bar{\psi }(y)\eta (x)=0, \\ \bar{\eta }(x)\psi (y)+\psi (y)\bar{\eta }(x)=0, \\ \bar{\eta }(x)\bar{\psi }(y)+\bar{\psi }(y)\bar{\eta }(x)=0. \\ \end{array}\end{eqnarray} \tag{ 56 }$$

Bosonic functional integration variable commutes with all Grassmann dynamical variables and Grassmann parameters

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \varphi (x)\psi (y)+\psi (y)\varphi (x)=0, \\ \varphi (x)\bar{\psi }(y)+\bar{\psi }(y)\varphi (x)=0, \\ \varphi (x)\eta (y)=\eta (y)\varphi (x), \\ \varphi (x)\bar{\eta }(y)=\bar{\eta }(y)\varphi (x). \\ \end{array}\end{eqnarray} \tag{ 57 }$$

In the calculations we often use the functional derivation operators $\frac{\delta }{\delta \eta (x)}$ and $\frac{\delta }{\delta \bar{\eta }(x)}.$ They anticommute each other

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \frac{\delta }{\delta \eta (x)}\frac{\delta }{\delta \eta (y)}+\frac{\delta }{\delta \eta (y)}\frac{\delta }{\delta \eta (x)}=0, \\ \frac{\delta }{\delta \eta (x)}\frac{\delta }{\delta \bar{\eta }(y)}+\frac{\delta }{\delta \bar{\eta }(y)}\frac{\delta }{\delta \eta (x)}=0, \\ \frac{\delta }{\delta \bar{\eta }(x)}\frac{\delta }{\delta \bar{\eta }(y)}+\frac{\delta }{\delta \bar{\eta }(y)}\frac{\delta }{\delta \bar{\eta }(x)}=0, \\ \end{array}\end{eqnarray} \tag{ 58 }$$

anticommute with fermionic integration variables

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \frac{\delta }{\delta \eta (x)}\psi (y)+\psi (y)\frac{\delta }{\delta \eta (x)}=0, \\ \frac{\delta }{\delta \eta (x)}\bar{\psi }(y)+\bar{\psi }(y)\frac{\delta }{\delta \eta (x)}=0, \\ \frac{\delta }{\delta \bar{\eta }(x)}\psi (y)+\psi (y)\frac{\delta }{\delta \bar{\eta }(x)}=0, \\ \frac{\delta }{\delta \bar{\eta }(x)}\bar{\psi }(y)+\bar{\psi }(y)\frac{\delta }{\delta \bar{\eta }(x)}=0, \\ \end{array}\end{eqnarray} \tag{ 59 }$$

but commute with the Bosonic integration variable

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \frac{\delta }{\delta \eta (x)}\varphi (y)=\varphi (y)\frac{\delta }{\delta \eta (x)}, \\ \frac{\delta }{\delta \bar{\eta }(x)}\varphi (y)=\varphi (y)\frac{\delta }{\delta \bar{\eta }(x)}\cdot \\ \end{array}\end{eqnarray} \tag{ 60 }$$

From the above-mentioned anticommutativity property of fermionic integration variables $\psi (x)$ and $\bar{\psi }(x),$ Grassmann parameters $\eta (x)$ and $\bar{\eta }(x)$ and functional derivation operators $\frac{\delta }{\delta \eta (x)}$ and $\frac{\delta }{\delta \bar{\eta }(x)},$ it is straightforward to derive the following formulae which are often used:

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \frac{\delta }{\delta \bar{\eta }(y)}{\rm exp} \left\{ {\rm i}\int {\rm d}x\bar{\eta }(x)\psi (x) \right\} \\ \quad ={\rm i}\psi (y){\rm exp} \left\{ {\rm i}\int {\rm d}x\bar{\eta }(x)\psi (x) \right\}, \\ \frac{\delta }{\delta \eta (z)}{\rm exp} \left\{ {\rm i}\int {\rm d}x\bar{\psi }(x)\eta (x) \right\} \\ \quad =-{\rm i}\bar{\psi }(z){\rm exp} \left\{ {\rm i}\int {\rm d}x\bar{\psi }(x)\eta (x) \right\}, \\ \frac{{{\delta }^{2}}}{\delta \eta (z)\delta \bar{\eta }(y)}{\rm exp} \left\{ {\rm i}\int {\rm d}x\left[ \bar{\psi }(x)\eta (x)+\bar{\eta }(x)\psi (x) \right] \right\} \\ \quad =\bar{\psi }(z)\psi (y){\rm exp} \left\{ {\rm i}\int {\rm d}x\left[ \bar{\psi }(x)\eta (x)+\bar{\eta }(x)\psi (x) \right] \right\}, \\ \frac{{{\delta }^{2}}}{\delta \bar{\eta }(z)\delta \bar{\eta }(y)}{\rm exp} \left\{ {\rm i}\int {\rm d}x\bar{\eta }(x)\psi (x) \right\} \\ \quad =-\psi (z)\psi (y){\rm exp} \left\{ {\rm i}\int {\rm d}x\bar{\eta }(x)\psi (x) \right\}, \\ \frac{{{\delta }^{2}}}{\delta \eta (z)\delta \eta (y)}{\rm exp} \left\{ {\rm i}\int {\rm d}x\bar{\psi }(x)\eta (x) \right\} \\ \quad =-\bar{\psi }(z)\bar{\psi }(y){\rm exp} \left\{ {\rm i}\int {\rm d}x\bar{\psi }(x)\eta (x) \right\}. \\ \end{array}\end{eqnarray} \tag{ 61 }$$

Denote $H\left( -{\rm i}\frac{\partial }{\partial {\bf x}},{\bf x} \right)$ the Hamiltonian of the electron with mass m moving in a confining potential U(x)

$$\begin{eqnarray}&&H\left( -{\rm i}\frac{\partial }{\partial {\bf x}},{\bf x} \right)=\frac{1}{2m}{{\left( -{\rm i}\frac{\partial }{\partial {\bf x}} \right)}^{2}}+U({\bf x}),\end{eqnarray} \tag{ 62 }$$

and

$$\begin{eqnarray}&&u(x-y)=u({\bf x}-{\bf y})\delta ({{x}_{0}}-{{y}_{0}}),\end{eqnarray} \tag{ 63 }$$

where u(x − y) is the potential energy of the Coulomb repulsion between two electrons

$$\begin{eqnarray}&&u({\bf x}-{\bf y})=\frac{{{e}^{2}}}{\left| {\bf x}-{\bf y} \right|},\end{eqnarray} \tag{ 64 }$$

e is the electron charge. The key mathematical tools of the functional integral method in quantum plasmonics are the functional integrals

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{Z}^{\psi }}\left[ \eta ,\bar{\eta } \right]=\int [D\psi ]\left[ D\bar{\psi } \right] \\ \quad \times {\rm exp} \left\{ {\rm i}\int {\rm d}x\left[ \bar{\psi }(x)\eta (x)+\bar{\eta }(x)\psi (x) \right] \right\} \\ \quad \times {\rm exp} \left\{ {\rm i}\int {\rm d}x\bar{\psi }(x)\left[ {\rm i}\frac{\partial }{\partial {{x}_{0}}}-H\left( -{\rm i}\frac{\partial }{\partial {\bf x}},{\bf x} \right) \right]\psi (x) \right\} \\ \quad \times {\rm exp} \left\{ -\frac{{\rm i}}{2}\int {\rm d}x\int {\rm d}y\bar{\psi }(x)\psi (x)u(x-y)\bar{\psi }(y)\psi (y) \right\} \\ \end{array}\end{eqnarray} \tag{ 65 }$$

and

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{Z}^{\psi }}={{Z}^{\psi }}\left[ 0,0 \right]=\int [D\psi ]\left[ D\bar{\psi } \right] \\ \quad \times {\rm exp} \left\{ {\rm i}\int {\rm d}x\bar{\psi }(x)\left[ {\rm i}\frac{\partial }{\partial {{x}_{0}}}-H\left( -{\rm i}\frac{\partial }{\partial {\bf x}},{\bf x} \right) \right]\psi (x) \right\} \\ \quad \times {\rm exp} \left\{ -\frac{{\rm i}}{2}\int {\rm d}x \right.\int {\rm d}y\bar{\psi }(x)\psi (x)u(x-y) \\ \quad \times \left. \bar{\psi }(y)\psi (y) \right\}. \\ \end{array}\end{eqnarray} \tag{ 66 }$$

When the electron–electron Coulomb interaction is neglected, instead of the functional integrals (65) and (66) we have the following ones

$$\begin{eqnarray}\begin{array}{lllllllllllllll} Z_{0}^{\psi }\left[ \eta ,\bar{\eta } \right] & = & \int \left[ D\psi \right]\left[ D\bar{\psi } \right]{\rm exp} \left\{ {\rm i}\int {\rm d}x\left[ \bar{\psi }(x)\eta (x) \right. \right. \\ {} & {} & +\left. \left. \bar{\eta }(x)\psi (x) \right] \right\}{\rm exp} \left\{ {\rm i}\int {\rm d}x\bar{\psi }(x)\left[ {\rm i}\frac{\partial }{\partial {{x}_{0}}} \right. \right. \\ {} & {} & -\left. \left. H\left( -{\rm i}\frac{\partial }{\partial {\bf x}},{\bf x} \right) \right]\psi (x) \right\} \\ \end{array}\end{eqnarray} \tag{ 67 }$$

and

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} Z_{0}^{\psi }=Z_{0}^{\psi }\left[ 0,0 \right]=\int [D\psi ]\left[ D\bar{\psi } \right]{\rm exp} \\ \quad \times \left\{ {\rm i}\int {\rm d}x\bar{\psi }(x)\left[ {\rm i}\frac{\partial }{\partial {{x}_{0}}}-H\left( -{\rm i}\frac{\partial }{\partial {\bf x}},{\bf x} \right) \right]\psi (x) \right\}\cdot \\ \end{array}\end{eqnarray} \tag{ 68 }$$

Expressions in formulae (65) and (67) are called the generating functionals.

The Grassmann dynamical variables $\psi (x)$ and $\bar{\psi }(x)$can be considered as a fermionic field and its hermitian conjugate. In the case of the electron gas with the electron–electron Coulomb repulsion, the action functional of the field $\psi (x)$ is

$$\begin{eqnarray}\begin{array}{lllllllllllllll} I\left[ \psi ,\bar{\psi } \right] & = & \int {\rm d}x\bar{\psi }(x)\left[ {\rm i}\frac{\partial }{\partial {{x}_{0}}}-H\left( -{\rm i}\frac{\partial }{\partial {\bf x}},{\bf x} \right) \right]\psi (x) \\ {} & {} & -\frac{1}{2}\int {\rm d}x\int {\rm d}y\bar{\psi }(x)\psi (x)(x-y)\bar{\psi }(y)\psi (y)\cdot \\ \end{array}\end{eqnarray} \tag{ 69 }$$

If the electron–electron Coulomb repulsion is neglected, the action functional of the system becomes

$$\begin{eqnarray}&&{{I}_{0}}\left[ \psi ,\bar{\psi } \right]=\int {\rm d}x\bar{\psi }(x)\left[ {\rm i}\frac{\partial }{\partial {{x}_{0}}}-H\left( -{\rm i}\frac{\partial }{\partial {\bf x}},{\bf x} \right) \right]\psi (x)\cdot \end{eqnarray} \tag{ 70 }$$

Thus, the functional integrals ${{Z}^{\psi }}$ and $Z_{0}^{\psi }$ of the electron gas are expressed in terms of its action functional $I\left[ \psi ,\bar{\psi } \right]$ and ${{I}_{0}}\left[ \psi ,\bar{\psi } \right]$ as follows

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{Z}^{\psi }}=\int \left[ D\psi \right]\ \;\left[ D\bar{\psi } \right]{\rm exp} \left\{ {\rm i}I\left[ \bar{\psi },\psi \right] \right\}, \\ Z_{0}^{\psi }=\int \left[ D\psi \right]\ \;\left[ D\bar{\psi } \right]{\rm exp} \left\{ {\rm i}{{I}_{0}}\left[ \bar{\psi },\psi \right] \right\}\cdot \\ \end{array}\end{eqnarray} \tag{ 71 }$$

In general, for any physical system described by a set of the fields ${{\varphi }_{\nu }}(x),\nu =1,\;2,\ldots ,$ between the action functional $I\left[ {{\varphi }_{\nu }} \right]$ of this system and its functional integral ${{Z}^{\{{{\varphi }_{\nu }}\}}},$ the relation of the form

$$\begin{eqnarray}&&{{Z}^{\{{{\varphi }_{\nu }}\}}}=\int \mathop{\Pi }\limits_{\nu }\left[ D{{\varphi }_{\nu }} \right]{\rm exp} \left\{ {\rm i}I\left[ {{\varphi }_{\nu }} \right] \right\}\end{eqnarray} \tag{ 72 }$$

always holds.

3.2. Green functions

Now we define the Green functions and derive the relations between them. Since these relations represent the dynamics of the corresponding quantum field, the functional integral method can be considered as a new method of the field quantization.

Considering Grassmann dynamical variables $\psi (x)$ and $\bar{\psi }(x)$ as a fermionic field and its hermitian conjugate, we define the average (also called expectation value) of any product $\psi ({{x}_{1}})\ldots \psi ({{x}_{n}})\bar{\psi }({{y}_{1}})\ldots \bar{\psi }({{y}_{n}})$ over all configurations of this field as follows:

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \left\langle \psi \left( {{x}_{1}} \right)\cdots \psi \left( {{x}_{n}} \right)\bar{\psi }\left( {{y}_{1}} \right)\cdots \bar{\psi }\left( {{y}_{n}} \right) \right\rangle \\ =\frac{1}{{{Z}^{\psi }}}\int [D\psi ]\left[ D\bar{\psi } \right]\psi \left( {{x}_{1}} \right)\cdots \psi \left( {{x}_{n}} \right)\bar{\psi }\left( {{y}_{1}} \right)\cdots \bar{\psi }\left( {{y}_{n}} \right) \\ \quad \times {\rm exp} \left\{ {\rm i}\int {\rm d}x\bar{\psi }(x)\left[ {\rm i}\frac{\partial }{\partial {{x}_{0}}}-H\left( -{\rm i}\frac{\partial }{\partial {\bf x}},{\bf x} \right) \right]\psi (x) \right\} \\ \quad \times {\rm exp} \left\{ -\frac{{\rm i}}{2}\int {\rm d}x\int {\rm d}y\bar{\psi }(x)\psi (x)u(x-y)\bar{\psi }(y)\psi (y) \right\} \\ \end{array}\end{eqnarray} \tag{ 73 }$$

when the electron–electron Coulomb interaction is taken into account, and

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{\left\langle \psi \left( {{x}_{1}} \right)\ldots \psi \left( {{x}_{n}} \right)\bar{\psi }\left( {{y}_{1}} \right)\ldots \bar{\psi }\left( {{y}_{n}} \right) \right\rangle }_{0}} \\ =\frac{1}{Z_{0}^{\psi }}\int [D\psi ]\left[ D\bar{\psi } \right]\psi \left( {{x}_{1}} \right)\ldots \psi \left( {{x}_{n}} \right)\bar{\psi }\left( {{y}_{1}} \right)\ldots \bar{\psi }\left( {{y}_{n}} \right) \\ \quad \times {\rm exp} \left\{ {\rm i}\int {\rm d}x\bar{\psi }(x)\left[ {\rm i}\frac{\partial }{\partial {{x}_{0}}}-H\left( -{\rm i}\frac{\partial }{\partial {\bf x}},{\bf x} \right) \right]\psi (x) \right\} \\ \end{array}\end{eqnarray} \tag{ 74 }$$

when the electron–electron Coulomb interaction is neglected and electron field is called the free field.

Let us consider in detail the case of free electron field. The average of the product $\psi (y)\bar{\psi }(z)$ over all field configurations is called two–point (one—particle) Green function

$$\begin{eqnarray}&&G(y,z)={{\left\langle \psi \left( y \right)\bar{\psi }\left( z \right) \right\rangle }_{0}}\cdot \end{eqnarray} \tag{ 75 }$$

From the expression (67) of the generating functional $Z_{0}^{\psi }\left[ \eta ,\bar{\eta } \right]$ it follows that

$$\begin{eqnarray}&&G(y,z)=-\frac{1}{Z_{0}^{\psi }}{{\left. \frac{{{\delta }^{2}}Z_{0}^{\psi }\left[ \eta ,\bar{\eta } \right]}{\delta \eta (z)\delta \bar{\eta }(z)} \right|}_{\eta =\bar{\eta }=0}}\cdot \end{eqnarray} \tag{ 76 }$$

Similarly, the average of the product $\psi ({{y}_{1}})\psi ({{y}_{2}})\bar{\psi }({{z}_{1}})\bar{\psi }({{z}_{2}})$ over all field configurations is called four-point (two-particle) Green function

$$\begin{eqnarray}&&G({{y}_{1}},{{y}_{2}},{{z}_{1}},{{z}_{2}})={{\left\langle \psi \left( {{y}_{1}} \right)\psi \left( {{y}_{2}} \right)\bar{\psi }\left( {{z}_{1}} \right)\bar{\psi }\left( {{z}_{2}} \right) \right\rangle }_{0}}\cdot \end{eqnarray} \tag{ 77 }$$

It is expressed in terms of $Z_{0}^{\psi }\left[ \eta ,\bar{\eta } \right]$ as follows

$$\begin{eqnarray}&&G({{y}_{1}},{{y}_{2}},{{z}_{1}},{{z}_{2}})=\frac{1}{Z_{0}^{\psi }}\frac{{{\delta }^{4}}Z_{0}^{\psi }\left[ \eta ,\bar{\eta } \right]}{\delta \eta ({{z}_{2}})\delta \eta ({{z}_{1}})\delta \bar{\eta }({{y}_{2}})\delta \bar{\eta }({{y}_{1}})}\cdot \end{eqnarray} \tag{ 78 }$$

In order to establish the explicit form of the generating functional (67) we consider the Schrӧdinger equation

$$\begin{eqnarray}&&H\left( -{\rm i}\frac{\partial }{\partial {\bf x}},{\bf x} \right){{u}_{\alpha }}({\bf x})={{E}_{\alpha }}{{u}_{\alpha }}({\bf x})\end{eqnarray} \tag{ 79 }$$

and introduce $S(x,y)=S({\bf x},{\bf y};{{x}_{0}}-{{y}_{0}})$ the solution of the inhomogeneous differential equation

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \left[ {\rm i}\frac{\partial }{\partial {{x}_{0}}}-H\left( -{\rm i}\frac{\partial }{\partial {\bf x}},{\bf x} \right) \right]S(x,y)=\delta (x-y) \\ \quad =\delta ({\bf x}-{\bf y})\delta ({{x}_{0}}-{{y}_{0}}). \\ \end{array}\end{eqnarray} \tag{ 80 }$$

S(x, y) has the general form

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} S({\bf x},{\bf y};{{x}_{0}}-{{y}_{0}})=\frac{1}{2\pi }\int {\rm d}\omega {{{\rm e}}^{-{\rm i}\omega ({{x}_{0}}-{{y}_{0}})}}\mathop \sum \limits_{\alpha } \frac{{{u}_{\alpha }}({\bf x}){{u}_{\alpha }}({\bf y})*}{\omega -{{E}_{\alpha }}+i0} \\ \quad \quad +{\rm i}\mathop \sum \limits_{\alpha } {{C}_{\alpha }}{{{\rm e}}^{-{\rm i}{{E}_{\alpha }}({{x}_{0}}-{{y}_{0}})}}{{u}_{\alpha }}({\bf x}){{u}_{\alpha }}({\bf y})* \\ \quad =\frac{1}{2\pi }\int {\rm d}\omega {{{\rm e}}^{-{\rm i}\omega ({{x}_{0}}-{{y}_{0}})}}\mathop \sum \limits_{\alpha } {{u}_{\alpha }}({\bf x}){{u}_{\alpha }}({\bf y})* \\ \quad \quad \times \left[ \frac{1-{{C}_{\alpha }}}{\omega -{{E}_{\alpha }}+{\rm i}0}+\frac{{{C}_{\alpha }}}{\omega -{{E}_{\alpha }}-{\rm i}0} \right], \\ \end{array}\end{eqnarray} \tag{ 81 }$$

where the constants ${{C}_{\alpha }}$ are related with the parameters of the electron gas. Their physical meanings will be clarified latter.

Now let us perform the shift of the functional integration variables

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \psi (x)\to \psi (x)+\int {\rm d}yS(x,y)\eta (y), \\ \bar{\psi }(x)\to \bar{\psi }(x)+\int {\rm d}y\bar{\eta }(y)S(x,y), \\ \end{array}\end{eqnarray} \tag{ 82 }$$

in the rhs of formula (68) for the constant $Z_{0}^{\psi }.$ The functional integral in this formula must be invariant under the shift (82) of the functional integration variables. After lengthy but standard transformations and calculations we obtain the following formula

$$\begin{eqnarray}\begin{array}{lllllllllllllll} Z_{0}^{\psi } & = & \int [D\psi ]\left[ D\bar{\psi } \right]{\rm exp} \left\{ {\rm i}\int dx\left[ \bar{\psi }(x)\eta (x) \right. \right. \\ {} & {} & +\left. \left. \bar{\eta }(x)\psi (x) \right] \right\} \\ {} & {} & \times {\rm exp} \left\{ {\rm i}\int {\rm d}x\bar{\psi }(x)\left[ {\rm i}\frac{\partial }{\partial {{x}_{0}}}-H \right. \right. \\ {} & {} & \times \left. \left. \left( -{\rm i}\frac{\partial }{\partial {\bf x}},{\bf x} \right) \right]\psi (x) \right\} \\ {} & {} & \times {\rm exp} \left\{ {\rm i}\int {\rm d}x\int {\rm d}y\bar{\eta }\;(x)S(x,y)\eta (y) \right\}\cdot \\ \end{array}\end{eqnarray} \tag{ 83 }$$

Comparing the rhs of relation (83) with the definition (67) of $Z_{0}^{\psi }\left[ \eta ,\bar{\eta } \right],$ we derive the explicit expression of this generating functional

$$\begin{eqnarray}&&Z_{0}^{\psi }\left[ \eta ,\bar{\eta } \right]=Z_{0}^{\psi }{\rm exp} \left\{ -{\rm i}\int {\rm d}x\int {\rm d}y\bar{\eta }(x)S(x,y)\eta (y) \right\}.\end{eqnarray} \tag{ 84 }$$

From expression (84) and formula (76) we obtain

$$\begin{eqnarray}&&G(y,z)={\rm i}S(y,z).\end{eqnarray} \tag{ 85 }$$

Similarly, from expression (84) and formula (78) we derive a relation between four-point and two-point Green functions

$$\begin{eqnarray}\begin{array}{lllllllllllllll} G({{y}_{1}},{{y}_{2}},{{z}_{1}},{{z}_{2}}) & = & G({{y}_{2}},{{z}_{1}})G({{y}_{1}},{{z}_{2}}) \\ {} & {} & -G({{y}_{1}},{{z}_{1}})G({{y}_{2}},{{z}_{2}}), \\ \end{array}\end{eqnarray} \tag{ 86 }$$

which is the well-known Wick theorem for the four-point Green function of the free electron field. By means of the same reasonnings as those above presented, we can verify the validity of the Wick theorem for any 2n point Green function of the free electron field.

Now we consider again formula (81) for the function $S({\bf x}-{\bf y};{{x}_{0}}-{{y}_{0}}).$ Note that the limit $\varepsilon \to +0$ of the average of the product $\bar{\psi }({\bf x},t+\varepsilon )\psi ({\bf x},t)$ is the particle density $n({\bf x})$:

$$\begin{eqnarray}\begin{array}{lllllllllllllll} n({\bf x}) & = & \mathop{{\rm lim} }\limits_{\varepsilon \to +0}\left\langle \bar{\psi }({\bf x},t+\varepsilon )\psi ({\bf x},t) \right\rangle \\ {} & = & -{\rm i}\mathop{{\rm lim} }\limits_{\varepsilon \to +0}S({\bf x},{\bf x};-\varepsilon ). \\ \end{array}\end{eqnarray} \tag{ 87 }$$

From formula (81) for $S(x,x;-\varepsilon )$ it follows that

$$\begin{eqnarray}&&n({\bf x})=\mathop \sum \limits_{\alpha } {{C}_{\alpha }}{{\left| {{u}_{\alpha }}({\bf x}) \right|}^{2}}.\end{eqnarray} \tag{ 88 }$$

Thus, C_α is the electron occupation number n_α at the quantum state with the wave function u_α(x): C_α = n_α.

3.3. Scalar field of collective oscillations in electron gas

Now we study the system of interacting electrons with the following functional integral

$$\begin{eqnarray}\begin{array}{lllllllllllllll} Z & = & \int \left[ D\psi \right]\left[ D\bar{\psi } \right]{\rm exp} \left\{ {\rm i}\int {\rm d}x\bar{\psi }(x)\left[ {\rm i}\frac{\partial }{\partial {{x}_{0}}} \right. \right. \\ {} & {} & -\left. \left. H\left( -{\rm i}\frac{\partial }{\partial {\bf x}},{\bf x} \right) \right]\psi (x) \right\}{\rm exp} \left\{ -\frac{{\rm i}}{2}\int {\rm d}x \right. \\ {} & {} & \times \left. \int {\rm d}x^{\prime} \bar{\psi }(x)\psi (x)u(x-x^{\prime} )\bar{\psi }(x^{\prime} )\psi (x^{\prime} ) \right\}\cdot \\ \end{array}\end{eqnarray} \tag{ 89 }$$

The Coulomb term in the functional integral (89) is bilinear with respect to the electron density $\bar{\psi }(x)\psi (x).$ We linearize this bilinear interaction Hamiltonian by introducing a scalar field $\varphi (x)$ playing the role of the order parameter of collective oscillations in the electron gas and using the following functional integral

$$\begin{eqnarray}\begin{array}{lllllllllllllll} Z_{0}^{\varphi } & = & \int [D\varphi ] \\ {} & {} & \times {\rm exp} \left\{ \frac{{\rm i}}{2}\int {\rm d}x\int {\rm d}x^{\prime} \varphi (x)u(x-x^{\prime} )\varphi (x^{\prime} ) \right\}\cdot \\ \end{array}\end{eqnarray} \tag{ 90 }$$

Performing a shift of the functional integration variable

$$\begin{eqnarray}&&\varphi (x)\to \varphi (x)-\bar{\psi }(x)\psi (x),\end{eqnarray} \tag{ 91 }$$

we rewrite $Z_{0}^{\varphi }$ in a new form

$$\begin{eqnarray}\begin{array}{lllllllllllllll} Z_{0}^{\varphi } & = & \int [D\varphi ]{\rm exp} \left\{ \frac{{\rm i}}{2} \right.\int {\rm d}x\int {\rm d}x^{\prime} \varphi (x)u(x-x^{\prime} ) \\ {} & {} & \times \left. \varphi (x^{\prime} ) \right\}{\rm exp} \left\{ -{\rm i} \right.\int {\rm d}x\int {\rm d}x^{\prime} \bar{\psi }(x)\psi (x)u \\ {} & {} & \times \left. (x-x^{\prime} )\varphi (x^{\prime} ) \right\}{\rm exp} \left\{ \frac{{\rm i}}{2} \right.\int {\rm d}x\int {\rm d}x^{\prime} \bar{\psi } \\ {} & {} & \times \left. (x)\psi (x)u(x-x^{\prime} )\bar{\psi }(x^{\prime} )\psi (x^{\prime} ) \right\}\cdot \\ \end{array}\end{eqnarray} \tag{ 92 }$$

From this relation we derive the celebrated Hubbard–Stratonovich transformation

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {\rm exp} \left\{ -\frac{{\rm i}}{2}\int {\rm d}x\int dx^{\prime} \bar{\psi }(x)\psi (x)u(x-x^{\prime} )\bar{\psi }(x^{\prime} )\psi (x^{\prime} ) \right\} \\ =\frac{1}{Z_{0}^{\varphi }}\int [D\varphi ]{\rm exp} \left\{ \frac{{\rm i}}{2}\int {\rm d}x\int {\rm d}x^{\prime} \varphi (x)u(x-x^{\prime} )\varphi (x^{\prime} ) \right\} \\ \quad \times {\rm exp} \left\{ -{\rm i}\int {\rm d}x\int {\rm d}x^{\prime} \bar{\psi }(x)\psi (x)u(x-x^{\prime} )\varphi (x^{\prime} ) \right\}\cdot \\ \end{array}\end{eqnarray} \tag{ 93 }$$

Substititing the expression in the rhs. of relation (93) instead of its lhs. which is a factor in the rhs. of formula (89), we transform the functional integral Z into the form

$$\begin{eqnarray}\begin{array}{lllllllllllllll} Z & = & \frac{1}{Z_{0}^{\varphi }}\int [D\varphi ]{\rm exp} \left\{ \frac{{\rm i}}{2}\int {\rm d}x\int {\rm d}x^{\prime} \varphi (x) \right. \\ {} & {} & \times u\left. (x-x^{\prime} )\varphi (x^{\prime} ) \right\}\int \left[ D\psi \right]\left[ D\bar{\psi } \right] \\ {} & {} & \times {\rm exp} \left\{ -{\rm i}\int {\rm d}x\int {\rm d}x^{\prime} \bar{\psi }(x)\psi (x)u(x-x^{\prime} )\varphi (x^{\prime} ) \right\} \\ {} & {} & \times {\rm exp} \left\{ {\rm i}\int {\rm d}x\bar{\psi }(x) \right.\left[ {\rm i}\frac{\partial }{\partial {{x}_{0}}}-H \right. \\ {} & {} & \times \left. \left. \left( -{\rm i}\frac{\partial }{\partial {\bf x}},{\bf x} \right) \right]\psi (x) \right\}. \\ \end{array}\end{eqnarray} \tag{ 94 }$$

According to the definition (74) for the average of an expression over all field configurations, formula (94) can be rewritten as follows

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} Z=\frac{Z_{0}^{\psi }}{Z_{0}^{\varphi }}\int [D\varphi ]{\rm exp} \left\{ \frac{{\rm i}}{2}\int {\rm d}x\int {\rm d}x^{\prime} \varphi (x)u(x-x^{\prime} )\varphi (x^{\prime} ) \right\} \\ \quad \times {{\left\langle {\rm exp} {\mkern 1mu} \left\{ -{\rm i}\int {\rm d}x\int {\rm d}x^{\prime} \bar{\psi }(x)\psi (x)u(x-x^{\prime} )\varphi (x^{\prime} ) \right\} \right\rangle }_{0}}\cdot \\ \end{array}\end{eqnarray} \tag{ 95 }$$

Expanding the exponential function

$$\begin{eqnarray*}&&{\rm exp} \left\{ -{\rm i}\int {\rm d}x\int {\rm d}x^{\prime} \bar{\psi }(x)\psi (x)u(x-x^{\prime} )\varphi (x^{\prime} ) \right\}\end{eqnarray*}$$

into a functional power series of the scalar field, we obtain

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{\left\langle {\rm exp} \left\{ -{\rm i}\int {\rm d}x\int {\rm d}x^{\prime} \bar{\psi }(x)\psi (x)u(x-x^{\prime} )\varphi (x^{\prime} ) \right\} \right\rangle }_{0}} \\ \quad =F[\varphi ]=\sum \limits_{n=0}^{\infty } {{F}^{n}}[\varphi ], \\ \end{array}\end{eqnarray} \tag{ 96 }$$

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{F}^{(n)}}[\varphi ] \\ \quad =\frac{1}{n!}{{\left\langle {{\left[ -{\rm i}\int {\rm d}x\int {\rm d}x^{\prime} \bar{\psi }(x)\psi (x)u(x-x^{\prime} )\varphi (x^{\prime} ) \right]}^{n}} \right\rangle }_{0}}. \\ \end{array}\end{eqnarray} \tag{ 97 }$$

In [30, 32] it was shown that

$$\begin{eqnarray}&&{{F}^{(0)}}[\phi ]=1,\end{eqnarray} \tag{ 98 }$$

$$\begin{eqnarray}&&{{F}^{(1)}}[\phi ]={\rm i}{{W}^{(1)}}[\phi ],\end{eqnarray} \tag{ 99 }$$

$$\begin{eqnarray}&&{{W}^{(1)}}[\varphi ]=-\int {\rm d}x\int {\rm d}x^{\prime} n({\bf x})u(x-x^{\prime} )\varphi (x^{\prime} ),\end{eqnarray} \tag{ 100 }$$

$$\begin{eqnarray}&&{{F}^{(2)}}[\varphi ]=\frac{1}{2!}{{\left( {\rm i}{{W}^{(1)}}[\varphi ] \right)}^{2}}+{\rm i}{{W}^{(2)}}[\varphi ],\end{eqnarray} \tag{ 101 }$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{W}^{(2)}}[\varphi ] & = & \frac{{\rm i}}{2}\int {\rm d}x\int {\rm d}x^{\prime} \int {\rm d}y\int {\rm d}y^{\prime} S(x,y)S(y,x) \\ {} & {} & \times u(x-x^{\prime} )u(y-y^{\prime} )\varphi (x^{\prime} )\varphi (y^{\prime} ), \\ \end{array}\end{eqnarray} \tag{ 102 }$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{F}^{(3)}}[\varphi ] & = & \frac{1}{3!}{{\left( {\rm i}{{W}^{(1)}}[\varphi ] \right)}^{3}}+\left( {\rm i}{{W}^{(1)}}[\varphi ] \right) \\ {} & {} & \times \left( {\rm i}{{W}^{(2)}}[\varphi ] \right)+{\rm i}{{W}^{(3)}}[\varphi ], \\ \end{array}\end{eqnarray} \tag{ 103 }$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{W}^{(3)}}[\varphi ] & = & \frac{{\rm i}}{2}\int {\rm d}x\int {\rm d}x^{\prime} \int {\rm d}y\int {\rm d}y^{\prime} \int {\rm d}z\int {\rm d}z^{\prime} S(x,y) \\ {} & {} & \times S(y,z)S(z,x)u(x-x^{\prime} )u(y-y^{\prime} ) \\ {} & {} & \times u(z-z^{\prime} )\varphi (x^{\prime} )\varphi (y^{\prime} )\varphi (z^{\prime} ), \\ \end{array}\end{eqnarray} \tag{ 104 }$$

and so on. Continuing similar calculations in higher orders, finally we obtain

$$\begin{eqnarray}&&F[\varphi ]={\rm exp} \left\{ {\rm i}W[\varphi ] \right\}\end{eqnarray} \tag{ 105 }$$

with

$$\begin{eqnarray}&&W[\varphi ]=\sum \limits_{n=1}^{\infty } {{W}^{(n)}}[\varphi ]\cdot \end{eqnarray} \tag{ 106 }$$

Using expression (95) and relations (96)–(106), we obtain the functional integral Z expressed in term of the scalar field $\varphi (x)$ only

$$\begin{eqnarray}&&Z=\frac{Z_{0}^{\psi }}{Z_{0}^{\varphi }}\int \left[ D\varphi \right]\;{\rm exp} \left\{ {\rm i}I[\varphi ] \right\},\end{eqnarray} \tag{ 107 }$$

with the effective action functional

$$\begin{eqnarray}\begin{array}{lllllllllllllll} I[\varphi ] & = & \frac{1}{2}\int {\rm d}x\int {\rm d}y\varphi (x)u(x-y)\varphi (y) \\ {} & {} & -\int {\rm d}x\int {\rm d}y\;n({\bf x})u(x-y)\varphi (y) \\ {} & {} & +\frac{{\rm i}}{2}\int {\rm d}x\int {\rm d}y\int {\rm d}x^{\prime} \int {\rm d}y^{\prime} S(x^{\prime} ,y^{\prime} )S(y^{\prime} ,x^{\prime} ) \\ {} & {} & \times u(x^{\prime} -x)u(y^{\prime} -y)\varphi (x)\varphi (y) \\ {} & {} & +\frac{{\rm i}}{3}\int {\rm d}x\int {\rm d}y\int {\rm d}z\int {\rm d}x^{\prime} \int {\rm d}y^{\prime} \int {\rm d}z^{\prime} \\ {} & {} & \times S(x^{\prime} ,y^{\prime} )S(y^{\prime} ,z^{\prime} )S(z^{\prime} ,x^{\prime} )u(x^{\prime} -x) \\ {} & {} & \times u(y^{\prime} -y)u(z^{\prime} -z)\varphi (x)\varphi (y)\varphi (z)+\cdots . \\ \end{array}\end{eqnarray} \tag{ 108 }$$

In the approximation of the second order with respect to the scalar field $\phi (x),$ the effective action can be written as follows

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{I}_{0}}[\varphi ] & = & -\int {\rm d}x\;\nu (x)\varphi (x)+\frac{1}{2}\int {\rm d}x\int {\rm d}y\varphi (x) \\ {} & {} & \times A(x,y)\varphi (y), \\ \end{array}\end{eqnarray} \tag{ 109 }$$

where

$$\begin{eqnarray}&&\nu (x)=\int {\rm d}y\;u(x-y)n({\bf y}),\end{eqnarray} \tag{ 110 }$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} A(x,y) & = & u(x-y)+{\rm i}\int {\rm d}x^{\prime} \int {\rm d}y^{\prime} \;S(x^{\prime} ,y^{\prime} ) \\ {} & {} & \times S(y^{\prime} ,x^{\prime} )u(x^{\prime} -x)u(y^{\prime} -y). \\ \end{array}\end{eqnarray} \tag{ 111 }$$

From the principle of extreme action

$$\begin{eqnarray}&&\frac{\delta {{I}_{0}}[\varphi ]}{\delta \varphi (x)}=0.\end{eqnarray} \tag{ 112 }$$

We derive the dynamical equation for the scalar field ${{\varphi }_{0}}(x)$ corresponding to the extreme action

$$\begin{eqnarray}&&\int {\rm d}y\;A(x,y){{\varphi }_{0}}(y)=\nu (x).\end{eqnarray} \tag{ 113 }$$

It has the solution

$$\begin{eqnarray}&&{{\varphi }_{0}}(y)=\int {\rm d}y\;{{A}^{-1}}(y,x)\nu (x),\end{eqnarray} \tag{ 114 }$$

where ${{A}^{-1}}(y,\;x)$ is the kernel of the integral operator inverse to that with the kernel $A(y,\;x)$:

$$\begin{eqnarray}\begin{array}{lllllllllllllll} \int {\rm d}z\;A(x,z){{A}^{-1}}(z,y) & = & \int {\rm d}z\;{{A}^{-1}}(x,z)A(z,y) \\ {} & = & \delta (x-y). \\ \end{array}\end{eqnarray} \tag{ 115 }$$

The extreme value of the action functional equals

$$\begin{eqnarray}&&{{I}_{0}}[{{\varphi }_{0}}]=-\frac{1}{2}\int {\rm d}x\int {\rm d}y{{\varphi }_{0}}(x)A(x,y){{\varphi }_{0}}(y).\end{eqnarray} \tag{ 116 }$$

The fluctuations of the scalar field $\varphi (x)$ around the extreme field ${{\varphi }_{0}}(x)$ are described by the difference

$$\begin{eqnarray}&&\zeta (x)=\varphi (x)-{{\varphi }_{0}}(x).\end{eqnarray} \tag{ 117 }$$

In terms of this new field $\zeta (x)$ the effective action ${{I}_{0}}[\varphi ]$ has the quadratic form

$$\begin{eqnarray}&&{{I}_{0}}[{{\varphi }_{0}}+\zeta ]={{I}_{0}}[{{\varphi }_{0}}]+\frac{1}{2}\int {\rm d}x\int {\rm d}y\zeta (x)A(x,y)\zeta (y).\end{eqnarray} \tag{ 118 }$$

The dynamical equation for the field $\zeta (x)$is

$$\begin{eqnarray}&&\int {\rm d}yA(x,y)\zeta (y)=0.\end{eqnarray} \tag{ 119 }$$

For testing the validity of above-presented functional integral formalism now we apply it to the simple case of a homogeneous electron gas in the three-dimensional space. In this case function S(x, y) depends only on the difference x–y of the coordinates and has the Fourier transformation

$$\begin{eqnarray}\begin{array}{lllllllllllllll} S(x-y) & = & S({\bf x}-{\bf y};{{x}_{0}}-{{y}_{0}}) \\ {} & = & \frac{1}{{{(2\pi )}^{4}}}\int {\rm d}{\bf k}\int {\rm d}\omega \;{{{\rm e}}^{{\rm i}[{\bf k}({\bf x}-{\bf y})-\omega ({{x}_{0}}-{{y}_{0}})]}}\tilde{S}({\bf k},\omega ) \\ \end{array}\end{eqnarray} \tag{ 120 }$$

with the Fourier transform

$$\begin{eqnarray}&&\tilde{S}({\bf k},\omega )=\frac{1-n({\bf k})}{\omega -E({\bf k})+{\rm i}0}+\frac{n({\bf k})}{\omega -E({\bf k})-{\rm i}0},\end{eqnarray} \tag{ 121 }$$

where E(k) is the kinetic energy E(k) = k²/2m of the electron with momentum k, n(k) is the electron occupation number at the state with momentum k, $0\leqslant n({\bf k})\leqslant 1.$ Function A(x, y) determined by formula (111) also depends only on the difference x–y and has following Fourier transformation

$$\begin{eqnarray}\begin{array}{lllllllllllllll} A(x-y) & = & A({\bf x}-{\bf y};{{x}_{0}}-{{y}_{0}}) \\ {} & = & \frac{1}{{{(2\pi )}^{4}}}\int {\rm d}{\bf k}\int {\rm d}\omega \ \;{{{\rm e}}^{{\rm i}[{\bf k}({\bf x}-{\bf y})-\omega ({{x}_{0}}-{{y}_{0}})]}} \\ {} & {} & \times \tilde{A}({\bf k},\omega ) \\ \end{array}\end{eqnarray} \tag{ 122 }$$

with the Fourier transform

$$\begin{eqnarray}&&\tilde{A}({\bf k},\omega )=\tilde{u}({\bf k})+\tilde{\Pi }({\bf k},\omega )\tilde{u}{{({\bf k})}^{2}},\end{eqnarray} \tag{ 123 }$$

where

$$\begin{eqnarray}&&\tilde{\Pi }({\bf k},\omega )=\frac{{\rm i}}{{{(2\pi )}^{4}}}\int {\rm d}{\bf p}\int {\rm d}\varepsilon \;\tilde{S}({\bf p}+{\bf k},\varepsilon +\omega )\tilde{S}({\bf p},\varepsilon )\end{eqnarray} \tag{ 124 }$$

and $\tilde{u}({\bf k})$ is the Fourier transform of $\tilde{u}({\bf x})$

$$\begin{eqnarray}&&\tilde{u}({\bf k})=\frac{4\pi {{e}^{2}}}{{{{\bf k}}^{2}}}\cdot \end{eqnarray} \tag{ 125 }$$

The dynamical equation (119) becomes

$$\begin{eqnarray}&&1+\tilde{u}({\bf k})\tilde{\Pi }({\bf k},\omega )=0.\end{eqnarray} \tag{ 126 }$$

This equation determines the dependence of ω on k- the plasmon dispersion ω(k). In order to establish the dispersion we must calculate $\tilde{\Pi }({\bf k},\omega ).$ After the integration of the rhs of formula (124) with respect to the variable , we obtain

$$\begin{eqnarray}&&\tilde{\Pi }({\bf k},\omega )={{\tilde{\Pi }}_{1}}({\bf k},\omega )-{{\tilde{\Pi }}_{2}}({\bf k},\omega ),\end{eqnarray} \tag{ 127 }$$

where

$$\begin{eqnarray}&&{{\tilde{\Pi }}_{1}}({\bf k},\omega )=\frac{1}{{{(2\pi )}^{3}}}\int {\rm d}{\bf p}\frac{\left[ 1-n({\bf p}+{\bf k}) \right]n({\bf p})}{E({\bf p}+{\bf k})-E({\bf p})-\omega },\end{eqnarray} \tag{ 128 }$$

$$\begin{eqnarray}&&{{\tilde{\Pi }}_{2}}({\bf k},\omega )=\frac{1}{{{(2\pi )}^{3}}}\int {\rm d}{\bf p}\frac{\left[ 1-n({\bf p}) \right]n({\bf p}+{\bf k})}{E({\bf p}+{\bf k})-E({\bf p})-\omega }\cdot \end{eqnarray} \tag{ 129 }$$

By means of the substitution ${\bf p}\to {\bf p}-{\bf k}/2$ we rewrite formula (128) in the form

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{\tilde{\Pi }}_{1}}({\bf k},\omega ) & = & \frac{1}{{{(2\pi )}^{3}}}\int {\rm d}{\bf p} \\ {} & {} & \times \frac{[1-n({\bf p}+{\bf k}/2)]n({\bf p}-{\bf k}/2)}{E({\bf p}+{\bf k}/2)-E({\bf p}-{\bf k}/2)-\omega }. \\ \end{array}\end{eqnarray} \tag{ 130 }$$

By changing the integration variable ${\bf p}\to -{\bf p}$ we rewrite expression (129) as follows

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{\tilde{\Pi }}_{2}}({\bf k},\omega ) & = & \frac{1}{{{(2\pi )}^{3}}}\int {\rm d}{\bf p} \\ {} & {} & \times \frac{[1-n(-{\bf p})]n(-{\bf p}+{\bf k})}{E(-{\bf p}+{\bf k})-E(-{\bf p})-\omega }\cdot \\ \end{array}\end{eqnarray} \tag{ 131 }$$

Because both E(p) and n(p) are the function of p², we have

$$\begin{eqnarray*}&&\begin{array}{lllllllllllllll} n(-{\bf p})=n({\bf p}),\;n(-{\bf p}+{\bf k})=n({\bf p}-{\bf k}), \\ E(-{\bf p})=E({\bf p}),\;E(-{\bf p}+{\bf k})=E({\bf p}-{\bf k}). \\ \end{array}\end{eqnarray*}$$

Therefore

$$\begin{eqnarray}&&{{\tilde{\Pi }}_{2}}({\bf k},\omega )=\frac{1}{{{(2\pi )}^{3}}}\int {\rm d}{\bf p}\frac{[1-n({\bf p})]n({\bf p}-{\bf k})}{E({\bf p}-{\bf k})-E({\bf p})-\omega }\cdot \end{eqnarray} \tag{ 132 }$$

By means of the substitution ${\bf p}\to {\bf p}+{\bf k}/2$ we obtain

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{\tilde{\Pi }}_{2}}({\bf k},\omega ) \\ \quad =\frac{1}{{{(2\pi )}^{3}}}\int {\rm d}{\bf p}\frac{[1-n({\bf p}+{\bf k}/2)]n({\bf p}-{\bf k}/2)}{E({\bf p}-{\bf k}/2)-E({\bf p}+{\bf k}/2)-\omega }\cdot \\ \end{array}\end{eqnarray} \tag{ 133 }$$

Combining (130) and (133), finally we derive the expression of $\tilde{\Pi }({\bf k},\omega )$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} \tilde{\Pi }({\bf k},\;\omega ) & = & \frac{1}{{{(2\pi )}^{3}}}\int {\rm d}{\bf p}\left[ 1-n({\bf p}+{\bf k}/2) \right]n({\bf p}-{\bf k}/2) \\ {} & {} & \times \left[ \frac{1}{E({\bf p}+{\bf k}/2)-E({\bf p}-{\bf k}/2)-\omega } \right. \\ {} & {} & +\left. \frac{1}{E({\bf p}+{\bf k}/2)-E({\bf p}-{\bf k}/2)+\omega } \right]. \\ \end{array}\end{eqnarray} \tag{ 134 }$$

Consider the case of the electron gas at vanishing absolute temperature T = 0 and denote p_F the magnitude of the momentum of electrons at the Fermi surface. Because n(p) is equal to unity inside and on the Fermi surface, i. e. at $p\leqslant {{p}_{{\rm F}}},$ and vanishes outside this surface, i. e. at $p\gt {{p}_{{\rm F}}}$, the domain of integration in the rhs of formula (1134) must be the common region of the volume inside the spherical surface $\left| {\bf p}-{\bf k}/2 \right|={{p}_{{\rm F}}}$ and the region outside the spherical surface $\left| {\bf p}+{\bf k}/2 \right|={{p}_{{\rm F}}}.$ At small values of k $(k\leqslant 2{{p}_{{\rm F}}})$ two spheres $\left| {\bf p}-{\bf k}/2 \right|\leqslant {{p}_{{\rm F}}}$ and $\left| {\bf p}+{\bf k}/2 \right|\leqslant {{p}_{{\rm F}}}$ are overlapping (figure 1) and the domain of integration is the region bounded by two above-mentioned spherical surfaces.

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Let us choose the direction of vector k to be that of the vertical axis Oz in the rectangular coordinate system. Then for small values of k, the domain of integration is the region bounded from above by the spherical surface $\left| {\bf p}-{\bf k}/2 \right|={{p}_{{\rm F}}}$ and bounded from below by the spherical surface $\left| {\bf p}+{\bf k}/2 \right|={{p}_{{\rm F}}}.$ In this case formula (134) becomes

$$\begin{eqnarray}\begin{array}{lllllllllllllll} \tilde{\Pi }({\bf k},\;\omega ) & = & -\frac{2k}{{{\omega }^{2}}m}\frac{1}{4{{\pi }^{2}}}\int \limits_{0}^{2\pi } {{{\rm cos} }^{2}}\theta \\ {} & {} & \times \left\{ \int \limits_{{{p}_{1}}(\theta )}^{{{p}_{2}}(\theta )} \frac{1}{1-{{(kp/\omega m)}^{2}}{{{\rm cos} }^{2}}\theta }{{p}^{3}}{\rm d}p \right\}{\rm d}\theta , \\ \end{array}\end{eqnarray} \tag{ 135 }$$

where

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{p}_{1}}(\theta ) & = & {{\left[ p_{{\rm F}}^{2}-\frac{{{k}^{2}}}{4}{{{\rm sin} }^{2}}\theta \right]}^{1/2}}+\frac{k}{2}\;{\rm cos} \;\theta , \\ {{p}_{2}}(\theta ) & = & {{\left[ p_{{\rm F}}^{2}-\frac{{{k}^{2}}}{4}{{{\rm sin} }^{2}}\theta \right]}^{1/2}}-\frac{k}{2}\;{\rm cos} \;\theta . \\ \end{array}\end{eqnarray} \tag{ 136 }$$

Expanding the rhs of formula (135) into the power series of the small variable k² and limiting at the second order approximation, we obtain

$$\begin{eqnarray}&&\tilde{\Pi }({\bf k},\omega )=-\frac{{{n}_{0}}{{k}^{2}}}{m{{\omega }^{2}}}\left( 1+\frac{3}{5}\frac{p_{{\rm F}}^{2}}{{{m}^{2}}}\frac{{{k}^{2}}}{{{\omega }^{2}}} \right),\end{eqnarray} \tag{ 137 }$$

where

$$\begin{eqnarray}&&{{n}_{0}}=\frac{1}{{{(2\pi )}^{3}}}\frac{4\pi }{3}p_{{\rm F}}^{3}\end{eqnarray} \tag{ 138 }$$

is the electron density. Substituting expression (137) into the rhs of formula (123), using formula (125) and setting

$$\begin{eqnarray}&&\omega _{0}^{2}=\frac{4\pi {{e}^{2}}{{n}_{0}}}{m},\end{eqnarray} \tag{ 139 }$$

we obtain

$$\begin{eqnarray}&&\tilde{A}({\bf k},\omega )=\frac{4\pi {{e}^{2}}}{{{k}^{2}}{{\omega }^{2}}}\left( {{\omega }^{2}}-\omega _{0}^{2}-\frac{3}{5}\frac{p_{{\rm F}}^{2}}{{{m}^{2}}}\frac{\omega _{0}^{2}}{{{\omega }^{2}}}{{k}^{2}} \right)\cdot \end{eqnarray} \tag{ 140 }$$

Finally we consider the dynamical equation (119). In this case kernel $A(x-y)=A({\bf x}-{\bf y};{{x}_{0}}-{{y}_{0}})$ has the form (122) with the Fourier transform (123) determined by formula (140). Introducing the Fourier transformation of the scalar field $\zeta ({\bf x},\;t)$

$$\begin{eqnarray}&&\zeta ({\bf x},t)=\frac{1}{{{(2\pi )}^{4}}}\int {\rm d}{\bf k}\int {\rm d}\omega \;{{{\rm e}}^{-{\rm i}({\bf kx}-\omega t)}}\tilde{\zeta }({\bf k},\;\omega ),\end{eqnarray} \tag{ 141 }$$

we can rewrite the dynamical equation (119) as follows

$$\begin{eqnarray}&&\left( {{\omega }^{2}}-\omega _{0}^{2}-\frac{3}{5}\frac{p_{{\rm F}}^{2}}{{{m}^{2}}}\frac{\omega _{0}^{2}}{{{\omega }^{2}}}{{k}^{2}} \right)\tilde{\zeta }({\bf k},\omega )=0\cdot \end{eqnarray} \tag{ 142 }$$

Therefore between the energy ω and the momentum k of the free plasmon we have following dispersion equation

$$\begin{eqnarray}&&{{\omega }^{2}}=\omega _{0}^{2}+\frac{3}{5}\frac{p_{{\rm F}}^{2}}{{{m}^{2}}}\frac{\omega _{0}^{2}}{{{\omega }^{2}}}{{k}^{2}}\cdot \end{eqnarray} \tag{ 143 }$$

In the first-order approximation with respect to the small ratio ${{k}^{2}}/\omega _{0}^{2}$ this dispersion equation has the following solution

$$\begin{eqnarray}&&{{\omega }^{2}}=\omega _{0}^{2}+\frac{3}{5}\frac{p_{{\rm F}}^{2}}{{{m}^{2}}}{{k}^{2}}\cdot \end{eqnarray} \tag{ 144 }$$

In this approximation the Fourier transform $\tilde{A}({\bf k},\omega )$ of the kernel of the dynamical equation for free plasmon has the expression

$$\begin{eqnarray}&&\tilde{A}({\bf k},\omega )=\frac{4\pi {{e}^{2}}}{{{k}^{2}}{{\omega }^{2}}}\left( {{\omega }^{2}}-\omega _{0}^{2}-{{\gamma }^{2}}{{k}^{2}} \right)\end{eqnarray} \tag{ 145 }$$

with

$$\begin{eqnarray}&&{{\gamma }^{2}}=\frac{3}{5}\frac{p_{{\rm F}}^{2}}{{{m}^{2}}}\cdot \end{eqnarray} \tag{ 146 }$$

Note that the dispersion formula (144) was previously by many authors in conventional theory of plasma oscillations [37, 38]. The agreement of the above-presented result with that of the conventional theory of plasma oscillation demonstrates the validity of the functional integral method.

3.4. Quantization of plasmon field

Lets us apply the presented results to the study of the effective action (118) of the homogeneous free electron gas in the three-dimensional space. Consider the quadratic part of the this effective action

$$\begin{eqnarray}&&{{I}^{(2)}}[\zeta ]=\frac{1}{2}\int {\rm d}x\int {\rm d}y\;\zeta (x)A(x-y)\zeta (y).\end{eqnarray} \tag{ 147 }$$

Using the Fourier transformation (122) of A(x−y) and Fourier transformation (141) of the wave function $\zeta (x)$ we rewrite the functional (147) in the momentum representation

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{I}^{(2)}}[\zeta ] & = & \frac{1}{{{(2\pi )}^{4}}}\int {\rm d}{\bf k}\int {\rm d}\omega \;\frac{1}{2}\tilde{\zeta }(-{\bf k},-\omega ) \\ {} & {} & \times \tilde{A}({\bf k},\omega )\tilde{\zeta }({\bf k},\omega ). \\ \end{array}\end{eqnarray} \tag{ 148 }$$

Then using expression (145) of $\tilde{A}({\bf k},\omega )$ and setting

$$\begin{eqnarray}&&\tilde{\sigma }({\bf k},\omega )=\sqrt{\frac{4\pi {{e}^{2}}}{{{k}^{2}}{{\omega }^{2}}}}\tilde{\zeta }({\bf k},\omega ),\end{eqnarray} \tag{ 149 }$$

we obtain the expression of the functional (148) in terms of the new function $\tilde{\sigma }({\bf k},\omega )$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{I}_{{\rm eff}}}[\sigma ] & = & \frac{1}{{{(2\pi )}^{4}}}\int {\rm d}{\bf k}\int {\rm d}\omega \;\frac{1}{2}\tilde{\sigma }(-{\bf k},-\omega ) \\ {} & {} & \times \left[ {{\omega }^{2}}-\omega _{0}^{2}-{{\gamma }^{2}}{{k}^{2}} \right]\;\tilde{\sigma }({\bf k},\omega ). \\ \end{array}\end{eqnarray} \tag{ 150 }$$

Considering $\tilde{\sigma }({\bf k},\omega )$ as the Fourier transforms of a new scalar field $\sigma ({\bf x},t)$

$$\begin{eqnarray}&&\sigma ({\bf x},t)=\frac{1}{{{(2\pi )}^{4}}}\int {\rm d}{\bf k}\int {\rm d}\omega \;{{{\rm e}}^{{\rm i}({\bf kx}-\omega \,t)}}\tilde{\sigma }({\bf k},\omega ),\end{eqnarray} \tag{ 151 }$$

we obtain the following formula for the effective action of this new scalar field

$$\begin{eqnarray}&&{{I}_{{\rm eff}}}[\sigma ]=\int {\rm d}t{{L}_{\sigma }}(t)\end{eqnarray} \tag{ 152 }$$

with the effective Lagrangian

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{L}_{\sigma }}(t) & = & \frac{1}{2}\int {\rm d}{\bf x}\left\{ {{\left[ \frac{\partial \sigma ({\bf x},t)}{\partial t} \right]}^{2}}-{{\gamma }^{2}} \right. \\ {} & {} & \times \left. {{\left[ \nabla \sigma ({\bf x},t) \right]}^{2}}-\omega _{0}^{2}\sigma {{({\bf x},t)}^{2}} \right\}. \\ \end{array}\end{eqnarray} \tag{ 153 }$$

It is straightforward to derive the Lagrange equation of this scalar field

$$\begin{eqnarray}&&\frac{{{\partial }^{2}}\sigma ({\bf x},t)}{\partial {{t}^{2}}}-{{\gamma }^{2}}{{\nabla }^{2}}\sigma ({\bf x},t)+\omega _{0}^{2}\sigma ({\bf x},t)=0.\end{eqnarray} \tag{ 154 }$$

It follows that between the frequency ω and the wave vector k of the plane waves of the scalar field $\sigma ({\bf x},t)$ there must exist the relation

$$\begin{eqnarray}&&{{\omega }^{2}}=\omega _{0}^{2}+{{\gamma }^{2}}{{k}^{2}}.\end{eqnarray} \tag{ 155 }$$

Finally we perform the quantization of the scalar field in the framework of the canonical quantization procedure. The canonical momentum $\hat{\pi }({\bf x},\;t)$ corresponding to the canonical coordinate $\hat{\sigma }({\bf x},\;t),$ by definition, is the operator

$$\begin{eqnarray}&&\hat{\pi }({\bf x},t)=\frac{\delta {{L}_{\sigma }}}{\delta \dot{\sigma }({\bf x},t)}=\widehat{{\dot{\sigma }}}({\bf x},t)\end{eqnarray} \tag{ 156 }$$

with the notation

$$\begin{eqnarray}&&\widehat{{\dot{\sigma }}}({\bf x},t)=\frac{\partial \hat{\sigma }({\bf x},t)}{\partial t}\cdot \end{eqnarray} \tag{ 157 }$$

The Hamiltonian of the system has the expression

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{H}_{\sigma }} & = & \int {\rm d}{\bf x}\;\hat{\pi }({\bf x},t)\widehat{{\dot{\sigma }}}({\bf x},t)-{{L}_{\sigma }}(t) \\ {} & = & \frac{1}{2}\int {\rm d}{\bf x}\left\{ {{\left[ \frac{\partial \hat{\sigma }({\bf x},t)}{\partial t} \right]}^{2}}+{{\gamma }^{2}}{{\left[ \nabla \hat{\sigma }({\bf x},t) \right]}^{2}} \right. \\ {} & {} & +\left. \omega _{0}^{2}\hat{\sigma }{{({\bf x},t)}^{2}} \right\}. \\ \end{array}\end{eqnarray} \tag{ 158 }$$

According to the canonical quantization rule, between operators $\hat{\sigma }({\bf x},t)$ and $\hat{\pi }({\bf x},t)$ there exist the following equal-time commutation relations

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \left[ \hat{\sigma }({\bf x},t),\hat{\sigma }({\bf x}^{\prime} ,t) \right]=\left[ \hat{\pi }({\bf x},t),\hat{\pi }({\bf x}^{\prime} ,t) \right]=0, \\ \left[ \hat{\pi }({\bf x},t),\hat{\sigma }({\bf x}^{\prime} ,t) \right]=-{\rm i}\delta ({\bf x}-{\bf x}^{\prime} ). \\ \end{array}\end{eqnarray} \tag{ 159 }$$

The expressions (153) and (158) for the Lagrangian and the Hamiltonian of the new scalar field $\sigma ({\bf x},t)$ look like those of the Klein–Gordon real (hermitian) scalar field in relativistic quantum field theory except for scaling factor γ of the spatial coordinates. For the interpretation of the physical meaning of this scalar field we can apply the reasonning of the relativistic quantum field theory. In order to simplify the presentation we quantize the scalar field $\sigma ({\bf x},t)$ in a cubic box with a very large volume V, using the periodic boundary conditions, and expand the scalar field $\hat{\sigma }({\bf x},t)$ as follows

$$\begin{eqnarray}&&\hat{\sigma }({\bf x},t)=\frac{1}{\sqrt{V}}\mathop \sum \limits_{{\bf k}} \frac{1}{\sqrt{2\omega }}\left\{ {{{\hat{a}}}_{{\bf k}}}{{{\rm e}}^{{\rm i}({\bf kx}-\omega t)}}+\hat{a}_{{\bf k}}^{+}{{{\rm e}}^{-{\rm i}({\bf kx}-\omega t)}} \right\}\cdot \end{eqnarray} \tag{ 160 }$$

The corresponding expansion of the canonical momentum is

$$\begin{eqnarray}&&\hat{\pi }({\bf x},t)=-\frac{{\rm i}}{\sqrt{V}}\mathop \sum \limits_{{\bf k}} \sqrt{\frac{\omega }{2}}\left\{ {{{\hat{a}}}_{{\bf k}}}{{{\rm e}}^{{\rm i}({\bf kx}-\omega t)}}-\hat{a}_{{\bf k}}^{+}{{{\rm e}}^{-{\rm i}({\bf kx}-\omega t)}} \right\}.\end{eqnarray} \tag{ 161 }$$

Inverting the expansions (160) and (161) to express ${{\hat{a}}_{{\bf k}}}$ and $\hat{a}_{{\bf k}}^{+}$ in terms of $\hat{\sigma }({\bf x},0)$ and $\hat{\pi }({\bf x},0),$ we obtain

$$\begin{eqnarray}&&{{\hat{a}}_{{\bf k}}}=\frac{1}{\sqrt{V}}\int {\rm d}{\bf x}{{{\rm e}}^{-{\rm i}{\bf kx}}}\left[ \sqrt{\frac{\omega }{2}}\hat{\sigma }({\bf x},0)+\frac{{\rm i}}{\sqrt{2\omega }}\hat{\pi }({\bf x},0) \right],\end{eqnarray} \tag{ 162 }$$

$$\begin{eqnarray}&&\hat{a}_{{\bf k}}^{+}=\frac{1}{\sqrt{V}}\int {\rm d}{\bf x}{{{\rm e}}^{{\rm i}{\bf kx}}}\left[ \sqrt{\frac{\omega }{2}}\hat{\sigma }({\bf x},0)-\frac{{\rm i}}{\sqrt{2\omega }}\hat{\pi }({\bf x},0) \right].\end{eqnarray} \tag{ 163 }$$

From the canonical quantization rules (159) it follows the canonical commutation relations between operators ${{\hat{a}}_{{\bf k}}}$ and $\hat{a}_{{\bf k}}^{+}.$ We obtain

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \left[ {{{\hat{a}}}_{{\bf k}}},{{{\hat{a}}}_{{\bf l}}} \right]=\left[ \hat{a}_{{\bf k}}^{+},\hat{a}_{{\bf l}}^{+} \right]=0, \\ \left[ {{{\hat{a}}}_{{\bf k}}},\hat{a}_{{\bf l}}^{+} \right]={{\delta }_{{\bf kl}}}. \\ \end{array}\end{eqnarray} \tag{ 164 }$$

The Hamiltonian (158) is expressed in terms of these operators as follows

$$\begin{eqnarray}&&{{H}_{\sigma }}=\frac{1}{2}\mathop \sum \limits_{{\bf k}} \left( \hat{a}_{{\bf k}}^{+}{{{\hat{a}}}_{{\bf k}}}+\frac{1}{2} \right)\omega .\end{eqnarray} \tag{ 165 }$$

Remember that between frequency ω and wave vector k there exists the relation (155).

Operators ${{\hat{a}}_{{\bf k}}}$ and $\hat{a}_{{\bf k}}^{+}$ are interpreted as the destruction and creation operators of the quanta of the quantized scalar field $\hat{\sigma }({\bf x},t).$ These quanta are called the plasmons, and formula (155) becomes the relation between energy and momentum of plasmon.

Consider now the interaction of the electron gas with some external electromagnetic field. We choose to work in the transverse gauge of the electromagnetic field and denote ${\bf A}({\bf x},t)$ its vector potential

$$\begin{eqnarray}&&\nabla {\bf A}({\bf x},t)=0.\end{eqnarray} \tag{ 166 }$$

It was known [39] that the interaction Lagrangian has the following expression

$$\begin{eqnarray}&&{{L}_{\operatorname{int}}}(t)=\frac{e}{c}\int {\rm d}{\bf x}\;n({\bf x},t)\;\delta {\bf \dot{r}}({\bf x},t)\;{\bf A}({\bf x},t),\end{eqnarray} \tag{ 167 }$$

where c is the light velocity in the vacuum. Using relation (1) between the electron density $n({\bf x},t)$ and the charge density $\rho ({\bf x},t)$ of the electron gas, we rewrite the expression in the rhs of equation (167) as follows

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{L}_{\operatorname{int}}}(t) & = & \frac{e{{n}_{0}}}{c}\int {\rm d}{\bf x}\delta {\bf \dot{r}}({\bf x},t){\bf A}({\bf x},t) \\ {} & {} & -\frac{1}{c}\int {\rm d}{\bf x}\;\rho ({\bf x},t)\;\delta {\bf \dot{r}}({\bf x},t){\bf A}({\bf x},t). \\ \end{array}\end{eqnarray} \tag{ 168 }$$

In section 2 we have established the relationship between $\delta {\bf r}({\bf x},t)$ and the plasmonic field $\sigma ({\bf x},t).$ Therefore in order to study the plasmon–photon interaction it is necessary to express $\rho ({\bf x},t)$ in term of the components of the vector $\delta {\bf r}({\bf x},t).$ For this purpose we use formula (12), a consequence of the conservation of the total electron number in the electron gas. Using relation (1), we derive the following differential equation for the function $\rho ({\bf x},t)$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} \rho ({\bf x},\;t) & = & e{{n}_{0}}\mathop \sum \limits_{i} {{\nabla }_{i}}\delta {{r}_{i}}({\bf x},t)-\rho ({\bf x},t)\mathop \sum \limits_{i} {{\nabla }_{i}}\delta {{r}_{i}} \\ {} & {} & \times ({\bf x},t)-\mathop \sum \limits_{i} \delta {{r}_{i}}({\bf x},t){{\nabla }_{i}}\rho ({\bf x},t). \\ \end{array}\end{eqnarray} \tag{ 169 }$$

Since this differential equation cannot be exactly solved, we look for $\rho ({\bf x},t)$ in the form of a functional power series of the components $\delta {{r}_{i}}({\bf x},t)$ and write the solution in the form

$$\begin{eqnarray}&&\rho ({\bf x},\;t)=\sum \limits_{\nu =1}^{\infty } {{\rho }^{(\nu )}}({\bf x},\;t),\end{eqnarray} \tag{ 170 }$$

where $\rho ({\bf x},t)$ is the term of the ν-th order with respect to the components $\delta {{r}_{i}}({\bf x},t).$ Let us limit to the third order. By means of standard calculations it can be shown that

$$\begin{eqnarray}&&{{\rho }^{(1)}}({\bf x},t)=e{{n}_{0}}\mathop \sum \limits_{i} {{\nabla }_{i}}\delta {{r}_{i}}({\bf x},t),\end{eqnarray} \tag{ 171 }$$

$$\begin{eqnarray}&&{{\rho }^{(2)}}({\bf x},t)=e{{n}_{0}}\mathop \sum \limits_{i} \mathop \sum \limits_{j} {{\nabla }_{i}}\delta {{r}_{i}}({\bf x},t)\ \;{{\nabla }_{j}}\delta {{r}_{j}}({\bf x},t),\end{eqnarray} \tag{ 172 }$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{\rho }^{(3)}}({\bf x},t) & = & e{{n}_{0}}\mathop \sum \limits_{i} \mathop \sum \limits_{j} \mathop \sum \limits_{k} {{\nabla }_{i}}\delta {{r}_{i}}({\bf x},t)\ \;{{\nabla }_{j}}\delta {{r}_{j}}({\bf x},t) \\ {} & {} & \times {{\nabla }_{k}}\delta {{r}_{k}}({\bf x},t). \\ \end{array}\end{eqnarray} \tag{ 173 }$$

Substituting expression (170)–(173) into the rhs of relation (168), we obtain the explicit expression of the interaction Lagrangian L_int(t) in term of the components $\delta {{r}_{i}}({\bf x},t)$

$$\begin{eqnarray}&&{{L}_{\operatorname{int}}}(t)=\sum \limits_{n=1}^{\infty } L_{\operatorname{int}}^{(n)}(t),\end{eqnarray} \tag{ 174 }$$

where up to the fourth order we have

$$\begin{eqnarray}&&L_{\operatorname{int}}^{(1)}(t)=\frac{e{{n}_{0}}}{c}\int {\rm d}{\bf x}\mathop \sum \limits_{i} {{A}_{i}}({\bf x},t)\;\delta {{\dot{r}}_{i}}({\bf x},t)\;,\end{eqnarray} \tag{ 175 }$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} L_{\operatorname{int}}^{(2)}(t) & = & -\frac{e{{n}_{0}}}{c}\int {\rm d}{\bf x}\ \;\mathop \sum \limits_{i} {{A}_{i}}({\bf x},t)\;\delta {{{\dot{r}}}_{i}}({\bf x},t) \\ {} & {} & \times \mathop \sum \limits_{j} {{\nabla }_{j}}\delta {{r}_{j}}({\bf x},t), \\ \end{array}\end{eqnarray} \tag{ 176 }$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} L_{\operatorname{int}}^{(3)}(t) & = & -\frac{e{{n}_{0}}}{c}\int {\rm d}{\bf x}\ \;\mathop \sum \limits_{i} {{A}_{i}}({\bf x},t)\;\delta {{{\dot{r}}}_{i}}({\bf x},t) \\ {} & {} & \times \mathop \sum \limits_{j} \mathop \sum \limits_{k} {{\nabla }_{j}}\delta {{r}_{j}}({\bf x},t)\ \;{{\nabla }_{k}}\delta {{r}_{k}}({\bf x},t), \\ \end{array}\end{eqnarray} \tag{ 177 }$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} L_{\operatorname{int}}^{(4)}(t) & = & -\frac{e{{n}_{0}}}{c}\int {\rm d}{\bf x}\mathop \sum \limits_{i} {{A}_{i}}({\bf x},t)\delta {{{\dot{r}}}_{i}}({\bf x},t) \\ {} & {} & \times \mathop \sum \limits_{j} \mathop \sum \limits_{k} \mathop \sum \limits_{l} {{\nabla }_{j}}\delta {{r}_{j}}({\bf x},t) \\ {} & {} & \times {{\nabla }_{k}}\delta {{r}_{k}}({\bf x},t){{\nabla }_{l}}\delta {{r}_{l}}({\bf x},t). \\ \end{array}\end{eqnarray} \tag{ 178 }$$

In order to derive the relation between components $\delta {{r}_{i}}({\bf x},t)$ and the plasmonic field $\sigma ({\bf x},t)$ we use formula (15) with the longitudinal $q_{{\bf k}}^{(3)}(t)={{q}_{{\bf k}}}(t)$ but without the transverse ones

$$\begin{eqnarray}&&\delta {\bf r}({\bf x},t)=\frac{1}{\sqrt{V}}\mathop \sum \limits_{{\bf k}} {{{\rm e}}^{{\rm i}{\bf kx}}}\frac{{\bf k}}{k}{{q}_{{\bf k}}}(t)\end{eqnarray} \tag{ 179 }$$

as well as relation (29) between the generalized coordinate ${{q}_{{\bf k}}}(t)$ and the Fourier component ${{\tilde{\sigma }}_{{\bf k}}}(t)$ of the plasmonic field $\sigma ({\bf x},t).$ Thus we have

$$\begin{eqnarray}&&\delta {\bf r}({\bf x},t)=\frac{1}{\sqrt{{{n}_{0}}m}}\frac{1}{\sqrt{V}}\mathop \sum \limits_{{\bf k}} {{{\rm e}}^{{\rm i}{\bf kx}}}\frac{{\bf k}}{k}{{\tilde{\sigma }}_{{\bf k}}}(t).\end{eqnarray} \tag{ 180 }$$

Expressing the Fourier component ${{\tilde{\sigma }}_{{\bf k}}}(t)$ of the plasmonic field $\sigma ({\bf x},t)$ in terms of this field

$$\begin{eqnarray}&&{{\tilde{\sigma }}_{{\bf k}}}(t)=\frac{1}{\sqrt{V}}\int {\rm d}{\bf x}\;{{{\rm e}}^{-{\rm i}{\bf kx}}}\sigma ({\bf x},t),\end{eqnarray} \tag{ 181 }$$

from equation (180) we obtain

$$\begin{eqnarray}&&\delta {{r}_{i}}({\bf x},t)=\int {\rm d}{\bf y}{{F}_{i}}({\bf x}-{\bf y})\sigma ({\bf y},t),\end{eqnarray} \tag{ 182 }$$

where

$$\begin{eqnarray}&&{{F}_{i}}\left( {\bf x}-{\bf y} \right)=\frac{1}{\sqrt{{{n}_{0}}m}}\frac{1}{\sqrt{V}}\mathop \sum \limits_{{\bf k}} \frac{{{k}_{i}}}{k}{{{\rm e}}^{{\rm i}{\bf k}({\bf x}-{\bf y})}}.\end{eqnarray} \tag{ 183 }$$

Vector function F(x−y) with components (183) can be called the characteristic function or characteristic factor of the plasmon. Substituting expression (182) of the components $\delta {{r}_{i}}({\bf x},\;t)$ into the rhs of relations (175)–(178), finally we obtain

$$\begin{eqnarray}\begin{array}{lllllllllllllll} L_{\operatorname{int}}^{(1)}(t) & = & \frac{e{{n}_{0}}}{c}\int {\rm d}{\bf x}\int {\rm d}{\bf y}\ \;\mathop \sum \limits_{i} {{A}_{i}}({\bf x},t) \\ {} & {} & \times {{F}_{i}}(x-y){{{\dot{\sigma }}}_{i}}({\bf y},\;t), \\ \end{array}\end{eqnarray} \tag{ 184 }$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} L_{\operatorname{int}}^{(2)}(t) & = & -\frac{e{{n}_{0}}}{c}\int {\rm d}{\bf x}\int {\rm d}{{{\bf y}}_{1}}\int {\rm d}{{{\bf y}}_{2}}\mathop \sum \limits_{i} {{A}_{i}}({\bf x},t) \\ {} & {} & \times {{F}_{i}}({\bf x}-{{{\bf y}}_{1}})\mathop \sum \limits_{j} {{\nabla }_{j}}{{F}_{j}}(x-{{{\bf y}}_{2}}) \\ {} & {} & \times {{{\dot{\sigma }}}_{i}}({{{\bf y}}_{1}},t){{\sigma }_{j}}({{{\bf y}}_{2}},t), \\ \end{array}\end{eqnarray} \tag{ 185 }$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} L_{\operatorname{int}}^{(3)}(t) & = & -\frac{e{{n}_{0}}}{c}\int {\rm d}{\bf x}\int {\rm d}{{{\bf y}}_{1}}\int {\rm d}{{{\bf y}}_{2}}\int {\rm d}{{{\bf y}}_{3}}\mathop \sum \limits_{i} {{A}_{i}}({\bf x},t) \\ {} & {} & \times {{F}_{i}}({\bf x}-{{{\bf y}}_{1}})\mathop \sum \limits_{j} \mathop \sum \limits_{k} {{\nabla }_{j}}{{F}_{j}}({\bf x}-{{{\bf y}}_{2}}){{\nabla }_{k}}{{F}_{k}} \\ {} & {} & \times ({\bf x}-{{{\bf y}}_{3}}){{{\dot{\sigma }}}_{i}}({{{\bf y}}_{1}},t){{\sigma }_{j}}({{{\bf y}}_{2}},t){{\sigma }_{k}}({{{\bf y}}_{3}},t), \\ \end{array}\end{eqnarray} \tag{ 186 }$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} L_{\operatorname{int}}^{(4)}(t) & = & -\frac{e{{n}_{0}}}{c}\int {\rm d}{\bf x}\int {\rm d}{{{\bf y}}_{1}}\int {\rm d}{{{\bf y}}_{2}}\int {\rm d}{{{\bf y}}_{3}}\int {\rm d}{{{\bf y}}_{4}} \\ {} & {} & \times \mathop \sum \limits_{i} {{A}_{i}}({\bf x},t){{F}_{i}}(x-{{{\bf y}}_{1}})\mathop \sum \limits_{j} \mathop \sum \limits_{k} \mathop \sum \limits_{l} {{\nabla }_{j}} \\ {} & {} & \times {{F}_{j}}(x-{{{\bf y}}_{2}}){{\nabla }_{k}}{{F}_{k}}(x-{{{\bf y}}_{3}}){{\nabla }_{l}}{{F}_{l}}(x-{{{\bf y}}_{4}}) \\ {} & {} & \times {{{\dot{\sigma }}}_{i}}({{{\bf y}}_{1}},t){{\sigma }_{j}}({{{\bf y}}_{2}},t){{\sigma }_{k}}({{{\bf y}}_{3}},t){{\sigma }_{l}}({{{\bf y}}_{4}},t). \\ \end{array}\end{eqnarray} \tag{ 187 }$$

These formulae show that the interaction processes are nonlocal. The physical origin of this nonlocality is the complex structure of plasmon. Each plasmon is not a point particle, but it has the spatial extension around its center.

The plasmon–photon interaction Lagrangian (174) with the terms of the form (175)–(178) was derived when we use the electron–photon interaction Lagrangian (167) which is linear with respect to the vector potential A(x, t). If the nonlinear optical processes are taken into account, the electron–photon interaction Lagrangian must include also the terms nonlinear with respect to A(x, t). In this case the plasmon-photon interaction Lagrangian contains nonlinear terms describing the interaction processes with the participation of many photons.

The most powerful tool for the theoretical study of plasmonic processes and phenomena is the functional integral technique. In order to apply this technique to the study of plasmon–photon interaction let us extend the reasoning presented in section 3 and substitute

$$\begin{eqnarray*}&&-{\rm i}\frac{\partial }{\partial {\bf x}}\to -{\rm i}\frac{\partial }{\partial {\bf x}}+\frac{e}{c}{\bf A}({\bf x},t),\end{eqnarray*}$$

e being the absolute value of the electron charge, in the expression (62) of the quantum mechanical Hamiltonian of the electron. Then instead of the functional integral Z of the plasmonic field defined by formula (94) we must start from following functional integral of the electron gas interacting with some external transverse electromagnetic field A(x):

$$\begin{eqnarray}\begin{array}{lllllllllllllll} Z & = & \frac{1}{Z_{0}^{\varphi }}\int [D\varphi ]{\rm exp} \left\{ \frac{{\rm i}}{2}\int {\rm d}x\int {\rm d}x^{\prime} \varphi (x)u \right. \\ {} & {} & \left. \times (x-x^{\prime} )\varphi (x^{\prime} ) \right\}\int [D\varphi ]\left[ D\bar{\psi } \right] \\ {} & {} & \times {\rm exp} \left\{ -{\rm i}\int {\rm d}x\int {\rm d}x^{\prime} \bar{\psi }(x)\psi (x)u(x-x^{\prime} )\varphi (x^{\prime} ) \right\} \\ {} & {} & \times {\rm exp} \left\{ {\rm i}\int {\rm d}x\bar{\psi }(x)\left[ {\rm i}\frac{\partial }{\partial {{x}_{0}}}-H\left( -{\rm i}\frac{\partial }{\partial {\bf x}} \right. \right. \right. \\ {} & {} & +\left. \left. \left. {\rm i}\frac{e}{c}{\bf A}(x),{\bf x} \right) \right]\psi (x) \right\}\cdot \\ \end{array}\end{eqnarray} \tag{ 188 }$$

Then instead of formulae (95), (96), (105) and (106) we have following expression

$$\begin{eqnarray}\begin{array}{lllllllllllllll} Z & = & \frac{Z_{0}^{\psi }}{Z_{0}^{\varphi }}\int \left[ D\varphi \right]{\rm exp} \left\{ \frac{{\rm i}}{2} \right.\int {\rm d}x\int {\rm d}x^{\prime} \varphi (x)u \\ {} & {} & \times \left. (x-x^{\prime} )\varphi (x^{\prime} ) \right\}{\rm exp} \left\{ {\rm i}\sum \limits_{n,\,m=0}^{\infty } {{W}^{(n,\,m)}}\left[ \varphi ,{\bf A} \right] \right\}, \\ \end{array}\end{eqnarray} \tag{ 189 }$$

where

$$\begin{eqnarray}&&{{W}^{(0,\,0)}}\left[ \varphi ,{\bf A} \right]=0,\end{eqnarray} \tag{ 190 }$$

$$\begin{eqnarray}&&{{W}^{(n,\,0)}}\left[ \varphi ,{\bf A} \right]={{W}^{(n)}}\left[ \varphi \right],\;n\geqslant 1,\end{eqnarray} \tag{ 191 }$$

$$\begin{eqnarray}&&{{W}^{(0,1)}}\left[ \varphi ,{\bf A} \right]=0,\end{eqnarray} \tag{ 192 }$$

$$\begin{eqnarray}&&{{W}^{(0,\,2)}}\left[ \varphi ,{\bf A} \right]=-\frac{{{e}^{2}}}{2m}\int {\rm d}x\;n({\bf x}){\bf A}{{(x)}^{2}},\end{eqnarray} \tag{ 193 }$$

$$\begin{eqnarray}&&{{W}^{(1,\,1)}}\left[ \varphi ,{\bf A} \right]=0.\end{eqnarray} \tag{ 194 }$$

${{W}^{(n)}}\left[ \varphi \right]$ with n = 1, 2 and 3 were given in formulae (100), (102) and (104). In general, ${{W}^{(n,m)}}\left[ \varphi ,{\bf A} \right]$ is a homogenerous functional of the nth order with respect to the field φ(x) and of the mth order with respect to the field A(x). For example

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{W}^{(1,2)}}\left[ \varphi ,\;{\bf A} \right] & = & \frac{{\rm i}{{e}^{2}}}{2m}\int {\rm d}x\int {\rm d}x^{\prime} \int {\rm d}y\phi (x^{\prime} )u(x^{\prime} -x) \\ {} & {} & \times S(x-y)S(y-z){\bf A}{{(y)}^{2}}, \\ \end{array}\end{eqnarray} \tag{ 195 }$$

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{W}^{(2,2)}}\left[ \varphi ,{\bf A} \right] & = & \frac{{\rm i}{{e}^{2}}}{4m}\int {\rm d}x\int {\rm d}x^{\prime} \int {\rm d}y\int {\rm d}y^{\prime} \int {\rm d}z\varphi (x^{\prime} )u \\ {} & {} & \times (x^{\prime} -x)\varphi (y^{\prime} )u(y^{\prime} -y) \\ {} & {} & \times \left[ S(x-y)S(y-z)S(z-x) \right. \\ {} & {} & +\left. S(y-x)S(x-z)S\left( z-y \right. \right]{\bf A}{{(z)}^{2}}. \\ \end{array}\end{eqnarray} \tag{ 196 }$$

The scalar field φ(x) is expressed in terms of the plasmonic field σ(x,t) by means of formulae (117), (141), (149) and (151).

Thus the effective action

$$\begin{eqnarray}&&I\left[ \varphi ,{\bf A} \right]=\sum \limits_{n,m=1}^{\infty } {{W}^{(n,m)}}\left[ \varphi ,{\bf A} \right]\end{eqnarray} \tag{ 197 }$$

of the interaction between the scalar plasmonic field σ(x, t) and the transverse electromagnetic field A(x, t) has been established. It shows that this interaction is nonlocal. Moreover, it is not instantaneous. From the above-presented formulae it is straightforward to derive the effective action corresponding to any plasmon–photon coupling vertex such as:

• -  
  photon absorption or emission by a plasmon (n = 2, m = 1),
• -  
  decay of a plasmon into two photons, stokes or anti-stokes Raman scattering with the creation or the destruction of a plasmon (n = 1, m = 2),
• -  
  photon–plasmon scattering, annihilation of two plasmons into two photons (n = 2, m = 2) and so on. This will be done in a subsequent work.

In the present work we have reviewed the main results of the rigorous theoretical study of the plasmonic processes in the interacting electron gas. Only general basic postulates (axioms), called also the first principles, of the quantum theory are accepted as the assumptions. Two different methods were applied: the quantum-mechanical canonical quantization method and the functional integral technique. The reasonings of both methods were presented in this review.

The motivation to the research on the subject of the present review was explained in the introduction. The content of section 2 is the presentation on the quantum-mechanical canonical quantization of the plasmonic field. The functional integral method in quantum theory of plasmonic field was elaborated in section 3. The study on the plasmon–photon interaction in the quantum-mechanical canonical theory is the content of section 4. In section 5 the functional integral methods was applied to the study of the plasmonic processes.

From the results of the study by both methods we can firmly conclude that all plasmonic interaction processes in the matter are nonlocal ones, in contrast to the local interaction Hamiltonian accepted in the phenomenological theories as the assumptions. Thus it is worth revising the interaction Hamiltonians accepted in references [5–28], and a lot of works should be done in the future.

However, the contents of the reviewed theoretical works are still not enough for the comparison of the theoretical predictions with the experimental data, because the electron–phonon interaction certainly plays the important role in the physical processes with the participation of plasmons, but until now there was no theoretical work on related subjects. We do hope that the present review will motivate the theoretical research on the contribution of electron–phonon interaction to the plasmonic processes.

We have seen that there exist two related but different methods for the study of the plasmon–photon interacting system: the canonical quantization method of quantum mechanics and the functional integral method of quantum field theory. In the simple case of the homogeneous and isotropic electron gas in bulk conducting solid the results of both methods are consistent. However, even for this simple case, in the framework of the functional integral method we can derive the formula determining the momentum dependence of the plasmon energy (the plasmon dispersion) by means of simple calculations, while in the canonical quantization method of quantum mechanics it is necessary to carry out complicated calculations of the perturbation theory and the renormalization procedure. Moreover, it is very difficult to apply the canonical quantization method of quantum mechanics to the study of many photonic processes and phenomena, while the functional integral technique is a very efficient theoretical tool for the study of all plasmonic processes and phenomena as well as the plasmonic devices and systems.

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 the support.