The Feynman-Vernon Influence Functional Approach in QED

In the path integral approach we describe evolution of interacting electromagnetic and fermionic fields by the use of density matrix formalism. The equation for density matrix and transitions probability for fermionic field is obtained as average of electromagnetic field influence functional. We obtain a formula for electromagnetic field influence functional calculating for its various initial and final state. We derive electromagnetic field influence functional when its initial and final states are vacuum. We present Lagrangian for relativistic fermionic field under influence of electromagnetic field vacuum.


Introduction
At the present time the development of non-perturbative methods of non-linear dynamics description is actual for quantum systems interacting with strong electromagnetic field [1][2][3][4][5].One of these methods is Feynman-Vernon influence functional approach in the path integral formalism [6].This method development was presented in many papers [7][8][9].In the Feynman-Vernon influence functional approach the main point is its calculation for the model of investigated system.In this paper we calculate Feynman-Vernon influence functional of quantum electromagnetic field vacuum on fermionic field.

The Feynman-Vernon Influence Functional of quantum electromagnetic field on fermionic field
The QED Lagrangian for a spin-1/2 field interacting with the electromagnetic field is given by where γ μ are Dirac matrices, ψ(x) is a bispinor field of spin-1/2 particles (e.g.electron-positron field), A μ (x) is electromagnetic four-potential; is the electromagnetic field tensor, is four-current.We use natural units system c = = 1 in formulas ( 1)-(3) and further.By the use of the second quantization formalism we present fields as field operators ψ(x, t), ψ(x, t), ĵμ (x, t), Âμ (x, t) where x and t describe coordinate of 3D space and time moment.Here we introduce the creation and annihilation operators: where â † kλ and âkλ are creation and annihilation operators of photon with wave vector k and polarization (the eigenvalue of the z-component of the photon spin) λ = ±1, ε μ λ is unitary 4-vector of polarization and ε μ * λ ε λμ = −1; b † pσ and bpσ are creation and annihilation operators of electron with wave vector p and polarization σ = ± 1 2 ; ĉ † pσ and ĉpσ are creation and annihilation operators of positron with wave vector p and polarization σ = ± 1 2 , V is volume of space.For evolution of the interacting systems we construct Hamiltonian of full system Ĥ f ull on the base of operators â † kλ , âkλ , b † pσ , bpσ , ĉ † pσ , ĉpσ . where -Hamiltonian of Dirac field; -Hamiltonian of electromagnetic field; -interaction part of Hamiltonian.The evolution equation of statistical operator ρ is the following where ρ(t in ) is statistical operator, describing initial state at moment t in and Û(t f , t in ) is operator of evolution from initial time moment t in to final time moment t f For density matrix calculation we need to formulate equation (13) in functional representation.We introduce coherent representation |α kλ for electromagnetic field by the following way where b p,σ and c p,σ -grassman variable.
In holomorphic representation |a kλ , b pσ , c pσ = |a kλ ⊗ |b pσ ⊗ |c pσ we present the evolution equation in the following where the density matrix: For simplification we relabel variables α kλ and θ pσ to α and θ.

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Be the use of amplitude transition properties [10], we present the kernel of evolution operator where full system action is the following where is the action S f of fermionic field, is the action S b of bosonic field, is the action S int of interaction part.Using (24)-(29) we present equation ( 22) for density matrix evolution in the following form: We can choose at initial moment t in This is true for many models.We consider the quantum transition from pure fermionic field state |n at initial time moment t in to pure fermionic field state |m at time moment t f : and to coherent state |χ at time moment The transition probability P(m, χ, t f ; n, ζ, t in ) is determined from (30) by the following formula By the use of ( 30)-(38) we present the transition probability in the path integral representation Eq. ( 40) for quantum transition probability we present in the following form where is influence functional of electromagnetic field on fermionic subsystems.QUARKS-2016 is electromagnetic field transition amplitude from initial state |a in to final state |a f .
is action of fermionic field under electromagnetic field influence.It describe electromagnetic field influence on fermionic field.The influence functional (42) present the influence of one mode of electromagnetic field with wave vector k and polarization λ.
We note it by index The influence functional of all electromagnetic field modes calculate as production of one mode influence functionals: For calculating influence functional by eq. ( 42) we need to calculate amplitude

Electromagnetic field influence functional calculation
In this section we calculate functional For this we present equation (43) as finite-multiple integral [9]. where time interval t is spliting on equal intervals Δτ.So nΔτ = t f and a * n+1 = a * f , a 0 = a in .We integrate through intermediate variables a 1 , . . ., a n and obtain the following result in continual limit Δt → 0, n → ∞: Now we calculate the influence functional of electromagnetic vacuum.For this we use equation ( 42) and specify initial and final state of electromagnetic field as vacuum: By the use of ( 42) and ( 48),(52) we calculate the influence functional of one mode k and λ electromagnetic field on one mode p and σ fermionic field: where functionals S in f vackλ , S in f vackλ is determined by (51).We define influence functional of miltimode electromagnetic vacuum using (45), ( 51) and (53). where functional S in f vac has the same structure as S in f vac with variables b * (τ), b (τ), c * (τ), c (τ) i.e. j + μ , j − ν .
In equation ( 59) we transform iterated integral to double integral.Then we consider the limit t f → ∞, t in → −∞.Now we obtain the action S in f vac [ψ(x), ψ(x)]: We transform equation (58) in the same form.So the effective action (58) for action of fermionic field under electromagnetic field vacuum influence is presenting in the following S f ullvac ψ(x), ψ(x) = L f ullvac (γ μ ∂ μ ψ(x), ψ(x))dx, where is density of effective Lagrangian, is density of Lagrangian, which describe interaction of spinor field and vacuum of electromagnetic field.
The transition amplitude (57) is equivalent to S-matrix.In addition it contain contribution of electromagnetic vacuum influence on fermionic field dynamics:  QUARKS-2016 a in , b in , c in , t in and to specify χ(a), ζ(a).