Functional Approach for the Description of Vacuum Influence on Electron States

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 represent the equations of the movement for the fermionic field and expression for energy of an electron taking into account influence of a vacuum of the electromagnetic field.

Coherent states for electromagetic field where α kλ complex value, which describe states k mode of quantum electromagnetic field. These states (|α ) are non-orthogonal: There is resolution of the identity operator: Grassman states for Dirac field where θ p,σ grassman variable. These states (|θ ) are non-orthogonal: There is resolution of the identity operator:

Grassman variables properties
For two grassman variables θ and η Then and Evolution equation for density matrix in holomorphic representation |θ pσ , α kλ = |θ pσ ⊗ |α kλ The density matrix: The kernel of evolution operator: The evolution equation: The kernel of evolution operator where action

Evolution of density matrix in paths integral formulation
We have Fermionic density matrix and influence functional where F [θ(τ ), θ (τ )] is influence functional of electromagnetic field on fermionic subsystems.

Functional integration over electromagnetic field paths
For multimode field and two polarizations without interaction between modes

Vacuum influence functional
For the case when initial and final states of electromagnetic field are vacuum: We define influence functional of electromagnetic vacuum Fermionic density matrix with electromagnetic vacuum influence Influence functional in coordinate representation For t f → ∞, t in → −∞ we have relativistic invariant influence functional of electromagnetic vacuum: Fermionic density matrix evolution in coordinate representation where S f ull ψ(x), ψ(x) is given by where Lagrangian density is Quantum transition amplitude of electron in this model presents as path integral We find an equation for ψ(x) by the use of quasiclassal approximation [Feynman] δS f ull ψ(τ ), ψ(τ ) = 0 We consider an electron in a bound state (with potential energy U ( r)). We find an equation for ψ(x) by the use of (54) in the following form The equation (55) is nonlinear.
We write the equation in the form 2 We present ψ( r, t) = e − ı Et ψ E ( r). Then The equation (57) allows us to find the energy E and wave function ψ E (r) of an electron with the influence of vacuum fluctuations.
In nonlinear equation (57) the second term is small. Then (57) we present as Eψ E ( r) = c αp + βmc 2 + U ( r) ψ E ( r) For a spherically symmetric potential U = − Ze 2 r we have E nj and ψ nj where j = l + 1/2, l orbital quantum number, n principal quantum number.