The Feynman-Vernon Influence Functional Approach in QED

The Feynman-Vernon Influence Functional Approach in QED Mark Shleenkov, Alexander Biryukov Samara State University General and Theoretical Physics Department The XXII International Workshop High Energy Physics and Quantum Field Theory June 24 July 1, 2015 Samara, Russia

The Feynman-Vernon Influence Functional approach R.P. Feynman, F.L. Vernon, Jr. The Theory of a General Quantum System Interacting with a Linear Dissipative System, Annals of Physics 24, 118 173. «... It is shown that the effect of the external systems in such a formalism [paths integral formalism] can always be included in a general class of functionals (influence functionals) of the coordinates of the system only...»

Quantum electrodynamics QED Lagrangian where L(x) = ψ(x)(ıγ µ µ m)ψ(x) 1 4 F µν(x)f µν (x) ej µ (x)a µ (x) (1) F µν (x) = µ A µ (x) ν A µ (x) (2) j µ (x) = ψ(x)γ µ ψ(x) (3)

Quantum electrodynamics ˆψ(x, t) = ˆ ψ(x, t) = p,σ=1,2 p,σ=1,2 Â µ (x, t) = 1 2V ω p (f) 1 2V ω (f) p k,λ=1,2 (ˆbpσ u σ (p)e ıpx + ĉ pσu σ ( p)e ıpx ), (4) (ˆb pσ u σ (p)e ıpx + ĉ pσ u σ ( p)e ıpx ), (5) ĵ µ (x, t) = ˆψ(x, t)γ µ ˆψ(x, t) (6) 1 ) ε µ 2V ω (b) λ (â kλ e ıkx + â kλ e ıkx (7) k

Quantum electrodynamics Second quantization where Ĥ full = p,σ=1,2 +e ĵ µ + (k, t) = k,λ=1,2 ω (f) p (ˆb pσˆbpσ + ĉ pσĉ pσ ) + ı 2ω (b) k V k,λ=1,2 ω (b) k â kλâkλ+ ( ) ε µ λĵ+ µ (k, t)â kλ + ε µ λ ĵ µ (k, t)â kλ ĵ µ (x, t)e ıkx dx, ĵµ (k, t) = (8) ĵ µ (x, t)e ıkx dx (9)

Evolution equation for statistical operator Evolution equation for statistical operator ˆρ(t f ) ˆρ(t f ) = Û(t f, t in )ˆρ(t in )Û (t f, t in ) (10) where ˆρ(t in ) is statistical operator, describing initial state at moment t in, Û(t f, t in ) evolution operator. where Ĥfull: Û(t f, t in ) = ˆT exp[ ı tf t in Ĥ full (τ)dτ]. (11) Ĥ full = Ĥsys + Ĥfield + Ĥint (12)

Holomorphic representation Evolution equation for density matrix in holomorphic representation θ pλ, α kλ = θ pλ α kλ The density matrix: The kernel of evolution operator: The evolution equation: ρ(α f, θ f, α f, θ f; t f ) = θ f, α f ˆρ(t f ) θ f, α f (13) U(α f, θ f, t f α in, θ in, t in ) = θ f, α f Û(t f, t in ) θ in, α in (14) ρ(α f, θ f, α f, θ f; t f ) = d 2 α in d 2 θ in d 2 α in d 2 θ in U(α f, θ f, t f α in, θ in, t in )ρ(α in, θ in, α in, θ in; t in )U (α f, θ f, t f α in, θ in; t in ) (15)

Holomorphic representation Coherent states for electromagetic field â kλ α kλ = α kλ α kλ, α kλ â kλ = α kλ α kλ, (16) where α kλ complex value, which describe states k mode of quantum electromagnetic field. These states ( α ) are non-orthogonal: { α k λ α kλ = δ k kδ λ λ exp 1 ( α 2 k λ 2 + α kλ 2 2α k λ α ) } kλ. (17) There is resolution of the identity operator: α kλ α kλ d2 α kλ = ˆ1. (18)

Holomorphic representation Grassman states for Dirac field ˆbp,σ θ p,σ = θ p,σ θ p,σ, θ p,σ ˆb p,σ = θ p,σ θ p,σ, (19) where θ p,σ grassman variable. These states ( θ ) are non-orthogonal: { θ p σ θ pσ = δ p pδ σ σ exp 1 ( θ p 2 σ θ p σ + θ pσθ pσ 2θ p σ pσ) } θ. (20) There is resolution of the identity operator: θ pσ θ pσ d2 θ pσ = ˆ1. (21)

The kernel of evolution operator as a path integral The kernel of evolution operator U(αf, θ f, t f α f, θ f, t in ) = Dα (τ)dα(τ)dθ(τ)dθ(τ) exp { ıs full [ α (τ), α(τ), θ(τ), θ(τ) ]}, (22) where action S full [ α (τ), α(τ), θ(τ), θ(τ) ] = = S f [ θ(τ), θ(τ) ] + Sb [α (τ), α(τ)] + S int [ α (τ), α(τ), θ(τ), θ(τ) ]. (23)

The kernel of evolution operator as a path integral Action of fermionic field: ( ) [ ] tf θ(τ)θ(τ) θ(τ) θ(τ) S f θ(τ), θ(τ) = ω (f) θ(τ)θ(τ) dτ (24) 2ı Action of bosonic field: S b [α (τ), α(τ)] = Action of interaction part: tf t in t in S int [ α (τ), α(τ), θ(τ), θ(τ) ] = ( α (τ)α(τ) α (τ) α(τ) tf 2ı ) ω (b) α (τ)α(τ) dτ (25) t in ej µ (θ(τ), θ(τ))(ε µα (τ) + ε µ α(τ))]dτ; (26)

The Feynman-Vernon influence functional Evolution of density matrix in paths integral formulation We have ρ(α f, θ f, α f, θ f; t f ) = d 2 α in d 2 θ in d 2 α in d 2 θ in ρ(α in, θ in, α in, θ in; t in ) Dα (τ)dα(τ)dθ(τ)dθ(τ)dα (τ)dα (τ)dθ (τ)dθ (τ) { [ exp ı (S full α (τ), α(τ), θ(τ), θ(τ) ] [ ])} S full α (τ), α (τ), θ (τ), θ (τ), (27)

The Feynman-Vernon influence functional Fermionic density matrix and influence functional ρ(θ f, θ f ; t f ) = Sp αf =α ρ(α f f, θ f, θ f, α f ; t f ) = dθ f dθ f Dθ(τ)Dθ(τ)Dθ (τ)dθ (τ)dθ indθ in { ( )} exp ı S f [θ(τ), θ(τ)] S f [θ (τ), θ (τ)] F [θ(τ), θ (τ)] (28) where F [θ(τ), θ (τ)] is influence functional of electromagnetic field on fermionic subsystems. F [θ(τ), θ (τ)] = Sp αf =α Dα (τ)dα(τ)dα (τ)dα (τ) d2 α in d 2 α in f ρ(α in, θ in, α in, θ in; t in) { exp ı (S b [α [ (τ), α(τ)] + S int α (τ), α(τ), θ(τ), θ(τ) ] [ S b α (τ), α (τ) ] [ ])} S int α (τ), α (τ), θ (τ), θ (τ) In many cases, we can choose at initial moment t in ρ(α in, θ in, α in, θ in; t in ) = ρ f (θ in, θ in; t in ) ρ b (α in, α in; t in ) (30) (29)

where S infl [ α (τ), α(τ), θ(τ), θ(τ) ] = S b [α (τ), α(τ)] + S int [ α (τ), α(τ), θ(τ), θ(τ) ]. In general, influence functional (26) describes the influence (action) of electromagnetic field on fermionic field. The Feynman-Vernon Influence Functional Approach in QED The Feynman-Vernon influence functional Influence functional of electromagnetic field F [θ(τ), θ (τ)] = Sp αf =α f d 2 α in d 2 α in U infl (α f, θ f, t f α in, θ in, t in )ρ(α in, θ in, α in, θ in; t in )U infl(α f, θ f, t f α in, θ in; t in ) (31) where U infl (αf, θ f, t f α in, θ in, t in ) is electromagnetic field transition amplitude from initial state α in to final state αf inducing by external source j: U infl (αf, θ f, t f α in, θ in, t in ) = Dα (τ)dα(τ) exp {ıs infl [α (τ), α(τ), x(τ)]} (32)

The Feynman-Vernon influence functional Functional integration over electromagnetic field paths U infl (αf, θ f, t f α in, θ in, t in ) = exp e ıω(t f t in) αf α in e 2 ıα in e For multimode field without interaction between modes t f t in t f τ t in t in ε µ j + µ (τ)e ıω(τ tin) dτ ıα f e ε µ j µ + (τ)ε ν jν (τ )e ıω(τ τ ) dτdτ t f t in ε µ jµ (τ)e ıω(tf τ) dτ (33) U infl = k U (k) infl (34)

The Feynman-Vernon influence functional = k,λ=1,2 ıα (in) kλ exp e ıω k(t f t in) α (f) kλ α(in) kλ e2 2ω k V e 2ωk V k,λ t f t in t f τ t in t in ε µ λ j+ µ (k, τ)e ıωk(τ tin) dτ ıα (f) kλ = exp e ıω k(t f t in) αf α in e2 2ω k V e ıα in 2ωk V t f t in t f τ t in t in U infl (α f, θ f, t f α in, θ in, t in ) = ε µ λ j+ µ (k, τ)ε ν λ j ν (k, τ )e ıω k(τ τ ) dτdτ e 2ωk V ε µ λ j+ µ (k, τ)e ıωk(τ tin) dτ ıαf e 2ωk V t f t in ε µ λ j µ (k, τ)e ıω k(t f τ) dτ = ε µ λ j+ µ (k, τ)ε ν λ j ν (k, τ )e ıω k(τ τ ) dτdτ t f t in ε µ λ j µ (k, τ)e ıω k(t f τ) dτ (35)

Influence functional of electromagnetic field vacuum Vacuum influence functional For the case when initial and final states of electromagnetic field are vacuum: φ in (α in ) = α in 0 = exp { 12 } α in 2, φ f(α f ) = 0 α f = exp { 12 } α f 2. (36) We define influence functional of electromagnetic vacuum = exp k,λ e 2 2ω k V t in t in F vac vac [θ(τ), θ (τ)] = d 2 α f d 2 α f d 2 α in d 2 α in ρ f (θ in, θ in; t in) φ f (α f )U infl (αf, θ f, t f α in, θ in, t in)φ in(α in)φ in(α in)u infl(α f, θ f, t f α in, θ in; t in)φ f (α f ) = t f τ ( ε µ λ j+ µ (k, τ)ε ν λ jν (k, τ )e ıω k(τ τ ) dτdτ + ε µ λ j + µ (k, τ)ε ν λ j ν (k, τ )e ıω k(τ τ ) dτdτ ) (37)

From sum over k to integral: V (2) dk 3 k = e2 4ı = e2 (2) 3 t f τ t in t in = e2 (2) 3 t f τ t in t in k t f τ t in t in 1 2ω k [ λ e 2 2ω k V 1 2ω k ε µ λ ε ν λ t f τ t in t in [ ] λ [ λ ε µ λ ε ν λ ] ε µ λ εnu λ [ ( ) ] 1 2ıdk (2) 3 ε µ λ ω ε ν λ e ık(x x ) e ıω(τ τ ) dk k λ }{{} D µν (x x,τ τ ) where D µν (x x, τ τ ) is photon propagator 1. 1 V.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii Quantum Electrodynamics ] j + µ (k, τ)j ν (k, τ )e ıω k(τ τ ) dτdτ = j + µ (k, τ)j ν (k, τ )e ıω(τ τ ) dxdx dkdτdτ = j µ (x, τ)j ν (x, τ )e ık(x x ) e ıω(τ τ ) dxdx dkdτdτ = j µ (x, τ)j ν (x, τ )dxdx dτdτ

Influence functional of electromagnetic field vacuum F vac vac [θ(τ), θ (τ)] = exp e2 4ı e2 4ı t f τ t in t in t f τ t in t in j µ (x, τ)d µν (x x, τ τ )j ν (x, τ )dxdx dτdτ j µ(x, τ)d µν (x x, τ τ )j ν(x, τ )dxdx dτdτ (38) For t f, t in we have relativistic invariant influence functional of electromagnetic vacuum: F vac vac [θ(τ), θ (τ)] = exp { e2 4ı (jµ (x)d µν (x x )j ν (x ) + j µ(x)d µν (x x )j ν(x ) ) d 4 xd 4 x } (39)

Influence functional of electromagnetic field vacuum So we have effective non-local Lagrangian L = ψ(x)(ıγ µ µ m)ψ(x) + e2 4 j µ(x) D µν (x x )j ν (x )dx (40)

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