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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 in QED

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, 118­173.

. . . 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 . . .


The Feynman-Vernon Influence Functional Approach in QED Quantum electro dynamics

QED Lagrangian

1 L(x) = (x)(i µ µ - m) (x) - Fµ (x)F µ (x) - ej µ (x)Aµ (x) 4 where Fµ (x) = µ Aµ (x) - Aµ (x) j µ (x) = (x) µ (x)

(1)

(2) (3)


The Feynman-Vernon Influence Functional Approach in QED Quantum electro dynamics

^ (x, t) =
p, =1,2

1 2V 1 2V
(f ) p

^p u (p)eipx + c u (-p)e- b ^p ^ u (p)e- bp
ipx

ipx

, ,

(4) (5) (6) (7)

^ (x, t) =
p, =1,2

(f ) p

+ cp u (-p)e ^

ipx

^ ^ ^µ (x, t) = (x, t)µ (x, t) j ^ Aµ (x, t) =
k,=1,2

1 2V
(b) k

^k µ ak eikx + a e- ^

ik x


The Feynman-Vernon Influence Functional Approach in QED Quantum electro dynamics

Second quantization ^ Hf ^ ^p + c cp + bp b ^p ^
k,=1,2

ull

=
p, =1,2



(f ) p

k a ak + ^k ^
(b) k

+e
k,=1,2

i 2 V
(b) k

+ - ^ ^ µ^µ (k, t)ak + µ^µ (k, t)a j j

(8)

where ^µ (k, t) = j+ ^µ (x, t)e j
ikx

dx,

^µ (k, t) = j-

^µ (x, t)e j

-ikx

dx

(9)


The Feynman-Vernon Influence Functional Approach in QED Evolution equation for statistical op erator

Evolution equation for statistical op erator (tf ) ^ ^ ^ (tf ) = U (tf , tin )(tin )U (tf , tin ) ^ ^ (10)

where (tin ) is statistical operator, describing initial state at moment tin , ^ ^ U (tf , tin ) evolution operator. i ^ ^ U (tf , tin ) = T exp[-
t t
f

^ Hf ull ( )d ].

(11)

in

^ where Hf ull : ^ Hf
ull

^ ^ = Hsys + Hf

ield

^ + Hint

(12)


The Feynman-Vernon Influence Functional Approach in QED Holomorphic representation

Evolution equation for density matrix in holomorphic representation |p , k = |p |k
The density matrix:
(f , f , f , f ; tf ) = f , f |(tf )|f , f ^

(13)

The kernel of evolution operator:
^ U (f , f , tf |in , in , tin ) = f , f |U (tf , tin )|in , in

(14)

The evolution equation:
(f , f , f , f ; tf ) =

d2 in d2 in d2 in d2 in â (15)

âU (f , f , tf |in , in , tin )(in , in , in , in ; tin )U (f , f , tf |in , in ; tin )


The Feynman-Vernon Influence Functional Approach in QED Holomorphic representation

Coherent states for electromagetic field ak | ^ = k | , k |a ^
k = k |k ,

k

k

(16)

where k complex value, which describe states k mode of quantum electromagnetic field. These states (| ) are non-orthogonal: 1 |k |2 + |k |2 - 2k 2 There is resolution of the identity operator: k |
k

= k k



exp -

k

.

(17)

|

k

d2 k ^ k | = 1.

(18)


The Feynman-Vernon Influence Functional Approach in QED Holomorphic representation

Grassman states for Dirac field ^p, |p, = b where
p p,

p,

|p, ,



p,

|^ , = bp

p,

|

p,

,

(19)

grassman variable. These states (| ) are non-orthogonal: 1 2
p

|p = p p exp -

p



+ p p - 2

p

p

. (20)

There is resolution of the identity operator: |p d2 p ^ p | = 1. (21)


The Feynman-Vernon Influence Functional Approach in QED The kernel of evolution op erator as a path integral

The kernel of evolution op erator

U (f , f , tf |f , f , tin ) =

D ( )D( )D( )D( )â
f ull

â exp iS where action

( ), ( ), ( ), ( )

,

(22)

S

f ull

( ), ( ), ( ), ( ) =

= Sf ( ), ( ) + Sb [ ( ), ( )] + Sint ( ), ( ), ( ), ( ) . (23)


The Feynman-Vernon Influence Functional Approach in QED The kernel of evolution op erator as a path integral

Action of fermionic field:
tf

S

f

( ), ( ) =
tin

( )( ) - ( )( ) - 2i

(f )

( )( ) d

(24)

Action of bosonic field: Sb [ ( ), ( )] = Action of interaction part: Sint ( ), ( ), ( ), ( ) =
tf tin tf tin

( )( ) - ( )( ) - (b) ( )( ) d 2i

(25)

ej µ (( ), ( ))( ( ) + µ ( ))]d ; µ

(26)


The Feynman-Vernon Influence Functional Approach in QED The Feynman-Vernon influence functional

Evolution of density matrix in paths integral formulation

We have
(f , f , f , f ; tf ) =

d2 in d2 in d2 in d2 in (in , in , in , in ; tin )â ( ), ( ), ( ), ( ) , (27)

âD ( )D( )D( )D( )D ( )D ( )D ( )D ( )â â exp i Sf
ull

( ), ( ), ( ), ( ) - S

f ull


The Feynman-Vernon Influence Functional Approach in QED The Feynman-Vernon influence functional

Fermionic density matrix and influence functional
(f , f ; tf ) = S p
f

=f

(f , f , f , f ; tf ) =

df df D( )D( )D ( )D ( )din din â F [( ), ( )] (28)

â exp i Sf [( ), ( )] - Sf [ ( ), ( )]

where F [( ), ( )] is influence functional of electromagnetic field on fermionic subsystems.
F [( ), ( )] = S p â exp i Sb [ ( ), ( )] + S
int
f

=f

D ( )D( )D ( )D ( )

d2 in d2 in (in , in , in , in ; tin )â
int

( ), ( ), ( ), ( ) - Sb ( ), ( ) - S

( ), ( ), ( ), ( ) (29)

In many cases, we can choose at initial moment t

in

(in , in , in , in ; tin ) = f (in , in ; tin ) â b (in , in ; tin )

(30)


The Feynman-Vernon Influence Functional Approach in QED The Feynman-Vernon influence functional

Influence functional of electromagnetic field
F [( ), ( )] = Sp d2 in d2 in â (31)

f =

f

âUinf l (f , f , tf |in , in , tin )(in , in , in , in ; tin )Uinf l (f , f , tf |in , in ; tin )

where Uinf l (f , f , tf |in , in , tin ) is electromagnetic field transition amplitude from initial state |in to final state |f inducing by external source j : Uinf l (f , f , tf |in , in , tin ) =

D ( )D( ) exp {iSinf l [ ( ), ( ), x( )]}

(32)

where Sinf l ( ), ( ), ( ), ( ) = Sb [ ( ), ( )] + Sint ( ), ( ), ( ), ( ) . 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

Functional integration over electromagnetic field paths

tf t
f

+ - µ jµ ( ) j ( )e i ( - )

Uinf l ( , f , tf |in , in , tin ) = exp

f

e

-i (tf -t

in

) f



in

-e

2 t
in

d d - (33)

tin tf d - if e t
in

- iin e
tin

+ µ jµ ( )e

-i ( -tin )

- µ jµ ( )e

-i (tf - )

d

For multimode field without interaction between modes Uinf l =
k

U

(k) inf l

(34)


The Feynman-Vernon Influence Functional Approach in QED The Feynman-Vernon influence functional

Uinf l (f , f , tf |in , in , tin ) =

=
k,=1,2

exp e

-ik (tf -t

in

) (f ) (in) k k

e2 - 2k V

tf

+ - µ jµ (k, ) j (k, )e ik ( - )

d d -

tin t tf

in

tf e e (in) (f ) + - - ik µ jµ (k, )e-ik ( -tin ) d - ik µ jµ (k, )e-ik (tf - ) d = 2k V 2k V tin tin tf e2 + - e-ik (tf -tin ) f in - µ jµ (k, ) j (k, )eik ( - ) d d - = exp 2k V
k, t
in

tin

t

f

tf + µ jµ (k, )e -ik ( -tin ) d - if

- iin

e 2k V

t

e 2k V

- µ jµ (k, )e tin

-ik (tf - )

in

d

(35)


The Feynman-Vernon Influence Functional Approach in QED Influence functional of electromagnetic field vacuum

Vacuum influence functional
For the case when initial and final states of electromagnetic field are vacuum: 1 in (in ) = in |0 = exp - |in |2 , 2 1 (f ) = 0|f = exp - |f |2 . (36) f 2

We define influence functional of electromagnetic vacuum
F
v ac|v ac

[( ), ( )] =

2 d2 f d f d2 in d2 in f (in , in ; tin )â

= exp -

k,

â (f )Uinf l (f , f , tf |in , in , tin )in (in ) (in )Uinf l (f , f , tf |in , in ; tin ) (f ) = f in f tf 2 e + - µ jµ (k, ) j (k, )eik ( - ) d d + µ jµ+ (k, ) j- (k, )e-ik ( - ) d d 2k V tin tin

(37)


From sum over k to integral:
k



V (2 )3

dk
t
f

-
k t
f

e2 2k V
t

+ - µ nu jµ (k, )j (k, )e ik ( - )

d d =

in

tin



=-
t
f

e2 (2 )3
t



1 2
in

k

µ




+ - jµ (k, )j (k, )e

i ( - )

dxdx dkd d =

tin

e2 =- (2 )3
t

1 2
tin t
in

k

µ jµ (x, )j (x , )e-


ik(x-x ) i ( - )

e

dxdx dkd d =

=-

e2 4 i
t

f



1 (2 )3
in

2 idk k

µ
D
µ



e-

ik(x-x ) i ( - )

e

dk jµ (x, )j (x , )dxdx d d

tin

(x-x , - )

where D
1

µ

(x - x , - ) is photon propagator 1 .

V.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii Quantum Electro dynamics


The Feynman-Vernon Influence Functional Approach in QED Influence functional of electromagnetic field vacuum

F

v ac|v ac

e2 [( ), ( )] = exp - 4 i

tf



jµ (x, )D
t
in

µ

(x - x , - )j (x , )dxdx d d -

tin µ

e2 - 4 i
t

tf

jµ (x, )D
in

(x - x , - )j (x , )dxdx d d

(38)

t

in

For tf , tin - we have relativistic invariant influence functional of electromagnetic vacuum: F
v ac|v ac

[( ), ( )] = exp -

e2 4 i

jµ (x)D

µ

(x - x )j (x ) + jµ (x)D

µ

(x - x )j (x ) d4 xd4 x (39)


The Feynman-Vernon Influence Functional Approach in QED Influence functional of electromagnetic field vacuum

So we have effective non-local Lagrangian e2 L = (x)(i µ - m) (x) + jµ (x) 4
µ

Dµ (x - x )j (x )dx

(40)


Thanks for your attention!