LORENTZ AND GAUGE INVARIANT SELF-LOCALIZED SOLUTION OF THE

QED EQUATIONS

I.D.Feranchuk and S.I.Feranchuk

Belarusian University, Minsk

10th International School-Seminar

GOMEL, 15-26 of July, 2009

“Observed” QED

Maxwell-Dirac equations; “physical” electron

“Fundamental” QED with the well-defined Hamiltonian

that is point “bare” electron

)(ˆ)(ˆ)(ˆˆ0int rrrrde

Local (!)

Result of renormalization00; mmee

1;;137/14/2 cme

??;4/)( 02

00 me

QED renormalization:

Any observed values is expressed only through e and m and

can be calculated as PT series as a power of e

MATHEMATICAL CONTRADICTIONS IN THE STANDARD RENORMALIZATION SCHEME:1. It doesn’t allow to calculate the fundamental parameters of QED

(characteristics of the “bare” electron): they are infinite.2. It requires additional parameter in the Hamiltonian (cut-off momentum L ) or change of the Hamiltonian QED (counter-terms).3. It leads to the non-locality of the fundamental QED4. The theory is not self-consistent: it is developed as the series as a power

of constant that is proved to be infinite.

P.A-M.Dirac: “ the calculation rules of QED are badly adjusted with thelogical foundations of quantum mechanics and they cannot be considered as the satisfactory solution of the difficulties”. R.Feynmann: “It is simply a way to sweep the difficulties under the rug”.

0e

FORMULATION OF THE PROBLEM

Is it possible to develop the renormalization scheme of QED with the following conditions ?

• - mathematically closed and self-consistent; • - relativistic invariant;• - without any additional parameters;• - allows one to calculate the certain values ;• - renormalized without any deformation of the Hamiltonian

00 mиe

Non-perturbative approaches in QED

1. Solution of Schwinger-Dyson equation for approximation and (partial summation) : - there is singularity ; - cut-off momentum is necessary.2. Quenched QED – analog of Tamm-Dankov approximation – calculation on the limit number of states: - includes non-defined parameters ; - cut-off momentum is necessary.3. Model Nambu-Jona-Lasino – analog of the BCS model – modification of system ground state with due to strong

interaction : - cut-off momentum is necessary; - there is the modification of the fundamental QED Hamiltonian.4. Model self-field QED (Barut A.O. et al) – QED with the electromagnetic potential expressed through spinor field - the closest analog to our approach, however, the solution of the self- consistent equations was not found

)( p

)0(3/ c

cL

NNe;

0 0m

Self-localized one-particle excitations

For the standard PT in the quantum field theory: zero orderapproximation - asymptotically free states – plane waves !

However, there are examples with the localized one-particle state asthe zero order approximation! They can’t be obtained by usual PT !

1. One-dimensional soliton

( ) 0ˆ x

2. Polaron problem

0; ( )k eka u r - localized wave function

Key points

• Does the solution in the form of the self-localized state (SLS) really exist for

the QED Hamiltonian ?• How can SLS be interpreted?• Can SLS be used as the zero order basis for PT?

QED Hamiltonian (Coulomb gauge)2

0 0 0

1ˆˆˆ ˆ ˆ ˆ ˆˆ: { ( )[ ( ( )) ] ( ) ( ) ( ) ( ( )) }2

H dr r p e A r m r e r r r

( ) :ˆ kk

k n

1

ˆ ˆ ˆˆ ˆ( ) [ ( ) ( ) ( ) ( )]2

r r r r r

3 2ˆ ( ) { }

(2 )ipr ipr

ps ps ps p ss

dpr a u e b v e

( )1ˆ( ) [ ]2

ikr ikr

k kk

A r c e c eek

( ) 1 2ˆ k k kk k c cn

ˆ ( )ˆ( )

| |

rr dr

r r

0m

0 0( ), 0e e

Mass and charge of the “bare” electron are indefinite parameters of QED!

Trial state vector for variational PT

( ) ( )1 1 1 10 0 0 0 0 0PT PT

Ps Pse a p b

One-particle excitation (plane waves) for usual perturbation theory:

(0)1 1 ( ; ) { } 0 0qs qs qs qs qs qsU V A dq U a V b A

One-particle wave packet for variational perturbation theory:

ˆ; ; | ( ) ; ( )

ˆ0;0; | | 0;0; ; 1, 2,3

qs qs qs qsA U V r U V A r

A A A A

2 2[ ] 1qs qss

dq U V (0) (0) 2 21 1 0

ˆ [ ] 0qs qss

Q e dq V U q

(0) (0) 2 21 1

ˆ [ ] 0qs qss

P dqq U V P

VARIATIONAL EQUATIONS

0 0 0{( ( )) ) ( ) (0)} ( ) 0i e A m e r E r

0 0 0{( ( )) ) ( ) (0)} ( ) 0ci e A m e r E r

;A j 0( ) [ ( ) ( ) ( ) ( )]4

c ce drr r r r r

r r

[ ( ) ( ) ( ) ( )] 1c cdr r r r r

3 2( )

(2 )iqr

qs qss

dqr U u e

3 2( )

(2 )c iqr

qs qss

dqr V v e

( )0( ) ( ) ( ) ( )i rr e r r e A r

( )( ) ( )c i r cr e r

SEPARATION OF VARIABLES

( )

( )jlM

jlMjl M

g r

if r

1

1

( )

( )jlMc

jlMjl M

if r

g r

0 0

( ) 1( ) ( ( ))( ) 0

d rgrg E m e r rf

dr r

0 0

( ) 1( ) ( ( ))( ) 0

d rfrf E m e r rg

dr r

11 0 0 1

( ) 1( ) ( ( ))( ) 0

d rgrg E m e r rf

dr r

11 0 0 1

( ) 1( ) ( ( ))( ) 0

d rfrf E m e r rg

dr r

22 2 2 20

1 12

2[ ]

4

ed df g f g

dr r dr

BOUNDARY CONDITIONS AND SCALING VARIABLES

( )4r

qr

r

2 2 2 2 201 1 1 1 1 1 1 10

[ ( ) ( ) ( ) ( )]4

er dr f r g r f r g r

r

2 2 2

0

1[ ( ) ( )]

1r dr f r g r

C

2 2 2

1 10[ ( ) ( )]

1

Cr dr f r g r

C

20

0 0 0 0 0( ) ( )4

ex rm E m e r m x

0 0 1 0 1 1 0 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )u x m rg r v x m rf r u x m rg r v x m rf r

0 0

1 1

1 1u u v v

C C

1 0 1 01 1

C Cu v v u

C C

DIMENSIONLESS EQUATIONS

00 0

1(1 ( )) 0

duu x v

dx x 0

0 0

1(1 ( )) 0

dvv x u

dx x

00 0 0 00

( )1 1( ) ( ) ( ) [ ( )]

1

x

x

yCx x x dy dy y

C y x

2 2 2 20 0 0 0 00

[ ( ) ( )] 1 ( ) ( ) ( )dx u x v x x u x v x

0 0

1 1 1(0) [ ]

1 2 1

C CE m T

C C

2 20 00 0 0 0 0 00

[( ) 2 ( )]u v

T dx u v v u u vx

2 2

0 0 00( )dx u v

0 0

1

1

Ca x m r

C

DIMENSIONLESS SOLUTIONS

0 00

3 531; (0)a

a E I m

0

1[ ] 0.749

2I T a

22

0 4

qa

Non-trivial SLS for QED exists !

PHYSICAL INTERPRETATION (1)

q e1) Let 2

0 0; 1707a

1 , ,( ) { } 0 0 ( );qs qsq P s q P sP dq U a V b r A

2) Consider SLS with nonzero total momentum

2 21 1 , ,

ˆ( ) ( ) [ ] 0q P s q P s

s

P P P dqq U V P

0 0{( ) ( ) ( )} ( ) 0P i m e r P E P r P

0 0{( ) ( ) ( )} ( ) 0cP i m e r P E P r P

PHYSICAL INTERPRETATION (2)

( ) ( ) ( ) ( ) ( )cr P L P r K P r

( ) ( ) ( ) ( ) ( )c cr P K P r L P r

2 2 1L K 2 2(0)e pE E P

2 2 2 2

(0)

( (0)) ( (0))

e e e

e e

E EPL K

P E E P E E

0 0 00 0

(0) ; 646a a

m E I m m m mI

3) Muon as the excited SLS

exp( ) 206m

m ( ) 54D

m

m

PHYSICAL INTERPRETATION (3)

0 1 0 1

0 0

( ) 2422loc

m a I I I Im I I

1 1 1

1[ ]

2I T a 1 0.19; ( ) 181loc

mI

m

4) Gauge invariance

' [ ]r r e r

[ ]r e r

0( ) ( )r e A r 1

;2

eg

0

1

3g e J

2 2

0

( )[ ( ) ( )]J xA x u x v x dx

2 2

10

1 [ ( ) ( )]( ) [ ( ) ( )] .

x

x

u y v yA x y u y v y dy x dy

x y

1

81

3aJ

1 0.907

PHYSICAL INTERPRETATION (4)

5) Consideration of the interaction with the transversal quantum field by means of the perturbation theory

1 11 1ˆ( ) ( ) ( ) 0 ( ) ( )IP k k P P K P kP PH

01 1( ) { ( )( ) ( ) ( )( ) ( )}

2c c ikre

P k dr r r P r r P ee eP Pk

02 1k m GeV If the vertex function reduces to weak coupling QED with “physical” charge

PHYSICAL INTERPRETATION (5)

11

1

( ) ( )( ) ( )[ ]

( )( )2

e Pe ePL P L PE mE P mk P

If 02k m40

2 2 20

( )( 4 )2

meP k

k mk

In a one loop approximation there is no Landau pole !

220

20

4 1 ln( )1 ln33

c amIm

1 0.993c

PHYSICAL INTERPRETATION (5)

SIMPLE SCALE ESTIMATION

PHYSICAL INTERPRETATION (6)

ne0

ne0

20

00 )(

2e

mmm

00

1)(

ennee

)0(ee rr

RENORMALIZED PERTURBATION THEORY

Interaction with the transversal electromagnetic field can be considered withperturbation theory on the parameter

1

0 Change of the vortex function:

0( , ) , ,2

ep k p k L

k

CONCLUSIONS

1. Self-localized state (SLS) for the one-particle excitation for QED with the strong ”bare” coupling is found out of the perturbation theory. 2. Lorentz invariance of SLS is proved.3. Physical interpretation of SLS as the “physical” electron leads to the finite

values of the “bare” electron mass and charge.4. Condition of the gauge invariance of SLS leads to the charge quantization

condition that is the same form as for the Dirac’s monopole.5. Reasonable value for the μ-meson mass can be calculated if the latter

considered as the excited SLS. 6. If SLS is considered as the one-particle excitation the rest part of the

interaction can be taken into account by means of the perturbation theory with the renormalized weak coupling.

Detailed calculations in arxiv:math-ph/0605028; hep-th/0309072

THANK YOU FOR THE ATTENTION !!!