LORENTZ AND GAUGE INVARIANT SELF-LOCALIZED SOLUTION OF THE QED EQUATIONS I.D.Feranchuk and S.I.Feranchuk Belarusian University, Minsk 10 th International School-Seminar GOMEL, of July, 2009

“Observed” QED Maxwell-Dirac equations; “physical” electron “Fundamental” QED with the well-defined Hamiltonian that is point “bare” electron Local (!) Result of renormalization

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 QED 4.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 the logical 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”.

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

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

Self-localized one-particle excitations For the standard PT in the quantum field theory: zero order approximation - asymptotically free states – plane waves ! However, there are examples with the localized one-particle state as the zero order approximation! They can’t be obtained by usual PT ! 1. One-dimensional soliton 2. Polaron problem - 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) Mass and charge of the “bare” electron are indefinite parameters of QED!

Trial state vector for variational PT One-particle excitation (plane waves) for usual perturbation theory: One-particle wave packet for variational perturbation theory:

VARIATIONAL EQUATIONS

SEPARATION OF VARIABLES

BOUNDARY CONDITIONS AND SCALING VARIABLES

DIMENSIONLESS EQUATIONS

DIMENSIONLESS SOLUTIONS Non-trivial SLS for QED exists !

PHYSICAL INTERPRETATION (1) 1) Let 2) Consider SLS with nonzero total momentum

PHYSICAL INTERPRETATION (2) 3) Muon as the excited SLS

PHYSICAL INTERPRETATION (3) 4) Gauge invariance

PHYSICAL INTERPRETATION (4) 5) Consideration of the interaction with the transversal quantum field by means of the perturbation theory If the vertex function reduces to weak coupling QED with “physical” charge

PHYSICAL INTERPRETATION (5) If In a one loop approximation there is no Landau pole !

PHYSICAL INTERPRETATION (5)

SIMPLE SCALE ESTIMATION

PHYSICAL INTERPRETATION (6)

RENORMALIZED PERTURBATION THEORY Interaction with the transversal electromagnetic field can be considered with perturbation theory on the parameter Change of the vortex function:

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/ ; hep-th/

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