On quantization of electromagnetic field

  D. A. Arbatsky

March, 2002


In the following six papers relativistic canonical quantization was first described (though, in 2002 it was not named so yet) and applied to quantization of electromagnetic field.

I. Classical electrodynamics


A technique for investigation of classical fields is developed on the base of invariant Hamiltonian formalism. Electromagnetic and scalar fields are considered as particular examples of using the general method. Poisson brackets for these fields are calculated. The necessity of introduction of “non-physical” degrees of freedom for electromagnetic field is explained.

II. Arbitrariness in choice of Lagrangian


Here we show that addition to Lagrangian a divergence of a function does not change the symplectic structure on invariant phase space.

III. Formula for generators of infinitesimal linear canonical transformations


Here we suggest a formula for generators of infinitesimal linear symplectic transformations of invariant phase space. We discuss applications of this formula to classical and quantum field theory. We show the existence of generators of the symmetry group for quantum case.

IV. Theory of field representations


We introduce a notion of induced symplectic representation of the Poincaré group. Classical relativistic fields are considered as such representations. We describe the method of investigation of these fields in the sense of their reducibility. We introduce the notion of the field oscillator as an inducing Hamiltonian system.

V. Vector representation of little Lorentz group for light-like momentum


Using elementary geometric methods we prove the isomorphism of the little Lorentz group for light-like momentum and the group of motions of a Euclidian plane. In accordance with Jordan-Hölder-Noether theorem we perform “reduction” of the real and complex vector representations of this group. We also prove indecomposability of these representations.

VI. Quantization


Here we describe a general method of quantization of linear fields. We introduce a conception of quantization invariant with respect to action of some group. A space of quantum states of relativistic fields is constructed in apparently relativistic-invariant way. A connection with quantization of the field oscillator is established. We substantiate the necessity of using an indefinite scalar product for electromagnetic field. We discuss additional condition for “physically allowed” states of electromagnetic field. We discuss properties of the space of states of electromagnetic field from the point of view of functional analysis. We consider the question about origin of anti-unitary transformations in quantum field theory.

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