March, 2002

Abstract

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.

Abstract

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.

Abstract

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

Abstract

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.

Abstract

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.

Abstract

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.

Abstract

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.