Internal Symmetry Group and Density Matrix of Fields with Spins 0, 1

TL;DR

This paper explores the internal symmetry group U(3,1) of neutral vector fields with spins 0 and 1, analyzing their properties, conservation laws, and the reduction of symmetry in electrodynamics, including the structure of solutions for free particles.

Contribution

It introduces the analysis of the internal symmetry group U(3,1) for fields with spins 0 and 1, and studies its implications in quantized theories and electrodynamics.

Findings

The symmetry transformations are integro-differential in coordinate space.

Quantized theory requires an indefinite metric.

In electrodynamics, U(3,1) reduces to U(2).

Abstract

The internal symmetry group U(3,1) of the neutral vector fields with two spins 0 and 1 is investigated. Massless fields correspond to the generalized Maxwell equations with the gradient term. The symmetry transformations in the coordinate space are integro-differential transformations. Using the method of the Hamiltonian formalism the conservation tensors are found, and the quantized theory is studied. The necessity to introduce an indefinite metric is shown. The internal symmetry group U(3,1) being considered, after the transition to electrodynamics, reduces to the U(2) group. It is shown that the group of dual transformations is the subgroup of the group under consideration. All the linearly independent solutions of the equation for a free particle obtained in terms of the projection matrix-dyads.