Questions in the Theory of the (1,0)+(0,1) Quantized Fields

TL;DR

This paper explores the mathematical structure of antisymmetric tensor fields in quantum field theory, establishing their relation to Weinberg's formalism, proposing a new scalar Hermitian Lagrangian, and demonstrating their connection to Maxwell's equations.

Contribution

It introduces a novel scalar Hermitian Lagrangian for Weinberg's theory and clarifies the relationship between antisymmetric tensor fields and the (1,0)+(0,1) Lorentz group representation.

Findings

Mapped antisymmetric tensor fields to Weinberg's bispinor fields

Proposed a new scalar, Hermitian Lagrangian for the Weinberg theory

Showed that massless Weinberg-Tucker-Hammer equations encompass Maxwell's equations

Abstract

We find a mapping between antisymmetric tensor matter fields and the Weinberg's 2(2j+1)- component "bispinor" fields. Equations which describe the j=1 antisymmetric tensor field coincide with the Hammer-Tucker equations entirely and with the Weinberg ones within a subsidiary condition, the Klein-Gordon equation. The new Lagrangian for the Weinberg theory is proposed which is scalar and Hermitian. It is built on the basis of the concept of the `Weinberg doubles'. Origins of a contradiction between the classical theory, the Weinberg theorem B-A=\lambda for quantum relativistic fields and the claimed `longitudity' of the antisymmetric tensor field (transformed on the (1,0)\oplus (0,1) Lorentz group representation) after quantization are clarified. Analogs of the j=1/2 Feynman-Dyson propagator are presented in the framework of the j=1 Weinberg theory. It is then shown that under the definite choice of field functions and initial and boundary conditions the massless j=1 Weinberg-Tucker-Hammer equations contain all information that the Maxwell equations for electromagnetic field have. Thus, the former appear to be of use in describing some physical processes for which that could be necessitated or be convenient.