Relativistic Particle Theories Without Canonical Quantization

TL;DR

This paper proposes an alternative to canonical quantization for relativistic particles, using group theory and physical principles, leading to new mathematical structures and theories for particles of various spins.

Contribution

It introduces a novel approach based on physical principles and group theory, identifying new classes of irreducible representations for the Poincaré group, and develops alternative theories for spin 0 and spin 1/2 particles.

Findings

Four inequivalent theories for spin 0 particles differing from Klein-Gordon.

For spin 1/2, only one consistent theory exists, related to Dirac theory.

New classes of irreducible representations are necessary for these theories.

Abstract

The diffculties of relativistic particle theories formulated my means of canonical quantization, such as Klein-Gordon and Dirac theories, ultimately led theoretical physicists to turn on quantum field theory to model elementary particle physics. The aim of the present work is to pursue a method alternative to canonical quantization that avoids these dfficulties. In order to guarantee this result, the present approach is constrained to be developed deductively from physical principles. The physical principles assumed for a free particle consist of the symmetry properties of the particle with respect to the Poincar\'e group and of the transformation properties of the position observable, expressed by means of a suitably conceived notion of quantum transformation. In so doing, the effectiveness of group theoretical methods is exploited. Our work has pointed out the necessity of new classes of irreducible representations of the Poincar\'e group the theory can be based on. For spin 0 particle, four inequivalent theories are completely determined, that differ from Klein-Gordon theory for the physical interpretation but also for the mathematical structures. In particular, two of these theories requires irreducible representations in the new classes. The present lack of the explicit transformation properties of position with respect to boosts forbids the complete determination of non zero spin particle theories. A particular form of these transformation properties was in past adopted by Jordan and Mukunda. We check its consistency within the present approach; it is found that for spin 1/2 particles there is only one consistent theory; it is unitarily related to Dirac theory, but, once again, it requires the new classes of irreducible representations.