Quantum mechanics with non-unitary symmetries

TL;DR

This paper introduces a nonunitary Lorentz group representation framework in quantum mechanics, providing an alternative to QFT that maintains wave functions and avoids negative energy states, with implications for particle-antiparticle symmetry.

Contribution

It develops a consistent nonunitary Lorentz group representation approach in quantum mechanics, preserving wave functions and avoiding negative energy states, offering new insights into particle-antiparticle properties.

Findings

Redefines scalar product for nonunitary Lorentz representations

Derives wave equations based on superposition and symmetry principles

Models particles with unified Lorentz group representations

Abstract

This article shows that one can consistently incorporate nonunitary representations of at least one group into the ``ordinary'' nonrelativistic quantum mechanics. This group turns out to be Lorentz group thus giving us an alternative approach to QFT for combining the quantum mechanics and special theory of relativity which keeps the concept of wave function (belonging to some representation of Lorentz group) through the whole theory. Scalar product has been redefined to take into the account the nonunitarity of representations of Lorentz group. Understanding parity symmetry turns out to be the key ingredient throughout the process. Instead of trying to guess an equation or a set of equations for some wave functions or fields (or equivalently trying to guess a Lagrangian for the same), one derives them based only on the superposition principle and properties of wave functions under Lorentz transformations and parity. The resulting model has striking similarities with the standard quantum field theory and yet has no negative energy states, no zitterbewegung effects, symmetric energy momentum tensor and angular momentum density tensor for \emph{all} representations of Lorentz group (unifying the theoretical description of all particles), as well as clear physical interpretation. It also offers a possible interpretation why particles and antiparticles have opposite quantum numbers.