Integration of Dirac's Efforts to construct Lorentz-covariant Quantum Mechanics

TL;DR

This paper reviews Dirac's foundational efforts to develop a Lorentz-covariant quantum mechanics framework, integrating his various approaches to construct localized, Lorentz-invariant quantum systems and their underlying symmetry groups.

Contribution

It synthesizes Dirac's multiple contributions into a unified approach for Lorentz-covariant quantum mechanics, highlighting the construction of harmonic oscillator wave functions and symmetry group contractions.

Findings

Lorentz-covariant harmonic oscillator wave functions can be constructed.

The Lie algebra of the inhomogeneous Lorentz group relates to uncertainty relations.

The $O(3,2)$ de Sitter group contracts to the Lorentz group, underpinning quantum mechanics.

Abstract

The lifelong efforts of Paul A. M. Dirac were to construct localized quantum systems in the Lorentz covariant world. In 1927, he noted that the time-energy uncertainty should be included in the Lorentz-covariant picture. In 1945, he attempted to construct a representation of the Lorentz group using a normalizable Gaussian function localized both in the space and time variables. In 1949, he introduced his instant form to exclude time-like oscillations. He also introduced the light-cone coordinate system for Lorentz boosts. Also in 1949, he stated the Lie algebra of the inhomogeneous Lorentz group can serve as the uncertainty relations in the Lorentz-covariant world. It is possible to integrate these three papers to produce the harmonic oscillator wave function which can be Lorentz-transformed. In addition, Dirac, in 1963, considered two coupled oscillators to derive the Lie algebra for the generators of the $O(3,\,2)$ de Sitter group, which has ten generators. It is proven possible to contract this group to the inhomogeneous Lorentz group with ten generators, which constitute the fundamental symmetry of quantum mechanics in Einstein's Lorentz-covariant world.