On the geometry pervading One Particle States

TL;DR

This paper explores the geometric structures underlying quantum states of relativistic particles, deriving explicit representations of the Poincaré group via Geometric Quantization, and relating classical and quantum state spaces through symplectic manifolds.

Contribution

It provides explicit constructions of quantum state spaces for relativistic particles using Geometric Quantization, linking classical symplectic manifolds to quantum representations.

Findings

Explicit representations of Poincaré group for relativistic particles

Descriptions of massless particles via Penrose equations and electromagnetic fields

Connection between symplectic manifolds and classical particle state spaces

Abstract

In this paper, a way is given to obtain explicitly the representations of the Poincar\'e group as can be prescribed by Geometric Quantization. Thus one obtains some forms of the Space of Quantum States of the different relativistic free particles, and I give explicitly these spaces and the corresponding operators for the usually accepted as realistic physical particles. The general description of the massless particles I obtain, is given in terms of solutions of Penrose equations. In the case of Photon, I also give other descriptions, one in terms of the Electromagnetic Field. Since the results are derived from Geometric Quantization, they are related to certain Contact and Symplectic manifols, that I study in detail. The symplectic manifold must be interpreted, according with Souriau, as the Movement Space of the corresponding classical particle, and that leads to propose one of the spaces I use as the State Space of the corresponding classical particle. These spaces are also described in each case.