Bundle Theoretic Descriptions of Massive Single-Particle State Spaces; With a view toward Relativistic Quantum Information Theory

Abstract

Relativistic Quantum Information Theory (RQI) is a flourishing research area of physics, yet, there has been no systematic mathematical treatment of the field. In this paper, we suggest bundle theoretic descriptions of massive single-particle state spaces, which are basic building blocks of RQI. In the language of bundle theory, one can construct a vector bundle over the set of all possible motion states of a massive particle, in whose fibers the moving particle's internal quantum state as perceived by a fixed inertial observer is encoded. A link between the usual Hilbert space description is provided by a generalized induced representation construction on the L 2 -section space of the bundle. The aim of this paper is two-fold. One is to communicate the basic ideas of RQI to mathematicians and the other is to suggest an improved formalism for single-particle state spaces that encompasses all known massive particles including those which have never been dealt with in the RQI literature. Some of the theoretical implications of the formalism will be explored at the end of the paper.

Basic Notions of Poisson and Symplectic Geometry in Local Coordinates, with Applications to Hamiltonian Systems

“…VI B). Note also that if two contravariant fields on the manifolds M n and N k are given, we can of course compare them using the mapping (28), see Eq. (68) below as an example.…”