Probability Collectives for Unstable Particles

TL;DR

This paper explores the mathematical structure of unstable particles as probability collectives within Hilbert spaces, emphasizing their non-decomposable nature and implications for probability predictions and unitarity in quantum mechanics.

Contribution

It introduces a novel framework for representing unstable particles as higher-dimensional probability collectives using Hilbert-bein structures, extending traditional quantum models.

Findings

Unstable particles form non-decomposable probability collectives in Hilbert spaces.

A Hilbert-bein relates particle bases to diagonal scalar product bases, incorporating decay parameters.

Unitarity of the S-matrix can be maintained through transformations involving the Hilbert-bein.

Abstract

Unstable particles, together with their stable decay products, constitute probability collectives which are defined as Hilbert spaces with dimension higher than one, nondecomposable in a particle basis. Their structure is considered in the framework of Birkhoff-von Neumann's Hilbert subspace lattices. Bases with particle states are related to bases with a diagonal scalar product by a Hilbert-bein involving the characteristic decay parameters (in some analogy to the $n$-bein structures of metrical manifolds). Probability predictions as expectation values, involving unstable particles, have to take into account all members of the higher dimensional collective. E.g., the unitarity structure of the $S$-matrix for an unstable particle collective can be established by a transformation with its Hilbert-bein.