Relativistic massive particle with spin-1/2, a vector bundle point of view

TL;DR

This paper introduces a vector bundle approach to clarify the nature of spin for relativistic massive particles, resolving ambiguities and establishing a covariant reduced density matrix based on the Pauli-Lubansky vector.

Contribution

It provides a novel vector bundle framework to distinguish physical spin from ambiguous definitions and proves the spin reduced density matrix's lack of meaning, proposing a covariant alternative.

Findings

The standard spin reduced density matrix is mathematically meaningless.

A vector bundle perspective clarifies the physical meaning of spin for relativistic particles.

A covariant Pauli-Lubansky reduced density matrix can be constructed as the unique meaningful alternative.

Abstract

Recently, in the context of Relativistic Quantum Information Theory (RQI) of massive spin-1/2 particles, it has been suggested that it is impossible to perform a momentum-independent spin measurement, showing the inadequacy of the spin reduced density matrix as a legitimate information resource. This is because there is an unavoidable ambiguity in the definition of the spin of a moving particle. In this paper, by introducing a vector bundle theoretic way to view the single-particle state space, we try to rule out this ambiguity. The discrete degree of freedom of the resulting representation space contains information about the Pauli-Lubansky four-vector of the particle instead of the ambiguous spin. Comparing this representation with the standard one used in the RQI literature, we show that the discrete degree of freedom of the standard representation space attains the meaning of the Newton-Wigner spin. Also using this viewpoint, we give a mathematical proof of why the spin reduced density matrix is meaningless, which is stronger than the previous claims in that it asserts that the matrix is void of any meaning at all, not just in terms of the impossibility of measurement or Lorentz non-covariance. We give a way (which turns out to be the only way) to modify it to obtain the Pauli-Lubansky reduced density matrix, which is covariant under Lorentz transformations.