Identity, Geometry, Permutation And The Spin-Statistics Theorem

TL;DR

This paper critically analyzes the foundations of the spin-statistics theorem, proposing a revised symmetrization postulate that clarifies the role of particle labels, permutation, and geometry in quantum state symmetry and the Pauli principle.

Contribution

It introduces a revised symmetrization postulate allowing symmetric state vectors under pure permutation for any spin, emphasizing the importance of geometric and label considerations.

Findings

Proves the Pauli Exclusion Principle for arbitrary spin.

Identifies a geometrical asymmetry in particle pairs with a common reference frame.

Computes exchange phases for various spin frames.

Abstract

We examine historic formulations of the spin-statistics theorem from a point of view that distinguishes between the observable consequences and the ``symmetrization postulate''. In particular, we make a critical analysis of concepts of particle identity, state distinguishability and permutation, and particle ``labels''. We discuss how to construct unique state vectors and the nature of the full state descriptions required for this -- in particular the elimination of arbitrary $2\pi$ rotations on fermion spin quantization frames and argue that the failure to do this renders the conventional symmetrization postulate (and previous ``proofs'' of it) at best {\em incomplete}. We discuss particle permutation in a general way for any pairs of particles, whether identical or not, and make an essential distinction between exchange and pure permutation. We prove a revised symmetrization postulate that allows us to construct state vectors that are naturally symmetric under pure permutation, {\em for any spin}. The significance of particle labels (which, in the exchange operation, are not permuted along with other variables) is then that they stand in for any asymmetry (order dependence) that is present in the full state descriptions necessary for unique state vectors but not explicit in the regular state variables. {\em The exchange operation is then the physical transformation that reverses any asymmetry implicit in the labels}. We point out a previously unremarked geometrical asymmetry between all pairs of particles that is present whenever we choose a common frame of reference. We compute the exchange phase for various state vectors using different spin quantization frames, and prove the Pauli Exclusion Principle and its generalization to arbitrary spin.