The anomalous spin-statistics connection arising from pseudo-Hermiticity

TL;DR

This work introduces a pseudo-Hermitian quantum-field-theory framework in which the spin-statistics connection is reversed: integer-spin fields obey fermionic statistics and half-integer-spin fields obey bosonic statistics, achieved by using a pseudo-Hermitian adjoint $\overset{\:{}^{\neg}}{[t]{\chi}}=\eta^{-1}\bar{\chi}\eta$ with $H^{\#}=\eta^{-1}H^{\dagger}\eta=H$. The free theory maintains locality, Lorentz covariance, and unitarity through Hermitian $H_{0}$ and $\mathbf{P}_{0}$, though the Belinfante-Rosenfeld tensor is pseudo-Hermitian; two local solutions for the intertwiner $\eta$ yield the NSST, including a nontrivial one that flips statistics. Extensions to scalar, spinor, and vector fields demonstrate consistent NSST-compatible constructions, though interactions generically render the full Hamiltonian non-Hermitian and necessitate a positive-definite metric $\eta_{+}$ for a probabilistic interpretation. The results offer a foundational view of non-Hermitian QFTs with potential beyond-Standard-Model physics, while highlighting the challenges and necessary steps to formulate fully consistent interacting theories.

Abstract

We establish a new spin-statistics theorem for a class of free pseudo-Hermitian quantum field theories whose particles furnish unitary irreducible representations of the Poincaré group. In this framework, free pseudo-Hermitian fields with integer spin exhibit fermionic statistics, whereas those with half-integer spin exhibit bosonic statistics, opposite to the conventional case. This reversal arises from defining canonical field operators using pseudo-Hermitian conjugation rather than Hermitian conjugation, thereby circumventing the conventional spin-statistics theorem. The free fields retain locality, Lorentz covariance, and unitary evolution. However, interactions may violate unitarity due to the intrinsically non-Hermitian nature of the full Hamiltonian. We discuss potential resolutions to restore unitarity in interacting theories.