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arXiv:2505.21708 · Universe

Second-Order Pseudo-Hermitian Spin-1/2 Bosons

Armando de la C. Rangel-Pantoja, I. Díaz-Saldaña, Carlos A. Vaquera-Araujo

Published Dec 5, 2025• 1 citation
arXivHTMLPDF

Abstract

The canonical quantization of a field theory for spin-1/2 massive bosons that satisfy the Klein–Gordon equation is presented. The breakdown of the usual spin–statistics connection is due to the redefinition of the dual field, rendering the theory pseudo-Hermitian. The normal-ordered Hamiltonian is bounded from below with real eigenvalues, and the theory is consistent with microcausality and invariant under parity, charge conjugation and time reversal.

DOI
10.3390/universe11120400

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The canonical quantization of a field theory for spin-1/2 massive bosons that satisfy the Klein–Gordon equation is presented. The breakdown of the usual spin–statistics connection is due to the redefinition of the dual field, rendering the theory pseudo-Hermitian. The normal-ordered Hamiltonian is bounded from below with real eigenvalues, and the theory is consistent with microcausality and invariant under parity, charge conjugation and time reversal.
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We introduce a class of pseudo-bosonic Klein-Gordon fields in 1+1 dimensions and we discuss some of their properties. This work originates from non Hermitian quantum mechanics and deformed canonical commutation relations. We show that, within this class of fields, there exist a specific subclass with the interesting feature of having finite equal space-time two-points function, contrarily to what happens for {\em standard} Klein-Gordon fields. This, in our opinion, is a relevant aspect of our proposal which is a good motivation to undertake a deeper analysis of this (and related) quantum fields.
Abstract We introduce a class of pseudo-bosonic Klein–Gordon fields (KGFs) in 1+1 dimensions, and we discuss some of their properties. This approach originates from non-Hermitian quantum mechanics and deformed canonical commutation relations (CCRs). We show that, within this class of fields, there exists a specific subclass with the quite interesting property of having finite equal space-time two-point function, contrarily to what happens for ordinary KGFs. This, in our opinion, is a relevant aspect of our proposal, which is a good motivation to undertake a deeper analysis of this (and related) quantum fields.
For a given pseudo-Hermitian Hamiltonian of the standard form: H=p2∕2m+v(x), we reduce the problem of finding the most general (pseudo-)metric operator η satisfying H†=ηHη−1 to the solution of a differential equation. If the configuration space is R, this is a Klein-Gordon equation with a nonconstant mass term. We obtain a general series solution of this equation that involves a pair of arbitrary functions. These characterize the arbitrariness in the choice of η. We apply our general results to calculate η for the PT-symmetric square well, an imaginary scattering potential, and a class of imaginary delta-function potentials. For the first two systems, our method reproduces the known results in a straightforward and extremely efficient manner. For all these systems we obtain the most general η up to second-order terms in the coupling constants.
We reexamine the connection between spin and statistics through the quantization of a complex scalar field, using the formulation with the property that the hermitian conjugate of canonical momentum for a variable is just the canonical momentum for the hermitian conjugate of the variable. Starting from an ordinary Lagrangian density and imposing the anti-commutation relations on the field, we find that the difficulty stems from not the ill-definiteness (or unboundedness) of the energy and the breakdown of the causality but the appearance of states with negative norms. It is overcome by introducing an ordinary scalar field to form a doublet of fermionic symmetries, although the system becomes empty leaving the vacuum state alone. These features also hold for the system with a spinor field imposing the commutation relations on.