Crypto-Hermiticity of nonanticommutative theories

TL;DR

Nonanticommutative Minkowski supersymmetric theories can be viewed as crypto-Hermitian, allowing their Hamiltonians to be transformed into Hermitian form, though their physical implications, especially for field theories, remain uncertain.

Contribution

This paper demonstrates that nonanticommutative deformations lead to crypto-Hermitian Hamiltonians, preserving supersymmetry algebra and enabling Hermitian reformulation in Minkowski theories.

Findings

Deformed models are crypto-Hermitian and can be made explicitly Hermitian.

Supersymmetry algebra remains intact under deformation.

Unitarity of the S-matrix in deformed Minkowski field theories is uncertain.

Abstract

We note that, though nonanticommutative deformations of Minkowski supersymmetric theories do not respect the reality condition and seem to lead to non-Hermitian Hamiltonians H, the latter belong to the class of crypto-Hermitian (or quasi-Hermitian) Hamiltonians having attracted recently a considerable attention. They can be made manifestly Hermitian via the similarity transformation H -> e^R H e^{-R} with a properly chosen R. The deformed model enjoys the same supersymmetry algebra as the undeformed one though it is difficult in some cases to write explicit expressions for a half of supercharges. The deformed SQM models make perfect sense. It is not clear whether it is also the case for NAC Minkowski field theories -- the conventionally defined S--matrix is not unitary there.