Identification of the metric for diagonalizable (anti-)pseudo-Hermitian Hamilton operators represented by two-dimensional matrices

TL;DR

This paper develops a general method to identify metrics for two-dimensional diagonalizable pseudo-Hermitian and anti-pseudo-Hermitian Hamiltonians, analyzing how eigenvalue permutations and basis normalizations influence the metric and energy spectra, with implications for quantum theory.

Contribution

It introduces a comprehensive strategy to determine metrics for 2D pseudo-Hermitian Hamiltonians, clarifies the role of eigenvalue permutations, and critiques existing constraints in PT-symmetric quantum theory.

Findings

The metric depends on eigenvalue permutation and basis normalization.

Involutive C-operator implies the Hamiltonian must be symmetric.

The formalism extends to higher-dimensional Hamiltonians.

Abstract

A general strategy is provided to identify the most general metric for diagonalizable pseudo-Hermitian and anti-pseudo-Hermitian Hamilton operators represented by two-dimensional matrices. It is investigated how a permutation of the eigen-values of the Hamilton operator in the process of its diagonalization influences the metric and how this permutation equivalence affects energy eigen-values. We try to understand on one hand, how the metric depends on the normalization of the chosen left and right eigen-basis of the matrix representing the diagonalizable pseudo-Hermitian or anti-pseudo-Hermitian Hamilton operator, on the other hand, whether there has to exist a positive semi-definite metric required to set up a meaningful Quantum Theory even for non-Hermitian Hamilton operators of this type. Using our general strategy we determine the metric with respect to the two elements of the two-dimensional permutation group for various topical examples of matrices representing two-dimensional Hamilton operators found in the literature assuming on one hand pseudo-Hermiticity, on the other hand anti-pseudo-Hermiticity. The (unnecessary) constraint inferred by C. M. Bender and collegues that the ${\cal C}$-operator of ${\cal PT}$-symmetric Quantum Theory should be an involution (${\cal C}^2=1$) is shown - in the unbroken phase of ${\cal PT}$-symmetry - to require the Hamilton operator to be symmetric. This inconvenient restriction had been already - with hesitation - noted by M. Znojil and H. B. Geyer in 2006 (arXiv:quant-ph/0607104). A Hamilton operator proposed by T. D. Lee and C. G. Wick is used to outline implications of the formalism to higher dimensional Hamilton operators.