PT Symmetry as a Generalization of Hermiticity

TL;DR

This paper demonstrates that PT-symmetric quantum mechanics extends traditional Hermitian quantum mechanics by allowing more general Hamiltonians with real spectra, broadening the theoretical framework of quantum physics.

Contribution

It constructs the most general PT-symmetric Hamiltonians for 2x2 and 3x3 matrices, showing Hermitian matrices as special cases within this broader class.

Findings

PT-symmetric Hamiltonians have real spectra

Hermitian matrices are special cases of PT-symmetric matrices

PT symmetry generalizes Hermitian quantum mechanics

Abstract

The Hilbert space in PT-symmetric quantum mechanics is formulated as a linear vector space with a dynamic inner product. The most general PT-symmetric matrix Hamiltonians are constructed for 2*2 and 3*3 cases. In the former case, the PT-symmetric Hamiltonian represents the most general matrix Hamiltonian with a real spectrum. In both cases, Hermitian matrices are shown to be special cases of PT-symmetric matrices. This finding confirms and strengthens the early belief that the PT-symmetric quantum mechanics is a generalization of the conventional Hermitian quantum mechanics.