Can PT-Symmetric Quantum Mechanics be a Viable Alternative Quantum Theory?

TL;DR

This paper critically examines PT-symmetric quantum mechanics, revealing mathematical properties and limitations, especially regarding unitarity and physical viability of complex PT-symmetric Hamiltonians, and compares it to Hermitian quantum mechanics.

Contribution

The paper clarifies the mathematical structure of PT-symmetric Hamiltonians and discusses their physical implications, highlighting issues with unitarity and the reality condition of potentials.

Findings

Diagonalizable PT-symmetric Hamiltonians have real eigenvalues.

Time-dependent PT-symmetric Hamiltonians violate unitarity.

Physically meaningful PT-symmetric Hamiltonians must have real potentials.

Abstract

Update: A time-independent $n\times n$ PT-symmetric (and symmetric) Hamiltonian is diagonalizable since it has all distinct real eigenvalues and the resulting diagonal matrix is a real symmetric matrix. The diagonalization results an isometry so there shouldn't be any issue with unitarity and unfortunately this very elementary mathematical fact somehow did not draw the authors' attention. However, PT-symmetric quantum mechanics is not out of trouble. For time-dependent PT-symmetric (and symmetric) Hamiltonians (even $2\times 2$ ones) the authors observed that there is a violation of unitarity. Moreover, the first named author showed in his recent article arXiv:1312.7738 that PT-symmetric quantum mechanics is indeed a certain kind of Hermitian quantum mechanics and that in order for time-evolution to be unitary with respect to $J$-inner product (one that gives rise to a Hilbert space structure on the space of state functions), the potential energy operator $V(x)$ must be real. This means that those complex PT-symmetric Hamiltonians that have been studied by physicists are unfortunately unphysical. The first named author discussed in a subsequent article arXiv:1401.5149 that while finite-state PT-symmetric quantum mechanics with time-independent Hamiltonians is not physically any different from Hermitian quantum mechanics, PT-symmetric quantum mechanics exhibits a distinctive symmetry from that of Hermitian quantum mechanics.