PT-Symmetric Versus Hermitian Formulations of Quantum Mechanics

TL;DR

This paper compares PT-symmetric non-Hermitian and Hermitian quantum mechanics formulations, demonstrating that calculations are simpler and more straightforward in the PT-symmetric form due to the finiteness of graphs, unlike the divergent graphs in the Hermitian form.

Contribution

It provides a detailed comparison showing that PT-symmetric non-Hermitian quantum mechanics simplifies calculations compared to Hermitian formulations, especially regarding divergent graphs.

Findings

Non-Hermitian PT-symmetric Hamiltonian graphs are finite and easier to evaluate.

Hermitian Hamiltonian perturbation series involve divergent graphs requiring complex regulation.

Calculations of ground-state energy are simpler in the PT-symmetric formulation.

Abstract

A non-Hermitian Hamiltonian that has an unbroken PT symmetry can be converted by means of a similarity transformation to a physically equivalent Hermitian Hamiltonian. This raises the following question: In which form of the quantum theory, the non-Hermitian or the Hermitian one, is it easier to perform calculations? This paper compares both forms of a non-Hermitian $ix^3$ quantum-mechanical Hamiltonian and demonstrates that it is much harder to perform calculations in the Hermitian theory because the perturbation series for the Hermitian Hamiltonian is constructed from divergent Feynman graphs. For the Hermitian version of the theory, dimensional continuation is used to regulate the divergent graphs that contribute to the ground-state energy and the one-point Green's function. The results that are obtained are identical to those found much more simply and without divergences in the non-Hermitian PT-symmetric Hamiltonian. The $\mathcal{O}(g^4)$ contribution to the ground-state energy of the Hermitian version of the theory involves graphs with overlapping divergences, and these graphs are extremely difficult to regulate. In contrast, the graphs for the non-Hermitian version of the theory are finite to all orders and they are very easy to evaluate.