Oscillators with imaginary coupling: spectral functions in quantum mechanics and quantum field theory

TL;DR

This paper investigates quantum oscillators with imaginary coupling, demonstrating they can have real spectra and unitary evolution despite non-hermiticity, and explores implications for spectral functions and phase transitions.

Contribution

It provides a detailed analysis of non-hermitian, ${ m PT}$-symmetric oscillators in quantum mechanics and field theory, showing their physical consistency and spectral properties.

Findings

Non-hermitian ${ m PT}$-symmetric Hamiltonians can yield real spectra.

Positivity violation in spectral functions may indicate ${ m PT}$-broken phases.

Imaginary coupling oscillators can produce physically consistent quantum theories.

Abstract

The axioms of Quantum Mechanics require that the hamiltonian of any closed system is self-adjoint, so that energy levels are real and time evolution preserves probability. On the other hand, non-hermitian hamiltonians with ${\cal{PT}}$-symmetry can have both real spectra and unitary time evolution. In this paper, we study in detail a pair of quantum oscillators coupled by an imaginary bilinear term, both in quantum mechanics and in quantum field theory. We discuss explicitly how such hamiltonians lead to perfectly sound physical theories with real spectra and unitary time evolution, in spite of their non-hermiticity. We also analyze two-point correlation functions and their associated K\"allen-Lehmann representation. In particular, we discuss the intimate relation between positivity violation of the spectral functions and the non-observability of operators in a given correlation function. Finally, we conjecture that positivity violation of some spectral functions of the theory could be a generic sign of the existence of complex pairs of energy eigenvalues (i.e., a ${\cal{PT}}$-broken phase) somewhere in its parameter space.