Infinitely many inequivalent field theories from one Lagrangian

TL;DR

This paper explores a logarithmic time-like Liouville quantum field theory with generalized PT invariance, revealing an infinite set of inequivalent sectors and analyzing their spectra using PT-symmetric quantum theory techniques.

Contribution

It demonstrates that a single Lagrangian can generate infinitely many unitarily inequivalent field theory sectors distinguished by an integer label.

Findings

Lagrangian defines infinite inequivalent sectors labeled by n

Energy spectrum in quantum mechanics sectors calculated semiclassically

Energy levels depend on quantum number m and sector label n

Abstract

Logarithmic time-like Liouville quantum field theory has a generalized PT invariance, where T is the time-reversal operator and P stands for an S-duality reflection of the Liouville field $\phi$. In Euclidean space the Lagrangian of such a theory, $L=\frac{1}{2}(\nabla\phi)^2-ig\phi\exp(ia\phi)$, is analyzed using the techniques of PT-symmetric quantum theory. It is shown that L defines an infinite number of unitarily inequivalent sectors of the theory labeled by the integer n. In one-dimensional space (quantum mechanics) the energy spectrum is calculated in the semiclassical limit and the mth energy level in the nth sector is given by $E_{m,n}\sim(m+1/2)^2a^2/(16n^2)$.