Double-Scaling Limit of the O(N)-Symmetric Anharmonic Oscillator

TL;DR

This paper demonstrates that using PT-symmetric quantum theory allows a consistent double-scaling limit for the O(N)-symmetric anharmonic oscillator, resolving issues present in the conventional approach.

Contribution

It introduces a PT-symmetric formulation for the O(N)-symmetric anharmonic oscillator, enabling a consistent double-scaling limit in quantum mechanics.

Findings

PT-symmetric approach resolves negative coupling issues

Double-scaling limit is consistent in PT-symmetric formulation

Equivalent Hermitian theory explains critical behavior

Abstract

In an earlier paper it was argued that the conventional double-scaling limit of an O(N)-symmetric quartic quantum field theory is inconsistent because the critical coupling constant is negative and thus the integral representing the partition function of the critical theory does not exist. In this earlier paper it was shown that for an O(N)-symmetric quantum field theory in zero-dimensional spacetime one can avoid this difficulty if one replaces the original quartic theory by its PT-symmetric analog. In the current paper an O(N)-symmetric quartic quantum field theory in one-dimensional spacetime [that is, O(N)-symmetric quantum mechanics] is studied using the Schroedinger equation. It is shown that the global PT-symmetric formulation of this differential equation provides a consistent way to perform the double-scaling limit of the O(N)-symmetric anharmonic oscillator. The physical nature of the critical behavior is explained by studying the PT-symmetric quantum theory and the corresponding and equivalent Hermitian quantum theory.