Double-scaling limit of the O(N)-symmetric anharmonic oscillator

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 a zero-dimensional O(N)-symmetric quantum field theory one can avoid this difficulty if one replaces the original quartic theory by its -symmetric analog. In the current paper an O(N)-symmetric quartic quantum field theory in one time dimension and zero space dimensions (that is, O(N)-symmetric quantum mechanics) is studied using the Schrödinger equation. It is shown that the global -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 -symmetric quantum theory and the corresponding and equivalent Hermitian quantum theory.