The planar spectrum in U(N)-invariant quantum mechanics by Fock space methods: I. The bosonic case

TL;DR

This paper investigates the planar spectrum of U(N)-invariant quantum mechanics using Fock space methods, focusing on the bosonic sector and comparing supersymmetric and anharmonic oscillator models, providing analytical and numerical insights.

Contribution

It introduces Fock space techniques to analyze the planar spectrum in U(N)-invariant quantum systems, highlighting the separability in the supersymmetric case and addressing complexities in anharmonic oscillators.

Findings

Exact planar spectrum expressed via single-trace spectrum in supersymmetric case

Fock space methods effectively analyze the supersymmetric model

Qualitative spectral information obtained for anharmonic oscillator

Abstract

Prompted by recent results on Susy-U(N)-invariant quantum mechanics in the large N limit by Veneziano and Wosiek, we have examined the planar spectrum in the full Hilbert space of U(N)-invariant states built on the Fock vacuum by applying any U(N)-invariant combinations of creation-operators. We present results about 1) the supersymmetric model in the bosonic sector, 2) the standard quartic Hamiltonian. This latter is useful to check our techniques against the exact result of Brezin et al. The SuSy case is where Fock space methods prove to be the most efficient: it turns out that the problem is separable and the exact planar spectrum can be expressed in terms of the single-trace spectrum. In the case of the anharmonic oscillator, on the other hand, the Fock space analysis is quite cumbersome due to the presence of large off-diagonal O(N) terms coupling subspaces with different number of traces; these terms should be absorbed before taking the planar limit and recovering the known planar spectrum. We give analytical and numerical evidence that good qualitative information on the spectrum can be obtained this way.