Optimized Fock space in the large N limit of quartic interactions in matrix models

Abstract

We consider the problem of quantization of the bosonic membrane via the large N limit of its matrix regularizations H N in Fock space. We prove that there exists a choice of the Fock space frequency such that H N can be written as a sum of a non-interacting Hamiltonian H 0,N and the original normal ordered quartic potential. Using this decomposition we obtain upper and lower bounds for the ground state energy in the planar limit, we study a perturbative expansion about the spectrum of H 0,N , and show that the spectral gap remains finite at N = ∞ at least up to the second order. We also apply the method to the U (N )-invariant anharmonic oscillator, and demonstrate that our bounds agree with the exact result of Brezin et al. * mkhynek@kth.se 1 See [6] for a discussion of the spectrum, including the supersymmetric version of the model and related issues 2 The power of n for a quartic interaction it is usually argued to be −1, (the t' Hooft coupling [8]), however note that due to the definition of f (n) abc which contains an explicit factor of n 3 2 , the coupling constant in front of our potential should be multiplied by n −4 . In Section 4 we will show that this also follows from our construction