Commuting quantum matrix models

Veselin G Filev,Denjoe O’Connor

doi:10.1007/jhep03(2015)024

Veselin G Filev, Denjoe O’Connor

Open Access

https://doi.org/10.1007/jhep03(2015)024

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Abstract

We study a quantum system of $p$ commuting matrices and find that such a quantum system requires an explicit curvature dependent potential in its Lagrangian for the system to have a finite energy ground state. In contrast it is possible to avoid such curvature dependence in the Hamiltonian. We study the eigenvalue distribution for such systems in the large matrix size limit. A critical r\^ole is played by $p=4$. For $p\ge4$ the competition between eigenvalue repulsion and the attractive potential forces the eigenvalues to form a sharp spherical shell.

Highlights

Many of these models will have regimes where commuting matrices play a role
We study a quantum system of p commuting matrices and find that such a quantum system requires an explicit curvature dependent potential in its Lagrangian for the system to have a finite energy ground state
It has been suggested by Berenstein [10] that, in order to count 1/8-BPS states, a matrix model of commuting matrices is needed

Summary

One matrix quantum model

We revisit the quantisation of a one matrix model first presented in [1]. It is most convenient to perform the quantisation in the “curved” coordinates obtained by diagonalising the matrices These complications are not necessary in the one matrix model, where the configuration space is flat and diagonalisation is just a change of coordinates. The key idea to obtain a closed form expression for the distribution ρ valid at large N , is to notice that at large N the semi-classical approximation to the individual wave function is well justified and in the limit N → ∞ becomes exact. Where ωi = 2π/Ti is the frequency of the classical motion depending on the energy εi, and we have used that for large N to leading order we have cos2(N λ dλ′p(λ′)) ≈ 1/2 under the integral in (2.21) With this normalisation, and approximating again the fast oscillating cosine function with one half, for the absolute value square of the wave function fi at large. Which is our final expression for the distribution ρand agrees with the result in ref. [1], as it should, since we have only made the semiclassical analysis more explicit

Gaussian potential

Commuting matrix model

Conclusions

A Scalar curvature of the space of commuting hermitian matrices