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arXiv:0808.2474 · Communications in Mathematical Physics

Making Almost Commuting Matrices Commute

M. B. Hastings

Published Jul 18, 2009• 53 citations
arXivHTMLPDF
DOI
10.1007/s00220-009-0877-2

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Geometric Brownian motion (GBM) is a standard model in stochastic differential equations. In this study, we consider a matrix-valued GBM with non-commutative matrices. Introduction of non-commutative matrices into the matrix-valued GBM makes it difficult to obtain an exact solution because the existence of noise terms prevents diagonalization. However, we show that the replica method enables us to overcome this difficulty. We map the trace of the time evolution operator of the matrix-valued GBM with non-commutative matrices into the partition function of the isotropic Lipkin-Meshkov-Glick model used in quantum spin systems. Then, solving the eigenvalue problem of the isotropic Lipkin-Meshkov-Glick model, we obtain an analytical expression of the free entropy. Numerical simulation is consistent with our analytical result. Thus, our expression is the exact solution of the free entropy for the matrix-valued GBM with non-commutative matrices.
A new device in operator calculus is introduced to treat expressions which contain non-commutable factors. By means of this technique the defining equation for the generator of transformation function of exponential type is derived. It is shown that this equation contains exclusively commutators of relevant quantities. Then, making use of this peculiarity to the exponential formula, a method is proposed for analyzing the structure of transformation function out of a given Hamiltonian of the system, and this is explained by some examples. This method consists of following procedures: First, decomposition of given Hamiltonian into parts of different character and abstraction of elementary matrices from it. Second, formation of commutators between these matrices and extension, if necessary, of the set of matrices enumerated at the beginning to include new ones which appear through operation of commutation. The matrix ring thus obtained constitutes the frame work of the transformation function. Through the course of this analysis one can find the ground why problems which are already solved exactly can be treated easily, and also can obtain some instructions how one should proceed in the solution of a given problem.