DiracQ: A Package for Algebraic Manipulation of Non-Commuting Quantum Variables

Abstract

In several problems of quantum many body physics, one is required to handle complex expressions originating in the non-commutative nature of quantum operators. Their manipulation requires precise ordering and application of simplification rules. This can be cumbersome, tedious and error prone, and often a challenge to the most expert researcher. In this paper we present a software package DiracQ to facilitate such manipulations. The package DiracQ consists of functions based upon and extending considerably the symbolic capabilities of Mathematica. With DiracQ, one can proceed with large scale algebraic manipulations of expressions containing combinations of ordinary numbers or symbols (the c-numbers) and arbitrary sets of non-commuting variables (the q-numbers) with user defined properties. The DiracQ package is user extendable and comes encoded with the algebraic properties of several standard operators in popular usage. These include Fermionic and Bosonic creation and annihilation operators, spin operators, and canonical position and momentum operators. An example book is provided with some suggestive calculations of large-scale algebraic manipulations.