A term-rewriting system for computer quantum algebra

TL;DR

This paper develops a foundational term-rewriting system to enable computer algebra packages to perform symbolic quantum mechanics calculations in Dirac's notation, addressing a gap in existing tools.

Contribution

It introduces a formal mathematical framework and rewrite rules for invariant quantum mechanics expressions, facilitating symbolic manipulation in computer algebra systems.

Findings

Defines the mathematical features of the Hilbert-space invariant formalism

Provides a formal characterization of algebraic expressions in quantum mechanics

Develops rewrite rules for symbolic manipulation of quantum expressions

Abstract

Existing computer algebra packages do not fully support quantum mechanics calculations in Dirac's notation. I present the foundation for building such support: a mathematical system for the symbolic manipulation of expressions used in the invariant formalism of quantum mechanics. I first describe the essential mathematical features of the Hilbert-space invariant formalism. This is followed by a formal characterisation of all possible algebraic expressions in this formalism. This characterisation is provided in the form of a set of terms. Rewrite rules over this set of terms are then developed that correspond to allowed manipulations of the algebraic expressions. This approach is contrasted with current attempts to build invariant quantum mechanics calculations into computer algebra systems.