A field-theory motivated approach to symbolic computer algebra

TL;DR

The paper identifies limitations of general-purpose, list-based computer algebra systems for field-theory problems, such as deriving equations of motion or verifying symmetries. It introduces Cadabra, a C++ prototype that accepts TeX-like input and uses graph-structured tensors plus a versatile property system to manage indices, symmetries, and dependencies. A key contribution is the use of Young-projector methods to handle multi-term tensor symmetries, enabling systematic basis construction and decomposition of complex tensor expressions. The work demonstrates practical benefits for field-theory calculations, including robust dummy-index relabelling, canonicalisation with multi-term symmetries, and automated SUSY invariance checks, with open-source availability and documentation.

Abstract

Field theory is an area in physics with a deceptively compact notation. Although general purpose computer algebra systems, built around generic list-based data structures, can be used to represent and manipulate field-theory expressions, this often leads to cumbersome input formats, unexpected side-effects, or the need for a lot of special-purpose code. This makes a direct translation of problems from paper to computer and back needlessly time-consuming and error-prone. A prototype computer algebra system is presented which features TeX-like input, graph data structures, lists with Young-tableaux symmetries and a multiple-inheritance property system. The usefulness of this approach is illustrated with a number of explicit field-theory problems.