Source-linked AI summary
Introducing Cadabra: a symbolic computer algebra system for field theory problems
Kasper Peeters
TL;DR
Field theorists lack convenient support for specialized symbolic operations that are difficult to perform or combine in existing systems. Cadabra addresses this gap with TeX-based interaction and built-in field-theory algorithms, illustrated through calculations in gravity, supergravity, and quantum field theory. The paper presents these capabilities as making Cadabra well-suited to field-theory problems, while acknowledging that its scope still leaves room for improvements and additions.
Problem
Existing symbolic computer algebra systems inadequately support or integrate field-theory operations such as tensor canonicalisation, fermionic manipulation, and Fierz transformations.
Method
The paper introduces Cadabra through representative calculations using TeX-like input and built-in facilities for tensor identities, fermions, gamma matrices, Fierz transformations, derivatives, and variational problems.
Results
Cadabra provides a field-theory-oriented system with familiar TeX input and output, specialized tensor simplification, and support for numerous field-theory concepts.
Takeaways & Limitations
The presented capabilities and examples make Cadabra particularly well-suited to solving symbolic field-theory problems.
Takeaways & Limitations
Given the program’s scope and size, Cadabra still has room for improvements and additions.
Abstract
from arXiv · showhide
Cadabra is a new computer algebra system designed specifically for the solution of problems encountered in field theory. It has extensive functionality for tensor polynomial simplification taking care of Bianchi and Schouten identities, for fermions and anti-commuting variables, Clifford algebras and Fierz transformations, implicit coordinate dependence, multiple index types and many other field theory related concepts. The input format is a subset of TeX and thus easy to learn. Both a command-line and a graphical interface are available. The present paper is an introduction to the program using several concrete problems from gravity, supergravity and quantum field theory.
1 Introduction
Cadabra addresses field theorists’ difficulty with specialized symbolic calculations by providing a purpose-built system whose TeX-like interface and domain-aware algorithms support complex expressions.
- 1 Introduction: Existing systems handle general symbolic tasks well but make field-theory operations such as tensor canonicalisation and Fierz transformations difficult.Examples also include symmetry transformations of Lagrangians, Poisson brackets, and deriving field equations.
- 1 Introduction: Combining specialized packages is often tedious because general-purpose systems require substantial adaptation and do not easily integrate tensor manipulation with fermionic calculations.The paper identifies technical design choices in general-purpose systems as the underlying reason.
- 1 Introduction: Cadabra is introduced as a computer algebra system designed from the start for field-theory problems and presented through representative explicit calculations.The paper is intended primarily as a user guide rather than a description for computer scientists.
- 1 Introduction: TeX-based input and output let users write tensor indices, derivatives, commutators, and fermion products in familiar notation, including notebooks that are also valid TeX files.Cadabra provides both graphical notebook and command-line interfaces.
- 1 Introduction: Cadabra simplifies tensor expressions using symmetries, Bianchi and Schouten identities, while supporting commuting and anti-commuting tensors, multiple index sets, and dummy indices.These capabilities target relations and index structures common in field-theory calculations.
- 1 Introduction: The system directly supports field-theory concepts including gamma matrices, Fierz identities, covariant derivatives, implicit coordinate dependence, and variational derivatives.Its substitution command also handles anti-commuting objects and dummy indices.
- 1 Introduction: Users can preserve preferred expression forms and explicitly control which simplification steps run, including the verbosity of intermediate results.Cadabra avoids automatic transformations unless instructed and can execute user-defined command lists during calculations.
2 Bosonic basics
Cadabra combines TeX-like tensor input with property declarations that support index management, tensor symmetries, derivatives, and field-theory calculations. Its Young tableau methods handle multi-term and dimension-dependent identities, enabling automated verification of nontrivial tensor relations.
- Notation and properties: Cadabra uses property declarations to define index sets, tensor symmetries, derivative behavior, and dependencies needed for expression manipulation.These declarations enable automatic dummy-index relabelling, canonicalisation, and inherited properties such as the symmetry of double partial derivatives.
- Notation and properties: Cadabra’s substitution mechanism automatically relabels dummy indices and recognizes equivalent indexed patterns without explicit wildcards.The behavior depends on declaring the relevant index set and supports substitutions such as replacing B_{a b} with an expression carrying new contracted indices.
- Bianchi, Ricci and Schouten identities: Young tableau projectors extend canonicalisation beyond mono-term symmetries to multi-term identities such as the Bianchi identity.Applying Young projection makes the Bianchi identity manifest, while shorthand properties can associate Riemann and covariant-derivative expressions with the required tableau symmetries.
- Bianchi, Ricci and Schouten identities: Dimension-dependent simplification distinguishes products that remain unchanged in four or more dimensions from those that vanish in three dimensions by the Schouten identity.For two antisymmetric rank-three tensors, Young-projector decomposition returns the original product in four dimensions or higher but produces zero when indices range over three dimensions.
- Applications: Cadabra makes a Riemann-tensor polynomial proof straightforward compared with the tedious manual use of Bianchi identities, and verifies a Kaluza–Klein expression as equivalent to equation (24).The examples rely on tensor-property declarations linking curvature tensors or covariant derivatives to Bianchi identities; the Kaluza–Klein result follows after writing out the covariant derivative.
- Bianchi, Ricci and Schouten identities: Young-projector expansion makes a derivative identity vanish and thereby proves the relation labeled (19).The calculation applies the Leibniz rule, canonicalises mono-term symmetries, expands tensors using Young projectors, and obtains the stated identity.
3 Fermions, Dirac algebra and Fierz transformations
Cadabra supports compact, field-theory-oriented manipulation of fermions, spinors, gamma matrices, and tensor expressions. Examples show automated gamma-matrix expansion, antisymmetry handling, Fierz transformations, and supersymmetry variations while retaining readable notation.
- Fermions and gamma matrices: Cadabra manipulates anti-commuting objects, spinors, and gamma matrices in arbitrary dimensions, with suppressed or explicit spinor indices.Fermionic objects require appropriate property declarations, while index brackets support explicit matrix indices.
- Simple gamma matrix algebra: Gamma-matrix products are expanded into irreducible antisymmetrised gamma-matrix and Kronecker-delta components.The join_gamma algorithm expands adjacent generalised gamma matrices into a basis of fully antisymmetrised products.
- Simple gamma matrix algebra: The resulting matrix operations can switch between abstract and explicit index notation through index brackets.Cadabra can act inside index brackets and distribute brackets over sums when implicit indices permit it.
- Fierz transformations: Fierz transformations reorder four-fermion terms, and the worked Majorana-spinor example is reduced to an expression equivalent to the target identity.The calculation requires symbol-property declarations, a Fierz step, gamma-matrix joining, and Kronecker-delta elimination.
- Other assorted topics: Self- and mutual anti-commutativity let Cadabra reorder fermionic products, apply the required signs, and simplify expressions using induced antisymmetry.A declared self-anticommuting ψ_μ is treated as antisymmetric in its indexed product, while mutually anti-commuting symbols acquire signs during substitution.
- Other assorted topics: A supersymmetry-variation example combines multiple index types, fermionic properties, pattern matching, user-defined ordering, and variational differentiation.The remaining calculation uses partial integration and a Fierz identity to show that the result vanishes.
4 Conclusions
The paper presents Cadabra as a field-theory-focused system whose TeX-like notation and built-in algebraic facilities support calculations resembling handwritten workflows. It is freely distributable and available across supported platforms, while the authors acknowledge ongoing room for improvement.
- Conclusions: Cadabra’s TeX-based input and field-theory facilities are illustrated through concrete calculations and designed to resemble pencil-and-paper problem solving.The paper highlights anti-commuting variables, gamma matrices, implicit coordinate dependence, multiple dummy-index sets, and tensor canonicalisation.
- Conclusions: The system uses freely distributable libraries, avoids proprietary computer algebra systems, and provides source code for supported platforms.Binary packages are also available for Linux and Windows.
- Conclusions: The authors acknowledge substantial room for improvements and additions, with a separate paper planned for changes in the 2.x series.This conclusion marks ongoing development as a scope boundary of the presented system.