WolframQuantumComputation/ DiracMatrix

Generalized Dirac gamma matrices for arbitrary real symmetric metrics in any dimension

Contributed by: Mads Bahrami

DiracMatrix constructs Dirac gamma matrices satisfying the Clifford anticommutation relation {γ^μ, γ^ν} = 2 η^μν𝕀 for an arbitrary real symmetric metric η in any dimension. Flat metrics use the Brauer–Weyl (Pauli–Kronecker) construction; non-flat metrics are reached through a vielbein decomposition g = e^T . η . e. The package also provides transformations into the Dirac and Weyl (chiral) bases, and the canonical graded Clifford operator basis built via a numerically stable antisymmetric recursion.

Installation Instructions

To install this paclet in your Wolfram Language environment, evaluate this code:
PacletInstall["WolframQuantumComputation/DiracMatrix"]

To load the code after installation, evaluate this code:
Needs["WolframQuantumComputation`DiracMatrix`"]

Details

DiracMatrix exports ten symbols organised in four groups: gamma-matrix constructors (GammaMatrices, EuclideanGammaMatrices), metric utilities (FlatMetric, RandomCurvedMetric, MetricVielbein), basis transformations (ToDiracBasis, ToWeylBasis), and Clifford-algebra basis builders (CliffordBasis, CliffordCanonicalBasis). The implementation is pure Wolfram Language 13.0+ with no external dependencies.

Examples

Basic Examples

In[1]:=

Out[1]=

Scope (2)

In[2]:=

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Table[\[CapitalGamma][[\[Mu]]] . \[CapitalGamma][[\[Nu]]] + \[CapitalGamma][[\[Nu]]] . \[CapitalGamma][[\[Mu]]] - 2 \[Eta][[\[Mu], \[Nu]]] IdentityMatrix[4], {\[Mu], 4}, {\[Nu], 4}]]

Out[2]=

The canonical graded Clifford operator basis on an n-dimensional metric has Binomial[n,k] elements at grade k. For the 4-dimensional Minkowski metric the grade dimensions are {1,4,6,4,1}:

In[3]:=