GeometricAlgebraForMathematica

A Wolfram Language/Mathematica package for working with Geometric Algebra

To use this package, put it in your working directory and import it using:

<<GeometricAlgebra`

The basics

Basis vectors are represented using e[i] where i is an index/label. In typical usage, i is a non-negative integer, e.g. e[3], e[0], but arbitrary symbols or expressions are permitted: e[x], e[∞], and e[time] would all be valid. Additionally, basis k-vectors may, for convenience, be given as sequences of arguments: e[1,2] (a basis bivector), e[1,3,5] (a basis trivector). These are expanded into geometric products of basis vectors.

Geometric products are represented using the built-in Wolfram Language function NonCommutativeMultiply, which has operator form **. These may then be expanded using ExpandNCM. Basis vectors automatically order themselves when multiplied: e[2]**e[1] becomes -e[1]**e[2].

This library presently assumes orthogonality between all basis vectors. What each one squares to may be changed, however, by assigning values to Metric[i], which may be arbitrary expressions. This can be useful for Spacetime Algebra, Projective Geometric Algebra, and working in alternative charts, manifolds, or spaces. For example:

In[2]:= {e[0]**e[0], e[1]**e[1], e[2]**e[2]}
Out[2]= {1, 1, 1}

In[3]:= Metric[0]=0; Metric[1]=-1; Metric[2]=r;
In[4]:= {e[0]**e[0], e[1]**e[1], e[2]**e[2]}
Out[4]= {0, -1, r}

Multivector decomposition

KVectorPart[mv,k] will extract the k-vector part of a multivector mv. ScalarPart[mv] will give the scalar part of a multivector mv (equivalent to KVectorPart[mv,0]). KVectorDecomposition is also provided which splits a multivector into a list of its parts:

In[2]:= KVectorDecomposition[1+e[1]-e[2,3]+4e[1,2,3,4]]                                                                
Out[2]= {1, e[1], -e[2] ** e[3], 0, 4 e[1] ** e[2] ** e[3] ** e[4]}

Products

This package includes the following products: GeometricProduct, OuterProduct, CommutatorProduct, LeftContraction, RightContraction, FatDotProduct, and ScalarProduct.

Other operations

The package includes various involutions: Reversion, Involution, and Conjugation. MultivectorNorm and MultivectorNormSquared are provided for calculating norms. Angle computes the angle between two blades.

Symbolic manipulation

Symbolic vectors may be represented using OverVector, which renders an arrow over a symbol. These behave properly as though they are vectors in functions like KVectorPart and ExpandNCM. I also plan on expanding symbolic capabilities further as I continue to work on this package.