Geometric Algebra

Mathematica packages for Clifford (geometric) algebra calculations:

CGAlgebra

CGAlgebra.m is a Mathematica package for the 5D Conformal Geometric Algebra.

This package contains declarations for calculations with Conformal Geometric Algebra. Basis vectors {e₀, e₁, e₂, e₃, e_Infinity} are denoted by e[0], e[1], e[2], e[3], e[Infinity]. Geometric products of basis elements are denoted as e[0,1,2] (= e₀ e₁ e₂), etc.

The results of any calculation is given in terms of the geometric product of basis elements, that is, the outer (Grassman) product of basis elements or multivectors is calculated by using OuterProduct[] and the output is given in terms of geometric product of basis vectors.

Examples:

The vector e₀ + 2 e₁ + e_Infinity is written as:

     A = e[0] + 2 e[1] + e[\[Infinity]];

The multivector a + 5 e₁ + e₁e₂e₃ is

     B = a + 5 e[1] + e[1,2,3];

The geometric product

    GeometricProduct[A,B]

yields:

 10 + a e[0] + 2a e[1] + a e[\[Infinity]] + 5 e[0,1] - 5 e[1,\[Infinity]] + 2 e[2,3] + e[0,1,2,3] - e[1,2,3,\[Infinity]]

The inner product

   InnerProduct[A,B]

yields

   10 + 2 e[2,3]

A tutorial can be downloaded from: https://arxiv.org/abs/1711.02513

Clifford

clifford.m is the most recent version of the package by G. Aragon-Camarasa, G. Aragon-Gonzalez, J.L. Aragon and M.A. Rodriguez-Andrade. A user guide (CliffordUserGuide) is available, as well as Mathematica palette (CliffordPalette). The fundamentals of the package are presented in:

https://arxiv.org/abs/0810.2412

CliffordBasic

CliffordBasic.m is a completely renewed but reduced version of clifford.m package by G. Aragon-Camarasa, G. Aragon-Gonzalez, J.L. Aragon and M.A. Rodriguez-Andrade: https://arxiv.org/abs/0810.2412

Using rule-base programming the algebra over R^p,q in arbitrary dimensions is constructed as in A. Macdonald "An Elementary Construction of Geometric Algebra", Adv. Appl. Cliff. Alg. 12 (2002) 1-6.

In CliffordBasic, the j-th basis vector is denoted by e[j] and the geometric product of basis vectors, such as e₁e₃e₄, as e[1,3,4].

Examples:

The vector e₁ + 2 e₂ - a e₃ is written as

    A = e[2] + 2 e[2] - a e[3];

The The multivector a + 5 e₁ + e₁ e₂ e₃ is written as

     B = a + 5 e[1] + e[1,2,3];

The geometric product AB is calculated as

    GeometricProduct[A,B]

yielding:

    3 a e[2] - a^2 e[3] - 15 e[1,2] - a e[1,2] - 3 e[1,3] + 5 a e[1,3]

which can be factored using

     GFactor[%]
     
     3 a e[2] - a^2 e[3] + (-15-a) e[1,2] + (-3 + 5 a)e[1,3]

The signature of the R^p,q is set by $SetSignature={p,q}. If not specified, the default value is:

     $SetSignature={20,0}