Arbitrary Dimensional Clifford Algebras.
The clifford package: Clifford algebra in R
The clifford
package provides R-centric functionality for working with Clifford algebras of arbitrary dimension and signature. A detailed vignette is provided in the package.
Installation
You can install the released version of the clifford package from CRAN with:
# install.packages("clifford") # uncomment this to install the package
library("clifford")
set.seed(0)
The clifford
package in use
The basic creation function is clifford()
, which takes a list of basis blades and a vector of coefficients:
(a <- clifford(list(1,2,1:4,2:3),1:4))
#> Element of a Clifford algebra, equal to
#> + 1e_1 + 2e_2 + 4e_23 + 3e_1234
(b <- clifford(list(2,2:3,1:2),c(-2,3,-3)))
#> Element of a Clifford algebra, equal to
#> - 2e_2 - 3e_12 + 3e_23
So a
and b
are multivectors. Clifford objects are a vector space and we can add them using +
:
a+b
#> Element of a Clifford algebra, equal to
#> + 1e_1 - 3e_12 + 7e_23 + 3e_1234
See how the e2
term vanishes and the e_23
term is summed. The package includes a large number of products:
a*b # geometric product (also "a % % b")
#> Element of a Clifford algebra, equal to
#> - 16 + 6e_1 - 3e_2 - 2e_12 + 14e_3 + 12e_13 + 3e_123 - 9e_14 + 9e_34 - 6e_134
a %^% b # outer product
#> Element of a Clifford algebra, equal to
#> - 2e_12 + 3e_123
a %.% b # inner product
#> Element of a Clifford algebra, equal to
#> - 16 + 6e_1 - 3e_2 + 14e_3 - 9e_14 + 9e_34 - 6e_134
a %star% b # scalar product
#> [1] -16
a %euc% b # Euclidean product
#> [1] 8
The package can deal with non positive-definite inner products. Suppose we wish to deal with an inner product of
where the diagonal is a number of terms followed by a number of terms. The package idiom for this would be to use signature()
:
signature(3)
Function signature()
is based on lorentz::sol()
and its argument specifes the number of basis blades that square to , the others squaring to . Thus and :
basis(1)
#> Element of a Clifford algebra, equal to
#> + 1e_1
basis(1)^2
#> Element of a Clifford algebra, equal to
#> scalar ( 1 )
basis(4)
#> Element of a Clifford algebra, equal to
#> + 1e_4
basis(4)^2
#> Element of a Clifford algebra, equal to
#> the zero clifford element (0)
The package uses the STL map class with dynamic bitset keys for efficiency and speed and can deal with objects of arbitrary dimensions. Thus:
options("basissep" = ",")
(x <- rcliff(d=20))
#> Element of a Clifford algebra, equal to
#> + 4 + 5e_2 + 1e_5 - 2e_4,7 + 2e_11 + 4e_14 - 1e_10,14 + 3e_5,9,15 - 3e_18,19
summary(x^3)
#> Element of a Clifford algebra
#> Typical terms: 364 ... + 54e_5,9,10,14,15,18,19
#> Number of terms: 40
#> Magnitude: 265721
References
- D. Hestenes 1987. Clifford algebra to geometric calculus, Kluwer.
- J. Snygg 2010. A new approach to differential geometry using Clifford’s geometric algebra. Berghauser.
- C. Perwass 2009. Geometric algebra with applications in engineering. Springer.
Further information
For more detail, see the package vignette
vignette("clifford")