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.
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)
clifford
package in useThe 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
specifies 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
#> + 5 - 1e_2 + 6e_5 + 2e_4,7 - 3e_10 - 5e_11 + 3e_14 + 1e_10,14 + 4e_5,9,15 +
#> 9e_18,19
summary(x^3)
#> Element of a Clifford algebra
#> Typical terms: 140 ... + 216e_5,9,10,14,15,18,19
#> Number of terms: 44
#> Magnitude: 25376
For more detail, see the package vignette
vignette("clifford")