Random clifford objects

Random Clifford algebra elements, intended as quick “get you going” examples of clifford objects

Usage

rcliff(n=9, d=6, g=4, include.fewer=TRUE)
rclifff(n=100, d=20, g=10, include.fewer=TRUE)
rblade(d=7, g=3)

Arguments

n: Number of terms
d: Dimensionality of underlying vector space
g: Maximum grade of any term
include.fewer: Boolean, with FALSE meaning to return a clifford object comprising only terms of grade g, and default TRUE meaning to include terms with grades less than g (including a term of grade zero, that is, a scalar)

Details

Function rcliff() gives a quick nontrivial Clifford object, typically with terms having a range of grades (see grade.Rd); argument include.fewer=FALSE ensures that all terms are of the same grade. Function rclifff() is the same but returns a more complicated object by default.

Function rblade() gives a Clifford object that is a blade (see term.Rd). It returns the wedge product of a number of 1-vectors, for example $\left(e_1+2e_2\right)\wedge\left(e_1+3e_5\right)$.

Perwass gives the following lemma:

Given blades $A_{\langle r\rangle}, B_{\langle s\rangle}, C_{\langle t\rangle}$, then

$$ \langle A_{\langle r\rangle} B_{\langle s\rangle} C_{\langle t\rangle} \rangle_0 = \langle C_{\langle t\rangle} A_{\langle r\rangle} B_{\langle s\rangle} \rangle_0 $$

In the proof he notes in an intermediate step that

$$ \langle A_{\langle r\rangle} B_{\langle s\rangle} \rangle_t * C_{\langle t\rangle} = C_{\langle t\rangle} * \langle A_{\langle r\rangle} B_{\langle s\rangle} \rangle_t = \langle C_{\langle t\rangle} A_{\langle r\rangle} B_{\langle s\rangle} \rangle_0. $$

Package idiom is shown in the examples.

Author

Robin K. S. Hankin

Note

If the grade exceeds the dimensionality, $g>d$, then the result is arguably zero; rcliff() returns an error.

Examples


rcliff()
#> Element of a Clifford algebra, equal to
#> + 6 - 3e_2 + 4e_134 - 5e_16 + 8e_36 + 5e_1236 - 1e_256 + 9e_2356 + 6e_3456
rcliff(d=3,g=2)
#> Element of a Clifford algebra, equal to
#> + 5 + 8e_1 - 3e_2 - 8e_12 + 2e_3 - 1e_13
rcliff(3,10,7)
#> Element of a Clifford algebra, equal to
#> + 3 - 2e_49 + 1e_1457910 - 3e_1578910
rcliff(3,10,7,include=TRUE)
#> Element of a Clifford algebra, equal to
#> + 3 + 3e_56 - 1e_234568 - 3e_1569

x1 <- rcliff()
x2 <- rcliff()
x3 <- rcliff()

x1*(x2*x3) == (x1*x2)*x3  # should be TRUE
#> [1] TRUE


rblade()
#> Element of a Clifford algebra, equal to
#> - 21e_123 - 4e_124 + 1e_134 - 9e_234 - 8e_125 + 2e_135 - 18e_235 + 40e_126 +
#> 32e_136 - 15e_236 + 8e_146 - 20e_246 - 13e_346 + 16e_156 - 40e_256 - 26e_356 -
#> 22e_127 - 5e_137 - 18e_237 - 2e_147 + 6e_247 + 3e_347 - 4e_157 + 12e_257 +
#> 6e_357 - 24e_167 + 50e_267 + 31e_367 - 2e_467 - 4e_567

# We can invert blades easily:
a <- rblade()
ainv <- rev(a)/scalprod(a)

zap(a*ainv)  # 1 (to numerical precision)
#> [1] 1
zap(ainv*a)  # 1 (to numerical precision)
#> [1] 1

# Perwass 2009, lemma 3.9:


A <- rblade(d=9, g=4)
B <- rblade(d=9, g=5)
C <- rblade(d=9, g=6)

grade(A*B*C,0) - grade(C*A*B,0)   # zero to numerical precision
#> [1] 0



# Intermediate step

x1 <- grade(A*B,3) %star% C
x2 <- C %star% grade(A*B,3)
x3 <- grade(C*A*B,0)

max(x1,x2,x3) - min(x1,x2,x3)   # zero to numerical precision
#> [1] 0