pseudoscalar

## function () 
## {
##     m <- getOption("maxdim")
##     if (is.null(m)) {
##         stop("pseudoscalar requires a finite value of maxdim; set it with something like options(maxdim = 6)")
##     }
##     else {
##         return(e(seq_len(m)))
##     }
## }

To cite the clifford package in publications please use Hankin (2022). This short document discusses the pseudoscalar I in the clifford R package. The behaviour of I depends on the dimension n and the signature of the space considered, and as such function pseudoscalar() fails if maxdim is not set:

pseudoscalar()

## Error in pseudoscalar(): pseudoscalar requires a finite value of maxdim; set it with something like options(maxdim = 6)

Function pseudoscalar() needs option maxdim to ascertain what object to return. Let us set maxdim to 7:

options(maxdim=7)
pseudoscalar()

## Element of a Clifford algebra, equal to
## + 1e_1234567

The example above makes it clear that pseudoscalar() returns the unit pseudoscalar, in whatever dimension we are working in. The usual workflow would be to define maxdim and a signature at the start of a session, then define an R object (conventionally I), as the pseudoscalar. However, in this vignette we will repeatedly redefine the signature and the maximum dimension to illustrate different aspects of pseudoscalar(). The first feature of I is that |I|² = 1. For standard ℝ² and ℝ³, and Minkowski space Cl (3, 1) we have I² = −1:

options(maxdim=3)
signature(3)       # Cl(3,0)
(I <- pseudoscalar())

## Element of a Clifford algebra, equal to
## + 1e_123

drop(I^2)

## [1] -1

And for Minkowski space:

options(maxdim=4)
signature(3,1)       # Cl(3,1)
I <- pseudoscalar()
drop(I^2)

## [1] -1

However, we can easily define other signatures in which I² = +1:

options(maxdim=4)
signature(2,2)       # Cl(2,2)
(I <- pseudoscalar())

## Element of a Clifford algebra, equal to
## + 1e_1234

drop(I^2)

## [1] 1

The pseudoscalar I defines an orientation in the sense that, for any ordered set of n linearly independent vectors a₁, …, a_n their outer product will have either the same or opposite sign as I. Because the orientation is negated by interchanging a pair of vectors, we see that the orientation is preserved by even permutations of 1, 2, …, n. Working in Cl (5, 0):

options(maxdim=5)
signature(5)
I <- pseudoscalar()
ai <- list(); for(i in 1:5){ai[[i]] <- as.1vector(rnorm(5))}
ai[[1]] # the other 5 look very similar

## Element of a Clifford algebra, equal to
## + 1.263e_1 - 0.32623e_2 + 1.3298e_3 + 1.2724e_4 + 0.41464e_5

Reduce(`^`,ai)

## Element of a Clifford algebra, equal to
## + 3.3202e_12345

Above we see, from the last line, that the vectors a₁ to a₅ are independent (the result is nonzero). Further, we see that the vectors are a right-handed set, for the wedge product is positive. We can permute the vectors using the permutations package (Hankin 2020):

(p <- permutation("(12)(345)"))

## [1] (12)(345)

is.even(p)

## [1] FALSE

Above, we see that p is an odd permutation, being a product of a transposition and a three-cycle.

c(drop(Reduce(`^`,ai)),drop(Reduce(`^`,ai[as.word(p)])))

## [1]  3.3202 -3.3202

Above, we see that the sign of the wedge product of the permuted list has changed, consistent with the permutation’s being odd. We know various things about the pseudoscalar; below we will verify that a ⋅ (AI) = a ∧ AI for vector a and multivector A:

options(maxdim=7)   
signature(7)
(I <- pseudoscalar())

## Element of a Clifford algebra, equal to
## + 1e_1234567

(a <- as.1vector(sample(1:10,5)))

## Element of a Clifford algebra, equal to
## + 7e_1 + 6e_2 + 1e_3 + 4e_4 + 8e_5

(A <- rcliff())

## Element of a Clifford algebra, equal to
## + 7 + 2e_4 + 7e_234 - 6e_1345 + 9e_16 - 8e_126 + 6e_236 + 3e_1236 - 1e_1356 - 9e_2456

Above we choose randomish values for a and A. Observe that A has terms of different grades; it is not homogeneous. Numerical verification is straightforward [NB: “%.%” breaks markdown documents]:

LHS <- cliffdotprod(a, A*I) # Usual idiom would be "a %.% (A*I)"
RHS <- (a^A)*I
LHS - RHS

## Element of a Clifford algebra, equal to
## the zero clifford element (0)