Function `pseudoscalar()` in the `clifford` package

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 \(\left|I\right|^2=1\). For standard \(\mathbb{R}^2\) and \(\mathbb{R}^3\), and Minkowski space \(\operatorname{Cl}(3,1)\) we have \(I^2=-1\):

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 \(I^2=+1\):

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 \(a_1,\ldots, a_n\) 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 \(1,2,\ldots,n\). Working in \(\operatorname{Cl}(5,0)\):

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
## + 1.262954e_1 - 0.3262334e_2 + 1.329799e_3 + 1.272429e_4 + 0.4146414e_5

Reduce(`^`,ai)

## Element of a Clifford algebra, equal to
## + 3.32019e_12345

Above we see, from the last line, that the vectors \(a_1\) to \(a_5\) 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]  3.32019 -3.32019

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 \(a\cdot\left(AI\right)=a\wedge AI\) and \(A\cdot\left(BI\right)=A\wedge BI\) for vector \(a\) and multivectors \(A,B\):

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

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

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

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

A  # B looks very similar

## 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 vector \(a\) and multivectors \(A,B\). Observe that \(A\) and \(B\) have terms of different grades; they are not homogenous. 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)

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

## Element of a Clifford algebra, equal to
## - 35e_127 - 35e_157 - 28e_1457 - 7e_367 + 21e_1467 + 63e_13467 + 56e_123467 +
## 42e_2567 - 21e_34567 - 42e_1234567

References

Hankin, R. K. S. 2020. “Introducing the Permutations R Package.” SoftwareX 11.

———. 2022. “Clifford Algebra in R.” arXiv. https://doi.org/10.48550/ARXIV.2209.13659.

Robin K. S. Hankin

References