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 $\mathrm{Cl}(3,1)$ 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 $a_1,\dots ,a_n$ 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 $1,2,\dots ,n$. Working in $\mathrm{Cl}(5,0)$:

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 $a\cdot \left(AI\right)=a\wedge AI$ 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)