The signature of the Clifford algebra — signature • clifford

Getting and setting the signature of the Clifford algebra

Usage

signature(p,q=0)
is_ok_sig(s)
showsig(s)
# S3 method for class 'sigobj'
print(x,...)

Arguments

s,p,q: Integers, specifying number of positive elements on the diagonal of the quadratic form, with s=c(p,q)
x: Object of class sigobj
...: Further arguments, currently ignored

Details

The signature functionality is modelled on the lorentz package; clifford::signature() operates in the same way as lorentz::sol() which gets and sets the speed of light. The idea is that both the speed of light and the signature of a Clifford algebra are generally set once, at the beginning of an R session, and subsequently change only very infrequently.

Clifford algebras require a bilinear form $\left\langle\cdot,\cdot\right\rangle$ on $\mathbb{R}^n$. If ${\mathbf x}=\left(x_1,\ldots,x_n\right)$ we define

$$\left\langle{\mathbf x},{\mathbf x}\right\rangle=x_1^2+x_2^2+\cdots +x_p^2-x_{p+1}^2-\cdots -x_{p+q}^2 $$

where $p+q=n$. With this quadratic form the vector space is denoted $\mathbb{R}^{p,q}$ and we say that $(p,q)$ is the signature of the bilinear form $\left\langle\cdot,\cdot\right\rangle$. This gives rise to the Clifford algebra $C_{p,q}$.

If the signature is $(p,q)$, then we have

$$ e_i e_i = +1\, (\mbox{if } 1\leq i\leq p), -1\, (\mbox{if } p+1\leq i\leq p+q), 0\, (\mbox{if } i>p+q). $$

Note that $(p,0)$ corresponds to a positive-semidefinite quadratic form in which $e_ie_i=+1$ for all $i\leq p$ and $e_ie_i=0$ for all $i > p$. Similarly, $(0,q)$ corresponds to a negative-semidefinite quadratic form in which $e_ie_i=-1$ for all $i\leq q$ and $e_ie_i=0$ for all $i > q$.

A strictly positive-definite quadratic form is specified by infinite $p$ [in which case $q$ is irrelevant], and signature(Inf) implements this. For a strictly negative-definite quadratic form we would have $p=0,q=\infty$ which would be signature(0,Inf).

If we specify $e_ie_i=0$ for all $i$, then the operation reduces to the wedge product of a Grassmann algebra. Package idiom for this is to set $p=q=0$ with signature(0,0), but this is not recommended: use the stokes package for Grassmann algebras, which is much more efficient and uses nicer idiom.

Function signature(p,q) returns the signature invisibly; but setting option show_signature to TRUE makes signature() have the side-effect of calling showsig(), which changes the default prompt to display the signature, much like showSOL in the lorentz package. There is special dispensation for “infinite” $p$ or $q$.

Calling signature() [that is, with no arguments] returns an object of class sigobj with elements corresponding to $p$ and $q$. The sigobj class ensures that a near-infinite integer such as .Machine$integer.max will be printed as “Inf” rather than, for example, “2147483647”.

Function is_ok_sig() is a helper function that checks for a proper signature. If we set signature(p,q), then technically $n>p+q$ implies $e_n^2=0$, but usually we are not interested in $e_n$ when $n>p+q$ and want this to be an error. Option maxdim specifies the maximum value of $n$, with default NULL corresponding to infinity. If $n$ exceeds this, is_ok_sig() throws an error. Note that it is fine to have maxdim > p+q [and indeed this is useful in the context of dual numbers]. This option is intended to be a super-strict safety measure.


> e(6)
Element of a Clifford algebra, equal to
+ 1e_6
> options(maxdim=5)
> e(5)
Element of a Clifford algebra, equal to
+ 1e_5
> e(6)
Error in is_ok_clifford(terms, coeffs) : option maxdim exceeded

Author

Robin K. S. Hankin

Examples



signature()
#> [1] Inf   0

e(1)^2
#> Element of a Clifford algebra, equal to
#> scalar ( 1 )
e(2)^2
#> Element of a Clifford algebra, equal to
#> scalar ( 1 )

signature(1)
e(1)^2
#> Element of a Clifford algebra, equal to
#> scalar ( 1 )
e(2)^2   # note sign
#> Element of a Clifford algebra, equal to
#> the zero clifford element (0)

signature(3,4)
sapply(1:10,function(i){drop(e(i)^2)})
#>  [1]  1  1  1 -1 -1 -1 -1  0  0  0


signature(Inf)   # restore default




# Nice mapping from Cl(0,2) to the quaternions (loading clifford and
# onion simultaneously is discouraged):

# library("onion")
# signature(0,2)
# Q1 <- rquat(1)
# Q2 <- rquat(1)
# f <- function(H){Re(H)+i(H)*e(1)+j(H)*e(2)+k(H)*e(1:2)}
# f(Q1)*f(Q2) - f(Q1*Q2) # zero to numerical precision
# signature(Inf)