signature: The signature of the Clifford algebra

In clifford: Arbitrary Dimensional Clifford Algebras

The signature of the Clifford algebra

Description

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 showsig() [which is called by signature()] change the default prompt so it displays the signature, much like showSOL in the lorentz package. Note that changing the signature changes the prompt immediately (if show_signature is TRUE), but changing option show_signature has no effect until showsig() is called.

Calling signature() [that is, with no arguments] returns an object of class sigobj with elements corresponding to p and q. There is special dispensation for “infinite” p or 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 maxdim, then is_ok_sig() throws an error. Note that it is sometimes 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(s)

Robin K. S. Hankin

Examples

signature()

e(1)^2
e(2)^2

signature(1)
e(1)^2
e(2)^2   # note sign

signature(3,4)
sapply(1:10,function(i){drop(e(i)^2)})


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)

clifford documentation built on June 8, 2025, 10:56 a.m.