The function Ops.clifford() passes unary and binary arithmetic
operators “+”, “-”, “*”,
“/” and “^” to the appropriate specialist
function. Function maxyterm() returns the maximum index in the
terms of its arguments.
The package has several binary operators:
| Geometric product | A*B = geoprod(A,B) |
| \(\displaystyle AB=\sum_{r,s}\left\langle A\right\rangle_r\left\langle
B\right\rangle_s\) | Inner product | A %.% B = cliffdotprod(A,B) |
| \(\displaystyle A\cdot B=\sum_{r\neq 0\atop s\ne
0}^{\vphantom{s\neq 0}}\left\langle\left\langle A\right\rangle_r\left\langle
B\right\rangle_s\right\rangle_{\left|s-r\right|}\) | Outer product | A %^% B = wedge(A,B) |
| \(\displaystyle A\wedge B=\sum_{r,s}\left\langle\left\langle A\right\rangle_r\left\langle
B\right\rangle_s\right\rangle_{s+r}\) | Fat dot product | A %o% B = fatdot(A,B) |
| \(\displaystyle A\bullet B=\sum_{r,s}\left\langle\left\langle A\right\rangle_r\left\langle
B\right\rangle_s\right\rangle_{\left|s-r\right|}\) | Left contraction | A %_|% B = lefttick(A,B) |
| \(\displaystyle A\rfloor B=\sum_{r,s}\left\langle\left\langle A\right\rangle_r\left\langle
B\right\rangle_s\right\rangle_{s-r}\) | Right contraction | A %|_% B = righttick(A,B) |
| \(\displaystyle A\lfloor B=\sum_{r,s}\left\langle\left\langle A\right\rangle_r\left\langle
B\right\rangle_s\right\rangle_{r-s}\) | Cross product | A %X% B = cross(A,B) |
| \(\displaystyle A\times
B=\frac{1}{2_{\vphantom{j}}}\left(AB-BA\right)\) | Scalar product | A %star% B = star(A,B) |
| \(\displaystyle A\ast B=\sum_{r,s}\left\langle\left\langle A\right\rangle_r\left\langle
B\right\rangle_s\right\rangle_0\) | Euclidean product | A %euc% B = eucprod(A,B) |
In R idiom, the geometric product geoprod(.,.) has to be
indicated with a “*” (as in A*B) and so the
binary operator must be %*%: we need a different idiom for
scalar product, which is why %star% is used.
Because geometric product is often denoted by juxtaposition, package
idiom includes a % % b for geometric product.
Binary operator %dot% is a synonym for %.%, which
causes problems for rmarkdown.
Function clifford_inverse() returns an inverse for nonnull
Clifford objects \(\operatorname{Cl}(p,q)\) for
\(p+q\leq 5\), and a few other special cases. The
functionality is problematic as nonnull blades always have an inverse;
but function is.blade() is not yet implemented. Blades
(including null blades) have a pseudoinverse, but this is not
implemented yet either.
The scalar product of two clifford objects is defined as the
zero-grade component of their geometric product:
$$
A\ast B=\left\langle AB\right\rangle_0\qquad{\mbox{NB: notation used by both Perwass and Hestenes}}
$$
In package idiom the scalar product is given by A %star% B or
scalprod(A,B). Hestenes and Perwass both use an asterisk for
scalar product as in “\(A*B\)”, but in package idiom, the
asterisk is reserved for geometric product.
Note: in the package, A*B is the geometric product.
The Euclidean product (or Euclidean scalar product) of two
clifford objects is defined as
$$
A\star B=
A\ast B^\dagger=
\left\langle AB^\dagger\right\rangle_0\qquad{\mbox{Perwass}}
$$
where \(B^\dagger\) denotes Conjugate [as in Conj(a)]. In
package idiom the Euclidean scalar product is given by
eucprod(A,B) or A %euc% B, both of which return
A * Conj(B).
Note that the scalar product \(A\ast A\) can be positive or
negative [that is, A %star% A may be any sign], but the
Euclidean product is guaranteed to be non-negative [that is, A
%euc% A is always positive or zero].
Dorst defines the left and right contraction (Chisholm calls these the
left and right inner product) as \(A\rfloor B\) and
\(A\lfloor B\). See the vignette for more details.
Division, as in idiom x/y, is defined as
x*clifford_inverse(y). Function clifford_inverse() uses
the method set out by Hitzer and Sangwine but is limited to
\(p+q\leq 5\).
The Lie bracket, \(\left[x,y\right]\) is implemented in the
package using idiom such as .[x,y], and this is documented at
dot.Rd.
Many of the functions documented here use low-level helper functions
that wrap C code. For example, fatdot() uses
c_fatdotprod(). These are documented at lowlevel.Rd.