The grade of a clifford object

The grade of a term is the number of basis vectors in it.

Usage

grade(C, n, drop=TRUE)
grade(C, n) <- value
grades(x)
gradesplus(x)
gradesminus(x)
gradeszero(x)

Arguments

C,x: Clifford object
n: Integer vector specifying grades to extract
value: Replacement value, a numeric vector
drop: Boolean, with default TRUE meaning to coerce a constant Clifford object to numeric, and FALSE meaning not to

Details

A term is a single expression in a Clifford object. It has a coefficient and is described by the basis vectors it comprises. Thus $4 e_{234}$ is a term but $e_{3} + e_{5}$ is not.

The grade of a term is the number of basis vectors in it. Thus the grade of $e_{1}$ is 1, and the grade of $e_{125} = e_{1} e_{2} e_{5}$ is 3. The grade operator ${⟨ \cdot ⟩}_{r}$ is used to extract terms of a particular grade, with

$A = {⟨ A ⟩}_{0} + {⟨ A ⟩}_{1} + {⟨ A ⟩}_{2} + \dots = \sum_{r} {⟨ A ⟩}_{r}$

for any Clifford object $A$ . Thus ${⟨ A ⟩}_{r}$ is said to be homogenous of grade $r$ . Hestenes sometimes writes subscripts that specify grades using an overbar as in ${⟨ A ⟩}_{\overset{―}{r}}$ . It is conventional to denote the zero-grade object ${⟨ A ⟩}_{0}$ as simply $⟨ A ⟩$ .

We have

${⟨ A + B ⟩}_{r} = {⟨ A ⟩}_{r} + {⟨ B ⟩}_{r} {⟨ λ A ⟩}_{r} = λ {⟨ A ⟩}_{r} {⟨ {⟨ A ⟩}_{r} ⟩}_{s} = {⟨ A ⟩}_{r} δ_{r s} .$

Function grades() returns an (unordered) vector specifying the grades of the constituent terms. Function grades<-() allows idiom such as grade(x,1:2) <- 7 to operate as expected [here to set all coefficients of terms with grades 1 or 2 to value 7].

Function gradesplus() returns the same but counting only basis vectors that square to $+ 1$ , and gradesminus() counts only basis vectors that square to $- 1$ . Function signature() controls which basis vectors square to $+ 1$ and which to $- 1$ .

From Perwass, page 57, given a bilinear form

$⟨ x, x ⟩ = x_{1}^{2} + x_{2}^{2} + \dots + x_{p}^{2} - x_{p + 1}^{2} - \dots - x_{p + q}^{2}$

and a basis blade $e_{A}$ with $A \subseteq {1, \dots, p + q}$ , then

$gr (e_{A}) = | {a \in A : 1 ⩽ a ⩽ p + q} |$

${gr}_{+} (e_{A}) = | {a \in A : 1 ⩽ a ⩽ p} |$

${gr}_{-} (e_{A}) = | {a \in A : p < a \leq p + q} |$

Function gradeszero() counts only the basis vectors squaring to zero (I have not seen this anywhere else, but it is a logical suggestion).

If the signature is zero, then the Clifford algebra reduces to a Grassmann algebra and products match the wedge product of exterior calculus. In this case, functions gradesplus() and gradesminus() return NA.

Function grade(C,n) returns a clifford object with just the elements of grade g, where g %in% n.

Idiom like grade(C,r) <- value, where r is a non-negative integer (or vector of non-negative integers) should behave as expected. It has two distinct cases: firstly, where value is a length-one numeric vector; and secondly, where value is a clifford object:

Firstly, grade(C,r) <- value with value a length-one numeric vector. This changes the coefficient of all grade-r terms to value. Note that disordR discipline must be respected, so if value has length exceeding one, a disordR consistency error might be raised.
Secondly, grade(C,r) <- value with value a clifford object. This should operate as expected: it will replace the grade-r components of C with value. If value has any grade component not in r, a “grade mismatch” error will be returned. Thus, only the grade-r components of C may be modified with this construction. It is semi vectorised: if r is a vector, it is interpreted as a set of grades to replace.

The zero grade term, grade(C,0), is given more naturally by const(C).

Function c_grade() is a helper function that is documented at Ops.clifford.Rd.

Author

Robin K. S. Hankin

Note

In the C code, “term” has a slightly different meaning, referring to the vectors without the associated coefficient.

References

C. Perwass 2009. “Geometric algebra with applications in engineering”. Springer.

Examples


a <- clifford(sapply(seq_len(7),seq_len),seq_len(7))
a
#> Element of a Clifford algebra, equal to
#> + 1e_1 + 2e_12 + 3e_123 + 4e_1234 + 5e_12345 + 6e_123456 + 7e_1234567
grades(a)
#> A disord object with hash 185ca524fb71fa2ae28566b137978e475c1aa00f and elements
#> [1] 1 2 3 4 5 6 7
#> (in some order)
grade(a,5)
#> Element of a Clifford algebra, equal to
#> + 5e_12345


a <- clifford(list(0,3,7,1:2,2:3,3:4,1:3,1:4),1:8)
b <- clifford(list(4,1:2,2:3),c(101,102,103))

grade(a,1) <- 13*e(6)
grade(a,2) <- grade(b,2)
grade(a,0:2) <- grade(b,0:2)*7


signature(2,2)
x <- rcliff()
drop(gradesplus(x) + gradesminus(x) + gradeszero(x) - grades(x))
#>  [1] 0 0 0 0 0 0 0 0 0 0

a <- rcliff()
a == Reduce(`+`,sapply(unique(grades(a)),function(g){grade(a,g)}))
#> [1] TRUE