Operations
Like with other numbers, standard mathematical operations are supported that relate Clifford numbers to elements of their scalar field and to each other.
Unary operations
Grade automorphisms
Grade automorphisms are operations which preserves the grades of each basis blade, but changes their sign depending on the grade. All of these operations are their own inverse.
All grade automorphisms are applicable to BitIndex objects, and the way they are implemented is through constructors that use TransformedBitIndices objects to alter each grade.
Reverse
The reverse is an operation which reverses the order of the wedge product that constructed each basis blade. This is implemented with methods for Base.reverse and Base.:~.
In the future, Base.:~ will no longer be used for this operation; instead Base.adjoint will be overloaded, providing ' as a syntax for the reverse.
This is the most commonly used automorphism, and in a sense can be thought of as equivalent to complex conjugation. When working with even elements of the algebras of 2D or 3D space, this behaves identically to complex conjugation and quaternion conjugation. However, this is not the case when working in the even subalgebras.
Base.reverse — Methodadjoint(i::BitIndex) = reverse(i::BitIndex) = i' -> BitIndex
adjoint(x::AbstractCliffordNumber) = reverse(x::AbstractCliffordNumber) = x' -> typeof(x)Performs the reverse operation on the basis blade indexed by b or the Clifford number x. The sign of the reverse depends on the grade of the basis blade g, and is positive for g % 4 in 0:1 and negative for g % 4 in 2:3.
Grade involution
Grade involution changes the sign of all odd grades, an operation equivalent to mirroring every basis vector of the space. This can be acheived with the grade_involution function.
When interpreting even multivectors as elements of the even subalgebra of the algebra of interest, the grade involution in the even subalgebra is equivalent to the reverse in the algebra of interest.
Grade involution is equivalent to complex conjugation in when dealing with the even subalgebra of 2D space, which is isomorphic to the complex numbers, but this is not true for quaternion conjugation. Instead, use the Clifford conjugate (described below).
CliffordNumbers.grade_involution — Methodgrade_involution(i::BitIndex) -> BitIndex
grade_involution(x::AbstractCliffordNumber) -> typeof(x)Calculates the grade involution of the basis blade indexed by b or the Clifford number x. This effectively reflects all of the basis vectors of the space along their own mirror operation, which makes elements of odd grade flip sign.
Clifford conjugation
The Clifford conjugate is the combination of the reverse and grade involution. This is available via an overload of Base.conj.
conj(::AbstractCliffordNumber) implements the Clifford conjugate, not the reverse!
When dealing with the even subalgebras of 2D and 3D VGAs, which are isomorphic to the complex numbers and quaternions, respectively, the Clifford conjugate is equivalent to complex conjugation or quaternion conjugation. Otherwise, this is a less widely used operation than the above two.
Base.conj — Methodconj(i::BitIndex) -> BitIndex
conj(x::AbstractCliffordNumber) -> typeof(x)Calculates the Clifford conjugate of the basis blade indexed by b or the Clifford number x. This is equal to grade_involution(reverse(x)).
Binary operations
Addition and subtraction
Addition and subtraction work as expected for Clifford numbers just as they do for other numbers. The promotion system handles all cases where objects of mixed type are added.
Products
Clifford algebras admit a variety of products. Common ones are implemented with infix operators.
Geometric product
The geometric product, or Clifford product, is the defining product of the Clifford algebra. This is implemented with the usual multiplication operator *, but it is also possible to use parenthetical notation as it is with real numbers.
Wedge product
The wedge product is the defining product of the exterior algebra. This is available with the wedge() function, or with the ∧ infix operator.
You can define elements of exterior algebras directly by using Metrics.Exterior(D), whose geometric product is equivalent to the wedge product.
Contractions and dot products
The contraction operations generalize the dot product of vectors to Clifford numbers. While it is possible to define a symmetric dot product (and one is provided in this package), the generalization of the dot product to Clifford numbers is naturally asymmetric in cases where the grade of one input blade is not equal to that of the other.
For Clifford numbers x and y, the left contraction x ⨼ y describes the result of projecting x onto the space spanned by y. If x and y are homogeneous in grade, this product is equal to the geometric product if grade(y) ≥ grade(x), and zero otherwise. For general multivectors, the left contraction can be calculated by applying this rule to the products of their basis blades.
The analogous right contraction is only nonzero if grade(x) ≥ grade(y), and it can be calculated with ⨽.
The dot product is a symmetric variation of the left and right contractions, and provides a looser constraint on the basis blades: grade(CliffordNumbers.dot(x,y)) must equal abs(grade(x) - grade(y)). The Hestenes dot product is equivalent to the dot product above, but is zero if either x or y is a scalar.
Currently, the dot product is implemented with the unexported function CliffordNumbers.dot. This package does not depend on LinearAlgebra, so there would be a name conflict if this method were exported and both this package and LinearAlgebra were loaded.
Contractions are generally favored over the dot products due to their nicer implementations and properties, which have fewer exceptions. It is generally recommended that the Hestenes dot product be avoided, though it is included in this library for the sake of completeness as CliffordNumber.hestenes_dot, which is also not exported.
Commutator and anticommutator products
The commutator product (or antisymmetric product) of Clifford numbers x and y, denoted x × y, is equal to 1//2 * (x*y - y*x). This product is nonzero if the geometric product of x and y does not commute, and the value represents the degree to which they fail to commute.
The commutator product is the building block of Lie algebras; in particular, the commutator products of bivectors, which are also bivectors. With the bivectors of 3D space, the Lie algebra is equivalent to that generated by the cross product, hence the × notation.
The analogous anticommutator product (or symmetric product) is 1//2 * (x*y + y*x). This uses the ⨰ operator, which is not an operator generally used for this purpose, but was selected as it looks similar to the commutator product, with the dot indicating the similarity with the dot product, which is also symmetric.
Defining new products: Multiplication internals
Products are implemented with the fast multiplication kernel CliffordNumbers.mul, which accepts two Clifford numbers with the same scalar type and a CliffordNumbers.GradeFilter object. This GradeFilter object defines a method that takes two or more BitIndex objects and returns false if their product is constrained to be zero.
CliffordNumbers.mul requires that the coefficient types of the numbers being multiplied are the same. Methods which leverage CliffordNumbers.mul should promote the coefficient types of the arguments to a common type using scalar_promote before passing them to the kernel. Any further promotion needed to return the final result is handled by the kernel.
In general, it is also strongly recommended to promote the types of the arguments to CliffordNumbers.Z2CliffordNumber or CliffordNumber for higher performance. Currently, the implementation of CliffordNumbers.mul is asymmetric, and does not consider which input is longer. Even in the preferred order, we find that KVector incurs a significant performance penalty.