Numeric types
This package exports a variety of types that represents elements of Clifford algebras.
AbstractCliffordNumber{Q,T} and subtypes
The AbstractCliffordNumber{Q,T} type is the supertype for all implmentations of Clifford numbers. Q is a QuadraticForm, which describes the number of dimensions with positive, negative, and zero square, and T is a Union{Real,Complex} type of the coefficients.
We may introduce a new abstract type, StaticCliffordNumber{Q,T,L}, for static implementations, like all of the ones provided by this package. These should be implemented as fixed length data structures (ideally an NTuple{L,T}).
CliffordNumber{Q,T,L}: full grade Clifford numbers
CliffordNumber{Q,T,L} is the largest possible representation of a Clifford number, and it explicitly includes the coefficients for all 2^dimension(Q) basis blades.
While this type is useful if working with objects that mix even and odd grades (for instance, projectors or left minimal ideals), it is often more efficient to work with a smaller type, like the ones described below.
EvenCliffordNumber{Q,T,L} and OddCliffordNumber{Q,T,L}: even and odd graded elements
These types represent Clifford numbers of exclusively even or odd grade, respectively. These are the workhorses of geometric algebra, as they are produced through products of even or odd numbers of 1-blades. In the majority of cases, you can rely entirely on these types.
Internally, these are the same type: they alias CliffordNumbers.Z2CliffordNumber{P,Q,T,L}, where P is a Boolean parameter which is false for EvenCliffordNumber and true for OddCliffordNumber.
KVector{K,Q,T,L}: elements of homogeneous grade
This type represents a k-vector, or a Clifford number of homogeneous grade, with the parameter K indicating the grade.
It should be noted that in general, this type is not as efficient as EvenCliffordNumber, OddCliffordNumber, or CliffordNumber when calculating products (though this may change in the future). Use this type if your primary operations are addition, or if you need compact storage.
However, there is one exception to this: KVector{0}, which represents a scalar. This type has been optimized so that operations with it are simply converted to scalar operations. Many operations on Clifford numbers that return scalars will return a KVector{0} to preserve the metric signature and other semantics associated with AbstractCliffordNumber.
It is important to note that k-vectors are not k-blades (the wedge product of k 1-vectors) or k-versors (the geometric product of k 1-vectors). In dimensions up to 3, all k-vectors are also k-blades, but this is not generally true: as a counterexample, $e_1 e_2 + e_3 e_4$ is not representable as a k-blade. However, all k-blades are k-vectors. k-versors usually consist of more than one grade.
In the future, we may consider adding a DualKVector or PseudoKVector type to more easily represent pseudoscalars, pseudovectors, and related objects.
Promotion and widening
This package provides a robust promotion system for converting Clifford numbers and scalars to common types before performing common numeric operations.
When promoting the types of Clifford numbers, there are two different types of promotions that can occur: scalar promotions, which promote all the scalar types of the arguments to a common scalar type, and grade promotions, which promote all the arguments to types which have a common set of grades. This package provides the scalar_promote function that allows for the scalar types of each argument to be promoted to a common type. Promote rules have been defined so that promote performs a scalar promotion and a grade promotion. No function currently promotes only the grades of the inputs.
The widen function in Julia Base widens an Number type to a type that can represent the result of addition or subtraction with the the input type without overflowing or losing precision. This functionality is passed through to Clifford numbers, but it only affects the scalar type, not the grades.
The widen_grade function performs an equivalent operation with the grades of a Clifford number, converting KVector to EvenCliffordNumber or OddCliffordNumber depending on the grade, and converting those to CliffordNumber. New AbstractCliffordNumber subtypes will widen directly to CliffordNumber by default, and this should be overridden for new types so that it widens to the smallest wider type.
Construction and conversion
From the constructors
The constructors of all AbstractCliffordNumber subtypes accept Tuple or Vararg arguments. In interactive use, you will probably use the latter. When defining a new type, you only need to define the (::Type{T})(::Tuple) constructors, as the Vararg constructors are automatically provided.
Some type parameters may be omitted in constructors, and the differences in behavior between these constructors is given below, using CliffordNumber as an example:
CliffordNumber{Q,T}(x...)converts all argumentsxto typeT.CliffordNumber{Q}(x...)promotes all argumentsxto a common typeT, so it is equivalent toCliffordNumber{Q,promote_type(typeof.(x)...)}(x).CliffordNumber(x...)is not a valid constructor, as an algebra must be specified.
For types that include grade information, such as KVector, this information must be included to produce a valid constructor.
Indices
In most literature, the components of multivectors are listed in grade order. However, this ordering is not used here: instead, the natural binary ordering of blades is used.For a computer, each basis blade of an $n$-dimensional algebra can be represented with an $n$-bit integer: each of its binary digits correspond to the presence or absence of a vector composing the blade.
For a concrete example, the coefficients of a CliffordNumber{VGA(3)} are ordered like so:
\[\left(1, e_1, e_2, e_1 e_2, e_3, e_1 e_3, e_2 e_3, e_1 e_2 e_3\right)\]
Note how $e_1 e_2$ precedes $e_3$ here, but also note how the first four elements and the second four elements only differ by the presence of an $e_3$ factor.
CliffordNumber and its backing Tuple can be indexed straightforwardly with this relationship. The basis blade order of all AbstractCliffordNumber instances are identical, with smaller types like KVector{2,VGA(3)} skipping over all basis blades not of grade 2 in the list above.
Many resources do not use a lexicographic order for the bivectors of the algebra of physical space or the spacetime algebra, opting for cyclic permutations so that $e_3 e_1$ is preferred over $e_1 e_3$. It's a good idea to check the convention before construction so you can include any necessary negative signs.
As a workaround for the possibly unintuitive ordering of coefficients, you can also use sums of KVector instances: the sum will automatically promote to EvenCliffordNumber, OddCliffordNumber, or CliffordNumber as needed to represent all grades.
Scalars
CliffordNumber{Q} and EvenCliffordNumber{Q} also accept a single scalar argument, and this constructs an object with all non-scalar blade coefficients being zero. By definition, KVector{0} does the same, and is the most efficient representation of a scalar associated with an algebra.
From other Clifford numbers
Clifford numbers can be constructed from other CliffordNumbers. This implicitly performs a grade projection operation, so this construction will always succeed, even if some of the basis blades of the input are lost. By contrast, conversion will throw an InexactError if the result does not contain all of the basis blades of the result.
julia> test = CliffordNumber{VGA(3)}(1, 2, 3, 4, 5, 6, 7, 8)
8-element CliffordNumber{VGA(3), Int64}:
1 + 2σ₁ + 3σ₂ + 5σ₃ + 4σ₁σ₂ + 6σ₁σ₃ + 7σ₂σ₃ + 8σ₁σ₂σ₃
julia> EvenCliffordNumber(test)
4-element EvenCliffordNumber{VGA(3), Int64}:
1 + 4σ₁σ₂ + 6σ₁σ₃ + 7σ₂σ₃
julia> convert(EvenCliffordNumber, test)
ERROR: InexactError: ...This is an extremely important point: construction of a Clifford number type with fewer grades than the input performs a grade projection operation without throwing an error. However, conversion will throw an error if the grades of the input value are not present in the input type.
This is not how other subtypes of Number defined by Julia Base behave, as their conversion operations are generally defined to be identical to the constructor, and always throw the same error for a given pair of type and value.
If converting an AbstractCliffordNumber to any other numeric type, construction and conversion behave identically, as expected.
Construction and conversion of Clifford numbers from other Clifford numbers the only time that the quadratic form type parameter can be omitted, as it can be inferred directly from the input. In the case of CliffordNumbers.Z2CliffordNumber, the parity type parameter can also be inferred from a KVector input.
Scalar conversion
It may be desirable to convert the scalar type of a Clifford number without having to specify the full typename of the desired output type. The function scalar_convert(T, x) takes a type T<:Union{Real,Complex} and any Clifford number x and converts its scalar type to T. If x is a Real or Complex, it just converts x to an instance of T.