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Metric signatures

Clifford algebras are characterized by the metric signatures of the spaces they represent. Some are very commonly used, such as the algebra of physical space (APS), or are generated as part of a family, such as the projective geometric algebras (PGAs), but in other cases you may need the flexibility to work with custom metric signatures.

The CliffordNumbers.Metrics submodule provides tools for working with metric signatures.

Interface

The type parameter Q of AbstractCliffordNumber{Q,T} is not constrained in any way, which means that any type or data consisting of pure bits may reside there. However, for the sake of correctness and fully defined behavior, Q must satisfy an informal interface.

Metric signature objects are treated like AbstractVector{Int8} instances, but with the elements constrained to be equal to +1, 0, or -1, corresponding to basis 1-blades squaring to positive values, negative values, or zero. In the future, we may support arbitrary values for this type.

This array is not constrained to be a 1-based array, and the values of eachindex for the array correspond to the indices of the basis 1-blades of the algebra.

The Metrics.AbstractSignature type

We define a type, Metrics.AbstractSignature <: AbstractVector{Int8}, for which this interface is already partially implemented.

Pre-defined signatures

There are many commonly used families of algebras, and for the sake of convenience, we provide four subtypes of Metrics.AbstractSignature to handles these cases:

  • Metrics.VGA represents vanilla geometric algebras.
  • Metrics.PGA represents projective geometric algebras.
  • Metrics.CGA represents conformal geometric algebras.
  • Metrics.LGA{C} represents Lorentzian geometric algebras:
    • Metrics.LGAEast uses the East Coast convention (timelike dimensions square to -1).
    • Metrics.LGAWest uses the West Coast convention (timelike dimensions square to +1).

To construct an instance of one of these types, call it with the number of modeled spatial dimensions:

  • Metrics.VGA(3) models 3 spatial dimensions with no extra dimensions.
  • Metrics.PGA(3) models 3 spatial dimensions with 1 degenerate (zero-squaring) dimension.
  • Metrics.CGA(3) models 3 spatial dimensions with 2 extra dimensions.
  • Metrics.LGAEast(3) models 3 spatial dimensions with an extra negative-squaring time dimension.