Pseudoscalar - Pseudoscalars in Geometric Algebra

Pseudoscalars in Geometric Algebra

See also: Pseudoscalar (Clifford algebra)

A pseudoscalar in a geometric algebra is a highest-grade element of the algebra. For example, in two dimensions there are two orthogonal basis vectors, and the associated highest-grade basis element is

So a pseudoscalar is a multiple of e₁₂. The element e₁₂ squares to −1 and commutes with all even elements – behaving therefore like the imaginary scalar i in the complex numbers. It is these scalar-like properties which give rise to its name.

In this setting, a pseudoscalar changes sign under a parity inversion, since if

(e₁, e₂) → (u₁, u₂)

is a change of basis representing an orthogonal transformation, then

e₁e₂ → u₁u₂ = ±e₁e₂,

where the sign depends on the determinant of the rotation. Pseudoscalars in geometric algebra thus correspond to the pseudoscalars in physics.