Question

What is the link between the density matrix and Hestenes' spinors in geometric algebra?

Answer 1

The density matrix (or state matrix) is a generalization of a wave function that is able to describe incoherent superpositions of an N-state system. It is often written as a matrix and observables are calculated by taking the trace of the product of itself with the operator in question.

Hestenes' spinors are a geometric algebra description of things like a wave function and Dirac 4-component spinor, but if one applies the matrix representation of the algebraic elements directly, it resembles a density matrix-like entity much more closely than a wave function, especially when looking at the mathematical expressions for observables. I know and understand the map from "normal" spinors to Hestenes' spinors, which kind of contradicts my proposition.

Nevertheless, a Hestenes spinor can be written as more general entities (a rotor and a scalar). Is this, in light of the above, a further "generalization"

Answer 2

The geometric algebra thus unifies Stokes parameters, the Poincaré sphere, Jones and Mueller matrices, and the coherency and density matrices in a single.