Relativistic Quantum Mechanics
The relativistic quantum wavefunction is sometimes expressed as a spinor field, i.e.
where ϕ is a bivector, and
where according to its derivation by David Hestenes, is an even multivector-valued function on spacetime, is a unimodular spinor (or “rotor”), and and are scalar-valued functions.
This equation is interpreted as connecting spin with the imaginary pseudoscalar. R is viewed as a Lorentz rotation which a frame of vectors into another frame of vectors by the operation, where the tilde symbol indicates the reverse (the reverse is often also denoted by the dagger symbol, see also Rotations in geometric algebra).
This has been extended to provide a framework for locally varying vector- and scalar-valued observables and support for the Zitterbewegung interpretation of quantum mechanics originally proposed by Schrödinger.
Hestenes has compared his expression for with Feynman's expression for it in the path integral formulation:
where is the classical action along the -path.
Spacetime algebra allows to describe the Dirac particle in terms of a real theory in place of a matrix theory. The matrix theory description of the Dirac particle is:
where are the Dirac matrices. In the spacetime algebra the Dirac particle is described by the equation:
Here, and are elements of the geometric algebra, and is the spacetime vector derivative.
Read more about this topic: Spacetime Algebra, Spacetime Algebra Description of Relativistic Physics
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