Spacetime Algebra and Electron Physics

Abstract

This paper surveys the application of geometric algebra to the physics of electrons. It first appeared in 1996 and is reproduced here with only minor modifications. Subjects covered include non-relativistic and relativistic spinors, the Dirac equation, operators and monogenics, the Hydrogen atom, propagators and scattering theory, spin precession, tunnelling times, spin measurement, multiparticle quantum mechanics, relativistic multiparticle wave equations, and semiclassical mechanics.

Comment on `Dirac theory in spacetime algebra'

“…It is therefore of no physical consequence that the particular bivector γ 1 γ 2 appears in the Hestenes equation (11). [17] 5 Plane-Wave Solutions…”

Application of Clifford algebra to analysis of spin properties of semiconductors

“…The ket-vector |ψ and its Clifford algebra counterpart ψ satisfy (4) it follows that the spin-up and spin-down states are related by rules | ↑ ←→ 1 and | ↓ ←→ −Iσ 2 . Also, additional replacement rules exist [2,5,19,20] i∂|ψ /∂t ←→ Iψe 3 ,…”

“…In Cl 3,0 the electron spin polarization components are determined by mapping [2,4,5] ψ|σ k |ψ ←→ S k = σ k · S, where The average spin polarization components S k (k = 1, 2, 3 or x, y, z) depend on the initial multivector ψ i and effective energies ε 1 , ε 2 , ε 3 only. Figure 1 illustrates spin polarization precession trajectories on the Bloch sphere calculated with equations (9)-(11).…”

“…ck (cos θ sin 2 θ + sin θ ×cos 2 θ e 2 ) cos 2ϕ− 3+cos 2θ 4 sin θ sin 2ϕ e 3 , (32) with the multivector (19), where all basis vectors appear in the Hamiltonian) that the Clifford algebra that describes hole spin dynamics in decoupled heavy-light band approximation can be reduced to lower dimensional Clifford algebra. Indeed, it can be shown rigorously that in this approximation the total angular momentum operator which determines the precession of the spin polarization vector also reduces to lower Clifford algebra, namely, to Cl 3,0 algebra.…”

“…Two of the algebras, Cl 3,0 and Cl 3,1 , which are also called the geometric algebras [2,4,5], are used to describe particles in 3D Euclidean and 4D Minkowski spaces. Up till now the Clifford algebra was applied to classical mechanics [6,7], Pauli and Dirac spinors in three and four dimensions [1,2], field theory [5], including classical electrodynamics [8][9][10], gravitation and cosmology [5], and recently in computer science [11]. The reader unfamiliar with the geometric algebra can refer to Hestenes paper [3] where one can find an accessible introduction, more references, as well as websites on the Clifford algebra.…”