Quantum/Classical Interface: Classical Geometric Origin of Fermion Spin

Abstract

Although intrinsic spin is usually viewed as a purely quantum property with no classical analog, we present evidence that fermion spin has a classical origin rooted in the geometry of three-dimensional physical space. Our approach to the quantum/classical interface is based on a formulation of relativistic classical mechanics that uses spinors. Spinors and projectors arise naturally in the Clifford's geometric algebra of physical space and not only provide powerful tools in classical electrodynamics, but also reproduce a number of quantum results. We show in particular that many properties of elementary fermions, as spin-1/2 particles, are obtained classically and relate spin, the associated g-factor, its coupling to an external magnetic field, Zitterbewegung, and de Broglie waves. Spinors are also amplitudes that can undergo quantum-like interference. The relationship of spin and geometry is further strengthened by the fact that physical space and its geometric algebra can be derived from fermion annihilation and creation operators. The approach is important because of the insights it provides about spin and quantum phenomena more generally. Mathematics Subject Classification (2000). 15A66, 81P15, 81R25, 83A05.

A Probabilistic Model of Spin and Spin Measurements

“…In another recent theoretical study, evidence is presented that "…fermion spin has a classical origin rooted in the geometry of three-dimensional physical space" [17]. The success of our classical model, therefore, seems to provide evidence that an autonomous motion, which causes the spin as an average over time dependent instantaneous spin vectors, as proposed in [3,6,9,10], is the physical origin of these theoretical results, and offers a "classical explanation" of the formal quantum mechanical Pauli-Schrödinger theory.…”

ACR announcements

“…But at r = 0.1(top) the large and small components have complicated temporal phases. We point out again that the outer part of the four‐space wave function satisfies the Lorentz relation between position and time, r 2 – ( ct ) 2 = 0, which is indeed a signature of a geometric spacetime solution 1. The causal behavior is visible in the top figure where the wave function is essentially absent for ct < r .…”

“…The algebra of physical space (APS)1 is a name for the Clifford or geometric algebra, which can be closely associated with the geometry of special relativity and relativistic spacetime. For example, Dirac's equation may be presented as the scalar product of the electron's four‐momentum and the four‐vector given by Dirac's 4 × 4 gamma matrices, the latter of which is a Clifford algebra, operating on Dirac's four‐component column wave function, ψ D .…”

Algebra of Complex Vectors and Applications in Electromagnetic Theory and Quantum Mechanics

“…The complex vectors Z = a + i b and its conjugateZ = a − i b defined in the Equations (13) and (14) actually form a physical space when the third direction is chosen normal to the plane containing the orthogonal vetors a and b. Let c = 2i (a ∧ b) be a vector normal to both a and b.…”

“…The physical space formed by the complex null vectors is also analogous to the physical space formed by the creation and annihilation operators. Such physical space from fermion creation and annihilation operators was introduced by Baylis [13].…”