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.

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A promising approach to the quantum/classical interface is described. It is based on a formulation of relativistic classical mechanics in the Cli¤ord algebra of physical space. Spinors and projectors arise naturally and provide powerful tools for solving problems in classical electrodynamics. They also reproduce many quantum results, allowing insight into quantum processes. I. INTRODUCTIONThe quantum/classical (Q/C) interface has long been a subject area of interest. Not only should it shed light on quantum processes, it may also hold keys to unifying quantum theory with classical relativity. Traditional studies of the interface have largely concentrated on quantum systems in states of large quantum numbers and the relation of solutions to classical chaos,[1] to quantum states in decohering interactions with the environment, [2] or in continuum states, where semiclassical approximations are useful. [3] Our approach[4-7] is fundamentally di¤erent. We start with a description of classical dynamics using Cli¤ord's geometric algebra of physical space (APS). An important tool in the algebra is the amplitude of the Lorentz transformation that boosts the system from its rest frame to the lab. This enters as a spinor in a classical context, one that satis…es linear equations of evolution suggesting superposition and interference, and that bears a close relation to the quantum wave function.Although APS is the Cli¤ord algebra generated by a three-dimensional Euclidean space, it contains a fourdimensional linear space with a Minkowski spacetime metric that allows a covariant description of relativistic phenomena. The relativistic, spinorial treatment of APS is crucial in relating the classical and quantum formalisms. Fermionic, spin-1/2, properties follow from the spinor description of spatial rotations, and the extension to Lorentz transformations yields immediately the momentum-space form of the Dirac equation.Closely related work has been reported using other Cli¤ord algebras, in particular, complex quaternions[8] and the spacetime algebra (STA) of Hestenes[9, 10] and coworkers.[11] While each has its particular advantages and drawbacks,[12] they share a common (isomorphic) spinorial basis for a geometric algebra describing Lorentz transformations in spacetime. Of the three, APS is most intimately tied to the spatial vectors and their associated geometry. It is half the size of STA on the one hand, but unlike complex quaternions, is readily extended by complexi…cation.This paper presents an introduction to APS and brief overview of its results and promise. In the next section, we see how APS arises as the natural algebra of vectors in three-dimensional physical space, but how it also includes a fourdimensional linear space that models spacetime of special relativity. In Section 3, the eigenspinor is introduced and shown to be a valuable tool for …nding the relativistic dynamics of particles. Although it is a classical object, it obeys linear equations of evolution as in quantum mechanics. Section 4 introduces the con...