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American Journal of Physics

Pauli principle in Euclidean geometry

R. W. Hartung

Published Oct 1, 1979• 16 citations
Publisher

Abstract

A macroscopic toy of Dirac and Stong shows how to detect 2π rotations in our world. A theorem in geometry shows the toy is maximal; no device in three dimensions can measure bigger angles than this toy. Assuming then that the toy correctly models motion of fermions in our world, a ’’hand waving’’ derivation of the Pauli principle follows. Finally algebra is brought in to express the geometric truth. A famous sign ambiguity goes away.

DOI
10.1119/1.11635

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A macroscopic toy of Dirac and Stong shows how to detect 2π rotations in our world. A theorem in geometry shows the toy is maximal; no device in three dimensions can measure bigger angles than this toy. Assuming then that the toy correctly models motion of fermions in our world, a ’’hand waving’’ derivation of the Pauli principle follows. Finally algebra is brought in to express the geometric truth. A famous sign ambiguity goes away.
The rotation of a spin-½ fermion by 2π produces a phase factor of −1, not +1. This fact, confirmed by neutron interferometry (Werner et al. 1975; Rauch et al. 1975) and routinely exploited in quantum information, precision metrology, and the chemistry of atoms, has no classical analogue. In the Standard Model it is accommodated by postulating that fermions transform under the SU(2) representations with half-integer label j, and the spin–statistics theorem — that integer-spin particles obey Bose–Einstein statistics and half-integer-spin particles Fermi–Dirac — is established by Pauli’s 1940 proof using Lorentz invariance, microcausality, and energy-positivity. The spin label, the −1 phase, and the statistics theorem arrive in the conventional picture from three separate arguments, each importing assumptions from a different layer of theory. We show that in the (3+3) spacetime framework (de Haan 2026, book manuscript), in which the third time dimension t₃ is compactified as a discrete two-sphere S² with 2^152 Planck-area cells and three mutually orthogonal great circles, all three results follow from the discrete geometry itself — specifically from the Berry phase of orbits on the three great circles (computed from their solid angles) and the quaternionic i^k phase that the yz circle carries as a Hopf fibre. The electron is the spin-½ mode on the yz circle; the −1 phase on 2π rotation is i^2 accumulated over the two equatorial crossings of one yz orbit; the spin–statistics connection reads off directly from the Berry-phase sign; Pauli exclusion follows from the forced antisymmetry. The derivation uses only the discrete cell structure, the three great circles, the solid angles they subtend, and the quaternionic phase from the Hopf fibration S³ → S². It does not invoke SU(2) representation theory, the Dirac equation, Lorentz invariance, microcausality, or energy-positivity — each of these is itself derived elsewhere in the framework and using them as inputs would import derived concepts into a derivation that should be upstream of them. The paper’s contribution is the explicit statement that three previously-independent features of quantum mechanics — the half-integer spin label, the −1 phase on 2π, and the spin–statistics theorem — reduce to consequences of a single geometric statement: the electron lives on the yz great circle of the discrete S² and carries the Hopf-fibre i^k quaternionic phase. The textbook SU(2)/SO(3) account of spin is recovered as a representation-theoretic label for what the geometry has already produced, not as an axiomatic input.
The Geometric Algebra formalism opens the door to developing a theory replacing conventional quantum mechanics. Generalizations, stemming from changing of complex numbers by geometrically feasible objects in three dimensions, followed by unambiguous definition of states, observables, measurements, bring into reality clear explanations of weird quantum mechanical features, for example primitively considering atoms as a kind of solar system. The three-sphere S3 becomes the playground of the torsion kind states eliminating abstract Hilbert space vectors. The S3 points evolve, governed by updated Schrodinger equation, and act in measurements on observable as operators.
We set up the Maxwell's equations and subsequently the classical wave equations for the electromagnetic waves which together with their generating source, an oscillatory charge of zero rest mass in general travelling, make up a particle travelling similarly as the source at velocity ν in the field of an external scalar and vector potentials. The direct solutions in constant external field are Doppler-displaced plane waves propagating at the velocity of light c ; at the de Broglie wavelength scale and expressed in terms of the dynamically equivalent and appropriate geometric mean wave variables, these render as functions identical to the space-time functions of a corresponding Dirac spinor, and in turn identical to de Broglie phase waves previously obtained from explicit superposition. For two spin-half particles of a common set of space-time functions constrained with antisymmetric spin functions as follows the Pauli principle for same charges and as separately indirectly induced based on experiment for opposite charges, the complete wave functions are identical to the Dirac spinor. The back-substitution of the so explicitly determined complete wave functions in the corresponding classical wave equations of the two particles, subjected further to reductions appropriate for the stationary-state particle motion and to rotation invariance when in three dimensions, give a Dirac equation set; the procedure and conclusion are directly extendible to arbitrarily varying potentials by use of the Furious theorem and to particle motions in three dimensions by virtue of the characteristics of de Broglie particle motion. Through the derivation of the Dirac equation, the study hopes to lend insight into the connections between the Dirac wave functions and the electrodynamic components of simple particles under the government by the well established basic laws of electrodynamics.
In "Part I: Vector Analysis of Spinors", the author studied the geometry of two component spinors as points on the Riemann sphere in the geometric algebra of three dimensional Euclidean space. Here, these ideas are generalized to apply to four component Dirac spinors on the complex Riemann sphere in the complexified geometric algebra of spacetime, which includes Lorentz transformations. The development of generalized Pauli matrices eliminate the need for the traditional Dirac gamma matrices. We give the discrete probability distribution of measuring a spin 1/2 particle in an arbitrary spin state, assuming that it was prepared in a given state immediately prior to the measurement, independent of the inertial system in which measurements are made. The Fierz identities between the physical observables of a Dirac spinor are discussed.