The Geometrical Meaning of Spinors Lights the Way to Make Sense of Quantum Mechanics

G Coddens

doi:10.3390/sym13040659

Abstract

This paper aims at explaining that a key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time R4 and its subgroup SO(3) of the rotations of three-dimensional Euclidean space R3. In the three-dimensional rotation group, the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding that is acquired to derive the free-space Dirac equation from scratch, proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine’s statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further, we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM.

Highlights

Feynman’s statement reflects an unprecedented, very unpleasant situation in physics. We find it enlightening to formulate the problem of the meaning of quantum mechanics (QM) exactly in the same terms as the problem of the meaning of spinors
The present paper proposes such a reconstruction in a way that is perhaps totally different from what a physicist might expect, because it starts the journey by digging into the mathematics of spinors, and derives the Dirac equation from scratch with the rigour of a mathematical proof
Where do we stand now? What we have learned is that the Dirac equation can be derived without introducing extraordinary assumptions, such as many worlds or signalling back in time

Summary

Introduction

Richard Feynman [1] (recipient of the Nobel prize of physics in 1965), is notorious for his statement:. On the other hand Michael Atiyah (winner of the Fields medal in 1966) is not less notorious for having stated:. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the “square root” of geometry and, just as understanding the square root of −1 took centuries, the same might be true of spinors” [2]. Unlike differential forms, which are related to areas and volumes, spinors have no such simple explanation. They appear out of some slick algebra, but the geometrical meaning is obscure . They appear out of some slick algebra, but the geometrical meaning is obscure . . . ” [3]

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