Pauli and the Spin-Statistics Theorem

The following sections are included:

• Table of Contents

• Authors

• Acknowledgements

The following sections are included:

• Introduction

• Preliminary Remarks

• Anticommutation Relations for Dirac Spin-½ Fields

• Commutation Relations for Klein-Gordon Spin-0 Fields

• Concluding Remarks

• Bibliography and References

We describe Pauli's deduction of the Exclusion Principle, based on Stoner's explanation based on four quantum numbers of the Periodic Table of the Elements and the magic numbers 2,8,18,32,⋯.

The discovery of the spin of the electron as the physical interpretation of the fourth quantum number introduced by Stoner is followed through the credited discovery work of Goudsmit and Uhlenbeck, back to a precursor in the suggestion of Compton, and finally to Kronig, whose primary work never reached publication.

The fundamental derivation of the Planck blackbody distribution in the work of Bose, and the immediate recognition by Einstein of the possibility of an ideal gas satisfying the same statistics, with the resulting consequence of Bose-Einstein condensation are presented with the original literature.

The discovery by Heisenberg and by Dirac that the wave functions for identical particles satisfying Bose-Einstein statistics must be symmetric in the interchange of any two particles, and conversely that those satisfying Fermi-Dirac statistics must be antisymmetric, is presented from its origin.

The derivation by Fermi and by Dirac of the Statistical Mechanics for an ideal gas of identical particles satisfying the Pauli Exclusion Principle is developed.

Quantum field theory is introduced following the original paper of Dirac, where Dirac invents creation and annihilation operators and a number of other fundamental concepts, and applies them to the problem of emission and absorption of photons.

The idea of anticommuting operators for quanta satisfying the Pauli Exclusion Principle is followed from Jordan's original proposal to Jordan and Wigner's final complete exposition.

Dirac's invention of the relativistic spin-½ equation with its successful predictions of the electron spin and magnetic moment as well as the correct hydrogen spectrum, led also to a brief period of confusion about the negative energy states. The resolution of this confusion led Dirac, with the prompting of Weyl, Oppenheimer, and others, to the prediction of both the positron and the antiproton.

The controversies over the interpretation of the relativistic scalar wave equation were not finally resolved until 1934 when Pauli and Weisskopf showed that the paradoxes were overcome by canonical relativistic quantum field theory and, necessarily, a many body interpretation of the wave equation.

Pauli made the first attempt to prove that the spin-statistics relation is a necessary result of the postulates of relativistic quantum field theory. Almost simultaneously, Iwanenko and Socolow showed that quantization of the Dirac equation using anticommutation relations led naturally to results analogous to those of Pauli and Weisskopf for the spin-0 equation. In their paper, Iwanenko and Socolow also point out that problems arise from the wrong commutation relations.

Fierz's proof of the Spin-Statistics Theorem is the first formal and in any sense rigorous proof. It is based initially on the premise that each elementary particle corresponds to an irreducible relativistic spinor. Starting from this premise, Fierz proves the Spin-Statistics Theorem from the basic requirements of relativistic covariance, and positive definite energy. He points out in retrospect that the initial premise of irreducibility is actually unessential, thereby paving the way for Pauli's more general proof.

Belinfante based his proof of the Spin-Statistics Theorem on the requirement of invariance under a charge-conjugation transformation that had recently been invented by Kramers. The uniqueness of this requirement was criticized and, by using a series of his familiar but now recognizably invalid non-local transformations, ceemingly refuted by Pauli, with Belinfante himself as an apparently futilely protesting coauthor.

deWet proved that Fermi-Dirac quantization is not possible for tensor equations, leaving Bose-Einstein quantization as the only possibility for integral spin fields. Both are possible for the Dirac equation but Bose-Einstein "leads to difficulties".

Pauli's proof of the Spin-Statistics Theorem is based on a classification of the spinor representations of the proper Lorentz group into four classes different under the strong-reflection transformation x_µ → -x_µ. From this very general beginning, Pauli obtains the spin-statistics connection for non-interacting particles of arbitrary spin which are not necessarily represented by irreducible spinors (as was the case with Fierz and Belinfante) or subject to canonical field theory (as was the case for Iwanenko and Socolow, and for deWet). He still had to assume (with all the above), without proof, the continuability of field products to zero separation.

In the very first paper introducing his novel contributions to relativistic quantum field theory, Feynman shows that the one-loop contribution to the vacuum-to-vacuum amplitude, calculated according to the new Feynman Rules, gives a vacuum survival probability greater than unity when calculated with the "wrong" spin-statistics relation. This impossible result is interpreted by Feynman as proving the necessity of the correct spin-statistics relation.

Pauli interprets the result as due to an indefinite metric in the Hilbert space of states when the "wrong" statistics is used, in violation of one of the basic postulates of his original proof.

Schwinger's proof is based on the postulate of invariance under strong-reflection, following Pauli's original observation. Schwinger requires an anticommutation between Dirac field operators and an interchange of initial and final states, to maintain the invariance. This logic will eventually be reversed - most explicitly by Lűders and Zumino - and the full TCP-invariance will be based on the the Spin-Statistics Theorem, although Schwinger persistently defended his point of view.

Lűders and Zumino, and independently Burgoyne, proved the Spin-Statistics Theorem from basic postulates on quantum field theory, including invariance under the proper Lorentz group with no reflections. The analytic continuation in the separation of two fields in a vacuum expectation value of their product, due to Hall and Wightman, as well as their ability to treat interacting fields, are the key ingredients which advance their proofs over those of Fierz, deWet, and Pauli.

The Hall-Wightman theorem establishes that the vacuum expectation value

The long pursuit by Schwinger of an understanding of the spin-statistics connection, based fundamentally on Pauli's strong-reflection invariance as realized in Euclidean field theory, is reviewed.

The responses to Neuenschwander's challenging question - Is there a simple, straightforward, accessible, intuitive, 'physical' proof of the Spin-Statistics Theorem? - are dealt with individually, and in all cases found to lack credibility. This conclusion is a reiteration and expansion of the original evaluation of the responses by Hilborn.

The proofs of the Spin-Statistics Theorem are reviewed and summarized. Following this, we present a new proof of the great theorem which comes closest to satisfying Neuenschwander's requirements: the proof is simple, physical, and intuitive, but still not completely free from the complications of relativistic quantum field theory. This is as much as one can hope for the proof of a theorem whose most dramatic physical consequences are manifestly extremely non-relativistic: the Exclusion Principle effects on electrons in atoms and metals, the Bose-Einstein condensation effects in superfluids and superconductors, in lasers, and most recently in trapped atoms condensed at tenths of a micro-Kelvin. Finally, the most fundamental possible understanding of the spin-statistics connection is presented, based on the fact that classical oscillators satisfying Bose-Einstein statistics must necessarily be identified with spin-0 fields; those satisfying Fermi-Dirac statistics must necessarily be identified with spin-½ fields.

The following sections are included:

• Index