﻿ The Proof of the Spin-Statistics Theorem by Nicholas Burgoyne
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The Proof of the Spin-Statistics Theorem
by Nicholas Burgoyne

In the judgement of Ian Duck and E.C.G. Sudarshan in their book, Pauli and the Spin-Statistics Theorem the article published in 1958 by Nicholas Burgoyne entitled On the Connection of Spin and Statistics, was one of the first rigorous proofs of the Spin-Statistics Theorem. An article published independently by Lüders and Zumino also in 1958 achieved the same result.

Previous attempts at mathematically rigorous proof had essentially relied upon some truncated version of the TCP Theorem. TCP stands for Time reversal, Charge conjugation and Parity reversal. Duck and Sudarshan note that the TCP Theorem and the Spin-Statistics Theorem are mutually interdependent and with the assumption of one the other can be proved. Other theorists had assumed the validity of invariance under time reversal or charge conjugation. These are not true independently; it is the invariance under the combination of T, C and P which is true. Lüders and Zumino noted that

The situation is rather unsatisfactory … no independent proof of either of these theorem has been given.

Burgoyne instead relied upon a new theorem in quantum field theory developed by Hall and Wightman. Burgoyne did not limit his analysis to the special cases of spin 0 and spin ½.

Burgoyne made the assumption of relativistically invariant field theory which has the following properties:

1. No negative energy states.
2. The metric function for the Hilbert space of states is positive definite.
3. Two operators of different fields at points separated by a spacelike interval either commute or anticommute.
4. The vacuum state is not identically annihilated by a field.

Burgoyne then shows that the assumption of the anticommutation relation for the integral spin case leads to a contradiction. Therefore the commutation relations holds for the case of integral spin.

For the half-integral case the assumption of the commutation leads to a contradiction and hence the anticommutaton relation for this case. With the commutation and anticommutataion relations established for the two cases it is easy to establish that no more than one particle of a half-integral spin field can occupy a particular state. Likewise it is easily shown that any number of integral spin particlels may occupy a particular state.

Peter Nicholas Burgoyne was born in 1932. He received his bachelor's degree from McGill University in Montreal. Canada in 1955 and spent a year as a student at the Bohr Institute in Copenhagen where he met Wightman. He received his Ph.D. from Princeton in 1961. He then joined the faculty at the University of California in Berkeley and continued there until 1966. In 1966 he went to the University of Illionois in Chicago for two years. In 1968 he returned to California to join the faculty at the University of California in Santa Cruz and remained there until 1995. (To be continued.)

Source:

Ian Duck and E.C.G. Sudarshan, Pauli and the Spin-Statistics Theorem,, World Scientific, Singapore, 1997.