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The Proof of the Spin-Statistics Theorem
by Gerhart Lüders and Bruno Zumino

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 Gerhart Lüders and Bruno Zumino, entitled Connection between Spin and Statistics, was the first rigorous proof of the Spin-Statistics Theorem. An article published independently by Nicholas Burgoyne 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.

Lüders and Zumino instead relied upon a new theorem in quantum field theory developed by Hall and Wightman. Lüders and Zumino limited their analysis to the cases of spin 0 and spin ½.

For the spin 0 case they made five assumptions:

1. Invariance under the proper inhomogeneous Lorentz group, one that does include any reversals (reflections).
2. Two operators of the same field at points separated by a spacelike interval either commute or anticommute.
3. The vacuum is the unique state of lowest energy.
4. The metric function for the Hilbert space of states is positive.
5. The vacuum state is not identically annihilated by a field.

Lüders and Zumino then show that the assumption of the anticommutation relation for the spin 0 leads to a contradiction. Therefore the commutation relations holds for this case.

For the spin ½ 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 spin ½ field can occupy a particular state. Likewise it is easily shown that any number of spin 0 particlels may occupy a particular state.

Gerhart Lüders was born in Germany in 1920 and received his training in physics in Hamburg. After receving his Ph.D. in Hamburg in 1947 he joined the faculty at Munich in 1950. In 1960 he left Munich to join the faculty at Göttingen where he stayed until his death in 1995.

Bruno Zumino was born in Italy in 1923. He received his Ph.D. in mathematical physics in Rome in 1945. He was a professor at a campus of New York University from 1950 to 1968. He became of senior research fellow at CERN from 1968 to 1982. In 1982 he joined the faculty at the University of California in Berkeley. (To be continued.)

Source:

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