Local fields with the wrong connection between spin and statistics

by R. F. Streater; Commun. in Mathematical Phys., 5, 88-96, 1967.

ABSTRACT

It is proved that there exist free field operators which satisfy local commutativity and for which the labels denoting the components transform according to certain unitary representations of the homogeneous Lorentz group. The fields satisfy axioms similar to the Wightman axioms, and give rise to local algebras of observables obeying postulates similar to those suggested by HAAG. They describe a mass-degenerate tower of particles with spins 1/2, 3/2, 5/2,..., but they commute at space-like separation, giving rise to Bose statistics for the particles. This shows that the well-known theorem on spin and statistics cannot be extended to general theories of local observables; it also shows that the assumptions made in `S-matrix theory' do not hold for the S-matrix of a theory of interacting infinite fields.

REMARKS

I really enjoyed writing a paper with this title, after having co-authored the book PCT, Spin and Statistics, and All That with A. S. Wightman in 1962-1963. Although fields with infinitely many components were studied earlier, by Majorana in 1932 or so, and by Gelfand et al. in the 1940's, these were not second quantised. I conjecture that Dirac had the mistaken belief that his (Dirac) equation did not give rise to a unitary representation of the inhomogeneous Lorentz group (the Poincaré group), because the 4 X 4 matrices appearing in it were not unitary. He may have realised the great importance of unitary representations after Wigner's book, Group Theory with applications to atomic spectroscopy. It might be that the fear that his equation were badly wrong urged Dirac to invent, in about 1945, single-handedly, some irreducible unitary representations of the Lorentz group, a task thought to be too hard for mathematicians at the time. If so, it was all to no avail, as the unitarity of the representation (of the Poincaré group, as opposed to the Lorentz group acting on the spinors) given by the original Dirac equation was shown by Wigner (1939) and by Bargmann and Wigner (1947). Some of the models studied by Gelfand had a tower of different masses of finite multiplicity, and their second quantised versions violate either positivity or local commutativity. This is a consequence of our "No-Go Theorem". My paper was inspired by contemporary work by C. Fronsdal, and by P. T. Matthews and G. Feldman; I thought that these papers would not convince mathematical physicists, since they did not formulate any theorems with proofs. In my paper I suggested that the root of the problem (the wrong statistics) was caused by the violation of the compactness condition of Haag and Swieca, which expresses the thermodynamic stability of the theory. Subsequently, the spin-statistics theorem as well as PCT invariance, has been proved in the Haag framework by adding a suitable extra condition. See Daniel Guido and Roberto Longo, An Algebraic Spin and Statistics Theorem. These authors remark that in my model, the action of the Lorentz group on the algebra is not uniquely defined by its action on space-time; they show that by choosing a different action, which they claim is more natural, the wrong connection can be ruled out. In my opinion, it is better to argue, as they do, that my model violates the axiom called the split property; this is related to thermodynamic stability and guarantees the uniqueness of the Poincaré action on the observables.

The present paper is mentioned in the book Causal Nets of Operator Albegras, by H. Baumgaertel and Manfred Wollenberg (Akademie Verlag, 1992; ISBN 3-05-501421-9). They argue that the result hints that the local algebraic theory is more general that the quantum field theory of Wightman. While this is true, a simple enlargement of the Wightman axioms, to include fields with infinitely many components, will include the models presented in my paper.

Their book discusses, but does not completely solve, the important question of when a Haag field has an underlying Wightman theory.

The Bisognano-Wichmann theorem is needed for an algebraic proof of PCT and spin-statistics. Bisognani and Wichmann proved it in the Wightman context. Jens Mund proved the theorem for massive theories obeying the Haag axioms, in which the stable massive particles have no multiplicity in the mass.