Spontaneously broken symmetry in axiomatic theory

by R. F. Streater; Proc. Royal Soc., A287, 510-518, 1965.

ABSTRACT

It is suggested that the heuristic notion of spontaneously broken symmetry, introduced by Nambu, Jona-Lasinio and Goldstone in terms of a non-invariant vacuum, might be posed rigorously as the failure of the symmetry automorphism to be spatial in the representation in question. This is shown to be the case for the C*-algebraic formulation of the free boson field of zero mass in 4 space-time dimensions. It is shown that there exist infinitely many inequivalent positive-energy representations of the C*-algebra in that case. It is shown that the representations of the Poincaré group in all the representations are unitarily equivalent.

The concept of broken symmetry given here retains a meaning even when the representation has no vacuum state; for example, in the charge one sector of a relativistic quantum field theory. In such a case, the traditional definition of spontaneously broken symmetry, the non-uniqueness of the vacuum, does not make sense.

Note. This work was presented to a seminar group (including Wigner) at the University of Wisconsin in Madison, July, 1964.