Support of a Field in p Space

Abstract

The notion of generalized free field is introduced, as an obvious extension of a discrete superposition of independent free fields with different masses. The following assumptions are also made: there is an underlying Hilbert space H (positive‐definite metric), the theory is Lorentz invariant, the vacuum belongs to H and is there unique, the spectrum of the energy‐momentum operator is—apart from the origin—completely contained within the region p 2≥ε2, p 0>0. It is then shown that a necessary condition for a cyclic field to have support in p 2 only on a finite interval of the positive real axis, is that A(x) be a generalized free field. In the Appendix a similar result is derived under slightly weaker conditions.