A classical PT theorem

The following argument is adapted from that in Bell (1955). I cannot quite follow Bell’s argument1, so I have changed it to generate a result I understand. In outline, the result is as follows. We consider a classical theory given by a system of partial differential equations on a specified set of fields. Let Φ be the space of kinematically allowed fields. (In the general case, we may be dealing with a theory containing a number of interacting fields — scalar fields, tensor fields, etc — so, for a given theory, an element of Φ will be an ordered m-tuple of specified numbers of scalar fields, vector fields, rank 2 tensor fields, etc.) We note that any PDE can be expressed as the vanishing of some functional F : Φ → RM of the fields. (That is, F encodes the dynamics in the sense that: φ ∈ Φ is dynamically allowed iff F (φ) is the zero map on M .) Let there be given a representation of the full Lorentz group L on Φ (that is, we know how each of our fields ‘transforms under Lorentz transformations’). We assume that F is a [‘local’?] polynomial in the fields, and that the fields transform as components of tensors under proper2 Lorentz transformations. (Each of these two assumptions is crucial to the theorem.) It can then be proved that, if the set S of solutions of the