Inequivalent observable algebras. Another ambiguity in field quantisation

It is shown that, already for source-free Maxwell fields, one can construct observable algebras which in the classical theory are on the "same footing" as the one normally used but which in the quantum theory cannot be represented by operators on the standard Fock space. Thus, in quantum field theory there is a genuine freedom in the choice of the operator algebra over and above the well-known freedom in the choice of the representation of a given algebra. This freedom is likely to play an important role in non-perturbative treatments of non-abelian theories.