Garding domains and analytic vectors for quantum fields

If one studies the canonical commutation relations (CCR's) of quantum field theory in the unitary Weyl form, one does not know if one can find a common dense domain for the field operators since their domain of definition depends on the test function. We consider here a general class of test function spaces including the spaces S and D of Schwartz and the space U0≃R(∞) of all finite linear combinations of a countable basis. It is shown that there exists an invariant Gårding domain D on which all fields are defined and strongly continuous. D consists of analytic vectors for the fields. It turns out that the test function space can be enlarged by continuity. For irreducible or factor representations it becomes even a Hilbert space. The basic idea of the proof is the same as in the Schrödinger representation for one degree of freedom and very transparent. We simply use rapidly decreasing functions in ``Q-space'' and ``P-space'' as smoothing factors. That this can be done in the infinite case also is due to a new and interesting measure theoretic result derived here. As an application of our results, we mention that the renormalized fields (after removing the cutoff) of the Φ22n model of Glimm and Jaffe possess a Gårding domain for test functions in S or D for each time.