Abstract
Some models of Many Body Theory and Quantum Statistical Mechanics are for mulated and sometimes solved in terms of variables or fields which formally satisfy the Canonical Commutation Relations (CCR), but which actually cannot be represented as operators in a Hilbert space because of their bad infrared behaviour1,2. Typical examples are: 1. the infinite quantum harmonic lattice in thermal equilibrium and the Free Bose gas, both in space dimensions d ≤ 2 3,4; 2. the electrons in a periodic potential (Bloch electrons)3; 3. the massless scalar field in two space-time dimensions5. The strategies usually adopted to discuss the above models fall essentially in two categories: to represent the singular fields as operators on an indefinite metric space5; or to use a restricted set of field variables and describe the remaining degrees of freedom as morphisms of the regular field algebra6.
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