The Metaplectic Representation of the Infinite-Dimensional Symplectic Group

Abstract

In this section we introduce the canonical commutation relations and the CCR-algebra. Although the CCR-algebra has many properties analogous to the CAR-algebra treated in Chap. 2, there is a lack of norm continuity in the definition of the CCR-algebra. This lack of norm continuity is related to the fact that creation and annihilation operators in any representation are unbounded. Physically, this reflects the main difference between fermions, related to the CAR-algebra, obeying the Pauli principle (see Chap. 2), and bosons, related to the CCR-algebra, which do not obey the Pauli principle. So in the case of bosons there is no bound on the number of particles which can occupy a given physical state. This is reflected by the unboundedness of the annihilation and creation operators mentioned above. Usually one treats the boson case from a slightly different viewpoint than the fermion case due to the previously mentioned qualitative difference between the two cases. One uses the so-called Weyl operators which, roughly speaking, are the unitary operators constructed from the self-adjoint closure of the sum of the annihilation and creation operators, to study a bosonic many-particle system. The Weyl operators fulfil the so-called Weyl form of the canonical commutation relations.