Structure of canonical variables in the theory of quantum systems with finitely and infinitely many degrees of freedom

N. V. Borisov


Abstract: It is shown that any representation Of canonical variables, (i.e., a representation of the canonical commutation relations in the Heisenberg form) is a direct integral of irreducible (factor) representations; no assumptions are made concerning the possibility of a transition to the Weyl form of the commutation relations. This theorem is applied to the construction of decompositions into irreducible (factor) representations of any finite-dimensional and some inifinitedimensional Lie algebras by unbounded operators in Hilbert space. The need for such decompositions arises in the harmonic analysis of unitary representations of the corresponding Lie groups.

Received: 09.04.1973

 English version: Theoretical and Mathematical Physics, 1974, 19:1, 325–331

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