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Journal of Physics A Mathematical and General

Displaced Fock representations of the canonical commutation relations

P. Basarab-Horwath

Published Jun 1, 1981• 4 citations
Publisher

Abstract

Defines displaced Fock representations of the canonical commutation relations in an algebraic framework. Then the author considers the problem of unitary implementability of the symmetry group which acts on the one-particle space of the theory. This action induces an automorphism of the algebra of the CCR and a condition is found to ensure that this automorphism be implemented by a unitary group representation in the space of the displaced Fock representation. Using this condition the author proves the existence of infinitely many displaced Fock representations in which the group automorphism is implemented by a unitary action. This is done for a specific choice of a subgroup of the Poincare group, and for all cases of integer value of spin.

DOI
10.1088/0305-4470/14/6/019

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Defines displaced Fock representations of the canonical commutation relations in an algebraic framework. Then the author considers the problem of unitary implementability of the symmetry group which acts on the one-particle space of the theory. This action induces an automorphism of the algebra of the CCR and a condition is found to ensure that this automorphism be implemented by a unitary group representation in the space of the displaced Fock representation. Using this condition the author proves the existence of infinitely many displaced Fock representations in which the group automorphism is implemented by a unitary action. This is done for a specific choice of a subgroup of the Poincare group, and for all cases of integer value of spin.
Necessary and sufficient conditions are given for the unitary implementability of one-parameter unitary groups of one-particle automorphisms of the CCR algebra in representations symplectically related to the Fock representation. The criteria become particularly simple when the one-particle generator of the unitary group is positive and bounded away from zero; in this case the automorphism group is unitarily implementable only in the representations unitarily equivalent to the Fock representation. If the spectrum of the generator includes zero, however, the situation is more complicated; there then exist representations inequivalent to the Fock representation which admit unitary implementation of the automorphism group. It is also shown that whenever implementation of the automorphism group is possible, the implementing operators can be chosen to be a strongly continuous unitary group, guaranteeing the existence of a ‘‘second-quantized’’ self-adjoint generator.
For the case of infinitely many photon (Boson) modes we investigate the unitary implementability of a class of symplectic one-parameter groups (more exactly, of the associated groups of Bogoliubov automorphisms on the CCR algebra) in the Fock representation and in representations of the CCR algebra, which are symplectically related and inequivalent to Fock. Furthermore, the existence of the associated (squeezing) quadratic Hamiltonians is discussed. Finally, applications in the theory of quantization in QED and in the Luttinger model are pointed out.
We consider Hilbert space representations of a generalization of canonical commutation relations , where 's are the elements of an algebra with identity , is the imaginary unit, and is a real number with antisymmetry . Some basic aspects on Hilbert space representations of the generalized CCR (GCCR) are discussed. We define a Schrödinger-type representation of the GCCR by an analogy with the usual Schrödinger representation of the CCR with degrees of freedom. Also, we introduce a Weyl-type representation of the GCCR. The main result of the present paper is a uniqueness theorem on Weyl representations of the GCCR.
This paper studies a class of representations (called quadratic) of the canonical commutation relations over symplectic spaces of arbitrary dimension, which naturally generalizes coherent and symplectic (i.e. quasifree) representations and which has previously been heuristically employed in the special case of finite degrees of freedom in the physics literature. An explicit characterization of canonical quadratic transformations in terms of a 'standard form' is given, and it is shown that they can be exponentiated to give representations of the Weyl algebra. Necessary and sufficient conditions are presented for the unitary equivalence of these representations with the Fock representation. Possible applications to quantum optics and quantum field theory are briefly indicated.