Particle representations of canonical commutation relations

The aim of the present paper is to find a necessary and sufficient condition for the existence of the operator of the total number of particles in a representation of canonical commutation relations. The result is formulated by means of generating functionals of representations. It is shown that an irreducible representation possesses a (generalized) number operator if and only if the representation obtained by averaging its generating functional over the group of phase transformations is a factor representation of type I.     

Exponential representations of the canonical commutation relations

A class of representations of the canonical commutation relations is investigated. These representations, which are called exponential representations, are given by explicit formulas. Exponential representations are thus comparable to tensor product representations in that one may compute useful criteria concerning various properties. In particular, they are all locally Fock, and non-trivial exponential representations are globally disjoint from the Fock representation. Also, a sufficient condition is obtained for two exponential representations not to be disjoint. An example is furnished by Glimm's model for the :⁴: interaction for boson fields in three space-time dimensions.     

On representations of the canonical commutation relations

In the measure space construction of a representation of the canonical commutation relations, the strong continuity of any one parameter subgroup is proved.All multipliers for the separable case are expressed in a constructive manner and an irreducibility criterion for a subset of multipliers is obtained.Preprint No. 1970-27.On leave from Research Institute for Mathematical Sciences Kyoto University, Kyoto, Japan.     

A note on product measures and representations of the canonical commutation relations

There is a well-known theorem which states that a non-zero -finite left quasi-invariant measure on a -compact locally compact groupG must be equivalent to left Haar measure. It is shown in this paper that there is a natural generalization of this fact to the case in which the groupG is replaced by a product space, one factor of which is a group. With the aid of this generalization, an easy proof of the following fact, due to H. Araki, is given: the representations of the canonical commutation relations constructed in the usual measure-theoretic manner are ray continuous.     

Number operators for representations of the canonical commutation relations

Anumber operator for a representation of the canonical commutation relations is defined as a self-adjoint operator satisfying an exponentiated form of the equationNa*=a*(N+I), wherea* is an arbitrary creation operator. WhenN exists it may be chosen to have spectrum {0, 1, 2, ...} (in a direct sum of Fock representations) or {0, ±1, ±2, ...} (otherwise). Examples are given of representations having number operators, and a necessary and sufficient condition is given for a direct-product representation to have a number operator.     

Topologies induced by representations of the canonical commutation relations

This paper is concerned with continuity properties of representations of the canonical commutation relations, and is mainly devoted to a detailed discussion of the topologies induced on the test function spaces. The notion of closability of a representation of the canonical commutation relations is introduced and studied. We also discuss the strong continuity of functions of self-adjoint operators, and use bounded functions to define an analogue of the strong operator topology on the set of all self-adjoint operators.     

Continuity properties of the representations of the canonical commutation relations

We prove that a given representation of the canonical commutation relations can be extended uniquely by continuity to larger test function spaces which are maximal in the sense that no further extension is possible. For irreducible tensor product representations of the canonical commutation relations we give a necessary and a sufficient condition for the admissible test functions. We consider the problem of finding topologies on the test function spaces such that this extension can be obtained by a topological completion. Various examples are discussed.Supported in part by the National Research Council of Canada.An earlier version of the present work was distributed as a preprint entitled Topologies for Test Function Spaces for Representations of the Canonical Commutation Relations.     

On factor representations and theC*-algebra of canonical commutation relations

A newC*-algebra,A, for canonical commutation relations, both in the case of finite and infinite number of degrees of freedom, is defined. It has the property that to each, not necessarily continuous, representation of CCR there corresponds a representation ofA. The definition ofA is based on the existence and uniqueness of the factor type II₁ representation. Some continuity properties of separable factor representations are proved.     

Deformations of the canonical commutation relations

Deformations of the canonical commutation relations which have the effect of altering the spectrum of a standard Hamiltonian, bilinear in creation and annihilation operators are described. The problem of going over from an eigenvalue situation, as is the case in the vast majority of papers in the literature, to a theory with time evolution is discussed, and a special example with deformation parameter an Nth root of unity is constructed which possesses a consistent time evolution. This work is an account of some recent studies of associative deformations of the Heisenberg algebra of several creation and annihilation algebras, with Jean Nuyts of the University of Mons, Hainaut, together with some observations of my own concerning the difficulty of implementing time evolution in a quantum group context. It builds on earlier work with Cosmas Zachos (Argonne National Laboratory, USA), which in turn is re;ated to work of Manin, and Wess, Zumino and collaborators. The main idea is that, if quantum groups have any role in physics, then they must manifest themselves at the level of the basic rules of quantisation.     

On representations of canonical commutation relations in indefinite metric quantum field theory

Representations of the abstract algebra of CCR in indefinite inner product space are investigated. It is shown that these representations are characterized by functions with some non-standard positive definiteness property.     

Bogolyubov transformations for regular representations of canonical commutation relations in space with an indefinite metric

Bogolyubov transformations for regular representations of the algebra of canonical commutation relations (CCRs) are considered in space with an indefinite metric.     

A theorem on canonical commutation and anticommutation relations

The aim of this note is to characterize representations of the canonical commutation or anticommutation relations which, on a subspace of the space of test-functions, reduce to a sum of copies of the Fock representation.     

The form of representations of the canonical commutation relations for Bose fields and connection with finitely many degrees of freedom

Given a representation of the canonical commutation relations (CCR) for Bose fields in a separable (or, under an additional assumption, nonseparable) Hilbert space it is shown that there exists a decreasing sequence of finite and quasi-invariant measures _n on the space of all linear functionals on the test function space, such that can be realized as the direct sum of the , the space of all _n -square-integrable functions on. In this realizationU(f) becomes multiplication by. The action ofV(g) is similar as in the case of cyclicU(f) which has been treated byAraki andGelfand. But different can be mixed now. Simply transcribing the results in terms of direct integrals one obtains a form of the representations which turns out to be essentially the direct integral form ofLew. All results are independent of the dimensionality of and hold in particular for dim. Thus one has obtained a form of the CCR which is the same for a finite and an infinite number of degrees of freedom. From this form it is in no way obvious why there is such a great distinction between the finite and infinite case. In order to explore this question we derive von Neumanns theorem about the uniqueness of the Schrödinger operators in a constructive way from this dimensionally independent form and show explicitly at which point the same procedure fails for the infinite case.Part of this paper is contained in Section IV of theHabilitationsschrift Aspekte der kanonischen Vertauschungsrelationen für Quantenfelder byG. C. Hegerfeldt, University of Marburg 1968.     

Time-energy uncertainty and relativistic canonical commutation relations in quantum spacetime

It is shown that the time operatorQ ⁰ appearing in the realization of the RCCR's [Q,P^v]=–jhg^v, on Minkowski quantum spacetime is a self adjoint operator on Hilbert space of square integrable functions over _m =×v _m , where is a timelike hyperplane. This result leads to time-energy uncertainty relations that match their space-momentum counterparts. The operators Q appearing in Born's metric operator in quantum spacetime emerge as internal spacetime operators for exciton states, and the condition that the metric operator should possess a ground exciton state assumes the significance of achieving minimal spacetime4-momentum uncertainty in fundamental standards for spacetime measurements.Supported in part by NSERC research grant No. A5206.     

Extension of Tomita-Takesaki theory to the unbounded algebra of the canonical commutation relations

We consider the unbounded CCR algebra in infinitely many degrees of freedom equipped with a suitable faithful state. We prove that this state satisfies the KMS condition with respect to a certain time evolution and the associated unbounded GNS representation π_β has the structure encountered in Tomita-Takesaki theory; what is more, the commutant π_β$\text{U}$′ is a standard von Neumann algebra, invariant under the time evolution.     

Complex extension of the representation of the symplectic group associated with the canonical commutation relations

The aim of the investigation is to extend the representation of the real symplectic group associated with the canonical commutation relations into the complex symplectic group. It is shown that an extension exists to a semigroup S such that Sp(2n, R) ? S ? Sp(2n, C). The construction of the extension is achieved by extensive use of Bargmann's reproducing kernel space. We are able to give a simple geometric model for the semigroup S: it is exactly the semigroup of all complex symplectic transformations which increase the U(n, n) “length”.     

On the generalization of canonical commutation relations with respect to the orthogonal group in even dimensions

A method is proposed to obtain a set of commutation relations for creation and annihilation operators. This method has to do with the Lie algebra of the orthogonal group of even dimensions. It is shown that the irreducible representations with unique vacuum state lead to the well-known para-Fermi commutation rules. A conjecture is made that this conclusion holds also for other proposals to quantize by means of commutation rules which are related to the orthogonal group in even dimensions.     

Application of commutator theorems to the integration of representations of Lie algebras and commutation relations

Sufficient conditions on unbounded, symmetric operatorsA andB which imply that $$\exp (itA)\exp (isB)\exp ( - itA)$$ satisfies the well known “multiple commutator” formula are derived. This formula is then applied to prove new necessary and sufficient conditions for the integrability of representations of Lie algebras and canonical commutation relations and the commutativity of the spectral projections of two commuting, unbounded, self-adjoint operators. A classic theorem of Nelson's is obtained as a corollary. Our results are useful in relativistic quantum field theory.     

Metrics on test function spaces for canonical field operators

In a canonical field theory, the field (f) and momentum (g) are assumed defined for test functionsf andg which are elements of linear vector spaces and, respectively. Generally, the continuity of the map onto the unitary Weyl operatorsU(f),V(g) is taken as ray continuity, the barest minimum to recover the field operators as their generators, i.e.,U(f)=e ^i(f) ,V(g)=e ^i(g) . This leaves open the question of whether any wider continuity properties follow and what form they would take. We show that much richer continuity properties do follow in a natural fashion for every cyclic representation of the canonical commutation relations. In particular, we show that the test function space may be taken as a metric space, that the space may be uniquely completed in this topology, and that the map into the unitary Weyl operators is strongly continuous in this topology. The topology induced by this metric is minimal in the sense that it is the weakest vector topology for which the mapsfU(f),gV(g) are strongly continuous. An expression for a suitable metric can easily be given in terms of a simple integral over a state on the Weyl operators.