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.     

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.     

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.     

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.     

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.     

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.     

Feynman's approach to quantum mechanics: Trajectories,commutation relations and uncertainty principle

Summary In this paper the meaning of trajectory for a quantum-mechanical particle is discussed, starting from the path integral expression of the propagator. By a direct method the trajectories mostly contributing to the total amplitude are found, but it seems impossible to interpret them as paths in the physical space-time; on the contrary, the position-momentum commutation relations directly follow. Moreover, we show that the Heisenberg uncertaint principle can be obtained from the path integral approach. In order to give a better understanding of the characteristic quantum-mechanical features and of the difference from the classical problems, the diffusion equation for a Brownian particle is considered in the first part of the paper. To speed up publication, the authors have agreed not to receive proofs which have been supervised by the Scientific Committee.     

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.     

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.     

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.     

Geometry,commutation relations and the quantum fictitious force

We express the commutation relation between the operators of the momentum and the radial unit vectors in D dimensions in differential and integral form. We connect this commutator with the quantum fictitious potential emerging in the radial Schr?dinger equation of an s-wave. Received: 6 August 2002 / Revised version: 30 October 2002 / Published online: 26 February 2003 RID="*" ID="*"Corresponding author. Fax: +49-731/502-3086, E-mail: markus.cirone@physik.uni-ulm.de     

Canonical commutation relations of quantum mechanics and stochastic regularity

In a previous publication (Boletin de la Sociedad Matematica Mexicana 1975) it was established that any weakly stationary linearly regular stochastic process is unitarily equivalent to a quantum mechanical momentum evolution. The object of the present note is, as promised in the previous publication, to amplify some of the details concerning the just mentioned interesting connection, giving in particular a direct proof of the Szego-Kolmolgorov-Krein characterization of regular stationary processes. We also show that although the so-called decaying states without regeneration do not exist for unstable quantum systems, they are natural for regular stationary processes.     

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.     

A uniqueness theorem for anticommutation relations and commutation relations of quantum spin systems

Under quite natural assumptions we prove that a classical spin lattice has essentially two natural quantum extensions: The quantum spin lattice and the Fermi system. Moreover we derive a transformation from a commuting pair of Fermi systems to a purely anticommuting Fermi system.     

Test function spaces for direct product representations of the canonical commutation relations

It is shown how the test function spaces for the field operator and its canonical conjugate are determined by a given irreducible direct product representation of the canonical commutation relations. An explicit characterization of the admissible test functions (so that the smeared out field operators are selfadjoint) is given in terms of any one product state of the representation space.     

Current commutation relations in the framework of general quantum field theory

In this paper we give a rigorous formulation of Gell-Mann's equal time commutation relations in the framework of general quantum field theory. We show that this can be achieved despite the nonexistence of charge operators for nonconserved currents. Starting from the properly formulated equal time commutation relations of generalized charges, we justify the application of the Gauss-Theorem and we discuss the limits for large times of time dependent generalized charges. The Jost-Lehmann-Dyson representation is used in order to show that the equal time commutation relations always lead to exactly one, frame independent, sum rule. We discuss the connection between properties of the Jost-Lehmann-Dyson spectral function and the convergence of Adler-Weisberger type sum rules.On leave of absence from University of Pittsburgh, Pittsburgh 13 Penna.     

Characterizing multiple solutions to the time-energy canonical commutation relation via quantum dynamics

We address the multiplicity of solutions to the time-energy canonical commutation relation for a given Hamiltonian. Specifically, we consider a particle spatially confined in a potential free interval, where it is known that two distinct self-adjoint and compact time operators conjugate to the system Hamiltonian exist. The dynamics of the eigenvectors of these operators indicate that different time operators posses distinguishing properties that can unambiguously associate them to specific aspects of the quantum time problem.     

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.