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.     

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.     

Harmonic Oscillator in Characteristic p

We construct an irreducible representation of the canonical commutation relations by operators on a certain Banach space over a local field of characteristic p. The Carlitz polynomials forming the basis of the space are shown to be the counterparts of the Hermite functions for this situation. The analogues of coherent states are related to the Carlitz exponential.     

Representations of the CAR generated by representations of the CCR. Fock case

We present a method of constructing the Fock representation of the canonical anti-communtation relations in the Fock representation of the canonical commutation relations. An explicit formula for Fermi creation and annihilation operators in terms of Bose ones is given.     

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.     

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.     

Self-adjoint algebras of unbounded operators

Unbounded *-representations of *-algebras are studied. Representations called self-adjoint representations are defined in analogy to the definition of a self-adjoint operator. It is shown that for self-adjoint representations certain pathologies associated with commutant and reducing subspaces are avoided. A class of well behaved self-adjoint representations, called standard representations, are defined for commutative *-algebras. It is shown that a strongly cyclic self-adjoint representation of a commutative *-algebra is standard if and only if the representation is strongly positive, i.e., the representations preserves a certain order relation. Similar results are obtained for *-representations of the canonical commutation relations for a finite number of degrees of freedom.Work supported in part by U.S. Atomic Energy Commission under Contract AT(30-1)-2171 and by the National Science Foundation.Alfred P. Sloan Foundation Fellow.     

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.     

Commutation superoperator of a state and its applications to the noncommutative statistics

For a normal state on a von Neumann algebra the space of square-integrable operators is introduced. As distinct from the L² space in the classical probability theory, it possesses an additional skew-symmetric form and the associated superoperator, which is a convenient tool to describe commutation properties in L². It is shown that a state on the algebra of canonical commutation relations is Gaussian (quasi-free) iff the space of canonical observables is an invariant subspace of the corresponding commutation superoperator. Basing on these ideas a new approach to some problems in the noncommutative statistic is developed.     

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.     

Dobiński relations and ordering of boson operators

We introduce a generalization of the Dobiński relation, through which we define a family of Bell-type numbers and polynomials. Such generalized Dobiński relations are coherent state matrix elements of expressions involving boson ladder operators. This may be used in order to obtain normally ordered forms of polynomials in creation and annihilation operators, both if the latter satisfy canonical and deformed commutation relations.     

Quantum mechanics of a constrained particle on an ellipsoid: Bein formalism and Geometric momentum

In this work we apply the Dirac method in order to obtain the classical relations for a particle on an ellipsoid. We also determine the quantum mechanical form of these relations by using Dirac quantization. Then by considering the canonical commutation relations between the position and momentum operators in terms of curved coordinates, we try to propose the suitable representations for momentum operator that satisfy the obtained commutators between position and momentum in Euclidean space. We see that our representations for momentum operators are the same as geometric one.     

Internal geometry of hadron resonances

Motivated by previous work on high-energy quantum mechanics, a simple model is devised to study the internal geometry of hadron resonances. In this model we assume new basic canonical commutation relations between the (internal) coordinate and momentum operators of the hadronic quantum system. By systematically imposing Lie algebra commutation relations between these and other observables, we discuss the free and bound particle problems, identifying in each case the corresponding internal symmetries. For the bound particle problem, which models quark confinement, this symmetry turns out to be characterized by Dirac's two-oscillator representation of theO(3, 2) de Sitter group.     

Tempered distributions in infinitely many dimensions

In the present paper we continue investigating spaces of tempered distributions in infinitely many dimensions. In particular, we prove that those linear homogeneous transformations of the canonical pair of field operators, which preserve the commutation relations, can be implemented by an essentially unique intertwining operator. The dependence of this operator on the transformation is studied.     

Fermions and associated bosons of one-dimensional model

The representation of the canonical commutation relations involved in the construction of boson operators from fermion operators according to the recipe of the neutrino theory of light is studied. Starting from a cyclic Fock-representation for the massless fermions the boson operators are reduced by the spectral projectors of two charge-operators and form an infinite direct sum of cyclic Fock-representations. Kronig's identity expressing the fermion kinetic energy in terms of the boson kinetic energy and the squares of the charge operators is verified as an identity for strictly selfadjoint operators. It provides the key to the solubility ofLuttinger's model. A simple sufficient condition is given for the unitary equivalence of the representations linked by the canonical transformation which diagonalizes the total Hamiltonian.Work supported by the National Science Foundation.     

Quantum electrodynamics on a space-time lattice

A Minkowski-lattice version of quantum electrodynamics (or rather its simplified version, with matter described by a scalar field) is constructed. Quantum fields are consequently described in a gauge-independent way, i.e. the algebra of quantum observables of the theory is generated by gauge-invariant operators assigned to zero-, one-, and two-dimensional elements of the lattice. The operators satisfy canonical commutation relations. The uniqueness of representation of this algebra is proved. Field dynamics is formulated in terms of difference equations imposed on the field operators. It is obtained from a discrete version of the path-integral. The theory is local and causal.     

Quantum Mechanics in Curved Configuration Space

Different approaches are compared to formulation of quantum mechanics of a particle on the curved spaces. At first, the canonical, quasiclassical, and path integration formalisms are considered for quantization of geodesic motion on the Riemannian configuration spaces. A unique rule of ordering of operators in the canonical formalism and a unique definition of the path integral are established and, thus, a part of ambiguities in the quantum counterpart of geodesic motion is removed. A geometric interpretation is proposed for noninvariance of the quantum mechanics on coordinate transformations. An approach alternative to the quantization of geodesic motion is surveyed, which starts with the quantum theory of a neutral scalar field. Consequences of this alternative approach and the three formalisms of quantization are compared. In particular, the field theoretical approach generates a deformation of the canonical commutation relations between operators of coordinates and momenta of a particle. A cosmological consequence of the deformation is presented in short.     

On infinite direct products of continuous unitary one-parameter groups

We discuss the infinite product of unitary operators in an incomplete direct product of Hilbert spaces. Necessary and sufficient conditions are derived under which this infinite product leads to a continuous unitary one-parameter group provided each factor is assumed to have this property. A certain minimal condition guarantees the existence of a renormalized unitary group. An application is made to product representations of the canonical commutation relations in order to determine the admissible test functions.     

Canonical representations of the Lie superalgebraosp(1,4)

The method for constructing infinite-dimensional representations of Lie superalgebras proposed by the authors recently is applied to the superalgebraosp (1, 4). Explicit formulae for its generators in terms of two or three pairs of operators fulfilling the canonical commutation relations, at most one pair of operators fulfilling the canonical anticommutation relations and at most one real parameter are obtained. The generators of the Lie subalgebrasp (4, ) osp (1,4) are represented skew-symmetrically and both the Casimir operators are equal to multiples of the unity operator.Dedicated to Professor Ivan Úlehla on the occasion of his sixtieth birthday.