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.     

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.     

The canonical variables,the symplectic structure and the initial value formulation of the generalized Einstein-Cartan theory of gravity

Canonical variables for the generalized (non-metric) Einstein-Cartan theory of gravity are defined. The space of solutions is equipped with a closed differential 2-form . The symplectic 2-form has a diagonal representation in terms of canonical variables. A geometric interpretation of the canonical variables is presented and the 3+1 formulation of the field equations is given.     

A Class of Representations of the ∗-Algebra of the Canonical Commutation Relations over a Hilbert Space and Instability of Embedded Eigenvalues in Quantum Field Models

Abstract

In models of a quantum harmonic oscillator coupled to a quantum field with a quadratic interaction, embedded eigenvalues of the unperturbed system may be unstable under the perturbation given by the interaction of the oscillator with the quantum field. A general mathematical structure underlying this phenomenon is clarified in terms of a class of Fock space representations of the ?-algebra of the canonical commutation relations over a Hilbert space. It is also shown that each of the representations is given as a composition of a proper Bogolyubov (canonical) transformation and a partial isometry on the Fock space of the representation.     

Quantized self-dual Maxwell field on a null surface

The canonical formalism for a self-dual Maxwell field on a null plane is reviewed. After solution of the second class constraints, the transition to the quantum theory is carried out using a representation in which the self-dual Maxwell field is diagonal. The Gauss law constraint allows us to consider the physical state vectors to be holomorphic functionals of one complex function. Application of reality conditions allows us to define an inner product such that the Hermitean adjoint operators are identified with the classical complex conjugate operators. In going over to the Fourier expansion of the operators, we find that the inner product is formally convergent for positive frequency functionals and formally divergent for the negative frequency functionals. Following similar results of Ashtekar, Rovelli, and Smolin, negative frequency states are functional distributions identified with the helicity opposite to that of the positive frequency states.     

Wave functions relative to a real polarization

The structure of the space of wave functions in the representation given by a complete everywhere independent set of commuting observables is analyzed in the framework of geometric quantization. Under the assumptions that the chosen real polarization of the classical phase space is locally trivial and complete, it is shown that the wave functions are generalized sections of an appropriate line bundle with supports determined by generalized Bohr-Sommerfeld conditions. There is a canonical Hilbert subspace of the space of the wave functions with the scalar product defined in terms of the same expressions which appear in the generalized Bohr-Sommerfeld conditions.     

力学系统的二阶梯度表示

研究力学系统运动微分方程的梯度表示以及二阶梯度表示. 将完整和非完整力学系统的微分方程在正则坐标下表出. 给出系统成为梯度系统以及二阶梯度系统的条件. 举例说明结果的应用.     

Construction of a Weyl Representation from a Weak Weyl Representation of the Canonical Commutation Relation

Weak Weyl representations of the canonical commutation relation (CCR) with one degree of freedom are considered in relation to the theory of time operator in quantum mechanics. It is proven that there exists a general structure through which a weak Weyl representation can be constructed from a given weak Weyl representation. As a corollary, it is shown that a Weyl representation of the CCR can be constructed from a weak Weyl representation which satisfies some additional property. Some examples are given. The work is supported by the Grant-in-Aid No.17340032 for Scientific Research from Japan Society for the Promotion of Science (JSPS).     

Solution of the Hartree-Fock-Bogoliubov problem in the canonical representation

The Hartree-Fock-Bogoliubov equations are solved with a new method using the canonical representation in each step of the iteration. This is achieved by a modification of the Mang-Samadi-Ring gradient method. The canonical representation is the ideal basis for various projection techniques. Expressions are developed for the unprojected case and for the case with particle number projection before the variation. As a first test, an HFBC calculation for ¹⁵⁸Dy is performed. The resulting yrast lines, multipole pair fields and gyromagnetic factors with and without number projection are presented and compared.     

Uninvertible linear canonical transformations

Reducible quasi-Fock representations of canonical commutation relations (CCR) are studied which result from uninvertible linear canonical transformations of the Fock representation.Necessary and sufficient conditions are proved for: (i) invertibility of the linear canonical transformation, (ii) quasi-Fock character of the obtained representation of CCR, (iii) coincidence of the vacuum subspaces of two representations of this type, (iv) coincidence of W¹-algebras of operators of these representations.The results obtained are interesting for constructive quantum field theory with asymptotic fields giving rise to the quasi-Fock representations of CCR.     

Infraparticles with Superselected Direction of Motion in Two-Dimensional Conformal Field Theory

Particle aspects of two-dimensional conformal field theories are investigated, using methods from algebraic quantum field theory. The results include asymptotic completeness in terms of (counterparts of) Wigner particles in any vacuum representation and the existence of (counterparts of) infraparticles in any charged irreducible product representation of a given chiral conformal field theory. Moreover, an interesting interplay between the infraparticle’s direction of motion and the superselection structure is demonstrated in a large class of examples. This phenomenon resembles the electron’s momentum superselection expected in quantum electrodynamics.     

Effective one-dimensional models from matrix product states

In this paper we present a method for deriving effective one-dimensional models based on the matrix product state formalism. It exploits translational invariance to work directly in the thermodynamic limit. We show, how a representation of the creation operator of single quasi-particles in both real and momentum space can be extracted from the dispersion calculation. The method is tested for the analytically solvable Ising model in a transverse magnetic field. Properties of the matrix product representation of the creation operator are discussed and validated by calculating the one-particle contribution to the spectral weight. Results are also given for the ground state energy and the dispersion.     

Representation of Canonical Commutation Relations in a Gauge Theory,the Aharonov-Bohm Effect,and the Dirac-Weyl Operator

Abstract

We consider a representation of canonical commutation relations (CCR) appearing in a (non-Abelian) gauge theory on a non-simply connected region in the two-dimensional Euclidean space. A necessary and sufficient condition for the representation to be equivalent to the Schrödinger representation of CCR is given in terms of Wilson loops. A representation inequivalent to the Schrödinger representation gives a mathematical expression for the (non-Abelian) Aharonov-Bohm effect. Some aspects of the Dirac-Weyl operator associated with the representation of CCR are discussed.     

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.     

Weyl transformation in Fermi systems

In the path integral representation, the Hamiltonian in a quantum system is associated with the Hamiltonian in a classical system through the Weyl transformation. From this, it is possible to describe the time evolution in a quantum system by the Hamiltonian in a classical system. In a Bose system, the Weyl transformation is defined by the eigenstates of the canonical operators, since the Hamiltonian is given by a function of the canonical operators. On the other hand, in a Fermi system, the Hamiltonian is usually described by a function of the creation and annihilation operators, and hence the Weyl transformation is defined by the coherent states which are the eigenstate of an annihilation operator. Here, we formulate the Weyl transformation in Fermi systems in terms of the eigenstates of the canonical operators so as to clarify the correspondence between both systems. Using this, we can derive the path integral representation in Fermi systems.     

On the state space of the dipole ghost

A particular representation of SO(4, 2) is identified with the state space of the free dipole ghost. This representation is then given an explicit realization as the solution space of a 4th-order wave equation on a spacetime locally isomorphic to Minkowski space. A discrete basis for this solution space is given, as well as an explicit expression for its SO(4, 2) invariant inner product. The connection between the modes of dipole field and those of the massless scalar field is clarified, and a recent conjecture concerning the restriction of the dipole representation to the Poincaré subgroup is confirmed. A particular coordinate transformation then reveals the theory of the dipole ghost in Minkowski space. Finally, it is shown that the solution space of the dipole equation is not unitarizable in a Poincaré invariant manner.     

Mappin class group actions on quantum doubles

We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non-semisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT's is circumvented. We find compact formulae for theS ^±1-matrices using the canonical, non-degenerate forms of Hopf algebras and the bicrossed structure of doubles rather than monodromy matrices. A rigorous proof of the modular relations and the computation of the projective phases is supplied using Radford's relations between the canonical forms and the moduli of integrals. We analyze the projectiveSL(2, Z)-action on the center ofU _q(sl₂) forq anl=2m+1^st root of unity. It appears that the 3m+1-dimensional representation decomposes into anm+1-dimensional finite representation and a2m-dimensional, irreducible representation. The latter is the tensor product of the two dimensional, standard representation ofSL(2, Z) and the finite,m-dimensional representation, obtained from the truncated TQFT of the semisimplified representation category ofU _q(sl₂).     

Dirac Field on Moyal-Minkowski Spacetime and Non-commutative Potential Scattering

The quantized free Dirac field is considered on Minkowski spacetime (of general dimension). The Dirac field is coupled to an external scalar potential whose support is finite in time and which acts by a Moyal-deformed multiplication with respect to the spatial variables. The Moyal-deformed multiplication corresponds to the product of the algebra of a Moyal plane described in the setting of spectral geometry. It will be explained how this leads to an interpretation of the Dirac field as a quantum field theory on Moyal-deformed Minkowski spacetime (with commutative time) in a setting of Lorentzian spectral geometries of which some basic aspects will be sketched. The scattering transformation will be shown to be unitarily implementable in the canonical vacuum representation of the Dirac field. Furthermore, it will be indicated how the functional derivatives of the ensuing unitary scattering operators with respect to the strength of the non-commutative potential induce, in the spirit of Bogoliubov’s formula, quantum field operators (corresponding to observables) depending on the elements of the non-commutative algebra of Moyal-Minkowski spacetime.     

Effective Approach to Constructing n-Mode Wigner Operator in the Entangled State Representation

By extending Fan-Klauder entangled state representation to multipartite case. We construct n-mode Wigner operator in the common eigenvector of the multipartite centre-of-mass coordinate and two mass-weighted relative momenta, and its canonical conjugate state, they are both more complicated entangled state of continuum variables. the technique of integration within an ordered product (IWOP) of operators is essential in our derivation.     

A canonical structure for classical field theories

A general scheme of constructing a canonical structure (i.e. Poisson bracket, canonical fields) in classical field theories is proposed. The theory is manifestly independent of the particular choice of an initial space-like surface in space-time. The connection between dynamics and canonical structure is established. Applications to theories with a gauge and constraints are of special interest. Several physical examples are given.