Noncanonical quantization of gravity. II. Constraints and the physical Hilbert space

The program of quantizing the gravitational field with the help of affine field variables is continued. For completeness, a review of the selection criteria that singles out the affine fields, the alternative treatment of constraints, and the choice of the initial (before imposition of the constraints) ultralocal representation of the field operators is initially presented. As analogous examples demonstrate, the introduction and enforcement of the gravitational constraints will cause sufficient changes in the operator representations so that all vestiges of the initial ultralocal field operator representation disappear. To achieve this introduction and enforcement of the constraints, a well characterized phase space functional integral representation for the reproducing kernel of a suitably regularized physical Hilbert space is developed and extensively analyzed.Comment: LaTeX, 42 pages, no figure

Metrical Quantization

Canonical quantization may be approached from several different starting points. The usual approaches involve promotion of c-numbers to q-numbers, or path integral constructs, each of which generally succeeds only in Cartesian coordinates. All quantization schemes that lead to Hilbert space vectors and Weyl operators---even those that eschew Cartesian coordinates---implicitly contain a metric on a flat phase space. This feature is demonstrated by studying the classical and quantum ``aggregations'', namely, the set of all facts and properties resident in all classical and quantum theories, respectively. Metrical quantization is an approach that elevates the flat phase space metric inherent in any canonical quantization to the level of a postulate. Far from being an unwanted structure, the flat phase space metric carries essential physical information. It is shown how the metric, when employed within a continuous-time regularization scheme, gives rise to an unambiguous quantization procedure that automatically leads to a canonical coherent state representation. Although attention in this paper is confined to canonical quantization we note that alternative, nonflat metrics may also be used, and they generally give rise to qualitatively different, noncanonical quantization schemes.Comment: 13 pages, LaTeX, no figures, to appear in Born X Proceeding

Impurity-induced dephasing of Andreev states

A study is presented concerning the influence of flicker noise in the junction transparency on coherent transport in Andreev states. The amount of dephasing is estimated for a microwave-activated quantum interferometer. Possibilities of experimentally investigating the coupling between a superconducting quantum point contact and its electromagnetic environment are discussed.Comment: 8 pages, 4 figure

Spin tunneling and topological selection rules for integer spins

We present topological interference effects for the tunneling of a single large spin, which are caused by the symmetry of a general class of magnetic anisotropies. The interference originates from spin Berry phases associated with different tunneling paths exposed to the same dynamics. Introducing a generalized path integral for coherent spin states, we evaluate transition amplitudes between ground as well as low-lying excited states. We show that these interference effects lead to topological selection rules and spin-parity effects for integer spins that agree with quantum selection rules and which thus provide a generalization of the Kramers degeneracy to integer spins. Our results apply to the molecular magnets Mn12 and Fe8.Comment: 4 pages, 3 EPS figures, REVTe

Gazeau-Klauder type coherent states for hypergeometric type operators

The hypergeometric type operators are shape invariant, and a factorization into a product of first order differential operators can be explicitly described in the general case. Some additional shape invariant operators depending on several parameters are defined in a natural way by starting from this general factorization. The mathematical properties of the eigenfunctions and eigenvalues of the operators thus obtained depend on the values of the involved parameters. We study the parameter dependence of orthogonality, square integrability and of the monotony of eigenvalue sequence. The obtained results allow us to define certain systems of Gazeau-Klauder coherent states and to describe some of their properties. Our systematic study recovers a number of well-known results in a natural unified way and also leads to new findings.Comment: An error occurring in Theorem 12 and Theorem 13 has been correcte

Smooth Paths on Three Dimensional Lattice

A particular class of random walks with a spin factor on a three dimensional cubic lattice is studied. This three dimensional random walk model is a simple generalization of random walk for the two dimensional Ising model. All critical diffusion constants and associated critical exponents are calculated. Continuum field theories such as Klein-Gordon, Dirac and massive Chern-Simons theories are constructed near several critical points.Comment: 7 pages,NUP-A-94-

Monte Carlo Simulations with Indefinite and Complex-Valued Measures

A method is presented to tackle the sign problem in the simulations of systems having indefinite or complex-valued measures. In general, this new approach is shown to yield statistical errors smaller than the crude Monte Carlo using absolute values of the original measures. Exactly solvable, one-dimensional Ising models with complex temperature and complex activity illustrate the considerable improvements and the workability of the new method even when the crude one fails.Comment: 10 A4 pages, postscript (140K), UM-P-93-7

Unitary relation between a harmonic oscillator of time-dependent frequency and a simple harmonic oscillator with and without an inverse-square potential

The unitary operator which transforms a harmonic oscillator system of time-dependent frequency into that of a simple harmonic oscillator of different time-scale is found, with and without an inverse-square potential. It is shown that for both cases, this operator can be used in finding complete sets of wave functions of a generalized harmonic oscillator system from the well-known sets of the simple harmonic oscillator. Exact invariants of the time-dependent systems can also be obtained from the constant Hamiltonians of unit mass and frequency by making use of this unitary transformation. The geometric phases for the wave functions of a generalized harmonic oscillator with an inverse-square potential are given.Comment: Phys. Rev. A (Brief Report), in pres

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

Coherent states for exactly solvable potentials

A general algebraic procedure for constructing coherent states of a wide class of exactly solvable potentials e.g., Morse and P{\"o}schl-Teller, is given. The method, {\it a priori}, is potential independent and connects with earlier developed ones, including the oscillator based approaches for coherent states and their generalizations. This approach can be straightforwardly extended to construct more general coherent states for the quantum mechanical potential problems, like the nonlinear coherent states for the oscillators. The time evolution properties of some of these coherent states, show revival and fractional revival, as manifested in the autocorrelation functions, as well as, in the quantum carpet structures.Comment: 11 pages, 4 eps figures, uses graphicx packag