arXiv subjects

John R. Klauder

Publications and source records attributed to John R. Klauder.

At least 91 records · Page 5Linked to original sources

arXiv2004

Signal Transmission in Passive Media

Under rather general assumptions, and in a relatively simple and straightforward manner, it is shown that the characteristics of signals which travel through homogeneous, as well as inhomogeneous, passive media have the principal features usually associated with the phenomena of precursors, as generally follows from more elaborate studies. The simplicity of the present arguments permit analytic studies to be made for a greater variety of media than is normally the case.

math.GM↗

arXiv2004

The Distance Between Classical and Quantum Systems

In a recent paper, a "distance" function, \cal D, was defined which measures the distance between pure classical and quantum systems. In this work, we present a new definition of a "distance", D, which measures the distance between either pure or impure classical and quantum states. We also compare the new distance formula with the previous formula, when the latter is applicable. To illustrate these distances, we have used 2 \times 2 matrix examples and 2-dimensional vectors for simplicity and clarity. Several specific examples are calculated.

quant-ph↗

arXiv2004

Overcoming Nonrenormalizability -- Part 2

The procedures to overcome nonrenormalizability of ϕ^4_n, n\ge5, quantum field theory models that were presented in a recent paper are extended to address nonrenormalizability of ϕ^p_3, p=8,10,12,..., models. The principles involved in these procedures are based on the hard-core picture of nonrenormalizability.

hep-th↗

arXiv2004

The Utility of Coherent States and other Mathematical Methods in the Foundations of Affine Quantum Gravity

Affine quantum gravity involves (i) affine commutation relations to ensure metric positivity, (ii) a regularized projection operator procedure to accomodate first- and second-class quantum constraints, and (iii) a hard-core interpretation of nonlinear interactions to understand and potentially overcome nonrenormalizability. In this program, some of the less traditional mathematical methods employed are (i) coherent state representations, (ii) reproducing kernel Hilbert spaces, and (iii) functional integral representations involving a continuous-time regularization. Of special importance is the profoundly different integration measure used for the Lagrange multiplier (shift and lapse) functions. These various concepts are first introduced on elementary systems to help motivate their application to affine quantum gravity.

hep-th↗

arXiv2003

Weak Coherent State Path Integrals

Weak coherent states share many properties of the usual coherent states, but do not admit a resolution of unity expressed in terms of a local integral. They arise e.g. in the case that a group acts on an inadmissible fiducial vector. Motivated by the recent Affine Quantum Gravity Program, the present article studies the path integral representation of the affine weak coherent state matrix elements of the unitary time-evolution operator. Since weak coherent states do not admit a resolution of unity, it is clear that the standard way of constructing a path integral, by time slicing, is predestined to fail. Instead a well-defined path integral with Wiener measure, based on a continuous-time regularization, is used to approach this problem. The dynamics is rigorously established for linear Hamiltonians, and the difficulties presented by more general Hamiltonians are addressed.

hep-th↗

arXiv2003

How far apart are classical and quantum systems?

As is well known, classical systems approximate quantum ones -- but how well? We introduce a definition of a "distance" on classical and quantum phase spaces that offers a measure of their separation. Such a distance scale provides a means to measure the quality of approximate solutions to various problems. A few simple applications are discussed.

quant-ph↗

arXiv2003

Generalization of the Darboux transformation and generalized harmonic oscillators

The Darbroux transformation is generalized for time-dependent Hamiltonian systems which include a term linear in momentum and a time-dependent mass. The formalism for the $N$-fold application of the transformation is also established, and these formalisms are applied for a general quadratic system (a generalized harmonic oscillator) and a quadratic system with an inverse-square interaction up to N=2. Among the new features found, it is shown, for the general quadratic system, that the shape of potential difference between the original system and the transformed system could oscillate according to a classical solution, which is related to the existence of coherent states in the system.

quant-ph↗

arXiv2003

Affine Quantum Gravity

A sketch of the affine quantum gravity program illustrates a different perspective on several difficult issues of principle: metric positivity; quantum anomalies; and nonrenormalizability.

gr-qc↗

arXiv2003

Overcoming Nonrenormalizability

A suitable counterterm for a Euclidean space lattice version of ϕ^4_n theories, n\ge 4, is combined with several additional procedures so that in the continuum limit the resultant quantum field theory is nontrivial. Arguments to support this unconventional choice are presented.

hep-th↗

arXiv2003

The Feynman Path Integral: An Historical Slice

Efforts to give an improved mathematical meaning to Feynman's path integral formulation of quantum mechanics started soon after its introduction and continue to this day. In the present paper, one common thread of development is followed over many years, with contributions made by various authors. The present version of this line of development involves a continuous-time regularization for a general phase space path integral and provides, in the author's opinion at least, perhaps the optimal formulation of the path integral.

quant-ph↗

arXiv2002

Metric and Curvature in Gravitational Phase Space

At a fixed point in spacetime (say, x_0), gravitational phase space consists of the space of symmetric matrices F^{ab} [corresponding to the canonical momentum pi^{ab}(x_0) and of symmetric matrices {G_{ab}}[corresponding to the canonical metric g_{ab}(x_0), where 1 \leq a,b \leq n, and, crucially, the matrix {G_{ab}} is necessarily positive definite, i.e. \sum u^a G_{ab}u^b > 0 whenever \sum (u^a)^2 > 0. In an alternative quantization procedure known as Metrical Quantization, the first and most important ingredient is the specification of a suitable metric on classical phase space. Our choice of phase space metrics, guided by a recent study of Affine Quantum Gravity, leads to gravitational phase space geometries which possess constant scalar curvature and may be regarded as higher dimensional analogs of the Poincare plane, which applies when n=1. This result is important because phase spaces endowed with such symmetry lead naturally via the procedures of Metrical Quantization to acceptable Hilbert spaces of high dimension.

gr-qc↗

arXiv2001

Phase Space Geometry in Classical and Quantum Mechanics

Phase space is the state space of classical mechanics, and this manifold is normally endowed only with a symplectic form. The geometry of quantum mechanics is necessarily more complicated. Arguments will be given to show that augmenting the symplectic manifold of classical phase space with a Riemannian metric is sufficient for describing quantum mechanics. In particular, using such spaces, a fully satisfactory geometric version of quantization will be developed and described.

quant-ph↗

arXiv2001

The Affine Quantum Gravity Program

The central principle of affine quantum gravity is securing and maintaining the strict positivity of the matrix $\{\hg_{ab}(x)\}$ composed of the spatial components of the local metric operator. On spectral grounds, canonical commutation relations are incompatible with this principle, and they must be replaced by noncanonical, affine commutation relations. Due to the partial second-class nature of the quantum gravitational constraints, it is advantageous to use the recently developed projection operator method, which treats all quantum constraints on an equal footing. Using this method, enforcement of regularized versions of the gravitational operator constraints is formulated quite naturally by means of a novel and relatively well-defined functional integral involving only the same set of variables that appears in the usual classical formulation. It is anticipated that skills and insight to study this formulation can be developed by studying special, reduced-variable models that still retain some basic characteristics of gravity, specifically a partial second-class constraint operator structure. Although perturbatively nonrenormalizable, gravity may possibly be understood nonperturbatively from a hard-core perspective that has proved valuable for specialized models. Finally, developing a procedure to pass to the genuine physical Hilbert space involves several interconnected steps that require careful coordination.

gr-qc↗

arXiv2001

The Current State of Coherent States

The original canonical coherent states could be defined in several ways. As applications for other sets of coherent states arose, the rules of definition were correspondingly changed. Among such rule changes were a change of group and relaxation of the analytic nature of the labels. Recent developments have done away with the group connections altogether and thereby allowed sets of coherent states to be defined that are temporally stable for a wide variety of dynamical systems including the hydrogen atom. This article outlines some of the current trends in the definitions and properties of present-day coherent states.

quant-ph↗

arXiv2001

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.

gr-qc↗

arXiv2000

Ultralocal Fields and their Relevance for Reparametrization Invariant Quantum Field Theory

Reparametrization invariant theories have a vanishing Hamiltonian and enforce their dynamics through a constraint. We specifically choose the Dirac procedure of quantization before the introduction of constraints. Consequently, for field theories, and prior to the introduction of any constraints, it is argued that the original field operator representation should be ultralocal in order to remain totally unbiased toward those field correlations that will be imposed by the constraints. It is shown that relativistic free and interacting theories can be completely recovered starting from ultralocal representations followed by a careful enforcement of the appropriate constraints. In so doing all unnecessary features of the original ultralocal representation disappear. The present discussion is germane to a recent theory of affine quantum gravity in which ultralocal field representations have been invoked before the imposition of constraints.

quant-ph↗

arXiv2000

Coherent State Path Integrals without Resolutions of Unity

From the very beginning, coherent state path integrals have always relied on a coherent state resolution of unity for their construction. By choosing an inadmissible fiducial vector, a set of ``coherent states'' spans the same space but loses its resolution of unity, and for that reason has been called a set of weak coherent states. Despite having no resolution of unity, it is nevertheless shown how the propagator in such a basis may admit a phase-space path integral representation in essentially the same form as if it had a resolution of unity. Our examples are toy models of similar situations that arise in current studies of quantum gravity.

quant-ph↗

arXiv2000

Quantization of Constrained Systems

The present article is primarily a review of the projection-operator approach to quantize systems with constraints. We study the quantization of systems with general first- and second-class constraints from the point of view of coherent-state, phase-space path integration, and show that all such cases may be treated, within the original classical phase space, by using suitable path-integral measures for the Lagrange multipliers which ensure that the quantum system satisfies the appropriate quantum constraint conditions. Unlike conventional methods, our procedures involve no $δ$-functionals of the classical constraints, no need for dynamical gauge fixing of first-class constraints nor any average thereover, no need to eliminate second-class constraints, no potentially ambiguous determinants, as well as no need to add auxiliary dynamical variables expanding the phase space beyond its original classical formulation, including no ghosts. Besides several pedagogical examples, we also study: (i) the quantization procedure for reparameterization invariant models, (ii) systems for which the original set of Lagrange mutipliers are elevated to the status of dynamical variables and used to define an extended dynamical system which is completed with the addition of suitable conjugates and new sets of constraints and their associated Lagrange multipliers, (iii) special examples of alternative but equivalent formulations of given first-class constraints, as well as (iv) a comparison of both regular and irregular constraints.

hep-th↗