1,046 research outputs found
Fundamentals of Quantum Gravity
The outline of a recent approach to quantum gravity is presented. Novel
ingredients include: (1) Affine kinematical variables; (2) Affine coherent
states; (3) Projection operator approach toward quantum constraints; (4)
Continuous-time regularized functional integral representation without/with
constraints; and (5) Hard core picture of nonrenormalizability. The ``diagonal
representation'' for operator representations, introduced by Sudarshan into
quantum optics, arises naturally within this program.Comment: 15 pages, conference proceeding
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.Comment: 16 pages, LaTeX, no figure
On the role of coherent states in quantum foundations
Coherent states, and the Hilbert space representations they generate, provide
ideal tools to discuss classical/quantum relationships. In this paper we
analyze three separate classical/quantum problems using coherent states, and
show that useful connections arise among them. The topics discussed are: (1) a
truly natural formulation of phase space path integrals; (2) how this analysis
implies that the usual classical formalism is ``simply a subset'' of the
quantum formalism, and thus demonstrates a universal coexistence of both the
classical and quantum formalisms; and (3) how these two insights lead to a
complete analytic solution of a formerly insoluble family of nonlinear quantum
field theory models.Comment: ICQOQI'2010, Kiev, Ukraine, May-June 2010, Conference Proceedings (9
pages
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.Comment: 15 pages, Presented at the X-International Conference on Symmetry
Methods in Physic
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.Comment: 21 pages, no figures, accepted by J. Math. Phy
Divergence-free Nonrenormalizable Models
A natural procedure is introduced to replace the traditional, perturbatively
generated counter terms to yield a formulation of covariant, self-interacting,
nonrenormalizable scalar quantum field theories that has the added virtue of
exhibiting a divergence-free perturbation analysis. To achieve this desirable
goal it is necessary to reexamine the meaning of the free theory about which
such a perturbation takes place.Comment: 22 pages. Version accepted for publication; involves modest addition
to the end of Sec.
Path Integral Quantization and Riemannian-Symplectic Manifolds
We develop a mathematically well-defined path integral formalism for general
symplectic manifolds. We argue that in order to make a path integral
quantization covariant under general coordinate transformations on the phase
space and involve a genuine functional measure that is both finite and
countably additive, the phase space manifold should be equipped with a
Riemannian structure (metric). A suitable method to calculate the metric is
also proposed.Comment: plain Latex, 9 pages, no figure
Enhanced quantization on the circle
We apply the quantization scheme introduced in [arXiv:1204.2870] to a
particle on a circle. We find that the quantum action functional restricted to
appropriate coherent states can be expressed as the classical action plus
-corrections. This result extends the examples presented in the cited
paper.Comment: 7 page
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.Comment: 17 pages, LaTeX, no figure
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
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