579 research outputs found
An Analysis of the Representations of the Mapping Class Group of a Multi-Geon Three-Manifold
It is well known that the inequivalent unitary irreducible representations
(UIR's) of the mapping class group of a 3-manifold give rise to ``theta
sectors'' in theories of quantum gravity with fixed spatial topology. In this
paper, we study several families of UIR's of and attempt to understand the
physical implications of the resulting quantum sectors. The mapping class group
of a three-manifold which is the connected sum of with a finite number
of identical irreducible primes is a semi-direct product group. Following
Mackey's theory of induced representations, we provide an analysis of the
structure of the general finite dimensional UIR of such a group. In the picture
of quantized primes as particles (topological geons), this general
group-theoretic analysis enables one to draw several interesting qualitative
conclusions about the geons' behavior in different quantum sectors, without
requiring an explicit knowledge of the UIR's corresponding to the individual
primes.Comment: 52 pages, harvmac, 2 postscript figures, epsf required. Added an
appendix proving the semi-direct product structure of the MCG, corrected an
error in the characterization of the slide subgroup, reworded extensively.
All our analysis and conclusions remain as befor
Unitary Equivalence of the Metric and Holonomy Formulations of 2+1 Dimensional Quantum Gravity on the Torus
Recent work on canonical transformations in quantum mechanics is applied to
transform between the Moncrief metric formulation and the Witten-Carlip
holonomy formulation of 2+1-dimensional quantum gravity on the torus. A
non-polynomial factor ordering of the classical canonical transformation
between the metric and holonomy variables is constructed which preserves their
classical modular transformation properties. An extension of the definition of
a unitary transformation is briefly discussed and is used to find the inner
product in the holonomy variables which makes the canonical transformation
unitary. This defines the Hilbert space in the Witten-Carlip formulation which
is unitarily equivalent to the natural Hilbert space in the Moncrief
formulation. In addition, gravitational theta-states arising from ``large''
diffeomorphisms are found in the theory.Comment: 31 pages LaTeX [Important Revision: a section is added constructing
the inner product/Hilbert space for the Witten-Carlip holonomy formulation;
the proof of unitary equivalence of the metric and holonomy formulations is
then completed. Other additions include discussion of relation of canonical
and unitary transformations. Title/abstract change.
Remarks on the Configuration Space Approach to Spin-Statistics
The angular momentum operators for a system of two spin-zero
indistinguishable particles are constructed, using Isham's Canonical Group
Quantization method. This mathematically rigorous method provides a hint at the
correct definition of (total) angular momentum operators, for arbitrary spin,
in a system of indistinguishable particles. The connection with other
configuration space approaches to spin-statistics is discussed, as well as the
relevance of the obtained results in view of a possible alternative proof of
the spin-statistics theorem.Comment: 18 page
Classical phase space and statistical mechanics of identical particles
Starting from the quantum theory of identical particles, we show how to
define a classical mechanics that retains information about the quantum
statistics. We consider two examples of relevance for the quantum Hall effect:
identical particles in the lowest Landau level, and vortices in the
Chern-Simons Ginzburg-Landau model. In both cases the resulting {\em classical}
statistical mechanics is shown to be a nontrivial classical limit of Haldane's
exclusion statistics.Comment: 40 pages, Late
Analytic Representation of Finite Quantum Systems
A transform between functions in R and functions in Zd is used to define the
analogue of number and coherent states in the context of finite d-dimensional
quantum systems. The coherent states are used to define an analytic
representation in terms of theta functions. All states are represented by
entire functions with growth of order 2, which have exactly d zeros in each
cell. The analytic function of a state is constructed from its zeros. Results
about the completeness of finite sets of coherent states within a cell are
derived
Topology, Decoherence, and Semiclassical Gravity
We address the issue of recovering the time-dependent Schr\"{o}dinger
equation from quantum gravity in a natural way. To reach this aim it is
necessary to understand the nonoccurrence of certain superpositions in quantum
gravity.
We explore various possible explanations and their relation. These are the
delocalisation of interference terms through interaction with irrelevant
degrees of freedom (decoherence), gravitational anomalies, and the possibility
of states. The discussion is carried out in both the geometrodynamical
and connection representation of canonical quantum gravity.Comment: 18 pages, ZU-TH 3/93, to appear in Phys. Rev.
Fock Representations of Quantum Fields with Generalized Statistic
We develop a rigorous framework for constructing Fock representations of
quantum fields obeying generalized statistics associated with certain solutions
of the spectral quantum Yang-Baxter equation. The main features of these
representations are investigated. Various aspects of the underlying
mathematical structure are illustrated by means of explicit examples.Comment: 26 pages, Te
Caustic Formation in Tachyon Effective Field Theories
Certain configurations of D-branes, for example wrong dimensional branes or
the brane-antibrane system, are unstable to decay. This instability is
described by the appearance of a tachyonic mode in the spectrum of open strings
ending on the brane(s). The decay of these unstable systems is described by the
rolling of the tachyon field from the unstable maximum to the minimum of its
potential. We analytically study the dynamics of the inhomogeneous tachyon
field as it rolls towards the true vacuum of the theory in the context of
several different tachyon effective actions. We find that the vacuum dynamics
of these theories is remarkably similar and in particular we show that in all
cases the tachyon field forms caustics where second and higher derivatives of
the field blow up. The formation of caustics signals a pathology in the
evolution since each of the effective actions considered is not reliable in the
vicinity of a caustic. We speculate that the formation of caustics is an
artifact of truncating the tachyon action, which should contain all orders of
derivatives acting on the field, to a finite number of derivatives. Finally, we
consider inhomogeneous solutions in p-adic string theory, a toy model of the
bosonic tachyon which contains derivatives of all orders acting on the field.
For a large class of initial conditions we conclusively show that the evolution
is well behaved in this case. It is unclear if these caustics are a genuine
prediction of string theory or not.Comment: 23 pages, 5 figures; accepted for publication in JHEP. Revised
derivation of eikonal equation for the DBI action. Added comments concerning
the relationship between p-adic string theory and tachyon matter. Added
second example of inhomogeneous evolution in p-adic string theory. Misleading
statements concerning caustic-free evolution removed, references adde
Thermodynamics of an Anyon System
We examine the thermal behavior of a relativistic anyon system, dynamically
realized by coupling a charged massive spin-1 field to a Chern-Simons gauge
field. We calculate the free energy (to the next leading order), from which all
thermodynamic quantities can be determined. As examples, the dependence of
particle density on the anyon statistics and the anyon anti-anyon interference
in the ideal gas are exhibited. We also calculate two and three-point
correlation functions, and uncover certain physical features of the system in
thermal equilibrium.Comment: 18 pages; in latex; to be published in Phys. Rev.
Two-Loop Analysis of Non-abelian Chern-Simons Theory
Perturbative renormalization of a non-Abelian Chern-Simons gauge theory is
examined. It is demonstrated by explicit calculation that, in the pure
Chern-Simons theory, the beta-function for the coefficient of the Chern-Simons
term vanishes to three loop order. Both dimensional regularization and
regularization by introducing a conventional Yang-Mills component in the action
are used. It is shown that dimensional regularization is not gauge invariant at
two loops. A variant of this procedure, similar to regularization by
dimensional reduction used in supersymmetric field theories is shown to obey
the Slavnov-Taylor identity to two loops and gives no renormalization of the
Chern-Simons term. Regularization with Yang-Mills term yields a finite
integer-valued renormalization of the coefficient of the Chern-Simons term at
one loop, and we conjecture no renormalization at higher order. We also examine
the renormalization of Chern-Simons theory coupled to matter. We show that in
the non-abelian case the Chern-Simons gauge field as well as the matter fields
require infinite renormalization at two loops and therefore obtain nontrivial
anomalous dimensions. We show that the beta function for the gauge coupling
constant is zero to two-loop order, consistent with the topological
quantization condition for this constant.Comment: 48 pages, UU/HEP/91/12; file format changed to standard Latex to
solve the problem with printin
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