9 research outputs found
An algebraic extension of Dirac quantization: Examples
An extension of the Dirac procedure for the quantization of constrained
systems is necessary to address certain issues that are left open in Dirac's
original proposal. These issues play an important role especially in the
context of non-linear, diffeomorphism invariant theories such as general
relativity. Recently, an extension of the required type was proposed by one of
us using algebraic quantization methods. In this paper, the key conceptual and
technical aspects of the algebraic program are illustrated through a number of
finite dimensional examples. The choice of examples and some of the analysis is
motivated by certain peculiar problems endemic to quantum gravity. However,
prior knowledge of general relativity is not assumed in the main discussion.
Indeed, the methods introduced and conclusions arrived at are applicable to any
system with first class constraints. In particular, they resolve certain
technical issues which are present also in the reduced phase space approach to
quantization of these systems.Comment: 43 pages, Latex, CGPG-94/6-1. (References added; particularly to
earlier work by C.J.Isham using group theoretic ideas, in the introduction.
An algebraic approach to the quantization of cosntrained systems: finite dimensional examples
From the point of view of canonical quantum gravity, it has become imperative
to find a framework for quantization which provides a {\em general}
prescription to find the physical inner product, and is flexible enough to
accommodate non-canonical variables. In this dissertation I consider an
algebraic formulation of the Dirac approach to the quantization of constrained
systems, due to A. Ashtekar. The Dirac quantization program is augmented by a
general principle to find the inner product on physical states. Essentially,
the Hermiticity conditions on physical operators determine this inner product.
I also clarify the role in quantum theory of possible algebraic identities
between the elementary variables. I use this approach to quantize various
finite dimensional systems. Some of these models test the new aspects of the
algebraic framework. Others bear qualitative similarities to \gr, and may give
some insight into the pitfalls lurking in \qg. In (spatially compact) general
relativity, the Hamiltonian is constrained to vanish. I present various
approaches one can take to obtain an interpretation of the quantum theory of
such ``dynamically constrained'' systems. I apply some of these ideas to the
Bianchi I cosmology, and analyze the issue of the initial singularity in
quantum theory.Comment: 124 pages, LaTeX (run twice before printing), SU-GP-92/8-1. (Minor
corruption (extra blank line at line 2994) hopefully fixed.
Minisuperspaces: Observables and Quantization
A canonical transformation is performed on the phase space of a number of
homogeneous cosmologies to simplify the form of the scalar (or, Hamiltonian)
constraint. Using the new canonical coordinates, it is then easy to obtain
explicit expressions of Dirac observables, i.e.\ phase space functions which
commute weakly with the constraint. This, in turn, enables us to carry out a
general quantization program to completion. We are also able to address the
issue of time through ``deparametrization'' and discuss physical questions such
as the fate of initial singularities in the quantum theory. We find that they
persist in the quantum theory {\it inspite of the fact that the evolution is
implemented by a 1-parameter family of unitary transformations}. Finally,
certain of these models admit conditional symmetries which are explicit already
prior to the canonical transformation. These can be used to pass to quantum
theory following an independent avenue. The two quantum theories --based,
respectively, on Dirac observables in the new canonical variables and
conditional symmetries in the original ADM variables-- are compared and shown
to be equivalent.Comment: 34 page
An Algebraic Approach to the Quantization of Cosntrained Systems: Finite Dimensional Examples
We discuss the statistical mechanics of magnetic flux lines in a finite-thickness slab of type-II superconductor. The long wavelength properties of a flux-line liquid in a slab geometry are described by a hydrodynamic free energy that incorporates the boundary conditions on the flux lines at the sample\u27s surface as a surface contribution to the free energy. Bulk and surface weak disorder are modeled via Gaussian impurity potentials. This free energy is used to evaluate the two-dimensional structure factor of the flux-line tips at the sample surface. We find that surface interaction always dominates in determining the decay of translational correlations in the asymptotic long-wavelength limit. On the other hand, such large length scales have not been probed by the decoration experiments. Our results indicate that the translational correlations extracted from the analysis of the Bitter patterns are indeed representative of behavior of flux lines in the bulk
Time-of-arrival in quantum mechanics
We study the problem of computing the probability for the time-of-arrival of
a quantum particle at a given spatial position. We consider a solution to this
problem based on the spectral decomposition of the particle's (Heisenberg)
state into the eigenstates of a suitable operator, which we denote as the
``time-of-arrival'' operator. We discuss the general properties of this
operator. We construct the operator explicitly in the simple case of a free
nonrelativistic particle, and compare the probabilities it yields with the ones
estimated indirectly in terms of the flux of the Schr\"odinger current. We
derive a well defined uncertainty relation between time-of-arrival and energy;
this result shows that the well known arguments against the existence of such a
relation can be circumvented. Finally, we define a ``time-representation'' of
the quantum mechanics of a free particle, in which the time-of-arrival is
diagonal. Our results suggest that, contrary to what is commonly assumed,
quantum mechanics exhibits a hidden equivalence between independent (time) and
dependent (position) variables, analogous to the one revealed by the
parametrized formalism in classical mechanics.Comment: Latex/Revtex, 20 pages. 2 figs included using epsf. Submitted to
Phys. Rev.
Extraordinary elevation of the glass transition temperature of thin polymer films grafted to silicon oxide substrates
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