42,195 research outputs found
Schrodinger Evolution for the Universe: Reparametrization
Starting from a generalized Hamilton-Jacobi formalism, we develop a new
framework for constructing observables and their evolution in theories
invariant under global time reparametrizations. Our proposal relaxes the usual
Dirac prescription for the observables of a totally constrained system
(`perennials') and allows one to recover the influential partial and complete
observables approach in a particular limit. Difficulties such as the
non-unitary evolution of the complete observables in terms of certain partial
observables are explained as a breakdown of this limit. Identification of our
observables (`mutables') relies upon a physical distinction between gauge
symmetries that exist at the level of histories and states (`Type 1'), and
those that exist at the level of histories and not states (`Type 2'). This
distinction resolves a tension in the literature concerning the physical
interpretation of the partial observables and allows for a richer class of
observables in the quantum theory. There is the potential for the application
of our proposal to the quantization of gravity when understood in terms of the
Shape Dynamics formalism.Comment: 25 pages (including title page and references), 1 figur
Coherent states and the quantization of 1+1-dimensional Yang-Mills theory
This paper discusses the canonical quantization of 1+1-dimensional Yang-Mills
theory on a spacetime cylinder, from the point of view of coherent states, or
equivalently, the Segal-Bargmann transform. Before gauge symmetry is imposed,
the coherent states are simply ordinary coherent states labeled by points in an
infinite-dimensional linear phase space. Gauge symmetry is imposed by
projecting the original coherent states onto the gauge-invariant subspace,
using a suitable regularization procedure. We obtain in this way a new family
of "reduced" coherent states labeled by points in the reduced phase space,
which in this case is simply the cotangent bundle of the structure group K.
The main result explained here, obtained originally in a joint work of the
author with B. Driver, is this: The reduced coherent states are precisely those
associated to the generalized Segal-Bargmann transform for K, as introduced by
the author from a different point of view. This result agrees with that of K.
Wren, who uses a different method of implementing the gauge symmetry. The
coherent states also provide a rigorous way of making sense out of the quantum
Hamiltonian for the unreduced system.
Various related issues are discussed, including the complex structure on the
reduced phase space and the question of whether quantization commutes with
reduction
The new vertices and canonical quantization
We present two results on the recently proposed new spin foam models. First,
we show how a (slightly modified) restriction on representations in the EPRL
model leads to the appearance of the Ashtekar-Barbero connection, thus bringing
this model even closer to LQG. Second, we however argue that the quantization
procedure used to derive the new models is inconsistent since it relies on the
symplectic structure of the unconstraint BF theory.Comment: 16 pages, 1 figure; added subsection on ordering of Casimir
operators, more details on imposing simplicity constraint
Dynamical Aspects of Lie--Poisson Structures
Quantum Groups can be constructed by applying the quantization by deformation
procedure to Lie groups endowed with a suitable Poisson bracket. Here we try to
develop an understanding of these structures by investigating dynamical systems
which are associated with this bracket. We look at and , as
submanifolds of a 4--dimensional phase space with constraints, and deal with
two classes of problems. In the first set of examples we consider some
hamiltonian systems associated with Lie-Poisson structures and we investigate
the equations of the motion. In the second set of examples we consider systems
which preserve the chosen bracket, but are dissipative. However in this
approach, they survive the quantization procedure.Comment: 17 pages, figures not include
Computation of the Chiral Anomaly in the Bulk Quantization
The bulk quantization method is used for regularizing a conventional four
dimensional theory of massless fermions coupled to an external non-Abelian
gauge field and for subsequently evaluating the associated Ward identity. As a
result one obtains the well-known chiral anomaly.Comment: 19 page
- âŠ