21 research outputs found
Constants of motion and the conformal anti - de Sitter algebra in (2+1)-Dimensional Gravity
Constants of motion are calculated for 2+1 dimensional gravity with topology
R x T^2 and negative cosmological constant. Certain linear combinations of them
satisfy the anti - de Sitter algebra so(2,2) in either ADM or holonomy
variables. Quantisation is straightforward in terms of the holonomy parameters.
On inclusion of the Hamiltonian three new global constants are derived and the
quantum algebra extends to that of the conformal algebra so(2,3). The modular
group appears as a discrete subgroup of the conformal group. Its quantum action
is generated by these conserved quantities.Comment: 22 pages, Plain Tex, No Figure
Global constants in (2+1)--dimensional gravity
The extended conformal algebra (so)(2,3) of global, quantum, constants of
motion in 2+1 dimensional gravity with topology R x T^2 and negative
cosmological constant is reviewed. It is shown that the 10 global constants
form a complete set by expressing them in terms of two commuting spinors and
the Dirac gamma matrices. The spinor components are the globally constant
holonomy parameters, and their respective spinor norms are their quantum
commutators.Comment: 14 pages, to appear in Classical and Quantum Gravity, Spacetime
Safari: Essays in Honor of Vincent Moncrief on the Classical Physics of
Strong Gravitational Field
The Quantum Modular Group in (2+1)-Dimensional Gravity
The role of the modular group in the holonomy representation of
(2+1)-dimensional quantum gravity is studied. This representation can be viewed
as a "Heisenberg picture", and for simple topologies, the transformation to the
ADM "Schr{\"o}dinger picture" may be found. For spacetimes with the spatial
topology of a torus, this transformation and an explicit operator
representation of the mapping class group are constructed. It is shown that the
quantum modular group splits the holonomy representation Hilbert space into
physically equivalent orthogonal ``fundamental regions'' that are interchanged
by modular transformations.Comment: 23 pages, LaTeX, no figures; minor changes and clarifications in
response to referee (basic argument and conclusions unaffected
Comparative Quantizations of (2+1)-Dimensional Gravity
We compare three approaches to the quantization of (2+1)-dimensional gravity
with a negative cosmological constant: reduced phase space quantization with
the York time slicing, quantization of the algebra of holonomies, and
quantization of the space of classical solutions. The relationships among these
quantum theories allow us to define and interpret time-dependent operators in
the ``frozen time'' holonomy formulation.Comment: 24 pages, LaTeX, no figure
Classical and Quantum Evolutions of the de Sitter and the anti-de Sitter Universes in 2+1 dimensions
Two canonical formulations of the Einstein gravity in 2+1 dimensions, namely,
the ADM formalism and the Chern-Simons gravity, are investigated in the case of
nonvanishing cosmological constant. General arguments for reducing phase spaces
of the two formalisms are given when spatial hypersurface is compact. In
particular when the space has the topology of a sphere or a torus
, the spacetimes constructed from these two formulations can be
identified and the classical equivalence between the ADM and the CSG is shown.
Moreover in the case the relations between their phase spaces, and
therefore between their quantizations, are given in almost the same form as
that in the case when the cosmological constant vanishes. There are, however,
some modifications, the most remarkable one of which is that the phase space of
the CSG is in 1 to 2 correspondence with the one of the ADM when the
cosmological constant is negative.Comment: 37pages Latex (6 figures not included
Large Diffeomorphisms in (2+1)-Quantum Gravity on the Torus
The issue of how to deal with the modular transformations -- large
diffeomorphisms -- in (2+1)-quantum gravity on the torus is discussed. I study
the Chern-Simons/connection representation and show that the behavior of the
modular transformations on the reduced configuration space is so bad that it is
possible to rule out all finite dimensional unitary representations of the
modular group on the Hilbert space of -functions on the reduced
configuration space. Furthermore, by assuming piecewise continuity for a dense
subset of the vectors in any Hilbert space based on the space of complex valued
functions on the reduced configuration space, it is shown that finite
dimensional representations are excluded no matter what inner-product we define
in this vector space. A brief discussion of the loop- and ADM-representations
is also included.Comment: The proof for the nonexistence of the one- and two-dimensional
representations of PSL(2,Z) in the relevant Hilbert space, has been extended
to cover all finite dimensional unitary representations. The notation is
slightly improved and a few references are added
Canonical Quantization of (2+1)-Dimensional Gravity
We consider the quantum dynamics of both open and closed two- dimensional
universes with ``wormholes'' and particles. The wave function is given as a sum
of freely propagating amplitudes, emitted from a network of mapping class
images of the initial state. Interference between these amplitudes gives
non-trivial scattering effects, formally analogous to the optical diffraction
by a multidimensional grating; the ``bright lines'' correspond to the most
probable geometries.Comment: 22 pages, Mexico preprint ICN-UNAM-93-1
Homotopy Invariants and Time Evolution in (2+1)-Dimensional Gravity
We establish the relation between the ISO(2,1) homotopy invariants and the
polygon representation of (2+1)-dimensional gravity. The polygon closure
conditions, together with the SO(2,1) cycle conditions, are equivalent to the
ISO(2,1) cycle conditions for the representa- tions of the fundamental group in
ISO(2,1). Also, the symplectic structure on the space of invariants is closely
related to that of the polygon representation. We choose one of the polygon
variables as internal time and compute the Hamiltonian, then perform the
Hamilton-Jacobi transformation explicitly. We make contact with other authors'
results for g = 1 and g = 2 (N = 0).Comment: 34 pages, Mexico preprint ICN-UNAM-93-1
Square Root Actions, Metric Signature, and the Path-Integral of Quantum Gravity
We consider quantization of the Baierlein-Sharp-Wheeler form of the
gravitational action, in which the lapse function is determined from the
Hamiltonian constraint. This action has a square root form, analogous to the
actions of the relativistic particle and Nambu string. We argue that
path-integral quantization of the gravitational action should be based on a
path integrand rather than the familiar Feynman expression
, and that unitarity requires integration over manifolds of both
Euclidean and Lorentzian signature. We discuss the relation of this path
integral to our previous considerations regarding the problem of time, and
extend our approach to include fermions.Comment: 32 pages, latex. The revision is a more general treatment of the
regulator. Local constraints are now derived from a requirement of regulator
independenc