arXiv subjects

J. E. Nelson

Publications and source records attributed to J. E. Nelson.

24 records · Page 2Linked to original sources

arXiv1995

$\La \to 0$ limit of 2+1 Quantum Gravity for arbitrary genus

The abstract quantum algebra of observables for 2+1 gravity is analysed in the limit of small cosmological constant. The algebra splits into two sets with an explicit phase space representation;~one set consists of $6g-6$ {\it commuting} elements which form a basis for an algebraic manifold defined by the trace and rank identities;~the other set consists of $6g-6$ tangent vectors to this manifold. The action of the quantum mapping class group leaves the algebra and algebraic manifold invariant. The previously presented representation for $g=2$ is analysed in this limit and reduced to a very simple form. The symplectic form for $g=2$ is computed.

gr-qc↗

arXiv1994

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.

gr-qc↗

arXiv1993

Quantisation of 2+1 gravity for genus 2

In [1,2] we established and discussed the algebra of observables for $2+1$ gravity at both the classical and quantum level, and gave a systematic discussion of the reduction of the expected number of independent observables to $6g - 6 (g > 1)$. In this paper the algebra of observables for the case $g=2$ is reduced to a very simple form. A Hilbert space of state vectors is defined and its representations are discussed using a deformation of the Euler-Gamma function. The deformation parameter $þ$ depends on the cosmological and Planck's constants.

gr-qc↗

arXiv1993

Equivalent Quantisations of (2+1)-Dimensional Gravity

For spacetimes with the topology $\IR\!\times\!T^2$, the action of (2+1)-dimensional gravity with negative cosmological constant $\La$ is written uniquely in terms of the time-independent traces of holonomies around two intersecting noncontractible paths on $T^2$. The holonomy parameters are related to the moduli on slices of constant mean curvature by a time-dependent canonical transformation which introduces an effective Hamiltonian. The quantisation of the two classically equivalent formulations differs by terms of order $O(\hbar^3)$, negligible for small $|\La|$.

gr-qc↗

arXiv1993

The Constraints of 2+1 Quantum Gravity

To appear in proceedings of II Workshop on ``Constraints Theory and Quantisation Methods''Montepulciano (Siena) 1993} General discussion of the constraints of 2+1 gravity, with emphasis on two approaches, namely the second order and first order formalisms, and comparison with the four dimensional theory wherever possible. Introduction to an operator algebra approach that has been developed in the last few years in collaboration with T.Regge, and discussion of the quantisation of the g=1 case.

gr-qc↗

arXiv1992

Invariants of 2+1 Quantum Gravity

In [1,2] we established and discussed the algebra of observables for 2+1 gravity at both the classical and quantum level. Here our treatment broadens and extends previous results to any genus $g$ with a systematic discussion of the centre of the algebra. The reduction of the number of independent observables to $6g-6 (g > 1)$ is treated in detail with a precise classification for $g = 1$ and $g = 2$.

hep-th↗