Non-perturbative 2 particle scattering amplitudes in 2+1 dimensional quantum gravity

A quantum theory for scalar particles interacting only gravitationally in 2+1 dimensions is considered. Since there are no real gravitons the interaction is entirely topological. Nevertheless, there is non-trivial scattering. We show that the two-particle amplitude can be computed exactly. Although the complete theory is not well understood we suggest an approach towards formulating theN particle problem.     

2 + 1 dimensional gravity as an exactly soluble system

By disentangling the hamiltonian constraint equations, 2 + 1 dimensional gravity (with or without a cosmological constant) is shown to be exactly soluble at the classical and quantum levels. Indeed, it is closely related to Yang-Mills theory with purely the Chern-Simons action, which recently has turned out to define a soluble quantum field theory. 2 + 1 dimensional gravity has a straightforward renormalized perturbation expansion, with vanishing beta function. 2 + 1 dimensional quantum gravity may provide a testing ground for understanding the role of classical singularities in quantum mechanics, may be related to the discrete series of Virasoro representations in 1 + 1 dimensions, and may be a useful tool in studying three-dimensional geometry.     

Canonical quantization of 1+1 dimensional gravity

Based on Polyakov's evaluation of the Fadeev-Popov determinant for (1+1)-dimensional gravity in the conformal gauge we formulate a canonical quantization in the synchronous gauge. We find that the system is describable as a quantum mechanical system of one degree of freedom. The quantization can be carried out and solved when any number of gauge fields are included. Scalar and spinor fields lead to new difficulties. For positive cosmological constant the geometry collapses as suggested by the classical system. For negative cosmological constant a more interesting behavior involving exponentially expanding and contracting universes occurs.Supported in part by NSF Grant No. PHy 78-26847.On leave from Tel Aviv University. Supported in part by the Israel Commission for Basic Research.     

Particle scattering in loop quantum gravity

We devise a technique for defining and computing -point functions in the context of a background-independent gravitational quantum field theory. We construct a tentative implementation of this technique in a perturbatively finite model defined using spin foam techniques in the context of loop quantum gravity.     

Exact solution of quantum gravity in 1 + 1 dimensions

Quantum gravity in 1 + 1 dimensions, with zero cosmological constant is formulated, including contributions from all possible topologies. The spectrum and the S-matrix are calculated exactly. Pure gravity is found to exist in a disordered phase, dominated by topologically non-trivial configurations. In the presence of fermionic matter fields, space-time can undergo a phase transition to an ordered phase.     

Cosmological sector for localized mass and spin in 2+1 dimensional topologically massive gravity

The cosmological sector to the full non-linear topologically massive gravity (TMG) is obtained for localized sources of mass m and spin σ besides the asymptotically spinning conical flat sector previously obtained [A. Edery, M.B. Paranjape, gr-qc/9606029, to be published]. In a small region near but outside the sources, the metric resembles the spinning conical flat metric but we find that the mass m creates a negative deficit angle of 3m as opposed to m. Furthermore, it is not possible to recover the results of pure Einstein gravity in the limit μ→∞ unlike the flat sector.     

λ→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–6commuting 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 forg=2 is analysed in this limit and reduced to a very simple form. The symplectic form forg=2 is computed.     

Unique effective action in two dimensional induced quantum gravity

The gauge dependence of the effective action in two dimensional induced gravity with non-local action is investigated. The unique effective action in this theory on flat background is also discussed. The configuration-space metric dependence of the unique effective action is studied.On leave from Tomsk Pedagogical Inst., Tomsk, Russia Address after 1.08.92: Dept. of Physics, Hiroshima Univ., Higashi-Hiroshima 724, Japan     

Invariants of 2+1 gravity

In [1, 2] we established and dicussed 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 genusg 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 forg=1 andg=2.     

Time-dependent quantum mechanical approach to reactive scattering and related processes

The time-dependent quantum mechanical approach has emerged as a powerful and a practical computational tool for studying a variety of gas-phase chemical problems in recent years. In this report, we discuss the various developments that have made this possible with special emphasis on methodology and application to reactive scattering, photo-excitation processes and gas-surface interaction.     

Theories of gravity in 2 + 1 dimensions

We discuss the failure of general relativity to provide a proper Newtonian limit when the spacetime dimensionality is reduced to 2+1 and try to bypass this difficulty by assuming alternative equations for the gravitational field. We investigate the properties of spacetimes generated by circularly symmetric matter distributions in two cases: weakening Einstein equations, and by considering the Brans-Dicke theory of gravity. A comparison with the corresponding Newtonian picture is made.     

2D gravity+1D matter

We consider a formulation of nonperturbative two-dimensional quantum gravity coupled to a single bosonic field (d=1 matter). Starting from a matrix realization of the discretized model, we express the continuum theory as a double scaling limit in which the 2D cosmological constant g tends towards a critical value g_c, and the string coupling 1/N→0, with the scaling parameter ∝1n (g-g_c)/(g-g_c)N held fixed. We find that in this formulation logarithmic corrections already present at tree level persist to all higher genus, suggesting a behavior different from the previously considered cases of d<1 matter.