2 The Hamilton-Jacobi method

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Also, Abdaala, etc., [4] deal with gauge independent analysis of ID— gravity and .... Let us consider the action function for 2D gravity in the light cone gauge as ...
IC/IR/2003/13 INTERNAL REPORT (Limited Distribution)

United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

PATH INTEGRAL QUANTIZATION OF 2 D- GRAVITY

Sami I. Muslih1 Department of Physics, Al-Azhar University-Gaza, P.O. Box 1277, Gaza, Gaza Strip, Palestinian Autonomous Territories and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract 2 D- gravity is investigated using the Hamilton-Jacobi formalism. The equations of motion and the action integral are obtained as total differential equations in many variables.

The

integrabilty conditions, lead us to obtain the path integral quantization without any need to introduce any extra un-physical variables.

MIRAMARE - TRIESTE August 2003

Regular Associate of the Abdus Salam ICTP. [email protected] it

1

Introduction

2D induced gravity has received much attention in the last few years [1-9]. The usual way to deal with this theory is to consider, Dirac's method which is used in a wide variety of situations [10,11]. One basic feature of this method is that the symmetries are related to first class constraints [10]. It was proven by Polyakov [1], that an induced theory of garvity in two dimensions (2D — gravity) exhibits a (hidden) SL(2,R) symmetry, in which this result appear to violate the Dirac's method treatment of constrained systems, i.e, there is symmetry, but (apparently) there are no first class constraints [2-8]. The treatment of ID— gravity by using Dirac's method has been the interest of many authors [2-9]. For example, Barcelos-Neto [8] , has studied the validity of Dirac's method to hidden symmetry by considering the expansion of constraints in Fourier modes in terms of infinite constraints. Also, Abdaala, etc., [4] deal with gauge independent analysis of ID— gravity and propose a solution of the Liouvile theory in terms of the SL(2, R) current. Recently, Baleanu and Guler [6] have studied the SL(2, R) of ID — gravity by using the Hamilton-Jacobi method [12-16] and obtain similar results as given in references [3, 4, 8]. Even though of this progress in the study of 2D — gravity, the problem of obtaining the path integral quatization is still missing. The aim of this paper is to consider the path integral quantization of 2D — gravity using the canonical path integral method [17-21].

2

The Hamilton-Jacobi method

Recently the Hamilton-Jacobi method [12-16] has been developed to investigate constrained Hamiltonian systems. In this method if we start with a singular Lagrangian L — L(qi,qt,t), i = 1,2, ...,n, with Hessian matrix of rank (n — r), r < n, then the generalized momenta can be written as

f\ -r-

Pa= .p-, o = l,2,...,n-r, /·) Τ

Ρμ = -^Γ, μ = π-Γ + 1,...,η, Οίμ

(1) (2)

where ql are divided into two sets, qa and ίμ. Since the rank of the Hessian matrix is (n — r), one can solve the expressible velocities from (1) and after substituting in (2), we get (3)

Ρμ = -Ημ( * ι / > 9 α ) >

μ, V = Π - Τ + Ι , ...,Π.

(4)

The set of Hamilton-Jacobi partial differential equations [HJPDE] is expressed as [12, 13] QC1