GROUP-THEORETICAL QUANTIZATION OF 2-GRAVITY IN THE METRIC-TORUS SECTOR(1)

A symmetry based quantization method of reparametrization invariant sy stems is described; it will work for all systems that possess complete sets of perennials whose Lie algebras close and those that generate s ufficiently large symmetry groups. The construction leads to a quantum theory including a Hilbert space, a complete system of operator obser vables, and a unitary time evolution. The method is applied to the 2+1 gravity. The paper is restricted to the metric-torus sector, zero cos mological constant Lambda and it makes strong use of the so-called hom ogeneous gauge; the chosen algebra of perennials is that of Martin. Tw o frequent problems are tackled. First, the Lie algebra of perennials does not generate a group of symmetries. The notion of group completio n of a reparametrization invariant system is introduced so that the gr oup does act; the group completion of the physical phase space of our model is shown to add only some limit points to it so that the ranges of observables are not unduly changed. Second, a relatively large numb er of relations between observables exists; they are transferred to th e quantum theory by the well-known methods of Kostant and Kirillov. In this way, a uniqueness of the physical representation of some extensi on of Martin's algebra is shown. The Hamiltonian is defined by a syste matic procedure due to Dirac; for the torus sector, the result coincid es with that by Moncrief. The construction may be extended to higher g enera and nonzero Lambda of the 2+1 gravity, because some complete set s of perennials are well-known and there are no obstructions to the cl osure of the algebra. (C) 1998 American Institute of Physics.