arXiv : hep - th / 9303083 v 1 15 Mar 1993 æ

We discuss the Dirac quantization of two dimensional gravity with bosonic matter fields. After defining the extended Hamiltonian it is possible to fix the gauge completely. The commutators can all be obtained in closed form; nevertheless, the results are not particularly simple. Universidade de São Paulo IFUSP-preprint-1041 March 1993 ∗ Present Address: Instituto de F́ısica, Universidade Federal do Rio Grande do Sul CP 15.051, Cep 91.500, Porto Alegre, R.S., Brazil Introduction Two dimensional gravity has been studied in detail during the last few years. Several important and clear results have been obtained for Liouville theory as well as for the light cone gauge pure gravity. Moreover correlation functions involving dressed operators have been systematically computed. In this plethora of results, it is however disappointing that very few methods can shed some light in the canonical structure of higher dimensional gravitation theory, since two dimensional space-time has revealed strength as a theoretical laboratory for higher dimensional (specially gauge) theories. Thus we shall follow the works [6] and [7] using the canonical method in order to study two dimensional gravity. We shall start with a brief discussion of pure two dimensional gravity. This is a very simple case, since the number of constraints is too large, and the canonical formalism fails to provide a non trivial result. However, working in an analogous way as in ref.[6], it is possible to calculate some fundamental quantities before the gauge fixing procedure. To begin with we consider the two dimensional gravity Lagrangian L = √ −g ( − 2 φ φ− α 2 Rφ+ α 2 β ) , (1) where R is the scalar curvature, β is the cosmological constant, φ is an scalar auxiliary field and α is the renormalized matter fields central charge. We proceed calculating the Hamiltonian structure for the model using light-cone variables, and after adding a suitable surface term to (1) we end up with a Hamiltonian system of four first class constraints π = ∂L ∂(∂−g−−) = 0 = Γ , (2a) π = ∂L ∂(∂−g−+) = 0 = Γ , (2b) φ1 = 1 2 [ (∂+φ) − 4 α2 (g++π )− 4 α (g++π ++)π−++++ g++ +2α∂ +φ+α βg++ ] ,(3a) φ2 = π∂+φ− 2g++π − π∂+g++ . (3b) Equations (2) are the primary constraints while equations (3) are secondary. At this level the model reveals the well-known SL(2, R) structure in a very clear way, when we construct the following set of variables J = 1 g++ (φ2 − φ1) + αβ 2 , (4a) J = j − xJ , (4b) j = [ g++ ( π + α 2 ∂+φ g++ ) + α 2 ( π − α 2 ∂+g++ g++ − ∂+φ ) ] , (4c) J = j − 2xJ − (x)J , (4d) j = α(g++ + 1) , (4e) b = π − α 2 ∂+g++ g++ + ∂+φ , (4f) 1 which classically satisfy the Poisson bracket SL(2, R) algebra {J(x), J(y)} = −2ǫηcdJ(x)δ(x− y) + αη∂+δ(x− y) . (5) Using this structure it is also possible to calculate the quantum BRST charge in a gauge independent way, expanding the energy momentum tensor in terms of Virasoro modes;we define the Virasoro constraint as τ = Tm + TS ≈ 0 (6a) where the b-field and gravity energy momentum tensor are respectively Tm = 1 4 b + α 2 ∂+b , (6b) TS = 1 2α2 ηabJ J − ∂+J . (6c) Imposing now the nilpotency (Q̂ = 0) for the BRST charge Q̂ = c0(L0 − a) + ∑ n6=0 : cnL−n : − 1 2 ∞ ∑ −∞ (m− n) : c−mc−nbm+n : , (7) we obtain the usual relation for the central charges in a gauge independent way cmat + 3k k + 2 − 6k − 28 = 0 , k = α 2 8 . (8) As we mentioned earlier, the next step in our work was to fix completely the gauge freedom for the model using the canonical formalism. This is a difficult task because the theory is diffeomorphism invariant; as a consequence the Hamiltonian is a linear combination of constraints Hc = 4 [ − √−g g++ φ1 + g−+ g++ φ2 ] . (9) In order to avoid this problem we use time dependent gauge fixing constraints. Our result shows that this procedure annihilates all the physical degrees of freedom for the pure gravity theory. However if we consider the model coupled to matter fields S = 1 2 ∫ dx √ −g[−φ φ−X X − αRφ− γRX + (γ + α)β] , (10) It is possible to find a non-trivial reduced phase space leaving the canonical procedure as a possible technique for quantization in this case.