Model of two-dimensional quantum gravity in the strong coupling regime.

A model of two-dimensional quantum gravity that is the analog of the tensionless string is proposed. The gravitational constant (k) is the analog of the Regge slope (α ′ ) and it shows that when k → ∞, 2D quantum gravity can be understood as a tensionless string theory embeded in a two-dimensional target space. The temporal coordinate of the target space play the role of the time and the wave function can be interpreted as in standard quantum mechanics. Typeset using REVTEX ∗Alexander von Humboldt Fellow †Address after October 17, 1994, E-mail: [email protected] 1 The quantization of gravity in comparison with other physical fields is a problem fundamentally different because in gravity the spacetime itself evolves dynamically and the concepts of time and wave function seem be very differents as is usually assumed in quantum mechanics [1]. In the last years many efforts has been realized in order to understand these problems using differents points of view [2], however the full problem it is still difficult and it seems to suggest that simplifications should be introduced in order to reach a more complete understanding. In this letter we would like to present a new route in the solution to these problems proposing a two-dimensional gravity model that is the analog of string theory in the limit α ′ → ∞ (i.e. tensionless string theory). This model can be more easily interpreted that 2D full gravity and could throw some light to these fundamental problems. More precisely, we will show below that; a) k is just the analog of the Regge slope α ′ and the limit k → ∞ correspond to the strong coupling regime of gravity, b) 2D quantum gravity in the strong coupling limit is exactly a tensionless string theory embeded in a twodimensional target space with the temporal coordinate playing the role of time, c) The wave function can be interpreted as in conventional quantum mechanics with the target space considered as genuine spacetime. The Model Before to discuss our problem, let us start considering some basic issues of string theory. The action of string theory is S = − 2 ∫ dσ √ −gg∂αx∂βxημν , (1) where T = 1/α ′ is the tension and it is a parameter that appear in (1) in order to have a dimensionless expression. Once the equation (1) is established one could try to take the limit T → 0 (tensionless limit) and to study the consequences involved, but infortunately, this action is not well defined and one must modify (1) in order to have a smooth behaviour. 2 The tensionless limit at the level of the action is a problem that was studied by Schild many years ago [3] and after considered by several other authors [4–8] in differents contexts. In essence the procedure delineated above is equivalent to make the hamiltonian formulation of (1) and to take directly the tensionless limit in the constraints H⊥ = 1 2 (p + T x ′2 ), H1 = px ′ (2) because in such limit one replace (2) by H⊥ = 1 2 p, H1 = px ′ (3) and the algebra of constraints becomes [H⊥(σ),H⊥(σ ′ )] = 0, [H⊥(σ),H1(σ ′ )] = (H⊥(σ) +H⊥(σ ′ ))δ ′ (σ − σ′), (4) [H1(σ),H1(σ ′ )] = (H1(σ) +H1(σ ′ ))δ ′ (σ − σ′). A careful analysis of the action in the tensionless limit reveal that g = det(gαβ) is vanishes and, as a consequence, all the points of the string moves with the velocity of the light. Thus, one see the constraints of the tensionless string as describing a set of infinite massless relativistic particles (one for every point along the string) suplemented by the transversality condition px ′ = 0. Now, we would like to construct an analog model for 2D gravity. Two-dimensional gravity is described by the following action S = − 2 ∫