Solution to the zero-Hamiltonian problem in 2D gravity

Abstract

The zero-Hamiltonian problem, present in reparametrization invariant systems, is solved for the 2D induced gravity model. Working with methods developed by Henneaux, Teitelboim, and Vergara we find, systematically, reduced phase-space physics, generated by an effective Hamiltonian obtained after complete gauge fixing.

Time Boundary Terms and Dirac Constraints

“…[13][14][15][16][17][18][19]. Moreover, these way to deal with boundary term could be applied into the zero-Hamiltonian problem in 2D gravity [20,21] and into topological field theories [15]. Finally, the approach developed here can extended to its complex counterpart and analyze complex canonical transformations [22].…”

“…On the other hand, we take the boundary term B = B(q, t, p, p t )| τ2 τ1 . By applying the method, we add this boundary term to the action principle (21), namely…”

An effective Hamiltonian for 2D-black-hole physics

“…where D denotes the Dirac bracket [1] operation. In the 2D-induced-gravity case we found [5] g 11 (x), π 11 (y…”

“…Although this is the canonical bracket relation the gravitational field and the corresponding momentum are not independent quantities in this case [5]. For the effective hamiltonian density we obtained [5] H ef f = g 11…”

“…Once determined the simplectic structure, after complete gauge fixing, an special time-dependent canonical transformation is performed, obtaining the generator of dynamics for the physical variables. A direct application of these techniques was done in the 2D induced gravity model of Polyakov [4]; here [5] it was possible to obtain the physical Hamiltonian and dynamics of the true degrees of freedom in a systematic way; without the complications found when other methods are used [11]. The ZH problem was also focused in the 2D black hole theory, using dilatonic gravity models [6] (in particular, the CGHS model).…”