Hamiltonian Dynamics in Extended Phase Space for Gravity and Its Consistency With Lagrangian Formalism: A Generalized Spherically Symmetric Model as an Example

Abstract

Among theoretical issues in General Relativity the problem of constructing its Hamiltonian formulation is still of interest. The most of attempts to quantize Gravity are based upon Dirac generalization of Hamiltonian dynamics for system with constraints. At the same time there exists another way to formulate Hamiltonian dynamics for constrained systems guided by the idea of extended phase space. We have already considered some features of this approach in the previous MG12 Meeting by the example of a simple isotropic model. Now we apply the approach to a generalized spherically symmetric model which imitates the structure of General Relativity much better. In particular, making use of a global BRST symmetry and the Noether theorem, we construct the BRST charge that generates correct gauge transformations for all gravitational degrees of freedom.As a rule, Hamiltonian formulation for gravity is constructed following to the Dirac scheme [1,2] and is a starting point for most attempts to quantize gravity. However, there are some reasons to doubt that Dirac Hamiltonian formulation for gravitational theory [3] can be thought as an equivalent one to the original General Relativity. Indeed, in Einstein formulation of General Relativity g 0µ components of metric tensor (µ = 0, 1, 2, 3) are treated on an equal basis with the rest of components, g ij (i, j = 1, 2, 3). The theory is invariant under gauge transformations that touch all metric components. In the Dirac approach only g ij components with their conjugate momenta are included into phase space, the transformations for this variables are generated by constraints. In fact, the original General Relativity and the Dirac formulation are theories with different groups of transformations. We can think of it as a considerable mathematical indication that Lagrangian and Hamiltonian formalism appear to be non-equivalent for the full theory of gravity. Moreover, the algebra of constraints in Dirac phase space does depend on parametrization of gravitational variables. It creates a serious obstacle to find an algorithm to construct a generator that would give correct transformations for all gravitational variables in 1