Hamiltonian formulation for the theory of gravity and canonical transformations in extended phase space

Abstract

A starting point for the present work was the statement recently discussed in the literature that two Hamiltonian formulations for the theory of gravity, the one proposed by Dirac and the other by Arnowitt - Deser - Misner, may not be related by a canonical transformation. In its turn, it raises a question about the equivalence of these two Hamiltonian formulations and their equivalence to the original formulation of General Relativity. We argue that, since the transformation from components of metric tensor to the ADM variables touches gauge degrees of freedom, which are non-canonical from the point of view of Dirac, the problem cannot be resolved in the limits of the Dirac approach. The proposed solution requires the extension of phase space by treating gauge degrees of freedom on an equal footing with other variables and introducing missing velocities into the Lagrangian by means of gauge conditions in differential form. We illustrate with a simple cosmological model the features of Hamiltonian dynamics in extended phase space. Then, we give a clear proof for the full gravitational theory that the ADM-like transformation is canonical in extended phase space in a wide enough class of possible parametrizations.

18 pages, accepted for publication in Class. Quantum Grav

Cited by in corpus (14)

• Field-dependent symmetries in Friedmann-Robertson-Walker models
• Is the Wheeler -- DeWitt equation more fundamental than the Schrödinger equation?
• Dirac prescription from BRST symmetry in FRW space-time
• Hamiltonian analysis of General Relativity and extended gravity from the iterative Faddeev-Jackiw symplectic approach
• Quantum Symmetries and Conserved Charges of the Cosmological Friedmann-Robertson-Walker Model
• A superspace description of Friedmann-Robertson-Walker models
• Solution of the noncanonicity puzzle in General Relativity: a new Hamiltonian formulation
• Is the Copenhagen interpretation inapplicable to Quantum Cosmology?
• Generalized spherically symmetric gravitational model: Hamiltonian dynamics in extended phase space and BRST charge
• Some features of the extended phase space approach to quantization of gravity
• Remarks on the "non-canonicity puzzle": Lagrangian symmetries of the Einstein-Hilbert action
• What may come beyond the Wheeler-DeWitt approach?
• (Anti-)BRST Symmetries in FLRW Model: Supervariable Approach
• A view on the problems of Quantum Gravity