Space-Time Foliation and Frozen Formalism. Note Quote.

Diffeomorphism

The idea is that one foliates space-time into space and time and considers as fundamental canonical variables the three metric qab and as canonically conjugate momentum a quantity that is closely related to the extrinsic curvature Kab. The time-time and the space-time portions of the space-time metric (known as the lapse and shift vector) appear as Lagrange multipliers in the action, which means that the theory has constraints. In total there are four constraints, that structure themselves into a vector and a scalar. These constraints are the imprint in the canonical theory of the diffeomorphism invariance of the four-dimensional theory. They also contain the dynamics of the theory, since the Hamiltonian identically vanishes. This is not surprising, it is the way in which the canonical formalism tells us that the split into space and time that we perform is a fiduciary one. If one attempts to quantize this theory one starts by choosing a polarization for the wavefunctions (usually functions of the three metric) and one has to implement the constraints as operator equations. These will assure that the wavefunctions embody the symmetries of the theory. The diffeomorphism constraint has a geometrical interpretation, demanding that the wavefunctions be functions of the “three-geometry” and not of the three-metric, that is, that they be invariant under diffeomorphisms of the three manifold. The Hamiltonian constraint does not admit a simple geometric interpretation and should be implemented as an operatorial equation. Unfortunately, it is a complicated non-polynomial function of the basic variables and little progress had been made towards realizing it as a quantum operator ever since De Witt considered the problem in the 60’s. Let us recall that in this context regularization is a highly non-trivial process, since most common regulators used in quantum field theory violate diffeomorphism invariance. Even if we ignore these technical details, the resulting theory appears as very difficult to interpret. The theory has no explicit dynamics, one is in the “frozen formalism”. Wavefunctions are annihilated by the constraints and observable quantities commute with the constraints. Observables are better described, as Kuchar emphasizes, as “perennials”. The expectation is that in physical situations some of the variables of the theory will play the role of “time” and in terms of them one would be able to define a “true” dynamics in a relational way, and a non-vanishing Hamiltonian. In contrast to superstring theory, canonical quantum gravity seeks a non-perturbative quantum theory of only the gravitational field. It aims for consistency between quantum mechanics and gravity, not unification of all the different fields. The main idea is to apply standard quantization procedures to the general theory of relativity. To apply these procedures, it is necessary to cast general relativity into canonical (Hamiltonian) form, and then quantize in the usual way. This was partially successfully done by Dirac. Since it puts relativity into a more familiar form, it makes an otherwise daunting task seem hard but manageable.

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