Outline and historical overview

In the first part of this book we have derived a canonical connection formulation of classical General Relativity. We have defined precisely what one means by the canonical quantisation of a field theory with constraints and have emphasised the importance of n-form fields for a background-independent quantisation of generally covariant theories. In this part we will systematically carry out the canonical quantisation programme step by step and almost complete it. In more detail we will show that:

1. There exists a mathematically rigorous and, under natural physical assumptions, unique kinematical platform from which constraint quantisation is launched.

2. There exists at least one, consistent, well-defined quantisation of the Wheeler–DeWitt constraint operator whose action is explicitly known.

3. A corresponding physical inner product is known to exist.

4. There is a concrete proposal for constructing Dirac observables and physical Hamiltonians.

What is left to do is to check whether this solution of the quantisation problem has the correct semiclassical limit (semiclassical states at the kinematical level are already under control, however, not yet on the space of solutions to the constraints) and to construct quantisations of the classical formula for complete Dirac observables explicitly. This will involve, besides the improvement of the already available semiclassical techniques, the development of appropriate approximation schemes because the exact theory is too complicated to be solvable explicitly. After these steps have been completed one is ready to make physical predictions from the theory. The task will then be to identify quantum gravity effects, which lie in the realm of today's experimental precision, and to falsify the theory.