Introduction

“Prolegomena” means “preliminary observations,” and my title is meant to recall Kant's celebrated Prolegomena to Any Future Metaphysics That Can Claim to be a Science. My words, like his:

To use another Kantian phrase, I shall discuss some “conditions of possibility” of a quantum theory of gravity, stressing the need for solutions to certain fundamental problems confronting any attempt to apply some method of quantization to the field equations of General Relativity (GR). Not for lack of interest but lack of space-time (S-T), other approaches to Quantum Gravity (QG) are not discussed here (but see).

Background dependence versus background independence

The first problem is the tension between “method of quantization” and “field equations of GR”. The methods of quantization of pre-general-relativistic theories have been based on the existence of some fixed S-T structure(s), needed both for the development of the formalism and – equally importantly – for its physical interpretation. This S-T structure provides a fixed kinematical background for dynamical theories: the equations for particle or fields must be invariant under all automorphisms of the S-T symmetry group. GR theory, on the other hand, is a background-independent theory, without any fixed, non-dynamical S-T structures.