Approaches to Quantum Gravity

4Prolegomena to any future Quantum GravityJ. STACHEL4.1 Introduction“Prolegomena” means “preliminary observations,” and my title is meant to recallKant’s celebrated Prolegomena to Any Future Metaphysics That Can Claim to bea Science. My words, like his:are not supposed to serve as the exposition of an already-existing science, but to help inthe invention of the science itself in the first place.To use another Kantian phrase, I shall discuss some “conditions of possibility” of aquantum theory of gravity, stressing the need for solutions to certain fundamentalproblems confronting any attempt to apply some method of quantization to the fieldequations of General Relativity (GR). Not for lack of interest but lack of spacetime(S-T), other approaches to Quantum Gravity (QG) are not discussed here (butsee [35]).4.1.1 Background dependence versus background independenceThe first problem is the tension between “method of quantization” and “field equationsof GR”. The methods of quantization of pre-general-relativistic theories 1have been based on the existence of some fixed S-T structure(s), needed bothfor the development of the formalism and – equally importantly – for its physicalinterpretation. This S-T structure provides a fixed kinematical background fordynamical theories: the equations for particle or fields must be invariant under allautomorphisms of the S-T symmetry group. GR theory, on the other hand, is abackground-independent theory, without any fixed, non-dynamical S-T structures.1 In particular, non-relativistic Quantum Mechanics (QM) based on Galilei–Newtonian S-T, special-relativisticQuantum Field Theory based on Minkowski S-T, and Quantum Field Theories in non-flat Riemannian S-Ts.But see [30] for a discussion of topological QFT.Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter, ed. Daniele Oriti.Published by Cambridge University Press. c○ Cambridge University Press 2009.