Approaches to Quantum Gravity

Prolegomena to any future Quantum Gravity 47electrodynamics, have both passed this test; and its use in QG is discussed inSection 4.4.4.1.4 Outline of the chapterIn QM and SRQFT, the choice of classical variables and of methods to describeprocesses they undergo played a major role in determining possible forms of thetransition to quantized versions of the theory, and sometimes even in the content ofthe quantized theory. 4 Section 4.2 discusses these problems for Maxwell’s theory,outlining three classical formalisms and corresponding quantizations. The Wilsonloops method, applied to GR, led to the development of a background-independentquantization procedure. Section 4.3 surveys possible choices of fundamental variablesin GR, and Section 4.4 discusses measurability analysis as a criterion forquantization. The classification of possible types of initial-value problems in GRis discussed in Sections 4.5 and 4.6. Section 4.7 treats various “mini-” and “midisuperspace”as examples of partially background-dependent S-Ts in GR, and thequantization of asymptotically flat S-Ts allowing a separation of kinematics anddynamics at null infinity. There is a brief Conclusion.4.2 Choice of variables and initial value problems in classicalelectromagnetic theoryIn view of the analogies between electromagnetism (EM) and GR (see Section4.3) – the only two classical long-range fields transmitting interactionsbetween their sources – I shall consider some of the issues arising in QG first inthe simpler context of EM theory. 5 Of course, there are also profound differencesbetween EM and GR – most notably, the former is background dependent andthe latter is not. One important similarity is that both theories are formulated withredundant variables. In any gauge-invariant theory, the number of degrees of freedomequals the number of field variables minus twice the number of gauge functions.For Maxwell’s theory, the count is four components of the electromagneticfour-potential A (symbols for geometric objects will often be abbreviated by droppingindices) minus two times one gauge function equals two degrees of freedom.For GR, the count is ten components of the pseudo-metric tensor g minus two timesfour “gauge” diffeomorphism functions, again equals two. There are two distinctanalogies between EM and GR. In the first, A is the analogue of g. In the second, itis the analogue of Ɣ, the inertio-gravitational connection. In comparisons between4 In SRQFT, inequivalent representations of the basic operator algebra are possible.5 This theory is simplest member of the class of gauge-invariant Yang–Mills theories, with gauge group U(1);most of the following discussion could be modified to include the entire class.