Quantum Gravity

134 QUANTUM GEOMETRODYNAMICSand its momentum p cd (x) (or, in the approach of reduced quantization, a subsetof them, see Section 5.2 below). In the connection formulation of Section 4.3one has the connection A i a (x) and the densitized triad Eb j (y), while in the loopspaceformulation one takes the holonomy U[A, α] and the flux of Ej b through atwo-dimensional surface.In this chapter, restriction will be made to the quantization of the geometrodynamicalformulation, while quantum connection dynamics and quantum loopdynamics will be discussed in Chapter 6. Application of (5.2) to (4.64) wouldyield[ĥab(x), ˆp cd (y)] = iδ(a c δd b) δ(x, y) , (5.3)plus vanishing commutators between, respectively, the metric components andthe momentum components. Since p cd is linearly related to the extrinsic curvature,describing the embedding of the three-geometry into the fourth dimension,the presence of the commutator (5.3) and the ensuing ‘uncertainty relation’ betweenintrinsic and extrinsic geometry means that the classical space–time picturehas completely dissolved in quantum gravity. This is fully analogous tothe disappearance of particle trajectories as fundamental concepts in quantummechanics and constitutes one of the central interpretational ingredients of quantumgravity. The fundamental variables form a vector space that is closed underPoisson brackets and complete in the sense that every dynamical variable canbe expressed as a sum of products of fundamental variables.Equation (5.3) does not implement the positivity requirement deth >0ofthe classical theory. But this could only be a problem if (the smeared version of)ˆp ab were self-adjoint and its exponentiation therefore a unitary operator, whichcould ‘shift’ the metric to negative values.The second step addresses the quantization of a general variable, F ,ofthefundamental variables. Does the rule (5.2) still apply? As Dirac writes (Dirac 1958,p. 87), 1The strong analogy between the quantum P.B. ...and the classical P.B. ...leads us tomake the assumption that the quantum P.B.s, or at any rate the simpler ones of them,have the same values as the corresponding classical P.B.s. The simplest P.B.s are thoseinvolving the canonical coordinates and momenta themselves . . .In fact, from general theorems of quantum theory (going back to Groenewaldand van Hove), one knows that it is impossible to respect the transformationrule (5.2) in the general case, while assuming an irreducible representation ofthe commutation rules; cf. Giulini (2003). In Dirac’s quote this is anticipated bythe statement ‘or at any rate the simple ones of them’. This failure is related tothe problem of ‘factor ordering’. Therefore, additional criteria must be invokedto find the ‘correct’ quantization, such as the demand for ‘Dirac consistency’ tobe discussed in Section 5.3.The third step concerns the construction of an appropriate representationspace, F, for the dynamical variables, on which they should act as operators. We1 P.B. stands for Poisson bracket.