100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

Gravity, Geometry and the Quantum 357Now, most of the techniques used in the familiar, Minkowskian quantumtheories are deeply rooted in the availability of a flat back-groundmetric. In particular, it is this structure that enables one to single out thevacuum state, perform Fourier transforms to decompose fields canonicallyinto creation and annihilation parts, define masses and spins of particles andcarry out regularizations of products of operators. Already when one passesto quantum field theory in curved space-times, extra work is needed toconstruct mathematical structures that can adequately capture underlyingphysics (see Ford’s article in this volume). In our case, the situation is muchmore drastic 12 : there is no background metric whatsoever! Therefore newphysical ideas and mathematical tools are now necessary. Fortunately, theywere constructed by a number of researchers in the mid-nineties and havegiven rise to a detailed quantum theory of Riemannian geometry. 24,25,38Because the situation is conceptually so novel and because there areno direct experiments to guide us, reliable results require a high degree ofmathematical precision to ensure that there are no hidden infinities. Achievingthis precision has been a priority in the program. Thus, while one isinevitably motivated by heuristic, physical ideas and formal manipulations,generally the final results are mathematically rigorous. In particular, duecare is taken in constructing function spaces, defining measures and functionalintegrals, regularizing products of field operators, and calculatingeigenvectors and eigenvalues of geometric operators. Consequently, the finalresults are all free of divergences, well-defined, and respect the backgroundindependence (diffeomorphism invariance).Let us now turn to specifics. For simplicity, I will focus on the gravitationalfield; matter couplings are discussed in references [11,13,25,38].The basic gravitational configuration variable is an SU(2)-connection, A i aon a 3-manifold M representing ‘space’. As in gauge theories, the momentaare the ‘electric fields’ Ei a.d However, in the present gravitationalcontext, they also acquire a space-time meaning: they can be naturallyinterpreted as orthonormal triads (with density weight 1) and determinethe dynamical, Riemannian geometry of M. Thus, in contrast to Wheeler’sgeometrodynamics 2,3,4 , the Riemannian structures, including the positivedefinitemetric on M, is now built from momentum variables.The basic kinematic objects are: i) holonomies h e (A) of A i a , which dictatehow spinors are parallel transported along curves or edges e; andd Throughout, indices a, b, .. will refer to the tangent space of M while the ‘internal’indices i, j, ... will refer to the Lie algebra of SU(2).