Approaches to Quantum Gravity

54 J. Stachelof a projective structure, with results that depend only on that structure, should alsobe studied.The Riemann or affine curvature tensor. DeWitt [6; 7], and Bergmann and Smith[4] studied the measurability of the components of the linearized Riemann tensorwith respect to an inertial frame of reference, and drew some tentative conclusionsabout the exact theory. Arguing that, in gauge theories, only gauge-invariantquantities should be subject to the commutation rules, they concluded that measurementanalysis should be carried out exclusively at the level of the Riemanntensor. However, this conclusion neglects three important factors.(i) It follows from the compatibility of chrono-geometry and inertio-gravitational fieldin GR that measurements of the former can be interpreted in terms of the latter. Asnoted, the interval ds between two neighboring events is gauge invariant, as is itsintegral along any closed world line. Indeed, all methods of measuring components ofthe Riemann tensor ultimately depend on measurement of such intervals, either spacelikeor time-like, which agree (up to a linear transformation) with the correspondingaffine parameters on geodesics.(ii) Introduction of additional geometrical structures on the S-T manifold to modelmacroscopic preparation and registration devices produces additional gauge-invariantquantities relative to these structures (see [26]).(iii) While a geometric object may not be gauge-invariant, some non-local integral of itmay be. The electromagnetic four-potential, for example, is not gauge invariant, but itsloop integrals are (see Section 4.2). Similarly, at the connection level, the holonomiesof the set of connection one-forms play an important role in LQG. (see, e.g., [2; 27]).In both EM and GR, one would like to have a method of loop quantization thatdoes not depend on singling out a family of spacelike hypersurfaces. The various“problems of time” said to arise in the canonical quantization of GR seem to beartifacts of the canonical technique rather than genuine physical problems. 19 Thenext section discusses some non-canonical possibilities.Some tensor abstracted from the Riemann tensor, such as the Weyl or conformalcurvature tensor. For example, measurability analysis of the Newman–Penroseformalism, based on the use of invariants constructed from the components of theWeyl tensor with respect to a null tetrad (see, e.g., [33], Chapter 7), might suggestnew candidate dynamical variables for quantization.19 That is, problems that arise from the attempt to attach physical meaning to some global time coordinateintroduced in the canonical formalism, the role of which in the formalism is purely as an ordering parameterwith no physical significance (see [26; 22; 23]). The real problem of time is the role in QG of the local orproper time, which is a measurable quantity classically.