Approaches to Quantum Gravity

50 J. Stachelwhere S is any 2-surface bounded by C. 10 The relation between D (momentum)and E (velocity) is determined by the constitutive relations of the medium, the analogueof the mass in particle mechanics, which relates a particle’s momentum andvelocity.In a four-dimensional formulation, the “dual momenta” are the integrals ∫∫ S Gover any 2-surface S. This suggests the possibility of extending the canonical loopapproach to arbitrary spacelike and null initial hypersurfaces. But it is also possibleto carry out a Feynman-type quantization of the theory: a classical S-T pathof a such loop is an extremal in the class of timelike world tubes S (oriented 2-surfaces with boundaries) bounded by the loop integral ∫ CA on the initial and finalhyperplanes. To quantize, one assigns a probability amplitude exp iI(S) to eachsuch S, where I (S) is the surface action. The total quantum transition amplitudebetween the initial and final loops is the sum of these amplitudes over all such2-surfaces. 11 More generally, loop integrals of the 1-form A for all possible typesof closed curves C ought to be considered, leading to a Feynman-type quantizationthat is based on arbitrary spacelike loops. Using null-loops, null-hypersurfacequantization techniques might be applicable (see Section 4.6).The position and momentum-space representations of EM theory are unitarilyequivalent; but they are not unitarily equivalent to the loop representation. In orderto secure unitary equivalence, one must introduce smeared loops, 12 suggesting thatmeasurement analysis (see the Introduction) might show that ideal measurementof loop variables requires “thickened” four-dimensional regions of S-T around aloop. The implications of measurement analysis for loop quantization of GR alsodeserve careful investigation (see Section 4.4).4.3 Choice of fundamental variables in classical GROne choice is well known: a pseudo-metric and a symmetric affine connection, andthe structures derived from them. Much less explored is the choice of the conformaland projective structures (see, e.g., [14], Section 2.1, Geometries). The two choicesare inter-related in a number of ways, only some of which will be discussed here. 1310 Reference [32] gives a Lagrangian density for arbitrary constitutive relations. When evaluated on t = const,the only term in the Lagrangian density containing a time derivative is (∂A/∂t)·D, from which the expressionfor the momentum follows. If a non-linear constitutive relation is used, the difference between D and Ebecomes significant.11 See [21; 22; 23].12 The loops are “smeared” with a one parameter family of Gaussian functions over the three-space surroundingthe loop.13 Mathematically, all of these structures are best understood as G-structures of the first and second order; i.e.reductions of the linear frame bundle group GL(4, R) over the S-T manifold with respect to various subgroups(see [28]). The metric and volume structures are first order reductions of the group with respect to the pseudoorthogonalsubgroup SO(3, 1) and unit-determinant subgroup SL(4, R), respectively. The projective structure