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3.10 Lorentzian path integrals

Up to now, I have mainly concentrated on approaches to quantum gravity that fall under the broad heading of canonical quantization. An alternative approach - already implicit in the discussion of discrete methods - starts with the Feynman path integral, or “sum over histories.” In an important sense, path integral methods are less precise than those of canonical quantization: The infinite-dimensional “integral” over histories can rarely be rigorously defined, we do not really know what classes of paths to sum over, and ordering ambiguities in the operator formalism reemerge as ambiguities in the integration measure. On the other hand, path integrals allow us to ask questions - for example, about amplitudes for spatial topology change - that are difficult or impossible to formulated in a canonical theory.

The simplest path integral approach to (2+1)-dimensional quantum gravity is the phase space path integral, in which the action is written in the ADM form (29View Equation, 30View Equation), and the spatial metric g ij and momentum ij p are treated as independent integration variables. The lapse N and shift i N appear as Lagrange multipliers, and the integrals over these quantities yield delta functionals for the constraints H and Hi . One might therefore expect the result to be equivalent to the canonical quantization of Section  3.1, in which the constraints are set to zero and solved for the physical degrees of freedom. This is indeed true, as shown in  [77245] for spatially closed universes and  [63] for geometries with point particles. The main subtlety comes from the appearance of many different determinants, arising from gauge-fixing and from the delta functionals, which must be shown to cancel. The phase space path integral for the first order formulation similarly reproduces the corresponding canonically quantized theory.

It is perhaps more interesting to look at the covariant metric path integral, in which one starts with the ordinary Einstein-Hilbert action and gauge-fixes the full (2+1)-dimensional diffeomorphism group. This approach does not require a topology R × S, and could potentially describe topology-changing amplitudes. Unfortunately, very little is yet understood about this approach. Section 9.2 of  [81] describes a partial gauge-fixing, which takes advantage of the fact that every metric on a three-manifold is conformal to one of constant scalar curvature. But while this leads to some simplification, we are still left with an infinite-dimensional integral about which very little can yet be said.

By far the most useful results in the path integral approach to (2+1)-dimensional quantum gravity have come from the covariant first-order action (14View Equation). The path integral for this action was first fully analyzed in two seminal papers by Witten  [277279Jump To The Next Citation Point], who showed that it reduced to a ratio of determinants that has an elegant topological interpretation as the analytic or Ray-Singer torsion  [230] . The partition function for a closed three-manifold with /\ = 0 takes the form

|| (3)||3/2 || (1)||1/2 |detD w | |detD w | ZM = -------||-----(2)||--------, (74) |detD w |
where (n) D w = Dw * Dw *+ * Dw *Dw is the gauge-covariant Laplacian acting on n -forms and w is a flat SO(2, 1) connection. When M admits more than one such flat connection, Equation (74View Equation) must be integrated over the moduli space of such connections. This integral sometimes diverges  [279] ; the significance of that divergence is not understood.

Although it was originally derived for closed manifolds, Equation (74View Equation) can be extended to manifolds with boundary in a straightforward manner. The path integral then gives a transition function that depends on specified boundary data - most simply, the induced spin connection w, with some additional restrictions on the normal component of w and the triad E   [28384Jump To The Next Citation Point] . For a manifold with the topology R × S, the results agree with those of covariant canonical quantization: The transition amplitude between two surfaces with prescribed spin connections is nonzero only if the holonomies agree.

But the path integral can also give transition amplitudes between states on surfaces Si and Sf with different topologies. If we demand that the initial and final surfaces be nondegenerate and spacelike, their topologies are severely restricted: Amano and Higuchi have shown that Si and Sf must have equal Euler numbers  [6Jump To The Next Citation Point] . For such manifolds, concrete computations can exploit the topological invariance of the Ray-Singer torsion. Carlip and Cosgrove  [84], for example, explicitly compute amplitudes for a transition between a genus three surface and a pair of genus two surfaces.


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