Richard Feynman, in the course of his doctoral work, developed the path integral formulation of quantum mechanics as an alternative, space-time covariant description of quantum mechanics, which is nevertheless equivalent to the canonical approach [1]. It is thus not surprising that the path integral formulation has been of interest in the quantization of general relativity, a theory where space-time covariance plays a key role. However, once one departs from the regime of free, unconstrained systems, the equivalence of the path integral approach and canonical approach becomes more subtle than originally described by Feynman in [1]. In particular, in Feynman’s original argument, the integration measure for the configuration path integral is a formal Lebesgue measure; in the interacting case, however, in order to have equivalence with the canonical theory, one cannot use the naive Lebesgue measure in the path integral, but must use a measure derived from the Liouville measure on the phase space [2]. Such a measure has yet to be incorporated into spin-foam models, which can be thought of as a path-integral version of loop quantum gravity (LQG) [3, 4]. Loop quantum gravity is an attempt to make a mathematically rigorous quantization of general relativity that preserves background independence— for reviews, see [8, 6, 7] and for books see [9, 10]. Spin-foams intend to be a path integral formulation for loop quantum gravity, directly motivated from the ideas of Feynman appropriately adapted to reparametrization-invariant theories [4, 5]. Only the kinematical structure of LQG is used in motivating the spin-foamframework. The dynamics one tries to encode in the amplitude factors appearing in the path integral which is being replaced by a sum in a regularisation step which depends on a triangulation of the spacetime manifold. Eventually one has to take a weighted average over these (generalised) triangulations for which the proposal at present is to use methods fromgroup field theory [3]. The current spin foam approach is independent from the dynamical theory of canonical LQG [11] because the dynamics of canonical LQG is rather complicated. It instead uses an apparently much simpler starting point: Namely, in the Plebanski formulation [14], GR can be considered as a constrained BF theory, and treating the so called simplicity constraints as a perturbation of BF theory, one can make use of the powerful toolbox that comes with topological QFT’s [12]. It is an unanswered question, however, and one of the most active research topics momentarily1, how canonical LQG and spin foams fit together. It is one the aims of this paper to make a contribution towards answering this question. In LQG one is compelled to introduce a 1-parameter quantization ambiguity — the so-called Immirzi parameter [15, 16]. This enters the action through a necessary extra ‘topological’ term added to the Palatini action; the full action is termed the Holst action [17]. To properly incorporate the Immirzi parameter into spin-foams, one should in fact not start from the usual Plebanski formulation but rather an analogous generalization, in which an analogous topological term is added to the action, leading to what we call the Plebanski-Holst formulation of gravity [19, 20, 21]. In [22] we have shown (and partly reviewed) for a rather general theory that different canonical quantisation techniques for gauge theories, specifically Dirac’s operator constraint method, the Master Constraint method and the reduced phase space method all lead to the same path integral. A prominent role in establishing this equivalence is played by what is called “the choice of gauge fixing” (from the reduced phase space point of view) or, equivalently, the choice of clocks (from the gauge invariant i.e. relational point of view [24]). After a long analysis, it transpires that the common basis for the path integral measure, no matter from which starting point it is derived, is the Liouville measure on the reduced phase, which can be defined via gauge fixing of the first class constraints. This measure can be extended to the full phase space and one shows that the dependence on the gauge fixing disappears when one integrates gauge invariant functions2. From this point of view, that is, the equivalence between path-integral formulation and the canonical theory, it is obvious that formal path-integrals derived from the various formulations of gravity should all be equivalent, because all of them have the same reduced phase space— that of general relativity. We thus apply the general reduced phase space framework to the Holst action as the starting point for deriving a formal path integral for both the Holst action and the Plebanski-Holst action. It turns out that the resulting path-integral for either the Holst action or the Plebanski-Holst action is not the naive Lebesgue measure integral of the exponentiated action. There are extra measure factors of spacetime volume element V and spatial volume element Vs. The presence of a spatial volume element is especially surprising because it breaks the manifest spacetime covariance of the path-integral when we are off shell. The origin of this lack of covariance is in the mixture of dynamics and gauge invariance inherent to generally covariant systems with propagating degrees of freedom and it is well known that the gauge symmetries generated by the constraints only coincide on shell with spacetime diffeomorphism invariance. The quantum theory chooses to preserve the gauge symmetries generated by the constraints rather than spacetime diffeomorphism invariance when we take quantum corrections into account (go off shell). This kind of extra measure factor (so called local measure) has appeared and been discussed in the literature since 1960s (see for instance [25, 26]) in the formalism of geometrodynamics and its background-dependent quantizations (stationary phase approximation). The outcome from the earlier investigations appears to be that in background-dependent, perturbative quantizations, these measure factors of V and Vs only contribute to the divergent part of the higher loop-order amplitudes. Thus their meanings essentially depend on the regularization scheme used. One can of course try to choose certain regularization schemes such that, either the local measure factors never contribute to the transition amplitude, or that their effect is canceled by the divergence from the action [25, 26]. However, the power of renormalisation and the very reason we trust it is that its predictions are independent of the regularisation technique chosen. Therefore the status of these measure factors is very much unsettled, especially for non perturbative quantisation techniques. We here take the point of view that the measure factors should be taken seriously because they take the off shell symmetry generated by the constraints properly into account. In which sense this so called Bergmann – Komar “group” [33] is preserved in the path integral is the subject of the research conducted in [28]. In this article we confine ourselves to a brief discussion. In the formalism of connection-dynamics, which is a preparation of background-independent quantization, a similar local measure factor also appears. It was first pointed out in [27], whose path-integral will be shown to be equivalent to our present formulation up to a discrepancy whose origin we resolve. When we perform background-independent quantization as in spin-foam models, therefore the local measure factor should not be simply ignored, because the regularization arguments in background-dependent quantization have no obvious bearing in the background-independent context anymore. For example, spin-foam models are defined on a triangulation of the spacetime manifold with finite number of vertices, where at each vertex the value of local measure is finite, and the action also does not show any divergence. However, so far none of the existing spin-foam models implements this non-trivial local measure factor in the quantization. The quantum effect implied by this measure factor has not been analyzed in the context of spin-foam models. But without it there is no chance to link spin foams with canonical LQG which at present is the only method we have in order to derive a path integral formulation of LQG from first principles. In ongoing work [32] we analyse the non-trivial effects caused by this measure factor in the context of spin-foam models, and try to give spin-foam amplitudes an unambiguous canonical interpretation by establishing a link between path-integral formulation and canonical quantization. In this article we also make a few comments on this. inspirehep.net/record/837228
Posted on: Thu, 07 Aug 2014 22:43:44 +0000
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