Spin foam models are an attempt at a covariant or path integral formulation of canonical loop quantum gravity. The construction of such models usually relies on the Plebanski formulation of general relativity as a constrained BF theory and is based on the discretization of the action on a simplicial triangulation, which may be viewed as an ultraviolet regulator. The triangulation dependence can be removed by means of group field theory techniques, which allows one to sum over all triangulations. The main tasks for these models are the correct quantum implementation of the Plebanski constraints, the existence of a semiclassical sector implementing additional 'Regge-like' constraints arising from simplicial triangulations and the definition of the physical inner product of loop quantum gravity via group field theory. Here we propose a new approach to tackle these issues stemming directly from the Holst action for general relativity, which is also a proper starting point for canonical loop quantum gravity. The discretization is performed by means of a 'cubulation' of the manifold rather than a triangulation. We give a direct interpretation of the resulting spin foam model as a generating functional for the n-point functions on the physical Hilbert space at finite regulator. This paper focuses on ideas and tasks to be performed before the model can be taken seriously. However, our analysis reveals some interesting features of this model: firstly, the structure of its amplitudes differs Content from this work may be used under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 9 In fact, as shown in recent work [8,33,34], SFM defined as constrained BF models take the form of noncommutative discrete path integrals making use of a star product on functionals of the B variables. It can also be shown that the generating group field theories are just a particular class of non-commutative field theories [35]. 10 See, however, [8,33,37] for discussions of this point. 11 See, however,[33] for a recent critical review of the various arguments raised against the Barrett-Crane model.
Sheaves are objects of a local nature: a global section is determined by how it looks locally. Hence, a sheaf cannot describe mathematical structures which contain global or nonlocal geometric information. To fill this gap, we introduce the theory of "gleaves", which are presheaves equipped with an additional "gluing operation" of compatible pairs of local sections. This generalizes the conditional product structures of Dawid and Studený, which correspond to gleaves on distributive lattices. Our examples include the gleaf of metric spaces and the gleaf of joint probability distributions. A result of Johnstone shows that a category of gleaves can have a subobject classifier despite not being cartesian closed.Gleaves over the simplex category ∆, which we call compositories, can be interpreted as a new kind of higher category in which the composition of an m-morphism and an n-morphism along a common k-morphism face results in an (m + n − k)-morphism. The distinctive feature of this composition operation is that the original morphisms can be recovered from the composite morphism as initial and final faces. Examples of compositories include nerves of categories and compositories of higher spans.
Topos theory has been suggested by Döring and Isham as an alternative mathematical structure with which to formulate physical theories. In particular, it has been used to reformulate standard quantum mechanics in such a way that a novel type of logic is used to represent propositions. In this paper we extend this formulation to include temporally ordered collections of propositions as opposed to single-time propositions. That is to say, we have developed a quantum history formalism in the language of topos theory where truth values can be assigned to temporal propositions. We analyze the extent to which such truth values can be derived from the truth values of the constituent, single-time propositions.
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