Approaches to Quantum Gravity

The causal set approach to Quantum Gravity 421This is discussed very clearly in Section 5 of J. B. Hartle: “SimplicialMinisuperspace I: General Discussion”, J. Math. Phys. 26 (1985) 804–814.• Q - D. Oriti - to R. Williams:Just for the sake of clarity, let me clarify my doubt a bit more. If I take a smoothmanifold, I can define diffeos as smooth maps between points in the manifold,right? This definition does not need any notion of geometry, action, etc., I think.In a piecewise linear or simplicial space, is there an analogue notion of “diffeos”,i.e. maps between points in the space, that does not require any extrainformation, like geometry or an embedding into the continuum, i.e. an “intrinsic”analogue of diffeos? Also, I am a bit puzzled, because I have always thoughtof edge lengths in Regge calculus as “spacetime distances”, i.e. as the discreteanalogue of integrals along geodesics of the line element (possibly, better as thesup or inf of such distances, according to whether the geodesic is timelike orspacelike). As such they would simply be invariant under diffeos in the continuumembedding, they would simply not transform at all under them. What is theinterpretation of them that you are using and that is used in defining diffeos?– A - R. M. Williams:If you want an analogue of continuum diffeomorphisms as smooth transformationsbetween points in the manifold (with no notion of preservinggeometry or action), then one can define piecewise diffeomorphisms as oneto-oneinvertible maps of the simplicial space into itself, which are smoothon each simplex (e.g. relabelling vertices, or smooth diffeomorphisms of theinteriors of simplices). For a general curved simplicial geometry, one expectsdiffeomorphisms in this sense to leave the edge lengths unchanged or changethem only according to a trivial relabelling of the vertices (I am quoting Hartlehere).As for the definition of edge lengths, it depends how one arrives at the simplicialcomplex. If it arises from the triangulation of a continuum manifold,then I would define the edge lengths by geodesic distances between verticesin the manifold. But if the complex is a “given”, with no notion of an embedding,then the edge lengths are just “given” too and I do not see that one hasa notion of invariant distance.• Q - D. Oriti - to R. Gambini and J. Pullin:What is the exact relation of your “consistent discretization” scheme with traditionalRegge calculus? I understand from your work that your scheme allowsfor a definition of a canonical (Hamiltonian) formulation of Regge calculus, thathad proven difficult to achieve in the usual formalism. But what are similaritiesand differences, advantages and disadvantages, with respect to the Lagrangiansetting?