Approaches to Quantum Gravity

The causal set approach to Quantum Gravity 421
This is discussed very clearly in Section 5 of J. B. Hartle: “Simplicial
Minisuperspace 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 smooth
manifold, 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 extra
information, 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 thought
of edge lengths in Regge calculus as “spacetime distances”, i.e. as the discrete
analogue of integrals along geodesics of the line element (possibly, better as the
sup or inf of such distances, according to whether the geodesic is timelike or
spacelike). As such they would simply be invariant under diffeos in the continuum
embedding, they would simply not transform at all under them. What is the
interpretation 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 transformations
between points in the manifold (with no notion of preserving
geometry or action), then one can define piecewise diffeomorphisms as oneto-one
invertible maps of the simplicial space into itself, which are smooth
on each simplex (e.g. relabelling vertices, or smooth diffeomorphisms of the
interiors of simplices). For a general curved simplicial geometry, one expects
diffeomorphisms in this sense to leave the edge lengths unchanged or change
them only according to a trivial relabelling of the vertices (I am quoting Hartle
here).
As for the definition of edge lengths, it depends how one arrives at the simplicial
complex. If it arises from the triangulation of a continuum manifold,
then I would define the edge lengths by geodesic distances between vertices
in 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 has
a notion of invariant distance.
• Q - D. Oriti - to R. Gambini and J. Pullin:
What is the exact relation of your “consistent discretization” scheme with traditional
Regge calculus? I understand from your work that your scheme allows
for a definition of a canonical (Hamiltonian) formulation of Regge calculus, that
had proven difficult to achieve in the usual formalism. But what are similarities
and differences, advantages and disadvantages, with respect to the Lagrangian
setting?