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Given a smooth manifold $\mathcal{M}$ with a simplicial complex embedding $\mathsf{S}$, what specific tools or methods can be used to give an analysis of the time evolution of the manifold given some initial conditions? More specifically, if I have a gauge group $G_{\phi}$ acting on the simplicial complex with subgroups $S, SO(3), T$ representing scaling, rotation, and translation on the manifold, how can I introduce time into this picture?

This is a general question that has been applied in the context of Causal Dynamical Triangulations, and I am essentially looking for a reference on how to get started reading about this field, and other questions relating manifold triangulations and time evolution.

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I'm a little confused about your question but I will do my best to answer some of the points I understand.

I imagine your primary interest in asking this question (since you mention CDT) is that you would like to learn about using simplicial approaches to study general relativity or quantum gravity. For this the primary tool is the Regge Calculus, my favorite reference for this is Rafael Sorkins PhD thesis:

This incidentally also deals with how to study electromagnetism and so in a sense corresponds to to your questions about gauge groups. Remember though that a gauge group usually acts on an internal space, and so a simplicial complex is not enough to study gauge theories you also need to look into how you define fields (vector, spinor) on these geometrical objects. This can be tricky for the spinors.

To study explicitly the question of time evolution of a simplicial complex (via the Regge equations) the best reference I know is:

Finally you should realize that CDT does not have a time evolution in this sense. Although "time" enters as a criteria for which combinations of simplices appear in the sum over paths there is no classical time evolution as it is meant to be a quantum theory. That said in two dimensions you can find a Hamiltonian operator which evolves "loop states" via a Schrodinger type equation. Good CDT references are:

There are loads of references for this stuff so just look around, but these should start you on your way.

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Thanks for the references, this is exactly what I was looking for. Maybe the next time I ask a question about this stuff, it will not be misusing terminology! – Samuel Reid Mar 29 '12 at 6:26
I just about cried when I saw the thesis you linked to Regge Calculus wasn't typed in LaTeX. – Samuel Reid Mar 29 '12 at 6:30
its from a different era, but still the best reference on Regge calculus in my opinion. – Kyle Mar 29 '12 at 6:58

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