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On another note, pertaining to the spacetime continuum hypothesis itself, I would like to digress from the Science vs. Religion theme to divulge a conundrum I have yet to find my way through with regard to a discrete spacetime.

Years ago, while doing my graduate studies, I was in an advanced solid state physics class. We were discussing (specific) heat capacities of solids, the Debye model, and continuum vs. the more realistic discrete nature of solids. An important take-away was that while the Debye model requires an artificial cutoff (the Debye temperature or energy) in order to obtain finite heat capacities, the more realistic discrete nature of solids, while involving a significantly more complicated density of states, provides its own, natural cutoff (with the cutoff and the overall density of states exhibiting a remarkable resemblance, though far more rough and random looking, to the smooth Debye model with its artificial cutoff).

At this point I was familiar with quantum mechanics (and we used it extensively in this solid state physics class), the "renormalizeable" infinities of quantum field theory, and General Relativity with its predilection to bend spacetime based on any and all energy densities (so the infinities of quantum field theory lead to infinitely curved spacetimes: The universe can only be a singularity [I'm sorry, but that conflicts with my experience/observations :-) ]). So the germ of an idea was born: What if spacetime was not a continuum, but discrete?

I'm sure others have asked the same question, though I have had a hard time finding good works that follow this hypothesis.

I have come up with many interesting consequences of this hypothesis, including the unavoidability/inevitability of a statistical nature, and the potential for the dimensionality of the "manifold" to vary with spacetime location (not "compactification", but actual different measures in different locations [with limitations]). However, as I alluded to above, I have come to a roadblock I have yet to surmount: I can go from discrete spacetime to "connections" (of the parallel transport type) like what's in both General Relativity and Quantum Field Theory, and I can go from "curvature" tensors to the Einstein-Hilbert and Yang-Mills actions (and some cross terms that may or may not be eliminateable). The problem is in jumping the gap from "connections" to "curvature" tensors.

In a continuum spacetime it's rather trivial to consider a "connection" (parallel transport) around a loop and look at the limit as the loop is shrunk to a point, in order to make this connection to the "curvature" tensor(s). However, discrete spacetime has no such ability to take any such limits (so derivatives don't exist either*).**

Does anyone, per chance, have any suggestions of good work others may have done in anything similar? Are there other serious works pursuing a discrete spacetime/manifold hypothesis, that aren't simply working from a continuum model (superstring theory, loop quantum gravity, or some such) and obtaining suggestive results that point, somehow, to the "idea" that the "reality" is, in some sense, "discrete"?

I would be greatly appreciative of any pointers.

Thanks.

David

* In discrete spacetime/manifolds we no longer have calculus as we know it (differential geometry, if you will). Instead all is simply algebra—incredibly huge algebra problems like ten to the one hundredth simultaneous high dimensional vector equations in the case of a single proton or neutron over a time span of a single femtosecond. :-)

** I know that Lattice Gauge theory uses discrete spacetime. It uses parallel transport "connections" along the lattice links/edges between the discrete points (sites/vertices), and assigns the "curvature" to the surfaces (plaquettes/faces) in between points bounded by the lattice links/edges. Unfortunately, this assignment of a quantity to "nowhere" is an anathema to my way of thinking, for any model of "reality". (As a computational model, I have no problem. It's only when in search of a model of "reality" that I have a problem with it.)

Of course, another problem with Lattice Gauge theory is the use of a uniform lattice: This imposes a preferential reference frame. "Reality", in my opinion, adheres to general "invariance" (general covariance plus).