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Consider a set of N binary-state nodes at "time" t, each of which is a (boolean) transition function of two nodes in the set, evaluated at time t-1. Thus there are N of these boolean functions of two arguments. The nodes undergo an evolution (ie, follow a trajectory in state space) driven by the transition functions.

Further assume, initially, that the set of transition functions can change arbitrarily from one time-step to the next; the evolution is unconstrained and non-deterministic as possible, given the grounding assumptions.

Now, consider that the node states have names or labels (eg, "0" and "1"). The labeling is arbitrary, so that a description of such a system has a dual, obtained by globally exchanging state label "0" for "1" and "1" for "0", and each transition function by its dual, given that the functions are specified in terms of the labels. For example, AND(x,y) means a mapping {0,0:0; 0,1:0; 1,0:0; 1,1:1}, and OR(x,y) means a mapping {1,1:1; 1,0:1; 0,1:1; 0,0:0}. Thus, an exchange of label in the given mappings amounts to an exchange of AND for OR and vice versa.

We have a global symmetry of the description, reflecting the purely conventional choice of labeling for the node states. This symmetry does not imply any constraint on the dynamics. Suppose we would like to replace this global symmetry by a local symmetry, that allows local or "selective" relabeling of node states, both across the node collection, and through "time". How does this constrain the dynamics, and what is or should be invariant in this dynamics? For that matter, what is required to make the problem well-posed?

My question: Does the above sketch correspond to a developed area of mathematical investigation? In asking this question, I have in mind the observation that the set of N boolean transition functions can be regarded as an abstract simplicial complex; the binary-state nodes are vertices, and the transition functions are edges. I am therefore implicitly asking a question about the dynamics of this simplicial complex when bound to the dynamics of the node states in the way I have described. (This is the real motivation for my interest in this question.)

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It might help if you work out an example more explicitly showing what you mean by dynamics, and what exactly you're interested in. For instance, what's the "local symmetry" you're talking about in the case of three vertices forming a triangle? –  j.c. Oct 19 '09 at 16:35

3 Answers 3

It sounds like what you're after is some type of a lattice gauge theory. I wasn't able to figure out what you wanted from your question or answer, but I think the area of study you want to learn about is basically this one.

There's a very readable review by Kogut here, which also discusses spin systems (2D Ising model in some detail) and the "QC" correspondence between statistical mechanics and quantum mechanics. I found it fascinating and highly recommend it to anybody interested in these types of models, some of which have recently come into vogue in the physics of topological quantum computation.

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This is a bad idea. First, causal sets are, to put it politely "not well-respected" in theoretical physics. These kinds of discrete structures tend to break a significant amount of the structure of the theories; enough that they typically can't reduce to anything that looks close to any existing theory in any limit.

In particular, they break Lorentz invariance pretty directly, which is tied up with energy and momentum conservation. The standard "hope" is that that this invariance is really "approximate" somehow, and becomes exact in some limit. But this can't really happen, because, in some sense the "errors" cannot average out, they can only add together.

In particular, you no longer have properties like $E = m$ in a particle's rest frame, because this formula is a consequence of local Lorentz invariance. So, what you end up with, is having to fine-tune because making this approximate means saying something like "$E=m+\delta$" for a particle. But what about when you have $10^{23}$ particles? Well, now, that $\delta$ term has to be fine-tuned by hand to be $10^{-23}$ to keep the microscopic deviations of order one (which is already way too big). Then you can ask what happens if you boost a collection of particles (e.g., a lead ion) to near the speed of light, etc....

There are many other problems with discreteness, too, but this is an immediate and fatal one.

Incidentally, due to the seriousness of making spacetime discrete, a lot of the people who try to do these things (causal sets, dynamical triangulations, LQG, etc) like to claim that their theories do not actually do this. But this is basically a lie; there's no politer way to put it. If you ever see someone claim they do not make spacetime discrete, but do sometime like that, walk away, because they're probably a crackpot.

Also, the idea that spacetime has to be "discrete" somehow is a huge (but, sadly, common) misunderstanding of what quantum mechanics means. There is nothing "inherently" discrete, or any need to introduce any discreteness into physics other than as a computational tool (e.g., lattice field theory) or as a physical system whose initial conditions make this a convenient approximation (e.g., a regular solid structure, crystals, etc).

The initial idea behind thinking about these kinds of things wasn't so bad, though. It comes from the thinking of, in GR, interpreting the abstract points on a spacetime manifold as corresponding to "events," and the geodesic structure as connecting events together (see, for example, the first few chapters of Misner, Thorne, and Wheeler's GR book). This lead to trying to think of, instead of the geometric structure, the structure of "events", not in the sense of the topological structure of the manifold, but as some other structure inherited by how one can connect things together with geodesics. This naturally partitions things into "all things that could have effected the point $x$" and "all things that the point $x$ can effect," with the hope that by studying the structure of these kinds of causal relations, one can reformulate GR in a new (but equivalent!) way that may fit together nicely with the "observables" formulation of quantum mechanics.

It turns out, this doesn't work so well. The structures that you get are so uncontrolled and pathological and unconstrained, that the only thing you can do to get anything sensible is by just doing GR. AFAIK, almost all physicists (and certainly all of the top people in GR) abandoned this approach by the '80s.

You can find some references for this kind of stuff if you look, but it does not really lead anywhere.

Unfortunately, in the past 10 or 15 years, a handful of people came along and misunderstood what these people were doing, and started building crazy discrete models analogous to these, sometimes in combination with taking "lattice GR" ideas way too seriously, and produced a bunch of nonsense.

And, really, the only thing stopping most physicists from calling these ideas outright crackpot nonsense, is the involvement of actual GR bigshots in the ancestors to these ideas decades ago! Although, there are certainly a few well known physicists out there who are famous for getting very angry when these ideas are brought up ;).

If you're really determined to discover in detail why these ideas are so wrong, other than working out stuff for yourself, or learning why things are the way they are in GR and QFT (which is tough!--but you should!) you may be in some trouble. Physicists aren't really in the habit of writing papers about theories that they think are obviously, fundamentally broken. We just tend to make fun of them when talking to our colleagues and otherwise ignore them! So you can't find much in the way of refutations of specifics in published literature. But if you search carefully you can find the occasionally angry rant published somewhere, and there are a few unpublished papers on the arxiv about why these theories are all broken. The only author I remember offhand doing this is Peeters about LQG's many problems, but there are others if you look.

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Thanks, jc. My timing turned out to be unfortunate; sorry for not following up sooner.

What I am considering is an analogy between the system described above and gauge fields well known in theoretical physics. Take the most familiar and oldest example, the electromagnetic field. A matter field is coupled to a potential field—the electromagnetic scalar and vector potentials. The matter field is a complex quantity, and is characterized in part by a phase. A global change of the phase leaves all physical observables associated with the matter field unchanged. Similarly, a global shift in the components of the potential—a global change of "reference" potential—results in no observable changes to the electromagnetic field components derived from the potential.

However, in order to allow invariance of observables with respect to local shifts of the phase and the potential, the evolution of the matter field and the electromagnetic potential must be coupled in a particular way that tightly constrains the dynamics of the combined system, in order to allow local shifts of the phase to be compensated by local shifts of the potential so as to leave physical observables unchanged.

So the correspondence I'm proposing is this:

  • matter field: the binary-state nodes
  • field phase redefinition: local re-labeling of the states of the nodes
  • potential field: the transition functions (one per node) which act on the binary-state nodes (generating state transitions at successive "time" steps)
  • potential field redefinition: local replacement of transition functions by their boolean algebraic duals

For the boolean network the question now becomes: What invariant or "absolute" structure is associated with the local gauge symmetry being outlined here? To put it another way, what is the nature of the field that we take to be "observable", and how is its dynamics constrained?

I would like to suggest, roughly, that the right answer to this question is a causal structure implicit in the boolean network's evolution that is "compatible" with a smooth 4-dimensional pseudo-Riemannian manifold. More specifically, I have in mind what has come to be known as a causal set—a discrete skeleton for a spacetime. In this context, recall the observation in my initial post, that "the set of N boolean transition functions can be regarded as an abstract simplicial complex; the binary-state nodes are vertices, and the transition functions are edges."

I might mention that I've derived considerable conceptual inspiration from a recent paper by Hendryk Pfeiffer (arxiv:gr-qc/0404088), which contains the following remark:

If quantum general relativity in d = 3+1 is indeed a PL-QFT, the following two statements which sound philosophically completely contrary,
  • Nature is fundamentally smooth.
  • Nature is fundamentally discrete.
are just two different points of view on the same underlying mathematical structure: equivalence classes of smooth manifolds up to diffeomorphism.
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