On the Foundational Assumptions of Modern Physics by Benjamin F. Dribus

Cristi,

"What remains then" is indeed a legitimate question about my setup, which is quite minimalistic in its most general form. It is also worth asking if the causal metric hypothesis trivializes deep and subtle issues. My view is that one of the principle reasons manifold models have dominated physics is because they are so convenient mathematically; once you know about the continuum and the complex numbers their lure is almost irresistible. Hence, more primitive and messy approaches may have been neglected.

Coming from a math background and working mostly with algebraic schemes and complex manifolds, it is hard for me to believe that the physical world behaves in such a convenient way. Conceptual simplicity and mathematical convenience are very different! This essay and all the unpublished work associated with it represent my attempt to "think physically" rather than just mathematically; my focus here is the basic physical principles, and the associated math is not nearly as convenient as the math encountered in mainstream physics. In any case, I think approaches like this deserve more attention.

You seem to have some of the same philosophical motivations, refusing to reject singularities just because they are "mathematically ugly."

Take care,

Ben

Bee,

Thanks for the kind remarks! When I started thinking about this a couple of years ago I didn't yet know about causal sets, and I was amazed when I found Rafael Sorkin's papers. I think he does an excellent job of explaining a lot of the motivating ideas. His students and coworkers have gone on to develop various aspects of the theory, but I still tend to prefer his qualitative considerations and careful explanations.

The causal set community is still relatively small from what I understand, and I come completely from the outside. There are certain assumptions most of them make that I can't seem to convince myself of, but I haven't had much chance to discuss these things with any of them in depth. In any case, I have the utmost respect for their work. I am hoping an expert causal set theorist will come along and say "that won't work because..." and help me sharpen these ideas further.

Take care,

Ben

Hi Steve,

Algebraic K-theory is something I didn't originally plan to specialize in, but it kept coming up in seemingly "purely geometric" situations; particularly involving groups of algebraic cycles and their equivalence relations, the Hodge conjecture, and so on. It also applies to physics via string theory, cyclic homology, noncommutative geometry, the theory of motives, and number-theoretic topics like the Langlands program.

Entropy is something I've studied a great deal over the last few years and still don't adequately understand. Just in the field of quantum information theory, there are a lot of different notions of entropy, and there seem to be added complications in incorporating this into a primitive causal theory like I describe in my essay.

You use some terminology that I don't quite understand, such as "evolution spherization." Also, I am not sure when you are referring to spheres as physical spaces and when you are referring to them as parameter spaces like the Bloch sphere etc. Do you have all this written down somewhere?

Take care,

Ben

Hi Harlan,

Thanks for the feedback! Those are good questions, and I can only partially answer them. Let me itemize.

1. Regarding the prediction of the dimension, the first question is how you even define the dimension of a causal relation. It will be emergent, only making sense at large enough scales, and it won't be an integer in general, although it must be very close to 4 at appropriate scales. Fractal dimension is relevant here. There is actually a fair bit of literature on the dimensions of causal sets, but these papers tend to use hypotheses that seem to obscure part of the structure. I have made some progress on this for structures I consider relevant, but it is not yet developed to my satisfaction.

Then, of course, you have to predict it. One of the greatest difficulties with causal theories like causal set theory and some versions of my own ideas is that there are a lot more "obviously nonphysical" universes than physical ones. This is usually described as an "entropy problem," in the sense that nonphysical solutions tend to dominate just like "disordered" solutions dominate in classical statistical thermodynamics. One way around this is to use a Lagrangian approach which (potentially, hopefully!) selects for "physical" behavior by means of an action principle and interference effects. The million-dollar question is then, "what is the 'correct' Lagrangian/action?" Again, I have some ideas about this, but I don't yet know the answer.

2. Regarding antimatter, I can understand it in the context of causal theory only in a very indirect way. In quantum field theory, the necessity for antiparticles "falls out" of the elementary representation theory of the Poincare group, which is the symmetry group of Minkowski space. In causal theory, the Poincare group is replaced with families of refinements of binary relations, and an analogous "representation theory" must be developed. If anyone has done this, I haven't been able to find it, so I am in the beginning stages of doing it myself. There are some aspects of causal theory that make me confident matter-antimatter asymmetry should ultimately be inevitable from this point of view, but I can't explain that at the moment.

3. Regarding the relativity of simultaneity, this is one of the most natural aspects of causal theory. Different frames of reference, rather than merely involving different orderings of spacelike-separated events, ARE different orderings of spacelike-separated events. This prunes away "imaginary geometry" governing what happens, and leaves behind only what actually does happen.

Take care,

Ben

Jeff,

Thanks for the feedback! Yes, that's my website... I don't know how you found it because its not ready for primetime yet and I've done nothing to try to promote it (no time; dissertation year!), but anyway...

GR is usually taken as the prototype for background independence (in contrast to QFT and most versions of string theory) because spacetime interacts dynamically with matter-energy in GR. However, the whole point of background-independence is not taking things for granted, particularly things that by their very nature can't be observed, and GR does retain some traces of this. For instance, spacetime is still viewed as "containing" matter-energy even though the two interact; it's not that there is no background, just a dynamical background. This is better than a static background, but it's still something you can't observe; you can only ascribe properties to it by the behavior of matter-energy inside it. The causal metric hypothesis says that there is no background at all; spacetime and matter energy (at the classical level) are two aspects of a single structure.

The potential for paradoxes in GR (time-travel etc.) comes from clashes between two a priori different structures: a metric structure and a causal structure. The causal metric hypothesis says that there is only one structure. In particular, causal cycles are still possible, but they're not paradoxical. Take care,

Ben

Jeff,

Well, I don't absolutely object to differentiable manifolds, though I find anything so uniform rather hard to believe in at the fundamental scale. However, one had better recover a Lorentzian structure at large scales and low energies, or the idea won't work. That's part of the task for my approach, but there is good reason to believe that it can be done. What worries me more (but also interests me more) is recovering the representation theory that describes the particles in the standard model. This requires some mathematics that appears to be very little developed and should be a lot of fun to get a handle on. In any case, tensor fields would be emergent, just like the geometry they refer to.

I find the whole rating thing a bit embarrassing, because I'd prefer to just learn about other people's ideas rather than presume to judge the quality of their work. However, I feel justified in giving high ratings to essays that lead me to think about things in new ways, and your essay certainly did. Take care,

Ben

Thanks for the thoughtful feedback! I re-read your section on time, and it does seem that we are in closer agreement than I thought at first. In particular, your concept of time seems to arise from local properties of the "fundamental energy units," while the overall order emerges from a tendency toward uniformity, which seems like a description of some sort of potential energy or entropic condition. I tried to suggest something similar in my essay, but only very briefly, since I don't know how to describe this condition precisely yet. You also describe "things in space" as a way of talking about alterations or defects of the structure, which I completely agree with.

Also, when I said "if spacetime has a sufficiently simple structure," I guess I was being lazy... what I meant was "if the underlying structure, from which what is commonly called spacetime emerges, is sufficiently simple..."

Take care,

Ben

Yuri,

Thanks for the feedback. I just read your essay, which I found interesting in several regards. I note that you mention the idea that space can be described in terms of angles. Julian Barbour suggests something similar with his "shape dynamics," but doesn't suggest quantization.

You point out that the strong, weak, and electromagnetic interactions are of similar strengths and that gravity is much weaker. This is true, of course, but it's also interesting to think about the size scales on which these interactions dominate. The strong and weak interactions have very short range, while electromagnetism dominates up to about the everyday scale, where gravity takes over.

You also point out some interesting numerical relationships. There is much speculation about the dimensionality of space and the number of particle generations, but the 18-degree thing is something I have not heard of before. Take care,

Ben

Peter,

Thanks for the feedback! I'll be sure to have a careful look at your essay when I get back from my trip. I won't be too discouraged if my approach "falls short of the path to the holy grail," as you put it; I believe it's fine to think, speculate, and theorize about the biggest questions, but I'm not quite that ambitious about my ideas; at best they're part of the story.

Your ideas sound interesting as you describe them here, though I haven't yet had a chance to read your submission. Of course I have studied the common aspects of mathematical logic and some of the particular ideas applied to quantum settings, but I'm by no means an expert on this. Hopefully I can at least understand what you propose. Take care,

Ben