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

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

Yuri,

Let me make sure I understand. So you think that the ratio c/G is constant, but neither G nor c are independently constant? Do you mean constant in "space" or constant in "time?" Take care,

Ben

Dear Mauro,

I appreciate the excellent analysis. I will have to break my response into a couple of segments, so I will post them as new posts rather than replies. To the points on which we seem to agree, I have little more to add, though I am interested in the "Planck-scale experiment" you referenced. One point is that the generalization of covariance I have in mind is much more general than semigroup representations. For the points on which we may disagree, I will itemize.

1. Regarding transitivity, I must insist on distinguishing between the "causal order" (of a classical universe) and the "binary relation generating the causal order." On large scales, the intransitivity I am talking about is as simple as the fact that the statement "Jane talked to Bill, then Bill talked to Susan," is not the same as the statement "Jane talked to Bill, then Jane and Bill talked to Susan." In either case, Susan received information from Jane, so the two statements are indistinguishable in their causal orders. However, in the first instance, the information is transmitted only through Bill, whereas in the second case it is transmitted both through Bill and directly. Thus, there are two different binary relations that generate the same causal order: the intransitive one in which information passes only through Bill, and the transitive one in which information also reaches Susan directly from Jane. These two are a priori different. At an ordinary scale, this is obvious to everyone.

For fundamental physics, the reasoning is as follows. Many scientists (by no means all!) agree that "causality," however you define it, is one of the most fundamental concepts in physics. The question then becomes: how do you define/describe causality? Well, a cause and effect certainly seem to define a direction; you can imagine an arrow pointing from the cause to the effect. This is completely local. Include lots of causes and effects (vertices), and arrows (directed edges) without yet imposing any other conditions, and you get a directed graph, which is equivalent to a binary relation on the set of vertices. At this stage, there is nothing to rule out cycles, and certainly nothing to impose transitivity, which are both generally nonlocal phenomena. There is a "causal order" generated by this directed graph, which is the relation defined by closing the graph relation under transitivity. I put "order" in quotes because this is still more general at this stage than the usual definition of a partial order; it may still have cycles, for instance.

This is all purely classical. To obtain a quantum theory, you need the superposition principle. The appropriate version of this in this case is a path sum over a configuration space of classical causal universes; i.e., directed graphs. I will explain why this is the appropriate version below. The question then becomes, "which graphs should be included in the configuration space?" This is the first real choice in the entire procedure, and involves a judgment about what types of graphs correspond to physical reality. My personal guess would be "acyclic locally finite directed graphs," but I want to make it clear that these are second-level assumptions that come further along in the development. I prefer acyclicity because we don't seem to observe causal cycles, and I choose local finiteness because I suspect that volume has something to do with counting (not necessarily as simple as Sorkin's "order plus number equals geometry", but in the same spirit).

I particular, it makes an a priori difference if you include only transitive graphs (graphs in which there is an edge between two vertices whenever there is a path between them). It's conceivable that this difference would fall out of the path sum, but I see no justification for assuming this at the outset.

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2. You make the very helpful analogy that "the causal links are the "wires" in the quantum circuit." If so, I don't see any disagreement on this point, because the directed graphs representing quantum circuits are not transitive graphs. Also, the classical causal networks in your arXiv paper seem not only intransitive, but almost "maximally so" in this sense.

3. Regarding my reasoning for not absolutely ruling out cycles, I actually think GR is very discouraging to fans of time travel, and I'm certainly not trying to rescue GR here. It's true that GR gives a sliver of hope to believers in causal cycles, but I don't take these solutions very seriously. My reasoning is partly caution and partly based on some potentially interesting or suggestive properties of graphs containing cycles. The models I have thought the most about are acyclic, however.

4. Of course you're correct that a binary relation doesn't determine a metric in general. Sorkin discusses this at length. His "order plus number equals geometry" motto is based on metric recovery theorems that take as input an appropriate binary relation together with some volume data. His choice of how to provide volume data is the simplest (counting), but there are other ways, defined by taking advantage of local data in the graphs. For a homogeneous graph, simple counting is probably the only option, but I don't prefer the assumption of homogeneity.

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5. I clearly did not explain my use of the sum over histories method adequately enough, and it is no wonder given the length constraints. First, in his 1948 paper Feynman discussed summing over particle trajectories in Euclidean spacetime and thereby recovered "standard" quantum theory, with its Hilbert spaces, operator algebras, Schrodinger equation, etc. Feynman was able to take all the trajectories to be in the same space because he was working with a background-dependent model; the ambient Euclidean space is unaffected by the particle moving in it. Now, if GR has taught us anything, it is that "spacetime" and "matter-energy" interact, so different particle trajectories mean different spacetimes. Hence, in a background-independent treatment, Feynman's sum over histories becomes a sum over "universes," with a different classical spacetime corresponding to each particle trajectory. His original version is a limiting case in which the effect of the particle on the spacetime is negligible.

What are the "classical spacetimes" in my approach? Well, they are directed graphs. However, it is not quite right to just sum over graphs. The reason why can be understood by looking at Feynman's method more carefully. He considered a region R of spacetime, and interpreted his path sum as the amplitude associated with measuring the particle somewhere on the upper (i.e., future) boundary given an initial measurement on the lower boundary. Hence, the path sum measures not the amplitude of a particular universe, but the amplitude of "transition" from one family of universes to another. A discrete approximation of this represents each particle trajectory as a sequence of directed segments in the corresponding configuration space, which inherits a partial order from the time-orders of the individual spacetimes. It is now clear how to generalize to the nonmanifold case: the appropriate sums are sums over paths in causal configuration space.

Take care,

Ben

Daniel,

I appreciate the feedback. I will itemize my reply:

1. The length requirement did limit my ability to explain my background thoughts. My emphasis on causality is principally motivated by two factors:

First, most of science, and particularly the experimental method, consists of establishing "causal" relationships between things ("whenever we do so-and-so, we observe such-and-such"). The success of this approach is obvious, and I feel that its full potential should be exploited. Remarkably, much of modern theoretical physics has relegated causality to a secondary and sometimes obscure role, and many recent theories reject its fundamental importance altogether. Instead, they are based on objects like continuum manifolds, which have undeniable mathematical advantages, but which are disturbingly idealistic and exhibit properties that are obviously physically irrelevant (like the least upper bound property, nonmeasurable subsets, etc.)

Second, it is startling how much of the structure of even such idealized theories can be recovered from causal relations, and how almost every "spacetime-related concept" you can think of has an analogous "causal interpretation" that is more natural and more general. For instance, a "light cone" in relativity is an abstract geometric locus of events in spacetime, and the rule that "information cannot escape its light cone" is viewed as secondary. In causal theory, the light cone is simply the scope of information flow, and the idealized view of a geometric locus is secondary. As another example, "frames of reference" in relativity are associated with abstract local coordinate systems on a manifold, and the relativity of simultaneity is viewed as secondary. In causal theory, frames of reference are different orderings of events compatible with the causal order, so the physical idea of relativity of simultaneity is direct, and the idealized view of a coordinate system is secondary. In both examples, I think it's obvious which is the more natural and physical point of view.

2. Regarding the recovery of established physics, I don't mean that every prediction of currently accepted models, including those that haven't been experimentally verified, must be reproduced; that would be pointless. What I mean is that a new theory must do at least as well as current theories in explaining and predicting what we actually see. For instance, general relativity modifies Newton's laws only very slightly for solar system dynamics; if general relativity had predicted an inverse cube force for the gravity between the earth and the moon, it would have been thrown out. Similarly, a new theory must look like general relativity and quantum theory in any situation where those theories have been proven to work.

3. Regarding the general requirements for a theory, yes, I agree completely. Experimental evidence is the final judge.

Take care,

Ben

Dear Pentcho,

Thanks for the feedback! By "recovering" relativity, I don't mean that I believe relativity in Einstein's original form is absolutely valid (otherwise, why would it need to be replaced or superseded?) What I mean is that in any case in which relativity makes good predictions, any theory that supersedes it must do at at least as well. Hence, a new theory must be able to describe/predict anything that relativity can describe/predict.

Regarding the constancy of the speed of light, my guess would be that a concept like this only makes sense at sufficiently large scales. "Speed" requires a notion of distance, and my view is that spatial distance (separation) is ultimately just a way of talking about the extent to which events are "unrelated." It begins to look like a traditional distance only at large enough scales.

Regarding photon mass, it was thought for a long time that neutrinos had no mass, but it was eventually discovered that they do have mass after all. Hence, I would be inclined to keep an open mind even about something as "sacred" as that. However, mass itself is again an emergent concept in my view, so the questions of what a "photon" really is and what "mass" really is are things that cannot be taken for granted.

You'll have to remember that my background is mostly mathematical, and therefore I'm inclined to consider the possibility of things that most physicists "know" are wrong. This might be useful in some cases; in others it only reflects my own ignorance. Take care,

Ben

Dear Yuri,

I did look through the comments on your thread, but I am afraid I don't quite understand. It seems you are suggesting there is a simple dimensional relationship that explains the observed value of the cosmological constant. This would be great, but it's not obvious to me. Do you mind explaining a little more? Take care,

Ben