John,

Thanks for the clarification. I'll have to take a look at your book. Like I said before, it may be that you address all these issues there, so any remarks I made weren't intended as serious criticism. I would have to understand the basis of your ideas much better before I would be qualified to make any definitive remarks of that nature. I am sure part of my confusion arises from differences in terminology; you will recall from my bio that I have a mostly mathematical background, and it sometimes takes me a few tries before I understand what scientists with different backgrounds are talking about. By "true force," I assumed you meant an "interaction" rather than an effect arising from geometry, which is usually how gravitation is distinguished from the other "forces" in my experience. If you are taking electromagnetism as the prototype of a "true force" and simply arguing that gravity is analogous, I have no quarrel with that. In any case, I had better look over your ideas more carefully before making any other remarks, or risk making a fool of myself.

Aren't gravitons as an Archimedes screw model of a force carrying particle a viable alternative to helical EM gravity waves? Otherwise I agree with a lot of what you say Frank.

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  • Post #13

Edwin,

Thanks for the kind remarks. I will reply in an itemized fashion for clarity.

1. Automata, and particularly the homological/homotopical techniques used to study them, are certainly relevant to the approach I outlined. However, there are too many differences (and too much contextual baggage) to describe it in those terms. Automata tend to be discrete, rely on some type of initialization, involve multiple or weighted edges, simplices, or cubes, and so on.

2. I certainly don't rule out continuum models, though I don't think we should take them for granted. Riemann certainly didn't. In order-theoretic terms, the continuum has properties (like the least upper bound property) that seem to have no direct relationship to physics. As far as measurement is concerned, you could never tell the difference between reals, rationals, dyadic rationals, etc. (dense subsets). The symmetry properties of flat real manifolds seem impressive in light the fact that fundamental particles do appear to correspond to representations of the Poincare group, but only until you realize that the same thing can be described much more generally in order-theoretic terms. There are also plenty of direct physical reasons to doubt the continuum such as black hole entropy and the holographic principle. A lot of the "paradoxes" of quantum theory arise from imagining little point-like particles moving around in a manifold over the continuum.

3. I take it you don't favor the sum-over-histories approach in quantum theory? Do you prefer Hilbert spaces? To me, they appear (like the continuum) to be a too-good-to-be-true idealization that likely arises from something more primitive.

4. By the way, where you get the vector "C-field" you use in your essay? I know people have experimented with hypothetical scalar fields called C-fields in general relativity in the past, and have derived tensor fields from these by differentiation, but I'm not sure where this Ampere-type equation fits into the picture.

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  • Post #15

Dear Benjamin,

Thanks for the extensive reply to my comment, and thanks for looking at my essay.

Your response concerning automata agrees roughly with what I had in mind.

I'm glad you don't rule out continuum models. I have my own doubts about reasons to doubt the continuum, ie, black hole entropy and holographic principle. And I do agree that many quantum problems derive from imagining point like particles (with emphasis on 'point'). My particle model is an extended particle plus induced wave.

Nor do I favor sum-over-histories (as physical reality -- mathematically they're fine). For bound (discrete energy) states I am happy with Hilbert spaces. I found your description of the continuum as "too-good-to-be-true" fascinating, and also your opinion that it probably arises from "something more primitive".

The C-field is my own term (with historical conflicts) for the gravito-magnetic field (with gravito-electric G-field). It is treated in the weak field approximation in most general relativity texts, although it doesn't seem to make an impression on most physicists. I did not recall learning about it until I "independently" stumbled over it. Good references to the equation and to experimental measurements of the field are given in my essay.

I'm always impressed by competent mathematicians who work in physics and I always find that we think quite differently about both math and physics. Viva la difference!

Best,

Edwin Eugene Klingman

Hi Ben,

I just finished reading your paper. I enjoyed your writing style, you express yourself very well. You mentioned many mathematical frameworks in your essay of which I know little, so it is possible that the answers to the questions I am going to ask may already be obvious to someone who knows about these, but it may be still of benefit to those of us who don't know.

So, to return the favor of asking serious questions:

1. You idea seems to me a lot like a (very mathematically oriented) variant of relationism. I would have appreciated some comments that would have differentiated it. How is it different from relationism (or is it)?

2. How does your theory account for the fact that we seem to be able to assign metric relations to even causally unrelated events?

3. How does your framework address the fact that the order of spacelike separated events is frame-dependent?

4. Is there such a thing as a "correspondence principle" between the quantum and classical version of your principle and what is it? I ask because it almost seems like it is inverted according to your idea, the quantum version is determined by the causal relations of "constituent universes" but the universe is defined by the classical version. While it is true that also in standard QM a quantum state is a superposition of classical states, I would have expected as a feature of a more fundamental theory that quantum states can be defined without recourse to classical states unless it offers a "deeper" explanation for that.

    Evidently I got cut off there, but anyway, I had only one more question:

    5. Can your principles help resolve some of the notorious difficulties that arise when one tries to describe causal relationships?

    Overall very well written, although it may be too specialized for many readers on this forum. I would have especially liked an expanded discussion of the short paragraph on how causality connects with our established theories.

    All the best,

    Armin

      Armin,

      Thanks for the remarks and questions. Since most of my formal education and my "official" academic work is mathematical, I wrote this essay in an effort to help me begin a dialogue with competent physicists on topics I have thought about a great deal. I knew I would not get the style and focus precisely right at first, but I was hoping that some people could point out obvious flaws and things that required more or different explanations. Let me itemize my reply to correspond to your questions.

      1. Binary relations on sets obviously play a central role in my approach, but there are a lot of "relational" theories, and I am not sure if you are referring to a particular one of these (or group of these) when you reference "relationism." For instance, prominent physicists like Rovelli, Thiemann, Baez, Smolin, Markopoulou, Loll, Ambjorn, Sorkin, Rideout, Bombelli, etc. all emphasize binary relations, but they all include assumptions in their work that I disagree with. These physicists work primarily on loop quantum gravity, causal set theory, causal dynamical triangulations, and a number of lesser known variants. Of these ideas, mine are most similar to causal set theory (Sorkin, Rideout, Bombelli, etc.) but there are multiple crucial distinctions that make the overall picture quite different.

      2. There are "metric recovery theorems" (for instance, by Malament) that allow recovery of the entire metric structure of Lorentzian spacetime (including spacelike separation, etc.) from the causal structure and appropriate volume information. These play a prominent role in causal set theory; they imply that an appropriate causal set "looks like" a Lorentzian spacetime on sufficiently large scales. At the fundamental scale, you would define spacelike distance by counting relations; for instance, two unrelated elements with a common direct descendant are one unit of distance apart. Only at larger scales does this begin to resemble an ordinary distance function. My framework is more general because I don't assume a constant discrete measure, but the simplest versions still involve counting.

      3. Frame-dependent order (relativity of simultaneity) is one of the most important points to understand because it highlights the new meaning of covariance (order rather than symmetry). In my approach (and also in some versions of the above theories), a frame of reference is a refinement of the causal order; i.e., an assignment of order to certain events which are not related in the causal order, just like a frame of reference in relativity assigns order to certain spacelike-separated events. The whole point is that the causal order carries the canonical information; the refined orders carry additional contextual information.

      4. I think you point out a good way of comparing the Hilbert space version of quantum theory, in which classical states arise as an appropriate limit (correspondence principle), with Feynman's sum-over-histories version, in which the quantum picture is built up from classical alternatives via superposition. It is an interesting objection to the sum-over-histories version that the "building blocks" are classical; my view is to be grateful to Feynman for making the presence of a Hilbert space physically comprehensible; they're beautiful mathematically, but I prefer to see them arise from something primitive like superposition, just as I prefer to see manifolds arise from something primitive like binary relations.

      Great questions; I hope that explanation at least somewhat answers them. Take care,

      Ben

      Oh, I just missed your last question.

      5. There are many philosophical issues related to causality, and I am not sure which you are primarily referring to. However, a lot of these issues result from assuming the existence of other types of structure besides the causal structure, for instance, independent metric structure, or independent matter, energy, etc. I believe most such difficulties (at least, most that I can think of) can be explained in terms of the causal metric hypothesis, but the question is whether or not the explanation is satisfying. For example, the causal metric hypothesis includes the assumption that what we call time is just a way of talking about causality, and what we call causality is just a way of talking about binary relations on sets. If it is right, then it simplifies and solves many things, but it may not be right. And if it is wrong, it ignores some very important philosophical questions.

      Dear Benjamin,

      I read your essay with great interest. It contains a lot of deep thoughts including a deep analysis of the current situation.

      We agree in many points except the importance of the concept 'manifold'. I agree with you about the importance of background-independence. General relativity reach us to consider a diffeomorphism-invariant theory. This property is very restrictive in dimension 3 (and lower). If one fixes the topology (or the binary relation between the subsets) then everything is determined (by using the Geometrization conjecture, you will also obtain a canonicaly metric). That is the reason why one considers the special graph (the spine) of a 3-manifold containing all information. But this fails in dimension 4. But one think remains: one needs countable many subsets to obtain the 4-manifold (or the triangulation and the smoothness structure agree). Among this technical thinks, one important fact troubles me more. You wrote about a substitute of a manifold (a poset etc) and about a configuration space (which you use for the sum-over histories). I would expect in a unified theory that there is only one entity not two. So, if you believe (like I do) in the full geometrization then you need only the spacetime, nothing more.

      Furthermore, your concept of causality is interesting but I do not fully understand it: there is a unique path in the past (back to the cause) but different paths in the future (the openess of the future). Does your binary relation reflect this fact?

      Good luck for the contest

      Torsten

        Torsten,

        Thanks for the insightful comments. I will try to clarify a couple of the points you raise.

        1. I'm not sure if you regard matter-energy to be auxiliary to spacetime, or if you regard the two to be part of a single fundamental structure. I far prefer to regard them as part of a single structure, which I describe at the classical level by a binary relation. Hence, I do not expect this structure to be manifold-like at sufficiently small scales. This forces the theory (if it ever becomes sufficiently developed to be called a theory) to predict 4-dimensionality (and many other things) at large scales, probably by means of action principles and entropy.

        2. The causal set theorists have done a lot of experimenting over the years with "sprinkling" points in 4-manifolds; as you point out, the binary relation doesn't determine the geometry, but the argument Rafael Sorkin makes with his "order plus number equals geometry" phrase is that you can recover 4-D geometry from a suitable order if you supply appropriate measure-theoretic information as well. I think this is true. If it is not, then my ideas probably don't contain enough information. Note that the causal set theorists make a lot of other assumptions I find dubious, however.

        3. Regarding fundamental theories and single entities: the desire to describe spacetime and matter-energy as part of the same structure is a lot of the motivation for my ideas. I call the classical "posets" (not really posets in general, of course) "universes" to emphasize background independence: in Feynman's sum over histories, one thinks of particle "trajectories" but generally ignores the obvious fact that the "underlying spacetime" actually ought to respond in different ways to different trajectories, so one is summing over entire "universes" in this sense, not over trajectories in a single "universe." However, the "Universe," which is quantum mechanical, is the entire family of posets with their induced order. After all, similar remarks could be made about manifolds; the etymology even reflects this. A manifold is a set with an atlas, but no one argues that the presence of multiple charts means that the manifold is not a single entity. This analogy is imperfect in multiple obvious ways, but the main point is just that different models partition information in different ways and it is not necessarily easy to uniquely define what "unified" means.

        4. Regarding your final point about the open future, a single classical universe contains its entire history, but such a universe may be regarded as the source of any number of different transitions. In this sense the future is open and the past is fixed. However, I suspect this may not entirely answer your question.

        Thanks again for the feedback,

        Ben

        Dear Benjamin,

        When reading at the beginning of your essay

        "In this essay [...] I reject the manifold structure of spacetime, the existence of an independent time parameter and static background structure, the symmetry interpretation of covariance, the commutativity of spacetime, and a number of related assumptions."

        one may wonder "what remains then?". How far can you go with your causal metric hypothesis? From your essay, it seems that you can do a lot starting from this, although it seems also to remain a lot to do.

        Congratulations, and good luck with the contest and your research,

        Cristi Stoica

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          • Post #23

          Hi Ben,

          I'm a great fan of Causal Sets, and think this is a very timely essay. I like that the assumptions are so minimalistic. It's a possibility that hasn't really been paid enough attention to. Good luck,

          B.

            hello Mr.Dribus,

            I d like learn more about this K theory, it seems very relevant considering the geometrical conjecture.

            The entropical arrow of times and its causality is proportional at all 3D scales in its pure fractalization.

            In all case the maxwell's equations are important considering the heat and thermodynamics.if we consioder a pure cooling, it becomes relevant considering the two main gauges in 3D and its walls separating this infinite light and its universal sphetre in evolution spherization.The internal Energy U is relavant woth the enthalpy with the finite groups. H=U+PV.tHE SUBSTITUTIONS CONSIDERING THEI UNIVETRSAL NUMBER becomes very relevant. The Helmholtz function F=U-TS and the gibbs function G=H-TS. We can insert the finite groups and my equations. The volumes of the entanglement and its number are essential. It is relevant considering the maximum volume of the universal sphere and so the correlated rotating spheres inside this physicality. The quantization of mass so permits to see better. The rotations around the universal central sphere also is relevant.Like is relevant thje volume of this central BH.The measurables quantities are seen with determinsim and rationalism and the unknown can be seen when the finite groups are inserted. See that this quantum number is the same at the cosmological scale that for the quantum uniqueness. So the universal sphere does not turn so it is the maximum mass at the present.

            On the other side, the quantum spheres them turn very quickly. It is relevant to see these correlations.The entropy principle is so spiritual in fact. The aim is to fractalize correctly the steps of disponible energies.See that the rotations and the volumes are very relevant. My equation mcosV=constant with this finite number, is relevant because this constant is for all physical spheres in 3D , so the quantum spheres, the cosmological spheres and the universal sphere and its central sphere !There is an interesting link when I extrapolate the maximum volume in 250 billions of years considering my calmculations.At this momment a contraction is correlated , so the volume decreases. But in logic the central sphere, it increases in density and volume logically speaking.So it is a kind of oscillation like a oscilaltion of heart. So the volumes are very complex in fact at all 3D scales.

            The differentials appear with a real universality when the roups are finite at all 3D scales. The Universal sphere and its cosmological spheres is like a foto of our quantum uniqueness.

            ps Good Physicists Have Studied Under Very Fine Teachers.

            ps 2 The entropy is maximum in all, paradoxal but so evident.The steps are fascinating before this planck scale !

            ps3 eureka :)

            Regards

              Ben,

              I want to pull out a couple of things that I think are good points

              "For example, Einstein's equations in general relativity predict the

              curvature of spacetime, but not the dimension; a theory whose dynamical laws also predicted the dimension would be superior in an obvious way."

              "Our present understanding of antimatter comes almost entirely from quantum eld theory,"

              I think these are good points, my question then would be how do causal metric hypothesis account for these and also, how does it account for the relativity when two observers can assign different ordering of two observed events?

                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

                Ben -- congrats with your essay. It places IMHO a healthy focus on the key question "How to get an emergent metric from a local causal relationship?"

                Two more opportunities I would like to stress: Firstly, seeking recovery of a Lorentzian manifold is indeed a key challenge, but an emergent De Sitter manifold might be the true target that would allow you to get 'dark energy' to be emergent. Secondly, you don't mention unitarity as a key assumption. You might get some further mileage from entertaining the inevitable question "Is unitarity really required?"

                Good luck at the contest, I would be disappointed if your contribution doesn't score well!