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!

              Hello Mr. Dribus,

              I am understanding. no I have no publications. I am isolated at home without job, without nothing, just my personal probelms. I have not published.I have understood Mr Tegmark, ok. You can make what you want afterr all.I have made my works me, I have shown my theory to the world.If people copies or wants the prizes, you can have them Mr Tegmark and Mr Aguire.I thought that Fqxi was there to help the real innovators.I see simply a strategy. I am sad simply. You can with your friends, have a good life, and travel in private airplanes and buy opulences.Make what you want, me I sleep quiely and serenity. Of course I have neurological probelms and also I have a kind of depression due to my difficult life.But I have faith in God me, I have faith in this universal kindness and universal love. I have faith in this universal sphere. I am going to continue to read and discuss on fqxi. I forgive you all after all, you are simply persons loving monney and vanity.Perhaps you can evolve in a pure universality.

              It is the life, it exists a little of all on this planet, good people, bad people, universal people,envious,vanitious,.....the real importance for me is my faith in God.

              ps Mr Dribus, the entropy , it is god ! It is simple you know the truth !

              Regards

              • [deleted]

              • Post #32

              Hi Ben,

              I want to understand the meaning of "causal metric" hypothesis better. I found on another web page "The causal-metric hypothesis, if correct, greatly simplifies and clarifies theoretical physics. In particular, it is the purest possible version of background independence. A theory is background independent if its entire structure is dynamical, rather than relying on a static embedding space in which the dynamical entities of the theory reside."

              I am trying to understand this in contrast to GR now (without Lambda). What is the background that the structure isn't independent from, spacetime?

              Regards,

              Jeff

                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

                • [deleted]

                • Post #34

                Ben,

                I can see now why you would like to do away with the manifold structure. It would seem that our essays run counter to each other, which is great for me to develop an understanding of your intended meaning. To me, the constant multiple of the metric represents a static potential for curvature (a potential for energy) whereas the tensor (i.e. Einstein tensor) represents the dynamic portion which gives rise to what we perceive as matter and energy moving in spacetime.

                As an analogy, for me it is the derivatives within the fabric and not the fabric itself that is important, but the fabric does exist, whereas you would like to propose that the fabric itself doesn't exist even if the derivatives do?

                BTW, I hope you rate my essay as highly as I have yours.

                Regards,

                Jeff

                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

                Johannes,

                Thanks for those suggestions... both of them are right on target. Take care,

                Ben

                • [deleted]

                • Post #37

                Benjamin, you wrote:

                "The fi rst few assumptions I reject are that spacetime is a manifold, that systems evolve with respect to an independent time parameter, and that the universe has a static background structure."

                I reject too.

                See my essay

                http://fqxi.org/community/forum/topic/1413

                  Dear Ben, I liked your idea of casual metric very much. You said in my thread that "you and I have perhaps different ideas on the nature of time" but I don't think so. In my mind, time is the expression of changes in energy state, and what can be more causative than that?

                  Our major difference lies in you regarding matter and space as a single structure --like a true mathematician!-- and on a certain scale and at certain energies this is right. But there is also an intermediate scale, at low everyday energies, where this approach is not well suited, imo.

                  Here are the quotes from your essay that especially resonated with me:

                  Re : "These phenomena suggest the promise of physical models that naturally incorporate scale-dependence,.."

                  Agree with you: scale is everything.

                  Re : "The first few assumptions I reject are that spacetime is a manifold, that systems evolve with respect to an independent time parameter, and that the universe has a static background structure."

                  Agree again: time as an independent parameter is suitable only on macro scales, while on the quantum scale, I believe, the micro-processes themselves (not 'particles'!) define spacetime volumes they trace, which can be mapped into time and distances at different scales. As for the universe having a static structure -- who actually thinks so? I can't even fathom it.

                  Re : "Dimension becomes an emergent property, and is no longer assumed to be constant, nondynamical, or an integer."

                  I see it exactly the same way.

                  Re : "If spacetime has a sufficiently simple structure, "...

                  Yeah, what is spacetime?

                  Re : "Finally, the dimension of space as well as its curvature might vary with energy density, "...

                  Just my thoughts. See, we have more in common than it seemed at first.