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Smolin's list of five physics problems

  1. Aug 18, 2008 #1


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    Any comment on his list? Items you would reformulate, combine, add or eliminate? Is this checklist a good schematic of the challenges facing theoretical physics at this point in history? I find it helps to have memorized so I can use it as an informal gauge of progress and relevance.

    One thing that helps in memorizing is to notice that there are TWO unifications
    a unification of laws
    a (possibly partial) unification of fields.

    The unification of laws could be seen as imperative: quantum mechanics and general relativity both describe nature and nature is ONE. There must exist a quantum mechanical version of GR and a general relativistic version of QM, which form an organic whole.

    At present they appear formally incompatible and limited in their applicability. Both suffer from infinities/singularities and are incomplete. There must be a single theory to replace them (or so the argument goes).

    But this unification of the basic laws does not logically require that all forces be aspects of a single force, or that all particles be versions of a single particle, or that any such unification of fields must be achieved. That is a different sense of unification and I guess people might risk confusion if they speak as if they equate the two and moosh their unification fantasies together. In any case, whether it is right or wrong, the checklist has two separate unification items: #1 and #3.

    Once you take account of that, it is easy to assimilate and remember:

    Problem 1: Combine general relativity and quantum theory into a single theory that can claim to be the complete theory of nature. (This is called the problem of quantum gravity.)

    Problem 2: Resolve the problems in the foundations of quantum mechanics, either by making sense of the theory as it stands or by inventing a new theory that does make sense.

    Problem 3: Determine whether or not the various particles and forces can be unified in a theory that explains them all as manifestations of a single, fundamental entity.
    Let us call this problem the unification of the particles and forces, to distinguish it from the unification of laws, the unification we discussed earlier.

    Problem 4: Explain how the values of the free constants in the standard model of particle physics are chosen in nature.

    Problem 5: Explain dark matter and dark energy. Or, if they don't exist, determine how and why gravity is modified on large scales. More generally, explain why the constants of the standard model of cosmology, including the dark energy, have the values they do.

    These quotes are excerpted from pages 5-16 of the first edition of Smolin's book TWP. It's the latter part of Chapter 1, and probably the same pages in the orange paperback edition.

    I think it might be worthwhile to discuss them, and to see what comment people have.
  2. jcsd
  3. Aug 19, 2008 #2


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    Formulating the question is interesting, and I think it often the choice of formulation reveals in part ones personal expectations.

    My personal expectations is that some the problems are possibly related and that some of them may cancel each other, even though I agree with Marcus that they are not at this point related due to logical necessity, so it's not something that has to be. But from just some initial reflection of some of these questions one by one, from my own perspective, I am lead to similar abstractions of the problem.

    Smolin has asked the question (somewhere, I forgot where, but I think somwhere in his notes/lectures or the problem of time) what is the difference between initial condition and law. Of course the laws are supposedly fixed, but the question is what is the basis and nature for such universality - from the epistemological point? I think that basis is fairly weak from a fundamental point of view, and that the difference between initial conditions and law are more to be seen as "initial condition" vs "initial law" and where the difference is that the inertia of law is far higher.

    I think this is one question to sort out, and it relates I think both to #1 and #2 in particular and my personal view is that this also to a high degree relates to the method of science.

    In particular the notion of statistics and probability is squeezed when comparing particles physics and cosmology. Clearly the concept of statistics and probability in the case of cosmology need clarification IMO as compared to more easily reproducable short lived events taking place in a small lab device. To a decent extent the sin here IMO is that assumption of unambigous limits. Limits that are never realized, but still used as fictive references. Something doesn't smell right here to me.

    So I'd like to add to the list also a more foundational and slightly philosophical general question of the problem of the scientific method. It's clear that some of these problems, can't be solved as one would wish. To experiment with reproducing the universe isn't doable, but perhaps that instead might suggest another way of asking questions and searching for answers.

  4. Aug 19, 2008 #3
    Why should understanding a putative mass in a remote place be more important than understanding your own ?
  5. Aug 19, 2008 #4


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    I can't speak for the author, of course, but I don't think he is saying more important. Understanding your own mass would be problem #4.

    That is, understanding the parameters of the standard model, including the masses of baryonic matter.

    #5 involves as you say a putative mass or masses, which helps explain why it is separate from problem #4. The dark matter and energy effects may have alternative explanations---it has not yet been shown they correspond to particles/fields, and the putative particles/fields are not yet part of an established standard model. Until they are, one can argue that problem #5 is separate from #4.

    It is an interesting comment. I hope I understood the point you raised.
  6. Aug 19, 2008 #5
    Let me see :
    To me this seems more like Higgs couplings and hierarchy. When I say we don't understand hadron masses, I mean massless-quarks pure QCD with exact conformal symmetry built in, dynamically broken. It is merely a belief (of mine) that if we were able to calculate that approximately (or exactly if you insist, but anyway I don't mean brute force here, although it might help), that is if we were able to pin down the physical mechanisms and the associated relevant variables, we would learn something of value on the nature of mass and matter altogether. But I might be wrong : it is widely believed that the gluonic potential merely grows, and that this is the energy stored in hadrons, end of the story, engineering (lattice) problem to actually calculate it.

    For me Higgs and hierarchy are indeed separated problem from Dark matter/energy, although I easily reckon that most people would link the two. More precisely, it seems easier and more logical to approach the hadronic mass problem first, and only once we are happy with our understanding of it should we address wider and more complicated problems.

    This entire post is only personal opinions.
  7. Aug 19, 2008 #6


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    Thanks humanino
    I don't hear enough educated personal opinions on this forum.
    Presently, I'm listening to the end of
    N=6 Chern Simons matter theories, M2 branes and supergravity
    http://webcast.cern.ch/live.py [Broken]
    It has a lot of personal opinions.
    Last edited by a moderator: May 3, 2017
  8. Aug 19, 2008 #7
    Personally, I think that solving the problem about the foundations of QM (problem #2) will solve automatically the four other problems. If the foundations are limpid, the solutions of the different problems should be deduced through careful reasoning on the single fundamental entity.

    These problems are also listed at http://en.wikiversity.org/wiki/Theory_of_Everything_Project as a challenge.

  9. Aug 19, 2008 #8
    I think it's a good list, something I'd note though is that it's a list very finely attuned to Lee Smolin's research program and/or the specific kind of philosophy-of-science analysis he was trying to perform in TWIP.

    For an LQG researcher or a string theory researcher these seem like the questions to be asking. But it seems like people in other areas of physics may have other concerns which are more practically or philosophically important but are downplayed or ignored by this list (What is inflation? is one question that comes to mind that isn't on this list but seems like kind of a big deal).
  10. Aug 24, 2008 #9
    I would agree with ArjenDijksman in that they are all indications of the fractured nature of fundamental
    theory. I think solving #3 (which I think ArjenDijksman meant by the last two words of his first paragraph)
    would likely solve all, if by solving we mean FIND the single entity.
    A single entity would necessarily define the fundamental conflict of discrete and continuous, particle and force(field)
    Being single, such an entity would be both discrete and continuous or explain why
    and how such an entity diverges to the discrete and continuous observables of modern physics.
  11. Aug 24, 2008 #10


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    I agree there is a sense in which the list is focused---it doesn't blanket all types of physics research---it isn't exhaustive---it is tuned , as you suggest, to curiosity about the most fundamental realities. Whatever that means. :-)
    Personally I'm happy with the focus, for the purposes of this thread of discussion. I wouldn't try to add items in order to make it represent all of physics. And again speaking only for myself, I can't think of anything else I would add, in the area of deep fundamentals.

    I know what you mean about big deal! But the inflation hypothesis, and the various inflation scenarios, have a strange status. Inflation is conjectured to explain certain puzzles, which may well be resolved in some other manner. The spectrum of fluctuations in CMB temp. The spectrum of inferred fluctuations in density. Approximately equal CMB temperature in all directions. Approximate flatness.

    Inflation, as I see it, is not yet a done deal. Alternative answers to those riddles might show up. So I would rephrase the question slightly and instead of asking "what is inflation?" I would ask how can we explain all those features that inflation addresses. Don't wnt to quibble however. Maybe just rephrasing, as for example:

    6. How about inflation? Is it necessary to suppose a fit of it occurred---or are there other answers to the same package of puzzles? If there was an episode of inflation at the start of expansion, how did it work, what caused it?
  12. Aug 24, 2008 #11


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    Arjen and Chris,
    I can see how all five problems may have one that is the key. There may be a strategic problem which one should attack above all because solving it will automatically bring down the others. It is interesting that you see different ones.

    Arjen, your idea to start a Wiki course using these five questions was excellent. I am glad you posted the link and I hope it takes off. I have all I can handle right now but urge others to contribute to Arjen's discussion at the Wiki "campus".

    In his book "Trouble...and What Comes Next" Smolin has a discussion of the problems with the foundations of Quantum Mechanics, and the different ways they could end up being resolved. Either by finding a reasonable interpretation that makes complete sense of it. Or by finding an underlying deterministic ground from which the probabilistic patterns arise.
    And he mentions other alternative outcomes. Finding a deterministic ground underlying QM is something Gerard 't Hooft has been talking a lot about. His view carries a lot of weight. there was an excellent piece in December 2005 PhysicsWorld by him about this very thing.

    So Arjen's strategy would be to focus on fixing QM----the problem #2----and I can well believe that if one could ever do this properly it would give a new light on all the rest and maybe quickly bring about solutions to the other problems! As I read what 't Hooft says, he seems to be thinking this.

    Fixing QM might involve constructing a new quantum spacetime continuum or finding something underlying from which the appearance of spacetime emerges. 't Hooft said something about that. I should get the link to his PhysicsWorld essay. EDIT: Here it is



    the practical drawback of treating #2 as the key problem is that it seems (to me at least) to be very hard----something young people can tackle if they think they have many years to work on it, and an independent income to feed their families meanwhile. it's tough.

    the good thing about what Chris says-----namely look at #3 is that it doesn't necessarily involve re-working the foundations of Quantum Mechanics. All you have to do is, within the conventional Quantum Theory framework, and whatever model of spacetime you have, unify all the fields. that is, if Nature allows this. We don't know ahead of time if Nature has them unified so they are all facets of a single entity.

    This is a point Smolin makes. There are two issues of unification: One is unification of laws---both GR and QM have things wrong with them, limited applicability, each only applies to part of nature, but nature is a Unity, so GR and QM must be reconciled into a single consistent set of laws that applies more completely. One can say that there must be unification of laws. (unless I am mistaken or Smolin's argument is wrong)

    But I think we have no guarantee that all the forces, for example, are aspects of a single force! Maybe nature is not constructed that way. It is an attractive idea and there are historical precedents, but I don't see a compelling logical argument that things must be that way. Maybe one could have a world in which gravity was exactly the same is our gravity, but instead of strong and weak nuclear forces there were other forces behaving somewhat differently. Excuse me if this sounds silly, I am just speculating for a moment.

    Anyway Chris and Arjen it is a real pleasure to have your initial thoughts on those five problems. I hope you will pursue this further. I will try to write down my reactions as well.
    You have said the key points are #2 and #3, and I will probably go for problem #1 and tell you what I think the solution there will entail.

    Meanwhile thanks to all and thanks in advance to anyone else who has thoughts about this they want to add to the discussion stewpot. :biggrin:
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  13. Aug 25, 2008 #12


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    I am particularly happy that Smolin formulated the Problem 2, because most physicists seem to think that this is not a problem at all.
  14. Aug 25, 2008 #13


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    Yes, and to point out the obvious, you have yourself worked on that problem and published papers in that area of research.

    One thing about that list is that he seems to have worked hard on it to make it complete, and judging from our reaction it does seem to be. Nobody except Coin has proposed adding something. It is hard to think of something to add---some problem with comparable generality and fundamentalness.

    Demy, how do you visualize a solution of problem #2 looking? I don't mean point to some approach that has already solved it, I mean point to something that a solution would look LIKE. I suspect a solution to #2 would look most like a quantum continuum, such as Ambjorn and Loll have developed. A new mathematical model of continuum, something chaotic and unsmooth at the microscopic level, from which a smooth classical spacetime emerges at a macroscopic level. In effect, something to replace the 1850 idea of a differentiable manifold. This is how I picture a solution to #2. I am curious to know how you would picture it. I am asking you to conjecture (hope this is all right.)
  15. Aug 25, 2008 #14
    Chris: Yes, I think finding the single fundamental entity would likely solve all problems. Problems #2 and #3 tackle the fundamental aspects of physics, foundations of QM for #2, fundamental entity for #3.

    Marcus: Treating #2 as the key problem is indeed very hard. I'm on that problem for some 10-15 years of spare time. It is very important to read, re-read, re-re-read... insightful explanations and views about the quantum fundamentals. 't Hooft has some, my favourites are Feynman, Bell, Dirac, De Broglie, Bohm, to keep a long list short. Anyhow, I now believe that problem #2 is only difficult because we are trained along classical lines, hindering us to see the simplicity of quantum feaures. There are many features of QM that are intrinsically simple and I try to show that on videos I post on youtube.

    I think that the fundamental entity that could make sense of QM is what we could call the 'ket-particle', needle-shaped particles. Nothing extraordinary. The ket-vector was introduced by Dirac, Feynman helped to make it more understandable ('all we do is draw arrows on a piece of paper... that's all) but we still miss a thorough understanding of it. The motion of a single needle obeys a generalized Schrodinger equation d|A>= i omega dt |A>, in the sense that the vector difference between two subsequent states of that needle is always perpendicular to the needle (right angle characterized by the imaginary i). When you let a cloud of needles interact through collisions, you get wave behaviour. A needle is therefore guided by its own pilot-wave (Bohm, de Broglie). And the probability of interaction (detection) will than be proportional to the cross-section of two colliding needles that spin in the same pilot-wave (amplitude squared). Seeing particles as tiny spinning arrow-shaped objects fits in the formal theory of QM. It helps me to make some sense of it.
  16. Aug 26, 2008 #15


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    I share this perception. It appeared to me when I was a student that the kind of physics you could do as a professional is somewhat biased due to commercial elements. After many things in this world are under the constraints of economy. One side effect is that some (not all of course) interesting fundamental and hard problems simply aren't encouraged in professional the system. My teachers at the time admitted this. The reality is that you most "produce" papers and get published or your grants are not renewed, meaning that difficult problems, or less popular problems simply wont compete. Not because they aren't important, but because they aren't fit for the system.

    I don't think there is a quick solution to this though - the commercial constraints are real - except to encourage anyone who has ideas they believe in to work on it. Special honour also to all those who spend their spare time to work on this, even these ideas are needed I think. I think we need in particular new input that is unconventional and perhaps the mainstream system lacks this.

  17. Aug 28, 2008 #16
    Actually, I would suggest adding one item. Lee Smolin mentions in closing his last book, (I'm paraphrasing here) he will get a good cup of coffee, put his feet up and think about time, as he thinks the next revolution in physics will begin with a new, deeper understanding of the nature of time.
    I think he's right.
    Time is still today an ambiguous philosophical premise at the heart of the first 4 problems. (I am assuming #5 is empirical evidence that will be explained in the resolution of 1-4) When tackling any of these problems you soon realize something very significant and sublime is missing. You cannot just rearrange existing ontological constructs in the hope they will all fit into a more powerful order or pattern that reveals a great unifying principle or grand symmetry.
    As Einstein said, you must eaves drop on nature to find the principles. You must look beyond the logical progression of present doctrine, take an intuitive leap that hopefully finds you in a place you can return from via reason, as todays models are no correct, no amount of reason will get you there in the first place.
    I have found the long list of dualities defining the philosophical conflicts in present day theory, all fall to one or the other side of our concept of time. Philosophically speaking, time is the axis of a symmetry defined by these dualities.
    Particle - Wave
    Discrete - Continuous
    Finite - Infinite
    Determinate - Indeterminate
    Local - Non-Local
    Quantitative - Qualitative
    Symmetry - Conservation
    Etc., etc., etc.
    Time sets these concepts against each other yet requires they coexist in the ontology of physics. Consider the role of time in each. On one side you will find time does not exist, on the other time is a necessary integral aspect of the concept itself.
    It is our "lack" of knowledge of the true nature of time that gives rise to these dualities.
    I think a model that solves any one of these dualities will likely point the way to solving them all in a unification of far more fundamental nature than those being considered today.
  18. Aug 28, 2008 #17


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    I agree the that problem of defining time is a key issue. It is indeed one of the keys to for example distinguish laws from states. Different meanings of "time" could certainly mix law and states. I think this is exactly what Smolin meant by raising that question in the context of the problem of time.

    Part of solving 1 and 2, will I think for sure contain a new view of "time" as well.

  19. Aug 29, 2008 #18


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    I expect that from the requirement of mathematical consistency and completeness one will find out that some new element (that has no a counterpart in the usual formulation of QM) is necessary. Almost by definition, this will be a hidden variable.

    For my attempt to do something like this see
    in which I derive the Bohmian equation of motion from the requirement of manifest general-covariance of quantum field theory.
  20. Aug 29, 2008 #19


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    Hi Chrisc,

    I'm interested in hearing more about this. Also, I'm not sure I understand the Quantitative-Qualitative part. What did you have in mind?
  21. Aug 30, 2008 #20
    Hi dx.
    It is a subtle and difficult relationship that is expressed and still debated by physicists and philosophers in notions such as mass energy equivalence.
    It is at first reasonable to assume that in order to quantify anything, one must first qualify what is being quantified.
    This quickly digresses to the notion(some argue the reality) that qualification may arise through quantification. i.e. if it can be measured, it "is" what is being measured.
    As space, time, mass and energy are relative measures, any quantifiable variance between observers leads to the question of qualification of dimension. If the universe is considered via the conservation laws as a finite yet variable quantification of these fundamental dimensions, it must likewise be considered to be comprised of the finite yet variable qualifications of these fundamental dimensions.
    It may be that the successful quantization of space -time will arise from the successful qualification of space-time.
    As with all these dualities, there appears to be an axis of symmetry, perhaps a physical constant, that until recognized, compels us to make distinctions between them, when they may in fact already be the evidence of unification we are seeking.
  22. Aug 31, 2008 #21


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    Hello Chrisc, I haven't followed any of your past posts but the focus you try to described here sounds somewhat familiar to what I have been considering too.

    I have considered a general problem of constructing intrinsic measures, meaning that the observer constrains the construction of the measures. In particular are there complexity bounds, that relates the degree of "qualification" (to use your terminilogy) of the measure itself, to the observers degree of "self qualifiation".

    So I have come to the standpoint there there exists a very general duality between intrinsic measures, and the resolution or precision of the measures. I am still working on this, but in this there degree of qualification or confidence in "a measure" (ie a function or distribution) is bounded by the "space where it lives", which is the observer. I am still trying to find the standard physics in this, but clearly it seems this degree of confidence does one way or the other related to energy, mass and inertia. Loosely I "think" in terms of "mass of a mesure", which really means it's degree of confidence, and this is bounded by the observers mass. Therefore different observers can not necessarily agree unambigously on universal measures of idential qualification. And my thesis is that it's exactly this "inconsistency" that is responsible for physical interactions. So the inconsistencies is so canceled.

    I've been trying to work from scratch, and see how the logic of these constructed measures look like, and see what basic parameteras that appear. The obvious ones are what I call the "mass" of the measure, and the second is the "resolution". In communication language it the resolution corresponds to the channel bandwith, and the mass is the memory capacity in the transceiver.

    This has at least in the early intuitive sense, alot in common with blackhole information stuff. There are various bounds that come from the limited complexity of the observers. Remotely similar things like Einsteins equation could be hinted, in that measure of state, measures of changes and measures of measures are related from the reconstruction of instrinsic measures.

    Somehow I agree with this. But I don't know if we mean the same thing.

    I'm curious to hear if you have any more ideas around this? or know of any papers in the direction?

  23. Sep 1, 2008 #22
    Fra, I think we might be heading toward a similar concept with a focus on different key ontological constructs.
    When Einstein asked, "what are the general covariant laws of nature" I think he hit on a concept beyond what is
    presently understood as the necessity of physical laws to have no prior geometry.
    I have wrestled with these dualities for a long time and only in the last eight years realized a physical modeling consistent with evidence and more importantly consistent with fundamental principles of physics.
    For physical law to uphold general covariance requires specific measures be subject to differentiable coordinate transformations. Thus in testing the general covariance of the laws, the coordinates of the observer determine qualitatively and quantitatively, the dimensions being measured. For the laws to remain unchanged through such transformations of dimension, they must express a continuous symmetry of dimension.
    In essence, the conservation of general covariance of physical law expresses a continuous symmetry of dimension.
    I have not found any papers dealing with this directly, but some such as the two by Demystifier above come close in dealing with the specific fundamental requirements of general covariance.

    If I understand your analogy, you are saying essentially the same thing.
    I may be reading my own bias into what you've said, but it appears you are considering the uncertainty principle in a more general form - Quantitative versus Qualitative.
    The more accurately we attempt to quantify dimension the more general our qualification must be.
    It may help to look at the relationship of continuous symmetries and uncertainty.
    Many of the principles of physics express different aspects or observations of the same fundamental concepts.
    While each can be defined in explicit terms, they all appear as conjugate principles of a deeper general principle.
  24. Sep 1, 2008 #23


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    In a way I think of the attempt to construct intrinsic measures as an extension of these ideas. I think we have a larger class of symmetry transformations that just the "spacetime coordinates". For example I think the geometry and topology is also a valid transformation. And I think those transformations is associated with the observers internal structure, because the external degrees of freedom, IMO at least, can only be constructed as measurable in terms of the images that exists in the internal structure. So the topology of external space can change by transforming the internal structure and tracing along the construction of the measures. This will cause twists and strange statistics. I consider the superposition to be explained by this.

    But on top of this, what I picture saves us from the idea that anything is possible, is that the internal structure of an observer, is subject to evolutionary selection from it's environment. So the whole idea of this large class of transformations must be supplemented with an evolutionary logic. Here I think time evolution comes into play. This secures stability, and a certain level of continous chain of evens are secured by the inertia in all structures.

    I don't think in terms of continuum since I consider the notion of distinguishable states, the discrete models seem more natural. But I have a feeling thta in the end equivalent continuum models might be constructable, by adding consistently ghost degrees of freedom. It seems if the contiuum is taken as a starting point, for myself at least, I have lost track of the correlations between the ghost degrees of freedom.

    Yes, something like that. I connect this to the logic of how measures are constructed. There is always an uncertainty in the identity of the measure itself. This abstraction of the connection between observables, measures, statistics and it's evolution towards self-preservation are my personal key guides. I think the dimensionality should come out as, not unique, but still the preferred choice at a given confidence. This duality also resolves the philosophical issue of creation of something out of nothing. Becuase this question itself is not constructible in the corresponding limit - so it's not a problem. This is similar to the idea that some of the "problems" with say the big bang, are resolved by the fact that complex observers were probably rare back then. It's the same logic.

  25. Sep 2, 2008 #24


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    Hello Chrisc, here are some more reflections on your continuum. Maybe I don't get your ideas all the way, but I'm curious how you defend your apparent continuum terminology.

    You seem to think in terms of the continuum. What disturbs me about that is that I lack a good physical basis for the continuum. The usual picture of spacetime, which is easily imagined mathematically doesn't make the same sense physically, if we require that it's measures are to be intrisically constructible. Then the continuum seems to contain a great overhead, that furthermore maybe isn't even unique. I have tried to get away from this.

    It seems to me we need to get back to the idea of spacetime beeing relations. Relations between what and defined where? This is where I picture that the relation is one between the observer (which really is any subsystem of the universe; the distinction is that an arbirary subsystem of the universe is not coherently evolving - this distinguishes stable observer from unstable ones) and it's environment. I think of spacetime as the observers creation of an internal map, of the unknown. So at first it seems subjective. Another observer might hold another map of the "same" environment. But then, that would cause interactions between the two systems. So the inconsistency due to subjectivity is not just a "problem", I think of it as a possible exploit to unification. This makes interactions in spacetime, and spacetime itself somehow inseparable. And their dynamics is entangled. I think this can thus be extended from GR, to all interactions. But this scheme I picture I think makes no sense if one maintains the concept of physical continuum. To me the continuum is nothing but a large number limit, and I think that alot of physics actually take place where the large number limit isn't valid.

    So there is a scale here, where we have discreteness at one end and the continuum at the other end. I think think thta the scale of observation here should be taken more into account in the first principles. No just work out continuum models and then afterwards try to do renormalisation. If the measures were constructed properly from the beginning I think this would be more transparent.

  26. Sep 2, 2008 #25
    Can any physical entity prove or disprove the existence of a continuum or not? I don't think any entity or observer can account for everything else in the universe. We don't know how many facts there are in the universe. All we can assume is that all facts are consistent with each other.

    I think you are right. It seems to me that the mass in the Lagrangian for a free particle can be generalized to the metric in the Lagrangian in the Einstein-Hilbert action that give GR as its equations of motion. This would not make particle theory entangled with gravity, but it would mean that particle theory is a local version of GR.
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