• [deleted]

  • Post #52

One can look at this according to how one axiomatizes a space. If you have a space in n dimensions one can represent the positive tensor dimension as ||| ...|•ε = 0, where | represents an element such as a vector or spinor and the set |||...| means an exterior product of these. The ε means a Levi-Civita symbol and this is a skew product. This can be seen equivalently as a skew symmetrization of the |||...| in a higher dimensional space. If this is zero, then the space of tensors is symmetric. This system however requires there to be the |||...|•g, where g is a symmetric tensor. This way of thinking is what might be called Penrose-ology. Again this is equivalent to a symmetric trace in a higher dimensional space. The "dimension of these tensors" are n and -n respectively. They correspond to the symmetric and antisymmetric sets of tensors, which have a duality.

This duality between symmetric and skew symmetric elements, or for two tensors products of the sort

{ψ^a, ψ^b} = g^{ab}

[φ^a, φ^b] = ω^{ab}

involves supersymmetry. In the case of spacetime the generators of supersymmetry Q_a and \bar-Q_b construct Lorentz boosts

[Q_a,\bar-Q_b] = iσ^μ_{ab}∂_μ.

The relationship between the symmetry and antisymmetric approaches, say shape dynamics and causal set theory, might then have functors to Fermi-Dirac fields and boson fields, and a system which includes both might then have a graded Lie algebra with Grassmann generators that connect the two.

Cheers LC

Daniel,

I will have to think carefully about what you said and wrote about semantic completeness and observation. One of the weak points in my knowledge that has come up over and over again on this forum is the original ideas of Mach, although it seems that much of this involves covariance/equivalence principles, which I have thought about. Part of my task is just understanding the terminology, since I come mostly from the math side. Take care,

Ben

Lawrence,

Again, thanks for the insight. I have been having a somewhat similar discussion about possible duality/complementarity with Sean Gryb over on my thread. I won't remark further about this particular idea here without Daniel's permission, since it's somewhat tangential to the focus of his essay, but such discussion is always welcome on my thread. Take care,

Ben

Dear Lawrence and Ben

Unfortunately, I must admit I still don´t have enough technical language to participate in the discussion you have proposed, but please, feel free to discuss those matters as you wish.

Ben, sorry, I think I was not clear enough, but tomorrow I will prepare a more elaborate reply to you to explain those machian and extensions of machian thoughts I mentioned.

Daniel

Dear Ben

I will now try to explain the point I was trying to make. Let´s place ourselves in the 17th century and try to build physics from the scratch, that is, for the sake of the argument, let´s ignore any complications due to modern physics. I must firts say that Mach´s thoughts are philosophical. For someone coming from a math background this may seem extremely vague. But Mach´s philosophy has tight relations with GR, and this makes it very important.

Due to our stable enviroment, we were led to think that there was an invisible space and time background. The rotation of the earth provided a parameter to which all motion should be labeled and the distant stars provided an infinite grid to which all distances should be measured. It was then very natural to take the mathematical gadgets such as R³ to model the physical world. Then you can build equations between 'stuff' defined on R³ such as ma=-grad(V) but this theory cannot be tested because this R³ spatial grid cannot be seen... a closer look reveals that the distant stars also seem to move! There is no epistemological way to identify the grid. The best that can be done is to find a visible object, measure distances upon it and check if ma=-grad(V) would hold (such an object would provide an inertial frame of reference). But even so we could not identify a grid defined this way with absolute space, because the inertial frame object can be moving with constant velocity in relation to absolute space.

So, was absolute space a historical mistake? Could we formulate a physical theory without unobservable structure, in a purely relational (because all we see and measure are relations) way? This is what Mach intended, though he never wrote down a complete physical theory. But then, Barbour has shown that, in a sense, it suffices to impose a relational framework to a 3-metric field to RECOVER general relativity from first principles.

If one recovers GR from relational first principles, it becomes compelling to study the origin of these first principles... what´s the origin of Mach´s thoughts? What I have been thinking is that maybe it is possible to see Mach´s procedure as a part of something bigger.

Mach´s unease with classical mechanics can be seen as coming from the following: what do we mean when we say an object´s position at a time t is (x,y,z)? Mach´s criterion of meaning is OBSERVATION. To say that the position is (x,y,z) at a time t can only MEAN that the relative distance between the object and a reference is (x,y,z) and that a clock (which is a physical object) has marked t. For Mach, all statements of classical mechanics should MEAN only what can be observed upon then. Unobservable statements should be cut off from the start, because they don´t MEAN anything. What Mach was ultimately searching was MEANING, using OBSERVATION as criterion. And remarkably, this leads to GR via Barbour´s argument!

But the relational philosophy is not complete: it gives time and position a meaning upon the concept of physical object. But the concept of physical object is still MEANINGLESS! This is why I proposed to try to build a dynamics where time, space, motion and objects all gain their meaning upon each other, to see what would be the correct mathematical structure amd what result we would achieve. I call a completely meaningful description of the universe ''semantically complete''. Now one of the mysteries of QM is the nature of observation: why is observation so different from other physical phenomena? What causes the wave function to collapse? How can we classify a physical process as an ''observation''? Category theory now comes into the game: imagine a category where objects are semantically complete descriptions of the universe and functors that send such descriptions to descrptions that MEAN the same (let´s call them semantic functors). Now it seems that the concept of ''observation'' could be given a precise mathematical meaning: it is ''that thing that is used to build the semantic functor''! And relational physics can be simply stated as ''the square commutes'' as I put in my essay! This is the outline of what I was thinking.

I think the first step should be to cast barbour´s relational physics in a category theoretic framework. Barbour´s procedure of eliminating absolute structures is his method of best-matching, as I explained in the essay. I was thinking of trying to find the origin of best matching via category theoretical considerations. I don´t know if this is possible, but my intuition tell me it is. Once we do that, then I think everything would be easier to understand. Maybe you will find that interesting. I´ve seen that Derek Wise was trying to find relations between Barbour´s theory and Cartan geometry, which has some relations to category theory, we could investigate that. Anyway, I will be waiting for your feedback.

Best Regards,

Daniel

    Dear Daniel,

    I have been thinking a bit more about the last few sections of your paper. I am in the process of trying to learn several new things at once (and also write a dissertation about something completely unrelated!) so you'll have to forgive my delayed response.

    First, I appreciate your explanation in the previous post; I think I have a better idea now of how you are using certain terminology. In particular, I realize now that a large part of what you are presenting is your own ideas, so it's not surprising I haven't heard of this view before. Now let me itemize a few remarks.

    1. Regarding the concepts "time, space, object, motion," it seems that you want to define each in terms of the others. Now it seems clear that any logical or mathematical system (at least any system satisfying a suitable finiteness assumption) will have either undefined concepts at its lowest level (in terms of which the remaining concepts are defined), or will have redundancy at its lowest level (where the fundamental concepts define each other). It seems that these two possibilities are interchangeable: if you have redundancy, you can eliminate concepts one by one until the redundancy disappears and the remaining concepts are undefined. Conversely, you can define new concepts in terms of the fundamental (undefined) concepts and take these to be "equally fundamental." I suppose this is analogous to finding a basis of a vector space from a spanning set, or augmenting a basis to a larger spanning set that is no longer linearly independent. There are plenty of situations in which a larger redundant set of concepts is useful, so parsimony is not the only consideration here.

    2. It seems that "semantic completeness" as you define it requires redundancy, because if every concept can be defined in terms of others, then some of these concepts can be eliminated (at least, if there are a finite number of fundamental concepts).

    3. We must be very careful about the use of the word "object," because it has more than one meaning. It has a precise, axiomatic, but very flexible meaning in the context of category theory; as you point out, categorical objects could be Hilbert spaces, or logical propositions, or whatever. It seems to have a vaguer but more specific meaning in the sense of "physical object." When you mention defining time in terms of objects, space, and motion, the objects you are talking about in this case must mean "physical objects," such as "particles" or "fields," and to define "time," they must somehow be indentifiable as "the same object" after undergoing the "change" that defines time. In other words, I don't think a pair of different structures by itself can define time in a Machian sense; rather, it is necessary to be able to identify the "second" structure as being the "result" of "changing" the "first" structure. I use quotation marks to indicate that I am not attempting to be precise at this point! What I am trying to get at is that the concepts of "change" or "motion" require that the "initial and final states" be identified as different states or configurations of the "same object" rather than two totally unrelated structures.

    4. I am glad you pointed out the work of Derek Wise. I have not read these notes yet, and they seem very relevant to what we have been discussing.

    5. There is much more to discuss, but no time at the present to do so. I see you have an email address on your paper, and I also have one on mine... that way we can keep in touch after the essay contest is over.

    Take care,

    Ben

    • [deleted]

    • Post #58

    Dear Ben,

    Thanks for your reply. I understand your lack of time. A few remarks to your remarks:

    ''It seems that these two possibilities are interchangeable: if you have redundancy, you can eliminate concepts one by one until the redundancy disappears and the remaining concepts are undefined. Conversely, you can define new concepts in terms of the fundamental (undefined) concepts and take these to be "equally fundamental."''

    That´s the point. I was thinking that maybe we can define ''motion'' using ''time'', but then times becomes undefined... or we can define ''time'' using motion, but then motion becomes undefined... and so on. In the end, maybe all fundamental terms: space, time, motion, physical objects could have this ''duality''. The reason that points me for thinking this is that machian philosophy (which leads to GR) can be seen as a PART of this duality!

    ''2.It seems that "semantic completeness" as you define it requires redundancy, because if every concept can be defined in terms of others, then some of these concepts can be eliminated (at least, if there are a finite number of fundamental concepts).''

    I don´t know if they could be eliminated... but the CHOICE of fundamental terms would not be unique! Again, we could consider that motion gains meaning from time, space and objects, OR that time gains meaning from motion, space and objects. Then, we could postulate that no matter how we choose to represent the universe, there should be no physical change upon a different choice of fundamental terms. Absolute and relational views of motion would be a part of the same structure... and there would also be the ''something else''!

    Ultimately, what I propose is that by investigating the ''meaning'' of classical statements about motion, we can find new ways to conceive motion, and then build physics using this conceptions.

    ''What I am trying to get at is that the concepts of "change" or "motion" require that the "initial and final states" be identified as different states or configurations of the "same object" rather than two totally unrelated structures.''

    We can also define motion without an absolute structure by introducing a ficticious ''background'' structure for the final and initial state and then eliminating it using barbour´s best matching: take for instance two configurations of point particles defined in cartesian frames, say (xi,yi,zi) and (x'i,y'i,z'i), representing distinct instants of time. Now hold the first frame fixed and perform rotations and translations of the second until some incongruence measure such as SQRT((xi-x'i)2+(zi-z'i)2+(yi-y'i)2) gets minimized. This is defined as the best-matched distance between these two configurations, and this value can be calculated using only information meaningful in the relational view of motion.

    I´ll keep in touch for further discussion.

    Best regards

    Daniel

    • [deleted]

    • Post #59

    and one redundance and one for the sortings of datas and informations, and now you are going to make some logarythms for the sortings, we know we know.

    and after a mthematical universe proof of course of course.

    • [deleted]

    • Post #60

    Dear Daniel,

    I completely agree that "if the appearance of observation in the semantic functor could bring us any closer to quantum mechanics" since I see "observation" as a mapping from physical states to recorded experimental outcomes, i.e. to descriptions encoded in some memory medium using classical information. So observation is itself a functor, from a category in which the objects are quantum states and the morphisms are unitary transformations to a category in which the objects are descriptions encoded in classical information and the morphisms are formal operations defined on those descriptions. The criterion for descriptive coherence is precisely diagram commutativity. But this is not your "semantic functor" which you have defined as a category automorphism. It is the function that tests whether two physical configurations "mean" the same thing from some observer's point of view.

    Such "observation" functors are very familiar: they define the semantics that we associate with computer hardware. We pretend that "classical" computers are classical. This is of course nonsense; they are quantum systems just like everything else. Nonetheless, when we look at them, we assign a semantics under which their physical dynamics is mapped to formally-specified execution traces of classical algorithms. In my view, this is what ALL observation is.

    My PhD advisor, Rob Cummins, used to tell us all to imagine that our PCs grew up overnight in our back yards. I agree: this forces us to think about the semantics we assign to physical events in a coherent way.

    Good luck with your research,

    Cheers,

    Chris

      • [deleted]

      • Post #61

      Hi Daniel,

      To be honest, I doubt that it is reasonable to follow Barbour and somehow replace time. Your age refers to the time of your birth which is certainly not likely to be chosen as reference point for an absolute time. May we conclude that there is no absolute zero of time? I do not suggest referring to the hypothetical moment of a Big Bang. Being an old engineer, I see only the actual border between past and future a natural fix point suited to refer to in an non-arbitrary manner. You might try and find some flaw in my criticism .

      As for mathematics, it would not be unreasonable to completely avoid non-zero integration constants by agreeing on a definition of integration that always refers to the lower border zero.

      Eckard

        Dear Chris

        Thanks for your comment. I´ve read your essay and, as I said, I deeply impressed!

        ''So observation is itself a functor, from a category in which the objects are quantum states and the morphisms are unitary transformations to a category in which the objects are descriptions encoded in classical information and the morphisms are formal operations defined on those descriptions.''

        I see we have slightly different views. To describe a ''quantum system'' or a ''classical system'' we need to use structures like space and time. However, these structures may come with a large degree of redundancy, depending on how we conceive motion in the first place, as I have explained in my essay. So there should be a functor connecting all those semantically ambiguous states, and the outcome of physical process should not depend on how we describe it. This is where the diagram commutes in my view. And the functor that connects all the semantically ambiguous states should be built by using a criterion of ''meaning upon observation'' to fulfill the principle that ''empirical indiscernibles are physical indiscernibles'' (as Robert Spekkens put). My hope is that this would have relational physics and GR as a sub-product. But I see the process I have in mind could be greatly enhanced by first characterizing observation as a functor in the first place, as you said. I will have to think more about that.

        Best regards,

        Daniel

        Dear Eckards

        Barbour´s research on time deals exactly with the following redundancy: if eveything speed up in the universe, including clocks, we could never tell the difference. So, why not consider time an abstraction from motion instead of an invisible parameter?

        This is the relational view of time, and by combining with a relational view of space, one gets a completely different mathematical and physical structure than those of absolute view of motion. So the point is that distinct conceptions of motion may lead to distinct physics, and I have argued in the essay that maybe we´re not limited to absolute or relational conceptions.

        Best regards

        Daniel

        • [deleted]

        • Post #64

        Dear Daniel,

        The contests solicited to reveal wrong assumptions. As an engineer I know how to describe an invention. Accordingly my essay begins with a list of obvious shortcomings. Will your suggestions provide solutions to such enigma and suspected flaws?

        I do not deny that Barbour's approach might be reasonable to some extent. I merely doubt that it will help for instance to overcome Einstein's tense-less view. I rather see it a step in the wrong direction. See 1364.

        Best regards,

        Eckard

        ''The contests solicited to reveal wrong assumptions. ''

        As I say in the abstract, ''It is argued that there may be other ways to conceive motion and that a systematic investigation of these different conceptions may produce new physics. ''

        ''Will your suggestions provide solutions to such enigma and suspected flaws?''

        Who knows. Everybody is speculating to some extent here.

        ''I rather see it a step in the wrong direction.''

        Even if it is a step in the wrong direction, my conclusion remains: different conceptions of motion at the classical level may lead to distinct physics.

        Best regards

        Daniel

        Dear Daniel

        Reading your essay I could see you have the talent, mental stamina and imagination to discover new physics. The problem of understanding motion is a very interesting one. You promote Barbour's theory which I value for relegating time to an emergant status, but think the SS idea unnecessary theoretical complication. Mach's ideas are very interesting, but there are many ways to interpret them.

        Starting from very different premises developed in my 2005 Beautiful Universe Theory on which I based my fqxi essay Fix Physics! . I concluded that motion is the translation of a pattern (a concept you use) in the energy and orientation of dipolar nodes making up a universal lattice. In my theory general relativity becomes very simple, but motion involvs a self-convolution of the pattern not only of the object involved but (a la Mach) the nodes making up the surrounding gravitational field, on to those of the entire universe. For example a force on matter causes the pattern to compress first, and then start moving with its length contracted, 'pushing' the external gravitational node pattern ahead as well. This combines Newtonian SR and GR notions. I attach a figure from the BU paper to explain this qualitatively.

        I wish you all success, and urge you to stick close to simple physical concepts - one can easily get too abstract using clever mathematics.

        VladimirAttachment #1: 2_BUFIG26.jpg

          Dear Vladmir

          Thank you so much for these encouraging comments! I have read your essay and though I must say there a lot of points of divergence between my thoughts and yours, you´ve written in a very clear and concise way. Good luck.

          Daniel

          4 days later

          After studying about 250 essays in this contest, I realize now, how can I assess the level of each submitted work. Accordingly, I rated some essays, including yours.

          Cood luck.

          Sergey Fedosin

            • [deleted]

            • Post #69

            Thanks!

            If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is [math]R_1 [/math] and [math]N_1 [/math] was the quantity of people which gave you ratings. Then you have [math]S_1=R_1 N_1 [/math] of points. After it anyone give you [math]dS [/math] of points so you have [math]S_2=S_1+ dS [/math] of points and [math]N_2=N_1+1 [/math] is the common quantity of the people which gave you ratings. At the same time you will have [math]S_2=R_2 N_2 [/math] of points. From here, if you want to be R2 > R1 there must be: [math]S_2/ N_2>S_1/ N_1 [/math] or [math] (S_1+ dS) / (N_1+1) >S_1/ N_1 [/math] or [math] dS >S_1/ N_1 =R_1[/math] In other words if you want to increase rating of anyone you must give him more points [math]dS [/math] then the participant`s rating [math]R_1 [/math] was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.

            Sergey Fedosin

              Daniel

              Motion seems at the heart of misunderstanding in physics. I'm now re-reading those I've not yet scored and am pleased to say yours still stands out. You did undertake to read mine, and I hope you will as I'm very interested in your comments (and score!). Very well done for yours. I hope you find we have a rich common vein.

              Best wishes

              Peter