Dear Lev,

Schrödinger and Einstein are two of my favorites. They both suggested at some point that nature may be fundamentally discrete and combinatorial. Yet, their most astonishing results are based on the continuum.

Einstein: Special and General Theories of Relativity are based on continuum. His explanation on the photoelectric effect shows indeed that there is something fundamentally discrete about photons. But what is that discrete aspect?

Schrödinger: his equation, based on continuum, answered Einstein's question and provided the mathematical formalism of de Broglie's wave mechanics. From an equation based on continuum, we can obtain discrete sets of eigenstates. There is no contradiction: discrete emerged from continuous. So, Schrödinger's "own child" said that the continuum is fundamental.

Einstein tried to unify the forces by using continuous means. He did not succeeded, but he was able to build, with Rosen, (The particle problem in the general theory of relativity) a topological model of a charged particle. Again, discrete emerged from continuous.

Einstein, in his pursuit for local realism, supported de Broglie's pilot wave theory. On this basis, he rejected Schrödinger's idea (also based on continuum) that the wavefunction's square is the charge density of the electron (interpreted by Born as the probability density). Schrödinger realized later that, in fact, the entanglement was the main enemy of his idea, because the wavefunction's square cannot be interpreted as charge density for entangled particles. The entanglement rejected the locality, but not the continuum, from which it was initially derived.

Schrödinger and Einstein were not comfortable with the idea that the particle is a singularity of de Broglie's pilot wave, so they started to hope for a discrete solution. Which they couldn't provide.

So, to your question:

"Did we learned anything fundamentally new which might have changed his opinion?"

I would answer: "Did we learn anything which might have confirmed his opinion?"

Best regards,

Cristi

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  • Post #22

Cristi,

Schrödinger would not have changes his opinion.

Now, to your question: "Did we learn anything which might have confirmed his opinion?"

In order to learn something that would confirm his opinion, as always, we need to get out of the old "kitchen", build a new one, and to verify the predictions of this fundamentally new formalism. We need to look at the reality through new discrete "glasses", which we have not had: we don't even know what these "glasses" look like. That is why, I believe, we need to focus our efforts on trying to construct true "discrete" models of reality, not the handicraft models we call discrete. As you know from my essay, I am suggesting we need to construct a fundamentally new formal language that would elucidate the presently nebulous concept of discreteness.

AM I IN THE CONTINUOUS OR DISCRETE PARTY?

Does it matter?

My personal opinion is not that important. I tried to prove something, and only the arguments should be important. My personal opinion can be a complementary information, which helps to put the things in a larger picture, but what matters is what I can support with arguments, what I can prove.

In this line, I would like to say that I am glad to see at this contest such a wide diversity of opinions. The Nature is one, indeed, but we are far from knowing how she really is. So we try to guess the principles guiding her. So far, they are incomplete, and although they complement one another, they are in contradiction. I have no intention to take a side and claim that this is the truth, because I don't know the truth. I am just happy to see each effort, and each progress made by us in various directions. Am I proposing a continuous theory? Sure, but this shouldn't blind me against the discrete approaches. So, if you see me liking your discrete explanation, you should not conclude that I should not like a continuous explanation as well. Important is to progress, to add a new viewpoint, to solve a problem. Hence, we should encourage all good ideas which solve, or have the potential to solve a problem, to enrich our understanding, to widen our vision.

Good luck to all in this contest

Cristi

    Dear Lev,

    I am glad that you and others are trying to construct this new vision. I sincerely believe that this is a good thing, and that it will enhance our understanding. And I also sincerely believe that there should remain some of us to work in the "old kitchen" as well, because we need food until the new kitchen is ready.

    Schrödinger needs to prove his opinions like anybody else. And he succeeded very well to do this for many of his opinions. Now, if in the quote you gave, he states that space and time are fundamentally discrete, but he never provided evidence for his statement, I am free to adhere to his opinion or not. But just because he said so, it is not enough. I prefer to adhere to Schrödinger's opinions which he managed to prove, and which incidentally opposed the one you quoted.

    Now, this doesn't mean that eventually it will not turn out that the spacetime is discrete. I don't know. When the new kitchen is ready, I would like to be invited at dinner.

    Best regards,

    Cristi

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    • Post #25

    Dear Cristi,

    Thanks for the good wishes! My best wishes to you too.

    Just one little note. You know, of course, that at least 99.99% of researchers prefer to work in the old kitchen: it is so much more comfortable, but ... our scientific intuition, as was Schrödinger's, should not be biased by such comfort. ;-)

    Besides, all the comfort may turn out to be illusory, and so in the long run, the research life might turn out to be wasted (which wouldn't be so comfortable after all). ;-) The latter possibility has always been my greatest fear.

    Dear Lev,

    I invite you to visit my "kitchen" and taste my recipe. Please don't judge me only for using honey instead of sugar for this recipe; taste instead the cookie. Aren't the 99.99% you mention said that this meal cannot be cooked? How many of them do you see in my kitchen, cooking the same recipe?

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    • Post #27

    Dear Dr. Stoica,

    I enjoyed reading your essay before you lost me in mathematical technicalities. Nevertheless, before you launch into your thesis, you ask the following 3 questions, which I hope to answer:

    1) Is reality discrete or continuous? At least 'matter' and 'radiation' are discrete.

    2) (Was) it possible to find the answer by experiment? With respect to 'matter,' no, because we did not know what to look for.

    3) Or at least from theoretical arguments? Yes.

    You also point out that the reason we cannot decide between the continuous or discrete is that: "(T)he theories we know so far don't seem to make use in an essential, irreducible way of the discrete or continuous nature of the reality they propose." And further: "In addition, this theory should be mathematically and logically consistent, very well corroborated by experiments, and as simple as possible."

    You then proceed to make your case for your 'version of General Relativity.'

    While I am unable to mathematically argue the merits of your thesis, in terms of foundations it would be pointless to do so, as I show in my essay. Furthermore, the derived foundations in my essay are "mathematically and logically consistent, very well corroborated by experiments, and as simple as possible."

    I merely wish to bring this to your attention, for unless I am fooling myself the derived foundations leave no room for debate.

    Kind regards,

    Robert

    P.S - If I am fooling myself, could you please leave me a post to let me know.

    Dear Cristi,

    Here is a concrete mathematical question. It may be trivial, or not.

    Given a semi-riemannian metric g with its covariant version curvature

    tensor field R, is there (locally, around a point, or generically, like almost everywhere) another, non degenerate metric g', with the same curvature field R?

    Marius

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    • Post #29

    Cristi: I'm even happier that you like my essay after reading yours. I appreciated your resolution of the singularity and information paradox -- I found myself wondering why nobody had thought of your degenerate metric idea. I want to look further into your smooth quantum mechanics by way of your paper. Anyway, I am glad to find new and innovative arguments for (fundamentally) continuous nature that supplements the work by Zeh et al., so thank you for that, and good luck!

      Dear Karl,

      Thank you for your appreciation. Yes, when I initially considered to apply degenerate metrics to singularities I though it will be simple. It turned out that it was not that simple, because of the divergences and other problems which occur by the normal methods. The hardest part was to find a way to avoid them.

      Good luck to you too,

      Cristi

      Dear Marius,

      sorry, I just saw your post. Nice question, and far from trivial. And I think important, because we may need to know how to obtain the metric from the stress-energy tensor. I don't know how to answer it. Searching the net I found this article, this and this, which seem to show that in general the solution is unique, up to one arbitrary scalar. I don't know if it applies to all cases, and if they apply to the singular case as well.

      Best regards,

      Cristi

      Dear Marius,

      indeed, it is hard to choose a side. I agree with you that there is a limit in the precision, this is why I think it is hard to decide :)

      My personal view is that the continuum is a differentiable manifold, and many of the discrete aspects occur from its topological properties.

      Best regards,

      Cristi

      "AM I IN THE CONTINUOUS OR DISCRETE PARTY?"

      There is obviously both the continuous and the discrete in reality. The empirical evidence for this is obviously that we are all able to see bits and pieces of reality somehow and some say they can divide the bits and pieces to infinitely ever-increasing entropy.

      It does not really take peculiar minds to see the obvious. But understanding the obvious takes peculiar minds, because pointing out the obvious for peculiar minds to understand is quite difficult.

      The difficulty comes because of the unclear picture of the foundational ideas. Once the foundational ideas are clarified, eveyone could readily adapt their interpretations and most everyone will see that we have been describing the same thing, albeit unclearly.

      The answer to "what of reality is continuous and what is discrete" can readily be clarified if we take the analysis down to the fundamentals.

      Perhaps you will agree with me regarding what is continuous and what is discrete in reality if you read my essay... http://www.fqxi.org/community/forum/topic/835

        Dear Christi,

        I don't exactly know how to put this. But what makes the "degenerate metrics" degenerate - what is the cause of the process? Does the degeneration bring about "particles" or "voids"?

        Just a thought...

        Castel

          Dear Castel,

          I think that most of us would agree with you that "There is obviously both the continuous and the discrete in reality". Probably the divergences occur when discussing what of the two features is fundamental, and what is derived. If "some say they can divide the bits and pieces to infinitely ever-increasing entropy", my essay doesn't support their claim.

          Dear Castel,

          the metric defines the distance, and it can be represented (by choosing a coordinate chart) as a matrix whose entries depend on the position in space and on time. As a matrix, it has a determinant, which also depends on the position in space and time. This matrix determines (curvature, who determines) gravity, and it is in turn determined by the fields of matter (particles). There is no law able to prevent its determinant to become 0. It can become 0 because it changes, and the change is governed by Einstein's equations, or if you prefer to emphasize the time evolution, by the ADM equations. The particles have their role, because matter determines (curvature, who determines) metric, and if matter is dense enough, it evolves in a black hole - leading to singularities. That's pure General Relativity.

          Cristi

          Dear Christi,

          Regarding "change," - you say "change is governed by Einstein's equations or, if emphasis is on time evolution, by the ADM equations."

          By the phrase "prefer emphasis on time evolution," there seems to be the clear implication that there are other possible preferences that can be emphasized on. The idea looks like it is in accordance with Einstein's "arbitratry transformations of space and time"...

          Is the idea of a "preferred emphasis on space evolution" also correct? And is "space evolution" what you mainly describe by the following?

          - metric defines distance

          - metric can be represented by a matrix

          - matrix depends on the position in space and on time

          - matrix has a determinant

          - determinant also depends on the position in space and time

          - matrix determines (curvature) gravity

          - gravity in turn is determined by the fields of matter (particles)

          - determinant cannot be prevented from becoming 0

          - determinant becomes 0 because it changes

          - matter (particles) determines (curvature) metric

          - metric evolves into a black hole if matter is dense enough

          If it is not "space evolution" that you mean, then what do you mean? Does the word "metric" describe "motion" that defines distance covered per unit time?

          Do the words "change," "evolution," and "transformation" convey the same meaning?

          You seem to be unclear regarding "matter." You say "if matter is dense enough", implying that matter can be of various densities. Do you also mean there can be the preferred emphasis on the evolution of matter (particles)?

          Oddly, Einstein made no mention of the transformations of matter in conjunction with the arbitrary transformations of space and time. But he gave us the famous formula that really looks like the definition of mass-energy (matter) transformations. It would be somewhat novel if there is the proposition of the arbitrary transformations of space, time and matter.

          The maths are always impressive. But the beauty of the maths depends on the good sense in the interpretation.

          It really looks like matter (mass, energy) appears to be the most discrete phenomena in nature. So, I prefer the idea of the evolution of matter, the transformations of mass-energy - which at the fundamental level is, for me, the motion transformations defined on the ethereal substance within the space dimension. Gravity, electromagnetism, fields, mass, energy, etc., are all motions to me, albeit of varying forms according to the transformations of motion.

          My view is unconventional. But this allows me the idea that there is this much motion confined as particulate mass in this much unit of volumetric space and I can say this much kinetic energy can be release via a nuclear explosion of this much particulate mass from this much unit of space. Quantized motions! These have clear meanings to me...

          My rather novel ideas have been very difficult to get through to the reigning counter-intuition establishment. So, it would be nothing new if you don't get my drift...

          I hope you will rate my essay, too - so that I may also get the chance to be read by the establishment.

          Castel

            Dear Castel,

            If it was unclear for you what I said when discussing Einstein's equation vs. the ADM formalism, and if you are genuinely interested in the subject, please refer to one of the many GR books presenting it, for example the classic "Gravitation" by Misner, Thorne and Wheeler, or "Gauge Fields, Knots and Gravity" by John Baez and Javier Muniain. For many, the book "Gravitation" is an "authority", but please do not take these readings this way. I give you these references to read them critically, to check their calculations, their arguments relating the theory with experiment, and not to blindly accept them. Many accept them blindly, and many reject them blindly just because they think that if they don't understand the math, it is because the math has no "good sense in the interpretation". I would suggest to those an easier reading, such as La Fontaine's "Le Renard et les Raisins".

            Cristi

            Dear Christi,

            You haven't answered my questions. Instead you have given me advice. Nevertheless, thank you for the advice. :)

            Castel

            Dear Castel,

            Presuming that you are genuinely interested in answers: why don't you ask one clear question, and then let me answer, and then continue with another clear question and so on?

            Cristi