Dear Cristi,
Glad to see you in the contest. I read the essay, it is interesting. I have a question regarding your notes from the references. Why don't you put them on the arxiv? You could get lots of potentially interesting suggestions and/or healthy critical remarks.
Concerning semi-riemannian metrics, do you think is there any way to set up an experiment which could allow to check this hypothesis?
Best,
Marius
Cristi,
I find your answers enlightening. Thanks for the explanations. You do have an excellent grasp of the issues. Without going over them point by point I will say that I agree with your comments almost wholly, and I now understand your paper even better. I will read your previous essay on time about state vector reduction in QM as I only vaguely remember what you said. I am even happier with your essay after your detailed response to my questions.
One point that you made was that, "in Quantum Theory the time evolution is unitary, hence the information is preserved." While I agree with the statement, I think the following is relevant.
Veltman notes that Feynman rules are derived using the U-matrix, even though formal proofs exist that the U-matrix does not exist. (Diagrammatica, p.183). The U-matrix is unitary by construction, and implies conservation of probability, probability being "the link between the formalism and observed data." In my mind, this leaves some room for 'free will' in the universe, (with consequences for information) but I have not pursued the U-matrix much farther than that. Veltman claims the U-matrix and the equations of motion are to be replaced with the S-matrix, in which the interaction Hamiltonian determines the vertex structure. I believe this is why Ray Munroe bases his work (as I understand it) on the Coleman-Mandela theorem, which (according to Wikipedia) states that "the only conserved quantities in a "realistic" theory with a mass gap, apart from the generators of the Poincare group, must be Lorentz scalars." But this seems to constrain only symmetries of the S-matrix itself, not spontaneously broken symmetries which don't show up directly on the S-matrix limit. As the 'scattering' matrix is used to make sense of particle collisions, this seems reasonable, but 'scattering' of particles is a very artificial (if necessary) way of studying particles, and, as I've noted in my essay, leads to a Lagrangian that is based on inventing fields, whether or not those fields actually exist in nature. If they can be solved for then they are considered in some way 'real', and this leads, IMHO, to much of the confusion today.
Veltman also says that "unitarity, Lorentz invariance, locality, etc, are in some sense interchangeable." It seems to me that this is problematical in light of today's push to banish locality from QM.
I don't claim to understand the solution to these problems, just to note that there seems to be some circular logic going on, and I'm not sure that logic is preserved around a complete loop of the circle.
This is part of the reason I start with the logic of one field, and work from there, ignoring, for the most part, the established formalism's of QM and GR if they don't map 100 percent into my model in a way that will satisfy experts in either field.
Thanks for the essay and the explanation, and best of luck in the contest,
Edwin Eugene Klingman
Dear Steve,
thank you for your appreciation and kind words. I'm happy to see you too.
Best Regards,
Cristi
Dear Marius,
thank you for reading and appreciating my essay, and for the suggestions. I agree with you and I plan to arxiv them soon.
About semi-riemannian metric. When it is non-degenerate, it is the core of Relativity, both special and general. All the tests of Relativity also test the semi-riemannian metric. The special relativistic effects show that the Poincare symmetry is valid. The predictions of General Relativity, such as deflection of light, perihelion Advance of Mercury, etc, show that this metric exists but it is curved. One of the problems is the prediction of singularities: under general conditions GR saids that they exist. And these conditions are satisfied in our world - for example the existence of black holes. What the singularity theorems show is that at some extreme places, such as the "core" of a black hole, the metric becomes degenerate. In this case, the important invariants of semi-riemannian geometry become divergent.
So, I would say that I do not propose new hypotheses or laws of nature. I just modify the formalism, to base it on other invariants. For example, when the metric becomes degenerate in a way I named semi-regular, the curvature Rabcd becomes divergent, but its covariant version Rabcd remains smooth. Normally, they have the same information, but not when the metric is degenerate. In my approach, it turns out that the fundamental invariant is the covariant version of the Riemann tensor, which is smooth.
As an analogy, think at curves in plane. If we decide to characterize the way they turn by the curvature radius, the straight lines - having infinite curvature radius - will turn out to be singular. So, the best choice was to use the curvature, which is 1/R. In this case, the straight lines have curvature 0, which is more manageable. Basically, this is similar to what I did - I chose other fundamental invariants than it is customary. They contain the same information for non-degenerate metric, but they are not divergent, and continuous, when the metric becomes degenerate.
How can we test directly that the singular semi-riemannian metric occurs somewhere in the universe? It is hard, because the places where it is predicted are at the big bang, and inside the black holes. But maybe we can send some information in a black hole, and check that it survived the evaporation. I think that this is very difficult to do, perhaps impossible - we need to know completely the values of the fields, and to compute the outcome (the information survives, but it is mixed). Another way would be if micro black holes (predicted by Hawking) really exist. In this case, they should occur and evaporate quickly all the time, from quantum fluctuations. If they violate the conservation of information, then we can detect violations of unitarity and of the entanglement, by quantum experiments. Personally, I don't think that these experiments are practical. I just wanted to show that it is premature to reject General Relativity because it predicts singularities, by showing that these singularities are not "malign". I could not find a simple way to show this, so I had to develop some mathematics for the singular semi-riemannian manifolds.
Thank you for your observations, and good luck with the contest,
Cristi
Dear Eugene,
I am glad I could answer to your questions in a satisfying way. I share your concerns about the problems of QFT. QFT is a great success, but its mathematics is very messy and not yet consistent in a satisfying way. From this viewpoint, I think that GR is simpler and more consistent. Of course, GR should be completed with the proper fields and initial data to solve the problems raised by cosmology and particle physics (if this is possible). I think that very much in QFT is correct, but because we cannot solve the problems directly, we are forced to make approximations and perturbation calculus, and this is how more inconsistencies appear. My hope is that someday we will have a clean and consistent QFT, without infinities. We both try to solve these problems, you by using a fundamental field, me by taking one issue at a time.
Good luck with the contest and your research, and thank you for the relevant questions you asked.
Cristi
AM I IN THE ANALOG OR DISCRETE PARTY?
Dear visitor,
I think that this contest stimulated in all of us the curiosity about the orientations of each of the participants: "is he or she on the discrete side, or on the continuous side?".
So I will try to clarify my position, which is somehow ambivalent.
At the present time, I put my hopes in a continuous spacetime, on which continuous fields "grow". This may be obvious from my essay, in which I stated that the solution I propose to the singularities requires a continuous spacetime. (Of course, I may be wrong and the spacetime be discrete.)
On the other hand, I do not exclude the possibility that, even in the conditions of continuous spacetime and fields, the world still may be discrete. I illustrate this with the example of vector graphics format in computer graphics. This type of format allows infinite resolution, but in the same time it is digital. Digital does not necessarily means pixelated, so discrete information still may describe a continuous spacetime.
Similarly, continuous spacetimes endowed with continuous fields may very well be describable by digital information. After all, all books on continuous mathematics and physics can be scanned into a computer.
I argued for the continuity of spacetime and fields, but I do not exclude the possibility that all the information contained in these fields can be compressed in a digital format. I wish I could write about this too in my essay, but I need to do more research in this direction.
Best regards,
Cristi
Dear Cristi,
Yes, this is a nice question! Choose your side!
My side is that reality is continuous, but it looks discrete sometimes because it has a differential structure which is scale-dependent. Then, making an experiment amounts to draw a map of the part of reality at one scale, to another part of reality, at another scale. There is a mathematical limit of the precision any such map could have.
Physicists somehow are lost in another dream. Despite many claims that physics so inspired modern mathematics, in fact since some decades this is only very limited so, while the most dynamic mathematical fields have little to do with physics, but more with computer science or (less now but more in the future) biology. Or just pure curiosity.
AM I IN THE ANALOG OR DISCRETE PARTY?
Part 2
Here is a link to some older work I did, named World Theory. It is a mathematical framework (based on sheaf theory) - a general mathematical structure which speaks about any possible world consisting in space, time, matter and laws of nature. It can be particularized to obtain many of our current theories in the foundational physics. In other words, to make abstraction of the particular solution we adopt, and to say the most general things we can say about the world. The intention was to write the laws in the most general possible form, so that we can compare them, and see which principles really contradict each other and which can be reconciliated on a higher level of generality. It was not a unified theory, just a unified framework.
The mathematical structure defined there can be particularized to most of the continuous, and of the discrete theories which are currently researched. In other words, two theories about space, time and matter, one which is discrete and another which is continuous, are both particular instantiation of the mathematical structure I named there "world". The matter is, in all cases, a section in a sheaf over space and time, and this notion works with both continuous and discrete spacetimes, as the examples I gave there show.
Although it would have been appropriate for the theme, because it brings under the same umbrella discrete and continuous, in the present essay I did not pursue this idea. The reason is that I spoke about the World Theory in my FQXi essay about time, Flowing with a Frozen River. There I used World Theory to discuss time, determinism and causality.
Best regards,
Cristi
Dear Cristi,
Thank you for clarifying your position. When I read your essay, I was under the impression that you were saying that space and fields are divisible ad infinitum, and thus continuous.
Some of your older work implies that Nature is simultaneously continuous and discrete, and that was also my essay's point.
It seems we have many similar ideas, and I need to read your older work.
Have Fun!
Dear Cristi,
In connection with your just stated position, can you comment on the following opinion of Schrödinger (which I quote on p.1 of my essay:
"If you envisage the development of physics in the last half-century, you get the impression that the discontinuous aspect of nature has been forced upon us very much against our will. We seemed to feel quite happy with the continuum. Max Planck was seriously frightened by the idea of a discontinuous exchange of energy ... Twenty-five years later the inventors of wave mechanics indulged for some time in the fond hope that they have paved the way of return to a classical continuous description, but again the hope was deceptive. Nature herself seemed to reject continuous description...
The observed facts (about particles and light and all sorts of radiation and their mutual interaction) appear to be repugnant to the classical ideal of continuous description in space and time. ... So the facts of observation are irreconcilable with a continuous description in space and time."
Did we learned anything fundamentally new which might have changed his opinion?
Dear Ray,
thank you for the feedback. You understood well what I said in my essay: my solution to the singularities in general relativity requires spacetime and fields to be divisible ad infinitum.
In World Theory I define a mathematical structure named "world", which describe possible matter fields over possible spacetimes, subject to possible laws, "possible" in the mathematical sense. Known theories in physics, discrete or continuous, are find to be particular cases of this structure, in the same way as the definition of group applies to both discrete and continuous groups. But World Theory does not make any implications about the real world, it is just a metatheoretical framework.
But there is enough room for discrete structures in continuous theories, and I will return to this subject soon.
Best regards,
Cristi
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
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
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?
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
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!