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Essay Abstract

We argue that principles of quantum theory inevitably imply that any fundamental physical theory can be based only on a finite mathematics. A version of a quantum theory based on a Galois field (GFQT) with a characteristic p is described. Since any Galois field is finite, there are no infinities in this theory and all operators are well defined. In a formal limit p->\infty GFQT reproduces the results of standard theory based on continuity. In GFQT the notion of particle-antiparticle and the conservation of such additive quantum numbers as the electric, baryon and lepton charges can be only approximate if de Sitter energies are much less than p.

Author Bio

Graduated from the Moscow Institute for Physics and Technology, got a PhD from the Institute of Theoretical and Experimental Physics (Moscow) and a Dr. Sci. degree from the Institute for High Energy Physics (also known as the Serpukhov Accelerator). In Russia Felix Lev worked at the Joint Institute for Nuclear Research (Dubna). His major area of research was relativistic quantum theory and a quantum theory over a Galois field. Since 1999 Felix Lev lives in Los Angeles, California and works at a software company on mathematical algorithms for the IC industry.

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Dear Felix,

Interesting essay. For some time I wanted to understand QM based on p-adic numbers. In the light of Jordan algebra classification, I am extremely skeptical of this approach. Yes, it may solve some infinite dimensional field theory problems, but how does it stack up with simple textbook QM problems? You mention ref 4. Is there any good archive paper on this?

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Author Felix M Lev replied on Dec. 19, 2010 @ 08:17 GMT

Dear Florin,

Thank you for your note. You say that you are skeptical about the p-adic approach. Does this imply that you are skeptical about my approach as well? One of the main reasons why I prefer a finite field approach is that it does not contain infinities at all. In my approach there is no problem with standard QM since, as explained in the essay (see also my cited papers and/or my papers in the arXiv), in the formal limit p->\infty my approach recovers the results of standard quantum theory. I am not the author of [4], so you would better ask them about problems bothering you. Let me only note that the authors of the p-adic approach also state that they have a correspondence principle when p->\infty.

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Florin Moldoveanu replied on Dec. 19, 2010 @ 14:22 GMT

Dear Felix,

Thank you for your reply. I suspect that in the p-adic approach, simple problems like harmonic oscillator, or hydrogen atom have solutions which differ from standard QM. I can see how when p goes to infinity the usual results are obtained, but in that limit you are effectivey doing complex QM, In this sense, this approach is not different than say a dimensional regularization approach, but the problem still remains of proving that the infinities go away nicely. What I am saying is that this seems to be a case of having the cake and eating too: on one hand the lack of infinities collides with simple results, on the other hand, taking the limit recoveres the standard case but you don't want to reach the limit. I see thus as no different than arbitrarily truncating the Taylor series in the quantum gravity case; in other words it is an attempt of having a regularization technique, but with no proof of renormalizability.

It is clear that for quantum gravity something has to give. String theory is one way. Noncommutative geometry is another. p-adic and Galois approaches to QM seem to be an unnecessarily radical departure from the standard approach compared with those two approaches. Also an unproven approach as I don't see any serious renormalization results in the literature (for only a few days of reading the references on the archive), but I could be mistaken about that.

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Author Felix M Lev replied on Dec. 20, 2010 @ 07:27 GMT

Dear Florin,

Thank you for your opinion. If you look at the beginning of the discussion section in [11] (in the arXiv this is the paper http://xxx.lanl.gov/abs/1011.1076 ) you will see that I am aware of this opinion. This is a typical opinion and of course, everyone has a right to have his or her own preferences (“De gustibus non disputandum est!”). I will try to comment this opinion....

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Dear Florin,

Thank you for your opinion. If you look at the beginning of the discussion section in [11] (in the arXiv this is the paper http://xxx.lanl.gov/abs/1011.1076 ) you will see that I am aware of this opinion. This is a typical opinion and of course, everyone has a right to have his or her own preferences (“De gustibus non disputandum est!”). I will try to comment this opinion.

You think that “p-adic and Galois approaches to QM seem to be an unnecessarily radical departure from the standard approach”. So you think that the existing problems can be solved without radical approaches and you believe that they can be solved by the string theory, noncommutative geometry or something like that. Of course, only the future can be the judge. Nevertheless, some of your remarks seem strange to me.

You say that standard problems look strange when they are considered from the point of view of the p-adic approach. In your opinion, p is only a cutoff parameter. It seems to me, this has something common with the following hypothetical situation. Suppose that Heisenberg or Schroedinger wrote a paper on QM and a referee says: “Since you do not want to take the limit \hbar->0, I have big technical problems describing the motion of the Moon by the Schroedinger equation.” And this referee is right! Of course, there is no need to describe the motion of the Moon by the Schroedinger equation. Since this motion is quasiclassical, we can take the limit \hbar->0 immediately and get the Hamilton-Jacobi equation. But now we know that there are other problems where we cannot take the limit \hbar->0. Moreover, the fact that \hbar is not zero leads to a dramatic change of our understanding of nature.

As I explain in my papers, p is not only a cutoff parameter since in Galois fields the rules of arithmetic are essentially different. As a result, some quantities (e.g. the Dirac vacuum energy) which in standard theory are infinite become not of order p (as one might think if p is treated only as a cutoff parameter) but exactly zero. Another new features are that in GFQT the notion of particle-antiparticle is only approximate, there can be no neutral elementary particles and such quantum numbers as the electric, baryon and lepton charges are only approximately conserved. In particular, this completely changes the status of the problem called “Baryon asymmetry of the Universe”.

My final remarks are as follows. The absolute majority of physicists treat mathematics only as a tool for solving physics problems. Those physicists don’t care much whether or not mathematics is beautiful, rigorous etc. A well known example is that when we subtract one infinity from the other and get correct 8 digits for the electron and muon magnetic moments, physicists typically treat this as a great success (what is true) and believe that there is no need to be bothered by the lack of mathematical rigor. However, several famous physicists did not think so. For example, Dirac wrote:

“The agreement with observation is presumably by coincidence, just like the original calculation of the hydrogen spectrum with Bohr orbits. Such coincidences are no reason for turning a blind eye to the faults of the theory. Quantum electrodynamics is rather like Klein-Gordon equation. It was built up from physical ideas that were not correctly incorporated into the theory and it has no sound mathematical foundation.”

Dirac’s advice is:

“I learned to distrust all physical concepts as a basis for a theory. Instead one should put one's trust in a mathematical scheme, even if the scheme does not appear at first sight to be connected with physics. One should concentrate on getting an interesting mathematics.”

For me the main reason why GFQT is much more appealing than standard theory is that it is based on extremely simple and beautiful mathematics. I also believe that this mathematics is more physical than standard one. I believe that for quantum physicists it should be obvious that in nature there are no infinitely small or infinitely large quantities and the notions of infinitely small and infinitely large are only idealizations. We have to use these notions because standard quantum physics is based on mathematics developed mainly in the 19th century when people did not know about the existence of elementary particles and believed that any object can be divided by any number of parts. My hope was that physicists should be excited to realize that quantum physics can be based on mathematics involving only finite sets without such unphysical notions as infinitely small and infinitely large. My observation is that, although basics of Galois fields can be taught even at elementary schools, physicists typically are not aware even of those basics as a result of the fact that Galois fields are not taught at Physics Departments. In addition, as I noted above, physicists typically treat mathematics only as a tool for solving physics problems. As a consequence, when a typical physicist encounters some unknown mathematics, he or she has a temptation not to try to understand whether or not this mathematics is more elegant, pertinent etc. but to think that this is an unnecessarily radical departure from the existing approach.

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If an assumption leads to a contradiction one generally concludes that there's a flaw somewhere in the assumption. Zero-probability events (per mathematics) nonetheless do in fact occur (per reality). This contradiction appears to result from the assumption that the concept of infinity is objectively true. Ergo the concept of infinity (and by extension the continuum and an analog reality) looks like it might have a problem. Or is there another possible explanation?

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Author Felix M Lev replied on Dec. 20, 2010 @ 23:20 GMT

In my essay and papers I argue that only a finite mathematics can describe reality. In particular, the notion of probability can be only approximate. Indeed, this notion implies that we should carry out an infinite number of experiments within an infinite time interval. In reality this can be never done and when one says about probabilities, he or she believes that a finite number of experiments gives a value close to a hypothetical limit when the number of experiments is infinite. For example, we can never guarantee that the probability is exactly zero since if some event has not been found even in a very large number of experiments, there is no guarantee that it will not be found in a greater number of experiments.

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Dear Rick P,

can you give an example for

"Zero-probability events (per mathematics) nonetheless do in fact occur (per reality"?

Thank you.

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Anonymous,

Say you spin a fair pointer from a randomly chosen starting position by applying a randomly selected force. In an analog world there'd be an infinity of possible directions for it to end up pointing and so probabilistically it couldn't end up pointing anywhere. (1/Infinity)=0. Yet it does end up pointing somewhere.

Or, per [ http://mathforum.org/dr.math/faq/faq.prob.intro.html ]:

"Note that when you're dealing with an infinite number of possible events, an event that could conceivably happen might have probability zero. Consider the example of picking a random number between 1 and 10 - what is the probability that you'll pick 5.0724? It's zero, but it could happen."

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In probability theory, probability is defined as a measure of sets belonging to a sigma-algebra. Those sets can have only a positive nonzero measure. In physics, a definition of a physical quantity is a description how this quantity should be measured. So a question arises whether in physics it is possible to define probability in accordance with mathematics. Mathematics prompts us that we cannot define such a quantity as “the probability to find 5.0724” since the set containing only the point 5.0724 has measure zero and does not belong to the sigma-algebra. But we can try to define the probability to find a number in some interval. As I noted in the previous note, the only known way of defining probability in physics is that we should carry out an infinite number of experiments within and infinite time interval and this is problematic. Also, in quantum physics probability can be zero if there are superselection rules.

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Steve Dufourny replied on Dec. 28, 2010 @ 16:52 GMT

Hi all,

Congratulations for your beautiful essay dear Felix,The finite groups of Galois are relevants in my humble opinion when we want calculate rationally the quantic number and all its proportionalities.This system has a finite serie at my opinion.

To all, very relevant discussions.Don't stop dear Friends, hhihihi Laplace, Poisson and Gauss shall be happy to see these discussions and they shall say,; don't forget the theory of errors and the dispersions.....a kind of precison and sorting appears in the same rational logic.Like an Occam Raozr applied to maths for rational physics.

That permits to see better the serie towards the Planck scale and its finite number.

The infinity , the 0 and the - must be rationalized in the pure physicality and its pure laws in 3 Dimensions and a time constant of evolution.I d say even ,they doesn't really exist, if we add them yes, but not in our pure uniqueness, and their finite system and their pure number.

We can for example add or multiplicate our cosmological spheres, that doesn't mean that their number changes...their pure number inside an evolutive Unievrse rests like it is.It's the same for our quantum number, we can add or multiplicate them ,their pure number rests.It's a little like a proportional approximation in fact with rational limits.

Regards

Steve

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Author Felix M Lev replied on Dec. 29, 2010 @ 02:23 GMT

Dear Steve,

Thank you for encouraging words about my essay. Some your remarks are not clear to me and, probably, we have different opinions on some issues (e.g. on the role of geometry, whether the theory should be based on finite groups or Lie algebras over finite fields etc.). We could discuss them via email if you are interested. Happy New Year!

Felix.

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Steve Dufourny replied on Dec. 30, 2010 @ 12:26 GMT

Hi all,

dear Felix,

You are welcome.

Indeed we have different points of vue(as many here on FQXi,the sharing of ideas seems essential), but the most important is this universality behind.

I like finite groups, and I think that maths must be analyzed with the biggest rationality when we analyze physics in its details.

I utilize algebras with an ocaam razzor,it exists several methods ,interestings and relevants.I add or superimpose them.

But I don't rest in one method.

In fact lie algebras, Clifford's alg.,.....are interestings when they respect the foundamental theorem of algebras.Now of course the physicality is the physicality.And the number is the number.

I see the quantum entanglement a little as our universe.Now if the entanglement of spheres is specific....the volumes are important and the number is the same and finite as the serie of volumes.The begining is a fractal of the main central sphere.Now I ask me how is the serie between 1 and our number of cosmological spheres.My problem is about the spheres between the center and our planets.And between 1 and 2 and 3........the volumes decrease on a specific harmonous serie.

Yes of course here is mine , a simple google mail.We can speak here you know I am transparent.

Ps sorry for my poor litteral english.

dufournybionature@gmail.com

Regards

Steve

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Steve Dufourny replied on Dec. 31, 2010 @ 16:53 GMT

Hi ,

Dear Felix,

Like I said you in private, I have no publications,it's not my aim.

I just work simply about my theory of spherization,and I improve it.

I haven't finished my universities, in fact I have studied a little of all,I was in medecine, after in geology(there a little problem of neurology apparently,a little coma and a kind of epilepsy),after agronomy,.....and I continued my classments and works.I have even created an enterprize in horticulture and vegetal multiplication(at the age of 23) but apparently I am not skilling in business.thus brankrupcy in 2004,oh my god.well it's the past.

I continue simply my spherization theory.Isolated I agree but that goes .

About your essays, I see a very good knowledge about our foundamentals and its whole,about also our international language about sciences and maths and physics.

I like also your rational pragmatism about our reality.Indedd only precise results are essential.That means a logical method.It's important, thanks for that.At this momment it's rare to see rationalists.

In your conclusion, you say

"We conclude that the very notion of particle-antiparticle is approximate

and the electric, baryon and lepton charges are only approximately conserved quantities.

The non-conservation of the baryon and lepton quantum numbers has been

already considered in models of Grand Unification but the electric charge has been

always believed to be a strictly conserved quantum number. The non-conservation

of these quantum numbers also completely changes the status of the problem known

as ”baryon asymmetry of the Universe” since at early stages of the Universe energies

were much greater than now and therefore transitions between particles and

antiparticles had a much greater probability."

Could you develop a little please why a much greater probability?

If we take the CMD at low energy ....curves of Planck.

At high energy ther origin is not thermic if I can say.

Now let's take the annihilation of matter/anti matter and we see them on graphics with waves lenghts and ray in MeV.

we can take also an other example with RX or gamma R.....See the interactions of relativistic electrons coming from galaxies and the photon in low energy in the cosmological deep sphere.ISOTROPISM

How can we have a correct formalism interpreting a space time quadridimentional if the pseudo euclidian system smiles to Gallilei and Minkowski.I like the evolution and it's a main parameter,Fiedman and Robertson have understood this point of evolution, relativistic.

3 DIMENSIONS AND A TIME CONSTANT BUT THIS SPACE TIME EVOLVES SIMPLY IT IS THE REAL RELATIVITY.

MY ROTATING SPINNING SPHERES ANSXER TO .........GRAVITATION? QUANTIC ALSO....G c and h are linked .why because the sense of rotation has two main senses!!! TO MEDITATE.

Steve

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Dr. Lev ... thank you very much for your thoughtful reply. One problem with being a nice person is that people tend to take advantage of you, as I shall do now. This is a quote from a paper by Brukner and Zeilinger (see cite below). I'm wondering if you agree or not, or agree/disagree in part ...

“Clearly, a number of important questions remain open. Of these, we mention here two. The first refers to continuous variables. The problem there is that with continuous variables, one has in principle an infinite number of complementary observables. One might tackle this question by generalizing the definition of (3.4) to infinite sets. This, while mathematically possible, leads to conceptually difficult situations. The conceptual problem is in our view related to the fact that we wish to define all notions on operationally verifiable bases or foundations, that is, on foundations which can be verified directly in experiment. In our opinion, it is therefore suggestive that the concept of an infinite number of complementary observables and therefore, indirectly, the assumption of continuous variables, are just mathematical constructions which might not have a place in a final formulation of quantum mechanics.

“This leads to the second question, namely, how to derive the Schrödinger equation. ….”

from: "Quantum Physics as a Science of Information" (2005)

http://tinyurl.com/26dwfel

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Author Felix M Lev replied on Dec. 21, 2010 @ 22:30 GMT

Dear Rick P,

Thank you for this reference. If you read my essay you could see that it is in the spirit of these remarks.

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Dear Dr. Lev,

Thanks again. I certainly thought so, which is why I made the connection. There's plenty of material out there to choose from but very little of it anywhere near as apposite. But of course the devil (or God, if you're Mies van der Rohe) is in the nevertheless-not-entirely-spiritual details.

I know that both of these guys are Community members and possible contest voters so not to press you further.

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Dear Felix,

You did a nice exploration and exposition of a possible application of finite fields in quantum theory. Your article on arxiv shows that you developed extensively this idea. Since I do not know any field in fundamental physics which is closed, or at least which accounts for all observations, I think that we should not demand new-born theories to be perfect and answer all questions. Let's let them grow up so that we can really compare them with others which were developed during one century by so many scientists. I think it is good to question them and to compare them with experiment even from the beginning, but I don't think that their value should be judged before their maturity.

I will ask some questions about your essay, if you don't mind. Please, if you feel that we disagree at some points, consider my questions as a proof of interest and curiosity.

Best regards,

Cristi Stoica

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Dear Felix,

I confess that I do not perceive the standard mathematics as being wrong, and the discrete or finite one as the only justified. I don't think I have enough information to decide whether our world is discrete or continuous. This is why I salute both directions of research, and I am interested in the arguments or evidence of each of them.

You said: "Standard mathematics is based on axioms about infinite sets (e.g., Zorn's lemma or Zermelo's axiom of choice), which are accepted without proof. Our belief that these axioms are correct is based on the fact that sciences using standard mathematics (physics, chemistry etc.) describe nature with a very high accuracy."

This triggered in my head the following questions (I would be pleased to receive answers from other readers too):

[?] As far as I know, Zorn's lemma and the axiom of choice are independent of the other axioms in set theory (although they are equivalent for example in Zermello-Fraenkel's system). Would it be possible to interpret one of the known experiments, or to devise a new one, so that we can check if they are valid from our world?

[?] All mathematical physics uses mathematics based on some axioms. Were some of these axioms tested directly, or only through their consequences (predictions)? Would it be possible, at least in principle, a physics based on axioms which are tested directly?

Best regards,

Cristi Stoica

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Author Felix M Lev replied on Dec. 22, 2010 @ 19:35 GMT

Dear Cristi,

Thank you for encouraging words about my works.

I am not saying that standard mathematics is wrong. The question is whether we

i) accept a principle that only those statements have a physical significance, which can be experimentally verified (at least in principle) or ii) we agree that some statements (axioms) can be accepted without proof (for some reasons). Since you pose two questions [?] in your note, you probably think that we should accept i), right? But then we should acknowledge that standard axioms cannot be verified. For example, how can we verify that a+b=b+a for any natural numbers a and b?

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Cristi Stoica replied on Dec. 22, 2010 @ 21:19 GMT

Dear Felix,

could my question "Would it be possible, at least in principle, a physics based on axioms which are tested directly?" be answered positively by a physics based on finite fields?

Could the habitants of a finite universe know everything about their world, just because there is a finite number of things to be known? It seems to me that they are "more finite" than the knowledge about their world. So, I would incline towards the second possibility you mentioned.

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Dear Felix M. Lev,

Mathematics is a blind tool designed mainly to follow physics and describe the products made by physisists, but not a tool to discover something in physics; There are a few examples only when mathematical methods discovered something in physics, but a many thousands of erroneous mathematical papers and false mathematical "proofs". It is dangerous for physics because a lot of people mask their false and erroneous papers under mathematical formulas and mathematical theories.

I have examined some your papers in order to find what this mathematical instrument can really discover in PHYSICS; However, the most of your papers deals mainly with Galois fields. Even in your paper "A POSSIBLE MECHANISM OF GRAVITY" I don't found any physical mechanism of gravitation - mathematics only; Gravity is a manifestation of Galois fields? Can you explain how this manifestation of Calois fields can curve spacetime and slow down time? Also some your papers repeats the same information, for example the figure "Relation between Fp and the ring of integers" I saw in 3 your different papers.

Sincerely,

Constantin

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Marius Buliga replied on Dec. 22, 2010 @ 23:27 GMT

Dear Constantin,

You write: "Mathematics is a blind tool designed mainly to follow physics and describe the products made by physisists, but not a tool to discover something in physics" and so on. This is really nonsense.

I noticed that sometimes the authority argument works better than a more rational one.

Isaac Newton was Lucasian Professor of Mathematics at the University of Cambridge. His contemporary and archenemy Gottfried Wilhelm Leibniz was a mathematician as well. His Monadology, considered to be utterly unphysical, started the research program which ended with the invention of the computer and of the www (namely the http).

Coming back to Lev paper, it is certainly nothing wrong, IN PRINCIPLE, to suggest that a physical phenomenon "is a manifestation of Galois fields", because, look, Newton himself had to invent differential calculus in order to establish his theory.

(Leibniz invented the same mathematical "blind instrument" simultaneously and better than Newton, for reasons which were independent of physics.)

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Author Felix M Lev replied on Dec. 23, 2010 @ 02:16 GMT

Dear Constantin Leshan,

In my understanding, you think that my works contain only mathematics but not physics, right? For example, in my essay I describe the following results of my works. In contrast with standard theory, in my approach based on finite fields there are no independent irreducible representations (IRs) for a particle and its antiparticle but one IR describes an object such that a particle and its antiparticle are different states of this object. As a consequence, there are no neutral elementary particles, the electric, baryon and lepton charges can be only approximately conserved and even the notion of particle-antiparticle is only approximate. In your opinion, these results are only mathematical or they have something to do with physics?

The problem of describing gravity is out of the scope of this contest; we could discuss this problem via email if you are interested.

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Cristi Stoica replied on Dec. 23, 2010 @ 07:45 GMT

In addition to what Marius said, I would mention:

- Riemannian geometry preceded its applications to general relativity

- Hilbert spaces preceded their applications to quantum theory

- Clifford algebras and spinors preceded their applications to relativistic quantum theory

- connections on fiber bundles preceded their applications to Yang-Mills theory

- holonomy groups preceded their applications to gauge theory, to Wilson loops and to loop quantum gravity

- representations of Lie group preceded their applications to particle physics

- topos theory preceded its applications to quantum theory obtained by Chris Isham

- the particular Kaehler manifolds named Calabi-Yau manifolds preceded their applications to string theory

I agree that mathematics originated from practical necessities, which come from the physical world. But mathematicians are playful species, and they like to explore platonic worlds as well. For some reason, their explorations anticipated many of the necessities of physics. Or maybe physicists find easier to borrow from mathematics, rather than making their own tools ;-). Or when they do, the tools are often full of divergences and singularities, and are inconsistent. As John Baez said once, it is the job of mathematicians to eliminate these inconsistencies. So I would say that both physicists and mathematicians have their equally important role.

___

Lebniz's monadology finds also applications in Haskell (programming language). It also influenced Whitehead, and through him some applications to quantum theory.

Best regards,

Cristi

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Dear Marius Buliga and Cristi Stoica,

I agree that there are a few examples of successful predictions in physics made by mathematics, against the thousands of wrong predictions on the other hand; I can show that about 70 percents of all theoretical papers made by mathematicians in physics are wrong. For example, about hundreds of different theories of gravitation have been published in the academic journals, but it is self-evident that one or two similar gravitational theories only can be true at the same time, but not hundreds of theories. It is self-evident that 99 percents of all published gravitational theories are erroneous. Meanwhile all these erroneous theories have the "PERFECT MATHEMATICS" and "mathematical proofs" and are accepted by peer reviewed journals and physics community. Also the same situation arise in other areas of physics: about 70 percents of all theoretical papers in physics made by mathematicians are wrong. For example, let us analyze this paper this paper published in Physical Review Letters: I found tens of errors here whereas this paper has been supported by Physical Review Letters and NASA. You see, the authors try to prove their erroneous papers by help of mathematics; Thus, the mathematical proofs in physical theories must be in doubt. All the Standard Model is a mathematical model only that can compute only but explain nothing. The invasion of mathematicians will stop the development of Physics. That situation arise because peer reviewed Journals accepts papers with mathematical content only. Journals should accept that a physical logic (reasoning) must have equal rights with mathematical proofs.

Dear Felix M Lev

Yes, your mathematical approach tries to describe the EXISTING already physical phenomena only. Your results confirm my point of view that mathematics must be a tool to describe quantitatively products (theories and phenomena) made/discovered by physicists only, but it is not a indicator for physicists what they must do. Because physics is directed by mathematics, it is a cause of modern crisis in physics.

Sincerely,

Constantin

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Cristi Stoica replied on Dec. 23, 2010 @ 13:16 GMT

[I apologize to Felix, I do not want to monopolize this thread. I would kindly ask a FQXi admin who validates the comments to move the discussion to a different thread, if it is off topic. Or perhaps to move this comment and the father comment as children to the discussion opened by Constantin Leshan, so that the discussion gets collapsed and does not occupy too much of this page.]

Dear...

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Constantin Leshan replied on Dec. 23, 2010 @ 17:54 GMT

Dear Cristi Stoica,

The border between mathematicians and physicists is very thin; therefore by papers of mathematicians I mean papers were the percentage of mathematics is more than 40 of volume, or all proofs are mathematical; From this point of view, almost all gravitational theories are mathematical theories, made by mathematicians. Also, I do not accuse all mathematical world; I say that all false physical theories have mathematical proofs, consequently mathematical proofs in physical theories must be in doubt. Therefore, it is mathematics' fault that the most of published physical papers are wrong.

It is because journals accept papers with mathematical content only. I'm sure that the percentage of false theories may fall, if the journals allow physical reasoning instead of mathematical proofs. Thus, Journals must allow to physicists to publish their papers with physical proofs instead of mathematical. Since the Standard Model is more mathematical model than physical, therefore I accuse mathematics. We'll never find any Higgs boson because SM is a mathematical model only that may fall in nearest future. The future Physics will be based on physical reasoning rather than on mathematical proofs.

About moving this comment to the discussion opened by Constantin Leshan: soon I'll send my essay to FQXI. I invite you to find logical errors in my theory.

Sincerely,

Constantin Leshan

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Cristi Stoica replied on Dec. 23, 2010 @ 19:31 GMT

Dear Constantin,

I am sympathetic to your viewpoint that these day physics is too much math without physical content. I stop now because I took too much space from Felix with my off topic comments. We can continue by email. Good luck with your forthcoming essay.

Cristi

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I like the idea of using the Galois fields in physics. Are you aware of the result that it is possible to construct a complete set of mutually unbiased basis for finite dimensional quantum systems if and only if the base is indexed by a Galois field?

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Author Felix M Lev replied on Dec. 31, 2010 @ 01:19 GMT

A direct generalization of mutually unbiased bases to the case of Galois fields is meaningless for several reasons. For example, ½ in Galois fields is (p+1)/2, i.e. a huge number if p is huge. In standard theory, probabilities are normalized to one but this is only a matter of convention since not the probability itself has a physical meaning but only ratios of probabilities of different experimental outcomes have (that’s why Hilbert spaces in quantum theory are projective). In addition, as noted in my essay and papers, in theories over Galois fields the notion of probability can be only approximate. However, I believe that in situations when probability is meaningful, it is not difficult to modify the definition of mutually unbiased bases such that the main idea of the definition will be implemented. But the question that the base should be indexed by a Galois field is not clear to me. For example, when we have a finite dimensional linear space over a Galois field, we don’t say that the basis elements are indexed by a Galois field, right?

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I read your paper with interest. I do have a couple of questions about the idea of Galois QFT. The cyclotomic numbers of F_4 z = e^{i2πn/3} describe the root space of D_4. F_4 is the Dynkin diagram for D_4 ~ SO(8). The D_4 root lattice is the dual of the F_4 and a subring of Hurwitz quaternions. In this way Galois groups can characterize the symmetries of a QFT, or a YM gauge theory.

Cheers LC

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Lawrence B. Crowell replied on Jan. 17, 2011 @ 19:36 GMT

PS

I realized I used F_4 in two different contexts. At first I use F_4 as the Galois field, but then in reference with the D_4 root lattice I am referring to the exceptional F_4.

Cheers LC

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Author Felix M Lev replied on Jan. 18, 2011 @ 06:52 GMT

Thank you for your interest to my essay. Here I argue that any fundamental physics can be based only on a finite math and consider an approach based on a Galois field. So I cannot have e^{i2πn/3} or SO(8). Also, I cannot use Dynkin diagrams for describing Lie algebras over Galois fields since the latter are not algebraically closed. Your first F_4 is not a Galois field; probably you mean a Galois group. Galois groups are used for describing field extensions and in general the fields are not assumed to be necessarily Galois ones. Since you are talking about cyclotomic numbers, you probably mean extensions of Q. So, in my understanding, your questions refer to standard theory but not to my approach.

Best regards, Felix Lev.

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Steve Dufourny replied on Jan. 18, 2011 @ 11:10 GMT

Hi to both of you,

Happy dear lev to see this rationality, indeed the finite systems must respected their own limits.

We can superimpose but with rationality of course.

Finite maths....galois field.very relevant indeed , very relevant.

Steve

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Dear Lev,

in your essay you make the clear point that finite mathematics (such as GFQT) is the most pertinent choice for describing physical reality.

However, you do not seem to take an equally clear position about the ultimate nature of reality: is the universe (discrete and) finite or infinite? In fact, are both possibilites still open, under GFQT?

Do you perhaps envisage a third possibility, namely that GFQT works very nicely for just making accurate experimental predictions in a QM setting, without still resolving this finite vs. infinite universe puzzle, which is perhaps only of philosophical relevance?

If the question sounds indeed too philosophical (but that's essentially the title of the contest...), I could reformulate it as follows: would your theory be compatible, incompatible, or neutral, with a statement such as 'there are 10^234 atoms of spacetime in the universe'?

A second question. You talk about parameters p and n, defining the size of the GF, as universal constants. Would it make any sense to rather imagine them as changing, I mean on a cosmological scale?

Thanks!

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Author Felix M Lev replied on Jan. 19, 2011 @ 06:05 GMT

Dear Tommaso,

Thank you very much for your very important questions. Probably I need a few days to describe what I think.

Best regards, Felix Lev.

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Author Felix M Lev replied on Jan. 21, 2011 @ 08:19 GMT

Dear Tomasso,

I will try to answer your questions.

First of all, let me note that in my understanding, the question “Is Reality Digital Or Analog?” is meaningful only if it is understood as a question about mathematics describing reality. Some contest participants argue that e.g. mathematics might be continuous but physics - discrete but I don’t understand such arguments. In my essay...

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Dear Tomasso,

I will try to answer your questions.

First of all, let me note that in my understanding, the question “Is Reality Digital Or Analog?” is meaningful only if it is understood as a question about mathematics describing reality. Some contest participants argue that e.g. mathematics might be continuous but physics - discrete but I don’t understand such arguments. In my essay I argue that if we accept a principle that only those statements are meaningful, which can be experimentally verified (at least in principle) then only a finite mathematics can describe reality. We have no experience in this field and so nothing can be stated for sure.

But if indeed only a finite math describes reality then it is reasonable to think that our Universe is finite. Indeed, in finite systems, consistent calculations can be performed only modulo some number. So if we find effects which can be explained only by finite math then it will be a strong argument that the Universe is finite. You mention a possibility that the Universe is infinite but a finite math gives a good description of reality in some areas.

Probably this possibility is not realistic since if the Universe is infinite then it is not clear why physics is described by a finite math, but of course our experience is not sufficient and maybe for some reasons this scenario takes place.

If we try to construct a quantum theory based on a finite math then probably many possibilities can be investigated. For example, I argue that standard division has a limited meaning but I also do not see why division in Galois fields has a fundamental meaning. As I note in my essay, Metod Saniga believes that a theory based on a finite ring is even more interesting. But technically it is convenient to work with a field; for example, a well known result in algebra is that the dimension of a linear space is well defined only if the space is over a field or body. I show that a case of the field with p elements contradicts experiment and so the field should be extended. A simplest extension is a Galois field F(p^2) with p^2 elements but of course we cannot exclude a possibility that there are reasons for a Galois field version of the theory where the field is more complicated that F(p^2) and the latter is only an approximation.

It is easy to show (see e.g. pp. 5 and 6 of my essay) that there exists a correspondence principle between projective complex Hilbert spaces and projective spaces over Galois fields F(p^2). However, even in this case GFQT and standard theory are considerably different. For example, in GFQT one irreducible representation of the symmetry algebra describes a particle and its antiparticle simultaneously and there are no neutral elementary particles (so even the photon cannot be elementary). These problems are discussed in [11], which can be found at http://www.mdpi.com/2073-8994/2/4/1810/ . I have also considered a hypothesis that gravity is simply a manifestation of finiteness of physics and this work is underway (see e.g. http://xxx.lanl.gov/abs/0905.0767 ). In summary, the key problem is to construct a theory relating GFQT with experiment and this cannot be done simply by analogy with standard theory. If this problem is solved then it will be clear whether nature is described by finite math or not.

You also ask whether p is a fundamental constant or it may change with time. In a wider context, I am aware of different opinions on GFQT. Mathematicians sometimes say that a version with only one Galois field is not attractive since it is not clear why a special value of p is chosen. For this reason some of them prefer an adelic approach but in this case we do not have finiteness. On the contrary, physicists typically believe that no new fundamental constant p is needed. I believe that this is rather strange since history of physics tells us that new theories arise when a parameter, which in the old theory was zero or \infty becomes finite. Indeed, classical physics has no parameters at all but relativity arises when c is not infinite but finite. Analogously, quantum theory arises when \hbar is not zero but finite. So I believe that it is rather attractive that GFQT arises when p is not infinite but finite. My assumption is that p is the characteristic of our Universe since it defines the laws of physics. If this is reasonable then p is a fundamental constant and it is reasonable to believe that it does not change with time. I believe that such dimensionful quantities as the gravitational and cosmological constants are not fundamental and it’s reasonable to think that they are changing with time. The arguments and my opinion on c and \hbar are given in [9], which can be found at http://www.mdpi.com/2073-8994/2/4/1945/ .

Finally, as far as your question about 10^234 atoms of spacetime is concerned, I would like to note that I fully agree with Heisenberg and others that a fundamental physical theory should not involve spacetime at all. A detailed description of my point of view can be found in [9,11].

Best regards, Felix.

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Steve Dufourny replied on Feb. 28, 2011 @ 15:16 GMT

If you work together, Lawrence, Tommasi and Hector and Florin...you shall ponder very interestings things...

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De brogle maths togther with Einxstins dice programmed to obey the rules 1 ODD THROW+ 1 EVEN THROW= 2 ODD THROWS.

And 2 ODD THROWS+ 2 EVEN THROWS= 4 EVEN THROWS.

Can be used to generate a simulated quantum universe on computer where everything is determined.

And we cna develop an algorythm to predict random numbers in our real universe.

This maths apples before the big bang where foru states are one and this determines the fact that the four forces are one.

Everthing is determined when the four forces are one.

And Einsteins thoery determines everything that happens before the big bang................

Do you appreciate this contribution let me know.

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Dear Felix ,

I will not download your paper to read as I can see from your discussion with Florin that it is concerned with the mathematical description of reality rather than the physics of what is real.I have read your argument that it is a valid approach to the essay question. As I have no mathematical background and am not even a physics specialist I could not begin to comprehend what you have written in your essay. Likewise I would be unable to decipher a physics paper written by Lawrence Crowell.

With respect Sir can I just ask, Is reality digital or analogue? Did you come to a conclusion? or did you argue something entirely different?

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Georgina Parry replied on Feb. 14, 2011 @ 08:57 GMT

Dear Felix,

I had not read the entire content of your thread, which is now rather lengthy. On looking to see if you had posted a reply to my question I noticed your reply to Tomasso above, which actually does answer my question very well.

You have said "First of all, let me note that in my understanding, the question "Is Reality Digital Or Analog?" is meaningful only if it is understood as a question about mathematics describing reality." Which is probably because you are mathematician and think like a mathematician. Which is not a criticism but a possible reason for your particular thinking style and approach to the problem.

I actually agree that all we can do is create models which we hope describe reality as we can not fully construct the reality itself or know what lies beneath our mental interpretation and the descriptions created by human minds, verbal or mathematical.

I can see that you have given a full explanation to Tomasso. I can not grasp all of it but the last paragraph caught my attention. You say you agree with Heisenburg that a fundamental physical theory should not involve space-time at all. I agree that at the most foundation level space-time does not exist but that it is an emergent reality produced subsequent to interception of data by an observer, whether that is a conscious entity or an inanimate reality interface such as a camera or other recording device.

I wish I was able to discuss your essay in more detail. I have found some of the conversation in your thread most interesting.

Kind regards and good luck, Georgina.

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Author Felix M Lev replied on Feb. 14, 2011 @ 19:04 GMT

Dear Georgina,

Thank you for your interesting remarks. The notion of spacetime is now one of the most debated questions of modern physics. I need probably a few days to describe my understanding of this notion.

Best regards, Felix.

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Author Felix M Lev replied on Feb. 16, 2011 @ 07:13 GMT

Dear Georgina,

Thank you for your interest to my essay. First of all, let me note that I am not a mathematician since am not working on mathematical theories. Mathematicians work with theories based on sets of axioms; typically they don’t discuss how their theories apply to reality. But physics cannot be without math. In my essay I argue that any fundamental physical theory can be based...

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We conclude that the very notion of particle-antiparticle is approximate

and the electric, baryon and lepton charges are only approximately conserved quantities.

The non-conservation of the baryon and lepton quantum numbers has been

already considered in models of Grand Unification but the electric charge has been

always believed to be a strictly conserved quantum number. The non-conservation

of these quantum numbers also completely changes the status of the problem known

as ”baryon asymmetry of the Universe” since at early stages of the Universe energies

were much greater than now and therefore transitions between particles and

antiparticles had a much greater probability.

Felix,

This quote from your essay seems to support the analogue nature of particles, the inclination to change energies and nature like the electron neutrinos changing to tau neutrinos after emerging from the sun's fusion.

Your argument is esoteric, yet convincing, especially to the mathematically challenged like myself.

Jim Hoover

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Author Felix M Lev replied on Feb. 23, 2011 @ 18:10 GMT

Jim,

It is not clear to me why, in your opinion, this quote from my essay "seems to support the analogue nature of particles". As shown in the essay, there is a correspondence principle between my approach and standard theory. In particular, there are no obstacles for recovering the results on neutrino oscillations in my approach.

Felix.

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Felix,

Fascinating essay. You give me new hope that a true algebraic theory can subsume field representations.

I hope you get a chance to read my entry as well.

Good luck!

Tom

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Author Felix M Lev replied on Mar. 8, 2011 @ 21:54 GMT

Tom,

Thank you for your compliments. The ideas that the ultimate theory will not be based on local fields were very popular in 60th (e.g. the Heisenberg S-matrix program) but now those ideas are almost forgotten.

I read your essay and tried to understand your position. You quote many well known scientists who had different opinions. However, so far I could not understand what your preferences are. Probably a more careful reading is required.

Felix.

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Felix,

Briefly, my personal preference is for quantum field theory in a continuous function model. I find this possible only in an extradimensional theory. I discussed my own preference only in the technical endnote, because my intent in the essay was to survey how subtle the question of continuous vs. discrete really is.

I agree with you on the problematic nature of Zorn's lemma (axiom of choice). On that issue, you might be interested in my ICCS 2006 paper in which I show (see particularly 5.6 - 5.9) how a well ordered sequence is derived without appeal to AC.

Best,

Tom

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T H Ray replied on Mar. 11, 2011 @ 14:38 GMT

Fixing link:

ICCS 2006

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Author Felix M Lev replied on Mar. 12, 2011 @ 07:24 GMT

Tom,

Thank you for your explanations.

Good luck, Felix.

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Sub: Possibility of manipulation in judging criteria – suggestions for improvement.

Sir,

We had filed a complaint to FQXi and Scienticfic American regarding Possibility of manipulation in judging criteria and giving some suggestions for improvement. Acopy of our letter is enclosed for your kind information.

“We are a non-professional and non-academic entrant to the Essay...

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