Dear Felix,

Sorry for the delayed answer, I was caught up in a lot of work recently. Let me start by clarifying my intention. First and foremost I am interested in understanding your approach because I work in a different number system for QM myself. Second, I am interested in your essay entry. Let me repeat that I find your essay interesting, otherwise I would not spend my time trying to understand your ideas. Also you have strong claims, and strong claims deserve strong scrutiny, IMHO.

I am puzzled by your statements: "Let me also repeat that if I understand you correctly and you are interested only in discussing physics then the present forum is not an appropriate place for this discussion. " and "I believe that my essay is fully in the spirit of this essay contest entitled "Is Reality Digital or Analog?" So I discuss mathematics." First, this is a physics contest and FQXi is mostly a physics organization. Second, mathematical statements without physics support are irrelevant to deciding if nature is digital or analog. Mathematical (or any other kinds of) statements without agreement with reality are just marks on paper.

To me, discussing from the physics point of view it is the only thing which makes sense and interests me. But if you find this inappropriate, I will respect your wishes and not continue to ask questions. But if you want to continue the physics discussion, I am available.

5 days later

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

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.

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

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?

    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?

    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

    Dear Florin,

    I have no problem in discussing any aspect of my approach; the only problem I have is whether the discussion is in the framework of this forum. If you think it is, let's continue and if we have a feeling that it goes out of the scope of this contest, we could continue the discussion via email if are interested.

    I appreciate that you are interested in my approach. Indeed I have strong claims and agree with your statement that "strong claims deserve strong scrutiny". So I would be grateful for any criticism. My expectation is that first of all any criticism should say whether or not MY RESULTS are correct and physical. However, so far you did not discuss my results at all. You expressed your opinion that the string theory, or something like that have greater chances to solve the existing problems and that so far you see the application of my approach only in the Landau pole problem. These questions fully satisfy a principle that "De gustibus non disputandum est" and here I don't see any point for discussion. So I propose to discuss the results of my approach.

    You advised me to calculate the 8 digits in my approach. Many years ago it was indeed one of my main motivations. The idea was to replace irreducible representations (IRs) for the electron, positron and photon by their modular analogs and obtain a finite theory. However, this naïve attempt fails for the following reason: in a theory over a Galois field, one IR describes a particle and its antiparticle simultaneously and there are no IRs for neutral particles. In particular, even the photon cannot be elementary. Could you, please tell me your opinion on this situation? For example, a possible point of view is that if even a photon cannot be elementary then GFQT is unphysical.

    Happy New Year! Felix.

    17 days later

    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

      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.

      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

      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!

        Cyclotomic numbers are roots of a cyclotomic field, as I recall the term, but you are right it is a Galois extension of the rational numbers.

        I was wondering if there is a connection to standard quantum field theory. It seemed to me there should be an underlying Lie group theory with this.

        Cheers LC

        Dear Steve Dufourny,

        Thank you for your comments. I wish you success in developing your approach.

        Best regards, Felix Lev.

        ==========================================================

        Dear Lawrence B. Crowell,

        In standard theory (over C) we typically have a one-to-one correspondence between representations of Lie algebras and representations of corresponding Lie groups (for finite dimensional representations this is well known; for infinite dimensional ones this is usually a case for reasonable assumptions). Representations of Lie algebras by Hermitian operators have a clear physical meaning since the representation operators describe physical quantities. At the same time, one might ask whether all representation operators for representations of Lie groups are meaningful. For example, are temporal translations by 10^{-1000}sec. or spatial translations by 10^{-1000}m meaningful? However, in the theory over complex numbers we must agree that they are meaningful since we have infinitesimal transformations and exp and so we can construct any representation operator of a Lie group from a representation operator of its Lie algebra.

        However, in a theory over Galois fields, we don't have infinitesimal transformations and exp. So in this case there is no notion of a Lie group over a Galois field corresponding to a Lie algebra over a Galois field. I believe that this by no means indicates that a theory over Galois fields is unphysical. In many cases we can have finite transformations, which approximate transformations over C with a high accuracy. However, the absence of Lie groups considerably changes the theory. In my essay and papers these problems are discussed in details.

        Best regards, Felix Lev.

        So even if we assume there are no infinitesimal gauge transformation does a GQFT recover something of that type of physics in the limit things are continuous? I generally like new theories which recover known or standard theory in some limit.

        Cheers LC

        A necessary condition for any new theory to be reasonable is the existence of a correspondence principle between this theory and the conventional one. Well known examples are the correspondence principles between classical and relativistic theories when c->\infty and classical and quantum theories when \hbar->0. On pages 5 and 6 of my essay I give a simple explanation that if the characteristic p of the Galois field is large then any element of the projective complex Hilbert space can be approximated with any desirable accuracy by elements of a projective space over a Galois field with p^2 elements. In other words, there exists a correspondence principle between standard theory and GFQT in a formal limit p->\infty.

        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.