What Mathematics Is Most Pertinent For Describing Nature? by Felix M Lev

Dear Steve Dufourny,

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

Best regards, Felix Lev.

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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.

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.

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.

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.

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 only on a finite math. In other words, I believe that reality is not only discrete (digital) but even finite.

I read your essay and tried to understand your approach to spacetime. Probably our approaches have much in common since you do not accept that spacetime is fundamental. But in my understanding you accept that spacetime can be an emergent notion and here we have disagreements. Let me describe my understanding of spacetime.

In physics there is a principle that a definition of a physical quantity is a description of how it should be measured. In quantum theory this principle is formalized by requiring that any physical quantity can be discussed only in conjunction with the operator defining this quantity. When we have an elementary particle or a macroscopic body, we can define operators charactering them; some of the operators can be called the coordinate operators and we can discuss whether the coordinates can be measured with a sufficient accuracy etc. But the notion of spacetime has nothing to do with coordinates of real bodies. The assumption is that spacetime is a manifold, which exists even if there are no bodies at all. It is obvious that the notion of spacetime fully contradicts the above principle since spacetime exists only in our imagination and is not measurable. In particular, a discussion whether the empty spacetime can be curved or flat has no physical meaning; in particular, the cosmological constant problem is not a problem at all [9]. However, the absolute majority of physicists accept spacetime. Their argument is that although spacetime is not measurable, it is only an auxiliary tool for constructing equations of motion for real bodies in General Relativity (GR) or Hilbert spaces in Quantum Field Theory (QFT) and since those theories in many cases give an excellent agreement with the data, this proves that spacetime is meaningful. In other words, a question is whether nonphysical notions can be used at intermediate stages of constructing physical theories.

I believe it is obvious that the notion of spacetime reflects our macrocopic experience that everything is continuous, can be divided into any number of parts etc. Physicists used this notion when they did not know about elementary particles, that matter is discrete, cannot be divided into any number of particles etc. One can say that we are using this notion since we don't have another math. As shown in my essay and papers, we do have another math, which is not using continuity, the notion of infinitely small etc. but can be used in physics.

Several authors treated GR as a theory where spacetime is replaced by a reference frame. For example, in a well known textbook by Landau and Lifshits "Classical Field Theory", the reference frame in GR is defined as a collection of weightless bodies, each of which is characterized by three numbers (coordinates) and is supplied by a (weightless) clock. Such a notion (which resembles ether) is not physical even on classical level and for sure it is meaningless on quantum level.

In 60th, the majority of quantum physicists came to a consensus that any future fundamental theory should not involve spacetime at all (in the spirit of the Heisenberg S-matrix program). In the introductory section of the well known textbook [8] the authors argue that local quantum fields and Lagrangians are rudimentary notions which will disappear in the future theory. Nobody has refuted those arguments but in view of successes of QCD and electroweak theory physicists returned to QFT. In string theory the notion of spacetime is used even in a greater extent than in QFT. Here it is discussed whether spacetime has 10, 11 or 26 dimensions; physics is defined by a choice of a Calabi-Yau manifold at Planck distances etc. I believe it is rather obvious that manifolds, geometry, topology, differential equations etc. have arisen from our macroscopic experience. For example, the water in the ocean can be described by equations of hydrodynamics but we know that this is only an approximation since matter is discrete. There is no reason to believe that continuity, geometry, topology etc. work even at Bohr distances, to say nothing about Planck distances.

In my papers I argue that theory should start not from spacetime but from a symmetry algebra. The idea is simple (in the spirit of Dirac's paper [13]): each system is described by a set of independent operators and they somehow commute with each other. By definition, the rules how they commute define a Lie algebra which is treated as a symmetry algebra. For example, if we choose a Poincare or de Sitter algebras then in quasiclassical approximation we obtain a description equivalent to that obtained from a four-dimensional spacetime. When we choose the de Sitter algebra, we first do not have the de Sitter space, Riemannian geometry, metric, connection etc. However, as shown in [9], in quasiclassical approximation we recover standard results of GR for the motion of particles in the de Sitter space.

My impression is that, although very slowly, physicists are returning to ideas of 60th that fundamental physics should not involve spacetime at all; several physicists note that on quantum level spacetime is meaningless etc. But the number of such physicists is very small.

My impression from your essay is that in general you accept the above ideas. However, there is a trend that spacetime is not fundamental but emergent. I tried to understand those papers; maybe I am wrong but my impression is that for physicists it is very difficult to abandon spacetime at all. Some of them may accept that spacetime might be discrete or emergent but for me it is not clear why we need spacetime at all. Again, maybe I am wrong, but it seems to me that when you are talking about the emergent spacetime, you in fact mean measurements of coordinates of real bodies; so maybe this is only a question of terminology but the coordinates of real bodies have nothing to do with spacetime coordinates.

I would appreciate your comments. Thank you.

Best regards, Felix.

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.