Towards a Grand Unified Theory of Mathematics and Physics by Peter Woit

Dear Peter Woit:

In your essay you wrote, "By the time of Wigner's 1959 talk, quantum mechanics and the theory of group representations had developed far beyond the initial insights of the 1920s, with a myriad of close connections between the two subjects." What role do the monster group and the 6 pariah groups play in the theory of group representations as it relates to quantum mechanics? What might be the most important connection between quantum mechanics and the monster group? Are there 6 quarks because there are 6 pariah groups? In your essay, why did you omit any reference to Milgrom? What are the objections to the following ideas?

MILGROM DENIAL HYPOTHESIS: The main problem with string theory is that string theorists fail to realize that Milgrom is the Kepler of contemporary cosmology. (If Milgrom's acceleration law were empirically invalid then there is no way whatsoever that Milgrom could have convinced McGaugh and Kroupa. One might object that Kepler is sui generis but I regard that objection as a minor quibble.) Why, specifically, might the Milgrom Denial Hypothesis be wrong? Consider some speculations based upon Milgrom's work.

String theorists might have attacked the wrong target -- they hope to generalize Heisenberg's uncertainty principle to a stringy uncertainty principle involving hbar and alpha-prime. However, Milgrom's MOdified Newtonian Dynamics (MOND), the space roar, and the photon underproduction crisis suggest that Newton-Einstein gravitational theory is significantly wrong and the following 5 ideas are correct: (1) The law of conservation of gravitational energy is wrong (in the Newtonian approximation) because dark matter does not consist of particles. (2) Newton's third law of motion is wrong (because gravitons have a nonzero probability of exiting our universe). (3) Newton's law of gravity is significantly wrong because Milgrom's MOND is empirically valid. (4) Einstein's general relativity theory is significantly wrong because of MOND, the space roar, and the photon underproduction crisis. (5) Einstein's equivalence principle is contradicted by dark matter and dark energy, because dark matter has positive gravitational mass-energy and zero inertial mass-energy while dark energy has negative gravitational mass-energy and zero inertial mass-energy.

String theory with the finite nature hypothesis implies the Fernández-Rañada-Milgrom effect, the Space Roar Profile Prediction, and the 64 Particles Hypothesis (because string vibrations are confined to 3 copies of the Leech lattice).

String theory with the infinite nature hypothesis implies that the string landscape exists in some form, that superpartners exist in some form, and that Einstein's general relativity theory is correct in the limit as Planck's constant approaches zero. (MY FUNDAMENTAL GUESS: Newton-Einstein gravitational theory is significantly wrong because Newton-Einstein gravitational theory implies that Milgrom's MOND is false; therefore, string theory with the infinite nature hypothesis is false. If my fundamental guess is wrong, then superpartners with some extremely weird new physical hypothesis should explain why Newtonian-Einstein gravitational theory is 100% correct but appears to be slightly wrong as PROVED BEYOND A REASONABLE DOUBT by the work of Milgrom, McGaugh, Kroupa, and Pawlowski.) -- David Brown

General relativity is sometimes cited as a physical advance without much experimental help, as you say. But it did have some: special relativity was driven by experimental evidence, and all that implied that we needed a relativistic theory of gravity. And an early goal of the general relativity work was to explain the Mercury orbit anomaly.

Why should there be "deep and unexpected unity between the fundamental structures of mathematics and of physics"? Theoretical physics starts with postulates, and if some of them are false (as is, for instance, Einstein's 1905 constant-speed-of-light postulate), the respective models are absurd ("not even wrong").

Pentcho Valev

Dear Dr. Woit,

your essay is very interesting, and I enjoyed reading it. However, it seems to me that you overstate the notion of the standard model as the victim of its own success, and therefore underestimate the prospects for going beyond the standard model. Consider the following: (1) in an abelian gauge theory, the charges can take any value, whereas in a non-abelian gauge theory the charges are integer multiples of a fundamental charge. It therefore seems mysterious that in the standard model the U(1) charge happens to have just the value needed to give the electron and proton equal and opposite electric charges. The most likely solution to this mystery is that the standard model arises through spontaneous symmetry breaking from some grand unified theory. Thus we should be fairly confident that there is some unified theory, and the question then becomes which unified theory is the correct one. (2) In the standard model, neutrinos are massless, but we now have plenty of experimental data on neutrino masses and mixing angles through neutrino oscillation experiments, so there is experimental data that goes beyond the standard model. (3) Standard model matter makes up only about 5% of the density of the universe, with the rest being dark matter and dark energy. It is not clear how to come up with a good theory of dark energy or how to test such a theory. However, dark matter is likely to be a particle, and there is at least a decent chance that that particle will be detected soon. If so, that would be another set of experimental data that go beyond the standard model. (4) The standard model does not include gravity. While experimental gravitational phenomena are well described by classical (i.e. non-quantum) general relativity, the cosmic microwave background B modes that the BICEP2 experiment at first claimed to detect, are due to quantum fluctuations of the gravitational field. If someone ever does detect B modes, this too will be experimental data beyond the standard model.

Dear Dr. Woit,

If new mathematical methods are imported into physical theory, one surely expects that they will prove useful for empirical predictions (cf. matrix mechanics in the 1920s). The ideas from geometric Langlands and various insights into QFT may yield a better understanding of the mathematics, but where should we expect their empirical consequences to show up? What sort of observables, measurements, constants in QM or QFT will, according to you, let us feel this impact?

Best wishes,

Alexei Grinbaum

Dear Peter Woit,

Somehow, Wigner's references to invariance and very simple methods of experimenting with it, his statement that "physics would be impossible if there were no phenomena which are independent of all but a manageably small set of conditions" always made me think that he refers to the overall, general power of mathematics in physics, which is actually the idea on which I framed my essay. Moreover, the feeling of timelessness given by the inability to pinpoint a date at which his essay was written without knowing it in advance can make one forget when his ideas were conceived. Now, after having read your insightful analysis and after things have been put into the right historical perspective, I must agree that, as you say, his discourse must have been marked more by the fundamental discoveries that were very fresh in his time and his generalization did not necessarily imply that he was as interested in the descriptive power of math for all (simpler) physical systems.

That being said, I would like to thank you for a very enjoyable read, which to be honest I expected your essay to be since you mention the Langlands program, a recurrent theme on your blog. Should you ever decide to write another book, I strongly hope you will dedicate at least a portion of it to speak about your vision and knowledge of the topic.

Dear Peter Woit,

You wrote, "In this essay I'll argue that unified theories of fundamental physics are closely linked with some of the great unifying structures that mathematicians have found to underlie much of modern mathematics." If nature is infinite, then G2, F4, E6, E7, and/or E8 might be the basis for a unified theory. If nature is finite, then the monster group and the 6 pariah groups might be the basis for a unified theory. The space roar and the photon underproduction crisis suggest that nature might be finite (or at least that our universe has a finite wavelength and undergoes cycles of expansion and instantaneous quantum collapse).

In "Is String Theory Even Wrong?", "American Scientist", March-April 2002, you wrote: "... string theory predicts that the world has 10 space-time dimensions, in serious disagreement with all the evidence of one's senses. Matching string theory with reality requires that one postulate six unobserved spatial dimensions of very small size wrapped up in one way or another. All the predictions of the theory depend on how you do this, but there are an infinite number of possible choices, and no one has any idea how to determine which is correct." Your objection does not apply to my physical interpretation of string theory. Suppose that string vibrations are confined to 3 copies of the Leech lattice. Imagine 36 different quarks moving in 36 different particle paths. These 36 dimensions might be approximately isomorphic to 26 dimensional bosonic string theory with 10 dimensions for a general relativistic model. Such an approximation scheme might yield a physical interpretation of string theory that takes the place of curling up of extra spatial dimensions. The idea is to replace supersymmetry with some version of Wolframian pseudo-supersymmetry.

"One can argue that Einstein's successful development of general relativity was an example of this. Little help came from experiment, but a great deal from mathematicians and the powerful new formalism of Riemannian geometry."

The mathematicians had to change and fudge the equations countless times until "excellent agreement with observation" was reached:

Michel Janssen: "It is not hard to imagine Einstein's excitement when he inserted the numbers for Mercury into the new expression he found and the result was 43", in excellent agreement with observation."

Other predictions turned out to be absurd. For instance, according to Newton's emission theory of light, the speed of falling photons INCREASES like the speed of ordinary falling bodies - their acceleration in the gravitational field of the Earth is g. According to Einstein's general relativity, the speed of falling photons DECREASES - their acceleration in the gravitational field of the Earth is (-2g) - which is more than absurd of course:

Relativity 3 - gravity and light

"Einstein wrote this paper in 1911 in German. (...) ...you will find in section 3 of that paper Einstein's derivation of the variable speed of light in a gravitational potential, eqn (3). The result is: c'=c0(1+phi/c^2) where phi is the gravitational potential relative to the point where the speed of light co is measured. Simply put: Light appears to travel slower in stronger gravitational fields (near bigger mass). (...) You can find a more sophisticated derivation later by Einstein (1955) from the full theory of general relativity in the weak field approximation. (...) Namely the 1955 approximation shows a variation in km/sec twice as much as first predicted in 1911."

"Specifically, Einstein wrote in 1911 that the speed of light at a place with the gravitational potential phi would be c(1+phi/c^2), where c is the nominal speed of light in the absence of gravity. In geometrical units we define c=1, so Einstein's 1911 formula can be written simply as c'=1+phi. However, this formula for the speed of light (not to mention this whole approach to gravity) turned out to be incorrect, as Einstein realized during the years leading up to 1915 and the completion of the general theory. (...) ...we have c_r =1+2phi, which corresponds to Einstein's 1911 equation, except that we have a factor of 2 instead of 1 on the potential term."

Pentcho Valev

Dear Peter Woit,

I wish you an ironical welcome to this mainly crackpot-dominated community of FQXI essay authors, which I also only joined this year, and where I feel rather isolated in my try to defend scientifically sound views, as you probably do if you cared to review other essays.

I found your essay to be among the best, so last week I gave it the 10 rate in a hopeless try to provide a little balance to the absurdly low rating you got from the senseless majority of this community.

I actually had my little part in this adventure you described, as my PhD thesis was dedicated to cleaning up the construction of the Vassiliev invariants of links in the Euclidean 3D space as obtained from the perturbative expansion of the generic Chern-Simons quantum field theory, before I left this field of research to dedicate myself to mathematical foundations and the restructuring of undergraduate-level concepts of maths and physics.

Now you may be interested with my selection of the best essays in this FQXI contest, with explanations on how things are going on here, which I just wrote to help the minority of proper scientists and other genuine science lovers to find their way in this mess.

Dear Dr. Woit,

I hope that you will have more to say about the conclusion that you reach. As you state, "the fundamental laws of physics point not to some randomly chosen mathematical structure, but to an exceptionally special one." Just before this you called the physically relevant mathematical structure "a distinguished point in the space of all mathematical structures." I interpret this to mean that the point in the space of mathematical structures is distinguished in terms of its mathematical properties alone, without reference to its relevance for physics. If there is a mathematically special structure, and if that structure also is embodied in the fundamental laws of physics, then the structure is doubly distinguished. It holds a position intrinsically distinguished among mathematical structures, and it also is fundamental for physics. More significantly, perhaps the former explains the latter. That is, according to this way of thinking, physical reality embodies a particular mathematical order, as opposed to some other mathematical order, because there is something special about the structure which applies to physical reality. In this way we might be able to understand what now seem to be arbitrary choices in the mathematical order of physical existence. From your essay, I understand that recent history gives us reasons to think this is so. The reasons are in the recent history of mathematics, the recent history of physics, and especially in the connections between the two historical processes. I think many people will be interested to see how physics and mathematics continue to advance together.

Laurence Hitterdale

Dear Peter,

I hope you may engage in this blog.

I appreciate your optimism but ask do you have any firm or 'new' direction to achieve the aim of a complete mathematical formalism? I suggest an apparently very valuable new approach in my own essay based simply on the hierarchical formalism of brackets in arithmetic. It seems very powerful but little understood yet. I hope you may look and evaluate.

What now seems certain to me is that, if using only the present approaches, your important aims, well described in your essay, may be unlikely to be met.

I do hope you may comment on my proposition.

Peter Jackson

Dear Peter Woit

Until now, your essay gives me the most important idea of all the essays, what to study in physics. I most like your first sentence in section 4.1 and I wish to understand it visually as much as possible. I want to understand why the names bundles and fibres, what in these calculations is meant by such prolonged structures? What there is also an analogy and visualization of curvature and of connections? Which components of curvature are B and E? Can you suggest which link to read, (beside yours) where this is also visually explained?

I like also that Weizsacker explained why space is three dimensional. With link, page 3, this is also consequence of SU(2), because there are three independent pauli spin matrices. This is also claimed by Brukner and Zeilinger. What is your opinion about this?

I suggests that three dimensional nature of a light ray is a consequence of three dimensional nature of space. What is your opinion? However, three dimensional nature of a light ray is a consequence U(1) symmetry for electromagnetic field and also due to this I am interested in backgroud of U(1).

Best regards

My essay

It would be very much appreciated if mr. Woit would respond to the comments.

Peter,

You have given us ample examples of unexpected and deep connections between mathematics and physics, but I can't help feeling that the fundamental question has not been answered. Why do you think these connections exist? Is this something we will eventually discover by exploring the connections themselves, or will it always appear to be a "miracle"?

Dear Professor Woit,

I am just a humble student but noticed a similarity of your proposed unification to the main thesis presented in my little opera "Map = Territory" where I ponder the possibility of an actual merger of the description and the described in fundamental physics.

Your excellent overview of the various remarkable connections between math and physics reminded me of my beginner's take on the subject and I would very much appreciate an opinion of an accomplished professional like you (if some time can be found for that). I would be honoured by your feedback and advice.

With deep respect and best wishes,

Martin (a frequent visitor to your blog)

Electromagnetism is expressed in the same formalism as General Relativity:

General Relativity is described by relating the 10-dimensional field of energy and pressures, to the 20-dimensional field of space-time curvature.

This space-time curvature is understood as the curvature of the tangent bundle of space-time, where the fibre of this bundle at each point is the tangent space (the space of small vectors around this point: this is the space of "speed vectors" for imaginary movements of an abstract point across space-time, in duality with the space of 1-forms that are the possible differentials of scalar fields at this point).

We have a connection on this fibre bundle, i.e. a correspondence between nearby fibres, so that when going around a closed curved in space-time, the fibre undergoes "parallel transport" and comes back to itself not identically but "rotated" by an element of the Poincare group at this point.

Now electromagnetism is the same, replacing the tangent bundle by an U(1) bundle, that is, the fibre is a 1-dimensional Hilbert space.

Whenever we choose a convention of trivialization of this bundle, the connection (which gives the correspondence between the fibres above 2 points along a line connecting these points) appears as a 1-form (see wikipedia "differential form") that is usually called "Electromagnetic four-potential". Gauge transformations (which modify the 4-potential by the differential of a scalar field, keeping the electromagnetic field unchanged) are conversions between different possible trivializations of this bundle (turning each fibre by an angle given by this scalar field). Thus, the electromagnetic field is the curvature of the connection of the U(1) bundle in the same way as the Riemann curvature tensor is the curvature of the tangent bundle.

Now you may ask : why is the group taken as the cyclic group U(1) (the group of rotations in a complex line) instead of the simple additive group of R ? After all, both would work as well for classical electrodynamics.

The answer is that we need this for quantum physics : all electric charges are multiples of an elementary charge. The contributions to the wavefunction of an electron at a given point of space-time from different possible worldlines, are calculated as exp(iS/в„Џ) where S is the action of the worldline. This action is given as the integral of the electromagnetic four-potential along the worldline, or more meaningfully, it is the parallel transport by this connection, so that the actions from different worldlines differ by the integral of the curvature of the connection in the space between these lines (any surface bordered by these lines).

Now the reason why the fibre is U(1) is that when the transport along 2 worldlines differ by 1 quantum of action (S2-S1=h), their contributions to the wavefunction are identical (exp(iS1/в„Џ)=exp(iS2/в„Џ)).

B is the curvature above space-like surfaces (contained in the simultaneity space of the given observer), while E is the curvature above surfaces extended in 1 space dimension + the time dimension.

If we want to enter the details of the wavefunction of the electron, it is a bit more complicated because it has a spin 1/2, so that the fibre has not 1 complex dimension but 4, that is the sum of dual spinor spaces, where each spinor space has 2 complex dimensions, and its space of Hermitian forms is identified with the tangent space at this point of space-time. But the point is that the gauge symmetry of electromagnetism operates there by multiplication by a field of unitary complex scalars.

Mr. Woit is sorry but he does not have the time to reply here (as he privately replied to me, he is "way too busy trying to keep up with writing notes for the class [he is] teaching and some other projects"...)

Dear Peter Woit,

You wrote, "In this essay I'll argue that unified theories of fundamental physics are closely linked with some of the great unifying structures that mathematicians have found to underlie much of modern mathematics." According to Fredkin, complete infinities, infinitesmals, local sources of randomness, and singularities do not occur in nature. What are LIe groups and Lie algebras? In order to answer the preceding question, it is necessary to consider the problem of infinity and the axiomatization of the real line, the complex plane, and so on. Is the axiom of choice true for physics, false for physics, or irrelevant for understanding physics and nature? According to the Wikipedia page for the mathematician Jerry L. Bona, he "is probably best known for his statement about equivalent statements of the Axiom of Choice: "The Axiom of Choice is obviously true, the Well-ordering theorem is obviously false; and who can tell about Zorn's Lemma?" According to Wolfram's conjecture, there are 4 or 5 simple rules that generate satisfactory approximations to quantum field theory and general relativity theory. If Wolfram's conjecture happens to be correct, then the basis for understanding nature might consist of the monster group and the 6 pariah groups. I conjecture that string theory needs 2 additional hypotheses: (1) String vibrations are confined to 3 copies of the Leech lattice. (2) Nature is finite and digital. (The maximum physical wavelength is the Planck length times the Fredkin-Wolfram constant.) If the two preceding conjectures are correct, then how might the 64 Particles Hypothesis be justified? Each dimension of spacetime has 4 dimensions of hbar uncertainty and 16 dimensions of hbar/alpha-prime uncertainty. If nature has Wolframian simplicity, it should be possible to identify the 64 dimensions of hbar/alpha-prime stringy vibrational uncertainty with 64 virtual particle paths that provide a phase space unifying the virtual mass-energies of the alternate universes of the multiverse. I call this idea, "Wolframian pseudo-supersymmetry." What is your best guess for the explanation of the GZK paradox?

Dear Peter Woit,

Enjoyed your essay, good job. Very thought provoking "A great mystery of the subject remains that of the explanation for this particular set of Lie groups and the relative normalization of the Yang-Mills action terms (why U(1) ﾃ-- SU(2) ﾃ-- SU(3)?, why the values of the three coupling constants?). Is there some more fundamental geometrical structure that would explain these choices?"

I think yes. It's fun to try and find objects with properties that match these groups. A lot can be learned. Would enjoy your comments on geometric structures you will find in my essay.

Regards and best of luck, Ed