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

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.

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

"The lesson drawn here from history is that the fundamental laws of
physics point not to some randomly chosen mathematical structure, but to an
exceptionally special one, requiring a deep understanding of the mathematical
world in order to fully appreciate it."
There is only a unique system of unities in which there is no contradiction between quatum mechanics & General Relativity. Bohr principle of correspondance play an importnat role to built this system. Also the image what is entropy in a multiple systems and its correspondant significance for one system is also the key of unification.