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FQXi FORUM

February 25, 2018

CATEGORY:
Trick or Truth Essay Contest (2015)
[back]

TOPIC: Towards a Grand Unified Theory of Mathematics and Physics by Peter Woit [refresh]

TOPIC: Towards a Grand Unified Theory of Mathematics and Physics by Peter Woit [refresh]

Wigner’s “unreasonable effectiveness of mathematics” in physics can be understood as a reflection of a deep and unexpected unity between the fundamental structures of mathematics and of physics. Some of the history of evidence for this is reviewed, emphasizing developments since Wigner’s time and still poorly understood analogies between number theory and quantum field theory.

Peter Woit is a Senior Lecturer in the Mathematics department at Columbia University. He is author of the book "Not Even Wrong" and maintains the blog "Not Even Wrong". His current main project is a textbook on quantum mechanics from the point of view of representation theory.

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...

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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...

view entire post

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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.

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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

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Pentcho Valev

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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...

view entire post

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...

view entire post

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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

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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

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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.

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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.

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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.

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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.

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"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...

view entire post

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

Michel Janssen: "It is not...

view entire post

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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.

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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.

report post as inappropriate

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

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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

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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

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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

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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

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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

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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...

view entire post

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...

view entire post

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

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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"...)

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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"?

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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"?

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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)

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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)

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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." According to Fredkin, complete infinities, infinitesmals, local sources of randomness, and singularities do not occur in nature. What are LIe groups and Lie...

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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." According to Fredkin, complete infinities, infinitesmals, local sources of randomness, and singularities do not occur in nature. What are LIe groups and Lie...

view entire post

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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) × SU(2) × 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

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Enjoyed your essay, good job. Very thought provoking "

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

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