Unifying Foundations for Physics and Mathematics

During recent travels I attended two conferences (in Paris and Berkeley) and met up with quite a few people. At the Paris conference I gave an intentionally provocative talk to the philosophers of physics there, slides are here. The argument I was trying to make is essentially that more attention should be paid to evidence for a deep unity in much of modern mathematics, which at the same time is connected to our best unified theory of physics (the Standard Model and GR). Edward Frenkel has made some similar points, referring to the Langlands program and its connections to physics as a “Grand Unified Theory of Mathematics”. The specific structures underlying this unification seem to me to deserve attention as providing an important way of thinking about what’s at the “foundations” of both math and physics.

Another motivation for this talk was to make an argument against what I see as having become a widespread and standard ideology about the search for a unified theory in physics. Talking to many physicists and mathematicians interested in physics, I noticed that the conventional wisdom, shared by the establishment and contrarians alike, is that the SM and GR are likely low energy emergent theories, that some completely different sort of theory is needed to describe very short distances such as the Planck scale. Physics establishment figures tend to believe that following the path started with string theory, then AdS/CFT, lately quantum error correction or whatever, will someday lead to a dramatically different sort of theory, replacing space, time and maybe quantum mechanics. Contrarians often have their own favorite idea for a radically different starting point. For an example of this, take a look at Figures 2 and 3 of Mike Freedman’s The Universe from a Single Particle (he spoke about this in Berkeley). Figure 2 is the “establishment” picture, with AdS/CFT the fundamental theory, well-decoupled from the emergent SM + GR (since no one has any idea how to relate them). His Figure 3 shows his own proposal, even better decoupled from any connection to the SM + GR.

Given the extreme level of experimental success of the SM + GR, the obvious conjecture is that these are close to a unified theory valid at all distances. That the mathematical framework they are built on is closely connected to unifying structures in mathematics provides yet more evidence that what one is looking for is not something completely different. The odd thing about the present moment is that arguing that our well-established successful theories can provide a solid basis for further unification makes one a contrarian, with the “establishment” position that a revolution sweeping such theories aside is needed.

I hope to find time in the next few weeks to write up what’s outlined in the slides as a more detailed article of some sort. More immediately, I plan to write a blog entry and perhaps some more detailed notes about the “twistor $P^1$” mentioned at the end of the talk, explaining how it shows up in Euclidean twistor theory as well as in recent work on the Langlands program.

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15 Responses to Unifying Foundations for Physics and Mathematics

  1. AP says:

    The Langlands program is important, but it is hardly a “Grand Unified Theory of Mathematics”. It does not even interact with the majority of areas of mathematics, and outside of number theory it is a pretty niche subject (e.g. it does make contact with algebraic geometry and representation theory, but most workers in those fields do not care about it). It’s even stranger to think of it as “foundational”, at least in the way that mathematicians usually use the word.

  2. Peter Woit says:


    The use of a different meaning than usual for “foundations” was meant to be provocative, drawing attention to a too-little recognized and poorly understood deep structure unifying different areas of mathematics. That what most mathematicians work on and care about has little to do with this is true, but also true of “foundations” in the usual sense (and also true in physics where most research has nothing to do with a possible unifying theory).

  3. DL says:

    “Given the extreme level of experimental success of the SM + GR, the obvious conjecture is that these are close to a unified theory valid at all distances. ” Does not the Dark Matter vs Mond problem indicate that something is seriously wrong?

  4. Peter Woit says:

    I don’t want to start a serious discussion of dark matter here (especially not of MOND vs. dark matter). It’s a huge and very complicated subject that is the main focus of attention for attempts to get beyond the SM, precisely because it’s the only significant source of a possible discrepancy between the SM and observations.

    Just note that there’s no discrepancy between the SM and observations in any earth (or solar system..)-bound experiment. The dark-matter problem shows up in very large scale astrophysical observations, as a conflict between these and best astrophysical modeling. While the evidence seems to be against this, in principle the problem could be in the astrophysical modeling, not the underlying theory (e.g. primordial black holes).

    Even if dark matter is a new non-SM particle, such a thing can be accommodated with a small change to the SM, adding one or more new fields with no SM, only gravitational interactions. You might even be able to do this just with right-handed neutrino fields, which fit very nicely into the SM, in some sense are expected.

  5. Low Math, Meekly Interacting says:

    I could be conflating issues relevant to this post with irrelevant ones, but hopefully not. Whether connecting unifying developments in mathematics and physics seems too mystical strikes me as a very superficial objection. What I wonder about is how such conjecture helps with the problem everyone typically cites, namely incurable infinities. One could reasonably say GM and QFT were unified back in the 50s or thereabouts, but such theories were “diseased”, as Feynman put it. The current approaches aspire to cure the disease with discretized space-time, extended objects, “asymptotic safety”. They give mathematically sensible answers that, so far, don’t agree with reality very well.

    So what does something akin to connecting number theory and algebraic geometry credibly do to make quantum gravity yield finite answers that resemble our world without introducing something very new at the microscopic level?

  6. Peter Woit says:

    The way things work is not that some particular piece of known mathematics directly solves deep problems in theoretical physics.

    My argument here is that the answer to the problems of quantum gravity are not going to be found by throwing out the ideas about geometry, gauge symmetry, spinors, the Dirac operator that our best theory is built on, that have been wildly successful and that are mathematically very deep. What’s needed is an extension of these ideas, not abandoning them for something completely different, arguing that somehow you’ll recover them later as emergent phenomena.

    I don’t want to host now a discussion of everyone’s favorite ideas about quantum gravity. Since it’s my blog, I will just mention that the work I’ve been doing points in the direction of fundamental space-time degrees of freedom being closely related to the SM ones, using spinors, twistors, and the chiral spin-connection. I don’t claim to have solved the problem, but do see new things to think about that may go somewhere: use of just one chirality of spin connection and a fundamentally conformally invariant formalism, working in Euclidean space time as primary, having a degree of freedom that distinguishes the imaginary time direction. For more details, see https://arxiv.org/abs/2104.05099

  7. WTW says:

    Following is a transcription excerpted from an IAS Video Lecture by Leonard Susskind on 29 March 2021. https://www.ias.edu/video/some-half-baked-thoughts-about-de-sitter-space
    (The transcription is my own, not from Google; the only AI used was that which is between my ears. Any errors are mine.)

    Let’s begin with a nice thought… In fact let’s imagine a group of very smart theorists. They know all about quantum mechanics and relativity, but have never heard about modern astronomy or particle physics.

    If they were anything like us, they would eventually discover black holes, quantum field theory, string theory, supersymmetry, the holographic principle, large-N matrix theory, AdS/CFT. In other words, from a pure theory point of view, they would be about where we are now.
    What would they take away from all of this?

    Here’s what I think they would say: I think they would say, first of all, if you want to put gravity and quantum mechanics together, you better have a frozen time-like boundary to anchor the theory, and to define observables. AdS is great in this respect. And maybe flat space is OK. To make the string scale much smaller than the cosmic radius, since in this case the cosmic radius would be the AdS scale, the boundary theory better be super-strongly coupled — but knowing as much as we do, they might believe that the only theories that can be pushed to super-strong coupling are supersymmetric. That would also fit with what they know about matrix theory, where without supersymmetry you can’t even seperate two particles. Asymptotic boundaries, super-strong coupling, supersymmetry — these would be their touch stones.

    What would happen, then, if some crazy man showed up and told them that he had seen the data, and could assure them that space has no boundary, that spacetime is more like deSitter space than Anti-deSitter space, that there’s no supersymmetry, and — despite that — locality works down to ultra-microscopic scales?

    I tell you what I think they would say. I think they would say, “Baloney! Your model is in the swampland.”
    They’d say this not because of some made-up input bound on the inflaton, but because the model violates the entire foundation of their mathematical understanding!

    OK, what should we take away from this parable?
    The answer is, I don’t really know. …”

    I’ve heard similar “parables”, with varying descriptions of what is or isn’t discrepant, from people like Arkami-Hamed and Ed Witten among others. I believe the general summary is “We’re missing something, and we don’t know what it is.” And we tend to think (or at least pretend) — since people in this field are so damn smart — that we know more than we actually do.

    The critique I would give of your presentation is that you are underestimating the fundamental inconsistencies between and among these theories, that have tried to be papered over with mathematical slight of hand. Rather than providing some deep underlying fundamental truth, it tends to be used as fairy dust. Separating the wheat from the chaff while navigating our desire for universality of mathematical form is one of the key issues that must be addressed if we are to make real progress.

  8. Paolo Bertozzini says:

    Dear Peter,

    thanks for pointing this out clearly. There is indeed an unusual and quite widespread consensus between most of the current approaches to quantum gravity: not only that geometry of space-time (and the standard model) is an emergent macroscopic feature, but that it is even impossible to talk about “geometry” at the fundamental quantum level (for example one can see the recent brief talks at the “1st Workshop of the International Society for Quantum Gravity” [https://isqg.org/first-isqg-meeting/] that are available on the ISQG YouTube channel). Surprisingly this “emergentism” seems relatively common even in areas, like non-commutative geometry, that in principle are already capable of “saying something” on space-time (and standard model) at the quantum level.

    One should probably make here a distinction between “macroscopic emergence” exclusively via phase-transitions (that is the position I was referring to in the previous paragraph) and an “operational/spectral” point of view (where space-time, commutative or not, is determined by other degrees of freedom, but is otherwise perfectly well defined also at the fundamental level).

    In the current trends in AdS/CFT, some geometric properties (related to the modular structure of the local operator algebras in CFT with respect to the vacuum or its perturbations) are being investigated; such results will likely survive even without the background propaganda of “emergence” and will continue to make sense also in situations that are different from the original AdS/CFT string-theoretic motivation (since they are just based on Reeh-Schlieder theorem in QFT and Tomita-Takesaki modular theory). The recovery of “space-time geometry” from states on local algebras of operators has actually a much older tradition, going back to works in Algebraic QFT by Haag, Bannier, Keyl, Buchholz, Summers (just to cite a few).

    Best Regards.

  9. Peter Woit says:

    “If they were anything like us, they would eventually discover black holes, quantum field theory, string theory, supersymmetry, the holographic principle, large-N matrix theory, AdS/CFT. In other words, from a pure theory point of view, they would be about where we are now.”
    This indicates the fundamental psychological problem of the field. He can’t even imagine the possibility that string theory, SUSY, etc were wrong turns, that he and others have gone down a blind alley for decades.

    A huge problem with the current relation of math and physics is that those with a deep belief in the unity of math and physics who can’t believe they’re in a blind alley are left searching for unity in an unpromising place (e.g. AdS/CFT), limiting what they can find. It’s really important for people to distinguish what is solid fundamental theory (SM +QG), concentrate on looking for deep mathematics there, being very wary about starting with unsuccessful speculative physics as a starting point.

    In some sense all mathematics is connected, and if you start at an unpromising place you may sooner or later make your way somewhere interesting (for instance, if you start by believing 6d Calabi-Yaus are fundamental). But it’s much, much harder to make progress that way and you’re likely to end up convincing others that you’re “Lost in Math”, that looking for unity through math is a fool’s errand.

  10. Peter Shor says:

    Peter, you say

    Physics establishment figures tend to believe that following the path started with string theory, then AdS/CFT, lately quantum error correction or whatever, will someday lead to a dramatically different sort of theory.

    It seems to me that trying to base a fundamental theory of quantum physics on quantum information and quantum error correction is an error on the order of trying to base a fundamental theory of classical physics on thermodynamics.

    Thermodynamics will tell you a lot about classical physics (and maybe you can even derive the ideal gas laws from it), but ultimately it will leave a lot of physics unspecified; thermodynamics will never give you the properties of electrons.

  11. d_b says:

    @Peter Shor
    Isn’t that precisely the idea, though? The claim (or speculation, rather — I’m not sure that I would elevate it to a “claim” just yet) is that classical gravity is part of the emergent, low energy behavior of complex quantum systems (or perhaps ensembles of quantum systems). The fundamental theory would just be quantum mechanics, and the quantum error correction is supposed to explain how a quantum mechanical system can robustly encode classical spacetime.

  12. Peter Shor says:


    My point is that I think it’s much more likely that some theory of quantum gravity is fundamental, and all the quantum information type stuff these researchers are looking at is emergent. And if this is true, there’s a limit to how far they can go unless they realize this.

  13. SRP says:

    Given the accessible experimental anomalies that have to do with the proton, notably the Krisch transverse-polarized collision data (repeatedly confirmed as showing spin effects not dying out as they are supposed to at higher energies), statements about how the SM is just too gosh-darn perfect to give theorists any clues strike the layman as avoidance.

  14. Peter Woit says:

    I’d be very interested (because of the euclidean twistor unification picture I’ve been working on) to see a discussion of spin-dependent effects in proton collisions that indicates experimental anomalies that cannot be plausibly attributed to our lack of a reliable non-perturbative calculational methods for those effects.

  15. SRP says:

    Here is Krisch in 2010, after telling the story of how from the beginning QCD theorists firmly predicted the wrong result based on general principles (I’ve found a few presentations by others since then saying yep, still an anomaly, along with some other problems):

    “To summarize, for the past 30 years QCD-based calculations have continued to disagree with the ZGS 2-spin and AGS 1-spin elastic data, and the ZGS, AGS, Fermilab and now RHIC [28] inclusive data. To be specific:
    * These large spin effects do not go to zero at high-energy or high-Pt, as was predicted.
    * No QCD-based model can yet explain simultaneously all these large spin effects. There is a BASIC PRINCIPLE OF SCIENCE:
    * If a theory disagrees with reproducible experimental data, then it must be modified. Precise spin experiments could provide experimental guidance for the required modifica- tion of the theory of Strong Interactions. New experiments at higher energy and higher Pt on the proton-proton elastic cross-section’s: dσ/dt, Ann and An could provide further guidance for these modifications, just as the RHIC inclusive An experiment [28] confirmed the earlier Fermilab experiments [27]. Elastic scattering is especially important because:
    * It is the only exclusive process large enough to be measured at TeV energy. This is probably because proton-proton elastic scattering is dominated by the diffrac- tion due to the millions of inelastic channels that compete for the total cross-section of only about 100 milibarns at TeV energies. Many people may have forgotten this simple but essential geometrical approach [1], which I learned from Prof. Serber’s optical model in 1963 [3]; perhaps it should now be learned or relearned by others.”


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