Some Math Items

Some math items that may be of interest:

Update: The Scholze review has been removed (temporarily?). A cached version is here.
Update: The review was temporarily removed just because what was posted wasn’t a finalized version, this is explained here. They should repost once Scholze has a chance to make any final edits.

Posted in Uncategorized | 5 Comments

More of the Same (Physics, Math and Unification)

I was going to just provide the following links with a some comments, but decided it would be a good idea to put them into what seems to me the larger context of where we are in fundamental physics and its relationship to mathematics.

For the latest on the conventional physics approach to unification (GUTS, SUSY, strings, M-theory), there’s:

  • The Lex Fridman podcast has an interview with Cumrun Vafa. Going to the section (1:19:48) – Skepticism regarding string theory) where Vafa answers the skeptics, he has just one argument for string theory as a predictive theory: it predicts that the number of spacetime dimensions is between 1 and 11.
  • A second edition of Gordon Kane’s String Theory and the Real World has just appeared. One learns there (page 1-19) that

    There is good reason, based on theory, to think discovery of the superpartners of Standard Model particles should occur at the CERN LHC in the next few years.

For the latest in mathematics and the interface of math and physics, there’s

About the first two links, I’m at a loss for words.

The second two are extremely interesting topics indicating a deep unity of number theory, geometry and physics. They’re also not topics easy to say much about in a blog posting. In the Fargues-Scholze case that’s partly because the new ideas they have come up with relating arithmetic and geometry are ones I don’t understand very well at all (although I hope to learn more about them in the future). The connections they have found between representation theory, arithmetic geometry, and geometric Langlands are very new and it will likely be quite a few years before they are well understood and their implications well-developed.

In the Gaiotto-Witten case, some of what they discuss is very familiar to me: geometric quantization has been a topic of fascination since my student days, and one major goal of my QM book was to work out in detail (for the case of $\mathbf R^{2d}$) some of the subtleties about quantization that they discuss. For co-adjoint orbits in Lie algebras, geometric quantization has a long history, and “brane quantization” may or may not have anything new to say about this. For moduli spaces of vector bundles on Riemann surfaces, and Hitchin moduli spaces of Higgs bundles on Riemann surfaces, “brane quantization” might come into its own.

There is a fairly short path now potentially connecting fundamental unifying ideas in number theory and geometry to our best fundamental theories in physics (and seminars on arithmetic geometry and QFT are now a thing). The Fargues-Scholze work relates arithmetic and the central objects in geometric Langlands involving categories of bundles over curves. These categories in turn are related (in work of Witten and collaborators) to 4d TQFTs based on twistings of N=4 super Yang-Mills. This sort of 4d QFT involves much the same ingredients as 4d QFTs describing the Standard Model and gravity. For some better indication of the relation of number theory to this sort of QFT, a good source is David Ben-Zvi’s lectures this past semester (see here and here). I’m hopeful that the ideas about twistors and QFT in Euclidean signature discussed here will provide a close connection of such 4d QFTs to the Standard Model and gravity (more to come on this topic in the near future).

Posted in Langlands, Twistor Unification | Leave a comment

Steven Weinberg 1933-2021

I heard this morning the news that Steven Weinberg passed away yesterday at the age of 88.  He was arguably the dominant figure in theoretical particle physics during its period of great success from the late sixties to the early eighties.  In particular, his 1967 work on unification of the weak and electromagnetic interactions was a huge breakthrough, and remains to this day at the center of the Standard Model, our best understanding of fundamental physics.

During the years 1975-79 when I was a student at Harvard,  I believe the hallway where Weinberg, Glashow and Coleman had offices close together  was the greatest concentration of the world’s major figures driving the field of particle theory, with Weinberg seen as the most prominent of the three.  From what I recall, in a meeting one of the graduate students (Eddie Farhi?) referred to “Shelly, Sidney and Weinberg”, indicating the way Weinberg was a special case even in that group.   I had the great fortune to attend not only Coleman’s QFT course, but also a course by Weinberg on the quantization of gauge theory.

Weinberg was the author of an influential text on general relativity, as well as a masterful three-volume set of textbooks on QFT.  The second volume roughly corresponds to the course I took from him, and the third is about supersymmetry.   While most QFT books cover the basics in much the same way, Weinberg’s first volume is a quite different, original and highly influential take on the subject. It’s not easy going, but the details are all there and his point of view is an important one.  When you hear Nima Arkani-Hamed preaching about the right way to understand how QFT comes out uniquely as the only sensible way to combine special relativity and quantum mechanics, he’s often referring specifically to what you’ll find in that first volume.

Besides his technical work, Weinberg also did a huge amount of writing of the highest quality about physics and science in general for wider audiences.  An early example is his 1977 The Search for Unity: Notes for a History of Quantum Field Theory (a copy is here). His 1992 Dreams of a Final Theory is perhaps the best statement anywhere of the goal of fundamental physical theory during the 20th century. His large collection of pieces written for the The New York Review of Books covers a wide variety of topics and all are well worth reading.

At the time of the 1984 “First Superstring Revolution”, Weinberg joined in and worked on string theory for a while, but after a few years turned to cosmology. In early 2002 he was one of several people I wrote to about the current state of string theory, and here’s what I heard back from him:

I share your disappointment about the lack of contact so far of string theory with nature, but I can’t see that anyone else (including those studying topological nontrivialities in gauge theories) is doing much better. I thinks that some theorists should go on pushing as hard as they can on string theory, and others should do something else, but it is not easy to see what. I have myself voted with my feet (if that is the appropriate organ here) and switched entirely to work in cosmology, which is as exciting now as particle physics was in the 1960s and 1970s. I wouldn’t criticize anyone for their choices: it’s a tough time for fundamental physics.

A couple years after that time, Weinberg’s 1987 “prediction” of the cosmological constant became the main argument for the string theory multiverse. This “prediction” was essentially the observation that if you have a theory in which all values of the cosmological constant are equally likely, and put this together with the “anthropic” constraint that only for some range will galaxy formation give what seem to be the conditions for life, then you expect a non-zero CC of very roughly the size later found. I’ve argued ad nauseam here that this can’t be used as a significant argument for string theory in its landscape incarnation. One way to see the problem is to notice that my own theory of the CC (which is that I have no idea what determines it, so any value is as likely as any other) is exactly equivalent to the string landscape theory of the CC (in which you don’t know either the measure on the space of possible vacua, or even what this space is, so you assume all CC equally likely). One place where Weinberg wrote about this issue is his essay Living in the Multiverse, which I wrote about here (the sad story of misinterpretation of a comment of mine there is told here).

Weinberg’s death yesterday, taking away from us the dominant figure of the period of particle theory’s greatest success is both a significant loss and marks the end of an era. His 2002 remark that “it’s a tough time” is even more true today.

Update: Scott Aaronson writes about Weinberg here, especially about getting to know him during the last part of his life.

Posted in Obituaries | 18 Comments

The Problem of Quantization

I’ve been watching Witten’s ongoing talks about geometric Langlands mentioned here, and wanted to recommend to everyone, mathematician or physicist, the first of them, on The Problem of Quantization (pdf here, video here, the question session is very worthwhile). For those very sensibly not interested in the intricacies of geometric Langlands, this talk is about the fundamental issue of “quantization”.

Hamiltonian mechanics gives a beautiful geometrical formulation of classical mechanics in terms of the Poisson bracket on functions, while quantum mechanics involves operators with non-trivial commutators. It was Dirac’s great insight that “quantization” takes functions to operators, taking the Poisson bracket to the commutator. In mathematician’s language, it’s supposed to be a unitary representation of the Lie algebra of the infinite dimensional group of canonical transformations of a symplectic manifold, so a homomorphism from functions with Poisson bracket to the Lie algebra of skew-adjoint operators on a complex vector space.

The problem with this is that you’d like to have an irreducible representation, but the only way to get this is to pick some extra structure on the symplectic manifold. The standard example is the phase space $\mathbf R^{2n}$, where you have to pick a decomposition into position and momentum coordinates. The state space will then be functions of just position, or just momentum. A different choice is to complexify, and look at functions of either holomorphic or anti-holomorphic coordinates. This choice is called a “polarization”. One aspect of the “problem” of quantization is that, given a phase space (symplectic manifold), there may not be an appropriate polarization. Or, there may be many different ones, with no obvious reason why they should give the same quantum theory.

Witten doesn’t mention one aspect of this that I find most fascinating. For relativistic quantum field theories the phase space is a space of solutions of a relativistic wave-equation. To get physically sensible results one must choose a polarization that distinguishes between positive and negative energy (or between functions which extend holomorphically in the positive or negative imaginary time direction).

In these lectures, Witten advertises a rather exotic quantization contruction, using (even for a finite dimensional symplectic manifold ) conformally invariant boundary conditions in a two-dimensional QFT. I’m not convinced that this is really a good way to deal with the case where what you’re doing is looking for representations of a finite-dimensional Lie algebra, but it’s plausible this is the right way to think about the geometric Langlands situation, where you’re trying to quantize a moduli space of Higgs bundles.

In the question section, someone asked about my favorite approach to this problem, essentially using fermionic variables and cohomology. This can be thought of in general as using spinors and the Dirac operator, with the Dolbeault operator a special case when the symplectic manifold is Kähler. Witten responded that he had only really looked at this in the Kähler special case.

Posted in Quantum Mechanics, Quantum Theory: The Book, Uncategorized | 11 Comments

Deterioration of the World’s Thinking About the Deepest Stringy Ideas

For quite a few years now, I’ve been mystified about what is going on in string theory, as the subject has become dominated by AdS/CFT inspired work which has nothing to do with either strings or any visible idea about a possible route to a unified fundamental theory. This work is very much dependent on choosing a special background, in tension with the idea that, whatever string theory is, it’s supposed to be a unique theory that relates all possible backgrounds. This issue came up in a discussion session at Strings 2021, and it turns out that others are wondering about this too. There’s this today from Lubos Motl:

Aside from more amazing things, the AdS/CFT correspondence became just a recipe for people to do rather uninspiring copies of the same work, in some AdS5/CFT4 map, and what they were actually thinking was always a quantum field theory, typically in D=4 (and it was likely to be lower, not higher, if it were a different dimension!) whose final answers admit some interpretation organized as a calculation in AdS5. But as Vafa correctly emphasized, this is just a tiny portion of the miracle of string/M-theory – and even the whole AdS/CFT correspondence is a tiny fraction of the string dualities.

This superficial approach – in which people reduced their understanding of string theory and its amazing properties to some mundane, constantly repetitive ideas about AdS/CFT, especially those that are just small superconstructions added on top of 4D quantum field theories – got even worse in the recent decade when the “quantum information” began to be treated as a part of “our field”. Quantum information is a legitimate set of ideas and laws but I think that in general, this field adds nothing to the fundamental physics so far which would go beyond the basic postulates of quantum mechanics…

When Cumrun correctly mentioned that the real depth of string theory is really being abandoned, Harlow responded by saying that there were some links of quantum information to AdS/CFT, the latter was a duality, and that was important. But that is a completely idiotic way of thinking, as Vafa politely pointed out, because string theory (and even string duality) is so much more than the AdS/CFT. In fact, even AdS/CFT is much more than the repetitive rituals that most people are doing 99% of their time when they are combining the methods and buzzwords of “AdS/CFT” and “quantum information”. Many people are really not getting deeper under the surface; they are remaining on the surface and I would say that they are getting more superficial every day.

According to Lubos, he’s not the only one who feels this way, with an “anonymous Princeton big shot” agreeing with him (hard to think of anyone else this could be other than Nima Arkani-Hamed):

There is a sociological problem – coming from the terrifying ideological developments in the whole society – that is responsible for this evolution. I have been saying this for a decade or two as well – and now some key folks at Princeton and elsewhere told me that they agreed. The new generation that entered the field remains on the surface because it really lacks the desire to arrive with new, deep, stunning, revolutionary ideas that will show that everyone else was blind. Instead, the Millennials are a generation that prefers to hide in a herd of stupid sheep and remain at the surface that is increasingly superficial…

So most of the stuff that is done in “quantum information within quantum gravity” is just the work of mediocre people who want to keep their entitlements but who don’t really have any more profound ambitions. As the aforementioned anonymous Princeton big shot told me, their standards have simply dropped significantly. The toy models in the “quantum information” only display a very superficial resemblance to the theories describing Nature. That big shot correctly told me that in the early 1980s, Witten was ready to abandon string theory because it had some technical problems with getting chiral fermions and their interactions correctly.

Harlow says that many of the people – who may be speakers at the annual Strings conference and who may call themselves “string theorists” when they are asked – don’t really know even the basics of string theory. And they can get away with it. Just like there is the “grade inflation” and the “inflation of degrees”, there is “inflation in the usage of the term string theorist”. Tons of people are using it who just shouldn’t because they are not experts in the field at all. Harlow said that many of those don’t understand supersymmetry, string theory etc. but it’s worse. I think that many of them don’t really understand things like chiral fermions, either. It’s implicitly clear from the direction of the “quantum information in quantum gravity” papers and their progress, or the absence of this progress to be more precise. They just don’t think it’s important to get their models to a level that would be competitive with the previous candidates for a theory of everything – like the perturbative heterotic string theory, M-theory on G2 manifolds, braneworlds, and a few more. They are OK with writing a toy model having “something that superficially resembles a spacetime” and they want to be satisfied with that forever.

I don’t want to start here an ad hominem discussion of Lubos and his often extreme and eccentric views. On the topic though of the devolution of string theory as a TOE to playing with toy models of AdS/CFT using quantum information, it seems quite plausible that not only the “anonymous Princeton big shot” but quite a few other theoretical physicists see the current situation as problematic.

Posted in Strings 2XXX | 4 Comments

Even More Langlands

Various news at least tangentially related to the Langlands program:

Posted in Langlands | 3 Comments

Strings 2021

Strings 2021 started today, program is available here. Since it’s online only, talks are much more accessible than usual (and since it’s free, over 2000 people have registered to in principle participate via Zoom). Talks are available for watching every day via Youtube, links are on the main page.

As has been the case for many years, it doesn’t look like there will be anything significantly new on the age-old problems of getting fundamental physics out of a string theory. But, as has also been the case for many years, the conference features many talks that have nothing to do with string theory and may be quite interesting. I notice that Roger Penrose, a well-know string theory skeptic, will be giving a talk on the last day of the conference next week.

Another series of talks that I took a look at and that I can recommend is Nima Arkani-Hamed’s lectures on Physics at Future Colliders at the ICTP summer school on particle physics. He never actually gets anywhere near discussing the topic of the title for the talks, but does give a very nice leisurely introduction to computing amplitudes for zero-mass particles. What he’s doing is emphasizing ideas that are often not taught in conventional QFT courses (although they should be). His second talk explains how to think of things in terms of classifying representations of the Poincare group, an old topic that unfortunately is often no longer taught (see chapter 42 of my QM textbook). His third talk emphasizes thinking of space-time vectors as two by two matrices (see section 40.4 of my QM book). This is a truly fundamental idea about space time geometry that gets too little attention in most physics courses.

Update: At String 2021, yesterday Nima Arkani-Hamed gave a talk on “Connecting String Theory to the Real World We See Outside Our Windows”, where he sometimes sounds like me, contrasting the pre-LHC claims of string theorists:

1. LHC will discover SUSY
2. String Theory Loves SUSY + Unification

to what they are saying now that the LHC has found no SUSY

CICADAS [i.e. crickets]. (Anyway, String Theory is mainly about Quantum Gravity).

He goes on to explain the “landscape philosophy”, which he sees string theorists (and himself) as now adopting. According to this philosophy, “connection to particle physics appear[s] hopeless/”parochial”/unimportant”. As a result, he sees the current situation as

  • String theorists are for the most part no longer actively pursuing connecting to particle physics of the real world.
  • Understandable as a short-term strategy
  • But in my view a real mistake in the long run…

One reason for this being a real mistake is that, divorced from input from the real world, theory becomes sterile:

Questions Posed by Nature are Vastly Deeper and more fruitful than ones we humans tend to pose for ourselves.

Unfortunately I don’t think Arkani-Hamed has any compelling argument against “string theory implies landscape implies nothing to say about particle physics”. He discusses the “swampland philosophy”, but gives as a challenge to theorists just making more precise the sort of empty question that this philosophy deals in (he asks whether D=9 SU(2021) to the power 2021 is in the swampland).

Update: In the final discussion section, Witten emphasizes that “What is string theory?” still has no answer, that we have “little idea what it really is”. He lists two main things we know about the supposed theory:

1. General string perturbation theory using 2d conformal field theory. He mentions that one basic problem with this is that there is no understanding of what happens in time-dependent backgrounds, so, in particular, this is useless for addressing the big bang, which is the one place people now point to as a possible connection to real world data.

2. AdS/CFT

He notes that to get at non-perturbative string theory we seem to need some more general understanding of quantum theories, going beyond the usual quantization starting with an action, and ends by saying maybe quantum information theory can help. In the discussion section, Vafa challenges him on this, saying he sees no indication that quantum information theory gives any insight into dualities he sees as the central aspect of the non-perturbative theory. Witten’s answer is that this was just a vague hope, that the duality ideas are now 25 years old, we haven’t progressed beyond them, need something new.

Posted in Strings 2XXX | 34 Comments

Various Math Items

Some math-research items:

  • Mura Yakerson has been doing a really wonderful series of interviews with mathematicians, available at her math-life balance web-page or Youtube channel. I’ve just started listening to some of them, including ones with Peter Scholze and Dustin Clausen (Clausen is John Tate’s grandson, the latest AMS Notices has a memorial article).
  • There’s a remarkable report out from Peter Scholze about the progress of the Liquid Tensor experiment. Back when I first heard about this, I figured it was a clever plot by Scholze to get other people to help with a very complicated part of a proof, by getting them to work out the details, with the excuse being that they would be doing a computer check of the proof. Seemed to me very unlikely you could check such a proof with a computer, but that by forcing humans to try to disambiguate things carefully enough in preparation for a computer proof, he’d get a human-checked proof. Looks like I was wrong.
  • For yet more Scholze news, the Fields Medal symposium this year will be devoted to his work.
  • Trying to find something of interest in math, that wasn’t Scholze-related, I noticed this site devoted to the case of Azat Miftakhov, where there will be an online Azat Miftakhov Day program. Foiled though on the Scholze front, since he’s a speaker there, talking about Condensed Mathematics.
  • The list of those giving plenary lectures at next years ICM is here.

Update: Kevin Hartnett at Quanta has a good new article up about quantum field theory and mathematics (an inexhaustible topic…)

Update: Also from the Simons Foundation, there’s a wonderful profile of my Columbia colleague Andrei Okounkov, who has been very active in bringing together mathematics and ideas from quantum field theory.

Update: Nature has a story about the Liquid Tensor Experiment.

Posted in Uncategorized | 17 Comments

Non-empirical Physics

I haven’t been paying much attention in recent years to the philosophers of science studying “Non-empirical” or “Post-empirical” physics or theory confirmation. At various times I did write fairly extensively about this, see for instance here, here and here. By 2015 there was a conference in Munich on the topic, which led in 2019 to a volume of papers entitled Why Trust a Theory?

There’s a new paper out along similar lines, String theory, Einstein, and the identity of physics: Theory assessment in absence of the empirical, evidently to appear in a journal special issue from a 2019 conference on Non-Empirical Physics from a Historical Perspective.

The reaction of most physicists to this sort of thing is exemplified by Will Kinney’s tweet about the paper:

WTAF

In the past few years I’ve been writing less and less here and elsewhere about the issue of evaluating string theory as physics, for several reasons:

  • String theory has effectively gone completely post-empirical, decoupling from any possible relation to experiment. This Week’s Hype used to be a regular feature here, devoted to debunking the numerous bogus claims regularly being made for how to “test string theory”. One rarely sees these anymore, with the string theory community now having given up on this and somehow comfortably moved into a completely post-empirical mode.
  • I’m actually much more sympathetic than most people to the idea that there is a serious and very interesting question about how to evaluate ideas about theoretical fundamental physics in the absence of viable experimental tests. But I haven’t had much luck finding others who share my views. The reaction to blogposts like this recent one tends to be pretty uniformly scornful, that I’m just Lost in Math. The post-empirical philosophers of science deal with me differently, pretty much doing their best to ignore me (I don’t make it into the extensive bibliography of the new paper on the arXiv).
  • There are two other projects that seem to be a much better way to spend my time (the twistor unification stuff, and improving the textbook on QM and representation theory).

By the way, I notice that there is an arXiv trackback already for another blog entry about this paper, wondering if trackbacks here are still censored.

Well, that’s all about this for now, best to take my own advice and go think about something else.

Update: I just ran across this AIP interview with John Schwarz from last year. Schwarz seems to feel that string theory unification is a huge success, despite the testability problem. On the failure to find the superpartners he and other string theorists expected, that’s a problem for experimental physics, not for string theory:

As I said, if supersymmetry is not discovered, there’s a danger that experimental particle physics will die. If that happens, it would be tragic, but it wouldn’t be the end of string theory. String theory will continue, regardless, and will continue to advance.

On the topic of answering those who argued that superpartners would not be found back in the 2000s, and who have put forward detailed criticisms of string theory unification, here’s what he has to say:

There were a couple popular books that attacked string theory about a decade or so ago. The authors clearly had chips on their shoulders. For people without a physics background it’s not possible to assess whether what they’re reading makes sense or not. But anyone with at least an undergraduate education in physics I think can recognize that they should not be taken seriously.

Posted in Uncategorized | 27 Comments

The Evolution of the Physicist’s Picture of Nature

Reading this Nautilus article about Julian Barbour led me recently to something I don’t think I’ve ever read before, Dirac’s 1963 Scientific American article The Evolution of the Physicist’s Picture of Nature. There is a very famous quote from this article that I’ve often seen:

It is more important to have beauty in one’s equations than to have them fit experiment

but I was unaware of the context of that quote, in which the famous part is prefaced by “I think there is a moral to this story, namely that…” The story that Dirac had in mind was that of the discovery of the Schrödinger equation.  Famously, Schrödinger first wrote down a relativistic wave equation (now known as the Klein-Gordon equation).  This equation is what one quickly gets if one follows de Broglie’s idea that matter is described by waves, and uses the relativistic energy-momentum relation.   Here’s the full story, as told by Dirac, giving his famous quote in context:

I might tell you the story I heard from Schrödinger of how, when he first got the idea for this equation, he immediately applied it to the behavior of the electron in the hydrogen atom, and then he got results that did not agree with experiment. The disagreement arose because at that time it was not known that the electron has a spin. That, of course, was a great disappointment to Schrödinger, and it caused him to abandon the work for some months. Then he noticed that if he applied the theory in a more approximate way, not taking into ac­ count the refinements required by relativity, to this rough approximation his work was in agreement with observation. He published his first paper with only this rough approximation, and in that way Schrödinger’s wave equation was presented to the world. Afterward, of course, when people found out how to take into account correctly the spin of the electron, the discrepancy between the results of applying Schrodinger’s relativistic equation and the experiments was completely cleared up.

I think there is a moral to this story, namely that it is more important to have beauty in one’s equations than to have them fit experiment. If Schrodinger had been more confident of his work, he could have published it some months earlier, and he could have published a more accurate equation. That equation is now known as the Klein-Gordon equation, although it was really discovered by Schrödinger, and in fact was discovered by Schrödinger before he discovered his nonrelativistic treatment of the hydrogen atom. It seems that if one is working from the point of view of getting beauty in one’s equations, and if one has really a sound insight, one is on a sure line of progress. If there is not complete agreement between the results of one’s work and experiment, one should not allow oneself to be too discouraged, because the discrepancy may well be due to minor features that are not properly taken into account and that will get cleared up with further developments of the theory.

There’s another remarkable aspect of this Scientific American article, something about it that would be completely inconceivable today: they write down three equations, including both the famous non-relativistic Schrödinger equation for the Coulomb potential, as well as the relativistic Klein-Gordon version.

Dirac notes that Schrödinger found his formulation of quantum mechanics in a very different way than Heisenberg found his:

Heisenberg worked keeping close to the experimental evidence about spectra that was being amassed at that time, and he found out how the experimental information could be fitted into a scheme that is now known as matrix mechanics. All the experimental data of spectroscopy fitted beautifully into the scheme of matrix mechanics, and this led to quite a different picture of the atomic world.

whereas

Schrödinger worked from a more mathematical point of view, trying to find a beautiful theory for describing atomic events, and was helped by De Broglie’s ideas of waves associated with particles. He was able to extend De Broglie’s ideas and to get a very beautiful equation, known as Schrödinger’s wave equation, for describing atomic processes. Schrodinger got this equation by pure thought, looking for some beautiful generalization of De Broglie’s ideas, and not by keeping close to the experimental development of the subject in the way Heisenberg did.

At the end of the article, Dirac makes the case that progress in fundamental physics may not come from a theorist like Heisenberg finding a scheme to match experimental results, but from a theorist like Schrödinger pursuing mathematical beauty:

It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better.

This view provides us with another way in which we can hope to make advances in our theories. Just by studying mathematics we can hope to make a guess at the kind of mathematics that will come into the physics of the future. A good many people are working on the mathematical basis of quantum theory, trying to understand the theory better and to make it more powerful and more beautiful. If someone can hit on the right lines along which to make this development, it may lead to a future advance in which people will first discover the equations and then, after examining them, gradually learn how to apply them. To some extent that corresponds with the line of development that occurred with Schrodinger’s discovery of his wave equation. Schrödinger discovered the equation simply by .looking for an equation with mathematical beauty. When the equation was first discovered, people saw that it fitted in certain ways, but the general principles according to which one should apply it were worked out only some two or three years later. It may well be that the next advance in physics will come about along these lines: people first discovering the equations and then needing a few years of development in order to find the physical ideas behind the equations. My own belief is that this is a more likely line of progress than trying to guess at physical pictures.

The context for his famous quote is thus an argument for pursuing fundamental physics by looking for mathematical beauty, not giving up on a beautiful equation just because it doesn’t seem to fit experiment. As in Schrödinger’s case, more effort may be needed to understand the actual relationship of the equation to reality.

Besides this argument, which I’ve always been well aware of and sympathetic to (despite not knowing the context in which Dirac was making it), there’s something else I found very striking about the 1963 article. Dirac begins by explaining that the four-dimensional Lorentz symmetry of relativity is in a sense broken by the choice of a way of describing the state of the world:

What appears to our consciousness is really a three-dimensional section of the four-dimensional picture. We must take a three-dimensional section to give us what appears to our consciousness at one time; at a later time we shall have a different three-dimensional section. The task of the physicist consists largely of relating events in one of these sections to events in another section referring to a later time. Thus the picture with four­ dimensional symmetry does not give us the whole situation. This becomes particularly important when one takes into account the developments that have been brought about by quantum theory. Quantum theory has taught us that we have to take the process of observation into account, and observations usually require us to bring in the three-dimensional sections of the four-dimensional picture of the universe.

The special theory of relativity, which Einstein introduced, requires us to put all the laws of physics into a form that displays four-dimensional symmetry. But when we use these laws to get results about observations, we have to bring in something additional to the four-dimensional symmetry, namely the three-dimensional sections that describe our consciousness of the universe at a certain time.

Dirac also refers to work on canonical formulations of general relativity, aimed at quantizing gravity:

… if one insists on preserving four-dimensional symmetry in the equations, one cannot adapt the theory of gravitation to a discussion of measurements in the way quantum theory requires without being forced to a more complicated description than is needed bv the physical situation. This result has led me to doubt how fundamental the four-dimensional requirement in physics is. A few decades ago it seemed quite certain that one had to express the whole of physics in four­-dimensional form. But now it seems that four-dimensional symmetry is not of such overriding importance, since the description of nature sometimes gets simplified when one departs from it.

Thinking about twistor unification has led me to some similar thoughts: a Euclidean formulation of quantum theory requires picking a choice of imaginary time direction and breaking SO(4) symmetry in order to define states. Dirac thinks of our consciousness as giving us access to the state of the universe defined on a 3d slice, but the twistor point of view is even more directly related to our conscious experience. A point in space time is defined by the sphere of light rays through the point, and it is this sphere that our vision gives us direct access to, with 3d space something we make up out of these spheres.

A common argument against Dirac’s point of view is that it’s engaging in mysticism. For another recent article that touches in a different way on the mystical nature of a discovery about fundamental physics, see this interview with Frank Wilczek, where he tells this story:

The mystical moment came while I was visiting Brookhaven National Laboratory, on Long Island. Somehow—I don’t remember how, exactly—I wound up alone, standing on a jerry-rigged observation platform above a haphazard mess of magnets, cables, and panels. This was a staging area for assembling detectors and renovating pieces of the main accelerator there, the Alternating Gradient Synchotron (AGS). I must have gotten separated from my host for a few minutes. In any case, there I was, alone inside an aircraft-hangar-sized metallic box, staring down at the kind of equipment that people use to explore the fundamentals of Nature experimentally.

And then it happened. It came to me, viscerally, that the intricate calculations I’d done using pen and paper (and wastebasket) might somehow describe this entirely different realm of existence—namely, a physical world of particles, tracks, and electronic signals, created by the kind of machinery I was looking at. There was no need to choose, as philosophers often struggled to do, between mind or matter. It was mind and matter. How could that be? Why should it be? Yet I somehow, I suddenly knew that it could be so, and should be so.

That was my mystical experience. I warned you that it was ineffable.

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