Why Steven Weinberg did and didn't put us on the right track?
A quantum maestro has passed away,
Long live the present and future quantum heroes !
Steven Weinberg 1933-2021
An american physicist of the highest stature, the father of the standard model of electro weak unification in particle physics & 1979 Nobelist Steven Weinberg has passed away last Friday at the age of 88.
I remember reading in 1994 “Towards the final laws of physics” (1986 Dirac Memorial Lectures) while I was a science student at college. It was my first encounter with a simple but dense & insightful expository text (42 pages in-octavo format) about STRING theory. I remember it well as his lecture was published in the same book with another one by Feynman ("Why antiparticles ?") which I found much less understandable!
String theory: a smell of necessity?
I read the Weinberg's lecture in French in a translation by great teachers & physicists : F. Balibar, J. Kaplan, A. Laverne, J-M. Levy-Leblond. One could appreciate the pedagogical annotations and epistemological remarks by them with some answers by Weinberg for the French edition. The 27th footnote below caught my eye as it tended to temper somehow the enthusiasm of Weinberg for string theory in 1986:
"In string theory, this means that you have to sum over the histories of all two-dimensional surfaces that could produce the particular collision process. Now, each history is characterized by ... two functions ... x and g... What you actually have to do to calculate the probability amplitude is to evaluate a numerical quantity I[x, g] for each history and then add up e^(-I[x,g]) for all possible surfaces. The quantity I[x, g] is called the action, and is a functional of x and g...
... This action is the integral of a Lagrangian... just like the one I wrote down for quantum electrodynamics ... You might very well say,' Who ordered that? Who said that that is the correct theory of the world? Maybe you know experimentally that that is the way ordinary strings behave, but so what? What has that got to do with ultimate elementary glitches in spacetime? Why not an infinite number of other terms? Why strings at all?'
This theory is actually rationally explicable in terms of its symmetries in a way that is not immediately obvious. The action (5) has a number of symmetries. Because of the introduction of the metric, just as in general relativity, there is a symmetry under changes in the definition of the coordinates... There is also a symmetry that is a little less obvious and is only true in two dimensions. This is symmetry under a change in the local length scale, a so-called Weyl transformation... And finally there is also the rather obvious symmetry under Lorentz transformations The first two of these symmetries seem to be indispensable to the theory. Without them, you find that when you try to calculate the sums over all surfaces, the results do not make sense. Without these two symmetries, then it seems that either you get negative probabilities or the probabilities don't add up to one. In fact, there are some very subtle quantum effects that can break the symmetries, and unless you put in just the right mix of coordinates and also spin one-half coordinates (which I have not mentioned here), then there will be quantum anomalies that will spoil the symmetries. But, if you're very careful, you can get the quantum anomalies to cancel so that these symmetries remain and then the sum over the histories of these surfaces do make sense.
There is a beautiful branch of mathematics, one of the most beautiful ever invented, that deals precisely with those properties of two-dimensional surfaces that are left unchanged by coordinate and Weyl transformations. This branch of mathematics goes right back to the work of Riemann in the first part of the nineteenth century. Many of its classic results turn out to be just what we need to understand the physics of strings...
It is these ancient theorems that allow you to calculate the sums over the histories of the surfaces. If the symmetries weren't present you couldn't do the calculations, and they would probably give nonsense if you could. So the symmetries seem to be indispensable. Furthermore, and now this is the point, this is the punch line, the symmetries determine the action. This action, this form of the dynamics, is the only one consistent with these symmetries. (There are qualifications*, but what I am saying is close enough to the truth.) In other words, there is nothing else you could have added. There are no other possible terms consistent with these symmetries. This, I think, is the first time this has happened in a dynamical theory: that the symmetries of the theory have completely determined the structure of the dynamics, i.e. have completely determined the quantity that produces the rate of change of the state vector with time. I think it is for this reason, more than any other, that some physicists are so excited. This theory has the smell of inevitability about it. It is a theory that cannot be altered without messing it up; that is, if you try to do anything to this theory, add additional terms, change any of the ways that you define things, then you find you lose these symmetries, and if you lose these symmetries the theory makes no sense. The sums over histories, the sums over surfaces break down and give nonsensical results. For this reason, quite apart from the fact string theory incorporates gravitation, we think that we have more reason for optimism now about approaching the final laws of nature than we have had for some time.
* footnote #27 (by translators): Since 1986, the situation has evolved. On the one hand, string theories have learned to live in a four-dimensional space-time; on the other hand, the number of theories that could be consistent has grown considerably: it may not be over - which somewhat dispels the "smell of inevitability" evoked by Weinberg a little later. (Weinberg's reply: it is true that since 1986, a large number of versions of string theory have appeared on the market. However, some theorists still believe (or at least hope) that these are only different solutions of a few fundamental theories).
[Original footnote in French : Depuis 1986, la situation a évolué. D'une part, les théories des cordes ont appris à vivre dans un espace-temps à quatre dimensions ; d'autres part, le nombre de théories qui pourraient être cohérentes s'est considérablement accru : il n'est peut-être pas fini - ce qui dissipe quelque peu le "parfum de nécessité" évoqué un peu plus loin par Weinberg. (Réponse de l'auteur : il est vrai que depuis 1986, un grand nombre de versions de la théorie des cordes sont apparues sur le marché. Cependant, certains théoriciens croient encore (ou au moins espèrent) qu'il ne s'agit que de solutions différentes d'un petit nombre de théories fondamentales.)]
The Standard Model: the most effective, if not fundamental, quantum gauge field theory
I think the smell of necessity associated with string theory as it was advocated by Weinberg in 1986 has become even more evanescent for most physicists at least, with the passing of time. Meanwhile his first model of electroweak interactions for leptons from 1967 completed with the building of the standard model for all particle physics thanks to the celebrated contributions of Salam, Glashow, and many more invaluable researchers) has enjoyed continued success since the 1972 weak neutral current discovery to 2012 Higgs boson detection announcement. So physicists will be forever grateful to Weinberg as one of the founding farther of the most effective quantum field theory. The word in italics can be taken in the layman sense but also in a technical sense as discussed for instance in this CERN lecture by Weinberg himself .
Steven Weinberg memory is rightly celebrated for his tremendous contribution to particle physics and his insightful expository texts not only on quantum field theories but also on astroparticle physics and cosmology, well on modern physics so to speak. I couldn't do as good a job as american bloggers like Scott Aaronson or Peter Woit did in recent posts, the more so as both had the privilege to interact with him personally. So in the rest of this post I will dwell on a different topic that I can't introduce with a better quotation than this one:
Julius Getman, (colleague of Steven Weinberg at The University of Texas at Austin)
When Iliopoulos asked: "Steve, why didn't you put us on the right track?"
I find it's worth wondering what were (or could have been) some of the conceptual or technical obstacles that (may have) hindered a brilliant and successful theoretical physicist like Weinberg. I don't aim at demeaning his merits of course but I 'd like to show how demanding it is to be a high level researcher, how long it can take to have a chance your theory proves to be useful for empirical science and how dependent one can be of the conceptual tools at ones disposal (not to mention the concrete ones required to see ones model tested)!
I will start with the following well documented difficulty met by Weinberg. It has to do with a crucial piece of the Standard Model, namely the Brout-Englert-Higgs mechanism. It's vividly reported in the following extract:
Weinberg was not the only one of course to have been dettered by the mathematical infinities inherent to the quantum field theories particle physicists were dealing with. One probably had to need the youth and originality of another giant (and alive) physicist (Gerard 't Hooft) to cope with these infinities namely demonstrating the renormalisability of the very specific Yang-Mill-Higgs gauge theory that makes the Standard Model, establishing (his PhD superviser)...
... Veltman’s legacy: when you are short of intuition, sit down and compute. If you are smart, ideas may be revealed from the computations.
I have always been puzzled by this fact: Weinberg being oblivious to his most important (at least in our current empirical context and conceptual mindset) theoretical work at such a crucial moment! Weinberg's own comment refers to a psychological barrier. But I can't help thinking there was another kind of barrier, having to do with a lack of conceptual understanding of the Higgs boson probably best epitomized in Seldon Glashow's colloquial remark ("Really, the Higgs is more like a plumber with duct tape, holding the standard model together") not to mention the former doubts of theoretical physicists on the existence of elementary scalar quantum fields.
The emergence of the spectral Standard Model as a mathematical necessity
Now I would like to remind you of the take (and evolution) of Weinberg about the role of geometry in general relativity and particle physics. It was discussed by Peter Woit in a 2007 post and interestingly commented by his (well ;-) educated physicist community of readers. I summarize the main points with the following extracts (emphasizing with bold type is mine):
In his well-known and influential book on GR, [Weinberg] explicitly tries to avoid using geometrical motivation, seeing this as historically important, but not fundamental. To him it is certain physical principles, like the principle of equivalence, that are fundamental, not geometry. There’s a famous passage at the beginning of the book that goes:
However, I believe that the geometrical approach has driven a wedge between general relativity and the theory of elementary particles. As long as it could be hoped, as Einstein did hope, that matter would eventually be understood in geometrical terms, it made sense to give Riemannian geometry a primary role in describing the theory of gravitation. But now the passage of time has taught us not to expect that the strong, weak and electromagnetic interactions can be understood in geometrical terms, and too great and emphasis on geometry can only obscure the deep connections between gravitation and the rest of physics.
This was written in 1972, just a few years before geometry really became influential in particle physics, first through the geometry of gauge fields, later through geometry of extra dimensions and string theory. I recall seeing a Usenet discussion of whether Weinberg had ever “retracted” these statements about particle physics and geometry. Here’s an extract from something written by Paul Ginsparg, who claims:
back to big steve w., when he wrote the gravitation book he was presumably just trying to get his own personal handle on it all by replacing any geometrical intuition with mechanial manipulation of tensor indices. but by the early 80’s he had effectively renounced this viewpoint in his work on kaluza-klein theories (i was there, and discussed all the harmonic analysis with him, so this isn’t conjecture…), one can look up his research papers from that period to see the change in viewpoint... also around '83-'84 (just before string theory), some particle theorists were thinking about the geometric formulation of anomalies in gauge theory and gravitation... so the viewpoint of particle theorists as resisting the geometrical interpretation is many decades out of date, and was never true of string theorists.
At the risk of being pedantic and conceited I think the 2021 reader should be informed that after more passage of time (and some energy consumed to fuel long computations and distillation of their proper conceptual meaning) one can safely state that strong, weak and electromagnetic interactions have been understood in a geometrical formalism informed by the quantum. I have no idea if Weinberg had any opportunity to get informed about this mathematical demonstration and I wonder what he would have think of its last possible particle physics and cosmological ramifications. My filling is that it can be awfully time consuming to educate oneself on a new mathematical formalism the more so when your time is precious because you are busy with other tasks or simply get older! This is why I imagine most theoretical physicists tend to trust or rely on the take of their close younger or more skilled work mates to decide if it's worth paying attention to a new line of research. It has been reported elsewhere how some people in the past where eager to know what was the last book a big shot of Princeton was supposed to be reading or what had been the reaction or comments of Weinberg to a specific lecture at a conference...
Incidentally it happens the mathematician (whom heart lies with physics too ;-) who initiated a spectral noncommutative framework to achieve a geometrization of the Standard Model in a different way a priori than the string theorists, completed his thesis* in the early 70'. (*A thesis which would offered him a Field medal in 1982 and was based interestingly on the thoughts of Von Neumann on entanglement in quantum physics). It's exactly the time when Weinberg was writing his book on general relativity!
Now let's introduce the so called spectral Standard Model through its most recent review article :
In 1988, at the height of the string revolution, there appeared an alternative
way to think about the structure of space-time, based on the breathtaking
progress in the new field of noncommutative geometry. Despite the success
of string theory in incorporating gravity, consistency of the theory depended
on the existence of supersymmetry as well as six or seven extra dimensions.
Enormous amount of research was carried out to obtain the Standard Model
from string compactification, which even up to day did not materialize. ... Alain Connes [28] laid down the blue
print of a new innovative approach to uncover the origin of the Standard Model and its symmetries... by making space slightly noncommutative
by tensoring the four dimensional space with a space of two points, one
gets a parallel universe where the distance between the two sheets is of the
order of 10^−16 cm, with the unexpected bonus of having the Higgs scalar
field mediating between them. Although this looked similar to the idea of
Kaluza–Klein, there were essential differences, mainly in avoiding the huge
number of the massive tower of states as well as obtaining the Higgs field
in a representation which is not the adjoint. Soon after this work inspired
similar approaches also based on extending the four-dimensional space to
become noncommutative [43, 44, 45, 46, 23].
In this survey we will review the key developments that allowed noncommutative geometry to deepen our understanding of the structure of spacetime and explain from first principles why and how nature dictates the
existence of the elementary particles and their fundamental interactions. In
Section 2 we will start by reviewing the pioneering work... introducing the basic mathematical definitions and structures needed to define a noncommutative space. We summarize the characteristic ingredients
in the construction of the Connes–Lott model and later generalizations by
others. We then consider how to develop the analogue of Riemannian geometry for non-commutative spaces, and to incorporate the gravitational field
in the Connes–Lott model. In Section 3 we present a breakthrough in the development of noncommutative geometry with the introduction of the reality
operator which led to the definition of KO dimension of a noncommutative
space. With this it became possible to present the reconstruction theorem
of Riemannian geometry from noncommutative geometry. Section 4 covers
the formulation and applications of the spectral action principle where the
spectrum of the Dirac operator plays a dominant role in the study of noncommutative spaces. This key development allowed to obtain the dynamics
of the Standard Model coupled to gravity in a non-ambiguous way, and to
study geometric invariants of noncommutative spaces. We then show that
incorporating right-handed neutrinos with the fundamental fermions forces
a change in the algebra of the noncommutative space and the use of real
structures to impose simultaneously the reality and chirality conditions on
fundamental states, singling out the KO dimension to be 6. We show in
detail how the few requirements about KO dimension, Majorana masses for
right-handed neutrinos and the first order condition on the Dirac operator,
singles out the geometry of the Standard Model. In Section 5 we present a classification of finite noncommutative spaces of KO dimension 6 showing the almost uniqueness of the Standard spectral model. In Section 6
we give a prescription of constructing spectral models from first principles
and show that the spectral Standard Model agrees with the available experimental limits, provided that the scale giving mass to the right-handed
neutrinos is promoted to a singlet scalar field. We then show that there exists a more general case where the first order condition on the Dirac operator
is removed, the singlet scalar fields become part of a larger representation
of the Pati–Salam model. The Standard Model becomes a special point
in the spontaneous breaking of the Pati–Salam symmetries. In Section 6
we show that a different starting point where a Heisenberg like quantization condition in terms of the Dirac operator considered as momenta and
two possible Clifford-algebra valued maps from the four-dimensional manifold to two four-spheres S4 result in noncommutative spaces with quantized
volumes. The Pati–Salam model and its various truncations are uniquely
determined as the symmetries of the spaces solving the constraint.
[Submitted on 28 Apr 2019 (v1), last revised 23 Jun 2019 (this version, v2)]
As I already sait I ignore if Weinberg had ever been informed about or got interested in this thread of successful mathematical physics research but one has to recognize the careful thus pretty slow pace of its developments (=theorem delivery) that contrast (in more than one way) with the development of string/ M-theory. So one can suppose that despite tremendous capabilities Weinberg hadn't enough time to study seriously this field. Moreover he probably didn't have the opportunity to get close to a convincing enough noncommutative geometer as the late Robert Brout did.
Let's focus for a while on another quite respected physicist: John Iliopoulos. He also played quite an important part in the emergence of the Standard Model as we saw before. He is well alive hopefully, being a bit younger than Weinberg, and he's quite unique as he provides us at the same time with a long enough experience on the Standard Model and an educated point of view on its spectral perspective. In the past he started with emphasizing the lack of stability under renormalization of any relation among the set of parameters of the Standard Model uncovered in the noncommutative geometric framework. He was also probably attentive early to the development of the spectral action principle (check note #7, page7 of the paper The uncanny precision of the spectral action). Nowadays his most eloquent personal statement is probably :
...non-commutative geometry has come to stay! It is part of gauge theories.
Whether it will turn out to be convenient for us to use, is still questionable. It will depend on our
ability to simplify the mathematics sufficiently, or to master them deeply, in order to get new insights.
Quite independently, let me point out that the spectacular accuracy reached by experiments, as well
as theoretical calculations, has made particle physics a precision science. Therefore, "approximate"
theories are no more sufficient. A discrepancy by a few percent implies that we do not have the right
theory! This strongly restricts the possible ways to go beyond the Standard Model.
On the other hand, the completion of the Standard Model strongly indicates that new and exciting Physics is around the corner. But for the moment, we see no corner! Could non-commutative
geometry show the way out of this dilemma?
As I already wrote, Iliopoulos has never hesitated to express his prejudices and also his doubts about the relevance of this abstract geometrical framework to better understand Higgs boson physics (see this pedagogical video of him from 2017 about that very subject). Nevertheless this hasn't prevented him to get recently involved in working for a proper quantization of the spectral standard model with van Suijlekom & Chamseddine.
Will the spectral Standard Model turn out to be convenient for physicists to use?
I haven't intended to make Weinberg's passing away an awkward opportunity to publicize "my" pet theory but I've tried to highlight one specific theoretical construction based on recently proved mathematical theorems that could deliver a better understanding of the quantum dynamics of the electroweak symmetry breaking and it's possible connection with gravity or spacetime structure. One should also remember that this dynamics has just started to be probed by the LHC!
To know more about what other physicists think about the spectral Standard Model I encourage the young or interested reader to start with the following one hour lecture by Chamseddine where he addresses in a very frank way the audience's questions about the relation similar to Heisenberg noncommutativity of coordinates and momenta in Quantum Mechanics he uncovered with the cosmologist Mukhanov and the crucial help of Connes. He also explains how this relation implies the quantization of the volume of space-time and leads uniquely to identify the underlying geometry, which turns out to be noncommutative. The resulting dynamical theory unifies all interactions including gravity with the Standard Model the effective symmetry at low energies.
It happens that Alain Connes too may be willing to answer inquisitive or naive questions as the following session (with one of the father of string theory in the audience: Gabriele Veneziano) below proves:
The following transcript of the end of the video above (1:51:02) will make a nice conclusion to this last paragraph:
“You should read Riemann ... he had already understood the limitation of his point, of course it should work beautifully in the large but not in the small, the reason was... everything was based on light rays and solid bodies... solid bodies don't make any sense in the very small ... neither do light rays because of the quantum...
We are born in Quantum mechanics... it has been verified, checked so many times you cannot say nature is classical. Nature is quantum and from this quantum stuff we need to understand how our ... very classical (because of natural selection) way of seeing things can emerge... it's very difficult of course...
It's crucial we have doubts and we manifest these doubts… if we are just preaching this is a catastrophe… I admire Gabriele [Veneziano] for that … we spend our life doubting , the chance we are right are tiny”.
Conclusion
I hope I brought enough material to my courageous reader to imagine [her, his] own answer to the original question of this long post.
