Why Steven Weinberg did and didn't put us on the right track?
Steven Weinberg 1933-2021
String theory: a smell of necessity?
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:
When Iliopoulos asked: "Steve, why didn't you put us on the right track?"
... Veltman’s legacy: when you are short of intuition, sit down and compute. If you are smart, ideas may be revealed from the computations.
E.G. Floratos, J. Iliopoulos (2021)Acta Phys. Polon. B 52 (2021) 745
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
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.
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)]
...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?
Will the spectral Standard Model turn out to be convenient for physicists to use?
“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.