The Great Yogurt Project: Models and Symmetry Principles in Early Particle Physics

Abstract

According to the received view of the development of particle physics, mathematics, and more specifically group theory, provided the key which, between the late 1950s and the early 1960s, allowed scientists to achieve both a deeper physical understanding and an empirically successful modeling of particle phenomena. Indeed, a posteriori it has even been suggested that just by looking at diagrams of observed particle properties (see Fig. 1) one could have recognized in them the structures of specific groups (see Fig. 2). However, a closer look at theoretical practices of the 1950s and early 1960s reveals a tension between the employment of advanced mathematical tools and the “modeling” of observation, if the term “model” is understood as a construction allowing for the fitting and predicting of phenomena. As we shall see, the most empirically successful schemes, such as the “Gell-Mann and Nishijima model” or the “eightfold way”, were mathematically very simple, made no use of group-theoretical notions and for quite a time resisted all attempts to transform them into more refined mathematical constructs. Indeed, the theorists who proposed them had little or no interest in abstract approaches to mathematical practice. On the other hand, there were a number of particle theorists who did care about and employ group-theoretical notions, yet not primarily as tools to fit phenomena, but rather as a guide to uncover the fundamental principles of particle interactions. Moreover, these theorists did not regard all groups as epistemically equivalent, and instead clearly preferred those transformations related to space-time invariances over all others. These authors also often made a distinction between purely descriptive “models” and the “theories” they were (unsuccessfully) trying to build and which in their opinion would provide a deeper understanding of nature. Nonetheless, they expected their “theories”, too, to be empirically successful in describing observation, and thus to also function as “models”. In this sense, like their less mathematically-inclined colleagues, they also saw no clear-cut distinction between “modeling” and “theorizing” particle phenomena. In my paper I will discuss the development of these theoretical practices between the 1950s and the early 1960s as examples of the complex relationship between mathematics and the conceptualization of physical phenomena, arguing that, at least in this case, no general statements are possible on the relationship of mathematics and models. At that time, very different mathematical practices coexisted and the epistemic attitudes of physicists towards theoretical constructs could depend both on the assumptions and goals of the individual authors and on the specific mathematical methods and concepts linked to the constructs.

Introduction: The Coral Gables Conferences on “Symmetry Principles at High Energy” and the Yogurt Project

Between the years 1964 and 1968 five conferences on “Symmetry Principles at High Energy” were held at the University of Miami in Coral Gables.¹ Behram Kursunoglu, particle theorist and professor in Miami, who, in 1965, became founding director of the local Center for Theoretical Studies, organized the meetings.² As explained by Kursunoglu and his co-editor Arnold Perlmutter in the short preface to the proceedings of the first conference, the motivation for the initiative was to reflect and discuss theoretical methods that had begun to be employed in particle physics “during the past few years” and which were “sufficiently novel to warrant frequent gatherings of experts in particle physics.”³ These methods, referred to as “symmetry principles,” were largely based on the use of group theoretical structures.

Born and raised in Turkey, Kursunoglu jokingly inserted in the proceedings of the second conference a series of anecdotes about Nasreddin Hoja, a traditional jester figure from Turkish folklore. Among them was this story:

One morning a woodcutter saw Hoja by the edge of a lake, throwing quantities of yeast into the water. ‘What the devil are you doing, Hoja?’ he asked. Hoja looked up sheepishly and replied, ‘I am trying to make all the lake into yogurt.’ The woodcutter laughed and said, ‘Fool, such a plan will never succeed.’ Hoja remained silent for a while, and stroked his beard. Then he replied, ‘But just imagine if it should work!’⁴

In 1968, at the fifth and last conference of the series, Israeli theorist Yuval Ne’eman was asked to hold a final talk summing up the results of all five meetings and, at the end of his speech, he expressed his appreciation for Hoja:

I also express general sentiment when I say that we have all especially appreciated the proceedings of the second conference, with Nasreddin Hoja’s contributions. I wonder whether Kursunoglu can sometime tell us where Hoja learned his physics. I particularly appreciated Hoja’s Yogurt project.⁵

Ne’eman did not explain why he especially liked the yogurt story among the various Hoja anecdotes scattered in the 1965 proceedings, but it certainly suggests a disillusioned attitude towards the research program brought forward in the first five Coral Gables conferences: a dream many theorists had believed in, but that so far had not paid out. This impression is strengthened by the words immediately following the mention of the yogurt project:

I only have to add now a piece of information which may not be known to you, and this is that I am not only the summarizer of these five conferences but I am also the undertaker responsible for their lying in peace forever. The series of conferences on ‘Symmetry Principles at High Energy’ is hereby closed. I think Prof. Kursunoglu has felt that they represented such a nice achievement that one should not risk stretching it too much.⁶

Ne’eman explained that the next Coral Gable conference would be devoted to a new topic: “Fundamental Interactions.”⁷ From today’s point of view it seems difficult to account for Ne’eman’s downbeat mood, or for the decision to put an end to the conferences on symmetries, since the 1960s appear in retrospect as the time in which symmetry principles, especially in the form of group theory, became one of the main driving forces behind theoretical research in particle physics. Indeed, recollections of theorists, reflections of philosophers, and even some works by historians of science have expressed the conviction that, in the 1950s and 1960s, symmetry principles and group theoretical methods, which at the time were not usually regarded as equivalent, proved to be an effective heuristic tool to mathematically represent the many newly observed particle phenomena and allow for the prediction of as yet unobserved ones.⁸ From this background, theoretical developments in early particle theory have often been framed in terms of a progressing, successful application of abstract principles of mathematical invariance to the representation and explanation of particle phenomena. As I have shown elsewhere for the early 1950s, however, a closer analysis of primary historical sources does not support this claim.⁹ In the present paper I will focus on the late 1950s and early 1960s, showing how in that period, too, the relationship between group theory and the mathematical conceptualization of phenomena was not as straightforward as usually presented when we look at it today.

The main thesis of this paper is that theorists engaging in the construction of models or theories of particle phenomena in the 1950s and early 1960s did not in general regard abstract group theoretical constructs as particularly effective tools for the job. Some authors made no use of the concepts of group theory at all, while others only employed them in a very selective way, recurring only to those group structures that could be related to the invariances of space or space-time, like rotations, mirroring or translations. Interestingly, authors following the latter approach tended to employ the term ‘model’ to indicate purely descriptive constructs, and ‘theory’ to characterize those constructs they hoped might eventually be seen as explaining the laws behind particle phenomena. It was in building such ‘theories’ that they believed space-time invariances would provide an effective guideline. Thus, in the 1950s and early 1960s notions of group theory were primarily employed in particle physics by a specific group of theorists who sought not just to fit observation, but also to mathematically capture the features of fundamental microphysical interactions. Incidentally, their efforts were largely unsuccessful. I will support my claim by discussing two historical constellations: proposals made in the years 1953–1956 aimed at expanding the Gell-Mann-Nishijima model of particle classification into a ‘theory’ whose invariances were similar to those of space-time, and the systematic attempts to connect internal particle symmetries (e.g. isospin, strangeness) to space-time invariances which were discussed at the Coral Gable Conferences in the early 1960s.

In the next two sections of this paper, I will address some methodological issues regarding, first, the terms ‘model’ and ‘theory’ and, second, the distinction between different mathematical practices. In context of the second topic I will sketch the practices linked to rotation phenomena in quantum physics, and the way in which the formalism of non-relativistic spin provided the template first in the 1930s for describing nuclear interaction and then in the 1950s for formulating the first empirically successful scheme of particle classification: the so-called Gell-Mann-Nishijima model. After that, I will discuss the (ultimately unsuccessful) attempts made during the 1950s of either replacing or embedding that model in a broader mathematical construct based on invariances similar to those of relativistic space-time. In section “The Path from SU(2) to SU(3), or: Did Particle Physicist Know Group Theory?,” I will summarize the way in which, in the early 1960s, the Gell-Mann-Nishijima model was embedded in an empirically successful classification based on the group SU(3) (the eightfold way and the quark model), and that this happened through a complex path essentially unrelated to the projects described in section “The Search for a Theory of Isospin and Strangeness in the 1950s.” Finally, sections “Beyond SU(3)—The Mathematical Marriage of Space-Time and Internal Symmetries” and “The Rise and Fall of SU(6)” discuss the research program unfolding in the five Coral Gables conferences on “Symmetry Principles at High Energy,” following the (ultimately unsuccessful) attempts to expand the eightfold way and the quark model by combining them with the symmetries of relativistic space-time.

‘Models’ and ‘Theories’ as Actors’ Categories in Early Theoretical Particle Physics

There is in philosophy of science a long tradition of trying to characterize both the distinction and the mutual relationship between models and theories, and in particular their specific functions within scientific practice.¹⁰ No consensus exists so far on whether and how a distinction between the two may be drawn, and in the present study I will assume that all theoretical practices analyzed are equally well characterized as modeling or theorizing. However, I will endeavor to trace an (admittedly vague) distinction made by some, though by far not all, historical actors between ‘models’ (or ‘schemes’) on the one hand and ‘theories’ on the other. I will argue that, while both ‘models’ and ‘theories’ were seen as mathematical constructions whose prediction were expected to fit observation, the latter could be associated with an additional explanatory function, or at least with the hope that the mathematical structures involved in some way captured features of natural order beyond observable phenomena. Theorists endorsing this position in general did not attempt to bring forward arguments to support it, but usually simply expressed their belief that certain mathematical principles had a closer relationship to fundamental physical laws.

Theorists who drew the distinction sketched above between ‘models’ and ‘theories’ were also the ones who made explicit use of group theoretical structures, and more in general of abstract mathematical notions. However, what is particularly interesting is that they did not regard all group-theoretical concepts as equally relevant from a physical point of view, but rather pinned their hopes on symmetries linked to invariances of three-dimensional space or relativistic four-dimensional space-time. Thus, the focus was for example on the group O(3) (rotations and mirror transformations in three dimensions), on the Lorenz group (rotations in four-dimensional space-time), or on the Poincaré group (rotations and translations in four-dimensional space-time). The authors did not offer any particular reason why these specific mathematical structures should be epistemically preferred and, in the end, their efforts in constructing theories along these lines were largely unsuccessful. Yet the research program which Ne’eman, and later on also Kursunoglu, compared to Hoja’s yogurt project was in so far heuristically fruitful, that it generated a number of new, more limited theoretical approaches to studying high energy phenomena by means of group-theoretical methods.

In conclusion, if one asks what role abstract mathematical notions, and more specifically group theory played in theoretical practices of early particle physics, the answer is that they were predominantly used by authors who made a distinction between ‘models’ and ‘theories,’ and expected their constructs to be ‘theories’ which expressed fundamental features of nature. However, these theorists did not regard all invariances as fit to serve that aim, and rather focussed on group theoretical structures that could be associated to the symmetries of space-time. As we shall see, from this point of view empirically successful constructs were perceived as in some sense arbitrary if they did not conform to the expectation that all of the symmetry in nature should be similar to that occurring in space-time. Other theorists instead, although in practice employing the same mathematical methods, did not refer to them in terms of abstract concepts like group representations. Because of this, it is important in the analysis to make a methodological distinction between the performance of certain mathematical operations and the explicit use of the relevant abstract notions. This is the topic of the next section.

Mathematical Practices of Rotations and the Emergence of the Gell-Mann-Nishijima Model of Particle Classification

In the following pages I will often speak not of mathematics, but rather of mathematical practices. This choice allows for much more flexibility when trying to reconstruct what historical actors were doing (or not doing) when making use of mathematical formalisms and notions, and what aim they were (or were not) pursuing. An example of this distinction that is of relevance for our topic is the rotations of objects in three-dimensional space. Rotations of solid objects around themselves were known and had been studied since Antiquity, and at the latest in the early modern period geometrical and analytical formalisms for modeling them were developed.¹¹ During the nineteenth century this formalism allowed both to study in greater detail rotating bodies, and to connect those experiences to more abstract notions of rotations, such as the interplay of electric and magnetic forces.¹² With the emergence of vector analysis around the end of the nineteenth century, the rules of rotation for solid bodies were interpreted as pertaining to the mathematical objects labeled as vectors, while other rules were valid for axial vectors or tensors.¹³ Another mathematical perspective was added by group theory, which also emerged in that period, and in whose context the rules for rotations of vectors, tensors and other objects could be conceptualized as different representations of a more abstract notion: the group of rotations in three-dimensional space.¹⁴

It is neither necessary nor possible to discuss here the details of these complex historical developments, but it is important to note that, despite the emergence of all these new mathematical reflections, the rules for rotating a vector in space remain quite simple and can be employed without any knowledge of vector analysis or group theory. Moreover, even if a person has (some) knowledge of those fields of mathematics, this does not imply that she or he is conceiving the practice of analytically representing the rotation of a force or a velocity as the employment of a representation of the group of rotations in three-dimensional space. This rather trivial observation becomes relevant for our topic when we consider the subject of particle spin, which is of paramount importance for the history of particle theory.

During the early 1920s, in the context of emerging quantum physics, the spectroscopic and chemical properties of atoms were described in terms of a new degree of freedom attributed to electrons, which came to be referred to as ‘spin,’ as it was initially tentatively thought of as linked to a rotation of the electron around its axis.¹⁵ On the basis of spectroscopic results, Wolfgang Pauli formally represented the new degree of freedom by means of a two-component object (Pauli spinor). Electrons were spin doublets, since they only had two possible spin states: up or down. The rules according to which Pauli spinors transformed under space rotations were expressed in the form of the so-called Pauli matrices, and Hermann Weyl later recognized these as a representation of the group SU(2).¹⁶ With the emergence of a Paul Dirac’s quantum relativistic theory of electrons, Pauli spinors came to be interpreted as non-relativistic versions of the transformation properties of the quantum fields associated to electrons (and other particles) under relativistic space-time rotations. These in turn could be thought of in terms of an ‘intrinsic angular momentum’ which formally took the value 1/2 for some particles (e.g. electrons, protons, neutrons) and 0 or 1 for others (e.g. pions, photons). In the non-relativistic limit, the intrinsic angular momentum determined the way in which particles transform with respect to rotations in three-dimensional space. Thus, what had started as a simple, ad-hoc formalism to represent spectroscopic data had eventually been interpreted as a manifestation of mathematically more complex invariance properties of relativistic space-time. However, at first only mathematicians took interest in the more refined group-theoretical implications of these developments, while physicists only employed those mathematical tools strictly necessary for their work. More specifically, from the 1940s onward, scientists working in atomic, molecular and nuclear physics became fully familiar with the Pauli spinor formalism, which played an important role in their fields, but they did not necessarily know or care much about the relevant group theoretical notions, such as SU(2). In turn, the Pauli formalism provided the template for isospin, a new particle property originally introduced in nuclear physics the 1930s and extended to the many new particles discovered in cosmic ray and accelerator experiments from the late 1940s onward.

The formalism of isospin had been developed during the 1930s and 1940s in formal analogy to intrinsic angular momentum to mathematically represent interactions in atomic nuclei.¹⁷ To this end, the proton and the neutron were represented as two components of an ‘isospinor’ formally, but not physically, analogous to a Pauli spinor. This formalism proved useful in nuclear physics, although until the 1950s it was not regarded as expressing a physical quantity. Later on, the three pions, which had positive, negative and zero charge, were assumed to form an isospin triplet transforming like a vector under rotations in ‘isospace.’ Of course, rotations in ‘isospace’ had nothing to do with rotations in real space, yet they were formally fully analogous to them, so that, to manipulate isospin, it was not necessary to know anything about group theory, but it sufficed to be familiar with the formalism of vectors and (two-component) spinors commonly used in quantum physics to represent angular momentum and spin. In the early 1950s it was confirmed that further particles existed and were capable of strong interactions. They were variously known as V-particles, tau-mesons, and later on even as ‘curious’ or ‘strange’ particles, and experimental results on their possible charges, masses and decay modes were in constant flux.¹⁸ Nonetheless, some theorists attempted to come up with schemes describing their properties and interactions, and in this context some authors tentatively attributed isospin to them.

The first proposals of this kind were made in 1951 by Japanese theorists, among them Kazuhiko Nishijima, and their approach was soon taken up by U.S. American physicists, among them Murray Gell-Mann.¹⁹ For the most part, these authors tried to fit the (rapidly shifting) experimental results on decay modes of the new particles by grouping them into isospin doublets, triplets or singlets without making use of abstract group-theoretical notions. In the years 1955–56, an approach developed independently by Nishijima and by Gell-Mann turned out to be empirically quite successful.²⁰ Gell-Mann was the first one to make use of graphic representations of the model, placing the symbols he had introduced for labeling the new particles in a diagram with mass and electric charge displayed on the vertical and horizontal scales (see Fig. 1).²¹ Gell-Mann described his construct by stating that the new particles formed isospin multiplets which were ‘displaced’ with respect to those of the old particles (protons and neutrons, pions) and the amount of displacement was equal to the value of a new property: strangeness.²² For our topic, the details of Gell-Mann’s proposal are not important: what counts is the fact that it was presented in a very simple mathematical form which left unchanged the isospin formalism, only ‘displacing’ some multiplets. This scheme became known as the ‘Gell-Mann-Nishijima model’ or ‘strangeness model’ and in the second half of the 1950s was accepted as an empirically valid classification of new and old particles.

Fig. 1

© Annual Reviews, Inc., all rights reserved

Diagram representing the properties of the “strange” particles in: Murray Gell-Mann and Arthur H. Rosenfeld, “Hyperons and Heavy Mesons (Systematics and Decay),” Annual Review of Nuclear Science 7 (December 1957): 407–78, here 415, Fig. 1; permission conveyed through Copyright Clearance Center Inc.

The Search for a Theory of Isospin and Strangeness in the 1950s

Both before and after the emergence of the strangeness scheme, a small number of theorists attempted to employ considerations based on mathematical invariances and group-theoretical notions to classify the new particles.²³ In this section I will discuss a few such proposals and in particular the motivation brought forward by their authors, showing how they preferred groups identical with or related to those of (relativistic or non-relativistic) space-time transformations. One of the earliest and most persistent supporters of this approach was Abraham Pais who, already in 1953, sought a way not only to attribute isospin to the new particles, but also to take the isospin variable “seriously.”²⁴ Significantly, this meant for him to connect isospin to space-time transformations described by the Lorentz group. However, he noted, “the exploration of the irreducible representations of the Lorentz group shows that there is no such freedom left,”²⁵ so he opted for regarding the isospin formalism as part of a three-dimensional “ω-space” in which, using the relevant group-theoretical structures, the formal equivalents of both angular momentum and parity could be defined. Pais described his proposal as a “theory”²⁶ and explained his key assumption so:

The element of space-time is not a point but is a manifold (“ω-space”) which is carried into itself by all transformations of a three-dimensional real orthogonal group. Now this is a fancy way to talk about a simple manifold namely a two-dimensional sphere. Yet the phrase is used advisedly to direct attention to the group rather than to the sphere seen as a metric object or as being embedded in a three-dimensional Euclidean space.²⁷

The details of Pais’ proposal cannot be summarized here, yet we note how, on the one hand, he underscored the abstract notion of group as opposed to the intuitive image of a two-dimensional sphere, while on the other hand presenting his “ω-space” as something physical, and not as a purely formal construct, as it had been usually characterized so far. However, stating that each point in space-time is associated to a manifold did not add anything from the mathematical point of view: It was only a way to take isospin “seriously,” attributing to it an epistemic status analogous to that of space-time and thus, at least in Pais’ view, constructing not just an empirically successful ‘model,’ but an explanatory “theory” of elementary interactions. As we shall see later on, the explicit employment of certain group-theoretical structures in theoretical practices was often related to the wish of connecting to space-time. In other words, Pais’ approach was as empirically motivated as those of Gell-Mann and Nishijima, but made use of more refined mathematical tools, which Pais tried to interpret as expressing underlying physical structures.

Pais’s paper was among the earliest proposals on how to attribute isospin to the new particles, and some of its features served as a starting point for the schemes of Nishijima and Gell-Mann, yet those authors only took up specific points of Pais’ theory, and not its general group-theoretical approach. In the same year, Pais developed his ideas further, both to relate to new experimental results and to explore the physical–mathematical implications of his earlier theory.²⁸ He presented his reflections as an initial move to approach a form of theorization guided by mathematical invariances: “The present work must, therefore, be viewed as a first step in employing new invariance principles. If this direction of approach proves fruitful it should be followed, by further refinement.”²⁹ As in the first paper, Pais spoke of his “theory,”³⁰ and described the proposals by other authors, like Nishijima or Gell-Mann as “models.”³¹

In 1954 Pais brought forward yet another, expanded version of his theory of ω-space. He explained that the previous scheme was “too narrow,”³² and that ω-space should have not three, but four dimensions, corresponding to the orthogonal group O(4) of rotations in four-dimensional Euclidean space, which Pais noted was closely related to the Lorentz group of rotations in relativistic space-time.³³ In fact, at the end of the paper Pais mentioned attempts to relate space-time with isospin more fundamentally then what he had proposed, suggesting that his efforts might provide indications on the properties of a “more complete theory.”³⁴ Interestingly, in this paper he referred both to his own scheme and to the simpler construct of Gell-Mann as “models,”³⁵ using the word theory only to describe the “more complete theory” he was still looking for. In conclusion, Pais saw his proposals not just as attempts to describe observations, but also as first steps in constructing a theoretical structure expressing fundamental features of nature using as a guideline invariance principles and group-theoretical structures similar to those of space-time. In conclusion, Pais, Gell-Mann and Nishijima followed the same research program, whose primary goal was to come up with a mathematical classification of observed particle phenomena. Pais, however, tried to interpret some of the mathematical tools used in this process as related to underlying physical features, while Gell-Mann and Nishijima did not raise any such claims. The interesting point is that, for some reason which is today difficult to grasp, both Pais and later authors saw as potentially physically significant only those group-theoretical structures linked to space-time transformations.

Another author who, like Pais, attempted early on to employ space-time symmetries to classify the new particles was the Polish theorist Jerzy Rayski, who in 1954 proposed that “for every irreducible representation of the group of rotations and reflections corresponds a particle type.”³⁶ If this hypothesis should be correct, he explained, “the situation will be more satisfactory from a group-theoretical viewpoint, moreover, we shall possess a clue for understanding the existence of various types of particles and their properties.”³⁷ Like Pais’ ideas, Rayski’s proposal was quite short-lived and soon, as described in the previous section, the Gell-Mann-Nishijima strangeness model was recognized as empirically successful. Yet the strangeness scheme was not regarded by physicists as expressing fundamental properties of nature, and in 1956 Robert Oppenheimer, in his introduction to a theoretical session at the sixth Rochester conference on “High Energy Nuclear Physics,” made a parallel between that model and the development of non-relativistic and relativistic spin from spectroscopy:³⁸

Perhaps, using an analogy, one may say that [with the strangeness model] we are at a stage corresponding to the finding of the duplexity of atomic spectra, but not yet at the point of the discovery of electron spin, and certainly not at the stage of Dirac’s theory of the electron.³⁹

In the late 1950s, various authors attempted to develop the Gell-Mann-Nishijima model into a construct expressing the structure of subatomic physics, and, once again, group-theoretical considerations were employed to that end. However, just like in the case of Pais and Rayski, the groups chosen as guideline in this enterprise were those connected to space-time invariances. One might assume that, as suggested by Oppenheimer, the reflections of theorists were shaped by the example of spin, which, as we saw above, had emerged as an ad-hoc formalism to be revealed as expressing transformation properties of relativistic space-time.

The proposal by Pais served as the starting point for a new approach by Rayski,⁴⁰ as well as for the work of two theorists, the French Bernard d’Espagnat and French-Polish Jacques Prentki, both based at CERN at that time, who published a paper entitled “Mathematical Formulation of the Model of Gell-Mann.”⁴¹ In their paper, they compared the “theory” by Pais with the “model” by Gell-Mann, explaining how the first one was mathematically satisfactory, but experimentally problematic, while the latter one described experimental data well, but lacked a theoretical basis.⁴² To solve this problem their idea was to employ the same isospin space as Gell-Mann, but consider also mirror transformations in it, and then use the requirement of invariance of strong interactions with respect to this larger group to determine all possible particle fields, deducing the Gell-Mann model from invariance principles. In this way, they concluded, Gell-Mann’s “model” would receive the support of a “theory.” The two authors motivated their research by characterizing Gell-Mann’s model as “arbitrary” and in need of a “more mathematical formulation.”⁴³ However, they gave no reason why the model should be regarded as arbitrary or not mathematical enough, since it was quite efficient in fitting phenomena. It was purely their conviction that an explanatory construct should have specific mathematical features that led them to dismiss Gell-Mann’s model as arbitrary.

The year 1956 also saw the publication of a paper by Julian Schwinger on a “Dynamical Theory of K Mesons” in which yet another extension of isospin space to four dimensions was proposed.⁴⁴ Starting from the Gell-Mann-Nishijima model, Schwinger listed experimental results which in his opinion supported the view that the (approximate) invariance under rotations in the usual three-dimensional isospin space was a leftover of a rotational symmetry in a higher, four-dimensional Euclidean space, a symmetry broken by the presence of pion interactions.⁴⁵ Another attempt at expanding the Gell-Mann-Nishijima classification was made by the Brazilian theorist Jayme Tiomno, who spoke of the “scheme” by Gell-Mann and Nishijima, referring to his construct as a “theory”:

It is shown that the usually accepted Gell-Mann-Nishijima scheme is not unique and that another scheme [...] is possible. A theory is developed based on general symmetry principles, with this new scheme, and it is shown that [...] it is equivalent to Schwinger’s 4-dimensional (in isotopic spin space) theory.⁴⁶

These examples should have shown how, during the 1950s, there was a small but active group of theorists who employed abstract group-theoretical concepts in their work, and who were convinced that the structures related to space-time invariances provided a privileged means not just to fit observation, but also to better understand fundamental interactions, It was not the first, but rather the latter goal which motivated the use of group theory. At the same time, though, empirically successful approaches to fit phenomena arose from the work of theorists like Gell-Mann or Nishijima, who neither made use of group-theoretical notions, nor expressed a belief that any specific invariance would possess a special epistemic status in the exploration of microphysical interactions.

Meanwhile, in 1956, a very important event had taken place: the discovery of the violation of left–right invariance in weak interactions.⁴⁷ This development plays a role in our story because it prompted theorists to devote more attention to both symmetry and symmetry-breaking, and because it lowered the (so far extremely high) status of space-time invariances, letting it appear more plausible that internal symmetries like isospin and strangeness might be somehow connected to space-time transformations.⁴⁸ Thus, approaches like those of Pais or d’Espagnat and Prentki became more popular in the theoretical community and, as we shall see in the following pages, in the early 1960s the main agenda of group-theoretically-minded theorists became the establishment of a close connection between internal symmetries, like isospin and strangeness, and space-time invariances.

The Path from SU(2) to SU(3), or: Did Particle Physicist Know Group Theory?

Among the proposals made in the 1950s for expanding the Gell-Mann-Nishijima model only a few made use of group-theoretical notions. A different, more successful approach was to tentatively regard strongly interacting particles as composites of a small number of more fundamental objects. In 1956, the Japanese theorist Shoichi Sakata made the most influential proposal in this direction regarding only protons, neutrons and the newly discovered Lambda particle as elementary particles.⁴⁹ The “Sakata model”, as it was (and still is) usually referred to, had an implicit group-theoretical structure of type SU(3), which Sakata’s colleagues and students from the Nagoya school soon explicitly discussed in attempts to expand the model and compute its implications.⁵⁰ These results were presented at the tenth Rochester Conference on High Energy Physics held at CERN in 1960, so that at the latest at that point they became well known to the international particle community.⁵¹

Because of the interest in the Sakata model, attention was drawn to the SU(3) structure, and in 1961 Gell-Mann and Ne’eman independently proposed a classification scheme for strongly interacting particles based on that group.⁵² The scheme combined the isospin singlets, doublets and triplets of the Gell-Mann-Nishijima model into two ‘octets’ and Gell-Mann, who was always good in coming up with catchy labels, called it “the eightfold way” in reference to Zen Buddhism. It is interesting to note that neither Gell-Mann nor Ne’eman represented their model in graphic diagrams. In fact, the earliest diagrams of the eightfold way I could locate appeared in the proceedings of the first Coral Gables Conference on Symmetry Principles at High Energy Physics (see Fig. 2).⁵³ The eightfold way is still valid today, albeit as part of a more complex overarching theory of strong interactions, and is represented diagrammatically in most manuals of particle physics as in Fig. 2. Both Gell-Mann and Ne’eman later claimed to have arrived at SU(3) fully independently from the Sakata model, although, as we saw, in 1961 that structure was well known to theorists.⁵⁴ Apart from issues of priority, though, Gell-Mann and Ne’eman in their papers indeed introduced SU(3) through a path which had little to do with Sakata’s idea of a small number of elementary building blocks, and was instead inspired by a new approach to invariance which had been developed in the later 1950s and was referred to as ‘local gauge invariance.’ It is beyond the scope of this paper to discuss Gell-Mann’s and Ne’eman’s route to SU(3), but a short description of local gauge invariance is necessary, as it was yet another kind of symmetry principle striving to combine internal degrees of freedom with space-time variables.

Fig. 2

© Macmillan Publishers, all rights reserved

Diagrams of strongly interacting particles as multiplets of SU(3) from two contributions to the proceedings of the first Coral Gable Conference on Symmetry Principles in High Energy Physics. a is from Robert Adair, David Barge, W. T. Chu, and Lawrence Leipuner, “The Hunting of the Quark,” in Coral Gables Conference on Symmetry Principles at High Energy, ed. Behram Kursunoglu and Arnold Perlmutter (San Francisco: Freeman, 1964), 36‒44, here 43. © Macmillan Publishers, all rights reserved. b is from Sidney Meshkov, “Comparison of Experimental Reaction Cross Sections with Various Relations Obtained from SU₃,” in Coral Gables Conference on Symmetry Principles at High Energy, eds., Behram Kursunoglu and Arnold Perlmutter (San Francisco: Freeman, 1964), 104‒22, here 114.

Although Pais had spoken of each point in space-time as being an isospace manifold, he and other authors had conceived the symmetries of particles as made out of two mutually independent rotations, one in space-time and the other one in isospace. In 1954 Chen Ning Yang and Robert Mills instead proposed an invariance with respect to rotations of the isospin variable dependent on the space-time coordinate (“local gauge invariance”).⁵⁵ This kind of transformation was called “local” because it changed with space-time location. Yang and Mills had proposed a local version of the usual isospin transformation with its SU(2) structure, but this approach turned out to be too simple to accommodate the variety of particles and interactions observed at the time, and later authors expanded it, for example through local versions of the extended isospace variables proposed earlier on, or adding local rotations with respect to the strangeness variable.⁵⁶ However, the groups employed in local gauge invariance were still those of rotations in space, space-time or in Euclidean spaces with more than three dimensions. Instead, in 1961, Gell-Mann and Ne’eman contemporarily but independently brought forward proposals in which the local gauge invariance had the same group-theoretical structure as the Sakata model: SU(3). They arrived at this scheme by setting aside the idea of turning isospin into a non-local space-time-like variable, and instead looking for a local gauge invariance fitting known particle multiplets. Because of this closeness to observation, it is not surprising that they arrived at a similar solution as Sakata, though not exactly the same one.⁵⁷ However, the main aim of their plan did not work out, and local gauge invariance soon dropped out of the model, to make a comeback only a few years later, in a quite different form. What remained, were the SU(3) octets: the eightfold way.

Today, when looking at the octet structure of SU(3) (Fig. 2) and at the diagram drawn in 1957 by Gell-Mann and Rosenfeld (Fig. 1), one is tempted to read into the 1957 image the octet structure of the eightfold way, and then wonder how come no-one had made the connection before Ne’eman and Gell-Mann did. This retrospective, anachronistic perspective is often found in the recollections of the historical actors, who blamed themselves and their colleagues for not having come up earlier with the SU(3) structure. Ne’eman explicitly marveled that his colleagues had not deployed group-theoretical knowledge already available since the nineteenth century, noting that most of them had attended the lectures Giulio Racah had given on that topic at Princeton in 1951.⁵⁸ Ne’eman explained he knew little of group theory, and spent the summer of 1960 in systematically going through lists of group representations searching for the one fitting his scope, and eventually discovering the octet representation of SU(3).⁵⁹ Gell-Mann was among those who had attended Racah’s lectures, but explained in his recollections that he could not pay attention to their content because of Racah’s strong accent:

In 1951 I attended the beautiful lectures by Giulio Racah [...] but I did not really understand the material. The reason was not that the lectures were not elegant, or that they were not explicit. The problem was his accent. Of course I understood the words; I have no trouble following English spoken with a foreign accent. But his accent was so remarkable that I could not hear the substance. Every English word was pronounced with a perfect Florentine accent. For example he would say: ‘Tay vah-loo-ay eess toh eeg’ (The value is too high). So I never learned about Lie algebras, and I had to rediscover them.⁶⁰

Gell-Mann discussed at length how he allegedly rediscovered SU(3) on his own. Moreover, he complained that mathematicians teaching to physicists “give only trivial examples” instead of discussing potentially useful methods.⁶¹ Perhaps because of the colorful descriptions of Gell-Mann’s and Ne’eman’s alleged path to group-theoretical knowledge, historians have assumed that SU(3) remained hidden so long because physicists knew too little group theory, and its use by Gell-Mann and Ne’eman came to be described as a success of the application of group-theoretical methods to quantum physics.⁶²

Yet this explanation is much too simplistic: in the 1950s the idea of employing SU(3) to fit particle phenomena was in no way straightforward, and the initial failure to do so was not due to theorists’ ignorance of group theory. As we saw in the previous section, theorists interested in group-theoretical extensions of the isospin schemes were quite knowledgeable and, had they had any interest in SU(3), they could certainly have informed themselves like Ne’eman had done. In 1956 even Racah himself wrote a short paper on d’Espagnat and Prentki’s theory, and it is extremely improbable that he did not know about the group SU(3).⁶³ Therefore, despite all of the claims of what theorists would have used SU(3) after the fact, if only they had known about it, this was not the case: SU(3) was known at least to some of them, who however did not choose to employ it. The reason why none of these authors took interest in SU(3) has rather to be sought in the specific aims and premises of their mathematical practices. As we saw in section “The Search for a Theory of Isospin and Strangeness in the 1950s,” they were not trying to fit particles to some symmetric scheme regardless of its type, but attempted to find an invariance which was not only empirically plausible, but also—in their perception—physically satisfying, in that they could regard it as expressing features of reality. For them, this meant a symmetry displaying formal analogies or direct connections to space-time transformations. Since SU(3) had no relationship to those transformations, authors like Pais, d’Espagnat and Prentki, who most probably knew about it, never thought of employing it. On the other hand, Sakata and his collaborators saw mathematical invariances rather as tools to fit phenomena than as means to explain them, and eventually employed SU(3) to classify particles.

So we see how speaking in general of an application of ‘symmetry principles’ or ‘group theoretical methods’ to physics is not a heuristically fruitful way to frame the historical-epistemological constellations discussed in the previous sections, as the situation is much more complex. For example, Gell-Mann and Ne’eman, in developing the eightfold way, were indeed following a symmetry principle, namely local gauge invariance, yet it was a thoroughly different one than those that had guided Pais or d’Espagnat and Prentki. All the same, both research programs were shaped by an interest in connecting internal and space-time symmetries, and not of deploying group theory in general to fit phenomena. For theorists in the 1950s and 1960s, not all groups were equal.

A few years after the emergence of the eightfold way, in 1964, the quark model was proposed on the basis of purely theoretical considerations.⁶⁴ As shown above, in the eightfold way particles had been grouped into octects (and later also decuplets), which were representations of SU(3). However, so far no particles had been assigned to the smallest SU(3) representation, which is the triplet. Thus, the question arose whether still unobserved particles might exist that corresponded to the triplet and might perhaps take up the role of fundamental building blocks, as Sakata had envisaged.⁶⁵ Proposals in this sense were advanced, once more, by Gell-Mann and by George Zweig, a young U.S. American theorist working at CERN at the time.⁶⁶ Gell-Mann suggested three ‘quarks,’ while Zweig spoke of three ‘aces,’ but the two schemes were otherwise largely equivalent, and eventually became known under Gell-Mann’s label as ‘quark model.’ However, as we shall see in the next sections, the eightfold way and the quark model were seen by many theorists only as a starting point for a long sought after merger of internal and space-time symmetries.

Beyond SU(3)—The Mathematical Marriage of Space-Time and Internal Symmetries

With the success of the eightfold way, a growing number of theorists took interest in abstract group-theoretical structures and their possible use in high-energy physics. The five conferences on “Symmetry Principles at High Energy” (1964–1968) held at Coral Gables were both evidence of and a meeting point for a growing community of theoretical physicists interested in symmetries. Participants of the meetings included names which already were or would soon become well known in the theoretical high energy physics community, like Asim Barut, Nicola Cabibbo, Gerson and Sulamita Goldhaber, Bernard D’Espagnat, Robert Marshak, Louis Michel, Yoichiro Nambu, Yuval Ne’eman, Robert Oppenheimer, Lochlainn O’Raifeartaigh, Abraham Pais, Abdus Salam, Julian Schwinger, George Sudarsahn and Bruno Zumino. Therefore, the five volumes of proceedings of those conferences provide a good guideline to follow the rise of symmetry principles in high-energy physics and the mathematical practices that were referred to under that expression.

Were abstract group-theoretical notions now finally being regarded as a generally effective tool to fit observations in particle physics? As I will argue in the next pages, the answer to this question is no. The participants to the Coral Gable conferences certainly regarded empirical adequacy as a necessary feature of any mathematical conceptualization of particle phenomena, and therefore took as a starting point for all of their reflections the empirically successful eightfold way and quark model and tried to expand them into an equally successful construct. However, their explicit use of group theory was not aimed at improving the fit to phenomena, but rather at ensuring that the mathematical construct produced would also in some way reflect a deeper structure of fundamental interactions. Once again, these authors made a distinction between ‘models’ and ‘theories’ which was not necessarily shared by the whole particle physics community. The scientists meeting at Coral Gables expected the symmetries of nature to take a form similar to space-time invariances. The shared tenet of the emerging symmetry community was the idea that internal symmetries like isospin and strangeness should be combined to space-time degrees of freedom. Kursunoglu expressed this goal very forcefully in his contribution to the first conference:

The study of elementary particle events from the point of view of symmetry properties of their interactions has had both qualitative and quantitative success [...] However, it is also known that the present picture of the description of elementary particles based on the experimentally tested symmetry concepts is not entirely satisfactory. [...] The introduction of fictitious spaces like isotopic spin or unitary spin space, distinct from the space-time structure of elementary events has long been recognized to be quite unsatisfactory for further progress towards a real understanding of the dynamical principles underlying elementary particles interactions.⁶⁷

Kursunoglu immediately stated that he had no solution for this problem, but went on to present his reflections:

There are a number of ways of introducing new groups, that is, new quantum numbers to describe some properties of elementary particles. Almost any finite or infinite group can provide some discrete quantum number. The physics of the things almost always emerge from a skillful bookkeeping of the correspondence between these discrete numbers and observed facts. In the absence of basic physical principles it is quite possible that the correlation of facts and some real numbers can be achieved in more than one way. For example if SU(3) is an invariance group for elementary particles, why not the so-called G₂ or SU(4) or for that matter O₇ or any other well suited finite dimensional group?⁶⁸

We see here how Kursunoglu did not consider success in fitting phenomena as a sufficient criterion for regarding a particular group-theoretical structure as physically significant. To that aim, more guidance was needed:

We must, therefore, seek some guidance from the most basic invariance principles of physics, meaning that any extra quantum degree of freedom for elementary particles must be based on the inhomogeneous Lorentz group. We must establish a bridge between space-time and unitary structure of microphysics.⁶⁹

Although Kursunoglu did not use here the terms model and theory, it is clear that he was making a sharp distinction between “some real numbers” fitting “observed facts” (models) and groups mirroring the principles of natural laws (theories). In a similar vein, Asim Barut started his paper by saying:

We shall take the point of view that there is an exact symmetry governing the quantum numbers and the mass states of elementary particles. It is clear that this symmetry has to encompass the space-time symmetry and go beyond to include all other internal degrees of freedom or quantum numbers. We shall call this larger symmetry the dynamical symmetry.⁷⁰

Barut explained that physics had proceeded in the past by describing systems in terms of increasingly larger groups, quoting the history of spin as an example that one should search for a symmetry connecting space-time and internal quantum numbers. In a joint paper “On the Origin of Symmetries,” Ne’eman, Nathan Rosen and Joe Rosen explored the possibility that internal symmetries might “emerge from space-time itself.”⁷¹ They even brought gravity into play and concluded: “In fact, we got a larger symmetry than the observed one, and contracted it by assuming that first order gravitational curvature should not be left out and conjecturing that it should have just that observed effect.”⁷² It is of course not possible to discuss here the details of this and other proposals, but it is important to see how, once again, group-theoretical structure linked to space-time—and only those—were seen as guidelines for fundamental reflection more than for fitting phenomena. As Schwinger put it in his contribution: “what I want to talk about is not some modification of existing classification schemes, but rather a fundamental field theory of matter, i.e. of everything.”⁷³

Despite their focus on explanatory constructs, the participants of the Coral Gable conferences regarded empirical adequacy as a prerequisite for any theory, and so were quite interested in experimental results, to which some talks were devoted.⁷⁴ Interestingly, these experimental talks contained the earliest diagrams geometrically representing strongly interacting particles as multiplets of SU(3), as shown in Fig. 2, which is today the standard representations of the eightfold way. In Gell-Mann and Ne’eman’s papers no graphic representation of that (or other) kind appeared and, as noted above, the diagrams of the strangeness model developed by Gell-Mann did not express any group theoretical properties, but rather the measurable features of particles, like masses and charges (see Fig. 1). This observation confirms that it was only when the eightfold way became relevant for experimenters that the observed properties of particles started being seen and diagrammatically represented through the mathematical lens of SU(3). Soon, such representations would become a standard means of introducing young physics students to elementary particles and, at that point, it started being assumed that, had one known more group theory, one might have read off the multiplets from experimental reports. Yet it was not so, and the construction of the Gell-Mann-Nishijima model and the eightfold was a highly non-trivial development at the intersection of theory and experimentation.

Now that the SU(3) structure was there, though, all particles appeared as potential bearers of quantum numbers representing higher symmetries, and it is in this sense that they were featured on the cover of the proceedings of the first Coral Gables conference (see Fig. 3). It was there that Nasreddin Hoja made his first appearance, sitting backward on his donkey surrounded by the symbols of the particles making out the eightfold way.⁷⁵ Hoja reprised this role on the cover of the proceedings of the second conference (see Fig. 4), but this time he was surrounded by the symbols of various groups that had been or might be used to classify particles. This image represented the work done by theorists during the last year and expressed their hopes of further progress. The second Coral Gable conference was held in January 1965, and its proceedings appeared during that same year.⁷⁶ As noted by the editors in their short preface, very important developments had taken place since the first conference and, from the point of view of the Coral Gables participants, the most important event had been the proposal to expand the SU(3) quark model into a classification based on SU(6), which at least partially combined internal and space-time symmetries:

Fig. 3

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Cover image of: Behram Kursunoglu and Arnold Perlmutter, eds., Coral Gables Conference on Symmetry Principles at High Energy (San Francisco: Freeman, 1964).

Fig. 4

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Cover image of: Behram Kursunoglu, Arnold Perlmutter, and Ismail Sakmar, eds., Coral Gables Conference on Symmetry Principles at High Energy. Second Conference (San Francisco: Freeman, 1965).

The subject of combining internal and space-time symmetries was one of the topics discussed at the First Coral Gables Conference, held in January 1964. The issues raised at that time have since been considered - both extensively and intensively - by many experts, and their efforts have culminated in the proposal that SU(6) be accepted as a possible symmetry group of hadrons.

At this year’s conference a major part of the discussion was concerned with the expansion and development of this proposal, and it was the opinion of the participants that considerable progress was achieved.⁷⁷

At the time, approaches based on SU(6) appeared as a great breakthrough. Today this is practically forgotten, and in the following section we will follow its rise and fall in context of the rapid diversification of mathematical practices linked to the search for symmetry principles at high energy.

The Rise and Fall of SU(6)

When the quark model appeared in 1964, the fact that quarks would have electric charge equal to 1/3 of that of the electron raised many doubts about their existence, since so far no objects with fractional charges had ever been observed.⁷⁸ However, it was noted by a number of theorists that, if one considered quarks with spin up and quarks with spin down as two distinct particles, and not as two states of the same particle, then one would have six independent objects out of which strongly interacting particles might be constructed according to a scheme based on representations of the group SU(6). As it turned out, this classification appeared empirically plausible, with the particles forming 35-dimensional (mesons) and 56-dimensional (hadrons) SU(6) multiplets.⁷⁹

For theorists with an interest in symmetry principles this scheme was extremely attractive, because SU(6) was not simply a product of SU(3) and SU(2), but intimately combined the degrees of freedom of internal symmetries with two-component spin. The scheme was of course non-relativistic, but, as we shall see, Kursunoglu and other theorists were convinced that its relativistic extension was possible and would soon follow. The various symbols surrounding Hoja on the cover of the 1965 proceedings were proposals in that direction.

In his opening remarks at the second Coral Gables conference, Robert Oppenheimer spoke of the “successes of SU(3) and the successes of SU(6),” and stated that a main topic of the conference would be “in what sense, for what purpose and at what cost can one combine relativity with the good things about SU(6).”⁸⁰ This topic was indeed central to many talks at the conference. Salam proposed a “Covariant theory of strong interaction symmetries” based on a group with 144 parameters which he labeled as Ũ(12).⁸¹ Bunji Sakita presented a relativistic SU(6) theory, noting at the beginning of his paper that its formulation and results were “almost identical to those of Salam’s,”⁸² Korkut Bardacki, John Cornwall, Peter Freund, and Benjamin Lee instead argued for an extension to U(6) × U(6),⁸³ while Yoichiro Nambu assumed the existence of three distinct quark triplets.⁸⁴ Some authors discussed more in general the possibilities and problems of the research program: Robert Marshak and Susumu Okubo made an inventory of possible higher symmetries embedding the quark model,⁸⁵ George Sudarshan discussed “Questions of Combining Internal Symmetries and Lorentz Group,”⁸⁶ Louis Michel addressed “The Problem of Group Extensions of the Poincaré Group and SU(6) Symmetry.”⁸⁷

A critical note came from Julian Schwinger, who presented a development of his earlier idea of a distinction between “fundamental,” non-observable quantum fields and “phenomenological” fields representing observed particles.⁸⁸ He could not spell out the details of how to connect the two kinds of fields, as “this problem defies direct dynamical attack, at the moment,”⁸⁹ but developed some general considerations on symmetry, and in the end explicitly underscored that they should be regarded as different form the SU(6) approach pursued by most other authors:

It would be quite erroneous, however, to identify the concepts described in this note with the ideas of the SU(6) workers. The latter view SU(6), or some larger group, as an idealized dynamical invariance group, despite the difficulties raised by relativistic considerations, or, more conservatively, regard it as valid in a non-relativistic limit. [In our theory] [t]he requirement of invariance under the inhomogeneous Lorentz group is met through the machinery of field theory, which was invented for that purpose, and not by forcing an unhappy union between physically incompatible partners.⁹⁰

Yet Kursunoglu pursued further his dream of a union of internal and space time symmetries, wondering whether one might have to consider infinite Lie groups to that aim, and speculating that “[i]n such a theory, one may end up with a single infinite supermultiplet of hadrons.”⁹¹

Not all authors at the conference focused on how to combine space-time and internal symmetries, and some theoretical contribution had a closer link to experiment.⁹² Moreover, two papers discussed how SU(6) and other symmetry principles might be connected to an approach which, at the time, was emerging as either a complement or an alternative to quantum field theory: the so-called bootstrap theory and the relevant techniques of dispersion relations, Regge poles and current algebra. The bootstrap approach was promoted by Geoffrey Chew as an alternative to quantum field theory, because it did not assume the existence of elementary particles, but only aimed at mathematically representing processes of mutual transformation between particles states, none of which was regarded as more elementary than the others.⁹³ At the second Coral Gables conference, both Roger Dashen and Ne’eman discussed how far SU(6) and other symmetries might be applied to the bootstrap, and Ne’eman concluded his summarizing talk of the meeting by discussing symmetries in a bootstrap theory “clean of elementary quarks.”⁹⁴

Despite the great expectations and the efforts of theorists present (and absent) at the second Coral Gables conference, the research program of a relativistic extension of SU(6) saw little progress in the following year, and at the third Coral Gable conference in January 1966 a large part of the papers were devoted either to the bootstrap approach or to attempts to theoretically fit new experimental results on weak and strong interactions. Once again, Ne’eman held a final, summarizing talk, and this time he devoted some remarks to “a discussion of the ‘Philosophy’ of what we are doing here, in the context of the general development of ‘elementary particle’ physics.”⁹⁵ It is significant that Ne’eman not only put the word philosophy in quotations, but also the expression elementary particles: an indication of his interest in the bootstrap approach. Ne’eman went on to offer his interpretation of the history of chemistry as a development from the grasping of “patterns” in the form of the periodic table of elements to the understanding of “structures” with quantum physics.⁹⁶ He then compared this development with the present situation in high-energy physics, which he claimed was still in the pattern identification stage:

Some day we may be able to go beyond this pattern identification stage. Quarks and their like may yet land us in a world of subparticle physics with a very different set of basic laws; alternatively we shall perhaps really learn to use the bootstrap as a working piece of physics, not just as an idea”.⁹⁷

Although Ne’eman did not speak of models and theories, we may interpret his distinction between “patterns” and “structures” as reflecting the idea often expressed at the time with those terms: an opposition between mathematical constructs which simply fit phenomena (models, patterns), and those expressing the (hidden) principles of their coming-to-be (theories, structures). Once again it must be noted that Ne’eman was expressing the beliefs of the community gathered in Coral Gables—beliefs which were not necessarily shared by all particle theorists at the time (not to mention experimentalists). In general, among high-energy physicists, the term ‘model’ was used in a rather neutral sense, without implying expectations that at some point a ‘theory’ would emerge. Therefore, the distinctions made by Ne’eman cannot be taken to reflect a general feature of theoretical practices of early particle physics. Moreover, all theorists agreed that any theoretical construct should be expected to fit phenomena and allow for successful predictions, so that any ‘theory’ would also be an empirically successful ‘model.’ In his conclusions Neeman stated:

The search for ‘relativistic’ SU(6) has been replaced by the elaboration of a Lorentz-frame dependent structure, relativistic, but not explicitly covariant. Let us hope that the coming year will bring about a more complete understanding of the operator structure involved [...] with better chances of some structural understanding resulting from these developments.⁹⁸

Contrary to Ne’eman’s hopes, however, the following year brought little clarity, and the difficulties of linking internal symmetries with space-time invariances were shown to be not just of a technical, but also rather of a fundamental character. At the fourth Coral Gables conference, the final, summarizing talk was held by Nicola Cabibbo, who started with a look back at the development of symmetry principles in particles physics during the last years:

The discovery of the octet scheme led to a substantial step forward in our understanding of hadrons [...] This was followed by attempts to introduce even higher symmetries. [...] After some ambitious attempts to have an intimate merger of internal and Lorentz symmetries, discussed in 1964 at this conference, we had, later in 1964, the explosive success of nonrelativistic SU(6). The initial attempts to make SU(6) into a completely relativistic scheme were frustrated by a series of beautiful impossibility theories, the latest version of which was presented here by Professor Coleman.⁹⁹

The proceedings of the conference do not contain any paper by Sidney Coleman, but in that year a joint paper by Coleman and Geoffrey Mandula appeared, in whose abstract the authors stated: “We prove a new theorem on the impossibility of combining space-time and internal symmetries in any but a trivial way.”¹⁰⁰ At the beginning of the paper, they summarized the hope and efforts leading up to the present no-go theorem:

Until a few years ago, most physicists believed that the exact or approximate symmetry groups of the world were (locally) isomorphic to direct products of the Poincaré group and compact Lie groups. This world-view changed drastically with the publication of the first papers on SU(6); these raised the dazzling possibility of a relativistic symmetry group that was not simply such a direct product. Unfortunately, all attempts to find such a group came to disastrous ends, and the situation was finally settled by the discovery of a set of theorems which showed that, for a wide class of Lie groups, any group which contained the Poincare group and admitted supermultiplets containing finite numbers of particles was necessarily a direct product.¹⁰¹

The authors explained that those theorems only dealt with specific cases, and that their own formulation provided a comprehensive argument covering all of them. The Coleman-Mandula theorem is today famous as the negative starting point for supersymmetry, since in the early 1970s a number of authors independently showed how it was, after all, possible to non-trivially extend the Poincare group by introducing transformations of fermions into bosons and vice-versa.¹⁰² Yet in that case, too, no combination of internal SU(3) with space-time invariance was possible.

In 1966 a NATO International Advanced Study Institute on “Symmetry Principles and Fundamental Particles” took place in Istanbul, co-organized by Kursunoglu, and in its proceedings, Coleman started his lectures by explaining the clues that had led to the attempts of expanding SU(6), and then stated:

We would have been extremely happy to find a relativistic generalization of SU(6), even if all particles in a supermultiplet had the same mass, and even if there were no good perturbative mass formulas. The remaining lectures will be devoted to explaining why even this modicum of happiness is denied us.¹⁰³

It is interesting to note how Coleman half-jokingly remarked that theorists would have been happy even with a ‘theory’ that provided a quite inadequate fit to observation, or did not allow computing any predictions. Because of the obstacles encountered by the main research line of the previous years, the mathematical practices of symmetry presented at the fourth Coral Gables conference were more varied than in earlier meetings and, to use Ne’eman’s terminology, they explored rather the patterns than the structure of particle phenomena. To this end, they employed approaches linked to the bootstrap, like current algebra and dispersion relations.¹⁰⁴ In his summary talk, Cabibbo also noted the role of experiment, stating that “the work on the application of SU(3) has, in the meantime, gone on, in particular through the painstaking experimental search for new resonant states and their possible classification in multiplets of SU(3).”¹⁰⁵ A new, more diverse phase of symmetry practices had begun.

Conclusion: The End of the Yogurt Project?

By 1968, Kursunoglu had accepted that the dream pursued so far in the Coral Gable conferences would not become reality, but that the approaches developed in trying to realize it still constituted powerful heuristic tools for theoretical research. Accordingly, he decided that the fifth Coral Gable conference in January 1968 would be the last one devoted to “Symmetry Principles at High Energy,” leaving the stage to meetings studying “Fundamental Interactions at High Energies.”¹⁰⁶ In the preface to the proceedings, the editors listed as themes of this last meeting, beside the use of symmetry principles “for the elucidation of the nature of the fundamental laws governing the behavior of elementary particles,” also “the collation of experimental data and the prediction of new effects” and “the need to explain the deeper mystery of broken symmetries.”¹⁰⁷ The search for a ‘theory’ of fundamental symmetries had given way to the use of a broad range of mathematical practices involving invariances and group theory that attempted rather to establish patterns and ‘model’ phenomena. As I have discussed in more detail elsewhere, techniques and notions of symmetry breaking had in the meantime come to play a main role in this context.¹⁰⁸ However, the contributions to the fifth Coral Gables conference acknowledged the importance of their original program, and Lochlainn O’Raifeartaigh gave a summary of attempts to link internal and space-time symmetries:

The most dramatic occurrence in the field of internal symmetry-space-time speculations, was, of course, SU(6), and it raised immediately the question of if and how it could be made relativistic. This question gave rise to ‘no-go’ theorems, this time of the Coleman, Jordan, Weinberg type and to a number of relativistic and semi-relativistic generalizations.¹⁰⁹

As discussed at the beginning of this paper, Ne’eman gave a final talk summarizing the progress in all five conferences.¹¹⁰ In a section titled “Matchmakers,” he praised O’Raifeartaigh’s presentation and reminded the audience of the productivity of the second conference, where so many proposals for combining internal and space-time symmetries had been presented, adding:

All this material was thrown into the mélée, it all got caught in some kind of mixmaster and I think that his was perhaps the most successful of the conferences. [...] It was out of this mess that [...] the algebra of the currents in its better understood form grew, so did the infinite component theories.¹¹¹

The image of the mixmaster is helpful to express the way in which the initial, rather simple and fundamental approach for deploying some group-theoretical structures in high energy physics became the starting point of a diversity of mathematical practices linked to symmetry principles which could be physically interpreted either as fundamental theories or as phenomenological models. Symmetry principles went on to a new life, or lives, yet the original ‘matchmaking’ dream seemed to be at an end. It is in this sense that Ne’eman mentioned his appreciation for Hoja’s yogurt project, as both a warning and a tribute to those who had believed—and still believed—that unity of internal and space-time symmetries might be possible.

Almost twenty years later, in 1997, Kursunoglu recalled the launching of the Coral Gables conferences, and, like Ne’eman, he praised especially the second one, recalling the theories presented there by Salam and adding: “Unfortunately it turned out to be another good example of Hodja’s [sic] ‘yogurt project’.”¹¹² We may thus conclude that the systematic employment of group-theoretical structures in early particle physics has to be seen not as a fruitful practice for fitting phenomena, but rather as an ambitious, ultimately unsuccessful attempt at mathematically grasping the inner structures of nature. While rather simple mathematical methods proved successful in the first task, the use of abstract, more refined mathematical notions failed to achieve the second goal. Those theorists who wished to make a distinction between purely descriptive ‘models’ and both descriptive and explanatory ‘theories’ were faced with the choice of either giving up their wish for a deeper understanding of particle interactions, or accepting that the fundamental structures of microphysics could be captured by symmetries only vaguely (or not at all) related to the invariances of space-time.

Looking at the developments of the 1970s, one might argue that some authors chose the second possibility, and that the great yogurt project of symmetry matchmaking lived on in Grand Unified Theories, which were based on SU(5) and other special unitary group, but were eventually ruled out by the failure to observe proton decay. From the 1980s onward, though, a third answer to the theoretical dilemma sketched above emerged, which had already been implicit in Coleman’s remark from 1967. It was the possibility that a ‘theory’ of fundamental interactions would not necessarily need to deliver new, successful empirical predictions, but only needed not to explicitly contradict established experimental results. In this spirit, increasingly speculative proposals of physics ‘Beyond the Standard Model’ were made, which were assumed to coincide with the Standard Model at low energy, and whose predictions, when falsified, were moved up to energies not yet reached, as in the case of supersymmetry, or even only obtained in an experimentally inaccessible range, as in the case of string theory. Interestingly, theorists bringing forward these research projects usually describe their activities as ‘model-building,’ and, as I have argued in detail elsewhere, they tend to use the term ‘theory’ to indicate hypothetical mathematical constructs which are expected to reflect the inner structure of fundamental interactions, yet do not at present exist.¹¹³ However, despite—or perhaps thanks to—their highly speculative character, these theoretical research programs are without doubt still capable of inspiring in mathematically-minded authors the same enthusiasms which led Hoja to exclaim: “But just imagine if it should work!”