Research supported by the Käte Hamburger Kolleg “Cultures of Research”, RWTH Aachen, a Centre for Advanced Study in the Humanities funded by the German Federal Ministry of Education Research.
- 1 Narratologists have provided different, more or less abstract definitions of narrative and its comp (...)
1Narratives first became subjects of research within literary studies, but approaches and results from that field were later appropriated and expanded in other areas, such as art, film and media studies, ethnography, religious study, and eventually also science and technology studies (Herman et al. 2005). This diffusion was often linked to a concept of narrative based on more or less sharply defined formal features,1 which could therefore also be applied to research objects whose content is non-fictional, such as newspaper articles, documentaries, biographical interviews and, as I will argue in the following pages, scientific theories (Fludernik, Ryan 2019a). However, there have been critiques to this approach, and in particular the extension of the concept of narrative to the scientific field has been regarded as problematic, because scientific communications and scientific knowledge have traditionally provided the template for non-narrative practices (Brandt 2009; Plotnisky 2005). The role of narratives in teaching and popularization is often acknowledged, but authors stop short of regarding scientific knowledge as in itself displaying narrative features (Schrempp 2002). Historians and sociologists of science and technology, on the other hand, have often welcomed the extended notion of narrative as a tool for analysing not only didactic and popularization, but more in general scientific communication and the production of scientific knowledge (Azzouni et al. 2015; Blume, Leitgeb 2015; Borrelli 2019; Brandt 2009; Doxiadis, Mazur 2012; Hartmann 1999; Morgan, Wise 2017a, 2017b; Rosales 2017; Wise 2004, 2011, 2017). Interestingly, in this context doubts have been raised on whether a clear-cut border may be drawn between “fictional” and “factual” narrative (Brandt 2009).
- 2 I have discussed case studies in Borrelli 2012, 2015a, 2019, 2021. For an overview on narrative in (...)
2This broad debate cannot be summarized here, but it constitutes the backdrop to research on the role of narrative knowing in the sciences, and in particular in the exact ones, where traditionally logical-mathematical rigour rather than story-telling is expected. In my research I have built upon the work of a number of historians and philosophers of science to argue that, approximately since the middle of the twentieth century, theoretical practices in fundamental physics increasingly often displayed features that can be productively understood in term of narrative knowing.2 This claim challenges the traditional border between narratives and logical-mathematical forms, but as I will argue it can be plausibly upheld, also in view of results in the history and philosophy of science which have critically reassessed the image of scientific theories as closed, coherent physical-mathematical constructs which seamlessly lead to empirically correct predictions (Cartwright 1983). To actually deliver successfully testable descriptions of phenomena, physical laws have to be combined with a broad range of additional assumption, simplifications, and approximations specific to the case at hand: the theories in themselves only refer to idealized objects, and not to the real world.
3The paper starts with some general reflection on narrative and science and then discusses a fundamental premise for my argument, namely that it is too restrictive to regard theories in the exact sciences primarily as mathematical formulas, and that other modes of knowledge mediation have to be taken into account. Because of this, I devote some pages to explaining the epistemic relevance of words, images codes, as well as non-rigorous mathematical expressions in scientific practice, arguing that they can be seen as forming narrative constructs. In the second part of the paper, I discuss more in detail one concrete example: spontaneous symmetry breaking. In contemporary science, this term indicates a notion which emerged in high energy physics in the late 1960s to describe a rather specific set of theoretical structures. Later on, spontaneous symmetry breaking came to be regarded as an explanatory model that could be applied also in other areas of physics, such as thermodynamics, solid state physics and cosmology, and more recently even in the life science (Brading 2003). I will argue that the concept of spontaneous symmetry breaking can be productively applied to conceptualize extremely different physical systems, and so explain their behaviour, because its characterization is based on a narrative structure to which a broad range of physical phenomena and theories can be related. Moreover, each extension of its range of applications reinforces the status of spontaneous symmetry breaking as a heuristically fruitful notion in contemporary science, paving the way for further applications.
4Deploying concepts of narrative when studying theoretical practices in the natural sciences faces not only the philosophical problems mentioned in the previous section, but is also challenging from a methodological point of view. Which notions of narrative or story-telling are fitting for such an endeavour, and how can they be operationalized? In what sense can scientific theories and especially those from the exact sciences, be interpreted in terms of story-telling? Even when assuming that theoretical constructs in the exact sciences do not always constitute rigorous logical-mathematical structures, this is not sufficient to positively characterize them as displaying narrative features. These questions can hardly be answered once and for all in full generality, but I will offer some reflections which have proven applicable and useful to approach specific case studies. First of all, the notion of narrative that is most often used for conceptualizing theoretical practices is a minimal one: a narrative, or a story, is the sequential connection of at least two events, usually, but not always, of temporal, but at times also of causal kind (Morgan, Wise 2017b). This characterization is very far from the elaborate definitions developed by narratologists, but has so far proven fruitful for analysing specific case studies. In general, I regard it as productive to set aside attempts at formulating broad overarching hypotheses of how scientific knowing may or not be characterized in its possible relationship to narrative structures. The goal of my research is rather an historical-epistemological one: zooming in and reconstructing specific historical developments, asking how certain elements of (scientific) knowing—in this case the notion of spontaneous symmetry breaking—emerged in a given context, leaving open the question of whether and how far the conclusions drawn from this specific case study can be extended to other ones. More specifically, I argue that the concept of spontaneous symmetry breaking is not associated to a logical-mathematical structure, but rather to a narrative one, yet I do not wish to claim that this is the case for all notions of high energy physics. However, I do claim that such narrative constructs can play a much more important role in the physical sciences than usually assumed.
5Despite, or perhaps thanks to, its simplicity the chosen definition of narrative is helpful to approach theoretical constructs, as it allows to regard them not as abstract, disembodied structures, but rather as epistemic objects with their own specific material and performative dimensions. From this perspective, physical theories can be regarded as multi-medial constructs comprising, beside mathematical formulas, also verbal statements, images, procedures, numbers, diagrams or computer code. These diverse elements are connected to each other as parts of an overarching narrative which in scientific practice can be implemented in different versions fitting the specific needs of the moment. For example, in the case of fundamental physics, the same narrative can take a simpler, often visual form when used by experimenters, and a more elaborate one when employed by theorists, thus fostering productive exchange between the two communities.
6What does it mean in practice to approach scientific theories as material and performative constructs? Considering that science is a collective enterprise, all theoretical structures have to be accessible to the relevant community in a sensually perceivable way: they have to be seen, heard or grasped. This material, bodily dimension, which may also be referred to as “aesthetic,” is necessary both to share knowledge and to deploy it in practice. While one may easily regard instruments, procedures, images and even words as having a constitutively material dimension, it is less obvious to understand how this is true also for mathematical knowledge. Yet mathematical notions and methods also need to be sensually expressed to be communicated and employed, and in particular mathematical notations are both visible and manipulable according to more or less rigid procedures (Doxiadis, Mazur 2012).
7As a simple example of the epistemic relevance of sensually perceivable forms of mathematical concepts, let us consider the circle, a geometrical figure whose history reaches back into Antiquity and which can be known through various material forms. First of all, a circle can be known as a visually perceived figure with no further information beside its name. Indeed, a relevant portion of the general public if asked what a circle is, would probably reply by drawing its shape with a finger. Yet a circle can also be defined as the figure drawn with the help of a compass, as was usual for example in medieval Europe. This definition will be helpful only to those who know what a compass is and how it works, but it has the advantage that it allows them not only to visually recognize a circle, but also to draw one. For purposes other than drawing it, a verbal definition of a circle may be needed, such as: given a point on a plane, a circle is the set of all points on the plane having the same given distance from the initial point. Here, too, the characterization of the circle will only be helpful to those who already have certain information, such as knowing what a plane, a set and a distance are. Finally, to persons who are familiar with algebra and Cartesian reference systems, a circle can be defined as made out of the points on a plane whose coordinates (x,y) satisfy the condition x²+y² = a², where a is a constant representing the circle’s radius. This characterization can be of use in many respects, for example when relating circles to more complex curves, but does not provide any immediate idea of how a circle looks like, and does not allow to draw one by hand. On the other hand, if one is using a computer, the formula will allow to visualize and manipulate a circle.
8The characterizations listed above can be linked to each other in unequivocal ways, but they still correspond to different ways of knowing the circle, and are potentially accessible to different groups of persons and capable of fulfilling different functions. In conclusion, even the seemingly trivial example of the circle reveals the manifold epistemic potential of material and performative forms, which is neglected when considering the circle as a disembodied mathematical notion with various equivalent representations. The situation is even more challenging when one is considering not a purely mathematical construct, but a physical-mathematical one, which is assumed to have a connection with experience. In that case, different forms expressing the overarching concept or theory can be connected to different phenomena, and the concept or theory can be regarded as having multi-medial character. Moreover, as I will show when discussing the case study, it is possible that the different expression cannot be unequivocally connected to each other, as was instead the case for the circle. Indeed, it may happen that the construct comprises various mathematical forms that cannot be reduced to one another, so that one may ask how the whole achieves an epistemic unity: Why is it regarded by the scientific community as a single concept or theory? It is at this point that the narrative character of the construct becomes important: its various expressions, although not equivalent in a logical-mathematical sense, are nonetheless connected to each other to form a story which scientists can narrate in different variations, but still perceive as a single one. It is in this sense that I argue that some theoretical constructs in fundamental physics can be productively approached as multi-medial narratives, reconstructing the emergence of the different medial elements which would eventually contribute to the narrative, and the way in which they came to be related to each other.
- 3 The different approaches of mathematicians and theoretical physicists to mathematical rigour in con (...)
9Traditional history and philosophy of science, and in particular studies of scientific concepts and theories, have long neglected the complex, multi-layered dynamics that emerges when paying attention to the material and performative dimensions of theoretical constructs. Instead, mathematical formulas have usually been regarded as privileged carriers of physical meaning, while diagram, images or instruments could be seen as either epistemically equivalent to formulas, or as less valuable approximations. A similar attitude applied when reflecting on cases in which historical actors constructed and manipulated mathematical notations in ways that can be seen as non-rigorous according to the standards of their time (and to present ones). However, there are plenty of such cases, as there is little doubt that the use of non-rigorous mathematical practices is a necessary component of scientific knowledge production, especially in the physical sciences.3 In fact, non-rigorous mathematics, images and diagrams and other epistemic media different from rigorous mathematical formulas have often been recognized as relevant factors in past and present production of scientific knowledge, but at the same time regarded as contingent elements which may play a situated role in the so-called contexts of discovery, but need not be taken into account when looking at established scientific practices. Only recently the essential epistemic relevance of theoretical constructs expressed in terms different from rigorous mathematical formulas has been discussed by a number of authors, some of whom also made use of notions of narrative to grasp the epistemic dynamics of the case studies explored. In particular, John Norton Wise has deployed the concept of narrative knowing to interpret theoretical practices in the exact sciences, while Peter Galison has noted the role of non-rigorous mathematical practices in creating new, hybrid disciplinary fields at the border between physics and mathematics (Galison 2004; Wise 2004, 2011, 2017). In the case study I present below, the use of non-rigorous mathematical methods plays an important role, and I will therefore now devote a few more passages to it. My aim is not to formulate any generic reflections on the topic, but rather make clear to readers not familiar with the subject how much it is present in past and present scientific practice.
10To appreciate both the pragmatic and the epistemic relevance of non-rigorous mathematical practices it is important to keep in mind that the notion of mathematical rigour, too, is historically and culturally situated. Within Western culture, the idea of overarching, rigid prescription on how to perform mathematical calculations and mathematical proofs emerged in the course of the nineteenth century, together with the notion of pure mathematics. Of course, also in earlier times, as well as in non-Western cultures, there were sets of more or less rigid guidelines for mathematical computations, and of what may or not count as a mathematical proof (Chemla 2015). However, systematic attempts to provide explicit rules to define and manipulate mathematical entities date back to the nineteenth century, and since then the standards of what is or not rigorous have constantly changed with time, adapting to conform to new research developments. The physical sciences have often provided impulses for expanding the definition of mathematical rigour, while mathematicians have constantly complained about physicists’ doubtful symbolic practices and the lack of recognition for those mathematicians who provided post factum a rigorous definition for physicists’ creations (Atiyah et al. 1994; Jaffe, Quinn 1993, 1994).
11For physicists, though, playing fast and loose with mathematics is not a deliberate choice, but a necessary step to try and formulate hypotheses on how to model observed or assumed physical regularities. If the results of the modelling process somehow happens to fit empirical data, this fact is regarded as evidence that the procedure, although non-rigorous, is in essence correct and might at a later time be formulated in a more formally satisfactory way—as was indeed often the case. In the same spirit, physicists have no qualms about using so-called folk theorems which, in the words of Physics Nobel laureate Steven Weinberg are “things that have never been proven, but are known to be true” (Weinberg 1985: 125).
12One case of particular interest is that of the so-called method of renormalization in theoretical high energy physics, which will play a role in the case study discussed below (Brown 1993). This method was developed in the late 1940s and allows to obtain extremely precise and empirically validated predictions of particle phenomena, but has so far resisted all attempts at turning it into a rigorous mathematical procedure. Without delving into the technical details, renormalization can be said to involve manipulation of expressions which, although built out of mathematical notation, are mathematically meaningless, as for example the expression 1/0 (i.e. one divided by zero): although all three symbols have a clear definition, they cannot be combined in that way, as already schoolchildren learn. In the method of renormalization expressions of that kind, though much more complex, are manipulated in such a way as to make them disappear from the final results, which therefore are mathematically meaningful, and lead to (empirically correct) numerical predictions. As noted above, this empirical success fully satisfies physicists of the legitimacy of their methods. For mathematicians, though, a procedure which contains even only one non-permissible step is not viable, no matter how many empirically correct results it produces: with no rigour there is no way to say whether correct results may simply arise by fortuitous chance. When using mathematical symbolic notation in a non-rigorous way, physicists often bridge the gaps in the strictly mathematical reasoning by making use of other media, like words or diagrams, and at times the multi-medial construct that emerges can be interpreted as having narrative character, as noted in this quote, which was written in context of a dispute between mathematicians and theoretical physicists on the use of non rigorous mathematics:
The non-rigorous use of mathematics by scientist, engineers, applied mathematicians and others, out of which rigorous mathematics sometimes develops, is in fact more complex than simple speculation. While sloppy proofs are all too common, deliberate presentation of unproved results as correct is fortunately rare. Much more frequent is the use of mathematics for narrative purposes. An author with a story to tell feels it can be expressed most clearly in mathematical language. In order to tell it coherently without the possibly infinite delay rigor might require, the author introduces certain assumptions, speculations and leaps of faiths [...] In such cases it is often irrelevant whether the mathematics can be rigorized, because the author’s goal is to persuade the reader of the plausibility or relevance of a certain view about how some real world system behaves (Atiyah et al. 1994: 186).
13In the second part of this paper I focus on the way in which narrative constructs can play a role in theoretical physics, and do so by using as an example a concept which has become increasingly prominent in the natural sciences during the last decades, namely that of spontaneous symmetry breaking. Based on previous research results, and avoiding as far as possible technicalities, I will sketch its main narrative features, give an idea of the way in which they are connected to a diverse set of medial forms and show how the process of narrative knowledge construction unfolds in scientific practice.
- 4 The following discussion is based on: Borrelli 2015b, 2021.
14In today’s natural sciences the term spontaneous symmetry breaking can be used to describe both observable phenomena and physical-mathematical structures which cannot be immediately related to experience.4 These uses are not always mutually coherent, but nonetheless relate to each other in such a way as to give rise to a construct that has demonstrated its heuristic productivity in scientific practice. Various material and performative forms are involved in this network, rigorous and non-rigorous mathematical expressions, verbal statements, images and diagrams and variously mediated references to experience.
15A standard reference used when speaking of spontaneous symmetry breaking as an observable phenomenon is the behaviour of ferromagnetic materials such as iron, which can be described in idealized form by a rigorous physical-mathematical construct. Visually, these materials are often schematically represented as a collection of (microscopic) magnetic rods that can freely rotate in a plane (or sometimes in three-dimensional space). At temperatures higher than a certain threshold value (the critical temperature), the rods have random mutual orientations due to thermal motion, so that the resulting average magnetization of the system is zero. This situation is verbally described as symmetric, since from a macroscopic point of view the system is invariant under rotation, not having any preferred direction of magnetization. When the temperature of the system is lowered beyond the critical value, instead, the force between the microscopic rods will be able to overcome thermal motion, and the rods will tend to orient themselves all in the same direction, giving rise to an overall non-zero magnetization in a specific direction, a situation in which the rotational symmetry is said to be broken. The breakdown of symmetry appears in absence of an external magnetic field, and the direction of magnetization depends on random fluctuations in the positions of the magnetic rods: as soon as a sufficient number of them points by chance in the same direction, they will generate a magnetic field strong enough to make the other ones align themselves to it.
16The process of symmetry breaking has been described above verbally, but it could also be expressed in images (drawings, photographs or movies), or by means of more or less complex analytical formulas from statistical mechanics. It can also be numerically and visually simulated with the help of computer code. It is a time-based process which can unfold in two directions in time: when lowering temperature the symmetry is broken, when increasing it, the symmetry is restored. In this sense, it can be seen as displaying basic narrative structure. However, so far the narrative would seem to be an unnecessary addition to traditional scientific descriptions in terms of formulas, words, or experimental measurements. Yet the story gains a life of its own when a further verbal element is introduced: the spontaneity. As noted above, the behaviour of ferromagnetic materials is not simply regarded as an instance of symmetry breaking, but of spontaneous symmetry breaking. The term is used to indicate that the direction of magnetization is determined by factors internal to the system and not by any external agency. In the case of ferromagnetism, describing the process as spontaneous makes no difference with respect to the phenomenon itself, but renders the narrative more interesting by suggesting agency of the material, and allows to construct analogies to other phenomena or abstract constructs.
17For example, another experience often used to visualize spontaneous symmetry breaking is that of a mechanical system in unstable equilibrium, such as a pencil vertically balanced on its tip that, without being touched, falls down into a state of stable equilibrium, lying flat on a surface pointing in a given direction. In this case, too, there is a time-based transition from a (rotationally) symmetrical to an asymmetrical state, and in this case, too, it is assumed that no external agency broke the symmetry. Yet, other than with the ferromagnetic system, in this case the symmetry breaking factor cannot be generally seen an internal to the system (i.e. the pencil): this description relies on the common experience of precarious balance, and leaves it open whether the pencil fell down because of an air current, a sudden movement of the table or possibly an earthquake. Moreover, the mathematical formalisms describing the two systems—the ferromagnet and the pencil—have little or nothing in common, one being statistical mechanics and the other classical mechanics, so that no formal equivalence can be established between the two experiences. Nonetheless, both the falling pen and the ferromagnet are presented by physicists and philosophers as different version of the narrative of spontaneous symmetry breaking (CERN 2024b, Brading et al. 2017). It is important to note that here I am not criticizing a sloppy or incorrect use of terminology, but rather the contrary: My aim is to demonstrate how an overarching, multimedial narrative allows to establish a relationship between two phenomena having little in common, and let them appear as individual instances of a more general class. As I will show further below, it is the flexible character of the spontaneous symmetry breaking narrative that grant it great heuristic power, allowing to see different phenomena or theories as instances of spontaneous symmetry breaking. As with all heuristically powerful tools, spontaneous symmetry breaking, too, brings with it a danger, in that the possibility of easily framing a phenomenon as instance of spontaneous symmetry breaking may lead scientists to neglect alternative conceptualizations. In short, spontaneous symmetry breaking comes with its heuristic bias.
18What is the benefit of this connection? In the case of the two experiences discussed above, there is indeed little benefit, and the two phenomena came to be regarded as instances of spontaneous symmetry breaking only during the 1960s, when they started being used as templates for conceptualizing more abstract theoretical constructs from solid state and particle physics. It was at that time that the overarching narrative of spontaneous symmetry breaking emerged, to later become a heuristically productive element of theoretical practices in various areas of physics. In the following pages I will try and demonstrate the productivity of the story of spontaneous symmetry breaking in fundamental physics and in cosmology.
19Before discussing spontaneous symmetry breaking in theoretical physics research, I would like to stress one point of great relevance, namely that no overarching definition of spontaneous symmetry breaking as a rigorous mathematical construct exists. Mathematicians and mathematical physicists have indeed provided rigorous formal characterizations of spontaneous symmetry breaking which can be applied, among other things, to the critical behaviour of ferromagnets as described above (Strocchi 2008). However, these definitions cannot be used to describe all phenomena and formal constructs which physicists refer to as instances of spontaneous symmetry breaking, and the most notable example is what high energy physicists call the Higgs mechanism, through which the Higgs boson is said to “generate” the masses of all particles. The Higgs mechanism is described in all relevant textbooks as a case of spontaneous symmetry breaking, although physicists are well aware that it does not fall under the rigorous mathematical definition (‘t Hooft 1997: 196 note 15; Strocchi 2008: 193-195). One might ask whether high energy physicists could simply stop referring to the Higgs mechanism as a spontaneous symmetry breakdown, but the answer is no, because doing so would deprive them of a fundamental explanatory scheme. In the following pages I will try and sketch the reasons why this is the case.
20The most empirically successful theory of microphysics today bears the modest name of Standard Model. It emerged during the late 1970s and at its core is a mathematical expression (the “Lagrangian”) whose elements are referred to as “quantum fields” and can be interpreted in terms of a well-defined set of elementary particles, among them electrons, quarks, photons and the Higgs boson (Hoddeson et al. 1997). These particles are characterized by features like masses and electric charges, whose numerical values can be determined by comparing the predictions of the Standard Model with the results of experiments. However, numerical predictions can only be extracted from the Standard Model by means of the procedure of renormalization which, as noted above, is non-rigorous, although very precisely defined. The fact that the values obtained in this way match experimental results is for physicists sufficient reason to regard the theory as essentially sound despite its formal defects.
21The Standard Model is characterized by a set of mathematical invariances (symmetries), but there are also a number of symmetries which it does not display at all, or only in a very approximate way. In particular, there are given symmetries that are absent from the Standard Model only due to the fact that the particles it represents have a mass different from zero. In other words, if all particles were massless, the Standard Model would possess a much larger set of symmetries than is the case in reality. This situation is in itself fully unproblematic: Indeed, why worry about symmetries that are not there, as long as the theory passes all experimental tests? I have no answer to this question, but it is a fact that particle physicists, and in particular theorists, have been worrying about this question already during the 1970s, when the Standard Model was just starting to establish itself as empirically successful. At that time, even the theorists who had contributed to formulate it speculated that it would probably turn out to be only the approximation of a theory which would display more symmetries, and also somehow provide an explanation for the fact that elementary particles possessed masses which could differ by as much as two or three orders of magnitude. The question of why particle masses have such diverse values remains unanswered to this day, despite the huge amount of speculative models produced by theorists in the last decades. However, an answer to why particles have masses at all, as opposed to being massless, is today usually given by referring to a process of spontaneous symmetry breaking in which the Higgs boson plays a central role. It is narrative knowledge expressed in words, formulas and images in academic textbooks, such as this one:
This mechanism, by which spontaneous symmetry breaking generates a mass for a gauge boson [...] is now known as the Higgs mechanism. [...] the scalar [i.e. Higgs] field that causes spontaneous breaking of the gauge symmetry is an important ingredient in the structure of the [Standard Model] (Peskin, Schroeder 1995: 692, 715-716, italics in the original).
22The textbook by Peskin and Schroeder of course connects the verbal statements to relevant formulas, as well as images, but the same story can also be told in a simpler, purely verbal form when addressing a broader audience:
You and everything around you are made of particles. But when the universe began, no particles had mass; they all sped around at the speed of light. Stars, planets and life could only emerge because particles gained their mass from a fundamental field associated with the Higgs boson. The existence of this mass-giving field was confirmed in 2012, when the Higgs boson particle was discovered at CERN (CERN public outreach webpage 2024a).
23The CERN outreach page on the Higgs boson explains that it gives mass to particles through a “trick” which is spontaneous symmetry breaking. The text defines that notion by referring to the simple classical example of the pencil:
A spontaneously broken symmetry is one that is present in the equations of a theory but broken in the physical system. Imagine a pencil standing on its tip at the centre of a table. A perfectly symmetrical situation, but only for a moment: the pencil would immediately fall, breaking the rotational symmetry by selecting a single direction in which the pencil would be pointing. The laws of Nature however would remain unchanged, without a predefined direction written into them. So, the lack of symmetry was essentially “tricked” into the picture, without upsetting the symmetry of physics (CERN public outreach webpage 2024b).
24In high energy physics the narrative of spontaneous symmetry breaking performs a key scientific function: it explains the origin of mass. I will now sketch how this narrative unfolds, and offer a few information on how it came to be.
25In the 1970s some theorists noted that, by taking the core formula (Lagrangian) of the Standard Model and performing a transformation of the field corresponding to the Higgs boson, one obtains a theory in which all particles have zero mass, and which therefore displays a larger amount of symmetry than the Standard Model. The transformed theory does not describe the particle phenomena actually observed, but, despite its lack of contact with real phenomena, becomes a main character in the narrative of spontaneous symmetry breaking in particle physics. In that story, the transformed, symmetric theory plays the part of a more fundamental theory than the Standard Model, while the formal transformation linking the two is described as bringing about a breakdown of some of the symmetries of the fundamental theory, notably the ones associated with massless particles. Since the transformation is a formal change of variable of the field associated to the Higgs boson, that particle can be presented in the narrative as the origin of the symmetry breaking, and therefore of all masses. As in the case of ferromagnetism or of the pencil standing upright, the symmetry breaking is “spontaneous” because it is not due to an external cause, but rather to a factor inherent to the system, in this case the Higgs boson.
26The narrative is therefore very simple: the Higgs boson gives rise to a spontaneous breakdown of various symmetries of the fundamental, massless theory, which is transformed into the Standard Model with its set of massive particles. In the passage quoted above, the process of spontaneous symmetry breaking is presented as analogous to the falling of a pencil, which breaks the symmetry of observed phenomena, but not that of the fundamental laws of physics: although the pencil fell in a specific direction, it could in principle have fallen in any other oner one. Here it is important to note that the analogy between the two cases—the pencil and the Higgs mechanism—however plausible, has no correspondence in mathematical terms. It is part of a narrative that connects two very different scientific systems. The words used when verbally relating the story of how the Higgs generates mass suggest causality, and possibly even a form of agency on the part of the Higgs boson. A peculiar expression used in some theoretical texts is that the Higgs boson generates mass by “eating” another particle (the “Goldstone boson”). The particle which is said to be “eaten” is indeed present in the fundamental theory, but not in the Standard Model, so its disappearance can be plausibly expressed in cannibalistic terms. In the textbook quoted above Peskin and Schroeder state: “It is tempting to say that the gauge boson acquired its extra degree of freedom by eating the Goldstone boson” (Peskin and Schroeder 1995: 692, italics in the original).
27The narrative sketched above in verbal terms constitutes the common core of a broad range of more or less elaborate stories of how the Higgs boson generates mass, and these can be expressed not only in words, but also in images, diagrams, mathematical formulas and especially combinations of the various media. Each version of the narrative is expressed in forms fitting to the community sharing it. Theoretical physicists connect words and images to mathematical constructs and highlight the subtleties of formal transformations, while experimenters rather work with images combined with simple formulas. Versions with appealing visual and verbal forms can be presented to the general public and, despite their variety and even occasional incoherence, all of these constructs can be regarded as versions of the same overarching narrative. Here I would like to underscore once again that the narrative is not a simplification or interpretation of a theory: the narrative is itself the theory, as it has a physical meaning which its individual elements—words, formulas, images, codes—do not carry. None of the elements can by itself explain the origin of mass, but the whole narrative can do so, and it is this explanation that was honoured with the 2013 Physics Nobel Prize, awarded for “the theoretical discovery of a mechanism that contributes to our understanding of the origin of mass of subatomic particles” (Nobel Prize 2013).
28Zooming into how this process of meaning-making works, I would like to highlight the role of non-rigorous mathematical practices in constructing narrative, scientific knowledge. As we saw above, the core of the story is the idea that two different mathematical expressions leading to different empirical predictions are actually two different manifestations of the same theory. More precisely, the more symmetric formula with massless particles is seen as the original “unbroken” form of the theory, while the other one, which defines the Standard Model and predicts observable phenomena, derives from it by way of spontaneous symmetry breaking. This way of conceiving the relationship between a physical theory and its experimental predictions may seem rather obscure, not to say incomprehensible, to laypersons, and mathematicians regard it as an example of the kind of bad mathematical practices common among theoretical physicists. Yet this is a most cherished result of today’s fundamental physical research. How did physicists come to accept such ideas as plausible conceptualization of microphysical principles? To understand this, it is necessary to look back in history.
29The idea that two different mathematical formulas that can be transformed into each other may be two alternative representation of the same physical-mathematical theory is a trivial observation. For example, any transformation of reference system will change the form of the equations expressing the gravitational forces in the Solar system, but it is expected that the observable predictions obtained by the two expressions will be physically equivalent. Instead, in our case it is assumed that the symmetric theory and the Standard Model would lead to different empirical predictions, of which only those of the Standard Model correspond to our reality. That such a situation could be realized was suggested during the 1950s by two influential particle theorists: Werner Heisenberg and Julian Schwinger (Dürr et al. 1959; Heisenberg 1957; Schwinger 1957). Their proposals were made independently from each other and were quite different in their mathematical and verbal expressions, but shared a common feature: the notion that the fundamental theory of microphysical phenomena was characterised by symmetries that were not manifested in observable phenomena. How could this apparent loss of symmetry be explained? None of both authors could offer a formal justification for their claim, but they made it plausible by referring to the above-mentioned non-rigorous mathematical procedures of renormalization used to extract predictions from theories like the Standard Model. As mathematicians know well, when using non-rigorous procedures two formally equivalent formulas may lead to non-equivalent predictions, and mathematicians regard this as an indication that an error has been made. Both Heisenberg and Schwinger instead assumed that the non-rigorous procedures did not lead astray, but rather mirrored physical effects that were not yet fully understood, and could give rise to a disappearance of symmetries in observable phenomena. In this sense, the symmetric and the less symmetric theory represented two possible states of the microphysical world, of which only one happened to be realized in the present. Sometimes physicists referred to the absent symmetries not as broken, but as hidden. In his theory, Heisenberg associated this situation with the existence of an infinite number of possible “vacuum states” of the universe, none of which is really as empty and the name would suggest.
30Although the mathematical expressions proposed in the 1950s by Heisenberg and Schwinger did not stand the test of time, their idea that observed, non-symmetric particle phenomena could be derived from a symmetric theory lived on, spurring theorists to not only try and model experimental results, but also to search for a theory with higher symmetries than those realized in practice. The spontaneous symmetry breaking due to the Higgs boson is a result of this line of research. Theorists who today work at developing supersymmetry or string theory stand in that tradition, and their creations usually display narrative character (Borrelli 2012). Moreover, the spontaneous symmetry breaking due to the Higgs boson today also plays a role in cosmology.
31When the story of the Higgs boson and the origin of mass emerged in the 1960s and early ‘70s there were no indications that the spontaneous symmetry breaking should be conceived as happening in real time. However, from the 1970s onward, the spontaneous symmetry breaking due to the Higgs boson began being connected to the cosmological theories of the time, which postulated the origin of the universe as a “Big Bang” from which time, space and matter emerged (Weinberg 1977). In this context, it was postulated that, immediately after the Big Bang, microphysical phenomena were ruled by the fundamental, symmetric, massless theory, and that only later on spontaneous symmetry breaking occurred. At that point, particles became massive and microphysical interactions took up the form still valid today. To put it into Heisenberg’s terms, which are still very much in use today, when the early universe shifted from one to another vacuum, some symmetries ceased to be manifest in observable phenomena, but remained present at a deeper level of reality.
32The narrative of spontaneous symmetry breaking in microphysics thus became a part of the story of the universe. As we read above in the CERN outreach pages: “Stars, planets and life could only emerge because particles gained their mass from a fundamental field associated with the Higgs boson” (CERN public outreach webpage 2024a). This is an example of scientific meaning that can only exist as narrative knowledge.