Ontic Structuralism and the Symmetries of Particle Physics

Aharon Kantorovich
School of Physics and Astronomy
Tel Aviv University, Israel
 (Forthcoming in Journal for General Philosophy of Science, 2009)

Abstract 
According to Structural Realism, in mature science there is structural continuity along theoretical change. A major counterexample to this thesis is the transition from the Eightfold Way to the Standard Model in particle physics. Nevertheless, the notion of structure is significantly important in comprehending the theoretical picture of particle physics, where particles change and undergo transmutations, while the only thing which remains unchanged is the basic structure, i.e. the symmetry group which controls the transmutations. This kind of view agrees with the paradigmatic case where the structure is an internal symmetry and the instantiations are the elementary particles. The metaphysical view which reflects this situation is a version of ontic structuralism.

1. Historical background: the scope of structural continuity 
By his version of structural realism, John Worrall (1989) intends to resolve the conflict between two major arguments concerning scientific realism, the 'no-miracles argument' and 'pessimistic meta-induction'. The no-miracles argument may be summarized as follows: if the unobservable entities postulated by successful scientific theories did not exist or if the theories were not true, then the high predictive power and explanatory success of modern science would be treated as a miracle or cosmic coincidence. If we believe that scientific progress is not miraculous, we may accept scientific realism since it provides us with the best explanation for the success of science. In contrast with the no-miracles argument, the pessimistic meta-induction argument seems to undermine scientific realism. According to the latter argument, we cannot commit ourselves to the belief in present theories since successful theories throughout the history of science were refuted or abandoned. From this we conclude that present successful theories are likely to be replaced or refuted too. This is a kind of second-order induction, since the inductive inference operates here on laws or theories that in their turn are products of first-order induction performed on natural phenomena. 
Worrall attempts to overcome the above apparent conflict by claiming that in 'mature' science the transition from the old to the new theory there is continuity in mathematical structure, whereas the content changes. He refers in particular to physics and treats, as an example, the case of the transition from Fresnel's theory of light to Maxwell's theory of the electromagnetic field. He also mentions the cases in the early 20th century where the “correspondence principle” applied (e.g. the transition from classical to quantum mechanics). The latter cases demonstrate only partial continuity of structure. 
However, we cannot generalize from these cases to the sweeping thesis that in mature science the mathematical structure does not change, or undergoes only non-radical changes, in the transition from one theory to its successor. I would like to point to a major counterexample to the thesis which appears in the forefront of modern physics. In the transition from the Eightfold Way (EW) to the Standard Model (SM) in particle physics we find the same underlying mathematical group – SU(3) – as one of the symmetries operating in the field. But in the SM the same mathematical group operates on entirely different physical magnitudes – on 'colors' instead of the old 'flavors' – not to mention other substantial changes (e.g. the new gauge symmetry is local, whereas the old symmetry was global). Thus, if a physical structure is defined as a mathematical structure plus the physical interpretation, the structure behind the SM is entirely different from that behind the EW, so that there is no continuity in the transition EWSM. The EW was by no means unsuccessful. It yielded at least two major achievements. First, the prediction of the omega-minus particle which is considered to be an exemplary achievement in modern physics; it can be compared to the prediction of new chemical elements by the Periodic Table or the prediction of the positron by Dirac. Secondly, the EW led directly to the discovery/construction of the quarks and, eventually to the SM. In addition, the EW generated many successful predictions about relation between cross-sections of interactions between hadrons and other processes (such as decay rates of resonances). So the EW (as well as the successful SM) was part of a mature science, which contributed considerably to the progress of particle physics. Thus, the EW cannot be equated in this respect, for instance, to phlogiston theory which was typically part of an immature science. But in the transition to the new theory – the SM – there was no continuity of structure, although the same mathematical group reappears in the SM as one of the underlying symmetries. Thus, we have here a case of scientific progress without structural continuity.  
Since EW had successful predictions, it would be a miracle if it did not reflect – at least approximately – some aspects of reality. But contrary to Worrall’s conception, in the EWSM transition the underlying structure did not remain the same. A transition which would exhibit structural continuity could take place if the original ('flavor') SU(3) would be replaced by SU(6) which is an extension of the original SU(3). Indeed, the latter structure was generated by the triplet of up, down and strange quarks, {u, d, s}, which form the fundamental representation of flavor SU(3). If we add the other triplet of charm, bottom and top quarks, {c, b, t},which appear in the SM table of quarks, we get all six quarks which may be assigned to the six-dimensional, fundamental representation of the group SU(6). The transition from SU(3) to SU(6) would be simpler from the group-theoretical perspective. However, it would exclude leptons from the picture and would not account, for instance, for electroweak unification. This hypothetical transition would exhibit structural continuity at the expense of significantly reduced unification of the forces. 
Thus, in trying to overcome pessimistic meta-induction with respect to scientific theories, Worrall employs too 'optimistic' induction with respect to structures. But an overall optimism is not justified here. Instead, cases of theoretical change may be positioned anywhere along the spectrum of continuity/discontinuity with respect to structures. The cases of partial continuity, such as in the big changes in early 20th century physics, deserve a special attention since they signify real progress, where the old theories can be treated as limiting cases of the new ones (with c, h0). But it is not clear what kind of 'structure' was kept invariant in these cases. To treat some mathematical results as approximately true does not justify the new title of 'structural continuity'; this subject have been discussed extensively in physics and in the philosophy of science under titles such as 'correspondence principle.'  Nevertheless, although the scope and validity of structural continuity are limited, the notion of structure that emerges from this debate and – more importantly – from the developments in the second half of 20th century particle physics leads to new insights into this notion, independently of the question of structural continuity. 

2. Structures and symmetry groups in particle physics
One of the motivations for adopting the structuralist metaphysics comes from our experience in particle physics, where particles very frequently change their identity, are created and annihilated in various processes, while the only thing which remains unchanged is the intangible structure. The most significant structure in this connection is the symmetry group which governs particle transmutations. This symmetry – the so-called 'internal symmetry' – is the deepest layer in the ontological hierarchy. In particular, the SU(3) symmetry of the EW was ontologically prior to hadrons, whereas in the SM the gauge symmetry of grand unification, such as SU(5) – or one of its improved alternatives, such as SO(10) – is ontologically prior only to baryons. By ontological priority I mean the following: If there is a physically possible world where A exists but B does not, but not vice versa, then A is ontologically prior to B. I discuss the role of internal symmetries in particle physics and the concept of ontological priority in detail in a previous paper (Kantorovich 2003, 673). Similarly, Holger Lyre has suggested an account of objects that “takes the group structure as primarily given, group representations are then construed from this structure and have a mere derivative status” (Lyre 2004, 663). Some implications of above two views are discussed by James Ladyman and Don Ross (2007, 147). Following these views, Bryan Roberts (2008) singles out a unique species of structural realism which he calls 'group structural realism' (GSR). According to GSR, "[T]he existing entities described by quantum theory are organized into a hierarchy, in which a particular symmetry group occupies the top, most fundamental position" (I preferred to use here the word 'deepest' instead of 'top'). GSR reflects the above notion of ontological priority. Steven French and Ladyman (2009) claim that “Because of the width of its embrace and its complex history, defining what is meant by structure … is problematic” – but in GSR we have a precise characterization of this notion.  
If we extend the scope of our discussion from symmetry groups to structures in general, we would say that structures are ontologically prior to objects. This view is very close to ontic structural realism (OSR) – the view which claims that, to a first approximation, "[t]here are no things. Structure is all there is" Ladyman and Don Ross (2007, 130).  A less radical view is epistemic structural realism which claims that "all we know is the structure of the relations between things and not the things themselves" (Ladyman, 2007). The ontological priority of structures can be viewed as a weaker version of OSR according to which there are some physically possible worlds where things or objects do exist. This view seems to be more realistic since individual particle tracks can be observed very vividly in cosmic rays and bubble-chamber photographs (for instance).

3. The concept of structure 
If we do not restrict ourselves to symmetry groups, structural realists characterize structure as a set, or a web, of relations. For instance, John Stachel notes that “the concept of structure refers to some set of relations between the things or entities that they relate, called the relata” (Stachel 2005). But there are other meanings to this word. For instance, physicists have in mind the concrete sense of structure which refers to a system of entities possibly hidden from our senses – as in the case of the molecular structure of gases, the structure of hydrogen atom or the quark composition of hadrons as it was conceived in the EW. A concrete structure yields a set of relations, which was mentioned in Stachel's formulation, such as in the case of the kinetic theory of gases or Bohr’s model of the hydrogen atom, which generate the laws connecting the pressure, volume and absolute temperature of ideal gas (Boyle’s law etc.), or the regularities of hydrogen line spectra (Balmer’s series etc.), respectively. Thus, whereas an abstract structure refers exclusively to a network of laws and relations, a concrete structure is a tangible system of objects and their properties from which such a network can be derived. Thus, what is common to both kinds of structures is that both incorporate such a network. However, structural realists refer in general to the abstract kind of structure. Therefore, the notion of structure employed in physics (and in popular physics) has a different connotation than the abstract or mathematical notion advocated by structural realists. For instance, when physicists speak about "the structure of matter" or "the structure of hydrogen atom", they refer to a concrete structure. With the advance of science – in particular physics – we encounter more and more the abstract kind of structures, such as the internal symmetries of particle physics.
The structure underlying a certain domain of phenomena, such as those referred to by particle physics (e.g. interactions between elementary particles) should cover all the phenomena in that area, linking them into a web of relations. It cannot be just a collection of unrelated or loosely-related relations or regularities that are expressed by mathematical formulas, such as the empirical gas laws before the kinetic theory or the empirical laws of electricity and magnetism before Maxwell’s theory. We would not call such an accidental collection of relations or regularities expressed mathematically a structure. Ideally, the structure underlying a given area should cover the whole area and introduce the following four requirements into its system of regularities: (1) compactness, (2) interconnectedness, (3) completeness and (4) unification. The first requirement means that the structure is simple and includes a minimal number of natural kinds. The second means that each regularity is connected directly or indirectly to all other regularities so that no regularity remains isolated. The third means that the web of regularities covers all the phenomena in the field. And the fourth means that in the epistemic process of structuralization, seemingly disparate phenomena become unified. The network of regularities is unified through the abovementioned fourth attribute that the structure introduces into the otherwise accidental tangle of regularities. These four requirements assure us that the structure forms a non-fragmented system in a given domain of reality. Indeed, to some extent, these requirements were satisfied by the structure described by the kinetic theory of gases or by Maxwell’s theory of electromagnetism. This seems to be a reasonable (partial) explication of what is meant by 'structure' in the present context. A similar approach in expressed in the ‘Mill-Ramsey- Lewis' method that weeds out mere regularities and leave only genuine laws of nature. 

4. From regularities to laws and from laws to structures 
In view of the great theories of 20th century physics, laws have gained new significance from the structures within which they were embedded, possibly with some modifications. I suggest to treat a law of nature as a regularity which is backed by, or can be derived from, a comprehensive structure which backs all other regularities in the field. This conception reflects the metaphysical view that nature is not fragmented. For instance, only after the kinetic theory has been established, Boyle’s empirical law (or regularity) of ideal gas could be treated as a law of nature since it was backed by an established (concrete) structure which covered other regularities in the field as well. Every such regularity contributed to the degree of confirmation of the theory, and the latter, in its turn, conferred high confirmation upon the regularities. This kind of mutual confirmation was an indication that these regularities might represent laws. Indeed, a statement expressing a regularity which is not supported by a law of nature cannot have a high degree of confirmation; it may be accidentally true only for a very limited domain of reality. Thus, we may adopt the view that a law is a regularity which is linked with a set of many interrelated regularities in a given domain (a similar view is expressed by Kosso 2000). According to the scheme outlined above, the set satisfies our four requirements which characterize a structure. The regularities covered in this way by the structure would not be considered by the realist to be 'accidental' anymore – they would be treated as laws of nature and the integrated network of laws may be treated as a “super-law”. It should be made clear that this view about the laws of nature – in particular laws of physics – is supposed to explicate the physicists’ attitude towards laws; for instance, John Earman (2002) brings a citation from Steven Weinberg that can be interpreted in this manner. However, this view does not reflect some logical or metaphysical tools for treating laws available in the philosophy of science. Such tools are, in general, extremely sophisticated and involve serious problems to be useful for comprehending the nature of laws in the present context. Thus, our approach is naturalistic rather than analytic or a-prioristic. I discuss the naturalistic view in detail elsewhere (Kantorovich 1993, Ch. 4).
 In modern theoretical physics the transition to abstract structures became evident. The instrumentalist would be content with this since the mathematical machinery leads directly to calculations and predictions of measurable results and there is no need for a covering story. But the committed realist would look for something more – for the “building blocks” of reality, so to speak. This time, however, the building blocks are intangible. The realist might look for these building blocks in the metaphysics of Platonic realism (see, for example, Michael Tooley 1977 and James Brown 1994). The Platonic narrative tells us that laws of nature are invariable relations among universals. Laws are not parasitic on objects, properties or events. Laws do not have to be instantiated in order to exist. 
 In order to comprehend the Platonic view we may employ the treatment of laws, developed independently by David Armstrong (1983), Fred Dretske (1977) and Tooley. “Each claims that laws of nature are relations among universals, i.e. among abstract entities which exist independently of physical objects, independently of us and outside of space and time” (Brown, 96). Tooley’s view is the most Platonic among the three. Armstrong maintains that when we have a law which states that all Fs are Gs, it means that there is a relation of necessitation between the universals F-ness and G-ness. He symbolizes the relation of nomic necessitation between the universals F and G as N(F,G). He tries to specify this lawmaking relation by claiming that it is a causal relation which “relates types not tokens” (1993, 422). But the nature of this causal relation remains unclear. However, we may dispense with the metaphysically- loaded notion of nomic necessitation and replace it by employing the quasi-mathematical notion of structure. According to this conception, the structure is what makes the law by integrating it within a compact network of regularities. 
According to the Platonic view, the relations among universals can be treated independently of particulars. Structures which are webs of these relations can be treated similarly, in compliance with OSR according to which "There are no things. Structure is all there is", to use Ladyman and Ross' quotation mentioned above. 
	
5. Ontic structuralism: the case of internal symmetries  
A view which can be categorized as ontic structuralism is advocated  by French and Ladyman (F&L). Ladyman maintains that structures are “primitive and ontologically subsistent” (Ladyman 1998, 420). This means that structures are independent of an ontology of objects and properties. Furthermore, individuals can be viewed as “different representations of the same structure.” This statement can be understood most clearly when the structure is a symmetry group. Indeed, F&L’s paradigm case for structure is the symmetry group. Regardless of their original intention, which are related to permutation symmetry, their position would become much sharper if we apply it specifically to internal symmetries in particle physics, which have a distinctive ontological status. 
Let us consider the relation between internal symmetries and particles, viewed as structures and their instantiations, respectively, in the case of the EW, when field theory was ignored. For the sake of demonstration let us employ the simple examples of isospin-SU(2) and unitary-spin-SU(3) before the SM. According to these symmetries, a family of particles, such as the pair of nucleons or the eight lower baryons (the nucleons, sigmas, cascades and lambda), was treated as a set of different members of a group-theoretical representation of the symmetry group – such as an iso-doublet and an SU(3)-octet, respectively. According to this picture, different observers may perceive reality differently, depending on their frame of reference. For instance, when one observer sees a proton in his/her frame of reference, another observer, whose frame of reference is rotated from the first in isospin space, may see a neutron. Now, since objective reality should be independent of the reference frame, objective status can be given only to the invariants of the group. For instance, under isospin transformations the particle states (e.g. proton or neutron) are not invariant, whereas total isospin, measured by the operator I2 = Ix2 + Iy2 + Iz2, remains invariant. The operator of total isospin is an example of a Casimir operator. The Casimir operators commute with all the generators of the group and its eigenvalues are therefore invariant under all the transformations of the group (which are spun by these generators). In general, the eigenvalues of the Casimir operators are the invariant quantum numbers associated with the symmetry group and as such they characterize each representation of the group. The commutation relations between the generators Gi of a Lie group are: [Gi,Gj] = icijkGk, where cijk are called – not unexpectedly – the “structure constants” of the group. The commutations relations played the role of interconnected relations. The integrated set of structure-constants entailed a set of interrelated laws. They could be viewed as generalized conservation laws that dictate what particles could be produced in a given interaction. Symmetry-breaking terms accounted for a non-exact symmetry. A related structure was the algebra of currents, where the laws referred to different magnitudes, such as electric currents and other kinds of currents related to weak interactions. 
When French employs group theory for demonstrating the claim that structures require no individuals in order to exist, he poses the following question: “how can we talk of a group if we have done away with the elements which are grouped?” His answer is that “the elements themselves, regarded as individuals, have only a heuristic role in allowing for the introduction of the structures which then carry the ontological weight” (French 1999, 204). French does not refer here to internal symmetries, but this claim happens to express the attitude of some leading theoreticians in the mid-1960s who focused their interest on internal symmetries of hadrons, whereas the hadrons were treated mainly as means of discovering and testing the symmetry. The reason for that attitude was the explosion in hadron population at the time – from the two nucleons and three pions to tens of new short-living hadron resonances. As a consequence, the discovery of new hadrons at higher and higher energies has become almost a routine. A more significant task was to find the deep structure behind this flood of the short-living hadrons. As a result, these particles were not treated as “elementary” any more. Indeed, the word “elementary” was dropped from the title “elementary particle physics.” The relation between particles and the (abstract) structure, i.e. between hadrons and the EW, may be compared with the relation between the spectral lines of hydrogen and Bohr’s planetary model of the atom. The ontological priority of the structure is evident here. The physicist was interested in the details of hydrogen’s spectra as means of uncovering hydrogen’s concrete structure. So the relation between structures and their instantiations remains the same for the concrete and the abstract cases: In both cases the 'spectrum' of instantiations is dictated by the structure, and the structure is ontologically prior to the instantiations, according to our definition. 
Thus, in particle physics we have witnessed a shift of interest from particles to structures (internal symmetry groups or current algebras). Physicists were interested in particles mainly as means for discovering or confirming the symmetry group; the latter became their main concern. Particles act as information channels to the structure – to the internal symmetry – and as such they indeed carry less “ontological weight.” Thus, as we saw, the symmetry is ontologically prior to the particles. This means that there are possible worlds where the particles do not exist (but the symmetry always exists). This is our version of OSR. 
One of the best ways to demonstrate this relation between structures and their instantiations is to examine the classification of elementary particles in representations of internal symmetry groups. In the early and mid-1960s, when field theory was almost ignored in particle physics, theoretical particle physicists were divided into two camps: those who dealt with the dynamics of the interactions between the particles and those who were engaged with internal symmetries. The latter saw the main object of their study the algebraic relations (i.e. the commutation relations) between the quantum-mechanical operators that measure the various (generalized) charges (e.g. electric charge, isospin or hypercharge). As was indicated above, the group-structure could be defined through the commutation relations between the generators of the group, without mentioning particle states. The representations of the group, which were determined by the group-structure, accommodated the particles, but the structure existed independently of the existence of the particles, i.e. independently of the instantiations of the structure. 
Thus, if we adopt this Platonic view of structures, we can say that the structural relations exist independently of the instantiations. Instantiation is not a necessary condition for the existence of laws. For instance, a law stating that all entities of kind R (e.g. ravens) have the property B (e.g. black) exists even in a world where no entity of the kind R exists, i.e. in a possible world where the universal R is not instantiated. A law is just a relation among universals (e.g. between “ravenhood” and “blackhood”).  Hence, the notion of structure, which is an extension of this concept of law, can be treated in a Platonic manner as a web of laws. Our treatment of the relation between internal symmetries and particles demonstrates the relation between structures and their instantiations.  
        Thus, according to Platonic structuralism, which we adopt here, structures (such as internal symmetries) exist independently of the existence of their instantiations (e.g. particles or particle phenomena), but the only solid information channel to the structures is attained through these instantiations. Let us now define a 'physically possible world' as a world where the physical laws must be the same as in our world but the contingent physical conditions may be different. Then we can imagine a physically possible world where a structure (such as unitary symmetry) is not instantiated (e.g. a world empty of hadrons) and without sufficient energy to produce these instantiations (e.g. a world which does not include energetic enough photons). The fact that an internal symmetry can exist as a structure in a possible world which is empty of its instantiations suggests that an internal symmetry is ontologically prior to the particles. 
When we turn to the world of quantum fields we have the following pattern of advance in unification which is based on an underlying internal symmetry as the basic structure. In QED, the underlying symmetry group, U(1), unifies and controls the dynamics. In the case of the electroweak fields the unification is provided by the larger symmetry U(1) X SU(2). In grand unification the dynamics is determined by still larger symmetry group, such as SU(5). In all these cases the symmetry determines the dynamics. Thus all gauge symmetries from U(1) to SU(5) (for instance) exhibit the same pattern: the symmetry is a central component of the structure and the gauge fields are the central dynamical variables in the structure (see Kantorovich 2003 for further details). 
Thus, the form of structuralism advocated here is based initially on the internal symmetries of particle physics in the mid-1960s, when field theory was partially ignored.  However, the structuralist picture can be retained in view of the field-theoretical developments since then. It should be stressed that structuralism is not a logical conclusion of these symmetries. Rather, as will be shown below in our filter model for structures, structuralism seems to provide us with some new insights into the metaphysical interpretation of particle physics.

6. The filter model for structures: Weak ontic structuralism
The Platonic realist relation between structures and their instantiations can be illustrated in a metaphorical way in the case when the structure is the internal symmetry and the instantiations are particles. Let us imagine a vertical cylindrical mold filled with some amorphous plastic stuff, such as plasticine or dough, which is compressed from the top by a piston. Inside the cylinder, near the bottom, there is an invisible filter (hidden within the opaque cylinder) consisting of holes of different shapes and sizes. Pieces of this stuff (“particles”) emerge outside the cylinder through the filter. The filter in this model has more fundamental significance than the particles since it determines the shapes and sizes of the emerging particles; the geometrical properties of each particle determine its identity. Furthermore, the particles would not be produced at all without the act of compression. The filter, on the other hand, exists independently of this act. 
Thus, the parallelism between this model and the above relation between the structure and the particles consists of the following elements. The filter corresponds to the structure or the internal symmetry behind the particle phenomena. Again, for the sake of simplicity let us consider the exemplary case of the unitary symmetry or the EW as the symmetry of hadrons. In this case the structure encompassed the collection of all (ordinary and generalized) conservation laws of the various physical magnitudes of hadrons, such as electric charge, hypercharge, total isospin and the two magnitudes measured by the two Casimir operators of SU(3). The human observer, equipped with the appropriate experimental tools, can detect only the particles. The act of compressing the piston corresponds to an investment of energy in the process, whether in the lab or in natural processes (e.g. in the stars). The most conspicuous example of such a process is particle-antiparticle pair production in a photon-photon interaction, where hadron pairs are generated subject to the restrictions and control of the symmetry. In a pair production, such as the production of proton-antiproton pair, a minimal energy of the incoming photons is required. Thus, energy is what drives the piston in our metaphor. And what is the plasticine or the amorphous matter? It seems that the plasticine corresponds to the vacuum which is the state with the lowest energy of the world, or the world with all excitations removed.  
The structure – represented by the filter – is hidden from our eyes. We have an epistemic access (through sophisticated experimental devices) only to the shadows-world – the world of particles – whereas the symmetries are the Platonic puppets which hide from our eyes. We are imprisoned in our cave (the world outside the cylinder), observe the shadows (the particles) and make use of our imagination and reason to construct theories about the underlying structure. Our hypothetical hadron-free world is a world without sufficient energy to produce hadrons. This corresponds to the case where the act of compressing the piston is not strong enough, so that no piece of stuff emerges through the filter. 
The filter model cannot represent ontic structuralism, since it does not imply that there is nothing else in the world beyond structures; it accommodates particles as instantiations of the structure. However, it demonstrates the following weaker thesis: there is a physically possible world where the instantiations of the structure do not exist. In this possible world the physical contingencies are such that no instantiation exists, but the structure exists independently of contingencies. As we saw, in such a hypothetical world there is not sufficient energy to produce the instantiations. This happens, for instance, in a world empty of massive particles and the photon energies are too low to produce such particles. Therefore, we can say that the structure is ontologically prior to its instantiations. The latter have lower ontological status; they are eliminated only in some possible worlds, whereas structures exist in all possible worlds. This view may be called “weak ontic structuralism” (WOS). Internal symmetries, such as the EW, can be treated as a paradigmatic case for this relation between structures and their instantiations. 
In this connection, John Busch (2003) claims that those who adopt the Platonic view mean that “if there is nothing in this world that can occupy the role of relata, the relation still exist” (see also the abovementioned Platonic view of laws). He rejects the elimination of individuals implied by this view, claiming that “[i]t is plausible to believe that ontology terminates somewhere”, i.e. “that somewhere there are individuals”.  Indeed, it is true that a structure may be “made of” substructures, but the chain of substructures does not necessarily terminate in individuals. For instance, current algebra was made of quantum-mechanical operators which are not “individuals” or physical objects; rather they are components of the structure. The operators measure physical magnitudes of particles which may exist or not – depending on what possible world we are talking about – whereas the current algebra or the internal symmetry group does exist in every possible world, including a world empty of physical objects. Hence according to WOS the existence of objects is optional, whereas according to Busch’s belief it is obligatory. 
There are physically possible worlds where the fundamental symmetry group is not instantiated by particles during a certain period of time, but the physical conditions may change. For instance, particles pairs may be created from the vacuum by photons with sufficient energy. In that case the symmetry, which exists as an abstract structure, will dictate what particles will be created. Our actual world is such a world: the symmetry of grand unification was present as the basic structure without spatiotemporal extension, and immediately after the Big Bang the physical world was occupied only by photons, pions, electrons and neutrinos. The baryons were created a split second afterwards under the dictate of the symmetry group. Thus, according to current theories, the fundamental symmetry/structure was present in the universe as an independent entity at the moment of creation of the bulk of matter. We'll revisit this story in the next section.

7. The role of structures in physical process of creation
Perhaps all current physical theories that describe fundamental structures are converging into a general scheme where internal symmetry is the essential part of the structure. See, for instance, QED, quantum chromodynamics, electroweak unification and grand unified theories (GUTs). 
Cosmologists tend to adopt the thesis that the early universe started at the Big Bang with baryon number zero, or without any baryons or anti-baryons at all, and then evolved into the present universe with baryon excess. It is estimated that about 10-35 s after the Big Bang the world was empty of baryon matter and immediately afterward the baryons were born. The birth of baryons took place in processes such as 0e+p. So the symmetry group played a crucial role in the world before the birth of baryons; it dictated the spectrum of possible particles that could be created – the particles should occupy definite representations of the group. Thus, until the birth of baryons we have a possible world with a physical situation in which the symmetry exists but the baryons do not. This possible world happens to be our actual world. On the other hand, there is no physically possible world where baryons exist and the symmetry does not. Thus, according to our definition, the symmetry of grand unification (SU(5) for instance) is ontologically prior to baryons,  i.e. to the fermions which comprise the bulk of matter.
It should be noted that gauge bosons, including photons, do not have an “inferior” ontological status with respect to their respective symmetries (or structures). For instance, in the case of QED, photons exist in all physically possible worlds. Indeed, photons are required to preserve the invariance of the free-electron QED Lagrangian under local phase transformations (gauge symmetry group). Namely, the U(1) gauge symmetry requires the existence of photons. Hence in every physically possible world photons exist together with the structure. This is true also of all gauge bosons which are the carriers of forces. 
Our filter model and the concept of ontological priority apply, with some modification, also to the SM and GUTs. The Platonic world is occupied by the abstract structures which govern the behavior and composition of the physical world. At super-high energies all gauge bosons were practically massless, and all forces were unified, since all the carriers of forces – the gauge bosons – looked alike, i.e. like photons. At this energy scale, the symmetry was the symmetry of grand unification. As the physical universe cooled down after the Big Bang, the structure changed: the perfect symmetry was spontaneously broken (e.g. SU(5) was broken into SU(3) ׳ SU(2) ׳ U(1)), and the world has gradually reached its present physical conditions with a different structure. 
Thus, we reached the conclusion that the symmetry of grand unification is ontologically prior to baryon matter, which is the bulk of matter. This is not too far from the conclusion that we reached for the old symmetries: the old SU(3) symmetry was ontologically prior to hadrons, i.e. to baryons and mesons, whereas the symmetry of GUTs is ontologically prior only to baryons. Returning to our filter model, the vacuum – which consists now of the sea of virtual fermions-pairs and gauge bosons – corresponds to the plasticine, while the photon pairs that supplies the energy to the processes of particle production from the vacuum, correspond to the driving piston. So, in general outline, SM and GUTs still reflects weak ontic structuralism. The above description encapsulates the story of creation of matter under the control or dictate of the symmetry. A more detailed exposition of this hypothetical story appears in my previous paper (2003). 
              
8. A Final Remark: Gell-Mann as an ontic structuralist
Tian Yu Cao mentions (2003, 7) the case of Gell-Mann who introduced quarks as mathematical objects for generating his algebra of currents as the basic “observable structure.” The existence of quarks was entailed by the old SU(3) symmetry since they corresponded to the fundamental, 3-dimensional, representation of the group. Since baryons occupied the octet and decuplet representations, it was expected that also the fundamental triplet would be occupied by physical particles too. But all attempts to detect the missing particles had failed. Gell-Mann, who coined the name 'quarks' for these particles, treated them as mathematical entities from which current algebra was constructed. Quarks therefore played only “a heuristic role in allowing for the introduction of the structure”, to use French’s words. This reflected an anti-realist sentiment which was common to many theoretical physicists at the time. However, Cao interprets this view as ontic structuralism. Indeed, quarks could be treated as mathematical building blocks of the structure which consisted of the SU(3) symmetry and current algebra. This structure encompassed relations between operators that measure charges and currents. However, the latter could be accessible to the physicist only through the instantiations of the structure, i.e. through hadronic transitions. 
Cao claims that Gell-Mann’s approach, interpreted as ontic structuralism, apparently hindered the progress of particle physics, and he uses this questionable fact as an argument against ontic structuralism. Even if we accept Cao’s assertion that the progress of particle physics was halted by Gell-Mann, this is of course a fallacious kind of argument against ontic structuralism (“guilt by association”, to use F&L’s expression 2003, 42). In any case, Gell-Mann's major contribution to particle physics, and the Nobel Prize he won as a result, is not considered by particle physicists as hindering the progress of the field – to say the least. His achievements,  such as (the original version of) the quark model, were among those which paved the way to the SM. Regardless of the question whether Gell-Mann was consciously an ontic structuralist, his current algebra can be interpreted as supporting ontic structuralism, since his mathematical quarks did not refer to real particles. This is not intended of course to be a logical or analytic argument for ontic structuralism. Gell-Mann's views were accepted by a large portion of the community of particle physicists in the 1960's, hence, if we adopt the naturalistic attitude, these views were part of the common practice in particle physics, and as such they lended naturalistic support to ontic structuralism.

References
Armstrong, D.M. (1983). What is a law of nature? Cambridge: Cambridge University Press.
 Armstrong, D.M. (1993). The identification problem and the inference problem. Philosophy and Phenomenological Research, 53: 421-422. 
Brown, J.R. (1994). Smoke and mirrors: How science reflects reality.  London and New York: Routledge.
Busch, J. (2003). What structures could not be. International Studies in the Philosophy of Science, 17, 211-223.
Dretske, F. (1977). Laws of nature. Philosophy of Science, 44, 248-268.
Earman, J. (2002). Laws, symmetry, and symmetry breaking; invariance, conservation principles, and objectivity. 18 Biennial Mtg, PSA 2002 Symposia. 
Esfeld, M. and V. Lam. (2006). Moderate structural realism about space-time. Synthese 
French, S. (1999). Models and mathematics in physics: the role of group theory. In J. Butterfield and C. Pagonis, From physics to philosophy (187-207). Cambridge: Cambridge University.
French, S. and J. Ladyman. (2003). Remodelling structural realism: quantum physics and the metaphysics of structure. Synthese 136, 31-56.
French, S. and J. Ladyman. (2009, forthcoming) In defence of ontic structural realism
Kantorovich, A. (1993). Scientific  Discovery: Logic and Tinkering, Albany: State University of New York Press.
Kantorovich, A. (2003). The priority of internal symmetries in particle    physics. Studies in History and Philosophy of Modern Physics, 34, 651-675.
Kosso, P. (2000). Fundamental and accidental symmetries.  International Studies in the Philosophy of Science.  14, 109-121.
Ladyman, J. (1998). What is structural realism? Studies in History and Philosophy of Science, 29, 409-424.
Ladyman, J. (2007). "Structural Realism" entry in Stanford Encyclopedia of Philosophy (SEP). Internet.
Ladyman, J and D. Ross (2007). Every Thing Must Go: Metaphysics Naturalised, Oxford University Press.
Lyre, H. (2004). Holism and structuralism in U(1) gauge theory.  Studies in History and Philosophy of Modern Physics 35, 643-670.
Psillos, S. (1999). Scientific Realism: How Science Tracks Truth, London and New York: Routledge.
Roberts, B. (2009). Group structural realism. Blog: SoulPhysics.org
Stachel, John. (2005). Structures, individuality and quantum gravity. In D.P. Rickles et. al. (Eds), Structural Foundations of Quantum Gravity. Oxford University Press.
Tooley, M. (1977). The nature of laws. Canadian Journal of Philosophy, 7, 667-698.
Worrall, J. (1989). Structural realism: The best of both worlds? Dialectica, 43, 99-124.

Footnotes