the priority of internal symmetries in particle physics

PERGAMONStudies in History and Philosophy ofModern Physics 34 (2003) 651-675

Studies in Historyand Philosophyof Modern Physics

The priority of internal symmetries in particlephysics

Aharon KantorovichPhilosophy of Science Foundation, P. O. Box 39860, Tel Aviv 61398, Israel

Abstract

In this paper, I try to decipher the role of internal symmetries in the ontological maze ofparticle physics. The relationship between internal symmetries and laws of nature is discussedwithin the framework of "Platonic realism." The notion of physical "structure" is introducedas representing a deeper ontological layer behind the observable world. I argue that an internalsymmetry is a structure encompassing laws of nature. The application of internal symmetrygroups to particle physics came about in two revolutionary steps. The first was theintroduction of the internal symmetries of hadrons in the early 1960s. These global andapproximate symmetries served as means of bypassing the dynamics. I argue that the realistcould interpret these symmetries as ontologically prior to the hadrons. The second step was thegauge revolution in the 1970s, where symmetries became local and exact and were integratedwith the dynamics. I argue that the symmetries of the second generation are fundamental inthe following two respects: (1) According to the so-called "gauge argument," gauge symmetrydictates the existence of gauge bosons, which determine the nature of the forces. This view,which has been recently criticized by some philosophers, is widely accepted in particle physicsat least as a heuristic principle. (2) In view of grand unified theories, the new symmetries can beinterpreted as ontologically prior to baryon matter.© 2003 Elsevier Ltd. All rights reserved.

Keywords: Internal symmetry; Laws of nature; Particles; Physical structure; Platonic realism

1. Introduction

Symmetries have always played an important role in physics; however, in particlephysics, they have become central players. In the early 1960s, the construction of adetailed dynamics of strong interactions was entangled with insurmountable

E-mail address:[email protected] (A. Kantorovich).

difficulties, and field theory was partially ignored. In this methodological situation,the power of symmetry considerations followed from the fact that they did notdepend on detailed dynamical knowledge. Physicists could treat strong interactionswithout having to determine the Lagrangian first. They could shortcut the dynamics,drawing observable conclusions directly from the symmetries. In particular, theoriesof internal symmetries of hadrons (strongly interacting particles) played a centralrole at that period. Later, as field theory regained its status, internal symmetries haveplayed an even more important role.Internal symmetry groups act on internal variables that are divorced from

space–time. Eugene Wigner employs a somewhat different terminology when hedistinguishes between geometrical and dynamical principles of symmetry(Wigner, 1967, pp. 17–18). The former is formulated in terms of events in spaceand time, the latter in terms of laws of nature that operate on dynamicalvariables. As its name implies, a dynamical symmetry is a symmetry of a particularinteraction or force. Wigner refers to dynamical invariance groups such as thegauge group Uð1Þ; which applies to the electromagnetic interactions, and theSUð3Þ group, which applies (at that time) to the strong interactions. He claims,for instance, that ‘‘the electromagnetic interaction is gauge invariant, referringto a specific law of nature which regulates the generation of the electromagneticfield by charges, and the influence of the electromagnetic field on the motionof charges’’ (1967, pp. 17–18). Wigner’s dynamical symmetries are essentiallythe same as the internal symmetries. The internal symmetries are wider entitiesthan laws of nature; laws of nature can be derived from these symmetries. In thisconnection, Wigner stresses that there is a parallelism between the relation ofsymmetry principles to the laws of nature, on the one hand, and the relationof the laws of nature to physical events on the other. In particular, as we will see,an internal symmetry entails laws regarding the physical properties of variousparticles. Moreover, he points to the metatheoretical character of symmetriesthat makes them suitable for constructing fundamental theories (see Falkenburg,1988, p. 115). By this, he foresaw the heuristic power of symmetries in constructingtheories (see Section 5 below).The first internal symmetry of strong interactions was the isospin symmetry group

SUð2Þ that was proposed by Werner Heisenberg (among others) who suggested thatthe proton and the neutron are two states of the same entity—the nucleon—transforming into each other under the group. Thus, the proton and neutron belongto the same two-dimensional representation or family of SUð2Þ: This was anapproximate symmetry since the proton and neutron have slightly different masses.In the limit of vanishing electromagnetic force, the proton–neutron small massdifference vanishes. So in case the electromagnetic force, is shut off, the symmetrybecomes exact. An extension of this symmetry was SUð3Þ; or the Eightfold Way,which was proposed independently by Murray Gell-Mann and Yuval Ne’eman (seeGell-Mann & Ne’eman, 1964). It contained the SUð2Þ of isospin as a sub-group (theSUð3Þ of hadrons was also called ‘‘unitary spin,’’ since it was treated as an extensionof isospin). The SUð3Þ symmetry was even less exact than isospin symmetry; theaverage mass difference between the different members of an SUð3Þ representation

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(such as the octet of baryons, which included the nucleons) was much larger thanbetween different members of an isomultiplet (an isospin family).Today, in retrospect, physicists claim that these approximate symmetries were

merely ‘‘phenomenological.’’ For example, Weinberg (1997, p. 40) claims that ‘‘thesesymmetries never were anything but accidents.’’ However, at the time, thesesymmetries were treated by some leading theoreticians as more fundamental thanaccidental regularities. (Later on, I will turn to the notion of ‘‘fundamental’’symmetry employed by Kosso, 2000.) Their status was compared with that of theregularities appearing in Mendeleev’s Periodic Table that were by no meansaccidental; they were eventually accounted for by theories of atomic structure. Someparticle physicists hypothesized in the 1960s that these symmetries could be derivedfrom a hadron sub-structure such as the composite quark model. There were alsoattempts to derive SUð3Þ from an abstract model of space consisting of extremelysmall spheres (replacing the dimensionless geometrical points of ordinary space)which ‘‘breathe’’ infinitesimally; the extra dimensions supplied by this modelprovided the source for the variables required for the SUð3Þ internal symmetry.1 Thismodel was marginal and unsuccessful, but it demonstrated the search for underlyingstructure.In order to comprehend this novel conception of internal symmetry and its

relation to the concept of law, I propose a view that may be categorized, followingTooley (1977), as ‘‘Platonic realism’’ (for a provisional exposition of this conception,see Kantorovich, 1996). This view can deal with the ontological status of the internalsymmetries. I will argue that, in a sense that will be specified below, the internalsymmetries could be interpreted as ‘‘ontologically prior’’ to the hadrons. Then I willturn to the local gauge symmetries that have dominated particle physics since the1970s. In this new context, where field theory regained its status, the priority ofsymmetries acquired a new meaning.I will start in Sections 2 and 3 with a realist distinction between accidental

regularities and laws of nature and propose a view that physical laws draw theirnomic status from what I call a physical structure that backs the laws. The internalsymmetry is an example for such a structure. It should be emphasized that thisnotion of structure differs from the notion employed in ‘‘structural realism.’’ Thelatter view2 refers to the formal or mathematical component of a theory, as opposedto the physical content. The notion of structure employed here is not mathematicallyoriented. As we will see, a physical structure may be either concrete or abstract, butin the latter case, it is the physical content that is abstract. In Section 4, I will suggestthat the old internal symmetries could be treated as ‘‘ontologically prior’’ to thehadrons. In Sections 5–7, I will argue that the new internal symmetries arefundamental in the following two respects: (1) Gauge symmetry dictates the existence

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1This was an extension of Kaluza–Klein model which was supposed to unify gravitation and

electromagnetism by adding a fifth dimension (corresponding to the electromagnetic force) to ordinary

space–time. See for example Hara and Goto, 1968.2The version of structural realism initiated by Worrall (1989) comes as a response to the problem of

continuity across revolutionary changes of theory: the (mathematical) structure is retained, while its bearer

changes.

A. Kantorovich / Studies in History and Philosophy of Modern Physics 34 (2003) 651–675 653

of gauge bosons, which determine the nature of the inter-particle forces. This view,which has been recently criticized by some philosophers, is widely accepted inparticle physics, at least as a heuristic principle. (2) In view of grand unified theories(GUTs), the new symmetries can be interpreted as ontologically prior to baryonmatter.

2. The standard model, laws of nature and Platonic realism

Particle physics provides an appropriate testing ground for confronting the viewsof empiricists and realists regarding laws of nature. I would like to start with JamesR. Brown’s treatment of the debate in his Smoke and Mirrors (Brown, 1994, p. 93).Brown tries to defend a realist view by recruiting the case of the Standard Model(SM) in particle physics. According to this model, the fundamental spin-halfparticles (fermions) (i.e., the quarks and leptons) are arranged in a series of families,where the masses of the particles increase along the series. As Brown remarks, thisclassification scheme embodies laws of nature, such as the laws stating the mass ofeach particle. Another hypothetical law that may be implied by the model is a massformula that would express the pattern of growing masses along the series, andpossibly a mass formula within each family. At present only the first three families—the electron, the muon and the tau families—are occupied by known particles, butthere is an open possibility that the series of families does not stop at the third level.Attempts have been made to discover particles belonging to higher families. But, asBrown notes, the crucial point is that due to the lack of energy and the finiteness ofthe universe, particles belonging to higher families above some level will never bediscovered or produced. Therefore, the laws stating, for example, the masses of theseheavier particles will never be instantiated in our world. And a hypothetical mass lawthat corresponds to an infinite number of particles will be applied only to the finitenumber of actual and possible particles.The above version of SM would encompass an unlimited number of laws

corresponding to heavier particles, e.g., ‘‘the mass of particle p is m;’’ or lawsregarding the values of the various quantum numbers of the particles. Most of theselaws are uninstantiated in our world but some of them may be instantiated in someother physically possible worlds. Empiricism treats laws as mere regularities;therefore, it cannot account for those laws corresponding to the heavier particles thatare uninstantiated in the actual world. This is, in a nutshell, Brown’s argumentagainst the adequacy of empiricism or the regularity view. However, the argumentjust shows that the belief in the existence of the uninstantiated particles isinconsistent with empiricism. Such a belief might arise if there is a firm theoretical

basis for the existence of these particles and the corresponding laws. The SM is notconclusive in this respect. But we can take an example from older theories ofsymmetry (which were studies amidst the symmetry-boom in the 1960s) based on‘‘non-compact’’ groups (such as Uð6; 6Þ) that definitely included infinite-dimensionalrepresentations. These so-called ‘‘ladder’’ representations consisted of a series ofrungs, where each rung accommodated a family of particles or resonances. In these


representations, the laws corresponding to hadrons above some mass would neverhave been instantiated in the actual world. Nevertheless, for theoretical reasons,some physicists tended to believe in the existence of an infinite series of hadrons inthe ladder representations. For instance, this was the case of the so-called ‘‘spectrumgenerating algebras,’’ where the infinite representation was treated as representing ahadron spectrum, in analogy with the hydrogen energy spectrum (governed by thenon-compact group Oð3; 1Þ), where the higher hadronic resonances were treated asthe higher energy levels of the spectrum. Of course, the theoretical considerationswere related also to the experimental evidence, where the lower rungs fitted the datato a certain degree, i.e., the lower rungs were occupied by observed hadrons, and thephysicists naturally extrapolated the data to higher resonances. The regularist cannotaccount for this situation of infinite representations, yielding an infinite series ofuninstantiated laws. Thus, empiricism cannot provide an explication for thephysicist’s intuition of extrapolating the data to higher energies.Platonic realism, on the other hand, does account for uninstantiated laws.

Different versions of this view are advocated, for example, by Armstrong (1983),Dretske (1977), and Tooley (1977), while the latter explicitly employs the name‘‘Platonic realism.’’ Brown expresses this view as follows:

[L]aws of nature are relations among universals; i.e., among abstract entitieswhich exist independently of physical objects, independently of us and outside ofspace and timey Laws on the platonic view are not parasitic on existing objectsand events. They have life of their own (1994, pp. 96–97).

In short, according to this view, a property or a law does not have to beinstantiated in order to exist, i.e., instantiation is not a necessary condition forexistence. As Brown notes, this stands against the central empiricist intuition asexpressed by John Earman’s words: ‘‘that laws are parasitic on occurrent facts’’(Earman, 1986, p. 85). However, as we will see below, this is not true forsophisticated empiricism.A formulation of the relations-among-universals view of laws is given by

Armstrong in terms of the relation of nomic necessitation, NðF ;GÞ; holding betweenthe universals F and G: Brown maintains that it is ‘‘a primitive notion, a theoreticalentity posited for explanatory reasons.’’ Presumably, it explains why the followinguniversal regularity about particulars holds: ðxÞðFx*GxÞ: But if we treat N as aprimitive notion, it sounds like an ad hoc explanation. So we may ask where N

comes from?

3. Structures vs. laws

In order to provide a genuine explanation for the relation N; we may hypothesizethe existence of a wider, or deeper, physical structure S that endows upon theregularity the status of nomic necessitation. According to a common conception, anecessary requirement from a theoretical explanation is that the explanans shouldinclude radically new theoretical terms that do not appear in the explanandum.

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Thus, for instance, Boyle’s ‘‘law’’ standing by itself is a mere regularity. It is thephysical structure described by the kinetic theory of gases (or a more advancedmolecular-statistical theory) that endows upon the regularity its special (nomic)status. The new theoretical entities required by the explanation in this case are themolecules. The relation of nomic ‘‘necessitation’’ means that there is a strong linkagebetween the regularity and many other regularities that cover a whole physicaldomain. In other words, these regularities are brought together under the umbrellaof a compact structure. This view is also reflected in Kosso’s approach. Hedistinguishes between fundamental and accidental regularities by the fact that‘‘[fundamental] laws are regularities upon which many other things in naturedepend,’’ whereas ‘‘nothing else depends on [accidental regularities]’’ (2000, p. 111).But this amounts to saying that a law of nature, as opposed to an accidentalregularity, is integrated within a deep structure, described by a comprehensivetheory. And the latter connects the law with ‘‘many other things’’ in the world, i.e.,with many phenomena, laws, etc.I have not yet specified the meaning of ‘‘structure’’ in the present context, but for

the time being, a preliminary remark should be made: In parallel to the realist’sdistinction between a statement that refers to a law and the law itself, it is importantto make a clear distinction between a structure S and the theory Ts that refers to S;or describes it. A structure is not a theory or a property of a theory, rather it residesin nature. Thus, Ts is a (set of) statement(s) that can be tested. If it is confirmed, itendows a strong confirmation upon all the interrelated laws covered or entailed by S:The confirmability of a law integrated within a structure is higher than that of anisolated regularity. The reason for this is that the law-statement draws itsconfirmation from the integrated network of laws that cover a wide range ofphenomena. In fact, this is a mutual confirmation: a confirmation of each law-statement raises the degree of confirmation of Ts that in its turn raises the degree ofconfirmation of the other law-statements.Thus, by introducing S we avoid the usage of the metaphysical notion of nomic

necessitation, which does not carry with it any explanatory role in modern science;nomic necessitation is not explained by S; rather S replaces it. The structure, asopposed to the relation of nomic necessitation, is described by a testable andconfirmable theory. If Ts is highly confirmed, the realist is justified to believe that S

indeed exists. It is the structure that endows upon the regularities covered by it, orderived from it, their special status. Let us refer to the relation of entailment orcovering between S and the laws as the S-L scheme, where L designates a whole setof laws. Since a variety of laws are entailed/covered by a common structure, the lawsare interrelated. In other words, laws belonging to the domain covered by S arecharacterized by the fact that they are not ontologically isolated.The view that laws are distinguished from mere regularities by the fact that they

can be derived from a comprehensive theory describing a compact structure can berelated to Frank Ramsey’s account of laws. According to this widely cited account,laws of nature are ‘‘consequences of those propositions which we should take asaxioms if we knew everything and organized it as simply as possible in a deductivesystem’’ (Ramsey, 1978, p. 138). Ramsey’s treatment of laws reflects a logico-


positivist approach that avoids any contact with metaphysics. Earman suggests asophisticated version of Ramsey’s empiricism that does account for highlytheoretical laws and uninstantiated laws. This version is linked to the dynamicmanner of handling the deductive system. Initially, the system is built on the basis ofa limited range of observed regularities. But when the basis expands to include newfacts and regularities, there might be some clashes with the existing axioms, and thesystem should be modified.3 In this process of axiomatization and reaxiomatization‘‘uninstantiated laws may emerge as consequences of the axioms of the bestdeductive system’’ (1986, p. 97).There are two kinds of structures: concrete and abstract. For example, the kinetic

theory describes a concrete structure. An example of an abstract structure iselectromagnetism, described by Maxwell’s theory. The latter is a system ofinterrelated laws that transcends the empirical laws of electricity and magnetismwith which Maxwell started. The electromagnetic field was the main new theoreticalentity in this structure. As a result, the empirical laws or regularities wereintegrated (possibly in a modified form) within a system of interrelated laws. Thus,for example, Coulomb’s law acquired the status of a law of nature due to itsintegration with other regularities under the umbrella of electromagnetism. Beforethis, it could be treated merely as an empirical regularity. The displacement current(which was introduced into the equations for symmetry considerations) and theelectromagnetic waves (the solutions of the equations) were additional theoreticalentities implied by the new structure. Both concrete and abstract structures yield anetwork of laws.Thus, a law is interpreted in this way as a regularity that is integrated within a

structure. The structure is more economical, and the theory describing it isconceptually and mathematically simpler than an accidental collection of regularitiesor empirical laws. Metaphysical realists who believe in the existence of laws, to whichlaw-like statements refer, may further believe that a structural theory as a widerlinguistic entity refers to something ontologically wider or deeper, i.e., to a structurein the world. A newly discovered structure consists of the already known laws plusnew ones, integrated within a network of interrelated laws. Furthermore, newentities, such as the electromagnetic waves, emerge in the epistemic process of‘‘structuralization.’’Finally, an internal symmetry, our main concern here, is an abstract structure.

It encompasses and entails a network of laws: laws concerning the relationsbetween various properties of particles (such as different kinds of charges,magnetic moments and perhaps masses) and relations between cross-sections,or relative probabilities, of different kinds of events. Quantum electrodynamics(QED) constitutes the simplest and the prototypic example of a structure inter-woven with (or governed by) a symmetry—as we will see, the requirement ofinvariance under local Uð1Þ gauge symmetry gives rise to the electromagneticfield. Currently, internal symmetries are embedded within a wide structure,described by the SM, which includes the theories of strong and electroweak forces,

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3For a related view of treating theory-construction as a dynamic process, see also Kantorovich (1979).


the relevant theories of symmetry and the classification table of leptons andquarks. Thus, we have here a structure that covers and entails a variety oflaws in the domain of particle properties and behavior. The fact that physicistsfeel that this model is less satisfactory than a fully fledged theory (perhaps thisis why they call it ‘‘model,’’ rather than ‘‘theory’’) is due to the missing slots inthe model, such as the lack of an explanation for the masses of the fundamentalfermions. Since, in this case, the degree of compactness of the purported structureis weaker than expected, the nomic status that S endows upon the laws covered byit is weaker too.In general, nomic ‘‘necessitation’’ is a function of the strength and compactness

of the structure and the associated network of laws. This leads to methodologicalcriteria for assessing structural theories. In this connection, Bas van Fraassenmentions three standards of comparison between theories: simplicity, strengthand balance (Van Fraassen, 1989, p. 41). Simplicity and compactness areclosely related. As to strength, according to van Fraassen, a theory is strongerif it is more informative. This is not the same as the sense attributed here tostrength. In relation to an abstract structure, we are dealing with the strengthand number of the ties between the laws in the network. Yet, stronger tiesmay be related to more information. For instance, a law may be tied morestrongly to the network if it is tied to more laws and physical magnitudes whereeach such a tie constitutes a piece of information. The third standard isneeded because of the tension between the other two: too many links increaseboth the amount of information within the network and its degree ofcomplexity. However, to a great extent judgments regarding the balancebetween simplicity and strength are left to the intuition of the scientists, or to theirtacit knowledge.Philosophers since David Hume have been preoccupied with the notion of

natural law. In the 18th century this notion embodied one of the top achievementsof natural science. However, as van Fraassen claims, ‘‘the concept of law ofnature is an anachronismy [it] is a vestigial concept in contemporary science,’’since ‘‘[m]odern physics argues from symmetry and continuity—not fromuniversality or necessityy.’’4 Later I will turn to the role of symmetryarguments in particle physics. However, my claim here is more general. Inmodern physics, the concept of law of nature is redundant: since Descartes,Newton, and Hume, science has progressed a bit, and laws have been reducedto abstract structures which have become the main focus of interest in physics.If we employ the terminology of reductionism, we may say that the laws we havestarted with are reduced to a structure. But then structures replace the laws; astructure entails (in specific conditions) the corresponding laws, so that the latterbecome superfluous. Now we are already familiar with several celebrated structures,described by physico-mathematical theories, such as the theory of electromagnetismor the SM.

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4These quotations are taken from van Fraassen’s (2001) abstract of his book Law and Symmetry (1989).

A. Kantorovich / Studies in History and Philosophy of Modern Physics 34 (2003) 651–675658

Realism with respect to concrete structures involves the belief in the existence ofphysical entities, such as molecules or quarks. In the case of the kinetic theory, forinstance, we may say that the bearer of the structure described by the theory is thegas, i.e., an aggregation of molecules in motion under certain conditions. So what isthe bearer of the abstract structure described by particle theory such as the SM? Orwhat is the thing particle theory is about? Is it a system of interacting lepton, quarkand mediator fields? As we will see, contemporary particle theories are modeled onQED. So we may ask first what is the bearer of Maxwell’s equations, if anything, orwhat is the thing Maxwell’s theory is about? van Fraassen asks this question andgives two alternative answers (Van Fraassen, 2000, pp. 4–5): (1) Reification: ‘‘it is theelectromagnetic field itself, which is a thing, and not a shape or form of somethingelse.’’ This answer complies with the rejection of the ether. (2) Structuralism: ‘‘Theequations only describe a form or structure—if that is the form or structure ofsomething, that is an unknown entity. The field is first of all an abstract entityythough we can of course also give the name ‘field’ to whatever it is—if anything—thatbears this structure.’’ Structure is viewed here as a dependent entity—depending onsomething else which is the bearer of the structure. This is not our sense of structure.van Fraassen suggests three versions of structuralism: moderate, radical, andintermediate. According to moderate structuralism ‘‘the theory describes only thestructure of a bearer, which has also non-structural features,’’ but science does notdeal with the latter. The intermediate version says that ‘‘the structure described byscience does have a bearer, but the bearer has no other features at all.’’ This versionis at best mysterious; certainly it does not belong to the scientific realm.Finally, according to the radical version ‘‘structure is all there is.’’ Radical

structuralism is very close to our notion of abstract structure. Yet, as van Fraassenadmits, the distinction between reification and radical structuralism is dubious; both‘‘alternatives’’ refer to a structure that hangs on no bearer, in compliance with theEinsteinian rejection of the ether. Radical structuralism seems to be the closest to thePlatonic conception of laws: universals or laws exist independently of anyinstantiation. Platonic realism with respect to laws and with respect to abstractstructures reifies relations among universals and networks of such relations,respectively.To sum up, according to our conception of structure, S should obey the

following three requirements: (1) a metaphysical requirement: S shouldentail a compact set of interrelated laws; (2) an epistemological requirement:S should include new kinds of universals referring to new properties and entitiesthat did not appear in the formulation of the empirical laws from which we havestarted; (3) a methodological requirement: the theory describing S should be muchmore confirmable (or falsifiable) than isolated law-statements. Structures may beeither concrete or abstract. Thus, in the case of pre-1970s internal symmetries thelaws concerning the properties and behavior of particles were derived in some casesfrom concrete models (e.g., composite quark model) or from abstract models (e.g.,the Eightfold Way or current algebras). As to contemporary SM and GUTs, the fieldtheories of electroweak and strong forces are integrated there with the theories ofsymmetry.


4. The first revolution in the conception of symmetry: the ontological priority of

internal symmetries

In this section, I would like to discuss the significance of what might be called the‘‘Platonic attitude’’ with respect to the internal symmetries in the 1960s.Instrumentalism is an easy refuge for those who cannot comprehend the reality ofabstract structures that deviate from traditional ontologies. Platonic realism withrespect to abstract structures can provide an alternative view to those who stillbelieve that science deals with reality and not merely with practical computations.Among the so-called higher symmetries which engaged particle physicists in the

1960s, we find the groups SUðNÞ; SUðNÞ � SUðNÞ and UðN;NÞ mainly with N ¼3; 6: SUð3Þ was the most successful and historically most influential. The originalSUð3Þ yielded a classification scheme of hadrons into families. The families weredictated by the mathematical structure of the underlying symmetry group; theycorresponded to the irreducible representations of the group. The N-dimensionalrepresentation is a space spun by N basis vectors that corresponded in the symmetrymodel to states with definite values of hypercharge,5 isospin, and the thirdcomponent of isospin quantum numbers (Y ; I , and I3; respectively). Each of theknown eight lower baryons had definite values of Y ; I , and I3; so that they wereassigned to an eight-dimensional representation (octet), where definite symmetryoperations transform the members (basis vectors) of the octet into one another.Similarly, the nine lower mesons were assigned to an octet plus an SUð3Þ singlet.Since the spin and parity were invariant under the transformations, they remainedthe same for all members of a given representation. Thus, the spin and parity of allmembers of the lower baryon octet were 1

2and +1, whereas the corresponding values

for the lower pseudo-scalar meson nonet (octet plus singlet) were 0 and –1,respectively. All this was summarized in the particle physicist’s jargon by theexpression that the two octets were ‘‘occupied’’ by the lower baryons and mesons.SUð3Þ has an infinite number of such representations, out of which only the one-,eight-, ten- and perhaps also the 27-dimensional representations were occupied byobserved hadrons—particles and resonances. In addition, since the symmetry wasapproximate, there were mass differences within each family and physicists employedsemi-phenomenological mass formulas describing the mass distribution in the mesonand baryon octets and in the baryon decuplet. The symmetry group could be viewedas an abstract structure S; whereas the relations between different hadronicproperties and processes that could be derived from the symmetry—or from S—formed the laws, according to our S-L scheme.Sticking to their traditional metaphysical habits, physicists hoped at first to find a

concrete structural model and dynamics behind the symmetry group. However, thefailure to provide a workable dynamical theory for the interactions of hadrons andcomposite models for their internal structure (the earlier quark model with apparentcomposite structure of hadrons was treated merely as a computational device at the

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5The hypercharge Y is defined as Y ¼ S þ B; where S is the strangeness and B is the baryon number,

which is constant within a given representation.


time) compelled some physicists in the 1960s to change their attitude towards thesymmetries. Instead of trying to derive the symmetries that govern the classificationand the dynamics of hadrons from traditional field-theoretical models, physicists sawas the object of their study the algebraic relations (i.e., the commutation relations)between the quantum-mechanical operators, which measure generalized charges andcurrents, as the ultimate physical reality. This agreed with our S-L scheme, wherethe current algebra (such as SUð2Þ � SUð2Þ or SUð3Þ � SUð3Þ), or the symmetrygroup, served as an S: This may be compared with the orthodox interpretation ofquantum mechanics that attributes reality of particulars only to properties or thingsthat are actually measured. However, in the present case we are talking about thereality of universals. Reality was attributed to universal relations betweenmeasurable quantities, or—translating this to a physical language—to algebraicrelations between operators that measure these quantities. According to the S-L

scheme, these individual relations constituted laws of nature, whereas the currentalgebras and symmetry groups (i.e., the integrated sets of relations) played the role ofabstract structures.Since particle physicists could not anchor their theoretical explanation on

enduring or stable material objects, that is, on sub-particles which would constitutecomposite structures, they anchored it on the algebra of conserved charges such aselectric charge, baryon charge (i.e., baryon number), and hypercharge orstrangeness. The symmetry of hadrons, such as SUð3Þ; yielded an integrated set ofconservation laws that were independent of space and time. This included also non-additive conserved magnitudes, such as total isospin, which characterized the isospinmultiplet (‘‘sub-family’’) to which a given hadron belonged, and the two magnitudesdefining an SUð3Þ super-multiplet (e.g. the octet or decuplet representation). As thenumber of hadrons became inflated, physicists became primarily interested in theconserved magnitudes and their symmetry group, rather than in the individualparticles. The discovery or production of new hadrons was not considered to be anaim in itself. It mainly served the purpose of discovering and confirming theories ofsymmetry. Such was, for example, the dramatic discovery of the omega-minus (O�)particle that was considered to be a great success of the Eightfold Way. This meansthat there was a shift of interest from the laws, or L; to S: And for the realist, thismeant that S; i.e., the symmetry (in that case), could be interpreted as morefundamental than the particles. When I say ‘‘particles,’’ I do not mean theirinstantiations. In these circumstances, claims about the proton, for instance, areuniversal statements about its properties and behavior, not claims about particulars.When we are talking, for instance, about the properties of the proton, we are talkingabout relations among universals, i.e., about laws. Thus, when I say that thesymmetry is more fundamental than the particles, I remain in the realm of universals.The proton, for instance, is characterized by a collection of laws, such as ‘‘the protonbelongs to an eight-dimensional representation of SUð3Þ;’’ or ‘‘the isospin of theproton is 1

2:’’ We may symbolize this list of laws as p � f

%8; I ¼ I3 ¼ 1

2;Y ¼ 1;P ¼

þ1;S ¼ 12; etc:g; referring to the proton’s SUð3Þ representation, total isospin, third

component of isospin, hypercharge, parity and spin, respectively. The proton as auniversal is instantiated by particulars that are characterized by their individual


spatio-temporal values (under the restrictions of quantum mechanics), in addition totheir universal properties.All the above-mentioned magnitudes, which are included in the characterization of

a particle’s state-vector, are universals. In particular, the universal N designates theirreducible representation to which the particle belongs (where N is thedimensionality of the representation), i.e., it determines the transformation proper-ties of the particle under the symmetry group. But unlike isospin symmetry, SUð3Þwas a broken symmetry. Thus, the isospin was an exact magnitude attached to eachhadron, whereas the SUð3Þ assignment N was approximate. For instance, the isospinof the Z0 meson is 0 and it definitely belongs to an iso-singlet. Thus, an Z0 instantiatesthe universal: ‘‘isospin zero.’’ Similarly, the SUð3Þ assignment of the Z0 could berepresented by the universal: ‘‘unitary-spin octet.’’ However, in this case thesymmetry was approximate and this was expressed by the fact that the wave functionof the meson was a mixture of two representations: the observed spin-zero mesonswith isospin and strangeness zero, Z0 and Z00; were mixtures of pure octet and singletstates. And this was supposed to be the case throughout the whole universe. So thesemesons could be described by the universals ‘‘ai1+bi8,’’ with i ¼ 1; 2; where the twovalues of i corresponded to Z0 and Z00; respectively. Thus, although the symmetrywas approximate, the representation mixings were perfect universals. For the sake ofillustration, let us consider the following examples of universals referring to ‘‘mixed’’entities: (1) ‘‘Mule’’ is a universal, although a mule is a ‘‘mixture’’ of horse and adonkey; (2) ‘‘White’’ is a perfect universal, although it consists of a mixture of allcolors; (3) ‘‘Chlorine’’ refers to a mixture of two isotopes. It may be claimed that thefirst example is not a natural kind, but at least examples (2) and (3) are natural enough.6

The fact that the internal symmetry was treated intuitively as more fundamentalthan the particles was reflected, therefore, by the fact that the particles were viewedas states of a more fundamental entity—the irreducible representation of thesymmetry group, a symmetry group which was determined by the commutationrelations of the operators which measure the charges or currents. Thus, the proton,for example, was treated as the quantum state jSUð3Þ;Y ; I ; I3S ¼ j

%8; 1; 1

2; 12S of the

unitary-spin octet, just as it was at the same time the state jSUð2Þ; I ; I3S ¼ j2; 12; 12S of

an isospin doublet. Thus, just as it could be viewed as one of the two states of thenucleon spectrum, it could also be viewed as one of the eight states of the lower-baryon spectrum, where the lower baryons are those belonging to the same octettogether with the proton. In other words, the two nucleons, three sigmas, twocascades, and the lambda could be viewed formally as eight ‘‘mass-states’’ of thesame ‘‘super-particle.’’7

ARTICLE IN PRESS

6The foregoing passage attempts to answer the following questions posed by one of the referees: (a) ‘‘It

is unclear what it means to say that group-theoretic irreducible representations are ‘universals’.’’ (b)

‘‘Moreover, the original SUðnÞy are not exacty Does this mean that their representations are

‘approximate’ universals?’’7 In practice, physicists did not treat the proton and the cascade, for example, as different states of the

same super-particle, since the mass difference between the two was too big. But there were models which

treated the N� and N�� resonances as recurrences of the nucleon with higher values of ‘‘orbital angularmomentum,’’ although the mass differences were even bigger than those in the proton octet.


In the Eightfold Way, the three-dimensional, fundamental, representation 3 ofSUð3Þ was not occupied by particles. This defect was remedied in the quark model,where the triplet was occupied by the quarks. The fact that quarks have not beendetected experimentally in isolation seems to provide a good reason in favor ofinstrumentalism: the group-theoretical machinery served here only as a device forcontrolling the data and predicting new experimental results. However, the indirectevidence was strong enough to support the belief in the existence of quarks, andphysicists (who otherwise may act as anti-realists) exhibited a realist sentiment withrespect to quarks.The symmetry group dictated via its representations the hadron spectrum (i.e., the

variety of charges and other quantum numbers, such as total isospin) anddetermined the possible outcomes of hadron interactions. Thus, hadrons were just

transient entities, whereas the internal symmetry could be seen as the permanent or

ultimate structure behind the flux of hadronic transmutations. Indeed all hadronsexcept the proton have very short lifetimes. The view which was intuitively held atthe time by some leading physicists (see for example Ne’eman (1974) and Heisenberg(1976)) that abstract symmetries, rather than concrete structures, stand at thebottom of physical reality (in the hadronic world), can be comprehended as aPlatonic view. This view could be seen an alternative to instrumentalism.The ontological status of the internal symmetry according to the above

conception, with the absence of underlying field theory, can be seen most clearlyin cases of hadron production from non-hadronic systems such as colliding electron–positron beams, where the electron–positron pair just supply the energy to theprocess in which hadrons are created. In the process of materialization the possibleoutcomes were determined by the conservation laws or, more generally, by thesymmetry group. The group was generated by a set of quantum-mechanicaloperators (the ‘‘generators’’ of the group) Gi which obeyed certain commutationrelations ½Gi;Gj ¼ icijkGk; where cijk are the ‘‘structure constants’’ characterizing thegroup. The commutation relations played the role of interconnected laws ofnature. They could be viewed as generalized conservation laws that dictated whathadrons could be produced in a given interaction. In the group-theoreticalterminology, ‘‘structure constants’’ happens to suit perfectly our conception ofabstract structure for the following two reasons. First, the integrated set of structureconstants entailed a set of interrelated laws. Secondly, it encompassed new kinds ofproperties and entities. For instance, the phenomenological laws that preceded theunitary symmetry attributed various physical properties, such as isospin, andstrangeness, to the particles, whereas the structure (the symmetry) included newkinds of properties, e.g. a particle having the transformation properties of a baryonoctet or decuplet, or a meson octet. Each of these properties determined the hadronspectra and the behavior of each particle in different interactions. The internalsymmetry could be interpreted as more fundamental than the hadrons since thesymmetry was a compact structure which encompassed many interrelated laws andtherefore it was more testable or confirmable than isolated laws, whereas without theunderlying symmetry, the (unindividuated) hadron consisted of an accidentalcollection of laws.


However, I would like to introduce a sharper concept, which will refer to theontological dimension of the symmetry: the concept of ontological priority. Theontological priority of the symmetry over the hadrons can be illustrated by thefollowing ‘‘thought experiment.’’ In a physically possible world with zero baryonnumber, there can be a moment in which the world is free of hadron matter (when allbaryons are annihilated and all mesons decay), and only photons and leptons areleft. In such a hadron-free situation, hadrons could subsequently be produced inhigh-energy processes, such as photon–photon or electron–positron collisions. Butthe symmetry dictates what possible hadrons will be produced. And in such asituation—whether it lasts forever (because of the lack of sufficient energy in theworld) or only for a certain period of time—the internal symmetry exists as an

underlying structure whereas hadrons are uninstantiated. Now, if for any two physicalentities A and B there is a possible world in which there is a physical situation whereA exists but B does not, but there can be no world in which B exists but A does not,we may say that A is ‘‘ontologically prior’’ to B. Thus, when an abstract structure S

exists but the actual particles pa do not, but not vice versa, we may say that S isontologically prior to pa:We therefore arrive at the following conclusion: the internal

symmetry is ontologically prior to hadrons; not to matter in general, which wouldinclude also leptons. When I say that a symmetry exists, I mean the existence of anabstract structure consisting of the algebraic relations expressed by the above-mentioned commutations relations that define the symmetry group. These relationshold between quantum-mechanical operators that measure the different hadroncharges. Thus, the set of commutation relations was a set of interrelated laws thatdetermined the spectrum of hadrons, i.e., the list of hadrons that can exist accordingto the symmetry group. Hence, the laws determined what hadrons could exist,presumably including their masses (which would be functions of the hadroncharges). We may call these laws ‘‘existential’’ laws. Thus, we have here a hadron-free world populated with abstract structures, where the latter determine whathadrons can be created, provided the right amount of energy is supplied. Thesymmetry dictated, therefore, both the spectrum of hadrons and what kinds ofhadrons will be produced in a given physical situation. Of course, we do not havehere an absolute dictate; once we choose a representation for a family of hadronsthat have some common properties, the classification of the rest of the family isdetermined. However, this kind of ‘‘dictate’’ is weaker than the dynamical dictatethat will be discussed when we turn to gauge dynamics.This picture entirely accords with the version of Platonism formulated by Brown

(1994). We have here a physically possible world stripped of hadron matter. In thisworld, conservation laws and symmetries are algebraic relations between quantum-mechanical operators that measure physical quantities. These relations and theunderlying structures ‘‘are not parasitic on existing objects and events. They have lifeof their own’’ and ‘‘exist independently of physical objects, independently of us andoutside space and time,’’ to use Brown’s words. This physically possible world istotally incompatible with the na.ıve empiricist’s intuition (as expressed by Earman)‘‘that laws are parasitic on occurrent facts.’’ Indeed, only empirical or phenomen-ological regularities, which are merely inductive generalizations of occurrent facts of


scientific experience, may be treated as ‘‘parasitic’’ on the facts. Highly theoreticallaws that are underdetermined by the facts cannot be described as parasitic on thefacts. They are formulated in terms of new kinds of concepts which do not appear inthe observational vocabulary and which describe a deeper level of reality, i.e., a deepstructure. Such laws and in particular the underlying structures are not summaries ofthe facts. They have extra conceptual content over the facts and represent a widerphysical story that is unacceptable to anti-realists. In our case, the abstract structurebehind hadron phenomena was the wider story that is hidden from direct scientificexperience. We have an experimental contact only with the world of particles, butthis is only the tip of the iceberg. The supra-empiricist world is the Platonic iceberg.To this point, I have discussed possible metaphysical implications of the first

generation of the internal symmetries in particle physics, symmetries that wereapproximate and global. This might be referred to as the first revolution in theconception of symmetry in particle physics: symmetries that dictated the existence ofparticles and not only their properties or behavior. In the following sections, I willexamine the new conception that arises from the second revolution: the transition toexact local8 symmetries, where the symmetries and the dynamics have becomeinseparable in an overall structure.

5. The second revolution: the gauge argument

In 1964 Wigner refers to the ‘‘dynamical invariance groups’’ corresponding to thefour ‘‘distinct types of interactions’’—gravitational, electromagnetic, weak andstrong—and complains that ‘‘the problem of interactions is still a mysteryy Thegroups seem to be quite disjointed, and there seems to be no connection between thevarious groups which characterize the various interactionsy’’ (Wigner 1967, p. 18).Yet, he mentions ‘‘a fruitful line of thinking about how the interaction itself may beguessed once its group is known.’’ He refers here mainly to Utiyama (1956) and alsoto Yang and Mills (1954). The problem of multiplicity of interactions was partiallyresolved in the second revolution: the electromagnetic and weak forces were unifiedin electroweak unification, and both were unified with the strong force throughgrand unification. Furthermore, Utiyama’s idea that the symmetry gives a clue to thenature of the interaction was fully realized.With the introduction of Yang and Mills (Y–M) gauge symmetries to the

mainstream of particle physics in the 1970s, the Heisenbergian tradition ofapproximate global symmetry of hadrons was replaced by an exact and local gaugesymmetry of all particles and interactions, except gravitation. This can be treated asthe second revolution in the conception of symmetry. The very presence of exactsymmetry means that the transformations of the symmetry group leave theLagrangian invariant. This implies that the particles must belong to representationsof the symmetry group, and that each representation is occupied by particlesidentical in their mass. Furthermore, one collection of particles, the gauge bosons—

ARTICLE IN PRESS

8 In general, an exact symmetry may be global, whereas an approximate symmetry may be local.


the mediators of the interaction—are forced by the symmetry to be massless, just likethe photon which is the single gauge boson in the case of QED. Gauge symmetrybrought field theory back to play a central role in particle physics. Furthermore,according to the so-called ‘‘gauge argument,’’ local gauge symmetries dictate the

nature of the forces.Some philosophers who discuss the gauge argument (for example, Auyang (1995),

Brown (1999), Teller, (2001) and Brading (2002)) refer to Lewis Ryder’s bookQuantum Field Theory, which is widely used in particle physics. Ryder demonstratesthe argument for the simplest case of the electromagnetic field along the followinglines. He starts with the Lagrangian L0 of a free complex scalar matter field f (Ryder1996, pp. 94–95). L0 is invariant under global gauge transformations (gaugetransformations of the first kind):

f-e�iyf; f�-eiyf�; ð1Þ

where y is a constant phase factor. This is an internal transformation, since it doesnot involve space–time. The gauge transformations generate the unitary group Uð1Þ:However, L0 is not invariant under local gauge transformations (gauge transforma-tions of the second kind), where y is replaced by yðxÞ which is an arbitrary function ofspace–time. To restore invariance under localized Uð1Þ; a new variable is introduced:a four-vector Am; which couples directly to the current:

Jm ¼ iðf�@mf� f@mf�Þ: ð2Þ

The Lagrangian is a Lorentz scalar, so the simplest scalar term is added as aninteraction term to the Lagrangian

L1 ¼ �eJmAm; ð3Þ

where e is the coupling constant. We want the Lagrangian to be invariant under thejoint local transformation of the f and Am fields. The latter undergoes the followinglocal gauge transformation:

AmðxÞ-A0mðxÞ ¼ AmðxÞ � 1=e@myðxÞ: ð4Þ

But the Lagrangian is still not invariant. In order to make it invariant, another scalarterm is added,

L2 ¼ e2AmAmf�f; ð5Þ

so that

dL0 þ dL1 þ dL2 ¼ 0: ð6Þ

Consequently, the total Lagrangian

Ltot ¼ L0 þ L1 þ L2 ð7Þ

becomes invariant due to the introduction of the field Am: However, a third term, L3;corresponding to the contribution of Am by itself to the Lagrangian should be added.Since Ltot is gauge invariant, L3 should be invariant too. Thus, a gauge invariant(anti-symmetric) tensor

Fmu ¼ @mAu � @uAm ð8Þ


is constructed. L3 is chosen to be the simplest scalar that can be constructed fromFmu; i.e.,

L3EFmu � Fmu: ð9Þ

At this stage Ryder points out that ‘‘[i]t will be recognized that Fmu defined [above] isthe electromagnetic field tensor, whose six components are the three electric and threemagnetic field components’’ (i.e., the six components of the anti-symmetric 4� 4tensor). From this he concludes: ‘‘What we have done, therefore, is to show how theelectromagnetic field arises naturally by demanding invariance of the action [orLagrangian] under gauge transformations of the second kind’’ (1996, p. 95).Thus, Ryder’s conclusion is that the requirement of invariance under local gauge

transformations leads to the introduction of the electromagnetic field. However, theLagrangian was already invariant before L3 was introduced. The term L3 was added‘‘by hand’’ in order to include a contribution of Am by itself to the Lagrangian, whereAm is dynamically independent of the matter field f; it is not required or entailed bylocal gauge symmetry.9 Yet, the requirement of gauge invariance serves as anecessary precondition in constructing L3: In this sense the electromagnetic fieldarises ‘‘naturally’’ from the requirement of gauge invariance. ‘‘So we arrive at a new

interpretation of the electromagnetic field: it is the gauge field which has to be

introduced to guarantee invariance under local Uð1Þ gauge transformations’’ (Ryder,1996, p. 96).The foregoing description reflects the standard account of the Y–M gauge theories

in particle physics. Some philosophers criticize the logic behind this approach.10

Harvey Brown, who criticizes the gauge argument, talks nevertheless about ‘‘theheuristic importance of treating electromagnetism as a gauge theory, which is ofcourse illustrated in the successful use of the gauge principle in the Standard Modelof particle physics’’ (1999, p. 53). Here we have a pragmatic justification of the gaugeargument. However, in the context of the present paper, which concentrates on somemetaphysical interpretations of the standard account—not on critical analysis of thephysicists’ claims—I assume that the gauge argument is valid, or at least that it has aheuristic or pragmatic value, as Brown admits. The metaphysical conclusions drawnhere are therefore conditional on this assumption. The gauge argument with respectto QED can be generalized to all local gauge theories, as the particle physicistAnthony Zee puts it: ‘‘symmetry dictates design’’ (SDD). This is the central messageof the book Fearful Symmetry (Zee, 1986).11

Local gauge symmetry demands the presence of a certain number of extra fieldsassociated with massless particles. The latter are the gauge bosons—the mediators ofthe forces. Given the symmetry, the principle SDD implies that the Lagrangian

ARTICLE IN PRESS

9 I would like to thank an anonymous referee for drawing my attention to this point.10See, for example, Brown, 1999, pp. 50–53, Earman, 2000, p. 1.11The above description of the gauge argument relies mainly on the line of thought initiated by Young

and Mills and Utiyama and highlighted by Wigner. This approach is reflected by Ryder’s account and

Zee’s book. It would not be a wild exaggeration to say that a significant number of leading particle

physicists adopt the SDD claim. I would like to thank an anonymous referee for drawing my attention to

the alternative approaches to the standard account.


should be constructed out of fields combined in such a way that it will be invariantunder the symmetry. Once physicists decide on a group, the number of gauge bosonsis determined completely. The number and properties of the gauge bosons aredictated by the gauge symmetry, whereas the identity and properties of the matterparticles that interact with the gauge bosons are not. However, all the particlesinteracting with the gauge bosons should belong to definite representations of thesymmetry group or to certain mixtures thereof. This is so since if two particlesinteract with each other, both particles should belong to representations of the samegroup; the couplings between the particles under the transformations of the groupcorresponds to the ‘‘direct product’’ of the corresponding representations.The dynamics of a gauge theory can then be described as follows: when a particle

emits or absorbs a gauge boson, it may change into another particle. By exchangingbosons, particles interact with each other, or exert forces on each other. Thus, thegauge bosons are represented by the operators of the group that transform thedifferent members of a representation (i.e., particles) into each other. In other words,the mathematical operators correspond to physical entities. Thus, SDD requires twomain things: (1) A definite number of massless gauge bosons—the mediators of theforces—are determined by the symmetry. (2) Particles that couple to the gaugebosons should belong to definite representations of the symmetry group; thisrequirement is common to the old and new symmetries. The first requirement ismuch stronger than the second one. The first requirement of SDD determinesunconditionally the existence of the fields that mediate the forces between theparticles, i.e., it dictates the structure of the Lagrangian and therefore the dynamics.This is so since the mediators should belong to a definite representation of thegroup—i.e., to the regular representation—and this dictates the number of mediatorsand their properties. The first requirement is stronger also because it further requiresthat the mediators should be massless.The old SUð3Þ symmetry that changed flavor (e.g. the quark labels: up, down, and

strange), but not color, was an approximate symmetry. The masses of members of arepresentation were not equal. In this case, the symmetry is said to be brokenexplicitly. However, if the symmetry that leaves the Lagrangian invariant is exact butit is not manifest in the physical system described by the theory, the symmetry is saidto be spontaneously broken by the physical states of the theory. We might say in thiscase that the symmetry is hidden in the physical system. The Higgs field (i.e., a fieldwhich is not zero at the ground state) causes a spontaneous breakdown of thesymmetry of the Lagrangian.Only at the super-high energy scale (about 1015 times the proton mass) is there a

perfect symmetry, and all gauge bosons are massless, like the photon. According topresent cosmological theory, this happened at the first nanoseconds of the Big Bang.The symmetry broke down as the world cooled, and the energies decreased. When agauge symmetry is spontaneously broken, the corresponding gauge bosons, exceptthe photon, acquire mass. Starting with a symmetrical Lagrangian, afterspontaneously symmetry breaking, the actual physical laws derived from theLagrangian are not symmetrical. The lower the energy, the farther we depart fromthe ideal world with perfect symmetry.


6. Unification induced by local gauge symmetries

The theory of electroweak unification was partially constructed along guiding-lines of simplicity. Physicists tried to find the simplest group that wouldaccommodate the known gauge bosons. The Y–M gauge bosons W ’s (intermediatevector bosons) are the mediators of weak interactions. A typical weak interactionprocess mediated by a W boson takes place when a down quark emits a W andchanges into an up quark thus effecting the ‘‘charged-current’’ process: nn-pe�: Aswe have seen, the simplest gauge theory is QED, with a single gauge boson g and thesymmetry group is Uð1Þ: After it was found that SUð2Þ does not fit the experimentaldata of weak interactions, Sheldon Glashow suggested the symmetry Uð1Þ � SUð2Þas the next simple candidate. Between the two ‘‘factors,’’ Uð1Þ is a natural candidatefor accommodating the photon. Now, according to Y–M theory, the regularrepresentation of the gauge group should accommodate the gauge bosons. Theregular representation of SUðNÞ is ðN221Þ dimensional. Thus, the regularrepresentation of SUð2Þ is the triplet 3. The W7 were assigned to this representationand the extra gauge boson Z0 was assigned to a certain mixture of the Uð1Þ singletand the neutral member of an SUð2Þ triplet, whereas the photon was assigned to thecomplementary combination of the two. An exchange of Z0 effects the ‘‘neutralcurrent’’ process: nn-nn: As gauge bosons, the W ’s and Z have to be massless in asituation of full symmetry. Thus, the photon, the W ’s, and the Z are related, as thegauge bosons of a Y–M symmetry Uð1Þ � SUð2Þ; transforming into each otherunder the symmetry group. When the gauge symmetry is spontaneously broken, theW ’s become massive, as the mediators of the short-range weak interactions shouldbe, whereas the photon remains massless.However, we can imagine very high-energy processes where the energies are

much larger than the masses of the W ’s and the Z; and the gauge bosons maybe treated effectively as massless. In these processes, the W ’s, Z, and the photonbecome similar again; they couple to particles at the same strength. Thus,the electromagnetic and weak interactions no longer exist as separate entities:they are unified into a single electroweak interaction. At low energies the W ’s and Z

are dragged down by their masses, so they behave as mediators of the short-rangeweak forces.The gauge bosons W7 transform particles into each other. For instance, in the

charged-current process nn-pe�; the neutrino emits a Wþ that is absorbed by adown quark ðdÞ within the neutron ðuddÞ; transforming it into an up quark ðuÞ; andconsequently the neutron is transformed into a proton ðuudÞ: The gauge bosons Z0

and g transform particles into themselves, e.g. in the neutral-current process nn-nn;the neutrino emits a Z0 which is absorbed by one of the quarks within the neutron,transforming it into itself. In general, particles are transformed into each other andinto themselves by the gauge bosons. The latter are therefore physical realizations ofmathematical operators. They play the role of the generators of the group thattransform the basis vectors of each representation into each other. The number ofgenerators of SUðNÞ is equal to the number of basis vectors of the regularrepresentation that is occupied by the gauge bosons. Thus, each generator (a


mathematical object) corresponds to a gauge boson (a physical object); there is adirect correspondence between the abstract structure—i.e., the symmetry—and thelaws operating in the observational level. After all, the distance between the abstractstructure and concrete physical objects is very short here.In QCD, the problem of renormalization and incomputability of strong

interactions was solved by asymptotic freedom: It was shown that at super-highenergies (asymptotic range of energies) the coupling strengths approach zero so thatthe particles move independent of each other. Furthermore, it turned out that Y–Mtheory is the only asymptotically free theory. This means that the strong forcebecomes weak when two quarks interact at very high energies. Thus, one can useperturbation methods to calculate those strong interaction processes in which quarksget close to each other at very high energies. As the quarks move away from eachother, the coupling strengths increase—the result being quark confinement or‘‘infrared slavery.’’ Since the results of these calculations agree with observations,QCD now is almost universally accepted.QCD is based on Y–M theory of color symmetry SUð3Þc: The group SUð3Þc

transforms quarks with the same flavor and mass and with different colors (yellow,red, and blue) into each other. The gauge bosons in the theory are the eight gluons

that mediate the inter-quark forces. And according to QCD the gluons are enslavedtoo. Here too, the symmetry dictates the identity of the mediators and requires thatthe gluons should belong to the regular representation of SUð3Þ; i.e., to the octet.Again, gluons play the role of the generators of SUð3Þc; transforming quarks withthe same flavor and different colors into each other.Thus, in addition to the W7; Z0, and g; there are eight gluons; all twelve are Y–M

gauge bosons. Asymptotic freedom has the effect that as the energy increases, thegluons become increasingly weaker. However, electromagnetism is not asymptoti-cally free, so at still higher energies, the electromagnetic force becomes stronger,while the strong force becomes weaker. At some energy, the strength of theelectromagnetic force will be equal to that of the strong force. Hence, furtherunification is possible; i.e., grand unification. The masses of the W ’s and Z resultfrom spontaneous symmetry breaking. At high enough energies, they are effectivelymassless, like the photon. At still higher energies, the W ’s and Z become stronger,while the gluons become weaker.Thus, in grand unification, the photon, W ’s, and Z and the eight gluons can be

treated as gauge bosons of a single Y–M theory. The photon, W ’s, and Z are thegauge bosons of a theory with the group Uð1Þ � SUð2Þ; while the gluons are thegauge bosons of a theory with the group SUð3Þc: So the minimal symmetry group forthe SM would be Uð1Þ � SUð2Þ � SUð3Þc: The simplest group in the SUðNÞ seriesthat contains this triple product is SUð5Þ; which is one of the major candidates forthe symmetry of grand unification. Once the group is specified, the number of gaugebosons is completely fixed by group theory. The number of generators of SUð5Þ is(52–1=) 24, and this is also the dimensionality of the regular representation thataccommodates the gauge bosons. This symmetry dictates, therefore, that in additionto the 12 electroweak and strong gauge bosons, there should be another set of 12gauge bosons: the X bosons.


If the symmetry at the grand unification scale is indeed SUð5Þ; then at lowerenergies it is spontaneously broken into the electroweak symmetry Uð1Þ � SUð2Þ andthe QCD symmetry SUð3Þc: At this stage, the X bosons acquire extremely high mass,while the gluons, photon, W ’s, and Z remain massless. As the energy furtherdecreases and reaches the electroweak energy scale, the Y–M theory based on Uð1Þ �SUð2Þ; in its turn, suffers spontaneous breaking, and as a result the W ’s and Z

become massive while the photon remains massless. Of all the gauge bosons inSUð5Þ; only the photon and the eight gluons appear as massless excitations at lowenergies. The gluons are enslaved in quark confinement, leaving only the photonsfree.Under the SUð5Þ transformations, the quarks and leptons are supposed to

transform into each other, i.e., to occupy representations of SUð5Þ: Before grandunification, quarks and leptons belong to different representations, so they did notmix. But in grand unification, the electron family consists of the following 15members or states: the three lepton states (consisting of the two spin components ofthe electron e; the single spin component of the electron-neutrino ne) and the 12quark fields u and d with three colors and two spin directions. All these are assignedto the five- and ten-dimensional representation of SUð5Þ: This arrangementtriplicates itself: The other two generations are—the muon family (including m; nm;the strange quark s; and the charmed quark c) and the tau family (t; nt; the top quarkt; and the bottom quark b). Thus, under the grand unification symmetry, quarks canbe transformed into leptons, and vice versa. The gauge bosons responsible for thesetransmutations are precisely the X bosons. Quarks can transmute into leptons byinteracting with X bosons. So disintegration of protons is possible: p-p0eþ (thelifetime of the proton is about 1030 yr) and p0 decays into an electron, positron, and aneutrino. The reverse process is also possible: protons may be created. Therefore,baryon number is not conserved. As we will see, in the next section, this conclusion hasmost significant metaphysical implications.

SUð5Þ faces some serious problems and there are other alternatives for thesymmetry of grand unification. However, the physicists who tended to believe in thisgroup did it because of the exact fit between the group and the known particles;according to SUð5Þ; the quarks and leptons of each family fit exactly into therepresentations 5+10 of SUð5Þ: So it is highly probable that SUð5Þ is at least part ofthe real picture.

7. Conclusion: the internal symmetry is ontologically prior to baryon matter

Symmetries induce a giant leap in the epistemic process of dematerialization ofmatter. According to SUð5Þ-based GUTs, all matter can disintegrate into electrons,positrons and neutrinos, but we cannot say that matter is made up of these leptons.In addition, particles can undergo all kinds of transmutations. Baryon number, themost ‘‘solid’’ quantity, is not conserved anymore. However, the symmetry, which isthe immutable structure behind the scenes, governs the patterns of transmutations inthis flux. Democritus’ dictum states that matter is made up of atoms and void. In


view of contemporary physical picture, this slogan may be updated: matter can beproduced out of the vacuum according to the dictate of symmetry. Contrary to theatomistic picture, this is a non-reductionist conception. Of course, the field-theoretical vacuum is far from being void or nothing; it is the state of the world withall the excitations removed. One commentator interpreted a newly discoveredprocess of producing eþe� pairs in the lab out of two colliding beams of photons as‘‘creating something out of nothing’’ (Browne, 1997). But in a field-theoreticalcontext, this kind of ‘‘nothing’’ includes everything in a virtual state.At the grand unification scale, before symmetry breakdown took place and the

world was governed by the maximal symmetry, baryons could vanish, and we getmaximal dematerialization of matter. Thus, the higher the symmetry, the fewer thekinds of immutable material particles in the world. The world starts at the Big Bangwith minimal matter and maximal symmetry. The process of materialization isaccompanied with spontaneous breaking of the symmetry. In general, the symmetryis the structure which dictates (1) the nature of the forces, (2) what possible particlesmay be produced in a given process, (3) the relative frequencies of the variousallowed processes, and (4) the classification of the particle (the ‘‘existential’’ laws).Thus, contemporary particle physics operates in a dual reality consisting of two

worlds: (1) The physical world which is filled with two kinds of entities: (a)Fundamental particles—quarks, leptons and gauge bosons; this is how theexperimental particle physicist sees the world. (b) Fields—matter fields (quark andlepton fields) and interaction fields (boson fields); the world as seen by thetheoretician. (2) The Platonic world that is occupied by an abstract structure thatgoverns the behavior and content of the physical world. At super-high energies allgauge bosons are massless, and all forces are unified (since all the carriers of forcelook alike). At this energy scale, the symmetry is the symmetry of grand unification.As the physical universe cooled down after the Big Bang, the perfect symmetry wasspontaneously broken, and we gradually reached our natural habitat, where allinteractions are non-symmetrical remnants of the perfectly symmetric world thatexisted in the past. In fact, we are one of the products of the broken symmetry. Thus,if the symmetry had not been broken, there would be no one around capable ofknowing the symmetry (or anything). So we have here an anthropic explanation forbreaking the symmetry: the breakdown of the symmetry is a necessary condition forthe emergence of an intelligent being capable of grasping the symmetry!Does the concept of ontological priority, as applied in Section 4 to the old internal

symmetries, apply also to the second revolution in the conception of symmetry? Forsymmetry considerations, cosmologists tend to adopt the thesis that the earlyuniverse started at the Big Bang with baryon number zero or, in fact, without anybaryons or anti-baryons at all, and then evolved into the present universe withbaryon excess. According to this picture, the birth of baryons, such as in the processp0eþ-p; came as a result of the mediation of X bosons. For instance, an anti-quarkfrom the incoming pion and the positron exchanged an X ; resulting in two outgoingquarks which joined the remaining quark from the pion to form a proton. Theviolation of the particles–anti-particles symmetry, or the CP violation, wasresponsible for the different decay rates of X ’s and anti-X ’s. So, as the initially


symmetric universe cooled down, it evolved into an asymmetric universe withdifferent numbers of X ’s and anti-X ’s. As a result, the quark population or thebaryon population evolved into a particle–anti-particle asymmetry.It is estimated that about 10�35 s after the Big Bang—let us denote this moment by

tG—the energy of every particle was of the order of the grand unification energy, andthe X bosons were generated and baryons were born. Shortly after this, theexpanding universe cooled down, and the energies of the X ’s decreased much belowtheir mass (about 1015 times the protons). At these energies, the X ’s becameextremely weak, and the protons could not decay anymore and ‘‘can live with quasi-immortality for the next 1030 yr’’ (Zee, p. 249). This is, of course, only an estimationfor the lower limit of the proton’s lifetime. Thus, the symmetry of grand unificationdictated the existence of the gauge bosons, including the X ’s, and consequently thenature of the forces, or the dynamics. In addition, it dictated through itsrepresentations what possible baryons will be generated, or instantiated, togetherwith other particles—mesons and leptons—and at what rate. Thus, according to thisview, before tG the world was baryon-free, whereas the symmetry of grandunification existed as an abstract structure. The world was populated by a ‘‘sea’’ offundamental fermions and gauge bosons in a quasi-virtual state. In other words, thissea played the role of the field-theoretical vacuum. The baryons were generatedsubsequently from the collisions of particles that were present, according to thesymmetry’s dictate. Quark–anti-quark pairs could be produced from the vacuum bycolliding gg pairs and then p0s could be produced by the former pairs. Protons couldbe created then through the reaction p0eþ-p; with the mediation of X ’s. Theprobability of the latter process is very low, but it is not zero. In this kind of process,the group dictates the spectrum of possible particles that would be created: theparticles should occupy definite representations of the group.Thus, we have a possible world with a physical situation (until tG) in which the

symmetry exists but the actual baryons do not. This possible world happens to beour actual world. On the other hand, there is no physically possible world wherebaryons exist and the symmetry does not. Thus, according to our definition, we canreach the following conclusion: the symmetry of grand unification is ontologically

prior to baryon matter—this is almost the same conclusion that was reached for theold symmetries: the flavor SUð3Þ symmetry was ontologically prior to hadrons, i.e.,to baryons and mesons, whereas SUð5Þ is ontologically prior only to baryons.However, as we have seen, SUð5Þ is not the only candidate for the symmetry ofgrand unification. It just demonstrates the ontological priority of the symmetry overa wide range of matter fields.According to the present conception, the internal symmetry is the deepest layer in

the ontological hierarchy. But where does the symmetry comes from? There havebeen attempts to explain the symmetries by super-string theory. This brings us backto a more ‘‘concrete’’ approach. Although ten-dimensional strings are far from beingsolid tangible objects, they are still graspable by our common spatial intuition.However, it is perfectly suitable to treat a symmetry as an abstract structure thatneeds no concrete structure as a deeper explanation. The question is why particlephysicists should choose a particular symmetry, such as SUð5Þ or one of its


competitors, such as SOð10Þ? It seems that we turn here back to phenomenology.Yet, as Falkenburg notes, there is no clear-cut distinction between phenomenologicaland deep-structural theories: the old hadron SUð3Þ was not necessarily aphenomenological theory. If we could derive it from a composite quark model, forinstance, ‘‘we have gone beyond pure phenomenology and laid the foundation fortheoretical explanation.’’ On the other hand, the new gauge symmetries ‘‘[have]phenomenological features, as long as the selection of the symmetry group itself andthe number of particle families comprised in it remains unexplained.’’ Thus,‘‘internal symmetries occupy an ambiguous position between phenomenologicaldescription and theoretical explanation’’ (Falkenburg, 1988, pp. 113, 127). However,this methodological situation is not exclusive to internal symmetries; many othercases in the history of physics ‘‘suffer’’ from this kind of ambiguity, where physicistsstart with a phenomenological approach and strive for what they think to be atheoretical explanation. Yet, the explanation need not be in terms of concretestructures. Since symmetry groups are abstract structures, physicists might choosebetween them by employing, for example, a standard of simplicity or compactness,in addition to strength and balance, as van Fraassen suggests; the group whichcomplies best with these standards might be treated as a theoretical explanation,rather than as a phenomenological model.

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