Philosophy of Science, 69 (September 2002) pp. S221-S234. 0031-8248/2002/69supp-0020 Copyright 2002 by The Philosophy of Science Association. All rights reserved. Gauge Principles, Gauge Arguments and the Logic of Nature Christopher A. Martin Indiana University http://www.journals.uchicago.edu/PHILSCI/journal/issues/v69nS3/693020/693020. html I consider the question of how literally one can construe the "gauge argument," which is the canonical means of understanding the putatively central import of local gauge symmetry principles for fundamental physics. As I argue, the gauge argument must be afforded (at best) a heuristic reading. Claims to the effect that the argument reflects a deep "logic of nature" must, for numerous reasons I discuss, be taken with a grain of salt. Symmetry principles have moved to a new level of importance in this century and especially in the last few decades: there are symmetry principles that dictate the very existence of all the known forces of nature. S. Weinberg Send requests for reprints to the author, 130 Goodbody Hall, Bloomington IN 47405; [email protected] I thank Jon Bain, Tony Duncan, John Earman and Chris Smeenk for their comments on an earlier draft. 1. Introduction. This sentiment expressed in the above quotation is characteristic of the canonical understanding of the structure of fundamental physical theory today. The symmetry principles that have come to prominence in recent decades are so-called local symmetry principles. Perhaps the most common way of presenting and understanding the workings or content of these symmetry principles is in the context of "the gauge argument," a fixture in the physics literature. Briefly, this argument starts from some known non-interacting charged matter field with a global symmetry and, through demanding that symmetry be made local, dictates the introduction of a new gauge field interacting in a prescribed way with the original matter field. The argument is taken to capture the operation of a fundamental physical principle, "the gauge principle": the principle holding that the form of the fundamental interactions is to be determined, or dictated, via requirements of local (gauge) symmetry. For the class of interacting field theories resulting in this way from the demand of local invariance, gauge field theories, contains those theories accurately describing what we find in nature! Given this, it might be hoped that the operations of the gauge argument and/or its purported universal applicability to all fundamental interactions, would provide the key to a proper understanding of the nature of gauge invariance and of the gauge structure of fundamental physical theory. Read literally, the gauge argument might appear to have an air of magic about it, to effect a sort of "epistemic miracle." For one seemingly gets a fair amount of "physical output" from but a modicum of "physical input." Faced with the argument presented in this schematic way, it is difficult not to be awed by what is apparently its sheer physical fecundity. However, the argument presented in this way seems to me just too good to be true, and indeed I claim it is. As the saying goes, "you don't get something for nothing," and not surprisingly the argument when looked at more critically is not quite so magical. In this paper, I warn against reading the gauge argument too literally in this way, as a sort of laying bare of the "logic of nature." There are three points at which a literal reading of the gauge argument falters. (1) The received wisdom concerning motivations for key parts of the argument is suspect, and (2) the argument does not proceed unambiguously. As I discuss in section 3, even if the initial requirement of local (as opposed to global) gauge invariance can be justified in some natural, nonquestion-begging way it is not clear that it can be this requirement does not alone uniquely dictate the form of the full theory. Moreover, (3) there are wholely other approaches to the logic of nature, ones that in effect assign a different origin to the gauge structure of fundamental theory. As I discuss in section 4, considerations surrounding renormalizability and effective field theories, for example, might be taken to supplant, indeed obviate, any principle of local gauge invariance as responsible for the "shape," including the gauge structure, of physical theory. Such considerations raise the interesting possibility that there is less to any fundamental "gauge principle" than the canonical understanding takes there to be. 2. The Gauge Argument. The canonical way of understanding the workings or content of local gauge symmetry principles is by pointing to the algorithm for producing interacting field theories from the demand of local gauge invariance the "gauge argument."1 Given the success of the gauge paradigm, it playing a key role in the developments culminating in the most far-reaching and successful physical theories we have ever known, some take the argument to represent the very logic of nature. Symmetry principles as embodied in the argument are then taken to express/reflect deep features of the physical world.2 In order to make some remarks concerning how literally we can construe the operations of the argument, we first look at a concrete example. Consider a field representing electrically charged matter. The free field obeys the Dirac equation which is just the Euler-Lagrange equation(s) for the Lagrangian (density) Dirac = (i - m) .3 The corresponding action is clearly invariant under so-called "global" U(1) phase transformations: eiq ; e-iq with a constant.4 It follows from Noether's first theorem that when the equations of motion are satisfied there will be a corresponding conserved current. Consider now "localizing" these phase transformations, i.e., letting become a function (x): eiq (x) ; e-iq (x) . The free field Lagrangian is clearly not invariant under such transformations since the in general non-vanishing derivatives of the arbitrary functions, i.e., (x), will now appear in the transformed Lagrangian. In order that the field admit the local transformations as variational symmetries, the Lagrangian must then be modified. In particular, we replace the free field Lagrangian with =q interacting = (i - m) - qA Dirac - J A , with J . The current J is in fact just the conserved Noether current associated with the global U(1) invariance. Towards securing local invariance we have introduced the field A , the gauge potential. The particular form of coupling of the matter field to this gauge potential in interacting is termed minimal coupling. This modified Lagrangian is now invariant under the local phase transformations provided that the vector field A is simultaneously transformed according to A (x) A (x) - (x). This transformation behavior is, of course, familiar as the covariant analog of the well-known electromagnetic gauge transformations. This suggests the possibility of viewing the new field A as representing the electromagnetic potential. Pursuing the idea of viewing the field A as representing the electromagnetic potential, we note that the Lagrangian interacting does not yield a fully interacting theory. Varying the Lagrangian with respect to the matter fields yields the latter's equations of motion the field now being coupled to the A field. But, it remains to add a "kinetic term" for the A field itself. Such a kinetic term, in effect, imbues the vector field with its own existence, accounting for the presence of non-zero electromagnetic fields, for the propagation of free photons. The Lagrangian kinetic = Maxwell =- F F , with the gauge field F defined as F := A - A gives (source-free) Maxwell's equations. Putting this all together yields the Lagrangian for the fully interacting theory: Maxwell-Dirac = interacting + kinetic = Dirac -J A - Maxwell. The inhomogeneous coupled equations of motion for the gauge field (the electromagnetic field) now follow from varying the full action with respect to A .5 Finally, consider that a mass term for the vector field A of standard form mass = photon- m2A A . is not gauge invariant. In keeping with the demand of local gauge invariance, the vector field A (i.e., the photon) must then be massless. Local gauge invariance thus necessitates a massless photon.6,7 3. Does Local Gauge Symmetry Dictate Interaction? Demystifying the Gauge Argument. Having provided a sketch of how the gauge argument works in a concrete example, let us proceed to consider how literally the argument can be construed. Does the argument truly dictate the introduction of new physics in a simple and direct manner? Not surprisingly, the gauge argument cannot proceed as simply as this schematic presentation might make it seem. 3.1. From Whence the Local Gauge Invariance? Most immediately what are we to make of the initial, central demand of local gauge invariance? The demand is anything but self-evident and presumably, in the context of the gauge argument, must be argued for on some basis. Unlike the global invariance, the demand for the corresponding local invariance does not have an immediate physical counterpart.8 Is it to be taken as a direct implementation of some sort of unassailable first principle? If so, is the demand (or principle) something with which we are already familiar only in a different form? A common justification for the demand of local gauge invariance in presenting the gauge argument is to present it as some sort of "locality" requirement. In outline, the "gauge locality argument" is that global gauge invariance is somehow at odds with the idea of a local field theory, and that to remedy this we must instead require local gauge invariance. This rather brief argument is just how Yang and Mills motivated the demand in their seminal 1954 paper, very much setting the tone for subsequent treatments.9 Just what to make of this argument is not clear, however, there are many interrelated senses of locality that might be at issue. At a general level, making precise the connection between the global/local distinction as it figures in classifying gauge transformations and in the spacetime sense is non-trivial. The fields and transformations ostensibly "live" in configuration space and need to be "brought down" to spacetime.10 Also, the local versus global distinction for gauge transformations as commonly employed is itself in general dependent on a choice of gauge this stems from the fact that in order to construe the fields (and transformations) as living in spacetime one must choose a gauge in the first place. This is made clear in treatments of gauge field theories in the principle fiber bundle framework, where gauge transformations are bundle automorphisms and the typical global/local distinction for gauge transformations, being gauge dependent, is seen to be of little use (Bleecker 1981). Generally, a "local" field theory is, according to the canonical understanding, just one specified by a Lagrangian depending on the values of the field and a finite number of its derivatives at a single point.11 This is understood to capture the idea that the equations of motion partial differential equations determined from this Lagrangian through a variational procedure should be such as to determine the behavior of the field at some point only in terms of what is happening at and around that point. The "locality" at issue here cannot be used to motivate the requirement of local gauge invariance over and above global invariance since it clearly does not exclude global symmetries. We have already seen that a field theory can be local in this sense and yet have a global gauge invariance. The free Dirac field described by the Lagrangian Dirac depends on the field and a finite number of its derivatives at a single point and thus is a "local" field theory in the above sense. Yet this Lagrangian is invariant under the group of global U(1) gauge transformations. Perhaps we could take the sense of "locality" at issue in the gauge argument to be tied more specifically to the dictates of special relativity (STR). Along these lines, the demand of local gauge invariance might be presented as providing a sort of "fix" for a global gauge invariance taken to be somehow at odds with the heart and spirit of STR.12 Let us call this particular form of the locality argument the "STR gauge locality argument." The claim/argument is presumably that the group of local gauge transformations loc and not the group of global gauge transformation glob are compatible with the dictates of STR. If this position could be maintained, then we would apparently have a clear physical basis for the demand of local gauge invariance and, with that, an underwriting for the gauge argument. One way of understanding the demands of STR in this context might be to take it as a prohibition of superluminal communication.13 Thinking of the demands of STR along these lines, one might argue that to perform a global gauge transformation would in effect be to perform "the same" transformation at each and every point, and that this seems to require/enable superluminal communication. But, I take it that this argument rests on blurring different senses of locality. It blurs matters of the (light-cone) structure of Minkowski spacetime in STR with that distinguishing from glob. loc And, it is clear that the former cannot be of direct significance to motivating the latter. For suppose that in the spirit of the "STR gauge locality argument," we eschew symmetry transformations wherein the same transformation is performed at points not in causal contact. Whatever its details, it is clear that the set of transformations compatible with STR along such lines, STR, is not the class of transformations which figures in and is crucial to the formulation of gauge theories, loc. In fact, the latter class of transformations contains as a subset (more precisely, as a rigid sub group) just those transformations gauge locality argument aims to exclude. glob which the STR In any event, it would seem that such a local-gauge-as-locality argument (whether of the specific STR variety or not), if it can be mounted, must rely on an active reading of the local gauge transformations. The force of any such argument, presumably, derives from the untenability of performing the "same" transformation everywhere at the same time (i.e., a global transformation), this violating some notion of locality. Yet if, as the received view holds, the transformations are viewed as merely passive coordinate relabelings even " ultra-passive" in the sense that there is no actual physical change of a physical reference frame but only a change in purely mathematical formal apparatus then it seems that global transformations cannot possibly pose any threat to locality. This because, by stipulation, there is nothing physical that gets changed under the transformations. Thus, I do not see how one can mount any argument for local gauge symmetry in the name of locality if one ascribes to the received view of gauge symmetry/invariance.14 I have considered only a (related) few of the most common ways of motivating the initial demand for local gauge invariance in running the gauge argument. For the remainder of the paper, I take it on faith that the demand can be justified in some non-question-begging way. This faith seems not too unreasonable given that the ascription of deep physical significance to local symmetry principles has come to such prominence in recent decades. The appeal to other elements figuring so deeply in the construction and/or structure of physical theories Lorentz invariance for example, has been well rationalized. Surely this one has too! On this faith (shaky as it may in fact be), let us continue to consider the prospects for a literal construal of the gauge argument. 3.2. "Fixing" the Lagrangian Guidance from Gauge Invariance. As we saw in section 2 the demand for local invariance of the action for the free field meant that one must necessarily modify the Lagrangian. This stems from the fact that sensible Lagrangians involve derivatives of the physical fields. These derivatives, though, do not transform covariantly (i.e., like the fields themselves) under local gauge transformations since the derivatives of the (arbitrary) functions specifying the local transformations enter. Thus one must modify the Lagrangian in order to cancel out these extra terms, and this modification includes the introduction of a new field, the gauge potential, coupled to the now interacting matter field. One important issue that I (purposely) glossed over earlier was the matter of the uniqueness of the modified Lagrangian under the imposition of local invariance. If we hope to make any strong sense of the gauge argument as truly "dictating" the form of the theory the correct theory we expect that the locally gauge invariant Lagrangian be unique. Not surprisingly, it is more complicated than this. First, if producing a gauge invariant action or Lagrangian were our only guide then there are clearly many other gauge invariant terms that could be added to the Lagrangian. For example, in addition to the "minimal term" qA which was added above, we could have added, say, higher order covariant derivatives, e.g., or F , etc. Or, one might include a dependence of interacting on F . More interestingly, one might consider the addition of a Pauli term, one proportional to [ , ] F , which is both gauge invariant and Lorentz invariant. Such a term would in fact have direct physical consequences making the magnetic moment of the electron an adjustable parameter (Weinberg 1995, 517). Such terms are thus apparently not included in the gauge invariant Lagrangian correctly describing nature.15 In order to pick out the minimal modification uniquely, we must bring in besides our general knowledge of field theories e.g., that only a certain number of derivatives should appear in any term of the Lagrangian if the equations of motion are to be of, say, second order in the derivatives of the fields the requirements of Lorentz invariance, gauge invariance, simplicity and, importantly, renormalizability. The minimal modification is then the simplest, renormalizable, Lorentz and gauge invariant Lagrangian yielding second order equations of motion for the coupled system (O'Raifeartaigh 1979). The key point is that, in the context of the gauge argument, the requirement of local gauge invariance gets a lot of its bite in combination with other formal and physical requirements. One might argue that, at the least, other things held fixed some requirement of formal simplicity selects the minimal modification as the unique gauge invariant modification. In this way, the demand of local gauge invariance might be equated with a principle of simplicity. While assumptions of simplicity have certainly proven valuable guides in past theorizing, there is no reason, though, to think that they provide unambiguous, let alone infallible guides in constructing theories and/or in construing any logic of nature.16 I suspect that these remarks are likely to little phase a good physicist, who would claim that the argument requires completion rather than critique. My point has been only that the demand of local gauge invariance is not the sole input to the gauge argument nor is it necessarily the most important. Setting aside this matter of the uniqueness of the gauge invariant minimal coupling, another important point is that, in contrast with how it is often portrayed, one does not strictly speaking "generate" a new interaction field in running the gauge argument. This gauge field, insofar as it is a physical field, is ultimately put in by hand. The gauge potentials (in our example, A ) generated in the gauge argument form a restricted class of all such A fields: since we start with a free matter field they are of course all gauge transformable to the zero field. Such potentials, though, correspond to zero F fields.17 Thus one generates only physically trivial gauge fields in running the gauge argument. It is thus not clear why in the context of the gauge argument one would necessarily add the kinetic term Maxwell for this gauge field to the Lagrangian. For it is this addition and the subsequent varying of the action with respect to the gauge potentials that "gives physical life" to the field. In the end, an important physical generalization is made in adding the kinetic term (by hand) to the Lagrangian. The generalization is from a non-physical, formal coupling of the matter field to trivial gauge fields (since F 0) to the physical coupling of the matter field to non-trivial gauge fields (F 0).18 In making this generalization, one puts in by hand much of the important physics of the new interacting theory.19 This point goes a long way toward explaining the easily acquired illusion of getting more physics out of the gauge argument than one puts in. The most I think we can safely say is that the form of the dynamics characteristic of successful physical (gauge) theories is suggested through running the gauge argument. This is not to say that the requirement of local gauge invariance cannot serve as a useful selection criterion for generating possible modifications of a free Lagrangian including the addition of (what we interpret as) a kinetic term associated with any newly introduced interaction field(s). It can and does serve such a purpose. In fact, historically, there was just such a pragmatic appeal to local symmetry principles.20 In any event, it is not how a straight-forward, literal reading of the gauge argument portrays it: i.e., it is not the case that by itself the demand of local gauge invariance (1) dictates uniquely the form of the interacting theory or (2) strictly speaking dictates the existence of, or accounts for, the origin of a new physical gauge field. In order to pick out the correct form of the theory, other considerations must ultimately enter. And, it is not at all clear that these other considerations or requirements are in any sense inferior conceptually or physically to that of local gauge invariance. In fact, quite the opposite. In lieu of any strong arguments for a physically based and uniquely implemented principle of local gauge invariance, perhaps the best chance we have of arguing for the uniqueness of the interacting Lagrangian Dirac Maxwell- is from a direction stressing certain of these other considerations. As we discussed, the demand of local invariance must be supplemented at various points of the gauge argument toward arriving at the fully interacting gauge theory correctly describing nature. Besides the prominent figuring of our prior knowledge of physical field theories, there are other fundamental requirements or constraints on theory such as Lorentz invariance and renormalizability. This last requirement in particular is a powerful one. For instance, a Pauli term, which is not excluded by the requirements of Lorentz invariance and local gauge invariance, and whose addition to the Lagrangian would drastically change the theory, is non-renormalizable and can be excluded on that basis. The next section pursues the idea that renormalization and/or other, even more fundamental, considerations, might bear a larger burden for the structure in particular, the gauge structure of our successful theories than does the operation of any fundamental gauge symmetry principle. 4. Turning it Around: Other Approaches to Gauge Invariance. Up to this point, I have followed the well-traveled path of treating the demand of local gauge invariance of the theory as part of the "physical input," the purportedly principled physical starting point, and the associated full-blown (gauge field) theory describing interactions as the "physical output." This way of viewing the gauge structure of fundamental theory is of course precisely that suggested by the canonical view of the gauge argument as the embodiment of a deep physical gauge principle. However, there are other ways of thinking about why our theories are the way they are, so to speak, and some of these have gauge invariance as more of an "output" than an "input." For example, there are arguments to the effect that various consistency requirements, mathematical and/or physical, require theories of, for example, self-interacting spin one particles to be of Yang-Mills form with its characteristic group properties.21 Such arguments clearly paint the gauge invariance of physical theory in a different light than does the canonical view. A more prominent approach which "in effect" turns the received view on its head is that of placing renormalizability (or, alternatively, perturbative unitarity) at the base of fundamental theory. It can be shown that the requirement of renormalizability (resp., perturbative unitarity) requires that theories have the characteristic form of (spontaneously broken) Yang Mills gauge theories.22 Renormalizability can be tied directly to a theory's being well behaved in the sense that it make sensible predictions for quantities of direct physical interest. Thus, one could reasonably argue that gauge invariance is but a feature of the class of wellbehaved (renormalizable) theories that happen to correctly describe the physics at hand. Interestingly, renormalizability has itself arguably been superseded in a certain sense. According to the currently prominent effective field theory program, the familiar renormalizable (quantum) field theories are actually low energy approximations to some more fundamental underlying theory. Besides the finite number of familiar renormalizable terms, such effective theories (rather, the actions) necessarily contain an infinite number of non-renormalizable terms. But as long as at high energies there is some underlying well defined theory (e.g., strings, loops, etc.), then at much lower energies these non-renormalizable interactions will be highly suppressed and thus will be calculationally insignificant, though not physically absent.23 That is, the low-energy "residue" will in fact "look like" a renormalizable theory. This, so the thinking goes, explains why we have gotten by so well focusing on the renormalizable terms. And, as I mentioned above, the renormalizable theories will necessarily be (spontaneously broken) Yang-Mills gauge theories. Such a view changes once again the whole picture with regard to what we take to be "fundamental." Given consideration of effective field theories, the appeal of ascribing any deeply fundamental significance to the gauge structure of our theories, especially as resulting from the operations of some deep physical gauge principle, is further diminished.24 This gauge structure, it could reasonably be argued, is but a direct consequence of (1) the empirical fact that there exist interacting spin one particles in nature and (2) the assumption that our theory of such particles is the residue of some more fundamental underlying theory (e.g., string theory). Under these assumptions, we arrive at the familiar Yang-Mills gauge theories describing such particles. So, we might avoid altogether any appeal to gauge symmetry principles in describing the shape or content of current theory. Gauge invariance arguably becomes an incidental, albeit interesting feature of successful renormalizable quantum field theories. The alternative approaches to the logic of nature discussed in this section call into question the very idea of a true physical "gauge principle." How, given this, might we account for the appeal of the more prevalent view of gauge invariance as just such a fundamental physical principle? There is of course the obvious answer that considerations of renormalizability and effective field theories is a relatively new business, and that historically there was in fact a firm heuristic basis for appeal to local symmetry principles.25 This is not to slight the significance of local gauge symmetry principles but rather to call attention to their true historical role. The history and the nature of the relevant physics seems to call out for little more than a heuristic reading of the gauge argument and of the role of gauge symmetry principles. Why, though, might flowery language concerning all-powerful gauge symmetry principles persist despite their contrasting and respectable historical role as heuristic guides? I think that a reasonable answer to this question might run as follows. Gauge invariance is indeed a very conspicuous, even powerful feature or property of theory: conspicuous and powerful just in that, formally, this feature is directly related to many characteristic properties of direct interest. To name but a few: the fact that the massless limit of the (quantum) propagator for vector particles is smooth, the fact that intricate cancellations occurring in certain calculations make the theories renormalizable, or even the fact that gauge theories can be given a geometrical formulation in terms of principle fiber bundles. The pitfall to be avoided, however, is that as a feature of theory so rich in interesting affiliations, it is perhaps too easy to uncritically elevate this feature to the defining "principle" of the theory, the one from which all, including the form of the theory itself, is to be derived. Indeed, this gauge invariance is related to many interesting features of the theories in which it figures. But, it does not necessarily follow that we should be so awed by these interesting relationships that we exalt them in the form of an invented physical gauge principle operating in nature.26 This is made especially clear in light of the considerations just discussed which take the associated gauge structure to be a "side-effect" of working with the low-energy approximations ("residues") of some more fundamental underlying physical theory. Conclusion. In this paper I have discussed three key points at which a literal reading of the gauge argument runs into difficulties. Consequently, the claim that the argument, as the embodiment of some fundamental physical gauge symmetry principle, "dictates" or "determines" the form of fundamental interactions as well as the existence of certain physical fields must be taken with a large grain of salt. The motivations for the demand of local gauge invariance in the first place are not unambiguous the burden remains to justify this demand in some non-questionbegging way. Moreover, the choice of modified Lagrangian given just this demand is not unique, and other fundamental concerns must enter toward specifying the correct physical Lagrangian. And, the claim to "generate" a new gauge field in running the argument is not quite right. The imbuing of this new field with its own substantial reality, as a physical, coupled gauge field, is, once again, something that is not dictated by the local gauge invariance requirement per se, and must be done by hand. Perhaps nothing calls more into question a literal reading of the gauge argument than does the existence of quite different approaches to the gauge structure of our fundamental theories, ones which have the "logic of nature" wholely different. Such approaches indeed raise the possibility that, strictly speaking, there is nothing deserving to be called a physical gauge principle at work in fundamental physics. References Aitchison, I. J. R., and A. J. G. Hey (1989), Gauge Theories in Particle Physics. Bristol: IOP Publishing. First citation in article Auyang, S. (1995), How is Quantum Field Theory Possible? Oxford: Oxford University Press. First citation in article Bleecker, D. (1981), Gauge Theory and Variational Principles. Reading: Addison Wesley. First citation in article Brown, H. (1999), "Aspects of Objectivity in Quantum Mechanics", in J. Butterfield and C. Pagonis (eds.), From Physics to Philosophy. Cambridge: Cambridge University Press. First citation in article Cornwall, J., D. Levin, and G. Tiktopoulos (1974), "Derivation of Gauge Invariance from High-energy Unitarity Bounds on the S-Matrix", Physical Review D 10(4): 1145 1167. First citation in article Deser, S. (1970), "Self-Interaction and Gauge Invariance", General Relativity and Gravitation 1(1): 9 18. First citation in article Deser, S. (1987), "Gravity from Self-Interaction in a Curved Background", Classical and Quantum Gravity 4: L99 L105. First citation in article Foerster D., H. Nielsen, and M. Ninomiya (1980), "Dynamical Stability of Local Gauge Symmetry", Physics Letters 94B(2): 135 140. First citation in article Froggatt, C., and H. Nielsen (1991), Origin of Symmetries. Singapore: World Scientific. First citation in article Kosso, P. (1999), "Symmetry Arguments in Physics", Studies in History and Philosophy of Science 30A(3): 479 492. First citation in article Martin, C. (2002), Gauging Gauge: Remarks on the Conceptual Foundations of Gauge Symmetry. Ph.D. dissertation Pittsburgh, University of Pittsburgh. First citation in article Mills, R. (1989), "Gauge Fields", American Journal of Physics 57(6): 493 507. First citation in article Norton, J. (2000), " `Nature is the Realisation of the Simplest Conceivable Mathematical Ideas': Einstein and the Canon of Mathematical Simplicity", Studies in History and Philosophy of Modern Physics 31(2): 135 170. First citation in article O'Raifeartaigh, L. (1979), "Hidden Gauge Symmetry", Reports on Progress in Physics 42: 159 223. First citation in article Ramond, P. (1990), Field Theory: A Modern Primer. Reading: Addison Wesley. First citation in article Redhead, M. (1975), "Symmetry in Intertheory Relations", Synthese 32: 77 112. First citation in article Ryder, L. H. (1996), Quantum Field Theory. Cambridge: Cambridge University Press. First citation in article `t Hooft, G. (1980), "Gauge Theories of the Forces between Elementary Particles", Scientific American 242: 104 138. First citation in article van Fraassen, B. (1989), Laws and Symmetry. Oxford: Oxford University Press. First citation in article Wald, R. (1986), "Spin-Two Fields and General Convariance", Physical Review D 33(12): 3613 3625. First citation in article Wald, R., and J. Lee (1990), "Local Symmetries and Constraints", Journal of Mathematical Physics 31(3): 725 743. First citation in article Weinberg, S. (1965), "Photons and Gravitons in Perturbation Theory: Derivation of Maxwell's and Einstein's Equations", Physical Review 138(4B): 988 1002. First citation in article 1 Weinberg, S. (1974a), "Gauge and Global Symmetries at High Temperature", Physical Review D 9(12): 3357 3378. First citation in article Weinberg, S. (1974b), "Recent Progress in Gauge Theories of the Weak, Electromagnetic and Strong Interactions", Reviews of Modern Physics 46(2). First citation in article Weinberg, S. (1995), The Quantum Theory of Fields: Volume I, Foundations. Cambridge: Cambridge University Press. First citation in article Yang, C., and R. Mills (1954), "Isotopic Spin Conservation and a Generalized Gauge Invariance", Physical Review 96: 191. First citation in article What I here call the "gauge argument" is in fact just a convenient label for what is an amalgam of the most common features figuring in many similar such arguments in the physics literature, both popular and technical/text-book. (For an example of the former see, Mills 1989 or `t Hooft 1980, and for the latter see Aitchison and Hey 1989.) Though there are certainly variations in the way the argument is presented and, most importantly, in the overall place and significance assigned to the argument relative to other features of gauge theory what we consider here are the most central elements, the ones to which most presentations make appeal in one way or another. 2 See van Fraassen 1989 and Kosso 1999 for further discussion of the types and place of symmetry arguments in physics. 3 Here, I suppress all spinor indices, and is just the Dirac conjugate of and are the usual Dirac matrices. 4 The action will be invariant, and thus the Euler-Lagrange equations determined from minimizing (extremizing) it covariant, if the Lagrangian is quasi-invariant, i.e., invariant up to an overall divergence, under the transformations. 5 The homogeneous field equations follow from the local (gauge) invariance of the action, in fact being just the identities (generalized Bianchi identities) following from Noether's 2nd theorem. 6 As was necessary in formulating the Standard Model, one can through a spontaneous breakdown of the vacuum arrive at gauge field(s) with effective mass(es) without this spoiling the gauge invariance of the underlying Lagrangian. 7 This argument generalizes to fields carrying symmetries associated with arbitrary (in particular, non-Abelian) Lie groups, this yielding further interesting gauge structure. The chief difference is that the non-Abelian gauge group has the result that the gauge field "generated" in the gauge argument carries its own charge and is thus self-interacting. Specifically, the kinetic term necessarily includes self interactions in the form of a term proportional to the commutator of the gauge potentials. The requirement of local gauge invariance similarly determines the form of the (self-) interactions in the non-Abelian case. 8 The gauge argument begins from a globally gauge invariant Lagrangian in general, one specifically chosen to describe a field with a known converved current, which in turn is related via Noether's 1st theorem to invariance of the action under some compact simple Lie group. 9 Auyang 1995 contains a more developed argument along what I believe are similar lines. Auyang's argument against global symmetries, and for local symmetries, hinges on the assumption that a global symmetry of some field system implies the appeal to a global convention which is in turn associated with a single internal state space for the entire field. I think this assumption can safely be rejected. 10 See Wald and Lee 1990 for further discussion. 11 See, for example, Ramond 1990. In quantum field theory, on the other hand, locality is commonly formalized in terms of micro-causality, the commuting of space-like separate local algebras of observables. Clearly, there are numerous notions of "locality," some appropriate to only certain formal contexts. 12 See, for example, Ryder 1996, 93. 13 One might also consider the apparently weaker requirement of the Lorentz invariance of the action and corresponding Lorentz covariance of the associated equations of motion. But, the requirement of Lorentz invariance does not exclude Lagrangians with global invariances: e.g., the Lagrangian for the free Dirac field. 14 This, of course, leaves open the possibility that a viable locality argument of the above sort could be mounted if the gauge transformations were afforded some non-trivial physical significance. 15 According to a prominent modern view this is not quite correct. I discuss below further matters relevant to the exclusion of such terms. 16 See Norton 2000 for some critical discussion of considerations of simplicity. 17 Strictly speaking, this is true only locally. There are potential global issues as are evidenced, for example, in the familiar Bohm-Aharonov effect. 18 Note that the q appearing in the gauge transformations are normalization constants for the generators of the associated Lie group and can be scaled away. This is not to be identified with the electric charge e until one adds the kinetic term which integrally contains e. 19 I take it that it is in recognition of essentially this same point that Auyang, considering the gauge principle in electrodynamics, remarks: "It does not stipulate an interaction field but rules against its a priori exclusion . . ." (Auyang 1995, 58). See also Brown 1999. 20 The appeal of local gauge symmetry principles took root in a historical context in which such heuristic guides for "generating" physical field theories were of immediate practical importance. See Martin 2002 for further discussion here. 21 See Wald 1986; Deser 1970, 1987. Weinberg 1995, chaps. 5 and 8, in contrast with the flow of the gauge argument, starts from a quantum theory of massless spin one fields and arrives at a gauge invariant coupling to matter. 22 See Froggatt and Nielsen 1991, 123; Foerster, Nielsen, and Ninomiya 1980; Cornwall, Levin, and Tiktopoulos 1974; Weinberg 1974a, 1974b and the references contained therein. See also Weinberg 1965. 23 Formally, this result goes by the name of the decoupling theorem. 24 Froggatt and Nielsen 1991, ch. vii discusses a "random dynamics" program wherein one desideratum is the derivation or explanation of (the usually assumed) Lorentz invariance and gauge invariance from underlying randomness (or, effectively, a lack of assumption about what is going on) at the fundamental level. 25 See footnote 21. Redhead 1975, for example, discusses gauge symmetries in this way as heuristic symmetries. 26 One might consider that much of the rhetoric surrounding gauge principles is firmly entrenched in a specifically axiomatic view of physical theory. And one might take it that the rhetoric here also commits the sin of taking the language of such an axiomatization too seriously, reifying features of it as features of the world. Perhaps a semantic view of theories that, moreover, resists such a reification would go some way toward providing a basis for a deflation of the rhetoric here.

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