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
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, non-question-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.”Footnote 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.Footnote 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 Ψ representing electrically charged matter. The free field obeys the Dirac equation which is just the Euler-Lagrange equation(s) for the Lagrangian (density) 
Dirac = (iγ μ∂μ − m) Ψ.Footnote 3 The corresponding action is clearly invariant under so-called “global” U(1) phase transformations: Ψ → e iqΛΨ ; → e −iqΛ with Λ a constant.Footnote 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 Λ become a function Λ(x): Ψ → e iqΛ(x) Ψ ; → e −iqΛ(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., ∂μΛ(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 
interacting = (iγ μ∂μ − m) Ψ − qA μ γμΨ ≡ 
Dirac − J μA μ, with J μ = q γ μΨ. The current J μ 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 μ, 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 μ is simultaneously transformed according to A μ(x) → A μ(x) − ∂μΛ(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 μ as representing the electromagnetic potential.
Pursuing the idea of viewing the field A μ 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 μ field. But, it remains to add a “kinetic term” for the A μ 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 = −
$\frac{1}{4}$
F μνF μν, with the gauge field F μν defined as F μν := ∂μA ν − ∂νA μ gives (source-free) Maxwell's equations. Putting this all together yields the Lagrangian for the fully interacting theory: 
Maxwell−Dirac = 
interacting + 
kinetic = 
Dirac − J μA μ − 
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 μ.Footnote 5
Finally, consider that a mass term for the vector field A μ of standard form 
photon−mass =
$\frac{1}{2}$
m 2A μA μ. is not gauge invariant. In keeping with the demand of local gauge invariance, the vector field A μ (i.e., the photon) must then be massless. Local gauge invariance thus necessitates a massless photon.Footnote 6, Footnote 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.Footnote 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 Reference Yang and Mills1954 paper, very much setting the tone for subsequent treatments.Footnote 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.Footnote 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 Reference Bleecker1981).
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.Footnote 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 Ψ 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.Footnote 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.Footnote 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 
loc from 
glob. 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 
glob which the STR gauge locality argument aims to exclude.
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.Footnote 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—for example, Lorentz invariance—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 μ γ μΨ which was added above, we could have added, say, higher order covariant derivatives, e.g., 
μ
νΨ or 
λF μν, etc. Or, one might include a dependence of 
interacting on F μν. More interestingly, one might consider the addition of a Pauli term, one proportional to [γμ, γν] Ψ F μν, 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 Reference Weinberg1995, 517). Such terms are thus apparently not included in the gauge invariant Lagrangian correctly describing nature.Footnote 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 Reference O’Raifeartaigh1979). 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.Footnote 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 μ) generated in the gauge argument form a restricted class of all such A μ 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 μν fields.Footnote 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 μν ≡ 0) to the physical coupling of the matter field to non-trivial gauge fields (F μν ≠ 0).Footnote 18 In making this generalization, one puts in by hand much of the important physics of the new interacting theory.Footnote 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.Footnote 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 
Maxwell−Dirac 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.Footnote 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.Footnote 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 well-behaved (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.Footnote 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.Footnote 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.Footnote 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.Footnote 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-question-begging 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.
I thank Jon Bain, Tony Duncan, John Earman and Chris Smeenk for their comments on an earlier draft.
