The paper investigates the spontaneous breaking of gauge symmetries in gauge theories from a philosophical angle, taking into account the fact that the notion of a spontaneously broken local gauge symmetry, though widely employed in textbook expositions of the Higgs mechanism, is not supported by our leading theoretical frameworks of gauge quantum theories. In the context of lattice gauge theory, the statement that local gauge symmetry cannot be spontaneously broken can even be made (...) rigorous in the form of Elitzur’s theorem. Nevertheless, gauge symmetry breaking does occur in gauge quantum field theories in the form of the breaking of remnant subgroups of the original local gauge group under which the theories typically remain invariant after gauge fixing. The paper discusses the relation between these instances of symmetry breaking and phase transitions and draws some more general conclusions for the philosophical interpretation of gauge symmetries and their breaking. (shrink)

In a recent paper in this journal, Kosso ([2000]) discussed the observational status of continuous symmetries of physics. While we are in broad agreement with his approach, we disagree with his analysis. In the discussion of the status of gauge symmetry, a set of examples offered by 't Hooft ([1980]) has influenced several philosophers, including Kosso; in all cases the interpretation of the examples is mistaken. In this paper, we present our preferred approach to the empirical significance of (...) symmetries, re-analysing the cases of gauge symmetry and general covariance. Direct and indirect empirical significance Global and local continuous symmetries Gauge symmetry 3.1 Local gauge symmetry 3.1.1 Discussion of the first claim 3.1.2 Discussion of the second claim 3.2 Global gauge symmetry Space-time symmetries Direct and indirect empirical significance again Conclusion. (shrink)

Gauge symmetries play a central role, both in the mathematical foundations as well as the conceptual construction of modern (particle) physics theories. However, it is yet unclear whether they form a necessary component of theories, or whether they can be eliminated. It is also unclear whether they are merely an auxiliary tool to simplify (and possibly localize) calculations or whether they contain independent information. Therefore their status, both in physics and philosophy of physics, remains to be fully clarified. (...) In this overview we review the current state of affairs on both the philosophy and the physics side. In particular, we focus on the circumstances in which the restriction of gauge theories to gauge invariant information on an observable level is warranted, using the Brout-Englert-Higgs theory as an example of particular current importance. Finally, we determine a set of yet to be answered questions to clarify the status of gauge symmetries. (shrink)

In this paper, we discuss the macroscopic quantum behavior of simple superconducting circuits. Starting from a Lagrangian for electromagnetic field with broken gauge symmetry, we construct a quantum circuit model for a superconducting weak link (SQUID) ring, together with the appropriate canonical commutation relations. We demonstrate that this model can be used to describe macroscopic excitations of the superconducting condensate and the localized charge states found in some ultrasmall-capacitance weak-link devices.

According to conventional wisdom, local gauge symmetry is not a symmetry of nature, but an artifact of how our theories represent nature. But a study of the so-called theta-vacuum appears to refute this view. The ground state of a quantized non-Abelian Yang-Mills gauge theory is characterized by a real-valued, dimensionless parameter theta—a fundamental new constant of nature. The structure of this vacuum state is often said to arise from a degeneracy of the vacuum of the corresponding classical theory, (...) which degeneracy allegedly arises from the fact that “large” (but not “small”) local gauge transformations connect physically distinct states of zero field energy. If that is right, then some local gauge transformations do generate empirical symmetries. In defending conventional wisdom against this challenge I hope to clarify the meaning of empirical symmetry while deepening our understanding of gauge transformations. I distinguish empirical from theoretical symmetries. Using Galileo’s ship and Faraday’s cube as illustrations, I say when an empirical symmetry is implied by a theoretical symmetry. I explain how the theta-vacuum arises, and how “large” gauge transformations differ from “small” ones. I then present two analogies from elementary quantum mechanics. By applying my analysis of the relation between empirical and theoretical symmetries, I show which analogy faithfully portrays the character of the vacuum state of a classical non-Abelian Yang-Mills gauge theory. The upshot is that “large” as well as “small” gauge transformations are purely formal symmetries of non-Abelian Yang-Mills gauge theories, whether classical or quantized. It is still worth distinguishing between these kinds of symmetries. An analysis of gauge within the constrained-Hamiltonian formalism yields the result that “large” gauge transformations should not be classified as gauge transformations; indeed, nor should “global” gauge transformations. In a theory in which boundary conditions are modeled dynamically, “global” gauge transformations may be associated with physical symmetries, corresponding to translations of these extra dynamical variables. Such translations are symmetries if and only if charge is conserved. But it is hard to argue that these symmetries are empirical, and in any case they do not correspond to any constant phase change in a quantum state. (shrink)

We discuss some aspects of the relation between dualities and gauge symmetries. Both of these ideas are of course multi-faceted, and we confine ourselves to making two points. Both points are about dualities in string theory, and both have the ‘flavour’ that two dual theories are ‘closer in content’ than you might think. For both points, we adopt a simple conception of a duality as an ‘isomorphism’ between theories: more precisely, as appropriate bijections between the two theories’ sets (...) of states and sets of quantities. The first point is that this conception of duality meshes with two dual theories being ‘gauge related’ in the general philosophical sense of being physically equivalent. For a string duality, such as T-duality and gauge/gravity duality, this means taking such features as the radius of a compact dimension, and the dimensionality of spacetime, to be ‘gauge’. The second point is much more specific. We give a result about gauge/gravity duality that shows its relation to gauge symmetries to be subtler than you might expect. For gauge theories, you might expect that the duality bijections relate only gauge-invariant quantities and states, in the sense that gauge symmetries in one theory will be unrelated to any symmetries in the other theory. This may be so in general; and indeed, it is suggested by discussions of Polchinski and Horowitz. But we show that in gauge/gravity duality, each of a certain class of gauge symmetries in the gravity/bulk theory, viz. diffeomorphisms, is related by the duality to a position-dependent symmetry of the gauge/boundary theory. (shrink)

To systematically answer the generalized Kretschmann objection, I propose a mean to make operational a criterion widely recognized as allowing one to decide whether the gauge symmetry of a theory is artificial or substantial. My proposition is based on the dressing field method of gauge symmetry reduction, a new simple tool from mathematical physics. This general scheme allows one in particular to straightforwardly argue that the notion of spontaneous symmetry breaking is superfluous to the empirical success of the (...) electroweak theory. Important questions regarding the context of justification of the theory then arise. (shrink)

This paper is an analysis of the geometrical interpretation of local gauge symmetry for theories of the Yang-Mills type.

This paper shows how the study of surpluses of structure is an interesting philosophical task. In particular I explore how local gauge symmetry in quantized Yang-Mills theories is the by-product of the specific dynamical structure of interaction. It is shown how in non relativistic quantum mechanics gauge symmetry corresponds to the freedom to locally define global features of gauge potentials. Also discussed is how in quantum field theory local gauge symmetry is replaced by BRST symmetry. This (...) last symmetry is apparently the result of the fact that we do not know how to define quantum Yang-Mills theories without unphysical gauge boson states. Since Yang-Mills theories describe successfully three of the four fundamental interactions the elucidation of this symmetry is a pressing philosophical question. (shrink)

We discuss the symmetry basis of unified field theories, i.e., the generalized concept of local gauge symmetry, and its physical implications. The generalized Ward-Takahashi identities and the explicit constraints among renormalization constants are derived by using the path integral in a specific model. These constraints are confirmed at the one-loop level.

Much recent philosophy of physics has investigated the process of symmetry breaking. Here, I critically assess the alleged symmetry restoration at the fundamental scale. I draw attention to the contingency that gauge symmetries exhibit, that is, the fact that they have been chosen from an infinite space of possibilities. I appeal to this feature of group theory to argue that any metaphysical account of fundamental laws that expects symmetry restoration up to the fundamental level is not fully satisfactory. (...) This is a symmetry argument in line with Curie’s first principle. Further, I argue that this same feature of group theory helps to explain the ‘unreasonable’ effectiveness of mathematics in physics, and that it reduces the philosophical significance that has been attributed to the objectivity of gauge symmetries. (shrink)

Mathematically, gauge theories are extraordinarily rich --- so rich, in fact, that it can become all too easy to lose track of the connections between results, and become lost in a mass of beautiful theorems and properties: indeterminism, constraints, Noether identities, local and global symmetries, and so on. -/- One purpose of this short article is to provide some sort of a guide through the mathematics, to the conceptual core of what is actually going on. Its focus is (...) on the Lagrangian, variational-problem description of classical mechanics, from which the link between gauge symmetry and the apparent violation of determinism is easy to understand; only towards the end will the Hamiltonian description be considered. -/- The other purpose is to warn against adopting too unified a perspective on gauge theories. It will be argued that the meaning of the gauge freedom in a theory like general relativity is (at least from the Lagrangian viewpoint) significantly different from its meaning in theories like electromagnetism. The Hamiltonian framework blurs this distinction, and orthodox methods of quantization obliterate it; this may, in fact, be genuine progress, but it is dangerous to be guided by mathematics into conflating two conceptually distinct notions without appreciating the physical consequences. (shrink)

This article investigates and resolves the question whether gauge symmetry can display analogs of the famous Galileo’s ship scenario. In doing so, it builds on and clarifies the work of Greaves and Wallace on this subject.

This article uncovers a foundational relationship between the ‘gauge symmetry’ of a Newton-Cartan theory and the celebrated Trautman Recovery Theorem and explores its implications for recent philosophical work on Newton-Cartan gravitation.

Gauge symmetries provide one of the most puzzling examples of the applicability of mathematics in physics. The presented work focuses on the role of analogical reasoning in the gauge argument, motivated by Mark Steiner’s claim that the application of the gauge principle relies on a Pythagorean analogy whose success undermines naturalist philosophy. In this paper, we present two different views concerning the analogy between gravity, electromagnetism, and nuclear interactions, each providing a different philosophical response to the (...) problem of the applicability of mathematics in the natural sciences. The first is based on an account of Weyl’s original work, which first gave rise to the gauge principle. Drawing on his later philosophical writings, we develop an idealist reading of the mathematical analogies in the gauge argument. On this view, mathematical analogies serve to ensure a conceptual harmony in our scientific account of nature. We further discuss the construction of Yang and Mills’s gauge theory in light of this idealist reading. The second account presents a naturalist alternative, formulated in terms of John Norton’s account of a material analogy, according to which the analogy succeeds in virtue of a physical similarity between the different interactions. This account is based on the methodological equivalence principle, a simple conceptual extension of the gauge principle that allows us to understand the relation between coordinate transformations and gravity as a manifestation of the same method. The physical similarity between the different cases is based on attributing the success of this method to the dependence of the coupling on relational physical quantities. We conclude by reflecting on the advantages and limits of the idealist, naturalist, and anthropocentric Pythagorean views, as three alternative ways to understand the puzzling relation between mathematics and physics. (shrink)

Pure quantum phenomena are characterized by intrinsic recurrences in space and time. We use this intrinsic periodicity as a quantization condition to derive a heuristic description of the essential quantum phenomenology of superconductivity. The resulting description is based on fundamental quantum dynamics and geometrical considerations, rather than on microscopical characteristics of the superconducting materials. This allows us to investigate the related gauge symmetry breaking in terms of the competition between quantum recurrence and thermal noise. We also test the validity (...) of this approach to describe the case of carbon nanotubes. (shrink)

We make two points about dualities in string theory. The first point is that the conception of duality, which we will discuss, meshes with two dual theories being ‘gauge related’ in the general philosophical sense of being physically equivalent. The second point is a result about gauge/gravity duality that shows its relation to gauge symmetries to be subtler than one might expect: each of a certain class of gauge symmetries in the gravity theory, that (...) is, diffeomorphisms, is related to a position-dependent symmetry of the gauge theory. (shrink)

We discuss the concept of spontaneous breaking of gauge symmetry in super-conductors and superfluids and, in particular, the circumstances under which the absolute phase of a superfluid can be physically meaningful and experimentally relevant. We argue that the study of this question pushes us toward the frontiers of what we understand about the quantum measurement process, and underline the need for a new theoretical framework that keeps pace with modern technological capabilities.

Using the parametrised representation of field theory I demonstrate that in both local and global cases, internal and spacetime symmetries can be treated precisely on a par, so that gravitational theories may be regarded as gauge theories in a completely standard sense.

Classical generators of one-dimensional reparametrization, and higher dimensional diffeomorphism symmetries are displayed for the relativistic free particle, relativistic particles in interaction, and general relativity in both Lagrangian and Hamiltonian frameworks. Projectability of these symmetries under the Legendre map is achieved only with dynamical variable-dependent transformations. When gauge symmetries are included, as in Einstein-Yang-Mills and a new reparametrization covariant pre-Maxwell model, pure coordinate symmetries are not projectable. They must be accompanied by internal gauge transformations.

The classical field theories that underlie the quantum treatments of the electromagnetic, weak, and strong forces share a peculiar feature: specifying the initial state of the field determines the evolution of some degrees of freedom of the theory while leaving the evolution of some others wholly arbitrary. This strongly suggests that some of the variables of the standard state space lack physical content-intuitively, the space of states of such a theory is of higher dimension than the corresponding space of genuine (...) physical possibilities. The structure of such theories can helpfully be characterized in terms of the action of symmetry groups on their space of states; and the conceptual problems surrounding their strange behavior can be sharpened in light of the observation that it is usually possible to eliminate the redundant variables associated with these symmetries-which turn out to be precisely those variables whose evolution is unconstrained by the dynamical laws of the theory. This paper discusses this approach, uses it to frame questions about the interpretation of classical gauge theories, and to reflect (pessimistically) on our prospects of reaching satisfactory answers to these questions. (shrink)

We describe how the choice of an appropriate (“physical”) gauge leads to the solution of a nonperturbative problem in quantum electrodynamics: dynamical chiral symmetry breaking in QED in a constant magnetic field.

The concepts in the title refer to properties of physical theories and this paper investigates their nature and relations. The first three concepts, especially gauge invariance and indeterminism, have been widely discussed in connection to spacetime theories and the hole argument. Since the gauge invariance principle is at the crux of the issue, this paper aims at clarifying the nature of gauge invariance. I first explore the following chain of relations: gauge invariance $\Rightarrow $ the conservation (...) laws $\Rightarrow $ the Cauchy problem $\Rightarrow $ indeterminism. Then I discuss gauge invariance in light of our understanding of the above relations and the possibility of spontaneous symmetry breaking. (shrink)

It is well known that the self-energy of the gauge bosons is quadratically divergent in the Standard Model when a simple cutoff is imposed. We demonstrate phenomenologically that the quadratic divergences in fact unify. The unification occurs at a surprisingly low scale, \(\Lambda _\mathrm {u}\approx 4\times 10^7\) GeV. Suppose now that there is a spontaneously broken rotational symmetry between the space-time coordinates and gauge theoretical phases. The symmetry-breaking pattern is such that the gauge bosons arise as the (...) massless Goldstone bosons, whereas the dilatonic mode acts as the massive (Higgs) boson, whose vacuum expectation value determines the gauge couplings. In this case, the quadratic divergences or the tadpoles of the gauge boson self-energy should indeed unify because these divergences need to be cancelled by a universal dilatonic contribution, assuming dynamical symmetry breaking. If there is dynamical symmetry breaking, we are in principle able to calculate the value of the gauge couplings as well as the scale hierarchy \(\Lambda _\mathrm {cut}/\Lambda _\mathrm {u}\) . We perform this calculation by adopting a naive quartic symmetry-breaking potential which unfortunately violates local gauge invariance. Using tadpole-cancellation and dilatonic self-energy conditions, the value of \(\Lambda _\mathrm {cut}\) is then found to be approximately \(4\times 10^{18}\) GeV in the Feynman gauge and \(5\times 10^{15}\) GeV in the Landau gauge. The cancellation of an anomaly in the dilaton self-energy requires that the number of fermionic generations equals three. The symmetry-breaking needs to be driven by some other mass-generating mechanism such as electroweak symmetry breaking. Our estimation for \(\Lambda _\mathrm {u}\) is of the correct order if \(\Lambda _\mathrm {cut}\approx 5\times 10^{15}\) GeV. (shrink)

We propose a group-theoretical interpretation of the fact that the transition from classical to quantum mechanics entails a reduction in the number of observables needed to define a physical state and \ to \ or \ in the simplest case). We argue that, in analogy to gauge theories, such a reduction results from the action of a symmetry group. To do so, we propose a conceptual analysis of formal tools coming from symplectic geometry and group representation theory, notably Souriau’s (...) moment map, the Mardsen–Weinstein symplectic reduction, the symplectic “category” introduced by Weinstein, and the conjecture according to which “quantization commutes with reduction”. By using the generalization of this conjecture to the non-zero coadjoint orbits of an abelian Hamiltonian action, we argue that phase invariance in quantum mechanics and gauge invariance have a common geometric underpinning, namely the symplectic reduction formalism. This stance points towards a gauge-theoretical interpretation of Heisenberg indeterminacy principle. We revisit this principle in the light of the difference between the set-theoretic points of a phase space and its category-theoretic symplectic points. (shrink)

We propose an operator constraint equation for the wavefunction of the Universe that admits genuine evolution. While the corresponding classical theory is equivalent to the canonical decomposition of General Relativity, the quantum theory contains an evolution equation distinct from standard Wheeler–DeWitt cosmology. Furthermore, the local symmetry principle—and corresponding observables—of the theory have a direct interpretation in terms of a conventional gauge theory, where the gauge symmetry group is that of spatial conformal diffeomorphisms (that preserve the spatial volume of (...) the Universe). The global evolution is in terms of an arbitrary parameter that serves only as an unobservable label for successive states of the Universe. Our proposal follows unambiguously from a suggestion of York whereby the independently specifiable initial data in the action principle of General Relativity is given by a conformal geometry and the spatial average of the York time on the spacelike hypersurfaces that bound the variation. Remarkably, such a variational principle uniquely selects the form of the constraints of the theory so that we can establish a precise notion of both symmetry and evolution in quantum gravity. (shrink)

I address the view that the classical electromagnetic potentials are shown by the Aharonov–Bohm effect to be physically real. I give a historico-philosophical presentation of this view and assess its prospects, more precisely than has so far been done in the literature. Taking the potential as physically real runs prima facie into ‘gauge-underdetermination’: different gauge choices represent different physical states of affairs and hence different theories. This fact is usually not acknowledged in the literature, neither by proponents nor (...) by opponents of the potentials view. I then illustrate this theme by what I take to be the basic insight of the AB effect for the potentials view, namely that the gauge equivalence class that directly corresponds to the electric and magnetic fields is too wide, i.e., the Narrow Equivalence Class encodes additional physical degrees of freedom: these only play a distinct role in a multiply-connected space. There is a trade-off between explanatory power and gauge symmetries. On the one hand, this narrower equivalence class gives a local explanation of the AB effect in the sense that the phase is incrementally picked up along the path of the electron. On the other hand, locality is not satisfied in the sense of signal locality, viz. the finite speed of propagation exhibited by electric and magnetic fields. It is therefore intellectually mandatory to seek desiderata that will distinguish even within these narrower equivalence classes, i.e. will prefer some elements of such an equivalence class over others. I consider various formulations of locality, such as Bell locality, local interaction Hamiltonians, and signal locality. I show that Bell locality can only be evaluated if one fixes the gauge freedom completely. Yet, an explanation in terms of signal locality can be accommodated by the Lorenz gauge: the potentials propagate in waves at finite speed. I therefore suggest the Lorenz gauge potentials theory—an even narrower gauge equivalence relation—as the ontology of electrodynamics. (shrink)

Why is gauge symmetry so important in modern physics, given that one must eliminate it when interpreting what the theory represents? In this paper we discuss the sense in which gauge symmetry can be fruitfully applied to constrain the space of possible dynamical models in such a way that forces and charges are appropriately coupled. We review the most well-known application of this kind, known as the 'gauge argument' or 'gauge principle', discuss its difficulties, and then (...) reconstruct the gauge argument as a valid theorem in quantum theory. We then present what we take to be a better and more general gauge argument, based on Noether's second theorem in classical Lagrangian field theory, and argue that this provides a more appropriate framework for understanding how gauge symmetry helps to constrain the dynamics of physical theories. (shrink)

The Higgs mechanism gives mass to Yang-Mills gauge bosons. According to the conventional wisdom, this happens through the spontaneous breaking of gauge symmetry. Yet, gauge symmetries merely reflect a redundancy in the state description and therefore the spontaneous breaking can not be an essential ingredient. Indeed, as already shown by Higgs and Kibble, the mechanism can be explained in terms of gauge invariant variables, without invoking spontaneous symmetry breaking. In this paper, we present a general (...) discussion of such gauge invariant treatments for the case of the Abelian Higgs model, in the context of classical field theory. We thereby distinguish between two different notions of gauge: one that takes all local transformations to be gauge and one that relates gauge to a failure of determinism. (shrink)

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 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.

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 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.

This article suggests a fresh look at gauge symmetries, with the aim of drawing a clear line between the a priori theoretical considerations involved, and some methodological and empirical non-deductive aspects that are often overlooked. The gauge argument is primarily based on a general symmetry principle expressing the idea that a change of mathematical representation should not change the form of the dynamical law. In addition, the ampliative part of the argument is based on the introduction of (...) new degrees of freedom into the theory according to a methodological principle that is formulated here in terms of correspondence between passive and active transformations. To demonstrate how the two kinds of considerations work together in a concrete context, I begin by considering spatial symmetries in mechanics. I suggest understanding Mach's principle as a similar combination of theoretical, methodological and empirical considerations, and demonstrate the claim with a simple toy model. I then examine gauge symmetries as a manifestation of the two principles in a quantum context. I further show that in all of these cases the relational nature of physically significant quantities can explain the relevance of the symmetry principle and the way the methodology is applied. In the quantum context, the relevant relational variables are quantum phases. (shrink)

Local gauge theories are in a complicated relationship with boundaries. Whereas fixing the gauge can often shave off unwanted redundancies, the coupling of different bounded regions requires the use of gauge-variant elements. Therefore, coupling is inimical to gauge-fixing, as usually understood. This resistance to gauge-fixing has led some to declare the coupling of subsystems to be the \textit{raison d'\^etre} of gauge \cite{RovelliGauge2013}. Indeed, while gauge-fixing is entirely unproblematic for a single region without boundary, (...) it introduces arbitrary boundary conditions on the gauge degrees of freedom themselves---these conditions lack a physical interpretation when they are not functionals of the original fields. Such arbitrary boundary choices creep into the calculation of charges through Noether's second theorem, muddling the assignment of physical charges to local gauge symmetries. The confusion brewn by gauge at boundaries is well-known, and must be contended with both conceptually and technically. It may seem natural to replace the arbitrary boundary choice with new degrees of freedom, for using such a device we resolve some of these confusions while leaving no gauge-dependence on the computation of Noether charges \cite{DonnellyFreidel}. But, concretely, such boundary degrees of freedom are rather arbitrary---they have no relation to the original field-content of the field theory. How should we conceive of them? Here I will explicate the problems mentioned above and illustrate a possible resolution: in a recent series of papers \cite{GomesRiello2016, GomesRiello2018, GomesHopfRiello} the notion of a connection-form was put forward and implemented in the field-space of gauge theories. Using this tool, a modified version of symplectic geometry---here called `horizontal'---is possible. Independently of boundary conditions, this formalism bestows to each region a physically salient, relational notion of charge: the horizontal Noether charge. Meanwhile, as required, the connection-form mediates an irenic composition of regions, one compatible with the attribution of horizontal Noether charges to each region. Thes aims of this paper are to highlight the boundary issues in the treatment of gauge, and to discuss how gauge theory may be conceptually clarified in light of a resolution to these issues. (shrink)

The world appears to be well described by gauge theories; why? I suggest that gauge is more than mathematical redundancy. Gauge-dependent quantities can not be predicted, but there is a sense in which they can be measured. They describe “handles” though which systems couple: they represent real relational structures to which the experimentalist has access in measurement by supplying one of the relata in the measurement procedure itself. This observation leads to a physical interpretation for the ubiquity (...) of gauge: it is a consequence of a relational structure of physical quantities. (shrink)

In the last 5 years, the controversy about whether or not gauge transformations can be empirically significant has intensified. On the one hand, Greaves and Wallace developed a framework according to which, under some circumstances, gauge transformations can be empirically significant—and Teh further supported this result by using the constrained Hamiltonian formalism. On the other hand, Friederich claims to have proved that gauge transformation can never be empirically significant. In this article, I accomplish two tasks. First, I (...) argue that there are strong reasons to resist Friederich’s proof because one of its assumptions is, at the very least, highly controversial. Second, I argue that despite criticism by Brading and Brown and Friederich, ‘t Hooft’s beam-splitter experiment is indeed a concrete example of a case where a local gauge symmetry has empirical significance. By shedding light on these two points, this article shows that recent arguments that claim gauge transformations cannot be empirically significant are not satisfactory. (shrink)

Symmetry considerations dominate modern fundamental physics, both in quantum theory and in relativity. Philosophers are now beginning to devote increasing attention to such issues as the significance of gauge symmetry, quantum particle identity in the light of permutation symmetry, how to make sense of parity violation, the role of symmetry breaking, the empirical status of symmetry principles, and so forth. These issues relate directly to traditional problems in the philosophy of science, including the status of the laws of nature, (...) the relationships between mathematics, physical theory, and the world, and the extent to which mathematics suggests new physics.This entry begins with a brief description of the historical roots and emergence of the concept of symmetry that is at work in modern science. It then turns to the application of this concept to physics, distinguishing between two different uses of symmetry: symmetry principles versus symmetry arguments. It mentions the different varieties of physical symmetries, outlining the ways in which they were introduced into physics. Then, stepping back from the details of the various symmetries, it makes some remarks of a general nature concerning the status and significance of symmetries in physics. (shrink)

I critically examine the assumption that the theoretical structure that varies under theoretical symmetries is redundant and should be eliminated from a metaphysical picture of the universe, following a ‘symmetry to reality’ inference. I do so by analysing the status of coordinate change symmetries taking a pragmatic approach. I argue that coordinate systems function as indexical devices, and play an important pragmatic role for representing concrete physical systems. I examine the implications of considering this pragmatic role seriously, taking (...) what I call a perspectivist stance. My conclusion is that under a perspectivist stance, all symmetries (including local gauge symmetries) potentially have a direct empirical status: they point to dynamical aspects that are invariant under changes of operationalisation, and they constitute a guide not to reality, but to nomology and kinship. (shrink)

The elucidation of the gauge principle ‘‘is the most pressing problem in current philosophy of physics’’ said Michael Redhead in 2003. This paper argues for two points that contribute to this elucidation in the context of Yang–Mills theories. (1) Yang–Mills theories, including quantum electrodynamics, form a class. They should be interpreted together. To focus on electrodynamics is potentially misleading. (2) The essential role of gauge and BRST symmetries is to provide a local field theory that can be (...) quantized and would be equivalent to the quantization of the non-local reduced theory. If this is correct, the gauge symmetry is significant, not so much because it implies ontological consequences, but because it allows us to quantize theories that we would not be able to quantize otherwise. Thus, in the context of Yang–Mills theories, it is essentially a pragmatic principle. This does not seem to be the case for the gauge symmetry in general relativity. (shrink)

We develop some ideas about gauge symmetry in the context of Maxwell’s theory of electromagnetism in the Hamiltonian formalism. One great benefit of this formalism is that it pairs momentum and configurational degrees of freedom, so that a decomposition of one side into subsets can be translated into a decomposition of the other. In the case of electromagnetism, this enables us to pair degrees of freedom of the electric field with degrees of freedom of the vector potential. Another benefit (...) is that the formalism algorithmically identifies subsets of the equations of motion that represent time-dependent symmetries. For electromagnetism, these two benefits allow us to define gauge-fixing in parallel to special decompositions of the electric field. More specifically, we apply the Helmholtz decomposition theorem to split the electric field into its Coulombic and radiative parts, and show how this gives a special role to the Coulomb gauge (i.e. div \((\mathbf{A}) = 0\) ). We relate this argument to Maudlin’s (Entropy, 2018. https://doi.org/10.3390/e20060465 ) discussion, which advocated the Coulomb gauge. (shrink)

According to the Standard Model of particle physics, some gauge transformations are physical symmetries. That is, they are mathematical transformations that relate representatives of distinct physical states of affairs. This is at odds with the standard philosophical position according to which gauge transformations are an eliminable redundancy in a gauge theory's representational framework. In this paper I defend the Standard Model's treatment of gauge from an objection due to Richard Healey. If we follow the Standard (...) Model in taking some gauge transformations to be physical symmetries then we face the “strong CP problem”, but if we adopt the standard philosophical position on gauge then the strong CP problem dissolves. Healey offers this as a reason in favor of the standard philosophical view. However, as I argue here, following Healey's recommendation gives a theory that makes bad empirical predictions. (shrink)

The paper proposes a combined account of identity for physical states and direct empirical significance for symmetries according to which symmetry-related state variables designate distinct physical states if and only if the symmetry that relates them has direct empirical significance. Strengthening an earlier result, I show that, given this combined account, the local gauge symmetries in our leading contemporary theories of particle physics do not have any direct empirical significance.

The effective dynamics of quarks is described by a nonperturbatively regularized NJL model equation with canonical quantization and probability interpretation. The quantum theory of this model is formulated in functional space and the gluons are considered as relativistic bound states of colored quark-antiquark pairs. Their wave functions are calculated as eigenstates of hardcore equations, and their effective dynamics is derived by weak mapping in functional space. This leads to the phenomenological SU(3) gauge invariant gluon equations in functional formulation, i.e., (...) the local gauge symmetry is a dynamical effect resulting from the dynamics of the quark model. (shrink)

In 1918, Emmy Noether published a (now famous) theorem establishing a general connection between continuous 'global' symmetries and conserved quantities. In fact, Noether's paper contains two theorems, and the second of these deals with 'local' symmetries; prima facie, this second theorem has nothing to do with conserved quantities. In the same year, Hermann Weyl independently made the first attempt to derive conservation of electric charge from a postulated gauge symmetry. In the light of Noether's work, it is (...) puzzling that Weyl's argument uses local gauge symmetry. This paper explores the relationships between Weyl's work, Noether's two theorems, and the modern connection between gauge symmetry and conservation of electric charge. This includes showing that Weyl's connection is essentially an application of Noether's second theorem, with a novel twist. (shrink)

Symmetries in physics are most commonly recognized and discussed in terms of their function in the mathematical formalism of the theories. Discussion of the observation of symmetries in nature is less common. This paper analyses the observation of particular symmetries such as Lorentz and gauge symmetries, distinguishing between direct observation of the symmetry itself and indirect evidence, the latter being the observation of some consequence of the symmetry are, in an important sense, directly observed, while (...) local symmetries such as gauge symmetry are not. (shrink)

A Lorentz-invariant gauge theory for massive fermions on R × S 3 spacetime is built up. Using the symmetry of S 3,we obtain Dirac-type equation and derive the expression of the fermionic propagator. Finally, starting from the SU(N) gauge-invariant Lagrangian, we obtain the set of Dirac-Yang-Mills equations on R × S 3 spacetime, pointing out major differences from the Minkowskian case.

In this paper, I try to decipher the role of internal symmetries in the ontological maze of particle physics. The relationship between internal symmetries and laws of nature is discussed within the framework of “Platonic realism.” The notion of physical “structure” is introduced as representing a deeper ontological layer behind the observable world. I argue that an internal symmetry is a structure encompassing laws of nature. The application of internal symmetry groups to particle physics came about in two (...) revolutionary steps. The first was the introduction of the internal symmetries of hadrons in the early 1960s. These global and approximate symmetries served as means of bypassing the dynamics. I argue that the realist could interpret these symmetries as ontologically prior to the hadrons. The second step was the gauge revolution in the 1970s, where symmetries became local and exact and were integrated with the dynamics. I argue that the symmetries of the second generation are fundamental in the following two respects: According to the so-called “gauge argument,” gauge symmetry dictates 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 physics at least as a heuristic principle. In view of grand unified theories, the new symmetries can be interpreted as ontologically prior to baryon matter. (shrink)

Einstein considered general covariance to characterize the novelty of his General Theory of Relativity (GTR), but Kretschmann thought it merely a formal feature that any theory could have. The claim that GTR is ``already parametrized'' suggests analyzing substantive general covariance as formal general covariance achieved without hiding preferred coordinates as scalar ``clock fields,'' much as Einstein construed general covariance as the lack of preferred coordinates. Physicists often install gauge symmetries artificially with additional fields, as in the transition from (...) Proca's to Stueckelberg's electromagnetism. Some post-positivist philosophers, due to realist sympathies, are committed to judging Stueckelberg's electromagnetism distinct from and inferior to Proca's. By contrast, physicists identify them, the differences being gauge-dependent and hence unreal. It is often useful to install gauge freedom in theories with broken gauge symmetries (second-class constraints) using a modified Batalin-Fradkin-Tyutin (BFT) procedure. Massive GTR, for which parametrization and a Lagrangian BFT-like procedure appear to coincide, mimics GTR's general covariance apart from telltale clock fields. A generalized procedure for installing artificial gauge freedom subsumes parametrization and BFT, while being more Lagrangian-friendly than BFT, leaving any primary constraints unchanged and using a non-BFT boundary condition. Artificial gauge freedom licenses a generalized Kretschmann objection. However, features of paradigm cases of artificial gauge freedom might help to demonstrate a principled distinction between substantive and merely formal gauge symmetry. (shrink)