2.1.1 Symmetries

It has almost become a cliché to emphasize that symmetry plays a central role in modern physics. Weyl declared that “[a]s far as I see, all a priori statements in physics have their origin in symmetry” (Reference WeylWeyl, 1952, 126), Yang argued that “symmetry dictates interactions” (Reference YangYang, 1980, 42), Weinberg famously said that “[s]ymmetry 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” (Reference WeinbergWeinberg, 1992, 142), and philosopher Christopher Martin called twentieth-century physics the “Century of Symmetry” (Reference Martin, Brading and CastellaniMartin, 2003). Cliché or not, it is simply a fact that our currently best physical theories exhibit symmetries. In some cases, symmetry considerations played an important heuristic role in formulating the respective physical theories (e.g., the theory of relativity). In other cases, symmetries were found retrospectively (e.g., classical electrodynamics). Due to the omnipresence of symmetries and the important heuristic role symmetry considerations play in modern physics, it is fair to say that understanding and interpreting symmetries is crucial for understanding our physical theories and what they tell us about the world. Accordingly, reflecting on the mechanism, nature, and heuristic significance of symmetries has become a central task of physicists and philosophers (see, e.g., Reference Brading and CastellaniBrading & Castellani, 2003; Reference Dasgupta, Knox and WilsonDasgupta, 2022; Reference Ismael, Knox and WilsonIsmael, 2022; Reference WeylWeyl, 1952; Reference WignerWigner, 1967b; Reference YangYang, 1996).

In particular, here we are interested in questions concerning the ontology of symmetries. Are certain symmetries mere mathematical artifacts or are they physically real transformations? Here we put a special focus on gauge symmetries, which are at the very heart of modern physics. Importantly, we do not only argue that understanding the nature of gauge symmetries helps us to better understand our physical theories. We also argue that critical conceptual reflection on the nature of gauge symmetries indicates that textbook accounts of the Brout–Englert–Higgs (BEH) mechanism are misleading.Footnote ¹ As we will elaborate shortly, this line of reasoning is not unfamiliar in the philosophy of physics community. It is one of our aims to present it in its strongest form and show how such considerations naturally lead to gauge-invariant approaches to the BEH mechanism. In this context, as mentioned in Section 1, we investigate some available methods for reducing the theories to their gauge-invariant syntax in Sections 5 and 6.

Before we turn to gauge symmetries, we shall begin with some general remarks about symmetries in physics. Symmetries can concern physical objects and states or physical theories and laws. We say that an object or theory possesses a symmetry if there are transformations that leave certain features of the objects or the theory to which the transformations are applied preserved or unchanged. With respect to these features, the object or theory is invariant concerning the respective transformations. In this sense, one can say that “[s]ymmetry is invariance under transformation” (Reference KossoKosso, 2000, 83). Transformations that leave certain aspects unchanged are referred to as symmetry transformations. Groups of such transformations are referred to as symmetry groups. Types of symmetries can be distinguished according to types of transformations. In physics, we often find the following distinctions: continuous versus discrete symmetries, external versus internal symmetries, and global versus local symmetries.

Continuous symmetry is invariance under continuous transformation. An example of a continuous transformation would be the rotation of a circle. Since the appearance of a circle does not change under continuous rotations, we say that circles possess a continuous symmetry. The appearance of snowflakes, on the other hand, remains unchanged by rotations of sixty degrees but not, say, fifty degrees. This would be an example of a discrete transformation and thus snowflakes possess a discrete symmetry. We are particularly concerned with continuous symmetries. Mathematically, continuous symmetries can be described by Lie groups.

External symmetry is invariance under transformations that involve a change of the space-time coordinates. Examples of external transformations are spatial rotations. Accordingly, circles possess a continuous external symmetry and snowflakes a discrete external symmetry. Internal symmetries are symmetries in which the respective transformations do not involve a change of the space-time coordinates. Examples of internal transformations would be permutations of particles or phase transformations. The symmetries we are interested in, namely, the gauge symmetries in quantum field theory, are internal symmetries.

For our purposes, the most important distinction is the one between global and local symmetries. It is standard (but slightly inaccurate) to characterize global transformations as those performed identically at each point in space-time. Similarly, local transformations are ones that are performed arbitrarily at each point in space-time. More accurately, we follow Brading and Brown in making a “distinction between symmetries that depend on constant parameters (global symmetries) and symmetries that depend on arbitrary smooth functions of space and time (local symmetries)” (Reference Brading and BrownBrading and Brown, 2004, 649).Footnote ²

Let us exemplify the difference between a global and a local transformation by considering a field $\Psi$ undergoing the following phase transformation:

$\Psi \left(x\right)\to {\Psi }^{\prime }\left(x\right)=e^{i\theta }\Psi \left(x\right),$

This is a (continuous, internal) global transformation because the phase change $\theta$ is independent of the space-time point ( $\theta$ does not depend on $x$ ). Now consider the phase transformation

$\Psi \left(x\right)\to {\Psi }^{\prime }\left(x\right)=e^{i\theta \left(x\right)}\Psi \left(x\right),$

This is a (continuous, internal) local transformation because the phase change $\theta \left(x\right)$ depends on the position in the field. The space of theories exhibiting local symmetries is much smaller – that is, more constrained – than the space of theories only exhibiting a global symmetry (since global symmetries, when they exist, are subgroups of local symmetries).

A prominent example of an external global transformation is the Lorentz transformation of special relativity. Accordingly, special relativity is based on an external global symmetry. Prominent examples of external local transformations are the arbitrary differentiable coordinate transformations we find in general relativity. Due to its general covariance, namely, the invariance with respect to such transformations, general relativity is based on an external local symmetry.

The symmetries that underlie modern particle physics are internal local symmetries. The Standard Model of particle physics, describing three of the four known fundamental interactions, is a non-Abelian gauge theory with an internal local $U\left(1\right)\times SU\left(2\right)\times SU\left(3\right)$ symmetry group. Concerning our terminology, it is to be noted that we use the terms “local symmetry” and “gauge symmetry” synonymously. Accordingly, any field theory in which the Lagrangian remains invariant under local transformations is a gauge theory. This means that all four fundamental interactions are described by gauge theories (Standard Model + general relativity).

Because gauge symmetries play such a prominent role in modern physics, they more than deserve due conceptual reflection. The central topic of this Element is born out of such reflection, namely, the ontological status of gauge symmetries. Should they be interpreted as merely the mathematical structure of our descriptions of reality or do they represent the structure of reality? To approach this question discussed in more detail in the following section, it is instructive to consider one of the finest examples of synergies between mathematics and physics: Noether’s results concerning the relationship between mathematical symmetries and physical theories.

The famous Noether theorem, also known as Noether’s first theorem, relates continuous global symmetries to conserved quantities. Stated informally, the theorem says that to every continuous global symmetry there corresponds a conservation law (see Reference Brading, Brown, Brading and CastellaniBrading and Brown, 2003). Conversely, every conserved quantity corresponds to a continuous global symmetry. Accordingly, global symmetries seem to be physical symmetries, symmetries of nature. And, indeed, there is some consensus that global symmetries are observable and that they have direct empirical significance (Reference Brading and BrownBrading & Brown, 2004; Reference FriederichFriederich, 2015; Reference GomesGomes, 2021; Reference HealeyHealey, 2009; Reference KossoKosso, 2000).

The situation is very different with respect to local symmetries. The difference can be illustrated by turning from Noether’s first theorem to what is sometimes called Noether’s second theorem (see Reference Brading, Brown, Brading and CastellaniBrading & Brown, 2003; Reference EarmanEarman, 2002, Reference Earman2004b; Reference RicklesRickles, 2008, 55f ).

It has been pointed out that Noether’s results imply that local symmetries impose “powerful restrictions on the possible form a theory can take” (Reference Brading, Brown, Brading and CastellaniBrading & Brown, 2003, 105). Specifically, the second Noether theorem ensures, through a so-called Gauss law, that the dynamics of the force fields are compatible with the dynamics of the charges that are their sources (Reference Gomes, Roberts and ButterfieldGomes, Roberts, & Butterfield, 2021). More broadly, “Noether’s second theorem tells us that in any theory with a local Noether symmetry there is always a prima facie case of underdetermination: more unknowns than there are independent equations of motion” Reference Brading, Brown, Brading and CastellaniBrading & Brown (2003: 104). As will be discussed in detail shortly, this underdetermination inherent to gauge theories implies “an apparent violation of determinism” Reference EarmanEarman (2002: 212).

2.1.2 Interpreting Gauge Symmetries: A First Look

Reference WignerWigner (1967b) compared the gauge invariance of electromagnetism to a theoretical ghost:Footnote ³

This metaphor seems to describe the accepted view that is also, at least officially and explicitly, reflected in physics textbooks as well as that of prominent voices in the philosophy of physics community, declaring gauge theories to contain “surplus structure” (Reference Redhead, Kuhlmann, Lyre and WayneRedhead, 2002), “formal redundancy” (Reference Martin, Brading and CastellaniMartin, 2003), or “descriptive fluff” (Reference EarmanEarman, 2004b, 1239). On one widespread understanding, the surplus structure of many theories is manifested in a multiplicity of mathematical representations for each physical state of affairs. Under this definition, surplus structure is ubiquitous in physics: the simple use of coordinates in space-time physics would count as surplus. In some cases, such a representational multiplicity may be simply understood as the capacity of the theory to accommodate the viewpoints of different observers, that is, what symmetries are often understood to be about (e.g., in special relativity). Therefore, a more useful definition of surplus structure should not encompass the kind of representational multiplicity that is epistemically unavoidable, or even a theoretical virtue.

Reference Redhead, Kuhlmann, Lyre and WayneRedhead (2002) provides a more precise characterization of surplus structure. Given the mathematical structure $M$ of a theory used to represent a physical structure $P$ , if $P$ actually maps isomorphically only onto a substructure $M^{\prime }$ of $M$ , then by definition the surplus structure of the theory is the complement of $M^{\prime }$ in $M$ . Situations in which the mathematical structure that is understood as correlating to a physical structure is embedded within a larger mathematical structure are fairly common in modern physics. It is therefore not hard to find examples that fit this description and present no special interpretive difficulties – the use of complex numbers in classical wave mechanics or in circuit analysis comes to mind.

But obviously there are instances in modern physics where capturing the notion of surplus structure is harder, as the boundary between surplus and essential theoretical structures is hard to draw and may evolve as more knowledge is gained. Generally, which of the mathematical structures of a given theory (if any) correlates to a physical structure is a matter of an interpretation of the theory, which would generally have to balance different theoretical virtues.

We take the relation of mathematical formalism to reality to have three layers: (i) the measurable, (ii) the ontological (or “real,” or physical), and (iii) the mathematical. With these layers in mind, we can stipulate three desiderata concerning interpretations of gauge theories:

1. (D1) To avoid ontological indeterminism.

2. (D2) To avoid ontological commitments to quantities that are not measurable even in principle.

3. (D3) To avoid surplus mathematical structure that has no direct ontological correspondence.

The first two desiderata motivate an interpretation of unobservable or underdetermined theoretical concepts as structure that has no bearing on the ontology and is in that sense “surplus.” On the other hand, considerations of localityFootnote ⁴ and explanatory capacity would often push in the opposite direction, supporting the indispensability of the surplus mathematical structure. In the context of space-time theories, the desiderata (D2) and (D3) are intimately related to symmetry principles aiming to bring together the symmetries of space-time and those of the dynamics (Reference EarmanEarman, 1989). These principles can be applied in interpreting physical theories as well as in constructing them. A possible way of applying analogous principles in the context of gauge theories is by requiring that the symmetries of the dynamics coincide with the kinematical symmetries, that is, with the automorphisms of the mathematical structure taken to represent the possible physical states (Reference HetzroniHetzroni, 2021).

Broadly speaking, interpretations of gauge theories that take gauge transformations to be physically real – that is, to relate physically distinct objects – support a realist commitment to gauge-dependent quantities. Let us call these T1 interpretations. A T1 interpretation is clearly in tension with D1 and D2, but not in conflict with D3 because, for T1, gauge transformations have ontological correspondence. In contrast, interpretations that take gauge symmetries to be manifestation of surplus mathematical structure may no longer be in conflict with either D1 or D2, since they restrict ontological commitments to gauge-invariant quantities whose evolution is deterministic (and which may even be measurable). Let us call these interpretations T2. According to T2, gauge theories, in their standard formulations, have mathematical surplus structure, in conflict with D3.Footnote ⁵

And in fact, D3, as it stands, is a matter of degree: all physical theories harbor some amount of surplus structure that is not directly measurable. After all, physics is not formulated solely in terms of – as brackets containing long conjunctions and disjunctions of – directly observable phenomena, like positions of dials and so on. And even if we weaken “measurable” to “ontological,” some amount of surplus structure – for instance, a choice of coordinates, units, and so on – will usually remain in our description.

Thus, these issues do not pertain merely to the interpretation of gauge theories, they can also motivate the reformulation and extension of existing theories that on the one hand remove superfluous structure to obtain a more parsimonious representation, or on the other hand, promote what initially seems like surplus structure to physical structure.

Therefore, to proceed, we need to separate the chaff from the wheat with regards to surplus structure, and this requires a more refined notion of the term, identifying it with theoretical or formal features that can be excised from a theory without incurring any detriment to its explanatory and pragmatic virtues. Of course, such criteria still leave open what should be counted as explanatory and pragmatic virtues, an issue that depends on one’s viewpoint and goals. Yet, on occasion the criteria are rather clear-cut, and even in more complicated realistic situations we suggest that some consensus should be pursued so as to make the criteria effective.

Here we will exemplify this point by the usage of one more criterion: locality. Thus, redundancy of representation in a theory will be counted as surplus structure if eliminating it still allows us to describe physical states via locally determined quantities, or local variables. This will be the motivation for some of the approaches presented in Section 5. The issue at hand is therefore not limited to whether gauge symmetries manifest surplus structure or not; the refined question is how to distinguish between genuinely surplus structure and that having a physical signature related to nonlocality or nonseparability of gauge physics. Answering this refined question provides a hard-and-fast criterion for a theory to meet all the desiderata D1–D3.

2.2.1 Gauge Invariance in Classical Electromagnetism

The formal property known today as gauge invariance already appeared in Maxwell’s 1856 On Faraday’s Lines of Force, in which he showed, inter alia, that the magnetic vector potential, introduced a few years earlier by William Thomson, can give rise to a unified mathematical description of different phenomena described by Faraday. This gauge invariance later allowed for the elimination of the vector potential from the equations in the modern formulation of Maxwell’s equations by Hertz and Heaviside. In classical electromagnetism the equations of motion of the fields are Maxwell’s equations:

$\begin{array}{cc}\nabla \cdot \vec{B}& =0\end{array}$(2.1)

$\begin{array}{cc}\nabla \times \vec{E}+\frac{\partial \vec{B}}{\partial t}& =0\end{array}$(2.2)

$\begin{array}{cc}\nabla \cdot \vec{E}& =\rho \end{array}$(2.3)

$\begin{array}{cc}\nabla \times \vec{B}-\frac{\partial \vec{E}}{\partial t}& =\vec{j}.\end{array}$(2.4)

The equations of motion of the particles include the Lorentz force $q\left(\vec{E}+\vec{v}\times B\right)$ derived from the fields.

This description appears to have a straightforward interpretation: the electric field $\vec{E}$ and the magnetic field $\vec{B}$ constitute the basic field ontology, and Maxwell’s equations determine their behavior. The local values of the field can be found empirically based on the action of Lorentz force on particles. This interpretation seems to yield a local understanding of the interactions and a picture of a continuous flow of energy in space through electromagnetic radiation. Yet, this theory is not free of conceptual problems.Footnote ⁶ During the first half of the twentieth century these problems raised the question of whether the electric and magnetic fields are real mediators of the interaction, or merely a mathematical tool that helps physicists to keep track of it. The central alternative to the field ontology was a picture of point particles directly interacting with each other at a distance. This kind of theory was famously advocated by Reference Wheeler and FeynmanWheeler and Feynman (1949) based on earlier theories. The gauge freedom of the theory is related to a third mathematical representation, based on the electric potential $V$ and the magnetic vector potential $\vec{A}$ . This representation is particularly convenient in various kinds of physical situations (such as those involving conductors). It also has the advantage of having (2.1) and (2.2) follow as identities from the kinematics rather than as additional dynamical equations. The potentials are defined such that the fields satisfy $\vec{E}=-\nabla V-\frac{\partial \vec{A}}{\partial t}$ and $\vec{B}=\nabla \times \vec{A}$ . Yet, the potentials are underdetermined by the fields: the same magnetic field can be represented by many mathematically distinct potentials. For given potentials $\vec{A}$ and $V$ , the potentials ${\vec{A}}^{\prime }=\vec{A}+\nabla f$ and $V^{\prime }=V-\frac{\partial f}{\partial t}$ represent the same values of the fields $\vec{E}$ and $\vec{B}$ for any arbitrary smooth function of space and time $f$ .

The following transformation is therefore considered to be the gauge transformation of the theory:

$\begin{array}{c}\vec{A}\to \vec{A}+\nabla f,\\ V\to V-\frac{\partial f}{\partial t}.\end{array}$(2.5)

Under this transformation the field values do not change, and Maxwell’s equations therefore remain invariant; Maxwell’s equations possess a gauge symmetry.

According to the field interpretation mentioned earlier, the electric and magnetic potentials are devoid of ontological importance. Gauge invariance is therefore naturally interpreted as a manifestation of a redundancy in the way the potential represents the physical situation, rendering them as mere mathematical auxiliaries. However, future developments in quantum theory and particle physics gave three reasons to think that things may be more complicated. The first is the formal indispensability of the potentials in the theory. The second is the Aharonov–Bohm effect. (These first two reasons are described in the next subsection.) The third reason is that the property of gauge invariance can be promoted to a gauge principle, using which the equations that govern the interaction can be derived without appealing to prior knowledge of the classical limit. This method is the basis for the “symmetry dictates interaction” conception of contemporary field theories (Subsection 2.2.3). This profound significance of gauge invariance appears to many to conflict with the view that regards this invariance as a mere matter of mathematical redundancy (Section 2.3.1). This tension is further sharpened in the context of spontaneous gauge symmetry breaking (see Sections 3 and 4).

2.2.2 The Aharonov-Bohm Effect

In classical electromagnetism the potentials are nonmeasurable quantities whose local values are not well defined and do not form an essential part of the mathematical description of the dynamics that can be expressed in terms of the fields using Maxwell’s equations and the Lorenz force equation. Yet, there is a significant place in which they become indispensable, at least from a formal point of view, and this is the Hamiltonian (and also the Lagrangian) formulation of the theory. In quantum mechanics the Hamiltonian formulation gains fundamental significance as the generator of the temporal dynamics. The theory does not include force as a fundamental entity, but only as a derived phenomenon at a classical limit. Accordingly, it is the electric and magnetic potentials, and not the fields, which appear in the Hamiltonian and thus in the Schrödinger equation.

Aharonov and Bohm were intrigued by the question of whether this theoretical difference between the quantum and the classical can make an observational difference, and proposed an experiment in which it does. The proposed experiment is an electron interference experiment, in which a beam is split into two branches that are brought back together to form an interference pattern (Fig. 2.1).Footnote ⁷ The electric and magnetic fields along the possible trajectories are zero. In addition, a conducting solenoid is placed between the two branches in an area of space that is shielded from the beam (the wave function at the neighborhood of the solenoid is zero). The current in the solenoid induces a magnetic field inside it, but the foil ensures that there is no overlap between the support of the wave function in space-time and the magnetic field. The surprising fact is that in this scenario the interference pattern would depend on the magnetic field.Footnote ⁸

Figure 2.1 The Aharonov–Bohm experiment.

Figure taken from Reference Aharonov and BohmAharonov and Bohm (1959).

The dependence of the interference pattern on the field can be expressed in terms of a phase difference between the two branches that is acquired in the process. The following calculation of the phase factor emphasizes the role of gauge invariance. Let us begin with the case $\vec{A}=0$ everywhere and at all times, which, in a particular gauge, describes the experiment conducted with a zero magnetic field (we shall also use $\varphi =0$ in all cases). Let us denote the solutions of the Schrödinger equation for the first branch ABF and for the second branch ACF by ${\psi }_1^0\left(\vec{r},t\right)$ and ${\psi }_2^0\left(\vec{r},t\right)$ respectively. An interference pattern is obtained in the interference region, the overlap of the support of these two wave functions, and in this case its form is given by ${\left({\psi }_1^0\left(\vec{r},t\right)+{\psi }_2^0\left(\vec{r},t\right)\right)}^2$ .

The experiment is described by the Hamiltonian:

$H=\frac{1}{2m}{\left(\vec{p}-\frac{q}{c}\vec{A}\right)}^2+q\varphi .$(2.6)

It is invariant under the gauge transformation:

$\vec{A}(\vec{r},t)\to \vec{A}(\vec{r},t)-\nabla \Lambda (\vec{r},t),$(2.7)

for arbitrary smooth $\Lambda$ . This invariance is helpful for the calculation of the effect of the magnetic field. Let us denote by ${\vec{A}}^{*}\left(\vec{r},t\right)$ the magnetic vector potential that describes the situation with a given nonzero magnetic flux ${\Phi }_B$ . For simplicity, the choice of gauge is such that $A^{*}$ is time independent and that at point A we have ${\vec{A}}^{*}=0$ . The magnetic field along the first branch is zero. That means that there is a certain gauge transformation ${\Lambda }_1\left(\vec{r}\right)$ such that the potential ${\vec{A}}_1\equiv \vec{\nabla }{\Lambda }_1$ corresponds to a situation with zero magnetic field everywhere (i.e. $\vec{\nabla }\times {\vec{A}}_1=0$ at all places), but along the first branch ${\vec{A}}_1={\vec{A}}^{*}$ . Similarly, since at any point along the second branch the magnetic field is zero and $\vec{\nabla }\times {\vec{A}}^{*}=0$ , there exist a gauge transformation ${\Lambda }_2\left(\vec{r}\right)$ and a corresponding potential ${\vec{A}}_2\equiv \vec{\nabla }{\Lambda }_2$ such that $\vec{\nabla }\times {\vec{A}}_2=0$ at all places, but along the second branch ${\vec{A}}_2={\vec{A}}^{*}$ .

Now, any electromagnetic gauge transformation (2.7) is a symmetry of the Hamiltonian (2.6) when it is accompanied by the local phase transformation $\psi \to e^{iq\Lambda \left(\vec{r},t\right)/\hslash c}\psi$ . As a consequence, if we start from $\vec{A}=0$ everywhere and apply the gauge transformation defined by ${\Lambda }_1\left(\vec{r}\right)$ that transforms the vector potential into ${\vec{A}}_1$ , the wave function ${\psi }_1^0\left(\vec{r},t\right)$ would be transformed into $e^{i{\Lambda }_1\left(\vec{r}\right)}{\psi }_1^0\left(\vec{r},t\right)$ . But since ${\vec{A}}^{*}={\vec{A}}_1$ along the first branch, the wave function $e^{i{\Lambda }_1\left(\vec{r}\right)}{\psi }_1^0\left(\vec{r},t\right)$ would also be the wave function of the wave-packet that travels along the first branch for the case of nonzero magnetic field. Similarly, if we start from $\vec{A}=0$ everywhere and apply the gauge transformation defined by ${\Lambda }_2\left(r\right)$ , the wave function ${\psi }_2^0\left(\vec{r},t\right)$ would be transformed into $e^{iq{\Lambda }_2\left(\vec{r}\right)/\hslash c}{\psi }_2^0\left(\vec{r},t\right)$ . This expression would also hold for the wave function that travels along the second branch when the magnetic field is nonzero. Therefore, in this case the interference pattern would be given by  ${\left(e^{iq{\Lambda }_1\left(\vec{r}\right)/\hslash c}{\psi }_1^0\left(\vec{r},t\right)+e^{iq{\Lambda }_2\left(\vec{r}\right)/\hslash c}{\psi }_2^0\left(\vec{r},t\right)\right)}^2={\left({\psi }_1^0\left(\vec{r},t\right)+e^{iq\left({\Lambda }_2\left(\vec{r}\right)-{\Lambda }_1\left(\vec{r}\right)\right)/\hslash c}{\psi }_2^0\left(\vec{r},t\right)\right)}^2$ .

The preceding definitions imply that at any point along the first branch ${\Lambda }_1\left(\vec{r}\right)={\int }_A^r{A}_1\left(\overrightarrow{r^{\prime }}\right)\cdot \overrightarrow{d{r}^{\prime }}$ , where the integral is taken from point $A$ along the path of the particle in the first branch. But since ${\vec{A}}_1$ equals ${\vec{A}}^{*}$ in that region, we get ${\Lambda }_1\left(\vec{r}\right)={\int }_A^r{A}^{*}\left({\vec{r}}^{\prime }\right)\cdot {\overrightarrow{dr}}^{\prime }$ . Similarly, along the second branch, ${\Lambda }_2\left(\vec{r}\right)={\int }_A^r{A}^{*}\left({\vec{r}}^{\prime }\right)\cdot {\overrightarrow{dr}}^{\prime }$ . In the interference region, the phase difference that is responsible for the change in the interference pattern is given by the difference between these two phase factors, which is exactly

$\Delta {\phi }_{AB}=\frac{q}{\hslash c}\oint A^{*}\left(\vec{r}\right)\cdot \overrightarrow{dr},$(2.8)

where the loop integral is over the closed loop ABFCA.

While this calculation was performed in a specific gauge, its final outcome is gauge independent. According to Stokes’ theorem, the loop integral equals the magnetic flux, such that the phase difference is $\Delta {\phi }_{AB}=q{\Phi }_B/\hslash c$ .

At its core, the effect seems to present a dilemma between local action of the gauge-dependent potential and nonlocal action of the field. Thus, the effect gave rise to new controversies concerning both the foundations of electromagnetism and those of quantum theory, and it also substantially outlines the interpretational possibilities in the context of gauge theories. One possible conclusion is that locality and gauge invariant ontology cannot be reconciled. A related issue is the role of topology in physics, which is often presented as the main foundational issue associated with the Aharonov–Bohm effect. The point is that it is possible to gauge away the potential in the domain of each of the two paths ABF and ACF, but not in the union of the domains, due to the fact that the latter domain is not simply connected.Footnote ⁹ This dependence of the effect on the topology is described (e.g., by Reference RyderRyder, 1996) as the simplest demonstration for the rich mathematical structure associated with the vacuum of field theories. Reference Nounou, Brading and CastellaniNounou (2003) portrays this description as underlying a distinct interpretational approach to the effect, an approach that can be extended to an interpretational approach toward gauge theories. We discuss these interpretational possibilities in more detail in Section 2.3.3.

2.2.3 The Gauge Principle

In the early twentieth century the gauge invariance of classical electromagnetism was not considered as bearing any fundamental significance. An original attempt to rethink the issue was made by Reference WeylWeyl (1918),Footnote ¹⁰ who aimed for a geometrical unification of gravitation and electromagnetism based on an extension of Einstein’s general theory of relativity, introducing the term “gauge transformations” for the first time.Footnote ¹¹ Weyl saw great importance in the notion of locality that is expressed in general relativity in the dependence of the metric on space-time points. However, the geometry of the theory does not fully manifest the desired form of locality, due to the invariance of the inner product under parallel transport, which defines a global standard of length. Relaxing this condition introduces (in a modern terminology) a connection $\varphi$ of the local scaling group, and a corresponding local scale factor $\lambda$ for the transformation of the metric $g\left(x\right)\to \tilde{g}\left(x\right)=\lambda \left(x\right)g\left(x\right)$ . Weyl then identified the components of the connection 1-form $\varphi$ with the components of the electromagnetic four-potential. Thus, the curvature of the scaling connection was identified with electromagnetism in the same way the curvature of the Levi–Civita connection is associated with gravity. The empirical adequacy of this identification was soon criticized by Einstein and others – see Reference O’RaifeartaighO’Raifeartaigh (1997).

In Weyl’s theory the length scale acquires a nonintegrable measure factor along a trajectory in space-time. Just after the introduction of quantum mechanics, Reference LondonLondon (1927) suggested to reinterpret Weyl’s theory based on the observation that the quantum phase factor can be seen as an imaginary version of Weyl’s measure factor.

This discrepancy suggested that the relationship revealed through the concept of gauge between electromagnetism and gravity is that of an analogy rather than a straightforward unification. This analogy was the basis of Weyl’s gauge principle – see Reference WeylWeyl (1929a, Reference Weyl1929b). In these papers Weyl presented a formulation of general relativity using tetrads (which had recently been used in Einstein’s theory of distant parallelism, rejected by Weyl) and used this formulation to emphasize similarities between the structure of gravity and electromagnetism. Weyl noted that the metric does not fully determine the tetrad: there is a freedom of local Lorentz transformations. Weyl’s theory is based on 2-spinors, whose rotation (in internal space) can be regarded as a representation of the same Lorentz group. The tetrads that define distant-parallelism thus determine not only space-time curvature and the connection but also those of the spinors’ internal space. The important point is that the choice of tetrad does not fully determine the state of the spinors, as there remains a freedom of a phase factor. By analogy to the gravitational case, this freedom should be manifested as an invariance under local phase transformations.

This desired local invariance motivated the introduction of a covariant derivative that includes a local quantity $f$ (the connection term in the covariant derivative) such that the action is invariant under the transformation:

$\begin{array}{ccc}\psi \to e^{i\lambda \left(x\right)}\psi & & f_p\to f_p-\frac{\partial \lambda }{\partial x_p}.\end{array}$(2.9)

Weyl then notes that the resultant $f$ term in the action is identical to “the manner $\dots$ that the electromagnetic potential interacts with matter according to experiment. This justifies the identification of the quantities $f_p$ introduced here with the electromagnetic potentials.” Weyl then identifies the electromagnetic field $f_{pq}=\frac{\partial f_q}{\partial x_p}-\frac{\partial f_p}{\partial x_q}$ . He further notes the connection of the transformation to conservation of electric charge and stresses the analogy to the connection between conservation of momentum and angular momentum in general relativity and the invariance under local Lorentz transformations (rotation of the tetrads in space-time). Weyl thus regards the gravitational interaction and the electromagnetic interaction as manifesting the same new principle of gauge invariance (see also Reference WeylWeyl, 1929b).

Weyl’s gauge terminology with respect to the electromagnetic interaction was embraced by Reference PauliPauli (1941), Reference Pauli(1980) in his influential writings on quantum physics. It allowed for a reconstruction of the electromagnetic interaction in a quantum context from simple principles that do not appeal to classical electromagnetism. For example, the electromagnetic interaction term is introduced into the free Dirac equation $i{\gamma }^{\mu }{\partial }_{\mu }\psi -m\psi =0$ using the gauge principle by replacing the derivative ${\partial }_{\mu }$ with a gauge covariant derivative $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$ . The resulting equation $i{\gamma }^{\mu }\left({\partial }_{\mu }+ie{A}_{\mu }\right)\psi -m\psi =0$ is invariant under local gauge transformations

$\begin{array}{ccc}\psi \to e^{-i\lambda \left(x\right)}\psi & & A_{\mu }\to A_{\mu }+\frac{1}{e}{\partial }_{\mu }\lambda \left(x\right).\end{array}$(2.10)

The analogy with the electromagnetic case motivated the successful attempt of Reference Yang and MillsYang and Mills (1954) to localize the $SU\left(2\right)$ isospin symmetry. The theory involves two Dirac spinors of equal mass that can be described using the Lagrangian $ℒ=i\overline{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi -\overline{\psi }m\psi$ with $\psi \equiv \left(\begin{array}{c}{\psi }_1\\ {\psi }_2\end{array}\right)$ . It is initially invariant under a global $SU\left(2\right)$ isospin symmetry. The Yang–Mills field $B_{\mu }$ (a $2\times 2$ matrix) is similarly introduced by replacing the derivative with a corresponding covariant derivative that renders the theory invariant under local $SU\left(2\right)$ transformations.

The significance of such gauge transformations is stressed through the fact that the invariance of the Lagrangian (or rather of the action) under some Lie group implies, via the famous Noether’s 1918 theorems (Reference Kosmann-Schwarzbach, Read and TehKosmann-Schwarzbach, 2022, Reference Kosmann-Schwarzbach and Schwarzbach2011), the conservation of associated physical charges. This was actually very significant in fueling Weyl’s enthusiasm regarding gauge invariance, from the very first inception of the concept in his 1918–19 papers.Footnote ¹³ This gives rise to an elegant symmetry argument “explaining” why the electric charge is conserved, in the same way that the energy-momentum tensor is conserved as a consequence of coordinate invariance.Footnote ¹⁴

Gauge theories of the Yang–Mills type were soon recognized as easy to renormalize. This provided a central motivation to pursue this line (Reference BrownBrown, 1993; Reference KibbleKibble, 2015). Thus, while Yang and Mills’s original theory failed to describe the strong nuclear interaction it was trying to account for, the method it presented soon became a template used in the construction of the theory of electroweak interactions (see Chapter 3) as a gauge theory of the group $SU\left(2\right)\times U\left(1\right)$ , and later also in the construction of the strong interaction. The development of the standard model was in this sense based on applying the same pattern of achieving local invariance by introducing gauge fields (Reference MillsMills, 1989). A retrospective description of these developments is given by Reference O’RaifeartaighO’Raifeartaigh (1997, emphasis in original):

Gauge theories obtained their modern geometric formulation from the late 1950s and on. In this formulation gauge fields are represented by differential forms that define connections of principle bundles: These are spaces for which to each point of space-time is attached a copy of an “internal” homogeneous space (the fiber) whose group of motions is the symmetry Lie group $G$ of the gauge theory – for example, in Maxwell’s theory, the fiber at each point is a circle along which one moves with the group $U\left(1\right)$ of rotations. The concept of gauge is often understood as being beyond a heuristic one, as a basic framework for fundamental field theories. This terminology and its application to the formulation of gauge theories is presented in Appendix A.

The gauge heuristics can be described in this language in the following way. Usually, one starts from a theory of free matter fields (without mutual interactions), say $\psi$ , which is only invariant under the action of the rigid, or “global,” Lie group $G$ : that is, $\psi \mapsto {\psi }^g:=\rho (g{)}^{-1}\psi$ , for $g\in G$ . This allows one to use their simple exterior derivatives to build the kinetic terms, which indeed transform likewise, $d\psi \mapsto d{\psi }^g=\rho (g{)}^{-1}d\psi$ : for example, the free Dirac Lagrangian $L_{\text{Free}}\left(\psi \right)=〈\psi ,\overline{)d}\psi 〉-m〈\psi ,*\psi 〉$ . Noticing then that the latter is not invariant under the substitution $g\to \gamma \in G$ , namely $L_{\text{Free}}\left(\psi \right)$ does not enjoy a local $G$ symmetry, one looks for the simplest modification of the free theory that does. This motivates the introduction of the potential $A$ minimally coupled to $\psi$ , together with its transformation (A-3) seen to compensate the inhomogeneous term arising from $d{\psi }^{\gamma }=d(\rho (\gamma {)}^{-1}\psi )$ . The rule of thumb being then that in the Lagrangian, one substitutes exterior derivatives by covariant exterior derivatives $d\mapsto D:=d+{\rho }_{*}\left(A\right)$ – thus named because $D\psi$ gauge transforms like $\psi$ : that is, $L_{\text{Free}}\left(\psi \right)\mapsto L_{\text{Dirac}}(\psi ,A)=〈\psi ,\overline{)D}\psi 〉-m〈\psi ,*\psi 〉$ . It is a theory of coupled matter fields whose interactions are mediated by a gauge potential  $A$ .

At this point, this gauge potential is still external as it has no dynamics of its own. To make it fully dynamical, one needs only to add a $G$ -invariant kinetic term involving a tensorial field built from $A$ : the most natural candidate is the field strength $F=dA+AA$ (as defined in Appendix A) and the simplest Lagrangian is the YM term in (A-8). The final theory $L_{\text{Final}}(\psi ,A)=L_{\text{YM}}\left(A\right)+L_{\text{Dirac}}(\psi ,A)$ is a $G$ -gauge theory of dynamical matter fields interacting via a dynamical gauge field.

It appears that by localizing the symmetry, $G\mapsto G$ , we have switched on the interaction field and transformed a free theory into a coupled theory, $L_{\text{Free}}\left(\psi \right)\mapsto L_{\text{Final}}(\psi ,A)$ . For $G=U\left(1\right)$ , one ends up with the Lagrangian for QED, $L_{\text{Final}}(\psi ,A)=L_{\text{QED}}(\psi ,A)$ , and for $G=SU\left(3\right)$ one ends up with the QCD Lagrangian $L_{\text{QCD}}(\psi ,A)$ . Thus, for the nongravitational interactions, the gold standard for the physics community is the fact that the gauge field theories for the nongravitational interactions can be combined and quantized to give a renormalizable QFT (see Section 2.2.4), the Standard Model (SM), with gauge group $U\left(1\right)\times SU\left(2\right)\times SU\left(3\right)$ , whose quantitative predictions are in agreement with experiments to an impressively high degree of precision.

In addition to the gauge symmetries in particle physics, the analogy with gravitation continued to play a role in various works aiming to go beyond general relativity in the description of gravity, or to provide a unified framework for all interactions. Reference UtiyamaRyoyu Utiyama (1956) fully articulated the modern understanding of the gauge argument in terms of “gauging” a global symmetry using a Lie group to obtain a gauge theory.Footnote ¹⁵ Utiyama applied his approach also to general relativity, showing for the first time that general relativity can be recovered from a gauge principle applied to the rigid Lorentz group.

The theory of Reference Brans and DickeBrans and Dicke (1961) generalized general relativity based on conformal transformations, similar to the ones presented in Reference WeylWeyl (1918). After the success of gauge theoretic approaches to the nuclear interactions in the mid to late 1960s and early 1970s, and after the inception of supersymmetry, explorations of gauge theories of gravity and supergravity blossomed in the second half of the 1970s (see Reference ScholzScholz, 2020 for a historical review and Reference Blagojević, Hehl and KibbleBlagojević, Hehl, & Kibble, 2013 for a detailed review of the theories). These theories describe gravity using different non-Riemannian geometries introduced through the process of gauging different groups larger than the Lorentz group. The aim is to bridge the language gap with the description of the other interactions so as to facilitate either their unification or the quantization of gravity.

In this framework, the gauge potentials of gravity were for a long time thought about in the same terms as Ehresmann connections on fiber bundles that describe nongravitational interactions. Only quite recently was it recognized that gauge gravity is better understood in terms of the geometry of Cartan connections on principal bundles; see, for example, Reference WiseWise (2009, Reference Wise2010). Cartan geometry is indeed the natural generalization of (pseudo) Riemannian geometry, as introduced by É. Cartan in the 1920s, and the direct precursor of the notion of connection introduced in the late 1940s by C. Ehresmann (who was a pupil of É. Cartan). See Reference Cap and SlovakCap and Slovak (2009) and Reference SharpeSharpe (1996) for modern introductions.

Interpretive issues surrounding gauge symmetries are thus pressing for gravitational and nongravitational theories alike. But we will focus on interpreting the gauge principle for nongravitational theories. In that respect, we will describe conceptual/philosophical aspects in Section 2.3.1.

2.2.4 Quantization

The advent of quantum field theory necessitated the reshaping of the understanding of gauge symmetries substantially. Especially, the role of gauge transformations moves from a transformation of solutions of the equations of motion to an integral part of the definition of the quantum theory. This also modifies, to some extent, how gauge transformations are perceived in a quantum field theory, as will be outlined here.

In principle, quantizing a gauge theory is performed similarly as with nongauge theories, but for a few subtleties. For the purpose of illustration, this will be done here using a path-integral approach. We select the vector potential as the integration variable.Footnote ¹⁶

A way (Reference Böhm, Denner and JoosBöhm, Denner, & Joos, 2001) to understand the origin of the ensuing subtleties when integrating over the vector potential is to note that any gauge transformation leaves the action invariant, and, as a shift, also does not influence the measure. Hence, there are flatFootnote ¹⁷ directions of the path integral, and thus the integral diverges when integrating along these directions.

There are only a few possibilities to deal with these divergencies. One is to perform the quantization on a discrete space-time grid in a finite volume. In this way the divergences become controllable and can be removed before taking the limit to the original theory (Reference Montvay and MünsterMontvay & Münster, 1994).

Another one is to transform to manifestly gauge-invariant variables. This would be achieved by a variable transformation to the dressed fields of Section 5. However, in most cases, this has been thus far found to be practically impossible.

The third option is to remove the divergences by fixing a gauge (Reference Böhm, Denner and JoosBöhm et al., 2001). This is achieved by sampling every gauge orbit only partially in such a way that the result is finite while gauge-invariant quantities are not altered. Even though gauge-variant information is removed, this is not equivalent to introducing a gauge-invariant formulation. Any gauge condition will define one distinct way of removing the superfluous degrees of freedom, but what is removed and what is left differs for every choice of gauge.

As gauge fixing plays a central role in contemporary particle physics, as well as in Section 6, it is worthwhile to detail it further. Gauge fixing proceeds by selecting a gauge condition $C_{\Lambda }$ , which may involve the gauge field as well as any other fields, and which can be parametrized by some quantity or function $\Lambda$ in an arbitrary way. Necessary conditions for this gauge condition are that every gauge orbit has at least one representative fulfilling this condition. We will furthermore assume, for reasons of practicality, that there is only one such representative. In non-Abelian gauge theories, the Gribov–Singer ambiguity makes this requirement very involved (Reference GribovGribov, 1978; Reference SingerSinger, 1978) to implement and formulate, even to the point of practical impossibility (Reference Lavelle and McMullanLavelle & McMullan, 1997; Reference MaasMaas, 2013; Reference Vandersickel and ZwanzigerVandersickel & Zwanziger, 2012). Conceptually, however, this will not be an issue for now.

The procedure for determining the amplitude of any gauge-invariant observable $f$ goes as follows:

$\frac{1}{N}{\int }_{\Omega }D{A}_{\mu }f\left(A_{\mu }\right)\exp \left(iS\left[A_{{}_{\mu }}\right]\right)$(2.11)

$\begin{array}{cc}& =\frac{1}{N}{\int }_{\Omega /{\Omega }_C}Dg{\int }_{{\Omega }_c}D{A}_{\mu }\Delta \left[A_{\mu }\right]\delta \left(C_{\Lambda }\right)f\left(A_{\mu }\right)exp\left(iS\right[A_{\mu }\left]\right)\end{array}$(2.12)

$\begin{array}{cc}& =\frac{1}{N^{\prime }}{\int }_{\Omega }D{A}_{\mu }\Delta \left[A_{\mu }\right]f\left(A_{\mu }\right)\delta \left(C_{\Lambda }\right)exp\left(iS\right[A_{\mu }\left]\right)\end{array}$(2.13)

$\begin{array}{cc}& =\frac{1}{N^{{}^{\prime \prime }}}{\int }_{{\Omega }_C}D{A}_{\mu }\Delta \left[A_{\mu }\right]f\left(A_{\mu }\right)exp\left(iS\right[A_{\mu }\left]\right),\end{array}$(2.14)

in which possible further fields are suppressed. The original expression (2.11) is an integral over the whole set $\Omega$ of all gauge orbits and all representatives on every gauge orbit. In (2.12) this set is split into the set ${\Omega }_C$ , which contains for all orbits only the representatives fulfilling $C_{\Lambda }$ , and the remainder $\Omega /{\Omega }_C$ . This allows us to write the integration along gauge orbits $g$ and over gauge orbits separately. This separation requires the introduction of a $\delta$ -function on the gauge condition and an additional weight factor, the Faddeev–Popov determinant $\Delta$ . The latter ensures that the weight of the representative is the same for all gauge orbits, and thus gauge-invariant, ensuring $⟨1⟩=1$ . Then the integrals along the gauge orbits are orbit-independent and can be absorbed in the normalization in (2.13). Finally, resolving the $\delta$ -function in (2.14) yields the gauge-fixed path integral, yielding yet another normalization. This approach is standard for perturbation theory (Reference Böhm, Denner and JoosBöhm et al., 2001).Footnote ¹⁸ As noted earlier, beyond perturbation theory the Gribov–Singer ambiguity makes this procedure in practice cumbersome. This manifests itself in the form and properties of both the Faddeev–Popov determinant $\Delta$ and the gauge condition $C_{\Lambda }$ (Reference Lavelle and McMullanLavelle & McMullan, 1997; Reference MaasMaas, 2013; Reference Vandersickel and ZwanzigerVandersickel & Zwanziger, 2012).

In principle, the Faddeev–Popov determinant is a simple functional extension of the following argument: for a function with a single root, $f\left(x_o\right)=0$ , the Dirac delta function obeys the following identity:

$\delta \left(f\right(x\left)\right)=\frac{\delta (x-x_o)}{|\det f\text{'}(x_o\left)\right|}.$(2.15)

Since the integral of $\delta (x-x_o)$ over $x$ gives unity, then

$|\det f\text{'}(x_o\left)\right|{\int }^{\text{}}dx\delta \left(f\right(x\left)\right)=1.$

The functional settic

$|\det J\text{'}({\phi }_o\left)\right|{\int }^{\text{}}D\phi \delta \left(F\right(\phi \left)\right)=1.$(2.16)

Usually the field $\phi$ is a gauge parameter, with (2.16) determining the factor for a gauge fixing.

Geometrically, the Faddeev–Popov determinant emerges as a functional Jacobian for a change of variables, since one now decomposes, as in Fubini’s theorem, a functional integration over all variables into an integration over a gauge-fixing section and an integration over the orbits (see e.g. Reference Babelon and VialletBabelon & Viallet, 1979 and Reference MottolaMottola, 1995). Of course, such a decomposition is also vulnerable to the Gribov ambiguity, and thus would not be available nonperturbatively. Nonetheless, this interpretation leads straightforwardly to the Faddeev–Popov determinant as follows: for the transformation $A_{\mu }\to A_{\mu }^{Cg}$ , where $A_{\mu }^C$ is the gauge-fixed connection, we obtain the respective Jacobian for the measure $DA\to det\left(J\right)D{A}^{C}Dg$ .Footnote ¹⁹

It is an interesting question to ask what happens if one tries to calculate a gauge-dependent amplitude. In fact, if done so by putting a gauge-dependent $f$ in (2.11), the answer is always zero, up to some $\delta$ -functions at coinciding arguments. Of course, such a calculation requires a method like the lattice regularization. The reason is that for any gauge field configuration with some value $A_{\mu }\left(x_0\right)$ at the fixed position $x_0$ , there exists a gauge transformation, which is only nonvanishing at $x_0$ , such that the value of the gauge-transformed gauge field is $-A_{\mu }\left(x_0\right)$ . In this way, any integration over the full gauge group yields zero. The only exception can happen if arguments coincide, yielding squares of the fields. On the other hand, evaluating a gauge-dependent quantity $f$ using (2.14) yields very nontrivial results.

The reason for this apparent disagreement is the step from (2.12) to (2.13). Here, the integral $\int Dg$ was absorbed in the normalization, because none of the remaining expressions depended on it, since all of them were gauge-invariant. This is no longer true, if $f$ is gauge-dependent. Then the integral over gauge transformations can no longer be separated as a factor and be removed.

Thus, from a purely mathematical point of view, the expressions (2.11–2.12) and (2.13–2.14) are distinct theories. From the point of view of physics, there is just an infinite number of equivalent quantum theories, the one without gauge fixing and the infinitely many choices of ${\Omega }_c$ , or equivalently $C_{\Lambda }$ , which all yield the same gauge-invariant observables, but differ for gauge-dependent ones.

Alternatively, this can also be taken to imply that any choice of theory with the action $S^{\prime }=S-iln\Delta$ gives the same gauge-invariant quantities, provided they are integrated over the corresponding set ${\Omega }_C$ , either directly implemented as integration range or by a $\delta$ -function. In either way, this leads ultimately to the expression (2.14).

This infinite degeneracy of quantum theories is a consequence of working with gauge freedom. This problem of infinite degeneracy would vanish if a transformation to gauge-invariant variables could be performed, as is aimed at in the dressing-field approach in Section 5. Alternatively, an approach like the one in Section 6 will take all these theories to be equivalent by defining the (gauge-invariant) observables to be only the ones that are the same for all. This yields the same observable result as the lattice regularization.

2.3.1 The Gauge Principle Meets Philosophy

In the late twentieth century, gauge symmetries began to attract the attention of philosophers. Many of them identified the gauge principle as a prominent example of a fundamental shift in the methodology of theoretical physics, intertwined with the elevation of abstract mathematics. Reference SteinerSteiner (1989; Reference Steiner1998) raised the issue in the context of the question of the applicability of mathematics to natural science. He regarded the gauge argument (especially in the version of Yang and Mills) as a Pythagorean analogy, namely one that can only be expressed in a mathematical language and is not based on a physical similarity. Such mathematical analogies, according to Steiner, are motivated by human values (such as aesthetics) that guide the development of mathematics, and their repeated success in physics puts physics at odds with naturalism.

A closely related issue is the apparent conflict between the standard understanding of gauge symmetries as a matter of a choice of convention and their great methodological significance. Yang and Mills described gauge freedom in terms of local conventions:

This view of gauge invariance became part of the received view, in part due to Reference WignerWigner (1967a), who emphasized the distinction between dynamical symmetries, such as gauge symmetries, and other symmetries such as Lorentz transformations. A philosophical view that adopts this approach was presented by Reference AuyangAuyang (1995) in her book that aimed to present quantum field theory in a Kantian categorical framework of objective knowledge. Reference TellerTeller (1997, Reference Teller2000) criticized this view, raising the question of “[h]ow can an apparently substantive conclusion follow from a fact about conventions?” (2000, p. 469).

This tension between the significance of gauge invariance as an essential part of the basis for the formulation of successful theories, and the appearance of gauge as essentially a manifestation of mathematical redundancy was dubbed by Reference Redhead, Kuhlmann, Lyre and WayneRedhead (2002) as “the most pressing problem in current philosophy of physics” (p. 299) and has been the subject of extensive philosophical inquiry. Many (Reference Martin, Brading and CastellaniMartin, 2003; Reference Norton, Brading and CastellaniNorton, 2003) have pointed out the similarity of the question to the controversial issue of the role of general covariance in general relativity (Reference NortonNorton, 1993), in which a mathematical constraint on the formulation of the theory is seen as fundamental to the construction of the theory.

One possible way to approach this question is to adopt a deflationary approach toward the gauge principle. Reference Brown, Butterfield and PagonisBrown (1999) noted that neither the curvature of the fiber bundle nor the back-reaction of the matter on the gauge field can be explained by the principle. Reference Martin, Brading and CastellaniMartin (2003) noted that in the gauge argument, the gauge principle is applied in conjunction with other principles such as Lorentz invariance, renormalizability, and simplicity. The requirement for a local symmetry does not by itself determine the form of the interaction, nor does it dictate its existence. While this criticism does amend misconceptions that appear in some textbooks, it does not claim to fully resolve the foundational and philosophical questions.

A recent interpretation of the gauge principle focuses on its methodological use in relation to Noether’s theorem: in Reference Gomes, Roberts and ButterfieldGomes et al. (2021), it is argued that, when constructing theories containing charges that are sources of fields that have their own dynamics – as happens in all known field theories – we have no way of ensuring that the conservation of charges is compatible with the fields’ dynamics, other than by explicit computation and trial and error. Unless, that is, the conservation of charges is associated to a rigid symmetry, through Noether’s first theorem. In that case, Reference Gomes, Roberts and ButterfieldGomes et al. (2021) show that converting that rigid symmetry into a malleable one enforces the compatibility between charge conservation and the dynamics of the corresponding fields. Thus, just as the GP guarantees that the equations governing the evolution of (for example) electromagnetic fields are compatible with the conservation of electric charge, so it would guarantee that the dynamics of any field are compatible with the conservation of its sources.

Other recent approaches seek for answers in the differential geometry of fiber bundles that underlies (classical) gauge field theory. (The valiant reader will find a short and dense overview of bundle geometry in Appendix B.) In this framework, gauge transformations arise in a manner very much analogous to coordinate changes in general relativistic physics: gauge transformations are seen to be generalized coordinate changes for an “enriched” space-time (the fiber bundle) whose points have an internal structure (the fibers). So, the gauge principle could be understood, like the principle of general covariance, as a principle of democratic epistemic access to the intrinsic geometry of an enriched space-time whose points are not structureless, and whose internal structure can be “probed” by the fundamental fields it contains.Footnote ²⁰ Embracing this as a satisfying explanation of the heuristic power of the GP means taking the mathematics seriously enough to accept that it may refer to an actual physical entity, therefore entertaining some degree of committment to the ontological character of the fiber bundle. See, for example, Reference CatrenCatren (2022) for a recent defence of this view.

Other approaches aim to relate the gauge heuristics to a more explicit physical content. One way to do so is by appealing to a notion of direct observability of gauge symmetries, in which the symmetry is restricted to observations performed on a certain subsystem (Section 2.3.4). A different approach (Reference LyreLyre, 2000, Reference Lyre2001, Reference Lyre2003; Reference MackMack, 1981) acknowledges gauge covariance as a formal requirement by itself and aims to account for its applicability by supporting it with an additional equivalence principle, formulated by analogy with Einstein’s presentation of general relativity based on general covariance and the equivalence principle. The principle roughly states that there is always a choice of gauge such that the gauge field vanishes at a given point.

Applying a generalized version of the equivalence principle can be beneficial in understanding gauge not only at the interpretational level but also at the level of heuristics. The “methodological equivalence principle” prescribes the introduction of an interaction based on an the way the dynamical law of a theory violates the invariance requirement. It provides a unified framework of understanding standard gauge theories, the use of general covariance in general relativity and tangent-space symmetries in gauge theories of gravity (Reference HetzroniHetzroni, 2020; Reference Hetzroni and ReadHetzroni & Read, 2023; Reference Hetzroni, Stemeroff, Posy and Ben-MenahemHetzroni & Stemeroff, 2023). What calls for an explanation is actually the local noninvariance of interaction-free theories (special relativity under general coordinate transformations, Dirac equation under change of local phase convention, etc.). The introduction of the gauge field (and similarly of the gravitational field) comes to provide a physical explanation. This approach is primarily motivated by existing evidence (supporting the locality of the preferred representations), but it can also motivate an interpretation of gauge theories as grounded in ontology of relational quantities, in harmony with a recent account by Reference RovelliRovelli (2014), and stressing a structural similarity between the gauge argument and Mach’s principle (Reference HetzroniHetzroni, 2021). The origin of the gap between gauge-independent physical phenomena and gauge-dependent theoretical representation is that while fundamental physical degrees of freedom amount to relations between pairs of physical entities, our theories use variables that refer to single objects, and not their relations.

2.3.2 Interpretation of Gauge Invariance

The appearance of a gauge symmetry in a given theory may have crucial implications for its possible interpretations and its ontology. The question of whether gauge symmetries reflect descriptive mathematical redundancy or some physical property is deeply entwined with issues of locality, separability, and determinism. Reconciling such theoretical virtues with each other (and in particular recall (D1)–(D3) introduced in Section 2.1.2) is a nontrivial matter.

Let us now introduce some terminology. We shall do so using the case study of electromagnetism (see Reference RicklesRickles, 2008: 48). Potentials related by the gauge transformation (2.5) (i.e., potentials that lead to the same fields) are gauge-equivalent. Gauge-equivalent configurations lie on the same gauge orbit. Accordingly, all potentials related by a gauge transformation are on the same gauge orbit. Our fields $\vec{E}$ and $\vec{B}$ remain constant under the gauge transformation (2.5), so they are gauge-invariant. The 4-vector potential $A^{\mu }$ , namely the gauge field, is gauge-variant. Many mathematically distinct potentials lie on the same gauge orbit and lead to the same field. The fact that it does not matter which vector potential from a given gauge orbit we choose to express the fields is referred to as gauge freedom.

Gauge freedom implies that our gauge theory is mathematically underdetermined: the values of the gauge field throughout space in a given moment of time together with the Maxwell equations do not determine the potentials in different times. In some interpretations, this underdetermination corresponds to a form of indeterminism. If we have a point $p_0$ on phase space as our starting point,Footnote ²¹ the system does not evolve into a state represented by a unique phase point $p_t$ ; instead “we now have an indeterministic time-evolution, with a unique $p_t$ replaced by a gauge orbit” (Reference Redhead, Kuhlmann, Lyre and WayneRedhead, 2002, 289). This is to say that if we take a gauge-variant quantity such as the vector potential to be physically real, and mathematically distinct vector potentials to be physically distinct, then not only is there no way to determine the state empirically, the state is also underdetermined by the equations of motion.

Of course, this problem of indeterminism vanishes if one’s realism is constrained to gauge-invariant quantities, namely by identifying all gauge-equivalent states with one physical state, an assumption that is often referred to as Leibniz Equivalence (see, for example, Reference Saunders, Brading and CastellaniSaunders, 2003). We may therefore make a very broad distinction between two kinds of interpretations of gauge theories. On the one hand, there are interpretations that deny Leibniz equivalence, therefore taking gauge transformations to be physical transformations, allowing for realist commitments to gauge variant quantities, and on the other, interpretations that adopt Leibniz equivalence. Such interpretations would usually regard gauge freedom as an expression of surplus mathematical structure and restrict realist commitments to gauge-invariant quantities.

Proponents of the former type of interpretations may argue for a one-to-one correspondence between points on phase space and physical states such that each point on phase space represents a distinct physical state. This would mean that different points within the same gauge orbit would correspond to physically distinct states. In our case of classical electrodynamics, this would mean that gauge-equivalent vector potentials would correspond to physically distinct states. These states would be empirically equivalent in the sense that no possible observation or measurement could reveal which gauge-equivalent potential is currently in effect. As Rickles puts it, “these physically distinct possibilities are qualitatively indistinguishable” (Reference RicklesRickles, 2008, 62). Accordingly, such interpretations that allow gauge-variant quantities to be physically real come with the great disadvantage of an awkward indeterminism that allows for physical quantities that are in principle unobservable. However, it is a virtue of such interpretations to avoid the existence of surplus mathematical structure that is typically considered a characteristic feature of gauge theories. Since mathematically distinct descriptions are related to physically distinct states, the problem of explaining surplus mathematical structure vanishes.

Proponents of the latter type of interpretations may argue for a many-to-one correspondence between points on phase space and physical states such that points within the same gauge orbit would correspond to one and the same physical state. In our case of classical electrodynamics, this would mean that vector potentials are unphysical and gauge-equivalent vector potentials would correspond to the same physical state. Accordingly, such interpretations that restrict realist commitments to gauge-invariant quantities come with the virtue of avoiding indeterminism and restricting realist commitments to quantities that are in principle measurable. However, such interpretations that opt for a many-to-one correspondence between mathematical descriptions and physical states still face the problem of dealing with all the mathematical redundancy. Many physicists and philosophers share the sentiment of Zee that “gauge theories are also deeply disturbing and unsatisfying in some sense: They are built on a redundancy of description” (Reference ZeeZee, 2010, 187). Instead, many would desire a more direct correspondence between the mathematics and the physics, such that every mathematical degree of freedom has observable effects.

2.3.3 Ontology of Gauge Theories

Interpretations of gauge theories are often presented as a response to the tension between gauge invariance and locality that is revealed in the Aharonov–Bohm effect (Section 2.2.2).

In their original paper, Aharonov and Bohm argued that

The radical suggestion that the actual physics behind electromagnetism is not gauge-invariant is supposed to maintain locality. (A straightforward way to interpret classical electromagnetism in this way is to revive the concept of the aether, and consider the vector potential as its velocity field, and the electric field as its acceleration. See Reference BelotBelot, 1998.)

The idea has several significant disadvantages. One problem is that the interpretation leads to the radical indeterminism discussed in the previous subsection. Such an interpretation is not necessarily local; it might also lead to a form of action at a distance that is more explicit than the nonlocality of the Aharonov–Bohm effect. An electric current that starts to flow, for example, can (depending on the actual gauge) immediately change the values of the vector potential throughout space. Furthermore, it is controversial whether the potentials can provide a truly local explanation for the Aharonov–Bohm effect, due to the issue of separability (Reference Eynck, Lyre and RummellEynck, Lyre, & Rummell, 2001; Reference HealeyHealey, 1997, Reference Healey1999; Reference MaudlinMaudlin, 1998). The potential’s approach was criticized by Aharonov himself, due to the unexplained gap it contains between the gauge-independent observed phenomena and their gauge-dependent theoretical description (Reference Aharonov and RohrlichAharonov & Rohrlich, 2008).

In light of the evident downsides of potentials’ ontology, it may be worthwhile to reconsider the standard field interpretation of electromagnetism and see if it can be adapted to account for the Aharonov–Bohm effect. Such an approach was presented by Reference DeWittDeWitt (1962), who concluded that

A different account of a nonlocal influence of fields is described by Reference Aharonov and RohrlichAharonov and Rohrlich (2008) (mainly Chapters 4–5) in terms of modular variables. Nevertheless, even in this nonlocal approach the potentials are an essential part of the theoretical description of the interaction.Footnote ²²

A third explanation was suggested by Reference Wu and YangWu and Yang (1975). It closes the gap between the observable properties and the postulated physical properties in a straightforward way. Wu and Yang noted that the actual observable quantity is not the AB phase difference $\Delta {\phi }_{AB}$ itself (Equation (2.8)), but rather the phase factor (or holonomy, or anholonomy):

${\Phi }_{AB}=exp\left(i\Delta {\phi }_{AB}\right)=exp\left(i\frac{q}{\hslash c}\oint A^{*}\left(\vec{r}\right)\cdot \overrightarrow{dr}\right).$(2.18)

Their approach, known today as the holonomies approach to the Aharonov–Bohm effect, is to promote this factor (defined for any nonintersecting closed curve in space-time) to a fundamental variable from which all other electromagnetic quantities can be derived. The description of electromagnetism based on holonomies is gauge-invariant and nonlocal. This kind of nonlocality is not a dynamical action at a distance, but a kinematic nonseparability. The electromagnetic field, Wu and Yang argue, “underdescribes electromagnetism,” while the phase (2.8) overdescribes it (p. 12). The phase factor (2.18) constitutes the variable that provides the complete description of electromagnetic phenomena. With the phase factors interpreted as a fundamental variable, the theory becomes nonseparable in a strong sense (Reference HealeyHealey, 2007): the physical processes are not supervenient on the assignment of local properties to space-time points. This interpretation was advocated by Reference BelotBelot (1998) as a fruitful interpretation and by Reference Eynck, Lyre and RummellEynck et al. (2001) because of the local action and measurability of the holonomies.

Forming an analogy with the ontology of space debate, Reference ArntzeniusArntzenius (2014) relates to holonomies interpretation as gauge relationism, which he contrasts with the fiber-bundle substantivalism he presents. The ontology in this approach is a literal reading of the fiber-bundle representation of gauge theories. It consists of a fiber bundle representing the possible states of a gauge field over a space-time manifold. The structure of the fiber (vector space, Abelian\non-Abelian group, etc.) determines the property of the interaction represented by the gauge fields, understood in terms of connections over the bundle. In the most literal reading, gauge transformations in these interpretations do reflect physical change (as in Reference MaudlinMaudlin, 1998). It is possible, however, to formulate a “sophisticated” version of this substantivalism, which denies the existence of possible worlds that are only connected by a gauge transformation. The main reason to favor substantivalist approaches to gauge, according to Arntzenius, is the availability and simplicity of theories formulated in terms of gauge-dependent variables, in contrast to the difficulties of constructing a dynamical theory in which holonomies are a fundamental variable.

2.3.4 Direct Empirical Significance

The debate about the ontological correlates of symmetries is sometimes linked to the question of which symmetries have “direct empirical significance” and which do not. The idea here is that if a symmetry is a mere descriptive redundancy, it cannot be said to have any direct empirical significance (though it may have indirect empirical significance: the very fact that the theory can be formulated in terms of these symmetries has specific further implications, e.g. conservation laws). In contrast, symmetries that connect states that are physically distinct yet empirically equivalent are said to have direct empirical significance.

This question is intimately related to the distinction between systems and subsystems. There is no physical point of view from which a state of the entire universe and a symmetry-related state could be distinguished empirically, and it is therefore common to regard symmetries of the entire universe as not carrying any empirical significance. For symmetries applied only to proper subsystems of the universe, the situation is different. Here the standard example is the thought experiment known as Galileo’s ship: the inertial state of motion of a ship is immaterial to how events unfold in the cabin but is registered in the values of relational quantities such as the distance and velocity of the ship relative to the shore.

The question at the center of the debate about direct empirical significance is whether the same holds for local symmetries as in gauge theories. The orthodox view, that local gauge symmetries do not carry empirical significance, has recently been argued to be derivable from three simple assumptions about direct empirical significance and physical identity of subsystem states (Reference FriederichFriederich, 2015, Reference Friederich, Massimi, Romeijn and Schurz2017); see Reference Murgueitio RamírezMurgueitio Ramirez (2021) for criticism of the assumptions used there. However, recently, dissonant voices have become more prominent, for example, Reference Greaves and WallaceGreaves and Wallace (2014), Reference TehTeh (2016), and Reference WallaceWallace (2022a, Reference Wallace2022b). The unorthodox view reproduces, for any subsystem, a common interpretation of asymptotic symmetries within physics: that any gauge transformation that preserves the state at the boundary and yet is not asymptotically the identity, acquires empirical significance.

But this transposition of asymptotic conclusions to compact subsystems has come under attack from many directions. For instance, in Reference GomesGomes (2021) counterexamples were shown to exist in vacuum electromagnetism; follow-up work gauge-fixes the local symmetry and shows that only global symmetries can carry empirical significance as subsystem transformations, and this occurs only when such global symmetries are associated with covariantly conserved quasi-local charges; see Reference Gomes, Read and TehGomes (2019b). Moreover, the common view of asymptotic symmetries can be recovered within such a gauge-fixed formalism (cf. Reference RielloRiello, 2020).

5.2.1 Residual Gauge Transformations

Let us indulge in a brief digression that is also a transition. In the BRST framework, infinitesimal gauge transformations are encoded in the BRST bigraded (in form and ghost degrees) differential algebra:

$\begin{array}{c}sA=-D^{A}v,sa=-{\rho }_{*}\left(v\right)a,\text{and}sv+1/2[v,v]=0,\end{array}$(5.5)

where $v$ the Lie $H$ -valued ghost field – which plays the role of the infinitesimal gauge parameter $\chi$ – and $s$ is the BRST differential increasing the ghost degree by 1. It satisfies $ds+sd=0$ and $s^2\equiv 0$ – due to the third relation defining its action on the ghost field. This relation is why $s$ is best interpreted geometrically as the de Rham derivative on the gauge group $H$ and $v$ as its Maurer–Cartan form (Reference Bonora and Cotta-RamusinoBonora & Cotta-Ramusino, 1983).

One shows that, at a purely formal level, the dressed variables satisfy a modified BRST ${}^{\text{u}}$ algebra: