2.1. The Basic Mathematical Setup of the Method

First, we need to lay the setup for a gauge theory. Consider a gauge theory on an m-dimensional space-time manifold (ℳ, g), whose gauge group of symmetry is $\mathscr{H}≔\{\gamma :ℳ\to H\}$, with H a Lie group with Lie algebra 𝔥, which by definition acts on itself via ${\gamma }_1^{{\gamma }_2}={\gamma }_2^{\mathrm{-1}}{\gamma }_1{\gamma }_2$.Footnote ⁴ The basic space of fields is $\Phi =\left\{A,F,\phi \right\}$, where F is the field strength (curvature $2\text{-form}\in {\Omega }^2\left(ℳ,𝔥\right)$) of the gauge potential A (connection $1\text{-form}\in {\Omega }^1\left(ℳ,𝔥\right)$) and φ is a matter field pertaining to a representation (ρ, V) of H. The gauge group acts on the space of fields, $\Phi \stackrel{\mathscr{H}}{\to }{\Phi }^{\gamma }$, as

(1)$\begin{array}{}& \begin{array}{cc}{A}^{\gamma }& ={\gamma }^{\mathrm{-1}}A\gamma +{\gamma }^{\mathrm{-1}}d\gamma ,F^{\gamma }={\gamma }^{\mathrm{-1}}F\gamma ,\\ {\phi }^{\gamma }& =\rho \left({\gamma }^{\mathrm{-1}}\right)\phi ,\text{\hspace{1em}\hspace{1em}and\hspace{1em}\hspace{1em}}D{\phi }^{\gamma }=\rho \left({\gamma }^{\mathrm{-1}}\right)D\phi \text{,}\end{array}\end{array}$

where $D≔d+{\rho }_{*}\left(A\right)$ is the covariant derivative implementing the minimal coupling between the matter field and the gauge potential.

Now, a physical theory is specified by its Lagrangian m-form L(A, φ). In the case of a gauge theory, the Lagrangian is required to be ℋ-gauge invariant: $L(A^{\gamma },{\phi }^{\gamma })=L(A,\phi )$. A prototypical and almost minimal Yang-Mills Lagrangian is

(2)$\begin{array}{}& \begin{array}{cc}L(A,\phi ,{\phi }^{\prime })& =L_{\mathrm{YM}}+L_{\mathrm{scalar}}+L_{\mathrm{Dirac}}\text{,}\\ & =\frac{1}{2}\mathrm{Tr}(F\wedge *F)+〈D\phi ,*D\phi 〉-m^2〈\phi ,*\phi 〉+〈{\phi }^{\prime },\gamma \wedge *D{\phi }^{\prime }〉-m^{\prime }〈{\phi }^{\prime },*{\phi }^{\prime }〉\text{,}\end{array}\end{array}$

where φ is a scalar field with mass m and φ′ is a spinor field with mass m′. Here ∧ is the wedge product of differential forms, $*:{\Omega }^p\left(ℳ\right)\to {\Omega }^{m-p}\left(ℳ\right)$ is the Hodge star operator, while Tr and 〈 , 〉 are bilinear forms on 𝔥 and V, respectively. As for $\gamma ={\gamma }_{\mu }{\mathrm{dx}}^{\mu }$, it is a 1-form whose components are Dirac gamma matrixes. A mass term for the gauge potential A, ${\mu }^2\mathrm{Tr}(A\wedge *A)$, failing to be gauge invariant by virtue of (1), is forbidden so that a gauge interaction is a priori long range.

The core idea of the dressing field method consists in the following simple observation. Suppose the structure group H has some normal subgroup J, so that the gauge group ℋ correspondingly has a normal subgroup 𝒥.Footnote ⁵ Suppose further that from the space of fields Φ of a gauge theory, one can extract a field $u:ℳ\to J$ defined by the transformation property under $\gamma \in 𝒥:u^{\gamma }={\gamma }^{\mathrm{-1}}u$. Such a field is called a dressing field. It allows one to perform a change of field variables, $\Phi \to {\Phi }^u$, by forming the following composite fields:

(3)$\begin{array}{}& \begin{array}{cc}{A}^u& ≔u^{\mathrm{-1}}\mathrm{Au}+u^{\mathrm{-1}}\mathrm{du},F^u=u^{\mathrm{-1}}\mathrm{Fu},\\ {\phi }^u& ≔\rho \left(u^{\mathrm{-1}}\right)\phi ,\text{\hspace{1em}\hspace{1em}and\hspace{1em}\hspace{1em}}{D}^u{\phi }^u=\rho \left(u^{\mathrm{-1}}\right)D\phi ,\end{array}\end{array}$

where $D^u≔d+{\rho }_{*}\left(A^u\right)$. These fields are 𝒥-invariant variables. Notice that despite a formal similarity with equation (1), the fields in equation (3) are not gauge transformations. Indeed, by virtue of its defining transformation property, the dressing field is not an element of the gauge group: $u\notin \mathscr{H}$.

Taking advantage of the ℋ-gauge invariance of the Lagrangian, which holds as a strictly formal property, and of the formal similarity between (1) and (3), we can rewrite the gauge theory in terms of the 𝒥-invariant variables:

$\begin{array}{cc}L\left(A,\phi ,{\phi }^{\prime }\right)& =L\left(A^u,{\phi }^u,{\phi }^{\prime u}\right),\\ & =\frac{1}{2}\mathrm{Tr}\left(F^u\wedge *F^u\right)+〈D^u{\phi }^u,*D^u{\phi }^u〉-m^2〈{\phi }^u,*{\phi }^u〉+〈{\phi }^{\prime u},\gamma \wedge *D^u{\phi }^{\prime u}〉-m^{\prime }〈{\phi }^{\prime u},*{\phi }^{\prime u}〉.\end{array}$

This theory is therefore not a ℋ-gauge theory, as the Lagrangian (2) would have us think, but a ℋ/𝒥-gauge theory. Clearly the transformation of the composite fields in (3) under the residual ℋ/𝒥-gauge symmetry depends on the behavior of the dressing field under this same symmetry. Interesting cases are described in Attard et al. (Reference Attard, François, Lazzarini, Masson and Kouneiher2018). Insofar as the genuine physical degrees of freedom (DOF) of a gauge theory are given by gauge-invariant quantities, the dressing field method helps to exhibit the physical content of a gauge theory.

A question naturally arises: Is the field redefinition in (3) harmless? As observed above, formally, the dressing operation exactly mimics gauge transformations in (1) that form a symmetry of the theory. So, it is expected that at the classical level the theories expressed with the original variables and the dressed ones are equivalent: the field equations for the composite fields {A^u, φ^u, φ′^u} obtained from L{A^u, φ^u, φ′^u} are the same as those of the original gauge variables {A, φ, φ′} obtained from L(A, ϕ, ϕ′). Both are in a relation of “covariance” with respect to (WRT) the dressing operation by u, in the same way that the field equations of gauge related fields {A, φ, φ′} and {A ^γ, φ^γ, φ′^γ} are in a relation of covariance WRT the action of the gauge group ℋ.

For the same reason, at the quantum level the redefinition in (3) should be harmless if the gauge symmetry is preserved in the quantized theory, that is, if it is free of gauge anomalies, which is the case for any viable candidate as a fundamental QFT (Bertlmann Reference Bertlmann1996). Thus, if a gauge QFT is anomaly free, it is equivalent to its dressed counterpart.Footnote ⁶

Now, even considering a quantized effective theory in which the gauge symmetry is broken, the “harm” done by the field redefinition in (3) is arguably still under control. An outline of the argument is as follows. In an effective QFT, the gauge symmetry breaking anomaly belongs to the cohomology of the BRST (Becchi-Rouet-Stora-Tuytin) nilpotent operator s, whose action on the gauge variables reproduces their infinitesimal gauge transformations, but with the infinitesimal gauge parameter replaced by the Fadeev-Popov ghost field v (formally one could write $\gamma =1+\upsilon$ in eq. [1]). The nilpotency, $s^2=0$, is secured via the relation $\mathrm{sv}=-(1/2)[v,v]=-v^2$ (the commutator is graded WRT the form and ghost degrees; v has ghost degree 1). An anomaly is thus an n-form 𝒜(v, A) linear in v, which is s-closed, $s𝒜\left(v,A\right)=0$, but not s-exact, $𝒜\left(v,A\right)\ne \mathrm{sB}\left(A\right)$. It is determined algebraically through the Stora-Zumino descent equations and receives no contribution from perturbation theory beyond one loop. Thus, in complete analogy, one could define a “mock BRST operator,” $\overline{s}$, whose action on the gauge variables produces the infinitesimal version of the composite fields. Formally, one could write $u=1+\upsilon$ in (3), where the infinitesimal dressing field satisfies $\overline{s}\upsilon =-(1/2)[\upsilon ,\upsilon ]=-{\upsilon }^2$, so that ${\overline{s}}^2=0$. The difference between the effective QFT and its dressed counterpart would be controlled by the “mock anomaly” $\overline{𝒜}\left(\upsilon ,A\right)$ linear in v, belonging to the $\overline{s}$-cohomology, obtained algebraically via analogues of the Stora-Zumino descent equations and receiving no contribution beyond one loop from perturbation theory.

2.2. Artificial versus Substantial Gauge Symmetry

Because of the many good properties of gauge theories (notably in relation with renormalizability), over time physicists have devised various ways to implement a gauge symmetry in a theory lacking it (Pitts Reference Pitts2008). The Stueckelberg trick is the forefather of these, and its generalization seems to be of some relevance still to contemporary studies (Ruegg and Ruiz-Altaba Reference Ruegg and Ruiz-Altaba2004). It is easily illustrated with the historic example of the Proca model for massive electromagnetism. Proca’s Lagrangian is

(4)$\begin{array}{}& L\left(A\right)=\frac{1}{2}F\wedge *F+{\mu }^{2}A\wedge *A\end{array}$

and describes a massive vector field, so that the theory has no 𝒰(1)-gauge symmetry. Now, it is possible to implement such a gauge symmetry by following the suggestion of Stueckelberg to add a compensating field, the Stueckelberg field B satisfying $B^{\gamma }=B-\mu \theta$, while the vector field becomes a gauge field transforming as $A^{\gamma }=A-d\theta$, with $\gamma =e^{i\theta }\in 𝒰\left(1\right)$. A minimal Stueckelberg Lagrangian is, then,

(5)$\begin{array}{}& L(A,B)=\frac{1}{2}F\wedge *F+{\mu }^2(A-\frac{1}{\mu }\mathrm{dB})\wedge *(A-\frac{1}{\mu }\mathrm{dB})\end{array}$

and is a 𝒰(1)-gauge theory.

In spite of what would be inferred from a superficial reading, the Lagrangians (4) and (5) actually describe the same theory. Indeed the 𝒰(1)-gauge symmetry is artificial, its presence being compensated by the DOF of the field B. In the case at hand, the Stueckelberg field is actually an abelian dressing field: $u≔e^{i/\mu B}$, so that $u^{\gamma }=e^{i/\mu (B-\mu \theta )}={\gamma }^{\mathrm{-1}}u$. The associated 𝒰(1)-invariant composite fields are then $A^u≔A+{\mathrm{iu}}^{\mathrm{-1}}\mathrm{du}=A-(1/\mu )\mathrm{dB}$, and the field strength $F^u=F$. So the Stueckelberg Lagrangian (5) is rewritten in terms of gauge-invariant composite fields as

(6)$\begin{array}{}& L(A,B)=L\left(A^u\right)=\frac{1}{2}{F}^u\wedge *F^u+{\mu }^2{A}^u\wedge *A^u\text{,}\end{array}$

which is nothing but the Proca Lagrangian, devoid of any gauge symmetry. One may think of the dressing field method as a reciprocal to the Stueckelberg trick: the latter aims at implementing an artificial gauge freedom, the former seeks to erase it to reveal the gauge-invariant content.

The above simple discussion is illustrative of an important point: if one can find in a gauge theory a local dressing field, meaning that its value at a space-time point depends only on this point and no others,Footnote ⁷ then the invariant composite fields in terms of which the theory can be rewritten are local variables. So, one pays no price in erasing the gauge symmetry, which is then fully dispensable. I therefore propose the following operational criterion:

Gauge theories present a number of conceptual as well as technical challenges. Among those is the fact that the gauge variables have a nondeterministic evolution, and the hindrance gauge symmetry poses a priori to the quantization of the theory. Dirac has pondered long and hard about these difficulties in the context of electromagnetism, an abelian 𝒰(1)-gauge theory. A solution he first proposed in a 1955 paper (DiracReference Dirac1955) and then developed in 1958 (Dirac Reference Dirac1958, sec. 80) was to reformulate the theory with gauge-invariant variables, which would qualify as physical variables.

In what follows, we use essentially the notations of Dirac (Reference Dirac1955) while setting all fundamental constants to 1. Let ψ be the electron spinor field and $A=\left(A_0,A_r\right)$ the EM gauge potential, subject to the 𝒰(1)-gauge transformations ${\psi }^{\prime }=e^{\mathrm{iS}}\psi$ and $A^{\prime }=A+\mathrm{dS}$. Dirac introduces the new variables ${\psi }^{*}=e^{\mathrm{iC}}\psi$ (eq. [16]) and the associated “covariant” derivative $d{\psi }^{*}-{\mathrm{iA}}^{*}{\psi }^{*}=e^{\mathrm{iC}}(d\psi -\mathrm{iA}\psi )$, with $A^{*}=A+\mathrm{dC}$ (eq. [21] and below). The phase factor is defined by $C\left(x\right)=\int c_r(x,x^{\prime })A^r\left(x^{\prime }\right)d^3{x}^{\prime }$, and in order for the new variables to be gauge invariant c_r(x, x′) must satisfy $(\partial /\partial {x^{\prime }}_r)c_r(x,x^{\prime })=\delta (x-x^{\prime })$ (eq. [18]). Dirac then notices that his equation (18) admits the Coulomb potential as a solution,Footnote ⁸ so that by proceeding with the quantization of the EM theory written in terms of his invariant variables, he interprets ψ* in the following way: “We can now see that the operator ψ*(x) is the operator of creation of an electron together with its Coulomb field, or possibly the operator of absorption of a positron together with its Coulomb field. It is to be contrasted with the operator ψ(x), which gives the creation or absorption of a bare particle. A theory that works entirely with gauge-invariant operators has its electrons and positrons always accompanied by Coulomb fields around them, which is very reasonable from the physical point of view” (657). An appealing conclusion indeed.

It is not hard to see that Dirac’s scheme is an instance of the dressing field method. Indeed, under gauge transformation of the gauge potential, the phase factor transforms as

$\begin{array}{cc}{C}^{\prime }\left(x\right)& =\int c_r(x,x^{\prime })A^{\prime r}\left(x^{\prime }\right)d^3{x}^{\prime }=C\left(x\right)+\int c_r(x,x^{\prime })\frac{\partial S}{\partial {x^{\prime }}_r}\left(x^{\prime }\right)d^3{x}^{\prime },\\ & =C\left(x\right)-\int \frac{\partial }{d{x^{\prime }}_r}{c}_r(x,x^{\prime })S\left(x^{\prime }\right)d^3{x}^{\prime }=C\left(x\right)-S\left(x\right)\text{.}\end{array}$

So $u=e^{\mathrm{iC}}$ transforms under $\gamma =e^{iS}\in 𝒰\left(1\right)$ as $u^{\prime }=\gamma u$ and is therefore an abelian dressing field, which means that ψ* and A* in Dirac’s equations (16) and (21) are abelian instances of the composite fields φ^u and A^u in (3) above.

Should we then conclude, on the basis of the criterion C1, that Dirac has revealed the 𝒰(1)-gauge symmetry of electromagnetism to be artificial? We must resist that conclusion because here, contrary to what happened in the Stueckelberg model, gauge invariance was not free; it could be achieved only at the price of locality. Indeed the dressing field $u=e^{\mathrm{iC}}$ is clearly nonlocal, so that the gauge-invariant composite fields ${\psi }^{*}={\psi }^u$ and $A^{*}=A^u$ in terms of which the theory is rewritten are also nonlocal variables. It appears then that in classical or quantum electrodynamics, there is a trade-off between gauge symmetry and locality: either one works with local gauge variables or with nonlocal gauge-invariant ones.

This conclusion extends to nonabelian Yang-Mills theories. A beautiful articulation is provided by Healey (Reference Healey2009), who argues that the physical content of gauge theories is best represented by the path-ordered trace holonomies of the connection, also known as Wilson loops, which are gauge-invariant nonlocal variables. The trade-off of gauge invariance versus locality is indeed a characteristic feature of genuine gauge theories (Guay Reference Guay2008; Dougherty Reference Dougherty2017; Nguyen et al. Reference Nguyen, Teh and Wells2018), so that one may argue that what is probed, indirectly, by a substantial gauge symmetry is the existence of nonlocal physical phenomena. In complement to C1, I therefore propose the following criterion:

In principle there is an asymmetry: it is easier to verify C1 than C2, since for the latter it seems hard to be sure to have exhausted all possibilities in any given case. But an important point should be stressed at this stage: the dressing field must be constructed out of the space of primary gauge variables of the theory. This means that in most relevant cases, the range of options is limited, and for all practical purpose the degree of confidence of having verified C2 can be quite high. This also highlights an obvious but crucial fact, namely, that the verdict on the nature of the gauge symmetry of a theory depends on its field content.

As a manner of illustration, consider the AB effect (Aharonov and Bohm Reference Aharonov and Bohm1959, Reference Aharonov and Bohm1961). We recall that one setup of the effect is a modified double slit experiment involving electrons, ψ, where a solenoid stands behind the first screen between the two slits. When a current traverses the solenoid, the interference pattern formed by the electrons on the second screen is shifted because of a phase factor depending only on the flux of magnetic field inside the solenoid: $e^{i{\int }_c^A}=e^{i{\int }_s^F}$ (c is a closed path from the source of the electron beam through the two slits to a point on the final screen, enclosing the surface s traversed by the solenoid). Yet, outside the solenoid—the only region accessible to the electrons—the EM field strength vanishes, $F=0$, and only the EM potential A is nonzero. So, the potential is the only local variable that is available to maintain a semblance of explanation of the alteration of the behavior of the electrons via a local interaction between two fields, A and ψ.

Of course the gauge noninvariance of A makes it a doubtful candidate as a genuine physical field, as many physicists and philosophers alike have pointed out. Curiously, it is not often stressed that it is also true for the spinor field ψ, seen either as a wave function for electrons or as the electron quantum field. Both field variables A and ψ should then be equally faulted for the difficulty in interpreting the AB effect in terms of local interactions of physical fields. Therefore, several authors did not shy away from concluding that the AB effect forces us to accept not the physicality of the gauge potential A (which was usually seen as a computational device in classical electromagnetism) as Aharonov and Bohm argued but rather that there are such things as nonlocal EM properties represented by gauge-invariant nonlocal variables.Footnote ⁹

But as pointed out by Wallace (Reference Wallace2014), in the semiclassical framework where ψ is a complex scalar field, a gauge-invariant local interpretation for the AB effect is available. As the reader now acquainted with the dressing field method will recognize, Wallace extracts a local dressing field from the polar decomposition $\psi =\rho u$, with $\rho =\left|\psi \right|$ and $u=e^{i\theta }$, since indeed ${\psi }^{\gamma }={\gamma }^{\mathrm{-1}}\psi$ implies $u^{\gamma }={\gamma }^{\mathrm{-1}}u$. He then proceeds to show that the theory can be expressed in terms of the gauge-invariant local composite fields ${\psi }^u=u^{\mathrm{-1}}\psi =\rho$ and $A^u=u^{\mathrm{-1}}\mathrm{Au}-{\mathrm{iu}}^{\mathrm{-1}}\mathrm{du}=A+d\theta$ (called “gauge-covariant derivative of θ” and noted Dθ in Wallace [Reference Wallace2014], 10), with field strength $F^u=F$, so that the phase factor is $e^{i{\int }_c{A}^u}$. Thus, from C1 one concludes that in the scalar-EM framework, 𝒰(1) is artificial, and the AB effect can be interpreted as resulting from the local interaction of the gauge-invariant fields ψ^u and A^u.Footnote ¹⁰

Notice however that such a (𝒰(1) adequate) polar decomposition is not available for a Dirac spinor field, so that a local dressing field cannot be thus constructed in the spinorial-EM framework. In this theoretical context, as shown in Dirac’s scheme, the only dressing field available is nonlocal and constructed out of the gauge potential. Hence, via C2 one concludes that the 𝒰(1)-gauge symmetry in spinorial EM is substantial and signals nonlocal EM properties, and the AB effect is interpreted as illustrating that fact.

The dependence of the nature of a gauge symmetry on the field content of the theory is further echoed in the theorization of gravitational physics. The gauge structure of GR is most apparent in the tetrad formulation. It is based on the bundle of pseudo-orthonormal frames with the Lorentz group as a structure group, 𝒫(ℳ,SO(1, 3)), endowed with the spin connection $A\in {\Lambda }^1(ℳ,𝔰𝔬(1,3\left)\right)$ whose curvature is the Riemann 2-form $R\left(A\right)=\mathrm{dA}+A\wedge A$. Furthermore, as a frame bundle it comes with a soldering, which is a linear isomorphism between the tangent space at each point $x\in ℳ$ and Minkowski space, $\theta :T_xℳ\to {ℝ}^{1,3}=\{{ℝ}^4,\eta \}$. It it thus represented by a 1-form $\theta \in {\Lambda }^1(ℳ,{ℝ}^{1,3})$ written explicitly as ${\theta }^a={e^a}_{\mu }{\mathrm{dx}}^{\mu }$, where ${e^a}_{\mu }$ is the vierbein, or tetrad field (hence the name of the formulation). The Lagrangian of pure gravity GR, L(θ, A), is invariant under the Lorentz gauge group $𝒮𝒪=\left\{\gamma :M\to \mathrm{SO}(1,3)\right\}$ acting on the variables as $A^{\gamma }={\gamma }^{\mathrm{-1}}A\gamma +{\gamma }^{\mathrm{-1}}d\gamma$ and ${\theta }^{\gamma }={\gamma }^{\mathrm{-1}}\theta$.

By inspection, it appears clearly that the tetrad field $e={e^a}_{\mu }$ is a local dressing field since it satisfies $e^{\gamma }={\gamma }^{\mathrm{-1}}e$. The 𝒮𝒪-invariant local composite fields are none other than the linear connection $\Gamma =e^{\mathrm{-1}}\mathrm{Ae}+e^{\mathrm{-1}}\mathrm{de}$ and its Riemann curvature $R\left(\Gamma \right)=e^{\mathrm{-1}}R\left(A\right)e$.Footnote ¹¹ The Lorentz gauge symmetry is then artificial and can be erased at no cost in the theory, whose Lagrangian reduces to the standard Einstein-Hilbert metric formulation: $L(\theta ,A)=L_{\mathrm{EH}}(g,\Gamma )$. The metric field, written as $g=e^T\eta e$, indeed appears as a natural 𝒮𝒪-invariant field in the Lagrangian. This remains true when gravity couples to radiation (EM), classically described matter (ponderable matter, gas of particles, dust, fluids), or scalar fields, collectively described as φ. So, according to C1, in general relativistic theories with Lagrangians $L(\theta ,A,\phi )=L(g,\Gamma ,{\phi }^e)$, the local Lorentz gauge symmetry 𝒮𝒪 is artificial and encodes no physics that is not already in the metric formulation (not even torsion or the gravitational AB effect) and does not restrict its interpretive resources.

Now, when matter is described by spinor fields, Lorentz gauge symmetry seems no longer dispensable. Indeed, the minimal coupling of spinors ψ to gravity is achieved via A, more precisely via ρ(A), where $\rho :\mathrm{SO}(1,3)\to \mathrm{SL}(2,ℂ)$ is a morphism from the Lorentz group to its universal cover: $D\psi =d\psi +\rho \left(A\right)\psi$. A Lorentz gauge transformation for the spinor field is then ${\psi }^{\gamma }=\rho \left({\gamma }^{\mathrm{-1}}\right)\psi$. There is an induced isomorphism $\overline{\rho }:{ℝ}^{1,3}\to \mathrm{Herm}(2,ℂ)$ between Minkowski space and the space of $2\times 2$ Hermitian matrixes, such that for any $v=v^a\in {ℝ}^{1,3}$ and $S\in \mathrm{SO}\left(1,3\right)$ one has $\overline{\rho }\left(\mathrm{Sv}\right)=\rho \left(S\right)\overline{\rho }\left(v\right)\rho {\left(S\right)}^{*}$, with * the complex conjugation and transposition. This means in particular that $\overline{\rho }\left({\gamma }^{\mathrm{-1}}\theta \right)=\rho \left({\gamma }^{\mathrm{-1}}\right)\overline{\rho }\left(\theta \right)\rho {\left({\gamma }^{\mathrm{-1}}\right)}^{*}$, so that the tetrad ρ(e) no longer satisfies the defining property of a dressing field WRT ρ(𝒮𝒪) and cannot be used to dress ψ. Thus, in a general relativistic theory with spinors L(θ, A, ψ), there is no local dressing field fitting for the whole theory, and only nonlocal dressing fields constructed via the holonomy of the spin connection are likely available. According to C2, the Lorentz gauge symmetry 𝒮𝒪 is then substantial.Footnote ¹²

It is interesting to notice that in both 𝒰(1)-EM and 𝒮𝒪-gravity, it is the spinorial nature of matter that obstructs the construction of a suitable local dressing field. So that it is the coupled systems “spinorial matter + EM” and “spinorial matter + gravitation” that display substantial gauge symmetries.

To wrap up this digression on gravitation, let us consider briefly a final example. In the tentative alternatives to GR known as massive gravity and bi-gravity, general coordinate invariance is broken because besides the usual dynamical metric g, a reference metric f_ab (often η_ab) is necessarily present in the mass term. Given any arbitrary coordinate system {x ^μ}, general covariance is restored via the introduction of scalar fields ϕ^a used to define a vielbein ${e^a}_{\mu }=(\partial {\varphi }^a/\partial x^{\mu })$ that allows one to promote the reference metric to a covariant tensor ${\overline{f}}_{\mu \nu }={e_{\mu }}^a{f}_{\mathrm{ab}}{e^b}_{\nu }$ (see de Rham [Reference de Rham2014] for a review). The scalar fields ϕ^a, or the vielbein e, are sometimes referred to as “clock-fields” (see Pitts [Reference Pitts2008] and references therein) or “Stueckelberg fields,” given that they play a role analogous to the Stueckelberg B field used to restore 𝒰(1) in massive EM. It is therefore no surprise that, beyond the mere analogy, the vielbein clock field is also a dressing field in the technical sense. Let $G={G^{\mu }}_{\nu }\in \mathrm{GL}\left(4\right)$ be the Jacobian of a coordinate change; the vielbein transforms as $e^{\prime }=eG$. So, $u=e^{\mathrm{-1}}$ satisfies $u^{\prime }=G^{\mathrm{-1}}u$ and is indeed a local dressing field, which indicates that the general covariance in massive gravity or bi-gravity is artificial and hides the existence of a “true gauge”: the preferred coordinates $x^{\mu }={\varphi }^a$.

The method presented here applies to various situations spanning a significant range of interesting physics. It suggests a criterion that allows one to adjudicate the nature of the gauge symmetry of a theory in an almost algorithmic way: inspect the field content, try to built a local dressing field. If you succeed, C1, then you have shown the gauge symmetry to be artificial and have found, in the form of the local gauge-invariant composite fields in (3), mathematical objects closer to adequate representatives of the true local physical DOF of the theory. If you fail and produce only nonlocal DF, C2, then you have shown (in all likelihood) the gauge symmetry to be substantial and have located in your theory nonlocal physical DOF that can be represented by the composite fields in (3).

The only remaining nonalgorithmic part of this scheme is the explicit construction of the dressing field. But given the clear specifications (1) to satisfy the defining gauge transformation property $u^{\gamma }={\gamma }^{\mathrm{-1}}u$ and (2) to be constructed out of the space of field variables of the theory, the problem is quite well circumscribed and should not pose much of a challenge in most interesting situations.

In any outcome, by systematically providing a “Ockhamized” version of the theory, the method clarifies its interpretive landscape. In particular, as shown by the discussion of the EM AB effect, given the observations of physical facts, slightly differing formalisms used to predict these facts may sharply differ in the metaphysical interpretations they allow.

As a case in point, the Glashow-Weinberg-Salam electroweak theory is of particular interest to philosophy of physics. It is therefore apt to analyze it in light of the dressing field method. We do so in the following section.

3.1. The Electroweak Model Treated via Dressing

The gauge group postulated a priori for the model is $\mathscr{H}=𝒰\left(1\right)\times 𝒮𝒰\left(2\right)=\left\{\alpha \times \beta :ℳ\to U\left(1\right)\times \mathrm{SU}\left(2\right)\right\}$. The space of field is $\Phi =\left\{A,F,\phi \right\}$, where $A=a+b$ is the gauge potential 1-form with curvature $F=f_a+g_b$, and φ is a ℂ²-scalar field. The scalar field couples minimally with the gauge potential via the covariant derivative $D\phi =d\phi +\left(g^{\prime }a+gb\right)\phi$, with g′, g the coupling constants of U(1) and SU(2), respectively. The gauge group acts as

$\begin{array}{ccc}{a}^a=a+\frac{1}{g^{\prime }}{\alpha }^{\mathrm{-1}}d\alpha & b^{\alpha }=b,& \text{and\hspace{1em}}{\phi }^{\alpha }={\alpha }^{\mathrm{-1}}\phi ,\\ a^{\beta }=a,& b^{\beta }={\beta }^{\mathrm{-1}}b\beta +\frac{1}{g}\beta d\beta ,& \text{and\hspace{1em}}{\phi }^{\beta }={\beta }^{\mathrm{-1}}\phi .\end{array}$

The ℋ-invariant Lagrangian form of the theory is

(7)$\begin{array}{}& \begin{array}{cc}L\left(a,b,\phi \right)& =\frac{1}{2}\mathrm{Tr}\left(F\wedge *F\right)+〈D\phi ,*D\phi 〉-U\left(\phi \right)\mathrm{vol}\text{,}\\ & =\frac{1}{2}\mathrm{Tr}\left(f_a\wedge *f_a\right)+\frac{1}{2}\mathrm{Tr}\left(g_b\wedge *g_b\right)+〈D\phi ,*D\phi 〉-\left({\mu }^2〈\phi ,\phi 〉+\lambda {〈\phi ,\phi 〉}^2\right)\mathrm{vol}\text{,}\end{array}\end{array}$

where μ, $\lambda \in ℝ$ and “vol” is the volume form on space-time ℳ. Neither a nor b can be massive, and indeed L contains no mass terms for them. While at least one massless field is expected in order to carry the EM interaction, the weak interaction is short range so its associated fields must be massive. Hence, it is necessary to reduce the 𝒮𝒰(2)-gauge symmetry in order to allow for mass terms for the weak fields.

As the usual narrative goes, this is achieved via SSB: if ${\mu }^2<0$, the electroweak vacuum given by $U\left(\varphi \right)=0$ seems degenerate as it appears to be an 𝒮𝒰(2)-orbit of nonvanishing vacuum expectation values for φ. When the scalar field settles randomly—spontaneously—on one of them, this breaks 𝒮𝒰(2) and generates mass terms for the weak fields with which it couples. Oddly, in order to exhibit the physical modes of the theory, it is claimed that a convenient choice of gauge is necessary, the so-called unitary gauge (see, e.g., Becchi and Ridolfi Reference Becchi and Ridolfi2006). But why are we allowed to use a gauge freedom if it is supposedly broken?

We suggest that a better approach and a more satisfactory interpretation is provided by the dressing field method. Indeed it is not hard to find a dressing field in the electroweak model. Considering the polar decomposition in ℂ² of the scalar field $\phi =u\eta$ with

(8)$\begin{array}{}& u\in \mathrm{SU}\left(2\right)\text{\hspace{1em}and\hspace{1em}}\eta ≔\left(\stackrel{0}{\left|\right|\phi \left|\right|}\right)\in {ℝ}^{+}\subset {ℂ}^2,\text{\hspace{1em}one has\hspace{1em}}{\phi }^{\beta }\Rightarrow u^{\beta }={\beta }^{\mathrm{-1}}u.\end{array}$

Thus, u is a 𝒮𝒰(2)-dressing field that can be used to construct the 𝒮𝒰(2)-invariant composite fields:

(9)$\begin{array}{}& \begin{array}{cc}{A}^u& =u^{\mathrm{-1}}\mathrm{Au}+\frac{1}{g}{u}^{\mathrm{-1}}\mathrm{du}=a+(u^{\mathrm{-1}}\mathrm{bu}+\frac{1}{g}{u}^{\mathrm{-1}}\mathrm{du})=a+B,\\ F^u& =u^{\mathrm{-1}}\mathrm{Fu}=f_a+u^{\mathrm{-1}}{g}_{b}u=f_a+G,\text{\hspace{1em}with }G=\mathrm{dB}+{\mathrm{gB}}^2,\\ {\phi }^u& =u^{\mathrm{-1}}\phi =\eta ,\text{\hspace{1em}and\hspace{1em}}{D}^u\eta =u^{\mathrm{-1}}D\phi =d\eta +(g^{\prime }a+gB)\eta .\end{array}\end{array}$

Since u is local, so are the above composite fields. Therefore, by virtue of criterion C1 we conclude that the 𝒮𝒰(2)-gauge symmetry of the model is artificial, so that the theory defined by the electroweak Lagrangian (7) is actually a 𝒰(1)-gauge theory, described in terms of local 𝒮𝒰(2)-invariant variables:

(10)$\begin{array}{}& L(a,B,\eta )=\frac{1}{2}\mathrm{Tr}(F^u\wedge *F^u)+〈D^u\eta ,*D^u\eta 〉-\left({\mu }^2{\eta }^2+\lambda {\eta }^4\right)\mathrm{vol}\text{.}\end{array}$

We have already reached our main conclusion: since the 𝒮𝒰(2)-gauge symmetry is artificial, the interpretation of the model in terms of SSB is superfluous and indeed impossible when expressed in equation (10). We could then stop here. But at this point it is not clear that our analysis reproduces all the phenomenology usually obtained via the standard interpretation. In what follows we show that it is indeed so: it is done simply by proceeding to the natural step of analyzing the residual and substantial 𝒰(1)-gauge symmetry of the model, which is very easily done from the viewpoint of the dressing field method.

Residual $𝒰\left(1\right)$ symmetry.—By its very definition ${\eta }^{\beta }={\eta }^{\alpha }=\eta$, so it is already a fully ℋ-gauge-invariant scalar field that then qualifies as an observable. As a rule, the 𝒰(1)-residual gauge transformation of the 𝒮𝒰(2)-invariant composite fields depends on the 𝒰(1)-gauge transformation of the dressing field u. One finds that

${\phi }^{\alpha }\Rightarrow u^{\alpha }=u\tilde{\alpha },\text{\hspace{1em}where\hspace{1em}}\tilde{\alpha }=\left(\begin{array}{cc}\alpha & 0\\ 0& {\alpha }^{\mathrm{-1}}\end{array}\right).$

Therefore, $B^{\alpha }={\left(b^{\alpha }\right)}^{u^{\alpha }}=b^{u\tilde{\alpha }}={\tilde{\alpha }}^{\mathrm{-1}}{u}^{\mathrm{-1}}\mathrm{bu}\tilde{\alpha }+(1/g){\tilde{\alpha }}^{\mathrm{-1}}\left(u^{\mathrm{-1}}\mathrm{du}\right)\tilde{\alpha }+(1/g){\tilde{\alpha }}^{\mathrm{-1}}d\tilde{\alpha }={\tilde{\alpha }}^{\mathrm{-1}}B\tilde{\alpha }+(1/g){\tilde{\alpha }}^{\mathrm{-1}}d\tilde{\alpha }$. Given the decomposition $B=B_a{\sigma }^a$, where the σ^a are the Hermitian Pauli matrixes and $B_a\in iℝ$, we have explicitly

$B=\left(\begin{array}{cc}{B}_3& B_1-{\mathrm{iB}}_2\\ B_1+i{B}_2& -B_3\end{array}\right)≕\left(\begin{array}{cc}{B}_3& W^{-}\\ W^{+}& -B_3\end{array}\right),\text{\hspace{1em}and\hspace{1em}}{B}^{\alpha }=\left(\begin{array}{cc}{B}_3+\frac{1}{g}{\alpha }^{\mathrm{-1}}d\alpha & {\alpha }^{\mathrm{-2}}{W}^{-}\\ {\alpha }^2{W}^{+}& -B_3-\frac{1}{g}{\alpha }^{\mathrm{-1}}d\alpha \end{array}\right)\text{.}$

The fields W ^± transform tensorially: it is then possible for these two fields to be massive. And B ₃ transforms as a 𝒰(1)-gauge potential, but with a different coupling constant, making it another massless field together with the initial 𝒰(1)-gauge potential a. But consider the 𝒰(1)-transformation of the 𝒮𝒰(2)-invariant covariant derivative:

$D^u\eta =d\eta +(g^{\prime }a+\mathrm{gB})\eta =\left(\begin{array}{c}{\mathrm{gW}}^{-}\eta \\ d\eta -g{B}_3\eta +g^{\prime }a\eta \end{array}\right),\text{\hspace{1em}and\hspace{1em}}{\left(D^u\eta \right)}^{\alpha }=\left(\stackrel{g{\alpha }^{\mathrm{-2}}{W}^{-}\eta }{d\eta -{\mathrm{gB}}_3\eta +g^{\prime }a\eta }\right)\text{.}$

We see that a 𝒰(1)-invariant combination of a and B ₃ appears. So, considering (a, B ₃) as a doublet in ℂ², one is invited to perform the natural change of variables:

$\left(\begin{array}{c}A\\ Z^0\end{array}\right)=\left(\begin{array}{cc}\cos {\theta }_W& \sin {\theta }_W\\ -\sin {\theta }_W& \cos {\theta }_W\end{array}\right)\left(\begin{array}{c}a\\ B_3\end{array}\right)=\left(\begin{array}{c}\cos {\theta }_{W}a+\sin {\theta }_W{B}_3\\ \cos {\theta }_W{B}_3-\sin {\theta }_{W}a\end{array}\right)\text{,}$

where $\cos {\theta }_W=g/\sqrt{g^2+{g^{\prime }}^2}$ and $\sin {\theta }_W=g^{\prime }/\sqrt{g^2+{g^{\prime }}^2}$ (θ_W is known as the Weinberg, or weak mixing, angle). By construction, the 1-form Z ⁰ is fully ℋ-gauge invariant and thus observable and potentially massive. Now, still by construction, $A^{\beta }=A$ and $A^{\alpha }=A+(1/e){\alpha }^{\mathrm{-1}}d\alpha$ with coupling constant $e={\mathrm{gg}}^{\prime }/\sqrt{g^2+{g^{\prime }}^2}$. So, A transforms as a 𝒰(1)-gauge potential; it can thus be interpreted as the massless mediator of the EM interaction whose coupling constant e is the elementary electric charge.

The electroweak theory (10) is then expressed in terms of the gauge-invariant fields η, Z ⁰ and of the 𝒰(1)-gauge fields W ^±, A:

(11)$\begin{array}{}& \begin{array}{cc}L(A,W^{\pm },Z^0,\eta )=& \frac{1}{2}\mathrm{Tr}(F^u\wedge *F^u)+d\eta \wedge *d\eta -g^2{\eta }^2{W}^{+}\wedge *W^{-}-(g^2+g{\prime }^2){\eta }^2{Z}^0\wedge *Z^0-\left({\mu }^2{\eta }^2+\lambda {\eta }^4\right)\mathrm{vol}\text{.}\end{array}\end{array}$

The next natural step is to expand the ℝ⁺-valued scalar field η around its unique ground state η₀, given by $U\left({\eta }_0\right)=0$, as $\eta ={\eta }_0+H$, where H is the gauge-invariant Higgs field.Footnote ¹³ Mass terms for Z ⁰, W ^±, and H depending on η₀ appear from the couplings of the electroweak fields with η and from the latter’s self-interaction.Footnote ¹⁴ The theory has two qualitatively distinct phases. In the phase in which ${\mu }^2<0$, η₀ vanishes and so do all masses. But in the phase in which ${\mu }^2<0$, the ground state is nonvanishing: ${\eta }_0=\sqrt{-{\mu }^2/2\lambda }$. The masses of the fields Z ⁰, W ^±, and H are then $m_{Z^0}={\eta }_0\sqrt{(g^2+{g^{\prime }}^2)}$, $m_{W^{\pm }}={\eta }_{0}g$, and $m_H={\eta }_0\sqrt{2\lambda }$. All physical predictions of the electroweak theory are indeed preserved in our treatment: masses are gained through a phase transition of the unique electroweak vacuum. Finally, notice that this approach to the electroweak model offers a satisfactory reconciliation with Elitzur’s (Reference Elitzur1975) theorem, stating that in lattice gauge theory a gauge symmetry cannot be spontaneously broken.

3.2. There Is No SSB in the Electroweak Model, and We Long Suspected It

It turns out that several authors were close to formulating such a gauge-invariant account of the electroweak theory. Even before the theory was proposed—barely 16 months after his celebrated paper (Higgs Reference Higgs1964)—Higgs hinted at a gauge-invariant formulation of the mechanism that ended up bearing his name, by working on an abelian toy model (see Higgs Reference Higgs1966, sec. 4). Two years after his own celebrated contribution with Guralnik and Hagen, Kibble (Reference Kibble1967) suggested a similar analysis working on both abelian and nonabelian models. Just before the conclusion of his paper, Kibble writes: “We note certain characteristic features of our model. It is perfectly possible to describe it without ever introducing the notion of symmetry breaking, merely by writing down the Lagrangian (66) [written with gauge-invariant variables]. Indeed if the physical world were really described by this model, it is (66) rather than (64) [i.e., the Lagrangian written in terms of gauge variables] to which we should be led by experiment” (1561). With insight, it is clear that both Higgs and Kibble were using instances of the dressing field method: see equation (22) and below (dressing and composite fields) in Higgs (Reference Higgs1966) as well as equations (9)–(59) (dressing fields) and (16)–(61) (composite fields) in Kibble (Reference Kibble1967).

But then, within the same year, the Glashow-Weinberg-Salam model was proposed, using the Brout-Englert-Higgs-Guralnik-Hagen-Kibble (BEHGHK) mechanism with its original interpretation in terms of SSB. So these important insights from Higgs and Kibble were eclipsed, and the view of SSB as a real physical phenomenon that had happened in the early universe gained currency. In a panel discussion during a large conference on the foundation of QFT gathering physicists as well as historians and philosophers of physics in Boston in 1996, the following exchange took place.

Yet, the treatment of the electroweak model through the bundle reduction theorem (see, e.g., Trautman Reference Trautman and Held1979; Westenholz Reference Westenholz1980; Sternberg Reference Sternberg1994) already cast some doubts on the interpretation of the SSB as a dynamical phenomenon. Indeed, thus formulated it appears that the model can naturally be rewritten on a U(1)-subbundle of the initial $U\left(1\right)\times \mathrm{SU}\left(2\right)\mathrm{-bundle}$.

As far as I know, the first to give a fully 𝒮𝒰(2)-gauge invariant formulation of the electroweak theory were Fröhlich, Morchio, and Strocchi (Reference Fröhlich, Morchio and Strocchi1981). Their account is actually fully equivalent to ours but much less synthetic and systematic: they are working on individual scalar components of all the fields involved. See their equation (6.1) describing the composites fields (including a dressed electron and neutrino). Subsequently, and especially in the last 10 years, several researchers independently rediscovered the gauge-invariant description formulated essentially as in the above treatment but without the conceptual clarity given by the dressing field method, as often the dressing field was mistaken for an element of the gauge group (Lavelle and McMullan Reference Lavelle and McMullan1995; see esp. eqq. [6] and [7] and comments in between) or the interpretive shift was not fully embraced (Chernodub, Faddeev, and Niemi Reference Chernodub, Faddeev and Niemi2008; Faddeev Reference Faddeev2009). Interestingly, the textbook by Rubakov gives essentially the dressing treatment of the abelian Higgs model but sticks to the usual treatment of the electroweak model using the unitary gauge (see Rubakov Reference Rubakov1999, chap. 6). Some improvement in conceptual clarity is found in Ilderton, Lavelle, and McMullan (Reference Ilderton, Lavelle and McMullan2010). But it is Masson and Wallet (Reference Masson and Wallet2011) who first really appreciated the interpretive shift that comes with the invariant formulation, and as a matter of fact their paper was a precursor to the development of the dressing field method. Unfortunately it never get published.

In parallel, in the last 15 years, philosophers of science have questioned the orthodoxy of the SSB in the electroweak model. Earman first raised the issue in striking terms:

Shortly after, Earman reiterated that “a genuine property like mass cannot be gained by eating descriptive fluff, which is just what gauge is. Philosophers of science should be asking the Nozick question: What is the objective (i.e., gauge invariant) structure of the world corresponding to the gauge theory presented in the Higgs mechanism?” (Reference Earman2004b, 1239). Emphasizing Dirac’s constrained Hamiltonian formalism as a systematic way to extract the gauge-invariant quantities of a gauge system Earman asks:

Since then, several authors have risen to the challenge, noticing or rediscovering for themselves the invariant formulation (Smeenk Reference Smeenk2006; Lyre Reference Lyre2008; Struyve Reference Struyve2011; Friederich Reference Friederich2013, Reference Friederich2014), and all essentially concluded that gauge SSB is indeed a dispensable notion. Since they rely to some significant degree on the dressing field method, the considerations presented in section 3.1 fully confirm the authors’ conclusion in synthesizing the core technical argument and vindicate Earman’s suspicion. Furthermore, the dressing field method places the specific question of gauge SSB in the electroweak model within the broader problem of distinguishing substantial from artificial gauge symmetries.