1.3.1 In Physics

• 1918–19: Weyl’s “unified theory”/infinitesimal geometry introduces gauge (Eichung) rescaling symmetry.

• 1929: Weyl introduces the gauge principle for the Abelian group U(1) in quantum mechanics, in order to “explain” Maxwell’s theory of electromagnetism.

• 1954: Yang & Mills, and Shaw, produce the first non-Abelian gauge theory for the group SU(2) (as an attempt to describe the strong interaction between proton/neutron as a doublet).

• 1954–55: Independently, R. Utiyama develops the framework of gauge theory for any Lie group $G$. He shows that general relativity is, in a certain sense, a gauge theory of the local Lorentz group $G=SO(1,3)$.

• 1960s–70s: By a series of rapid developments, the Standard Model of particle theory arises (electroweak unification by Glashow in 1961; spontaneous symmetry breaking (SSB) mechanism by Englert-Brout-Higgs in 1964, Quarks by Gell-Mann and Zweig in 1964, electroweak theory with SSB by Weinberg and Salam in 1967, renormalizability by ’t Hooft 1971; asymptotic freedom by Politzer (and Gross and Wilczek) in 1973, etc.). Particle physics is described as a gauge field theory of with gauge group $G=U\left(1\right)\times$ $SU\left(2\right)\times SU\left(3\right)$.

1.3.2 In Mathematics

• 1916–17: Theory of connections on manifolds by Levi-Civita, Schouten.

• 1918–19: Weyl’s infinitesimal geometry falls within this current of ideas.

• 1920s: Cartan’s “espaces generalisés,” a vast synthesis of (pseudo-) Riemannian and Klein geometries.

• 1930s: Whitney’s first definition of fibered spaces, or fiber bundles: spaces with “structured points”.

• 1950s–60s: Mature theory of connections on fiber bundles (Ehresmann, 1950; Steenrod, 1951; Kobayashi, 1957)

The two strands finally converged in the mid-1970s. In 1975, T. T. Wu and C.N. Yang published a paper about the physicist’s electromagnetic field theory and its relationship to the mathematician’s fiber bundle theory (Wu & Yang, Reference Wu and Yang1975). To clarify the deep – and precise – relation between these two strands, they constructed a dictionary. In 1976 Isadore Singer visited Stony Brook and Yang gave him a copy of the Wu-Yang preprint, which Singer took to Oxford. There, Michael Atiyah and other mathematicians studied the paper and began to work on gauge fields and related topics, leading to a period of close collaboration between mathematicians and physicists. Figure 1 is a table taken from a paper in this period.

Figure 1 The Wu-Yang Dictionary, as described by Isadore Singer, in an article about Weyl in Wells (Reference Wells1988).

2.1.1 Beware: Dubious Arguments Ahead

We begin by describing a quantum system with the Hilbert space $L^2\left(R^3\right)$ of wavefunctions, recalling that a unique pure quantum state is represented not by vector, but by a “rays” of vectors related by a complex unit. This implies that the transformation $\psi \left(\vec{x}\right)\mapsto e^{i\theta }\psi \left(\vec{x}\right)$ for some $\theta \in R$, referred to as a “global phase” transformation, acts identically on rays, and is in this sense an invariance of the quantum system.Footnote ⁵ This invariance is incorporated in the specification of the dynamics of the system, either via the Hamiltonian or the action, since either contains only real valued functions such as $|\psi {|}^2$ and ${\partial }_i\psi {\partial }_i\bar{\psi }$.

But now, the story goes, suppose we replace this global phase with a “local phase” transformation $\psi \left(\vec{x}\right)\mapsto e^{i\varphi \left(\vec{x}\right)}\psi \left(\vec{x}\right)$, in which the constant $\theta$ is replaced with a function $\varphi :R^3\to R$, or indeed with a smooth one-parameter family of such functions ${\varphi }_t\left(\vec{x}\right)$ for each $t\in R$; or, adopting the covariant notation in which $x=(t,\vec{x})$, we write as $\varphi \left(x\right)$. This transformation is “local” in the sense that its values vary smoothly across space and time.

The corresponding Hilbert space map $R_{\varphi }:\psi \mapsto e^{i\varphi }\psi$ does not act identically on rays. As to the dynamics, whereas $|\psi {|}^2$ would remain invariant under such a transformation, that would not be the case for terms involving derivatives, such as ${\partial }_i\psi {\partial }_i\bar{\psi }$, which under such a transformation acquires terms depending on ${\partial }_i\varphi$ such as $({\partial }_i\varphi {)}^2\psi \bar{\psi }$.

However, one might still wish to postulate that this transformation has no “physical effect” on the system, or is “gauge.” Various motivations for this step are given in the textbooks, often with vague references to general covariance of the kind found in general relativity. But to mimic the standard presentation, we will simply press forward, referring to $R_{\varphi }:\psi \mapsto e^{i\varphi }\psi$ as a local or malleable gauge transformation.

With respect to the dynamics, we still need to say something about the noninvariant terms involving derivatives. The big move of the gauge argument is to first introduce a vector $A=(A_1,A_2,A_3)$ and a scalar $V$, which are assumed to behave under the gauge transformation as,

$\begin{array}{ccc}A\mapsto A+\nabla {\varphi }_t,& & V\mapsto V-\frac{d{\varphi }_t}{dt}.\end{array}$(2.1.1)

To restore invariance of the dynamics under gauge transformations, and with an eye toward a modern gauge theory formulated as a vector bundle with a derivative operator, writing ${\partial }_{\mu }:=(\frac{d}{dt},\nabla )$ and $A_{\mu }=(V,A)$, one finds that one can restore gauge-invariance by replacing ${\partial }_{\mu }$ with,

$D_{\mu }:={\partial }_{\mu }+i{A}_{\mu }=(\frac{d}{dt}+iV,\nabla +iA)=(D_t,D).$(2.1.2)

This is commonly referred to as a “covariant derivative,” and it has the form of the familiar gauge freedom of the electromagnetic four-potential. That is, if we call $A_0:=V$, these transformations leave invariant the following tensor:

$F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\mu }{A}_{\nu },$(2.1.3)

where the electric and magnetic field are recovered in a given coordinate system as $E_i=F_{0i}$ and $B^i={ϵ}^{ijk}{F}_{jk}$, where $x^0$ are the time coordinates, $i,j,k$ are spatial indices, and ${ϵ}^{ijk}$ is the totally anti-symmetric tensor in space. In short, it appears as if minimal electromagnetic coupling has been derived out of nothing: or at least, from an assumption of gauge invariance.

2.1.2 Criticisms of the Gauge Argument

That is how the story is usually presented. I agree: it is far from water-tight. The argument begins with a system with a global symmetry, gratuitously generalizes it to a local symmetry – which, to emphasize, was not required for mathematical consistency or for empirical adequacy – and then, in order to fix the ensuing noninvariance of the governing equations, proceeds to conjecture a new force of nature, which, so far, has no reason to be dynamical at all. Ultimately, the argument gives us no reason to think of the field $A_{\mu }$ as being related to Maxwell’s equation. To put it uncharitably: the argument fixes a problem that didn’t exist by conjecturing a redundant field, and then turns this game around, claiming to come out successfully by “retrodicting” the existence of electromagnetism. More charitably: the gauge argument suffers from at least three categories of concerns. I will set out each of these three concerns here and in Section 2.2 present an alternative Noether gauge argument that answers them entirely.

The first category of concerns is the gauge argument’s claim to have derived a dynamics that is specifically electromagnetic in nature. Although a formal set of operators $A_{\mu }=(V,A)$ have been included in the dynamics, no evidence is given that these operators take the form required for any specific electromagnetic potential, or that the coupling to $A_{\mu }$ will be proportional to a particle’s charge $e$, or even that $A_{\mu }$ is nonzero. And if they could be shown to be nonzero, then as Wallace (Reference Wallace2009, p. 210) rightly asks: “how do neutral particles fit into the argument?” A minimally coupled dynamics does not apply to neutral particles, and yet since the gauge argument never mentioned or assumed anything about charge, it presumably is intended to apply to them.

A second category of problems arises out of the free-wheeling argumentative style of the gauge argument. For example, it is not a strict deductive derivation of either the electromagnetic potential or the dynamics. At best, the gauge argument appears to show that one can adopt a minimally coupled Hamiltonian in order to assure gauge invariance. But this does not ensure that one must do so: the door appears to be left open for other dynamics to be gauge invariant, but without taking the minimally coupled form that the gauge argument advocates. As Martin (Reference Martin2002, p. S230) writes: “The most I think we can safely say is that the form of the dynamics characteristic of successful physical (gauge) theories is suggested through running the gauge argument.”

Another example of free-wheeling argumentation is in the motivation for requiring the local gauge transformations $R_{\varphi }:\psi \mapsto e^{i\varphi \left(x\right)}\psi$ to be symmetries. Sometimes a preference for this transformation over global phase transformations is dubiously motivated by a desire to avoid superluminal signaling.Footnote ⁶ In other cases it is motivated by the coordinate invariance of a spatial coordinate system. But as Wallace (Reference Wallace2009, p. 210) points out, no reason is given as to why we do not similarly consider local transformations of configuration space, momentum space, or any other space, to be symmetries. Nor is there any clear reason why the $U\left(1\right)$ symmetry of electromagnetism is chosen as the global symmetry motivating the move to the local symmetry, as opposed (say) the $SU\left(3\right)$ symmetry of the strong nuclear force.

Regarding the generalization of the gauge argument to other global symmetry groups beyond electromagnetism, I wholeheartedly agree with Wallace: one should expect, and indeed I will argue in Section 2.2, that an appropriate generalization of the gauge argument can also be applied to these more general gauge groups.

My approach here speaks to a third category of concerns, that the gauge argument is awkwardly placed as an argument for a quantum theory of electromagnetism. Here too I agree with Wallace:

In Section 2.2, we will switch perspectives from the verdammten Quantenspringerei to the context of classical Lagrangian field theory, and propose a framework that substantially clarifies the roles of global gauge symmetries, of local gauge symmetries, and of their relationship, which I will call the “Noether gauge argument.”

2.2.1 Overview

For a more general view of how gauge symmetries constrain the dynamics of a physical theory, I will now, as announced in Section 2.1.2, make a two-step use of the theorems of Emmy Noether (Reference Noether1918): the first, and then the second. I will refer to this as the Noether gauge argument. Agreed: this is by no means a new observation, since practicing physicists use this property of gauge frequently!Footnote ⁷ But I believe it is worth highlighting and clarifying exactly the kind of information that can be extracted in various cases, as part of my advocacy that philosophical discussions of gauge should better recognize gauge’s significance for theory construction.

To recall the sketch of the argument: the Noether gauge argument proceeds in two steps. First, we choose a rigid gauge symmetry associated with an arbitrary global gauge group, and propose that its action produces a variational symmetry: by Noether’s first theorem, this guarantees the presence of a collection of conserved quantities. But matter fields do not exist in isolation: they couple to other “force” fields. Thus, in the second step, we introduce such a field and apply Noether’s second theorem, “loosening” the rigid global symmetries to local, malleable ones.

About the generality and applicability of Noether’s theorems, there are several issues that I will not address, but which should bridle undue enthusiasm (cf. Brown (Reference Brown, Read and Teh2022)). First, Noether’s theorems apply only to those theories that admit a Lagrangian (variational) formulation. But there are mathematical models that are useful and which do not admit such a formulation: Fourier’s heat equation and Navier-Stokes equations are well-known examples. Second, if one takes the equations of motion and not the Lagrangian as fundamental, there are many Lagrangians that give the same equations of motion, and there is, in general no unique symmetry associated with a given conservation law, or vice versa; (though most of these ambiguities can be accounted for by different boundary terms or boundary conditions, which don’t affect the fundamental meaning of the conservation law). Third, the meaning of the conserved quantity obtained from a given symmetry could be theory dependent. For instance, one can obtain a conserved quantity associated with time translation symmetry for a damped oscillator, but this is not energy as usually construed (for nondamped systems). There is also the matter of explanatory priority between conservation laws and symmetries, which I will not address here.

Thus, I will not only assume the minimum conditions under which the theorems apply, but in the interest of clarity and pedagogy, I will make several simplifying assumptions, both about the Lagrangian density and about the action of the gauge group, some of which are not strictly speaking necessary but which simplify my argument. So there are mathematically more general and more abstract ways to formulate this argument (see e.g. Gomes (Reference Gomes, Read and Teh2022)), but for our analysis, it is worthwhile to be specific about the field content of the theory, and show how local, or malleable symmetries provide three concrete constraints on the dynamics (namely, the vanishing of the three lines in Equation (2.2.8)). The interpretation of these constraints can be seen on a case-by-case, or sector-by-sector, basis: we will consider their implications for global versus local symmetries, as well as for theories that contain a force field that transforms under the transformation versus those that contain no such force field. Thus in the following sections we will spell out the consequences of the three constraints for four different sectors of the theory.Footnote ⁸

Throughout this discussion, I will follow standard practice and distinguish two equivalence relations for classical fields on a manifold. First, I will write “=” to denote ordinary equality between fields, irrespective of the satisfaction of the equations of motion, and refer to this as strong or off-shell equality. Second, given a fixed Lagrangian, I will write “$\approx$” to denote equality between fields that holds if the Euler-Lagrange equations are satisfied for that Lagrangian, and refer to this as weak or on-shell equality.Footnote ⁹

2.2.2 Mathematical Setup*

Now I will introduce, without much explanation, some of the mathematical objects that will be the focus of Section 3. Here, the reader should just take the definitions at face value: they will be explained and motivated in that section.

We start by assuming that $\phi$ is some field on spacetime, whose dynamically possible models are determined by a Lagrangian scalar function $L\left(\phi \right)\in C^{\infty }\left(M\right)$, as those that extremize the integral of this function over $M$ (called the action functional), a condition which we write as:Footnote ¹⁰

$\delta {\int }_{M}L\left(\phi \right)=0.$(2.2.1)

Using Leibniz-linearity of $\delta$, Equivalently, after successive integration by parts, we isolate $\delta \phi$ and write the conditions (2.2.1) as yielding equations of motion, up to boundary terms:

$\delta L=EL\cdot \delta \phi +d\theta \left(\delta \phi \right),$(2.2.2)

where EL is the Euler-Lagrange functional (the left-hand part of the Euler-Lagrange equations) which has one component for each direction of $\delta \phi$, and $\theta$ is a linear operator on variations of the fields, but it is a differential form of codimension one on spacetime (i.e. it is a boundary term).Footnote ¹¹

Suppose that, for any value of $\phi$, there is a family of transformations ${\delta }_{\xi }\phi$, whose parameters $\xi$ form an algebra, which is such that ${\delta }_{\xi }{\delta }_{{\xi }^{\prime }}\phi -{\delta }_{{\xi }^{\prime }}{\delta }_{\xi }\phi ={\delta }_{[\xi ,{\xi }^{\prime }]}\phi$,Footnote ¹² and so that ${\delta }_{\xi }L=0$. So from (2.2.2):

${\delta }_{\xi }L=EL\cdot {\delta }_{\xi }\phi +d{\theta }_{\xi }=0,$(2.2.3)

and so, for dynamically possible models, for which $EL=0$, we get

$d{\theta }_{\xi }\approx 0,$(2.2.4)

where ${\theta }_{\xi }$ is the Noether charge associated to the symmetry $\xi$, and where ${\theta }_{\xi }:=\theta \left({\delta }_{\xi }\phi \right)$. The Noether charge inherits the ambiguity $\theta \to \theta +d\kappa$ in the boundary term, but the ambiguity does not matter for conserved quantities, since $d\circ d=0$ (or, equivalently, the boundary of a boundary is empty: $\partial \partial B=\varnothing$).

Noether’s second theorem also follows from (2.2.3). Assume that the symmetries are malleable, or local, so that we can restrict to those $\xi$ such that ${\xi }_{|B}=0$. Now, there are inner products $⟨\bullet ,\bullet ⟩$ and $⟨⟨\bullet ,\bullet ⟩⟩$, on the appropriate function spaces of $\phi$ and $\xi$, respectively, so that

${\delta }_{\xi }\int L\left(\phi \right)=\int EL\cdot {\delta }_{\xi }\phi =\int ⟨EL,{\delta }_{\xi }\phi ⟩=\int ⟨⟨{\Delta }^{†}EL,\xi ⟩⟩=0$(2.2.5)

where ${\Delta }^{†}$ is the formal adjoint of ${\delta }_{\xi }\phi$, seen as a linear operator on $\xi$ (see Fischer & Marsden (Reference Fischer, Marsden and Hawking1979) for a thorough, geometric formulation of such inner products and adjoints in the space of fields of gauge theory and general relativity). Since (2.2.3) must vanish for all such $\xi$, it implies a local equation that the Euler-Lagrange equations must satisfy everywhere, and which is valid off-shell:

${\Delta }^{†}EL=0.$(2.2.6)

Because these constraints are valid off-shell, they reflect a kinematic property of the variables of the theory. For instance, the Bianchi identities for the curvature tensors (which applies both to (semi-)Riemannian geometry and principal fiber bundles; cf. Proposition 4 in “The Curvature” section) is a purely geometric property that will give rise to such constraints. These are geometrical identities, that tell us that not all components of the curvature tensors are independent: they satisfy differential and algebraic constraints. In all theories under consideration here, local, covariant tensors that involve derivatives of the fundamental variables – either the gauge potential or the metric – must be written in terms of curvature (cf. e.g. Lovelock (Reference Lovelock1972)), and so obey such geometric constraints. Thus, the equations of motion obtained from Lagrangians that involve derivatives of the fundamental variables, will in some way or another inherit these constraints and so cannot all be independent. In other words, there are more independent variables than there are equations of motion. For this reason such equations of motion cannot be used to uniquely determine the evolution of all the fundamental variables: they do so only for a constrained subset. I will get back to this topic in Section 5.3.1, when we discuss non-locality.

Now let us be more specific. Let us start with the material charges, which I will represent as fields with no spacetime index, but, assuming the value space is a vector space $V$, with indices $a$; so that ${\psi }^a\left(x\right)\in V$ for $x\in M$, where $M$ is spacetime and $V$ is a vector space. In relativity, $M$ is a smooth Lorentzian manifold, that is, equipped with a nondegenerate symmetric bilinear product that is not positive definite, having a signature of $(3,1)$. To simplify the notation, when discussing gauge theory, I’ll consider the case of a Minkowski metric, whose Levi-Civita covariant derivative I’ll denote as $\partial$ (the generalization to another metric would amount to a minimal replacement $\partial \to \nabla$). From the pragmatic, nongeometric standpoint of this section, assume that the symmetries associated to the conservation of charges arise from the action of a Lie group, $G$ on $V$ (cf. footnote 5). We define this action pointwise as $g\cdot \psi \left(x\right)=g\left(x\right)\cdot \psi \left(x\right)\in V$. Let $t_I^{ij}$ be the $n$-dimensional Hermitean matrix representation on $V$ of $g$, that is, $t:g\to GL\left(V\right)$, where the $I$ are indices of the Lie algebra space, in the domain of the map, and $i,j$ denote the matrix indices in the image of the map, acting linearly on $V$.Footnote ¹³

I will also assume that the forces that are sourced by $\psi$ have direction in space – as forces are prone to have – and take value again in some internal vector space. For reasons to be clarified in Section 3, at this point I will take these value spaces as being linearly isomorphic to the Lie algebra: so the force fields are labeled $A_{\mu }^I$, they take vectors of $M$ to $g$, with $\mu$ representing the spacetime components of the vector.Footnote ¹⁴ These fields are associated with a dynamics by postulating a real-valued action functional $S({\psi }_i,A_{\mu }^I)$, whose extremal values provide the equations of motion.

We take the (malleable, or local) gauge transformations, infinitesimally parametrized by $\xi \in g$, to act on our fundamental variables as:

$\left(\begin{array}{c}{\delta }_{\xi }{\psi }_i={\xi }^I{t}_I^{ij}{\psi }_j=(\xi t\psi {)}_i\\ {\delta }_{\xi }{A}_{\mu }^I=D_{\mu }{\xi }^I={\partial }_{\mu }{\xi }^I+[\xi ,A_{\mu }{]}^I\end{array}\right).$(2.2.7)

where the square brackets are the Lie algebra commutators (again, we will justify these transformations in Section 3). These transformation rules are not as general as they could be, but neither are they arbitrary: they are highly constrained by the theory of representations of Lie groups on vector spaces! But even without going into representation theory, the reader can recognize that these are the first-order terms of the Lie algebra action on the respective vector spaces – in particular, “first-order” in the derivatives of $\xi$ and in powers of $A$ and $\psi$ – and in this sense provide an appropriate approximation of any malleable gauge transformation.

Our aim now is to constrain how the matter fields $\psi$ couple to force fields. Let $L(\psi ,\partial \psi ,A,\partial A)$ be the Lagrangian defining our action $S(\psi ,A)$, which we assume for simplicity does not depend on derivatives of higher order than two.Footnote ¹⁵ Variation along the directions of the gauge transformations previously yields (with summation convention on all indices):

$\begin{array}{c}\begin{array}{c}\left(\frac{\delta L}{\delta {\psi }_i}(t^I\psi {)}_i+\frac{\delta L}{\delta {\partial }_{\mu }{\psi }_i}(t^I{\partial }_{\mu }\psi {)}_i+{\left(\frac{\delta L}{\delta A_{\nu }},A_{\nu }\right)}^I+{\left(\frac{\delta L}{\delta {\partial }_{\nu }{A}_{\mu }},{\partial }_{\mu }{A}_{\nu }\right)}^I\right){\xi }_I+\\ \left(\frac{\delta L}{\delta {\partial }_{\mu }{\psi }_i}(t^I\psi {)}_i+\frac{\delta L}{\delta A_{\mu }^I}+{\left(\frac{\delta L}{\delta {\partial }_{\nu }{A}_{\mu }},A_{\nu }\right)}^I\right){\partial }_{\mu }{\xi }_I+\\ \frac{\delta L}{\delta {\partial }_{\nu }{A}_{\mu }^I}{\partial }_{\mu }{\partial }_{\nu }{\xi }^I=0\end{array}\end{array}$(2.2.8)

Since the derivatives of $\xi$ are functionally independent, this equation implies that each line must vanish separately: the first line is a consequence of global symmetries – the equation would have to be satisfied even if the symmetry was independent of the spacetime point – and the remaining two are consequences of local symmetries. These are the fundamental constraints on the dynamics that we propose to analyze, and the task of the remainder of this section will be to unpack them.

The requirement that each of these lines vanishes provides a strong constraint on the form of the Lagrangian, and hence on the dynamics. This, I claim, provides the core of the Noether gauge argument.

2.2.3 The Four Different Cases

To extract interesting physical information from the constraint given by (2.2.8), there are four sectors to compare, arising from the use of either global or local symmetries, and either $A$-independent or $A$-dependent Lagrangians. We treat each sector in turn.

The results will be: when $A$ does not figure in the Lagrangian, a theory with global symmetries can be dynamically nontrivial and complete. In such a theory the charges don’t couple to forces, so it will not require further constraints for consistency. With local symmetries and no $A$-dependence, the constraints demand that the dynamics be trivial, that is, no kinetic term for the matter field can appear in the Lagrangian. When forces have their own dynamics, that is, when the Lagrangian is $A$-dependent, a theory with global symmetries may be incomplete, and require further constraints to render the dynamics of $A$ compatible with charge conservation; an example will be given. It is only in the last case, where we have local symmetries and $A$-dependence, that the equations of motion coupling forces to charges is automatically consistent with the conservation of charges (and so no further constraints are required). Thus, we will see the power of malleable symmetries and $A$-dependence together to secure an interacting dynamics that conserves charge. And this will be our Noether gauge argument.

Force-Independent Lagrangian, with Global Symmetries

First, suppose we are as in the first step of the textbook gauge argument: there is no $A$ in sight, and the symmetry is global, so that ${\partial }_{\mu }{\xi }^I=0={\partial }_{\mu }{\partial }_{\nu }{\xi }^I$. Then the vanishing of the first line of Equation (2.2.8) reduces to

$\frac{\delta L}{\delta {\psi }_i}(t^I\psi {)}_i+\frac{\delta L}{\delta {\partial }_{\mu }{\psi }_i}(t^I{\partial }_{\mu }\psi {)}_i=0.$(2.2.9)

But by the Euler-Lagrange equations $EL(\psi {)}_i\approx 0$, where $EL(\psi {)}_i=\frac{\delta L}{\delta {\psi }_i}-{\partial }_{\mu }\frac{\delta L}{\delta {\partial }_{\mu }{\psi }_i}$, we have

$\frac{\delta L}{\delta {\psi }_i}\approx {\partial }_{\mu }\frac{\delta L}{\delta {\partial }_{\mu }{\psi }_i},$(2.2.10)

where we again are using “$\approx$” to denote “on-shell” equality. Applying this to Equation (2.2.9) we find that

${\partial }_{\mu }\left(\frac{\delta L}{\delta {\partial }_{\mu }{\psi }_i}(t^I\psi {)}_i\right)={\partial }^{\mu }{J}_{\mu }^I\left(\psi \right)\approx 0$(2.2.11)

where we have defined the part that is conserved as the matter current:

$J_{\mu }^I\left(\psi \right):=\frac{\delta L}{\delta {\partial }_{\mu }{\psi }_i}(t^I\psi {)}_i,$(2.2.12)

as is customary.

In summary, we have derived what is guaranteed by Noether’s first theorem, that the current $J_{\mu }^I\left(\psi \right)$ is conserved on-shell. Or, turning this around: symmetry requires the Lagrangian to be restricted so that $J_{\mu }^I\left(\psi \right)$ defined in Equation (2.2.12) is divergenceless. Having constrained the space of theories in this manner, there are no more equations to satisfy: conservation of charge is consistent with the dynamics and no further constraints need to be imposed.

Force-Independent Lagrangian, with Local Symmetries

In the next case, suppose that we allow – in addition to “Force-independent Lagrangian, with global symmetries” section’s equations – the ones arising from a $\partial \xi \ne 0$, while still not allowing for an $A$ in the theory. We get, in addition to equations (2.2.12) and (2.2.11), from the vanishing of the second line of Equation (2.2.8):

$\frac{\delta L}{\delta {\partial }_{\mu }{\psi }_i}(t^I\psi {)}_i=J_{\mu }^I(\psi )=0.$(2.2.13)

So here the conserved currents are forced to vanish. Clearly this condition is guaranteed for all field values if $\frac{\delta L}{\delta {\partial }_{\mu }{\psi }_i}=0$, which requires a vanishing kinetic term. A careful analysis of more general cases reveals this is the only generic solution.Footnote ¹⁶

This analysis pinpoints the obstacle appearing in the textbook gauge argument that we rehearsed in Section 2.1.1. When the matter field Lagrangian has a nontrivial kinetic term, local transformations cannot be variational symmetries. That is: if we impose local symmetries without introducing a gauge potential, we cannot consistently also allow a term in the Lagrangian including ${\partial }_{\mu }{\psi }_i$. It is to allow such terms and still retain the local symmetries that the next two sections will introduce the gauge potential.

Force-Dependent Lagrangian, with Global Symmetries

We first proceed precisely as in the first case, introducing the $A$ field, but still keeping the symmetries global. Using the equations of motion for $A$ as well as those of $\psi$, that is, $EL\left[A\right]=0$ as well as $EL\left[\psi \right]=0$, we get, in direct analogy to (2.2.11), a conserved current that is a sum of two currents:Footnote ¹⁷

${\partial }_{\mu }\left(\frac{\delta L}{\delta {\partial }_{\mu }{\psi }_i}(t^I\psi {)}_i+{\left(\frac{\delta L}{\delta {\partial }_{\nu }{A}_{\mu }},A_{\nu }\right)}^I\right)={\partial }^{\mu }\left(J_{\mu }^I\right(\psi )+{\tilde{J}}_{\mu }^I(A\left)\right)\approx 0$(2.2.14)

and nothing more; there are no further conditions that the terms of the Lagrangian need to obey. (Here, the definition of ${\tilde{J}}_{\mu }^I\left(A\right))$ is given implicitly by (2.2.14).)

So, unlike the previous case, which admitted only a trivial kinetic term for the matter field $\psi$, this sector will admit many possible dynamics. The problem here is of a different nature: the theories are not sufficiently constrained; the equations of motion do not automatically guarantee conservation of charges.

Let us look at an example of how things can go wrong in this intermediate sector containing forces but only global symmetries, for the simple, Abelian theory. In the Abelian theory, $\tilde{J}\left(A\right)\equiv 0$, since quantities trivially commute. Thus, Equation (2.2.14) only contains the standard conservation of the matter charges and the symmetries are silent about the relationship between this charge and the dynamics of the forces.

Consider a kinetic term of the form ${\partial }_{(\mu }{A}_{\nu )}{\partial }^{(\mu }{A}^{\nu )}$ where round brackets denote symmetrization. So this differs from the standard Maxwell theory kinetic term for the gauge potential: namely, $V_{\mu \nu }{F}^{\mu \nu }:={\partial }_{[\mu }{A}_{\nu ]}{\partial }^{[\mu }{A}^{\nu ]}$ where square brackets denote anti-symmetrization. But the symmetrized version is nonetheless gauge-invariant (under global transformations). Now, the Euler-Lagrange equations for this theory differ only very slightly from the Maxwell-Klein-Gordon equations. The equations of motion for $A$ yield:

${\partial }^{\mu }\left({\partial }_{(\mu }{A}_{\nu )}\right)=J_{\nu }$(2.2.15)

in contrast with the usual ${\partial }^{\mu }\left({\partial }_{[\mu }{A}_{\nu ]}\right)=J_{\nu }$. But while the divergence of the right-hand side automatically vanishes, unlike the usual case the divergence of the left-hand side does not:

${\partial }^{\nu }{\partial }^{\mu }\left({\partial }_{(\mu }{A}_{\nu )}\right)={\partial }^{\mu }{\partial }_{\mu }{\partial }^{\nu }{A}_{\nu }=\square {\partial }^{\nu }{A}_{\nu }̸\equiv 0.$(2.2.16)

At this point, we would have to go back to the drawing board and introduce more constraints on the theory: this theory does not couple forces to charges in a manner that guarantees charge conservation.

Thus we glimpse my overall thesis: only by introducing local gauge symmetries do we restrict interactions between forces and their sources so that they are consistent with the conservation of the matter current.

Of course, in this example the culprit is easily found: the kinetic term ${\partial }_{(\mu }{A}_{\nu )}{\partial }^{(\mu }{A}^{\nu )}$ is not invariant under local transformations. According to the next section – our fourth sector – requiring this stronger form of invariance will restrict us to the space of consistent interactions. No tweaking required.

Force-Dependent Lagrangian, with Local Symmetries

In this fourth sector, we again obtain (2.2.14), from the vanishing of the first line of (2.2.8) – the constraint for the global symmetry – since nothing changes at that level. But, from the vanishing of the second line in Equation (2.2.8), we have:

$-\frac{\delta L}{\delta A_{\mu }^I}=\frac{\delta L}{\delta {\partial }_{\mu }{\psi }_i}(t{}^I\psi {)}_i+{\left(\frac{\delta L}{\delta {\partial }_{\nu }{A}_{\mu }},A_{\nu }\right)}^I=J{}_{\mu }^I(\psi )+{\tilde{J}}_{\mu }^I(A).$(2.2.17)

Once again using the Euler-Lagrange equations for $A$ to substitute the left-hand side, we find that

$EL(A{)}_{\mu }^I=\frac{\delta L}{\delta A_{\mu }^I}-{\partial }_{\nu }\frac{\delta L}{\delta {\partial }_{\nu }{A}_{\mu }^I}\approx 0.$(2.2.18)

Defining $\frac{\delta L}{\delta {\partial }_{\nu }{A}_{\mu }^I}=:k_{\mu \nu }^I$, we now obtain:

$J_{\mu }^I\left(\psi \right)+{\tilde{J}}_{\mu }^I\left(A\right)=-{\partial }^{\mu }{k}_{\mu \nu }^I+EL(A{)}_{\mu }^I\approx -{\partial }^{\mu }{k}_{\mu \nu }^I$(2.2.19)

This equation links both the matter and force currents to the dynamics of the force field, given in $EL(A{)}_{\mu }^I$.

We already know from the constraint for the global symmetry, Equation (2.2.8), that the sum of the currents is divergence-free on shell (cf. Equation 2.2.14). Thus, taking the divergence of (2.2.19), the left side vanishes, thus consistency between charge conservation and the dynamics of the force fields demand that the right-hand side must also vanish, implying that ${\partial }^{\nu }{\partial }^{\mu }{k}_{\mu \nu }^I=0$. Since partial derivatives are necessarily symmetric, all we need in order to satisfy conservation is that:

$k_{\mu \nu }^I=-k_{\nu \mu }^I\text{or}{k}_{\mu \nu }^I=k_{\left[\nu \mu \right]}^I,$(2.2.20)

which is just what we have from the vanishing of the third line of Equation (2.2.8). So the condition was automatically satisfied. The result of including local symmetries, in this simple case, restricts us to consider Lagrangians in which the derivatives of $A_{\mu }^I$ only enter in anti-symmetrized form: ${\partial }_{[\mu }{A}_{\nu ]}^I$. This restriction excludes the previous example of Equation (2.2.15).

More generally, if we try to find a Lagrangian that includes force fields without obeying the relations obtained from the local symmetries, the equations of motion of the force fields and those relating force fields and matter may require further constraints to be compatible with charge conservation, as we saw in the counter-example in the previous section. This is one superpower of local gauge symmetries: they link charge conservation – taken as empirical fact or on a priori grounds – with the form of the Lagrangian for the force fields.

3.2.1 The Intuition Behind Fiber Bundles

To gather intuition about principal fiber bundles (PFBs) as the “organizers” of symmetry principles, as described in Section 3.1, it is worthwhile to introduce them in the context of the familiar tangent vector fields on $M$.

Fiber bundles are spaces that locally look like a product; that is, they form a ‘bundle’ of fibers over a base manifold (usually spacetime). Let us denote fiber bundles by $E$; they are smooth manifolds that admit the action of a surjective projection ${\pi }_E:E\to M$ so that any point of $M$ has a neighborhood, $U\subset M$, such that $E$ is locally of the form ${\pi }_E^{-1}\left(U\right)\simeq U\times V$, where $V$, as previously, is isomorphic to some “fiber”: a space over each point of $M$ and in which the fields take their values, and similarly for all subsets of $U$, which ensures that ${\pi }_E^{-1}\left(x\right)\simeq V$. But the isomorphism between ${\pi }_E^{-1}\left(U\right)$ and $U\times V$ is not unique, which is why there is no canonical identification of elements of fibers over different points of spacetime. Each choice of isomorphism is called “a trivialization” of the bundle: it is basically a coordinate system that makes the local product structure explicit. It is standard to denote a fiber bundle $E$ over $M$, with typical fiber $V$, with the triple $(E,M,V)$.

Sections replace the functions $\tilde{\kappa }:M\to V$, that we would employ if the fields had a fixed, or “absolute” – that is, spacetime independent – space of values.

There are essentially two kinds of bundles that we will encounter here: a vector bundle and a principal fiber bundle; a third type, an associated vector bundle, is a vector bundle that is associated to a principal bundle.

Each matter field in a gauge theory is described by a section on a vector bundle, corresponding to that field. Indeed, given a vector bundle $E$ over $M$ (which we will describe next in more depth), we can directly define an affine connection $D$ as:

$D:\Gamma \left(E\right)\to \Gamma (T^{*}M\otimes E)$(3.2.1)

such that the product rule

$D\left(fs\right)=df\otimes s+fDs$(3.2.2)

is satisfied for all smooth, real (or complex)-valued functions $f$. But then the reader should ask: aren’t we essentially done? If we can define a covariant derivative for different matter fields directly, why introduce any other kind of bundle, what further structure do we need, for example, in order to write down a Lagrangian?

The problem, as it stands, is that each vector bundle has its own covariant derivative, and so the covariant derivatives of different matter fields are “uncoordinated.” Without such a coordination, covariant derivatives of different, but interacting fields would not “march-in-step.” This would imply that the notion of relative “charge,” for example, of electric charge of different matter fields, would be extremely history dependent, and unhelpful. The role of principal and associated bundle is to provide a mechanism for the coordination of covariant derivative among fields that have charges of the same type. In other words, associated vector bundles inherit their covariant derivatives from a single principal bundle, and so, if we tie each force to a principal bundle, we solve this coordination problem. I will discuss this further in Section 4.

3.2.2 The Mathematics of Principal Bundles*

A principal fiber bundle is, in short, just a manifold where some group acts, and whose equivalence classes under the group action correspond 1-1 to points of spacetime. In detail:

Naturally, we construct a projection $\pi :P\to M$ onto equivalence classes, given by $p\sim q\iff p=g\cdot q$ for some $g\in G$. That is: the base space $M$ is the orbit space of $P$, $M=P/G$, with the quotient topology; that is, it is characterized by an open and continuous $\pi :P\to M$.Footnote ²³ By definition, $G$ acts transitively on each fiber, that is, on each orbit of the group. Here, unlike in the general definition of a fiber bundle, we don’t need to postulate the local product structure: ${\pi }^{-1}\left(U\right)\simeq U\times G$; it is easy to prove that this follows from Definition 2 (see the section “Local sections” for the proof).

The automorphism group of $P$ are fiber-preserving diffeomorphisms, that is:

Definition 3 Diffeomorphisms

$\tau :P\to P\text{such that}\tau (g\cdot p)=g\cdot \tau \left(p\right).$(3.2.3)

Vertical automorphisms are those fiber-preserving diffeomorphisms for which $\pi \circ \tau =\pi$; that is, they are purely “vertical” automorphisms of the bundle.

But to link fibers, we need to postulate more structure than just $P$: we need a connection.

The Ehresmann Connection-Form

Given an element $\xi$ of the Lie-algebra $g$, and the action of $G$ on $P$, we use the exponential to find an action of $g$ on $P$. This defines an embedding of the Lie algebra into the tangent space at each point, given by the hash operator: ${\#}_p:g\to T_{p}P$. The image of this embedding we call the vertical space $V_p$ at a point $p\in P$: it is tangent to the orbits of the group, and is linearly spanned by vectors of the form

$\text{for}\xi \in g:{\xi }^{\#}\left(p\right):=\frac{d}{dt}{}_{|t=0}(exp(t\xi )\cdot p)\in V_p\subset T_{p}P.$(3.2.4)

Vector fields of the form ${\xi }^{\#}$ for $\xi \in g$ are called fundamental vector fields.Footnote ²⁴ The vertical spaces are defined canonically from the group action, as in (3.2.4). But we can define an “orthogonal” projection operator, $\hat{V}$ such that:

$\hat{V}{|}_V=Id{|}_V,\hat{V}\circ \hat{V}=\hat{V},$(3.2.5)

and defining $H\subset TP$ as $H:=ker\left(\hat{V}\right)$. It follows that $\hat{H}=Id-\hat{V}$ and so $\hat{V}\circ \hat{H}=\hat{H}\circ \hat{V}=0$.Footnote ²⁵ Moreover, since ${\pi }_{*}\circ \hat{V}=0$ it follows that:

${\pi }_{*}\circ \hat{H}={\pi }_{*}.$(3.2.6)

As I said in the previous section, the connection-form should be visualized essentially as the projection onto the vertical spaces: given some infinitesimal direction, or change of frames, the vertical projection picks out the part of that change that was due solely to a different choice of frames, and the connection-form tells us what that change of frame was. The only difference between $\hat{V}$ and $\omega$ is that the latter is $g$-valued, Thus we get it via the isomorphism between $V_p$ and $g$ ($\omega$’s inverse is $\#:g\mapsto V\subset TP$).

One often defines the connection directly, without appeal to vertical spaces:

Definition 4 (An Ehresmann connection-form) $\omega$ is defined as a Lie-algebra valued one form on $P$, satisfying the following properties:

$\omega \left({\xi }^{\#}\right)=\xi \text{and}{L_g}^{*}\omega ={Ad}_g\omega ,$(3.2.7)

where the adjoint representation of $G$ on $g$ is defined as ${Ad}_g\xi =g\xi g^{-1}$, for $\xi \in g$; ${L_g}^{*}$ is the pull-back of $TP$ induced by the diffeomorphism $g:P\to P$.

But it is possible to show that

Proposition 1 A Lie-algebra-valued one form on $P$ satisfies (3.2.7) if and only if $\omega ={\#}^{-1}\circ \hat{V}$ (where ${\#}^{-1}$ is only defined in the restriction to the vertical subspace $V\subset TP$).

The relationship between the connection, the Lie-algebra, and the vertical projection is illustrated in Figure 2.

Figure 2 The relation between the Ehresmann connection form $\omega$ and a vertical projection, on the principal fiber bundle with structure group $G$. Taken from Wikipedia, under Creative Commons License.

If, on the second condition in (3.2.7), we take the infinitesimal pull-back, we get the Lie derivative along a vector ${\xi }^{\#}$ on the left-hand side, and a Lie-algebra commutator on the right-hand side, that is,

$L_{{\xi }^{\#}}\omega =[\omega ,\xi ].$(3.2.8)

This equation is only valid for fundamental vector fields, ${\xi }^{\#}$. But a vertical field may be vertical without being fundamental: we could take different $\xi \in g$ at different orbits (as discussed in footnote 24), that is, $Z_p^v:=\left(\xi \right(\pi \left(p\right){)}^{\#}{)}_p\in T_{p}P$, which we abbreviate to $\left(\xi \right(x{)}^{\#}{)}_p$. The Lie derivative in (3.2.8) is not $C^{\infty }$-linear on ${\xi }^{\#}$, and so we expect some difference when we compute $L_{\xi (x{)}^{\#}}\omega$. To see what that is, we first define the inner derivative (or alternating contraction operator) on differential forms, $\iota$, so that the contraction of ${\iota }_{{\xi }^{\#}\left(\pi \right(p\left)\right)}\Lambda$ is $C^{\infty }$-linear in ${\xi }^{\#}\left(x\right)$, for any form $\Lambda$. Thus:

${\iota }_{{\xi }^{\#}\left(x\right)}d{\omega }_p=[{\omega }_p,\xi (x{)}_p]$(3.2.9)

Now, I will merely state Cartan’s Magic Formula, describing the relation between the Lie derivative and inner and exterior derivative (which is proven inductively)

$L_{{\xi }^{\#}\left(x\right)}(\bullet )=({\iota }_{{\xi }^{\#}\left(x\right)}d+d{\iota }_{{\xi }^{\#}\left(x\right)})(\bullet ).$(3.2.10)

This equation is extremely useful in differential calculus, and here it can be used to compute:

$L_{{\xi }^{\#}\left(x\right)}\omega ={\iota }_{{\xi }^{\#}\left(x\right)}d\omega +d\left(\omega \right({\xi }^{\#}\left(x\right)\left)\right)=[\omega ,\xi (x\left)\right]+d\xi \left(x\right),$(3.2.11)

where, reinstating the $\pi \left(p\right)$ in place of $x$, we read the action of the second term on $Z\in \Gamma \left(TP\right)$ as $d\xi \left(\pi \right(p\left)\right)\left(Z\right)={\pi }_{*}\left(Z\right)\left[\xi \right(\pi \left(p\right)\left)\right]$, which, in a local trivialization takes the derivative of the spacetime function and leaves the Lie-algebra values intact. Equation (3.2.11) will be useful to compute the change of the connection under a change of gauge.

Let us pause here for a second to describe an important related notion, of horizontal lift. The horizontal lift of a vector $X_x\in T_{x}M$ through $p\in {\pi }^{-1}\left(x\right)\subset P$ is a horizontal vector $X_p^h$ such that ${\pi }_{*}{X}_p^h=X_x$.

Let $\left[\right[\bullet ,\bullet ]{]}_N$ be the commutator of vector fields on a smooth manifold $N$. Then it is easy to show that (see Kobayashi & Nomizu (Reference Kobayashi and Nomizu1963, Prop 1.3): (i) the lift of $X+Y$ is $X^h+Y^h$, (ii) $f^h{X}^h$ is the lift of $fX$, where $f^h:=f\circ \pi$, (iii) $\hat{H}\left(\right[[X^h,Y^h]\left]\right)$ is the horizontal lift of $\left[\right[X,Y]{]}_M$ (this is the only nontrivial item, but it is easy to prove: for $\hat{H}\left(\right[[X^h,Y^h]{]}_P)$ is horizontal, and ${\pi }_{*}\hat{H}\left(\right[[X^h,Y^h]{]}_P)={\pi }_{*}(\left[\right[X^h,Y^h\left]{]}_P\right)=\left[\right[X,Y]{]}_M$ from the first two items.

In the principal bundle formalism parallel transport along a curve $\gamma :[0,1]\to M$, with $\gamma \left(0\right)=x$ and $\gamma \left(1\right)=y$, is described via a horizontal lift ${\gamma }^h$ of $\gamma$ through a particular initial point, or frame, ${\gamma }^h\left(0\right)=p\in {\pi }^{-1}\left(x\right)$. So one might be inclined to think that parallel transport requires the stipulation of an initial $p\in {\pi }^{-1}\left(x\right)$, for example, an initial frame. But the horizontal lift commutes with the group action, ${\gamma }^h\circ L_g=L_g\circ {\gamma }^h$ (which follows from horizontal curves being sent to horizontal curves by translation of the origin; cf. Kobayashi & Nomizu (Reference Kobayashi and Nomizu1963, Ch. II Prop. 3.2). That means we can think of parallel transport as an isomorphism of an initial to a final fiber, for example, for the path $\gamma$:

${\tau }_{\gamma }:{\pi }^{-1}\left(x\right)\to {\pi }^{-1}\left(y\right).$(3.2.12)

Given two different curves, $\gamma ,{\gamma }^{\prime }$, both between $x$ and $y$, it follows that there exists a $g\in G$ such that ${\tau }_{\gamma }=g\cdot {\tau }_{{\gamma }^{\prime }}$. By the composition properties of parallel transport, it is customary to focus only on closed curves $\gamma$, starting (and ending) at $x\in M$. For a path-connected $M$, the subgroup generated by such elements for all such closed paths depends on the base point $x$, only up to conjugation in $G$. So usually, the total group generated by parallel transport around closed curves is called the holonomy group, denoted by $Hol\left(\omega \right)$.

It is clear that, since parallel transport can be thought of at the level of entire fibers, as in (3.2.12), there is a frame-independent abstract mathematical object that corresponds to the Ehresmann connection form, sometimes called the Atyiah-Lie connection. This is a section of the vector bundle $T^{*}P/G$.Footnote ²⁶ In other words, if we know what parallel transport is at $p$, we know what it is at $g\cdot p$. By getting rid of this redundancy, we can find a global spacetime representation of the connection $\omega$. This Atyiah-Lie connection is a section on the bundle of connections, that is, ${\rm Y}\in \Gamma \left(T^{*}P/G\right)$, where $T^{*}P/G$ is a vector bundle over spacetime.Footnote ²⁷

The Curvature

To define curvature, we note that an infinitesimally small parallelogram with horizontal sides that projects onto a closed parallelogram on $M$, may not close on $P$: if a horizontal parallelogram starts at $p\in P$, it may end at $g\cdot p$. In other words, the horizontal distributions need not be involutive.

Definition 5 (The curvature Ω of ω) is a Lie-algebra valued two-form on $P$:

$\Omega (\bullet ,\bullet ):=\omega \left(\right[\left[\hat{H}\right(\bullet ),\hat{H}(\bullet \left)\right]{]}_P)$(3.2.14)

Let $P(M,G)$ be a principal fiber bundle and $\rho$ a representation of $G$ on a finite-dimensional vector space $V$; $\rho \left(a\right)$ is a linear transformation of $V$ for each $a\in G$ and $\rho \left(ab\right)=\rho \left(a\right)\rho \left(b\right)$ for $a,b\in G$. And let ${\Lambda }^n\left(N\right)$ be the space of alternating n-forms on a smooth manifold $N$.

Definition 6 (Pseudo-tensorial and tensorial forms.) A pseudotensorial form of degree $r$ on $P$ of type $(\rho ,V)$ is a $V$-valued $r$-form $\phi$ on $P$ such that

$L_g^{*}\phi =\rho \left(g\right)\cdot \phi \text{for}g\in G.$(3.2.15)

Such a form $\phi$ is called a tensorial form if it is horizontal, that is, $\phi (X_1,\dots ,X_r)=0$ whenever at least one of the tangent vectors $X_i$ of $P$ is vertical, that is, tangent to a fiber.

In other words, a pseudo-tensorial form is covariant under the group action, but not necessarily horizontal: it is only when it is also horizontal that we call it tensorial.

Then we define

Definition 7 (The gauge covariant exterior derivative) of pseudo-tensorial forms as:

$D\phi :=\left(d\phi \right)\circ \hat{H}.$(3.2.16)

It is easy to show that, whereas $d\phi$ is still only pseudo-tensorial, $D\phi$ is tensorial, and not only pseudo-tensorial. In the language of principal fiber bundles, this is why minimal coupling, $d\to D$, renders functions “coordinate-independent”: they acquire trivial dependence on the vertical directions along the fiber (which represent “coordinate” or frame changes).

We can rewrite (3.2.16) as:

$D\phi (\bullet )=d\phi (\bullet )-d\phi \left(\omega \right(\bullet {)}^{\#}),$(3.2.17)

where the second term is linear in $\omega (\bullet {)}^{\#}$ (i.e. can be read as ${\iota }_{\omega (\bullet {)}^{\#}}d\phi$) and can be understood as the vertical correction to the ‘gradient’ of $\phi$, so that this ‘gradient’ stays horizontal. We have:

${\iota }_{\omega (\bullet {)}^{\#}}d\phi =\rho \left(\omega \right(\bullet \left)\right)\phi .$(3.2.18)

Locally, in a trivialization of $P$, we write ${\phi }_{|U}\in \Gamma \left({\Lambda }^n\right({\pi }^{-1}\left(U\right))\otimes V)$, so ${\phi }_{|U}={\phi }_i{e}^i$, where ${\phi }_i\in {\Lambda }^n\left({\pi }^{-1}\right(U\left)\right)$ is a real-valued pseudo-tensorial n-form, and $\left\{e_i\right\}$ is a basis for $V$. Then $\rho (\omega {)}_{|U}\in \Gamma ({\Lambda }^n\left({\pi }^{-1}\right(U\left)\right)\otimes GL\left(V\right))$ and we can write, in this basis:

$\left(\rho \right(\omega )\phi {)}_{|U}=\rho (\omega {)}_j^i\wedge {\phi }^j{e}_i,$(3.2.19)

which gives the usual expression for the action of the covariant derivative in (3.2.17). This action on other Lie-algebra valued forms will usually be written just as $[\omega ,\phi ]$, with the understanding that, on a trivialization, we apply the $\wedge$ to the differential forms and the Lie bracket to the Lie algebra elements.

As with $\omega$, pseudo-tensorial forms are only required to be equivariant under the pointwise action of the group action, as in (3.2.15). Under spacetime dependent transformations, pseudo-tensorial forms are not necessarily equivariant (satisfying something like (3.2.15)). But the gauge-covariant derivative corrects for that: that is its role. In the infinitesimal case, we now prove:

Proposition 2 For a pseudo-tensorial form $\phi$ as per Definition 6, under an infinitesimal vertical autormorphism, $\xi (x{)}^{\#}$, we have the equivariance property:

$L_{{\xi }^{\#}\left(x\right)}D\phi =\rho \left(\xi \right)D\phi$

We use Cartan’s Magic Formula (3.2.10) and the Lie derivative of the Ehresmann connection, given in (3.2.11), in Equation (3.2.17), written as: $D\phi =d\phi -\rho \left(\omega \right)\phi$. Applying $d{\iota }_{{\xi }^{\#}\left(x\right)}$ to anything horizontal, like $D\phi$, vanishes completely, since it first linearly contracts the horizontal form with a vertical vector. Now we apply ${\iota }_{{\xi }^{\#}\left(x\right)}d$ to $D\phi$, obtaining (first term vanishes since $dd=0$):

$\begin{array}{c}{\iota }_{{\xi }^{\#}\left(x\right)}d\left(D\phi \right)=-{\iota }_{{\xi }^{\#}\left(x\right)}d\left(\rho \right(\omega \left)\phi \right)\end{array}$(3.2.21)

So, first, as will be shown in Proposition 3 just below:Footnote ²⁸

$-{\iota }_{{\xi }^{\#}\left(x\right)}d\omega =-{\iota }_{{\xi }^{\#}\left(x\right)}(-[\omega ,\omega \left]\right)=-\left[\xi \right(x),\omega ]+[\omega ,\xi (x\left)\right],$(3.2.22)

and so:

$\begin{array}{cc}{\iota }_{{\xi }^{\#}}d\left(D\phi \right)=& \rho (-[\xi ,\omega ]+[\omega ,\xi \left]\right)\phi -{\iota }_{{\xi }^{\#}}\left(\rho \right(\omega \left)d\phi \right)\end{array}$(3.2.23)

$\begin{array}{cc}=& (-[\rho \left(\xi \right),\rho \left(\omega \right)]+[\rho \left(\omega \right),\rho \left(\xi \right)\left]\right)\phi -\rho \left(\xi \right)d\phi \\ & +\rho \left(\omega \right)\rho \left(\xi \right)\phi \end{array}$(3.2.24)

$\begin{array}{cc}=& \rho \left(\xi \right)(d\phi -\rho (\omega \left)\phi \right)=\rho \left(\xi \right)D\phi ,\end{array}$(3.2.25)

where in going from the first to the second line, I used (3.2.18), and ${\iota }_{{\xi }^{\#}}\omega =\xi$, and going from the second to the third I used $[a,b]=\frac{1}{2}(ab-ba)$. □.

From Proposition 2, it follows that:

$L_{{\xi }^{\#}\left(x\right)}\Omega =[\xi ,\Omega ]{.}^{29}$(3.2.26)

Footnote ²⁹

Proposition 3 The two next definitions of curvature are equivalent to (3.2.14):

$\begin{array}{cc}\Omega & =D\Omega \\ \Omega & =d\omega +[\omega ,\omega ],\end{array}$(3.2.30)

where $d$ is the exterior derivative on $P$.

The proof proceeds through explicit insertion of horizontal and fundamental vertical vector fields and multi-linearity. First, one writes

$d\omega (X,Y)=X\left[\omega \right(Y\left)\right]-Y\left[\omega \right(X\left)\right]-\omega \left(\right[[X,Y]{]}_P),$(3.2.31)

the standard formula for the exterior derivative of a 1-form. (Note: this differs from the formula in some textbooks, such as in Kobayashi & Nomizu [Reference Kobayashi and Nomizu1963], by a factor of 2 on the left-hand-side; this gives a difference of 1/2 on the second term on (3.2.31)). So for two horizontal fields $X_H,Y_H$, since $\omega \left(X_H\right)=0=\left[\omega \right(X_H),\omega (Y_H\left)\right]$, it is immediate that :

$\left(d\omega \right)\circ \hat{H}(X_H,Y_H)=d\omega (X^h,Y^h)=d\omega (X^h,Y^h)+\left[\omega \right(X^h),\omega (Y^h\left)\right].$(3.2.32)

And using (3.2.31) it is immediate that $d\omega (X^h,Y^h)=\omega \left(\right[\left[\hat{H}\right(X^h),\hat{H}(Y^h\left)\right]{]}_P).$ This is the only case which has nonvanishing curvature. Now, for two vertical fields, the only nontrivial part of the equalities is to show that:

$d\omega ({\xi }^{\#},{\eta }^{\#})=-\left[\omega \right({\xi }^{\#}),\omega ({\eta }^{\#}\left)\right].$

We write, for $\xi ,\eta \in g$, $X={\xi }^{\#},Y={\eta }^{\#}$, and note that, because the orbits form integral submanifolds, commutators of vertical vector fields are vertical, and $[\xi ,\eta {]}^{\#}=[[{\xi }^{\#},{\eta }^{\#}]{]}_P$. So it follows from (3.2.31) that $d\omega ({\xi }^{\#},{\eta }^{\#})=-[\xi ,\eta ]$. I will only sketch the case for one vertical and one horizontal field (cf. Kobayashi & Nomizu (Reference Kobayashi and Nomizu1963, Theo. 5.2) for more detail). The idea is to show that the commutator $\left[\right[{\xi }^{\#},X^h]{]}_P$ between a fundamental vector field ${\xi }^{\#}$ and a horizontal vector field $X^h$ (that is covariant under $G$) is horizontal as well. To show that, we define the horizontal field $X_{g\cdot p}^h=L_g{}_{*}{X}_p^h$, and note that the Lie derivative $L_{{\xi }^{\#}}{X}^h=\underset{t\to 0}{lim}\frac{1}{t}(X^h-L_{exp\left(t\xi \right)}{}_{*}{X}_h)$ is also horizontal, since it is the difference between two horizontal vectors at $p$, and so this ensures that the right-hand side of (3.2.31) vanishes. □

Proposition 4 (the Bianchi identity) $D\Omega =0$ – this is called the Bianchi identity.

By the definition, it is sufficient to compute its value on three horizontal vectors (the others vanish). The gauge-covariant exterior derivative is (anti)linear, so $\hat{H}(X^h,Y^h,Z^h)=(X^h,Y^h,Z^h)$ and:

$D\Omega (X^h,Y^h,Z^h)=\left(d\Omega \right)(X^h,Y^h,Z^h)=(dd\omega +[d\omega ,\omega ]-[\omega ,d\omega \left]\right)(X^h,Y^h,Z^h)=0,$(3.2.33)

since every term has at least one contraction of $\omega$ with a horizontal vector. The Bianchi identity is a nontrivial condition that any curvature satisfies (see Baez & Munian (Reference Baez and Munian1994, p. 278) for the geometric interpretation of this identity). □

In order to connect the definitions previously to the usual definition of $\Omega$ in terms of the exterior product, we pick out a basis for the Lie-algebra, $\{{ϵ}_I\in g\}$, with structure constants $c_{JK}^I$ defined by ${ϵ}_i{c}_{JK}^I=[{ϵ}_J,{ϵ}_K]$. In terms of this basis, we write $\omega ={\omega }^I{ϵ}_i$ and (3.2.30) becomes:

${\Omega }^I=d{\omega }^I-c_{JK}^I{\omega }^J\wedge {\omega }^K.$(3.2.34)