Electromagnetism as a U(1) Yang-Mills theory — the geometric dictionary

Intuition Beginner

Electricity and magnetism look like two different forces, but a single rulebook controls both. That rulebook can be written in two styles. One style lists the electric and magnetic fields directly. The other style starts from a hidden quantity called the potential and produces the fields from it. The potential is more basic, and it carries a surprise: you can change it in many ways without changing any measurable field.

This freedom to redraw the potential is called gauge freedom. It is the same kind of freedom you have when you choose where to put the zero mark on a thermometer. The choice does not change real temperature differences. In the same way, the gauge choice does not change the real electric and magnetic fields.

The geometric picture takes this seriously. A potential is a rule for comparing a tiny internal phase from point to point. The fields are the curvature of that rule.

Visual Beginner

The picture shows a small loop in space and a clock-hand phase carried around it. The connection is the rule that turns the hand as you move. After one trip around the loop, the hand may not return to where it started. That mismatch is the field strength.

When the loop encloses magnetic flux, the phase mismatch is exactly that flux. Curvature and field are the same thing seen from two sides.

Worked example Beginner

Take a region where the magnetic field points straight up with strength $2$, filling a flat square patch of area $3$. The magnetic flux through the patch is field times area, so the flux is $2\times 3=6$.

Now carry the internal phase hand once around the boundary of the patch. The geometric rule says the total turning of the hand equals the enclosed flux. So the hand turns by $6$ units. If a second patch of area $5$ sits in the same field, its flux is $2\times 5=10$, and the hand turns by $10$ there.

What this tells us: the field is not an extra ingredient added to the potential. It is the bookkeeping of how the potential fails to close up around loops. Larger enclosed flux means more turning.

Check your understanding Beginner

Formal definition Intermediate+

Let $M$ be a smooth four-manifold, usually Minkowski space or a region of it, and let $P\to M$ be a principal $U(1)$-bundle 03.05.01. Since the Lie algebra $\mathfrak{u}(1)\cong i\mathbb{R}$ is one-dimensional and commutative, a connection on $P$ is described in a local trivialization by an ordinary real one-form, the gauge potential

$$A=A_{\mu }d{x}^{\mu }\in {\Omega }^1(M),$$

with the convention that the connection one-form is $iA$ valued in $\mathfrak{u}(1)$ 03.05.07. The field strength is the curvature of this connection 03.05.09. For a general structure group the curvature is $F=dA+\frac{1}{2}[A,A]$, but $U(1)$ is abelian, so the bracket term vanishes and

$$F=dA,F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }.$$

A gauge transformation is a change of local trivialization by a circle-valued function $g=e^{i\lambda }$ with $\lambda \in C^{\infty }(M)$. The potential transforms by

$$A\mapsto A+d\lambda ,$$

and because $F=dA$ with $d^2=0$, the field strength is gauge-invariant: $F\mapsto F+dd\lambda =F$. This is the geometric content of the statement that the electric and magnetic fields do not depend on the gauge ^{[tong §1]}.

The decomposition of $F$ into electric and magnetic parts reads $F=E_{i}d{x}^i\wedge dt+\frac{1}{2}{ϵ}_{ijk}{B}_{k}d{x}^i\wedge d{x}^j$ with $E_i=F_{i0}$ and $B_k=\frac{1}{2}{ϵ}_{kij}{F}_{ij}$. A non-example worth flagging: a one-form is not a gauge potential unless one specifies the bundle and trivialization; over a topologically non-product base the field strength $F$ can be globally defined while $A$ exists only locally, which is exactly the monopole situation treated below ^{[Wu-Yang 1975]}.

Key theorem with proof Intermediate+

Theorem (Maxwell's equations as Bianchi plus Yang-Mills). Let $A$ be a $U(1)$ gauge potential on $M$ with curvature $F=dA$, and let $J$ be the electric current one-form. Then the homogeneous Maxwell equations are equivalent to the Bianchi identity

$$dF=0,$$

and the inhomogeneous Maxwell equations are equivalent to the abelian Yang-Mills equation

$$d\star F=\star J,$$

which is the Euler-Lagrange equation of the action $S=-\frac{1}{4}{\int }_{M}F\wedge \star F+{\int }_{M}A\wedge \star J$.

Proof. Since $F=dA$, applying the exterior derivative gives $dF=d(dA)=0$ by $d^2=0$. In components this is ${\partial }_{[\lambda }{F}_{\mu \nu ]}=0$. Writing out the four-dimensional antisymmetrization separates into the two homogeneous laws: the purely spatial component yields $\nabla \cdot B=0$, and the mixed time-space components yield ${\partial }_{t}B+\nabla \times E=0$. The detailed component computation is carried out in the worked example below.

For the inhomogeneous laws, vary the action along $A\mapsto A+ta$ with $a\in {\Omega }^1(M)$ of compact support. Because $F=dA$ is linear in $A$, the curvature varies by $\delta F=da$. The kinetic term varies as

$$\delta \left(-\frac{1}{4}{\int }_{M}F\wedge \star F\right)=-\frac{1}{2}{\int }_{M}da\wedge \star F.$$

Integration by parts moves $d$ off $a$, using $d(a\wedge \star F)=da\wedge \star F+(-1{)}^{1}a\wedge d\star F$ and Stokes' theorem to discard the boundary term:

$$\delta \left(-\frac{1}{4}{\int }_{M}F\wedge \star F\right)=-\frac{1}{2}{\int }_{M}a\wedge d\star F.$$

The source term varies as $\delta {\int }_{M}A\wedge \star J={\int }_{M}a\wedge \star J$. Setting the total first variation to zero for every compactly supported $a$ forces

$$-\frac{1}{2}d\star F+\star J=0,$$

and absorbing the constant into the normalization of the action gives $d\star F=\star J$. Unpacking with the Hodge star on Minkowski space, the time component is Gauss' law $\nabla \cdot E=\rho$ and the spatial components are Ampere's law ${\partial }_{t}E-\nabla \times B=-j$. Conservation of charge $d\star J=0$ follows because $dd\star F=0$.

Bridge. This identification builds toward the nonabelian theory of the Yang-Mills action 03.07.05, where the same variational computation produces $d_A\star F=\star J$ with the gauge-covariant derivative $d_A$ replacing $d$, and the curvature acquires the bracket term that vanishes here; it appears again in the anti-self-dual equation 03.07.06, where the four-dimensional Hodge split of $F$ that we used to separate $E$ and $B$ becomes the splitting into self-dual and anti-self-dual parts; it connects to the Chern-Weil homomorphism 03.06.06, which turns the same curvature $F$ into the topological first Chern class that quantises magnetic charge; and it underlies the Atiyah-Singer index theorem 03.09.10, where the integral $\int F$ that measures monopole charge is the simplest characteristic number.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none — the principal-bundle and Hodge-star infrastructure required to state this dictionary intrinsically is not yet present in Mathlib.

-- Pseudocode only: U(1) gauge theory as a principal connection is not yet
-- available in Mathlib.
axiom GaugePotential (M : Type*) [Manifold M] : Type*    -- a 1-form A
axiom fieldStrength {M : Type*} [Manifold M] :
    GaugePotential M → TwoForm M                         -- F = dA

axiom bianchi {M : Type*} [Manifold M] (A : GaugePotential M) :
    exteriorDeriv (fieldStrength A) = 0                  -- dF = 0

axiom gauge_invariance {M : Type*} [Manifold M]
    (A : GaugePotential M) (lambda : SmoothFunction M) :
    fieldStrength (A + exteriorDeriv lambda) = fieldStrength A

The missing formalization work includes principal $U(1)$-bundles with connection one-forms, curvature as the exterior derivative, the Hodge star and codifferential on a (pseudo-)Riemannian four-manifold, the variational derivation of $d\star F=\star J$, and the first Chern number as $\frac{1}{2\pi }\int F$.

Advanced results Master

The Dirac monopole realizes magnetic charge as the topology of a $U(1)$-bundle over the sphere surrounding the source. Remove the monopole's location and the configuration space retracts onto $S^2$. The field strength is $F=g\sin \theta d\theta \wedge d\varphi$, the curvature of a connection that has no global potential. Cover $S^2$ by the northern and southern hemispheres, each minus the opposite pole, with potentials

$$A_N=g(1-\cos \theta )d\varphi ,A_S=-g(1+\cos \theta )d\varphi ,$$

each smooth on its patch. On the equatorial overlap they differ by a gauge transformation, $A_N-A_S=2gd\varphi =d(2g\varphi )$, with transition function $g_{NS}=e^{2ig\varphi }$. Single-valuedness of $g_{NS}$ as one traverses the equator forces $2g\in \mathbb{Z}$, which is the Dirac quantisation condition relating magnetic charge to a discrete invariant ^{[Dirac 1931]}.

The invariant is the first Chern number. Chern-Weil theory assigns to the curvature of the connection the class $c_1=\frac{1}{2\pi }[F]\in H^2(S^2;\mathbb{Z})$ 03.06.06, 03.06.04, and the integral

$$\frac{1}{2\pi }{\int }_{S^2}F=2g\in \mathbb{Z}$$

is the monopole charge. The Wu-Yang two-patch description showed that the apparent string singularity of the single-potential monopole is an artifact of trying to use one trivialization where the bundle is genuinely non-product ^{[Wu-Yang 1975]}.

The Aharonov-Bohm effect is the holonomy of the connection. For a loop $\gamma$ encircling a confined flux $\Phi$, the holonomy is $\exp \left(i{\oint }_{\gamma }A\right)=\exp (i\Phi )$, an element of $U(1)$ depending only on the enclosed flux. The connection carries information that the curvature alone, evaluated where the particle travels, does not.

In the nonabelian generalisation the structure group $U(1)$ is replaced by a compact Lie group $G$ with curvature $F=dA+\frac{1}{2}[A,A]$, and the Yang-Mills equation becomes $d_A\star F=\star J$ 03.07.05. The abelian case is the linearization in which the bracket term drops and the field equations become linear in $A$. The same Chern-Weil mechanism that produces $c_1$ for $U(1)$ produces the second Chern class $c_2$ for $SU(2)$, whose integral counts instanton number; that integer organizes the moduli theory of self-dual connections 03.07.06.

Synthesis. The geometric dictionary identifies the four-potential with a $U(1)$ connection, the field strength with its curvature $F=dA$, the homogeneous Maxwell equations with the Bianchi identity $dF=0$, and the inhomogeneous equations with the abelian Yang-Mills equation $d\star F=\star J$ extremising $S=-\frac{1}{4}\int F\wedge \star F$; this dictionary builds toward the nonabelian Yang-Mills action 03.07.05 by exhibiting the abelian theory as the case where the curvature bracket vanishes, appears again in the anti-self-dual equation 03.07.06 where the Hodge split of $F$ becomes the instanton self-duality condition, connects to the Chern-Weil homomorphism 03.06.06 which converts the curvature into the first Chern number that quantises magnetic charge, and underlies the Atiyah-Singer index theorem 03.09.10 for which $\frac{1}{2\pi }\int F$ is the simplest characteristic number. The single structural fact organizing magnetic charge is that a $U(1)$-bundle over $S^2$ is classified by an integer, so charge quantisation is a topological theorem rather than a dynamical assumption, and the Dirac monopole is its physical witness.

Full proof set Master

Proposition (homogeneous Maxwell equations from the Bianchi identity). Let $A$ be a $U(1)$ gauge potential on Minkowski space with curvature $F=dA$ and components $E_i=F_{i0}$, $B_k=\frac{1}{2}{ϵ}_{kij}{F}_{ij}$. Then $dF=0$ is equivalent to the pair $\nabla \cdot B=0$ and ${\partial }_{t}B+\nabla \times E=0$.

Proof. Since $F$ is a two-form, $dF$ is a three-form with components $(dF{)}_{\lambda \mu \nu }={\partial }_{\lambda }{F}_{\mu \nu }+{\partial }_{\mu }{F}_{\nu \lambda }+{\partial }_{\nu }{F}_{\lambda \mu }$, the cyclic sum. The condition $dF=0$ is ${\partial }_{[\lambda }{F}_{\mu \nu ]}=0$ for all index triples.

Take the purely spatial triple $(\lambda ,\mu ,\nu )=(1,2,3)$. Then

$${\partial }_1{F}_{23}+{\partial }_2{F}_{31}+{\partial }_3{F}_{12}=0.$$

Using $F_{ij}={ϵ}_{ijk}{B}_k$, this is ${\partial }_1{B}_1+{\partial }_2{B}_2+{\partial }_3{B}_3=\nabla \cdot B=0$.

Now take a triple with one time index, say $(\lambda ,\mu ,\nu )=(0,i,j)$ with $i,j$ spatial:

$${\partial }_0{F}_{ij}+{\partial }_i{F}_{j0}+{\partial }_j{F}_{0i}=0.$$

Substitute $F_{ij}={ϵ}_{ijk}{B}_k$, $F_{j0}=-E_j$, and $F_{0i}=E_i$:

$${ϵ}_{ijk}{\partial }_t{B}_k+{\partial }_j{E}_i-{\partial }_i{E}_j=0.$$

Contract with ${ϵ}_{lij}$ and use ${ϵ}_{lij}{ϵ}_{ijk}=2{\delta }_{lk}$ together with ${ϵ}_{lij}({\partial }_j{E}_i-{\partial }_i{E}_j)=-2(\nabla \times E{)}_l$:

$$2{\partial }_t{B}_l-2(\nabla \times E{)}_l=0,$$

so ${\partial }_{t}B+\nabla \times E=0$. The two families of triples exhaust the independent components of the three-form $dF$, so $dF=0$ is equivalent to the stated pair. $\square$

Proposition (integrality of the first Chern number). Let $L\to S^2$ be a complex line bundle associated to a principal $U(1)$-bundle carrying a connection with curvature $F$. Then $\frac{1}{2\pi }{\int }_{S^2}F\in \mathbb{Z}$.

Proof. Cover $S^2$ by two caps $D_{+}\supset$ north pole and $D_{-}\supset$ south pole overlapping in an annulus containing the equator $\gamma$. On each cap the bundle is product and the connection has a potential, $F=d{A}_{+}$ on $D_{+}$ and $F=d{A}_{-}$ on $D_{-}$. By Stokes,

$${\int }_{S^2}F={\int }_{D_{+}}d{A}_{+}+{\int }_{D_{-}}d{A}_{-}={\oint }_{\gamma }{A}_{+}-{\oint }_{\gamma }{A}_{-},$$

where the orientation of $\gamma$ as the boundary of $D_{-}$ is opposite to its orientation as the boundary of $D_{+}$. On the overlap the two potentials differ by the transition: $A_{+}-A_{-}=-i{g}_{+-}^{-1}d{g}_{+-}=d\chi$, where $g_{+-}=e^{i\chi }$ is the $U(1)$-valued transition function. Therefore

$${\int }_{S^2}F={\oint }_{\gamma }d\chi =\chi (\text{end})-\chi (\text{start}).$$

Since $g_{+-}=e^{i\chi }$ is single-valued around $\gamma$, the function $\chi$ changes by an integer multiple of $2\pi$ over the loop: ${\oint }_{\gamma }d\chi =2\pi n$ with $n$ the winding number of $g_{+-}:\gamma \to U(1)$. Hence $\frac{1}{2\pi }{\int }_{S^2}F=n\in \mathbb{Z}$. This integer is the first Chern number $c_1(L)$, and the Chern-Weil construction 03.06.06 shows it is independent of the chosen connection. $\square$

Proposition (Aharonov-Bohm holonomy). Let $A$ be a connection with $F=0$ on a tube-complement region, and let $\gamma$ be a loop with winding number one around a confined flux $\Phi ={\int }_{S}F$, where $S$ is any surface bounded by $\gamma$. Then the holonomy of $A$ around $\gamma$ is $e^{i\Phi }$, independent of the choice of $S$ and of the basepoint.

Proof. The holonomy is $\mathrm{hol}(\gamma )=\exp (i{\oint }_{\gamma }A)\in U(1)$. For a surface $S$ with $\partial S=\gamma$, Stokes gives ${\oint }_{\gamma }A={\int }_{S}dA={\int }_{S}F=\Phi$. Independence of $S$: if $S^{\prime }$ is another capping surface, $S-S^{\prime }$ is a closed surface enclosing the flux tube, and ${\int }_{S-S^{\prime }}F$ equals an integer multiple of $2\pi$ times the enclosed charge by the previous proposition; for a single confined flux line the two surfaces both cross it once, so ${\int }_{S}F={\int }_{S^{\prime }}F=\Phi$ and the holonomies agree as elements of $U(1)$. Basepoint independence follows because the holonomy of a loop is conjugation-invariant, and $U(1)$ is abelian so conjugation acts as the identity. $\square$

Connections Master

• Yang-Mills action 03.07.05 — the abelian theory developed here is the linearization of the nonabelian Yang-Mills functional in which the curvature bracket $[A,A]$ vanishes and the field equation $d_A\star F=\star J$ reduces to the linear Maxwell equation $d\star F=\star J$.

• Anti-self-dual equation 03.07.06 — the four-dimensional Hodge split of the field strength that separates the electric and magnetic parts of $F$ is the same operator whose eigenspaces define self-dual and anti-self-dual curvature, where the abelian dictionary becomes the instanton equation.

• Chern-Weil homomorphism 03.06.06 — the construction that converts the curvature $F$ of the $U(1)$ connection into the first Chern class is the engine behind charge quantisation, identifying $\frac{1}{2\pi }{\int }_{S^2}F$ with an integer cohomology class.

• Pontryagin and Chern classes 03.06.04 — the magnetic charge of the Dirac monopole is the first Chern number of the line bundle over the enclosing sphere, the lowest case of the characteristic-class invariants of complex bundles.

• Atiyah-Singer index theorem 03.09.10 — the curvature integral that measures monopole charge is the simplest characteristic number entering the index formula, where the same integrals of curvature polynomials compute analytic indices of elliptic operators.

Historical & philosophical context Master

Weyl introduced the idea of a local scale or phase factor in his 1918 attempt to unify gravity and electromagnetism, and in 1929 recast it as the gauge invariance of the quantum-mechanical phase, fixing the modern meaning of the word "gauge" ^{[Weyl 1929]}. Yang and Mills extended the abelian phase symmetry to the nonabelian group $SU(2)$ in 1954, producing the field strength with its self-interaction bracket and the equation that the abelian Maxwell theory specializes ^{[tong §2]}. Dirac argued in 1931 that the consistency of quantum mechanics in the presence of a magnetic monopole forces electric charge to be quantised, deriving the condition $2g\in \mathbb{Z}$ from the single-valuedness of the wavefunction ^{[Dirac 1931]}. Wu and Yang reformulated the monopole in 1975 as a connection on a non-product $U(1)$-bundle described by two patches and a transition function, removing the apparent string singularity and exhibiting the magnetic charge as the first Chern number ^{[Wu-Yang 1975]}. Aharonov and Bohm predicted in 1959 that the vector potential produces an observable phase shift even where the field strength vanishes, which the dictionary reads as the holonomy of the connection.

Bibliography Master

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  author  = {Weyl, Hermann},
  title   = {Elektron und Gravitation. I},
  journal = {Zeitschrift f\"ur Physik},
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  year    = {1929}
}

@article{YangMills1954,
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  journal = {Physical Review},
  volume  = {96},
  pages   = {191--195},
  year    = {1954}
}

@article{Dirac1931,
  author  = {Dirac, P. A. M.},
  title   = {Quantised Singularities in the Electromagnetic Field},
  journal = {Proceedings of the Royal Society of London A},
  volume  = {133},
  pages   = {60--72},
  year    = {1931}
}

@article{WuYang1975,
  author  = {Wu, Tai Tsun and Yang, Chen Ning},
  title   = {Concept of Nonintegrable Phase Factors and Global Formulation of Gauge Fields},
  journal = {Physical Review D},
  volume  = {12},
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  year    = {1975}
}

@article{AharonovBohm1959,
  author  = {Aharonov, Y. and Bohm, D.},
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}

@book{Sternberg2012,
  author    = {Sternberg, Shlomo},
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  publisher = {Dover Publications},
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}