2026-10-01 · chapter Yang–Mills and the mass gapPart 2 Physical preparationgauge transformations and U(1)

Electromagnetism is a gauge theory — the freedom in writing the potentials becomes an element of U(1) chosen at each point

The protagonists of electromagnetism are two fields carrying a three-dimensional vector at each point, the electric field $\mathbf E(t,\mathbf x)$ and the magnetic field $\mathbf B(t,\mathbf x)$. A particle of charge $q$ and velocity $\mathbf v$ feels the Lorentz force $\mathbf F=q(\mathbf E+\mathbf v\times\mathbf B)$, and the fields obey Maxwell's equations (SI units; $\rho$ is the charge density, $\mathbf J$ the current density, $c^2=1/\varepsilon_0\mu_0$).

The two equations of (M1) (Gauss's law and the Ampère–Maxwell law) say how charges and currents make the fields; the two of (M2) (the absence of magnetic charge and Faraday's law of induction) are constraints on the form of the fields. Example. Where there are no sources, (M1)(M2) give $\partial_t^2\mathbf E=c^2\nabla^2\mathbf E$ (the three-dimensional version of the wave equation of 2-02), and this wave is light. The dispersion $\omega=c|\mathbf k|$ has no lower bound.

Since (M2) contains no sources, the form of $\mathbf E,\mathbf B$ can be narrowed down before solving the equations. That is the next section.

With the identities $\nabla\cdot(\nabla\times\mathbf A)=0$ and $\nabla\times\nabla f=\mathbf 0$, (M2) can be satisfied automatically. Writing $\mathbf B$ as a curl makes the first equation hold; the second becomes $\nabla\times(\mathbf E+\partial_t\mathbf A)=\mathbf 0$, so we write $\mathbf E+\partial_t\mathbf A$ as a gradient.

We call $\mathbf A$ the vector potential and $\phi$ the scalar potential. On the whole of $\mathbb R^3$, if $\nabla\cdot\mathbf B=0$ then $\mathbf B$ can always be written as a curl Known (1-06).

Example 1. For a uniform $\mathbf B=B_0\mathbf e_z$, both $\mathbf A_1=\frac{B_0}{2}(-y,x,0)$ and $\mathbf A_2=B_0(0,x,0)$ can be used. Example 2. A long solenoid of radius $a$, with a uniform $B_0\mathbf e_z$ inside, no magnetic field outside, and flux $\Phi=\pi a^2B_0$. With the distance from the axis $r$ and the unit vector $\mathbf e_\theta$ pointing around the axis,

Outside, $\nabla\times\mathbf A=\frac1r\partial_r(rA_\theta)\mathbf e_z=\mathbf 0$, and yet $\mathbf A\ne\mathbf 0$. The line integral once round a circle of radius $r$ is $\Phi$, which by Stokes's theorem is the flux passing through it. Even for a path that runs only where the curl is 0, the value of one circuit is not 0. This is because the path encloses a hole (the solenoid), and section 06 is where quantum mechanics reads off this value.

(P) does not determine $(\phi,\mathbf A)$ uniquely from $\mathbf E,\mathbf B$. With a smooth function $\chi(t,\mathbf x)$,

this replacement leaves $\mathbf E,\mathbf B$ unchanged ($\nabla\times\nabla\chi=\mathbf 0$, $\nabla\partial_t\chi=\partial_t\nabla\chi$). We call this a gauge transformation, and $\chi$ the gauge function. Example. The two of Example 1 are carried into each other by $\chi=B_0xy/2$ ($\mathbf A_2-\mathbf A_1=\nabla\chi$).

Reducing the freedom by a condition (such as $\nabla\cdot\mathbf A=0$) is called gauge fixing (the quantum version is in 4-08).

Seen on a lattice

Cut the plane into a square lattice and hold $\mathbf A$ as a number $A_{ij}=\int_i^j\mathbf A\cdot d\mathbf l$ on each edge ($A_{ji}=-A_{ij}$). Label the corners of a square face $p$ 1 to 4 anticlockwise; by Stokes's theorem the flux $B_p$ through the face is the sum over the four edges, and the gauge function becomes the values $\chi_i$ at the lattice points.

The $\chi$ of each corner appears once in $+$ and once in $-$ and cancels, so $B_p$ does not change. Example. The sum of the edge values $(0.3,\,0.5,\,-0.1,\,-0.2)$ is 0.5. Transforming by $\chi=(1,0,2,-1)$ gives $(-0.7,\,2.5,\,-3.1,\,1.8)$, but the sum stays 0.5. Putting group elements on the edges gives the starting point of Wilson's lattice gauge theory (5-01).

ComputedThe numbers in the figure are computed on the spot from the sums over four edges and the addition of $\chi_j-\chi_i$.

"Flux through a single face only" is the lattice version of the solenoid of section 06. The row of non-zero edges changes shape under gauge transformations but does not disappear (it corresponds to the cut where the angle $\theta$ of section 06 jumps). Under "No magnetic field" only the edges fluctuate; this is a pure gauge.

Maxwell's equations too are the stationarity conditions of an action (2-01, 2-02). The unknowns are $(\phi,\mathbf A)$, and $\mathbf E,\mathbf B$ are understood as written by (P).

(M2) holds automatically from (P), and (M1) comes out as the Euler–Lagrange equations Known. $\mathcal L$ contains $\nabla\phi$ but not $\partial_t\phi$, so the $\varepsilon_0\nabla\cdot\mathbf E=\rho$ obtained by varying $\phi$ is a condition at equal times containing no second time derivative: a constraint. It corresponds to the equation for $\lambda$ in the toy of 2-01. In this way, of the four components of $(\phi,\mathbf A)$, one disappears through the gauge transformation (G) and one through this constraint, and the remaining $4-1-1=2$ is the number of polarisations of light Known (the quantum version is in 4-01).

Under (G) the first two terms do not change, and the last two increase by just $\rho\,\partial_t\chi+\mathbf J\cdot\nabla\chi$. Integrating by parts ($\chi$ is 0 far away),

That the action does not change for any $\chi$ is the same as the conservation of charge $\partial_t\rho+\nabla\cdot\mathbf J=0$. Seen from the side of the equations, the two equations of (M1) are not independent. Combining the time derivative of Gauss's law with the divergence of the Ampère–Maxwell law, the field terms vanish identically by $\nabla\cdot(\nabla\times\mathbf B)=0$, and only $\partial_t\rho+\nabla\cdot\mathbf J=0$ is left. If charge is conserved, part of the equations follows from the rest. This is the form of the second theorem of 2-01.

Example: a mass term cannot be put in. Following 2-02, adding $-\frac{\varepsilon_0\Omega^2}{2}|\mathbf A|^2$ makes the dispersion of transverse waves $\omega^2=c^2|\mathbf k|^2+\Omega^2$, with a lower bound $\Omega$. But this term changes under (G). Under gauge invariance, a mass term cannot be added directly to the action. For the electromagnetic field without interactions no lower bound arises, the mass of the photon is 0 in quantum theory too, and the free electromagnetic field has no mass gap Known.

Whether mass can arise from interactions while keeping gauge invariance is a separate problem. The Meissner effect in a superconductor and the Higgs mechanism (3-07) are examples Known. For Yang–Mills, the question is whether this happens without adding another field (section 09, 4-11).

A classical particle feels only $\mathbf E,\mathbf B$, but in quantum mechanics the potentials enter the formulas. We use three things (2-05). (i) A state is represented by a complex-valued wave function $\psi(t,\mathbf x)$, and $|\psi|^2$ is the probability density of position. (ii) For a constant $\alpha$, $\psi$ and $e^{i\alpha}\psi$ are the same state. (iii) A particle of mass $m$ and charge $q$ obeys the following Schrödinger equation ($\hbar=h/2\pi$).

(S) seems to break under (G), but if $\psi$ is also changed as

then (S) holds as it is. The key is the covariant derivative $D=\nabla-\frac{iq}{\hbar}\mathbf A$: the $\frac{iq}{\hbar}(\nabla\chi)\psi$ coming from the derivative of the product cancels the change in $\mathbf A$.

The time side works in the same way. Fock (1926) noticed this invariance, London (1927) connected it with the theory of scale (Eich) that Weyl had set up earlier, and Weyl (1928–29) raised it as a principle under the name of gauge invariance, it is said (Jackson–Okun 2001).

$e^{iq\chi/\hbar}$ is a complex number of absolute value 1, and the set of all of them forms a group under multiplication. This is U(1) (the unit circle; 1-02). A gauge transformation is the choice of one element of U(1) at each point. If the element is the same at every point, it is (ii) itself; if it differs from point to point, $\mathbf A$ makes things consistent (the argument in the opposite direction is in section 09).

The phase factor along a path

If the reference is chosen point by point, how are phases at distant points compared? For a path $C$ from $\mathbf x$ to $\mathbf y$,

Since it cancels the replacements at both ends, $W(C)\in U(1)$ is a tool that carries the phase from $\mathbf x$ to $\mathbf y$ (parallel transport, 1-07). For a closed path it is gauge invariant, and with the enclosed flux $\Phi_C$, $W(C)=e^{iq\Phi_C/\hbar}$. On a lattice it is the product of the $e^{iqA_{ij}/\hbar}$ of the edges, and the addition of section 03 becomes multiplication in the group.

Consider a double slit in which electrons pass on either side of the solenoid of Example 2. Where the electrons pass, $\mathbf B=\mathbf 0$, and classically nothing changes. In quantum mechanics the contributions of the path $C_1$ through slit 1 and the path $C_2$ through slit 2 are added, and the phase difference makes the fringes. Joining $C_1$ with the reversed $C_2$ goes once round the solenoid, so the phase difference due to $\mathbf A$ is

and the fringes shift by $\Phi/\Phi_0$ of the fringe spacing. If $\Phi$ is an integer multiple of $\Phi_0$, the result cannot be told apart from the original. The prediction is Aharonov–Bohm (1959). Ehrenberg–Siday (1949) had reached the same conclusion earlier, and a clue from 1939 has also been pointed out Known (Hiley 2013).

ComputedA schematic in which the fringes $I(y)\propto\operatorname{sinc}^2(0.6y)\cos^2(3.5\pi y+\pi\Phi/\Phi_0)$ ($\operatorname{sinc}u=\sin(\pi u)/(\pi u)$) are computed on the spot (the number and width of the fringes are chosen for legibility).

Why it cannot be removed by a gauge transformation. Outside, $\mathbf A=\nabla(\Phi\theta/2\pi)$ ($\theta$ is the angle around the axis), so it looks removable by $\chi=-\Phi\theta/2\pi$, but $\theta$ increases by $2\pi$ in one circuit and is not determined as a single function. What needs to be single-valued is the phase $e^{iq\chi/\hbar}=e^{-iq\Phi\theta/2\pi\hbar}$, and that holds only when $\Phi/\Phi_0$ is an integer; only then does $\mathbf A$ vanish and the effect disappear. The period $\Phi_0$ comes from here.

ComputedTable 1 Numerical integration of $\oint\mathbf A\cdot d\mathbf l$ around a solenoid of radius 1 with $\Phi=1$. The right two columns are the $\mathbf A$ of Example 2 and the $\mathbf A'$ obtained by transforming with $\chi=\sin x\cos2y+x^2y/3$. The difference from enclosed flux × winding number is within $4\times10^{-10}$.

ComputedFor electrons $\Phi_0=h/e=4.1357\times10^{-15}$ Wb; the magnetic field that sends this through a circle of diameter 1 µm is 5.27 mT. KnownThe first experimental test is said to be Chambers (1960), and Tonomura et al. (1986) measured it with the magnetic field confined in a superconductor.

The shift of the fringes also appears in the levels. Consider a particle of charge $q\gt0$ confined to a ring of radius $R$, with a flux $\Phi$ through the centre of the ring. On the ring $\mathbf B=\mathbf 0$, $A_\theta=\Phi/2\pi R$, and $\nabla$ reduces to the $\frac1R\partial_\theta$ along the ring. Putting a stationary state $\psi=e^{-iEt/\hbar}e^{in\theta}$ (a state in which $|\psi|^2$ does not depend on time; since it returns after one circuit, $n$ is an integer) into (S), the left side becomes $E\psi$ and the right side $\frac{1}{2mR^2}(\hbar n-q\Phi/2\pi)^2\psi$, and the energy is

For $\Phi\mapsto\Phi+\Phi_0$ the labels are merely renumbered as $n\mapsto n+1$, so the set of levels is a periodic function with period $\Phi_0$, and at $\Phi=\Phi_0/2$ the lowest two coincide. There is no magnetic field on the ring, and yet the levels know the flux. Byers–Yang (1961) is said to have discussed from this periodicity the quantisation of flux in a superconducting cylinder (period $h/2e$) Known.

From an examination of the AB effect, Wu–Yang (1975) summarised that the field strength describes electromagnetism too little, the phase $\frac{q}{\hbar}\oint\mathbf A\cdot d\mathbf l$ describes it too much, and the phase factor $e^{\frac{iq}{\hbar}\oint\mathbf A\cdot d\mathbf l}$ is what describes it neither more nor less Known. These levels are one example.

On a ring of $N$ points, consider the Hermitian matrix $H_{j+1,j}=-t\,e^{i\theta_j}$, $H_{j,j+1}=\overline{H_{j+1,j}}$ (indices modulo $N$), in which the amplitude $-t$ for moving to a neighbour ($t\gt0$) is multiplied by an element of U(1) for each edge. A gauge transformation is $\theta_j\mapsto\theta_j+\chi_{j+1}-\chi_j$, and the only thing that does not change is the product once round, $\prod_je^{i\theta_j}=e^{2\pi if}$ ($f=\Phi/\Phi_0$). For a uniform $\theta_j=2\pi f/N$ the plane waves are the eigenvectors, and

For $|n-f|\ll N$ this is $E_n\approx-2t+t(2\pi/N)^2(n-f)^2$, the same form as the continuous ring of section 07.

ComputedThe levels are computed on the spot from the closed formula ($t=1$).

ComputedTable 2 The lowest two levels (eigenvalues of the matrix) for $N=12$, $t=1$. The rightmost column is the maximum of the differences among three gauges (uniform / gathered on one edge / with a random $\chi$ added) and of the difference from the closed formula. Pressing a row moves to Figure 3.

The levels agree among the three gauges within rounding error, and are the same for $f$, $1-f$, $f+1$. For $N=192$, $(E+2t)/t(2\pi/N)^2$ approaches the $(n-f)^2$ of the continuous ring, and for the lowest two levels the relative difference was at most $5.0\times10^{-5}$ ($E_1$ of $f=0.25$) Computed.

Yang–Mills theory replaces the element chosen at each point, from U(1) to SU(N) (the $N\times N$ unitary matrices of determinant 1, 1-04) (3-02). Because the result depends on the order of multiplication, a commutator term is added to the field strength, the field itself carries colour charge (the counterpart of electric charge), and the equations become non-linear (3-03). Classical waves have no lower bound, as with light, but in the quantum theory a gap is thought to arise Physics (5-08). The difference from U(1) is in 5-06.

The next bridge is 3-01. This chapter put $\mathbf A$ first and found (G') afterwards. 3-01 goes the other way: it demands that (S) not change even if the phase is turned arbitrarily at each point, and derives $\mathbf A$ as the field that makes things consistent. Making the same demand with SU(N) gives the Yang–Mills field. This is the gauge principle, and the parallel transport of section 05 and the edge elements of section 08 are its parts.

What this chapter can say

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Next chapter: 2-04 Special relativity and spacetime. To go on to the gauge principle, 3-01.

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