2026-10-01 · chapter Yang–Mills and the mass gapPart 3 Yang–Mills theory (classical)the gauge principle

The gauge principle — demanding a local symmetry requires a tool for comparison, and that tool carries the force

In 2-03 the potentials $\mathbf A$ came first, and the replacement of phase (G') was found afterwards. This chapter goes the other way: it derives the electromagnetic field by demanding that "the theory does not change even if the phase is turned arbitrarily point by point". This procedure is the gauge principle.

The starting point is a field $\psi(t,\mathbf x)$ with complex values, the simplest model of a charged particle. Take the Lagrangian density (2-01, 2-02) to be

(for example $V=M^2|\psi|^2$, where $M$ is the mass of the particle). Turning by a constant $\alpha$ as $\psi\mapsto e^{i\alpha}\psi$ leaves $\mathcal L_0$ invariant (a global U(1) symmetry; U(1) is the group of complex numbers of absolute value 1, 1-02). Noether's theorem (2-01) gives a conservation law, written with a constant $q$ multiplied in.

Example. For a plane wave $\psi=e^{i(\mathbf k\cdot\mathbf x-\omega t)}$ ($\omega\gt0$), $\rho=2q\omega$, $\mathbf J=2q\mathbf k$. $q$ becomes the charge in section 05. A global symmetry says that "nothing changes if the reference of phase at every point is shifted all at once".

Let the angle of turning change from point to point, $\psi\mapsto e^{i\alpha(t,\mathbf x)}\psi$ (a local transformation). $|\psi|^2$ does not change, but the derivative acquires an extra term ($\partial_\mu$ are the components of $\partial_t$ and $\nabla$, $\mu=0,1,2,3$).

Example. The gradient energy $|\nabla\psi|^2$ of a constant field $\psi=1$ is 0, but turning by $\alpha=kx$ gives $\psi=e^{ikx}$, $|\nabla\psi|^2=k^2$. Only the reference was changed, yet (1) counts it as a different state.

The derivative is the limit of the difference $\psi(x+\varepsilon n)-\psi(x)$ of two neighbouring points divided by $\varepsilon$ ($n$ is the unit vector of the direction). If the two values are turned by different angles, the difference mixes "the change in the field" with "the change in the reference". Values at different points cannot be subtracted as they stand.

Why demand local transformations? The abstract of Yang and Mills (1954) says that the usual invariance under isospin rotations (section 06) is not consistent with the idea of localised fields (the content of the abstract is in 3-02 §01) Known. This is a demand (a principle), not a theorem, and whether nature follows it is decided by experiment (section 07).

To compare values at different points, we need a correspondence: "which reference of phase at the point $y$ corresponds to the reference at the point $x$". We call this the comparator $U(y,x)\in U(1)$, write $U(y,x)\psi(x)$ for $\psi(x)$ carried to $y$, and stipulate that under transformations

Carrying in the opposite direction is the inverse element, $U(x,y)=U(y,x)^{-1}$. Then the difference $\psi(y)-U(y,x)\psi(x)$ is merely multiplied by $e^{i\alpha(y)}$. Carry before comparing: that is the point.

For two nearby points the comparator is close to 1, and we name the rate of its deviation the field $A_\mu$: $U(x+\varepsilon n,x)=\exp\bigl(iq\varepsilon\,n^\mu A_\mu(x)\bigr)+O(\varepsilon^2)$. The opposite direction is its inverse, so $U(x,x+\varepsilon n)=\exp\bigl(-iq\varepsilon\,n^\mu A_\mu(x)\bigr)+O(\varepsilon^2)$. Carrying the value at $x+\varepsilon n$ to $x$ with this before subtracting, and taking the limit of the difference quotient with $n$ in the direction $\mu$,

This is the covariant derivative. Expanding (3) in $y=x+\varepsilon n$ determines how $A_\mu$ changes.

Example. Putting $A_0=-\phi$, $(A_1,A_2,A_3)=\mathbf A$ (the notation of 2-04), $\alpha=q\chi$, (5) becomes $\mathbf A\mapsto\mathbf A+\nabla\chi$, $\phi\mapsto\phi-\partial_t\chi$, $\psi\mapsto e^{iq\chi}\psi$. These are (G)(G') of 2-03. From the demand we have arrived at the electromagnetic potentials and their gauge transformations. Multiplying comparators along a path gives the phase factor of 2-03 §05, in geometric language parallel transport (1-07 §03).

Seen on a chain

If space is cut into a sequence of points $j=0,1,2,\dots$, the comparator becomes a $U_j=U(j{+}1,j)=e^{i\theta_j}$ on each edge (2-03 §08, the link variables of 5-01), and instead of the derivative we use the re-compared difference. (3) becomes $\theta_j\mapsto\theta_j+\alpha_{j+1}-\alpha_j$, and

The sum of squared absolute values does not change, while the sum of naive differences $\psi_{j+1}-\psi_j$ does.

ComputedThe numbers in the figure are computed on the spot. Point by point with the connection on, moving $s$ from 0 to 1 in steps of 0.25, the sum of naive differences is 1.317, 8.070, 19.536, 22.541, 17.700, while the sum of re-compared differences stays 1.317 (with the connection off both are the former; turning everything by the same amount, both are 1.317).

ComputedTable 1 A ring of 12 points (random field and comparators) before and after transformation. $W$ is the product of the comparators once round (SU(2) is in section 06), and $W'$ is the value after the transformation. "Max. entry of $W'-W$" is the maximum absolute value of the entries of the difference matrix.

The change in the sum of re-compared differences is within $3\times10^{-14}$ in every case (rounding error). Turning U(1) by the same amount everywhere, no column changes.

To make $A_\mu$ a field obeying its own equation, we add to the action a term containing derivatives of $A$ that does not change under (5). The material is the comparator of a closed path, in which, by (3), all the $\alpha$ cancel. Going from $x$ in the direction $\mu$, then in the direction $\nu$, once round a square of side $\varepsilon$, and back to $x$,

The same quantity also appears in $[D_\mu,D_\nu]\psi=-iqF_{\mu\nu}\psi$. The comparator carries from start to end, like the parallel transport of 1-07, and rewriting in the geometric notation $A_{\text{geom}}=-iqA$, $F_{\text{geom}}=-iqF$, (6) becomes $\exp(-\varepsilon^2F_{\text{geom}})$, agreeing with (10) of 1-07 and $U_p\approx\exp(-a^2F_{\mu\nu})$ down to the sign (in 5-01, which reads the lattice links in the opposite direction, the sign of the exponent is reversed). Example. The components are $F_{0i}=-E_i$, $F_{ij}=\varepsilon_{ijk}B_k$ (2-04), and $F$ is the electric and magnetic fields themselves. The electric and magnetic fields were the curvature of the comparator (the deviation after going round a small loop, 1-07 §05). Listing the invariant terms with the lowest order of derivatives,

The last term is $-\tfrac14F_{\mu\nu}F^{\mu\nu}$ ((8) of 2-04). The way of coupling that merely replaces the $\partial$ of (1) by $D$ is called minimal coupling.

The term that cannot be written. The term $-\tfrac12m^2A_\mu A^\mu=\tfrac12m^2(\phi^2-|\mathbf A|^2)$ that would give $A$ a mass changes under (5) by terms in $-\tfrac{m^2}{q}\bigl(\phi\,\partial_t\alpha+\mathbf A\cdot\nabla\alpha\bigr)$ and the second order of $\alpha$. A mass term cannot be written for a field that comes from the gauge principle (as in 2-03 §04) Known. This affects the distance a force reaches in section 05, and becomes a difficulty in section 06.

Expanding (7) in $q$, the quantities of section 01 appear as they are.

The gauge field couples to the conserved quantity (2) of the global symmetry, with the same $q$ that appeared in the transformation. This is exactly the form of (L) of 2-03.

Varying (8) with respect to $\phi$ gives Gauss's law.

The right side is not the $\rho$ of (2) itself but that with its $\partial_t$ replaced by $D_t$ (terms containing $\phi$ are added). For a point charge $Q$ at rest at the origin, we put this right side equal to $Q\,\delta^3(\mathbf x)$. In the static case $\mathbf E=-\nabla\phi$, and from $-\nabla^2\phi=Q\,\delta^3(\mathbf x)$, $\phi=Q\,G_0(r)$. The energy of a charge $Q'$ at distance $r$ is

Example. For $m=0$, charges of the same sign repel, and the force reaches far away like $1/r^2$. Merely demanding a local symmetry has produced a force between distant charges. In the theory (7) (when the minimum of $V$ is at $\psi=0$), adding the forbidden mass term provisionally would give $(-\nabla^2+m^2)\phi=\rho$, and $G_m$ (Yukawa type) does not reach much further than about $1/m$. Without it, the force is of Coulomb type and reaches far.

However, gauge invariance does not guarantee that a force reaches far. Even classically, with a Higgs field the distance reached becomes finite while gauge invariance is kept (3-07) Known. In terms of (8), when the minimum of $V$ is at $|\psi|^2=v^2\gt0$ and $|\psi|^2$ stays there, the term $q^2(\phi^2-|\mathbf A|^2)|\psi|^2$ has the same form as the mass term $\tfrac12m^2(\phi^2-|\mathbf A|^2)$ of $m^2=2q^2v^2$ ((9) of 3-07 puts the minimum of the size of the field at $v/\sqrt2$, so the same mass is written $m=qv$).

Figure 2 is $(-\Delta_{\mathrm{lat}}+m^2)G_m=\delta$ found numerically on a cubic lattice (spacing 1) ($\Delta_{\mathrm{lat}}f$ is the sum of the values at the six neighbours minus $6f$).

ComputedThe dots are computations on this machine, the solid line is computed on the spot. Static values: for $r=1,\,4,\,12$ the lattice $4\pi r\,G_0$ is 1.082, 1.020, 1.002, approaching Coulomb further than a few lattice spacings away. $4\pi r\,G_{0.5}$ is 0.670, 0.147, 0.0027 (the continuous $e^{-0.5r}$ is 0.607, 0.135, 0.0025), and the rate of decay along the lattice axis is $\mu=0.4949$, from $\cosh\mu=1+m^2/2$.

The proton and the neutron have almost equal masses and feel the nuclear force in almost the same way. The isospin view gathers the two into a two-component field $\psi=(\psi_1,\psi_2)$, and the theory of the form (1) does not change under the mixing rotations $\psi\mapsto\Omega\psi$ ($\Omega\in SU(2)$, 1-04; to distinguish it from the $V$ of (1), the matrix is written $\Omega$, as in 3-02). Yang–Mills (1954) imposed the demand of section 02 on this symmetry.

Making the comparator an SU(2) matrix with $U(y,x)\mapsto\Omega(y)\,U(y,x)\,\Omega(x)^{-1}$, and for two nearby points $U(x+\varepsilon n,x)=\exp(ig\varepsilon\,n^\mu A_\mu)+O(\varepsilon^2)$, $A_\mu=A^a_\mu T^a$ ($T^a=\sigma^a/2$; relative to the geometric notation of 1-07, $A_{\text{geom}}=-igA=-i\hat A$; $\hat A=gA$ is the "coupling absorbed" notation of 3-02 §02),

Example. For $\Omega=e^{i\alpha}$ ($1\times1$) we return to (5) ($g$ playing the role of $q$), and in geometric notation this is the $A\mapsto\Omega A\Omega^{-1}-(d\Omega)\Omega^{-1}$ of 1-07 §07 (1-07 writes the transformation matrix as $g$, but here $g$ is the coupling constant). There are three differences from U(1).

1. The number of fields. As many as the dimension of the Lie algebra are needed: 1 for U(1) (the photon), 3 for SU(2) ($A^1,A^2,A^3$), 8 for SU(3) (gluons), $N^2-1$ for SU(N).
2. The gauge field itself is turned. The $\Omega A\Omega^{-1}$ of (10) turns $A$ even when $\Omega$ is constant (the adjoint representation, 1-04 §06). Unlike the photon, it carries itself the "charge" it couples to. This is why in Table 1 the $W$ of SU(2) changes as $\Omega_0W\Omega_0^{-1}$ even under a global rotation, and only $\mathrm{tr}\,W$ remains; self-interaction comes from here (3-02).
3. There is one coupling constant. For U(1) a different $q$ can be put freely for each field. For a simple non-abelian group (1-03 §08) there is only one coupling constant $g$, and the strength of the coupling to each matter field is determined by $g$ and the generators $T^a$ of that field's representation ($\sigma^a/2$ for a two-component field, $(T^a)_{bc}=-i\varepsilon_{abc}$ for a three-component field in the adjoint representation) Known. The outline of the reason is that the non-linear term of the field strength $-ig[A_\mu,A_\nu]$ (section 09, 3-02) ties the normalisation of $A$ to $g$ and fixes it, and that if the group is simple the invariant form of the kinetic term is unique up to a constant multiple (2-04 §07, 1-03).

The problem of mass. For SU(2) too, no mass term can be written for the gauge field. In Yang's recollection, at a seminar in Princeton in late February 1954, Pauli asked "what is the mass of this field?" (the story is in the detour of 3-02). The nuclear force is short-range, which does not fit the Coulomb-type reach of section 05. §1 of the Jaffe–Witten problem statement also writes that the absence of mass in the classical waves was a serious obstacle Known, and says it was overcome for the weak force by adding a Higgs field (3-07), and for the strong force by a property of the quantum theory itself (asymptotic freedom, 4-06) Physics. Showing as mathematics the mass gap expected in the latter (4-11) is the task of the problem statement.

The gauge principle is a strong constraint, but it does not determine the theory uniquely (Table 2).

An example for the second row of the table. If the group is a circle, the charge of every field is an integer multiple of a single unit. If the group is the line $\mathbb R$, the ratios of charges can be any real numbers, and the principle alone gives no reason for charges to line up as integer multiples of one unit. An example for the third row. A term like $F^4$ is of higher order in the fields than $F^2$, and in four dimensions its coefficient has the dimension of length to the fourth power, so it is thought to have less effect the longer the distance (2-04 §08, 4-04 §06) Physics.

A local "symmetry" differs from an ordinary symmetry in producing a constraint (Gauss's law) and a redundancy of description (2-01 §08). How to divide out the redundancy is the subject of 3-04 and 4-08.

There is one more difference from an ordinary symmetry. A symmetry is spontaneously broken when the action (in statistical mechanics, the energy) is symmetric but the equilibrium state (the expectation values) is not. An example is the spontaneous magnetisation of 2-08 §04. Elitzur (1975) argued that in lattice gauge theory without gauge fixing a local symmetry is not spontaneously broken (the expectation values of quantities that are not gauge invariant are 0), and showed it concretely in simple systems on abelian lattices Known.

In 3-02 the square of (6) is turned with SU(N). Because comparators depend on the order of multiplication, a commutator is added to the field strength, and self-interaction appears.

Imposing Gauss's law in the quantum theory is 4-01. On the lattice of 5-01 the comparators $U_j$ are the variables, and the action is built from the products around the squares of (6) (plaquettes). That $g$ becomes dimensionless in four dimensions is in 2-04 §08 and the four-dimensions article.

To sum up: the demand for a local symmetry calls in a gauge field, that field couples to the conserved charge and carries a force, and classically its mass term cannot be written by hand (adding a Higgs field changes matters; 3-07). The problem statement asks to show that, in the four-dimensional non-abelian case, a mass gap nevertheless arises in the quantum theory (0-03).

What this chapter can say

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