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

Non-abelian gauge fields — when the order of multiplication matters, the field becomes its own source

In 3-01 §06, we imposed the demand that turning point by point be allowed on the symmetry (isospin) that mixes, by SU(2) matrices, the two-component field $\psi$ combining the proton and the neutron, and obtained the covariant derivative and the transformation law (10). This chapter extends that to $\mathrm{SU}(N)$ and follows in turn what arises from the commutator.

The abstract of the paper of Yang and Mills (Phys. Rev. 1954) points out that the usual invariance, turning by the same matrix at every point, is not consistent with the idea of localised fields, and continues: “The possibility is explored of having invariance under local isotopic spin rotations.” Rephrasing the first sentence of the abstract in this book's reading: that the choice of which direction to call "proton" at one point should fix the naming at distant points as well does not fit the laws of fields, which are written in terms of relations between nearby points. So we allow the transformation matrix $\Omega=\Omega(x)$ to be chosen point by point.

According to the abstract, the consequence is a $\mathbf b$ field that plays the same role for isospin as the electromagnetic field, obeys non-linear differential equations, and whose quanta are particles of spin 1, isospin 1 and charge $\pm e$ or 0 Known. §03–§06 turn this summary into formulas. The group carrying the strong force later turned out to be SU(3) (Jaffe–Witten's problem description: “the gauge group is G = SU(3)”). However, what was gauged was not isospin but SU(3) of colour, a different degree of freedom (isospin remains an approximate symmetry, not turned point by point). Mathematically, one just replaces the group (3-07, the detour of 1-04).

Detour: the spring of 1954

In Part 3, $\hbar=c=1$, and the signature of the metric is $(-,+,+,+)$ (2-04). $\mu,\nu=0,\dots,3$ are spacetime indices, $i,j,k=1,2,3$ spatial, and $a,b,c$ indices in the direction of the group (colour); indices appearing twice are summed. The group is $G=\mathrm{SU}(N)$, the generators are Hermitian $T^a$ ($a=1,\dots,N^2-1$), and the structure constants and normalisation are as follows (1-04).

Example. For SU(2), $T^a=\sigma^a/2$ (half the Pauli matrices) with $f^{abc}=\varepsilon^{abc}$; for SU(3), $T^a=\lambda^a/2$ (half the Gell-Mann matrices). The gauge transformation matrix is written $\Omega(x)\in G$ (the same letter as in 3-01), and the letter $g$ is kept for the coupling constant. A matter field $\psi(x)\in\mathbb C^N$ is taken to change as $\psi\mapsto\Omega(x)\psi$ (the fundamental representation).

This chapter follows the physicists' style, writing the Hermitian $A_\mu=A^a_\mu T^a$ and the coupling $g$ separately. The $\epsilon$ and $B_\mu$ on Yang's blackboard in the detour of §01 correspond to this $g$ and $A_\mu$ (the conventions for coefficients depend on the paper). The other notations in the book can be translated by the following lines.

We extend (10) of 3-01 to $\mathrm{SU}(N)$ and follow the derivation again. The trouble is the derivative. If $\Omega$ depends on the point, the derivative of the product produces an extra term.

The second term appears because values at neighbouring points with different frames were subtracted as they stand (1-07 §01). So, as in 3-01, we demand that the derivative with a correcting field $A_\mu$ added change in the same way as $\psi$.

The left side is $\Omega\partial_\mu\psi+(\partial_\mu\Omega)\psi-igA'_\mu\Omega\psi$, the right side $\Omega\partial_\mu\psi-ig\,\Omega A_\mu\psi$. Equality for every $\psi$ determines $A'_\mu$.

The first term is the rotation by the adjoint representation (1-04 §06), and the second is an inhomogeneous term coming from the slope of $\Omega$. Differentiating $\Omega\Omega^\dagger=1$, $(\partial\Omega)\Omega^{-1}$ is anti-Hermitian, and from $\det\Omega=1$ its trace is 0, so $A'$ also stays in the same Lie algebra. Example. For $\Omega=e^{i\alpha(x)}$ (U(1)) the first term stays $A_\mu$, and $A'_\mu=A_\mu+\partial_\mu\alpha/g$, putting $\chi=\alpha/g$, is exactly (G) of 2-03 (with the convention $A_\mu=(-\phi,\mathbf A)$ of 2-04). What is new in the non-abelian case is the first term, which sandwiches $A$.

Infinitesimal transformations — the gauge field itself turns

Putting $\Omega=1+i\alpha^aT^a$ ($\alpha$ small) and expanding (3) to first order, in components

The $(D_\mu\phi)^a=\partial_\mu\phi^a+g f^{abc}A^b_\mu\phi^c$ at the right end is the covariant derivative acting on fields in the adjoint representation. Example. For SU(2), taking $\alpha$ constant with $\alpha^3=\theta$ (others 0), $\delta A^1=\theta A^2$, $\delta A^2=-\theta A^1$, $\delta A^3=0$. Even under a rotation independent of position the components of the gauge field turn into each other, so the gauge field itself carries non-abelian charge (this is "the gauge field itself is turned" of 3-01 §06 seen in components). This is in contrast to the photon, which carries no charge, and it is what "isospin 1" in the abstract means.

Seen on a chain

Arranging points in a row makes the meaning of (2) visible. At each point $n$ put a field in the adjoint representation (a real three-component vector) $\phi_n$, and on each edge put the rotation $U_n$ that carries the value at the point $n+1$ into the frame of the point $n$. This is the non-abelian version of the U(1) elements on the edges of 2-03, and a component of the lattice of 5-01. Written with the comparator of 3-01 it is $U_n=U(n,n{+}1)$, the opposite direction from the $U_j=U(j{+}1,j)$ of the chain of 3-01. The $\Omega_n$ and $U_n$ here denote, by the same letters, the rotations of $3\times3$ given by SU(2) elements in the adjoint representation (1-04 §06).

The "covariant difference" at the right end is multiplied by only one rotation, so its length does not change. The naive difference $\phi'_{n+1}-\phi'_n$ has different rotations at the two points, so its length changes. With the spacing of the points $h$, putting $U_n\approx1-ighA$ and letting $h\to0$, the second formula of (5) becomes (3), and the covariant difference becomes $hD_\mu\phi$. Here $A$ acts as $A=A^aT^a_{\rm adj}$ using the matrices $(T^a_{\rm adj})_{bc}=-if^{abc}$ of the adjoint representation, and the components of $D_\mu\phi$ have the same form as the right end of (4).

ComputedThe numbers in this figure were computed on this machine (the same formulas and the same configuration as Table 1, computed in the browser). If JavaScript is disabled, Table 1 is the content of the figure.

The change in the covariant difference on each edge was at most $3\times10^{-16}$ (rounding error) Computed. That the two differ at $s=0$ is because $U_n$ is taken to be a small rotation.

In electromagnetism $\partial_\mu A_\nu-\partial_\nu A_\mu$ was a quantity unchanged by gauge transformations. In the non-abelian case, because of the first term of (3), this combination is not even covariant. The correct combination comes from the difference of covariant derivatives multiplied in the two orders.

Expanding the left side, the terms in $\partial_\mu\partial_\nu\psi$ and $A\,\partial\psi$ are symmetric in $\mu\nu$ and cancel, so no derivatives of $\psi$ remain. $F_{\mu\nu}$ is a matrix (an element of the Lie algebra) at each point, and in components it is as follows.

For SU(2), regarding the three components $(A^1_\mu,A^2_\mu,A^3_\mu)$ as a vector in the internal space, the last term is the cross product $g\,A_\mu\times A_\nu$. (2) means $D'_\mu=\Omega D_\mu\Omega^{-1}$ as an operator, so the commutator is sandwiched in the same way.

$F$ is not invariant; it turns at each point by the adjoint representation (it is covariant). This is a large difference from electromagnetism, where the electric and magnetic fields were themselves measurable quantities.

Example 1 (pure gauge). The $A_\mu=-\tfrac ig(\partial_\mu\Omega)\Omega^{-1}$ obtained by transforming $A=0$ by (3) has $F=0$, by (8). Conversely, if $F=0$ on a simply connected region such as $\mathbb R^4$, then $A$ can be written in this form Known. The notes Pauli attached to his letter to Pais of July 1953, as presented by Straumann, state this as “The vanishing of the field strength is necessary and sufficient for the $A^a_\mu(x)$ in the whole space to be transformable to zero.” It fails on regions with holes, and the Aharonov–Bohm effect of 2-03 is an example (the global story is in 3-04).

Example 2 (commutator only). For constant fields $A_x=a\,T^1$, $A_y=b\,T^2$, $F_{xy}=-ig\,ab\,[T^1,T^2]=g\,ab\,T^3\ne0$. In electromagnetism a constant potential makes no field. A figure showing this as the deviation after going round a small square is Figure 2 of 1-07.

ComputedGiving smooth $A$ and $\Omega$ for SU(2) and SU(3) and comparing by finite differences, the difference between the two sides of (8) was $5\times10^{-9}$ (the error of step $10^{-4}$); dropping the commutator gave 0.78, 1.74; and the $F$ of a pure gauge was at most $10^{-9}$.

Following electromagnetism, we define electric and magnetic fields for each direction of the group. With the same assignment as $F_{0i}=-E_i$, $F_{ij}=\varepsilon_{ijk}B_k$ of 2-04, the three spatial components are written in bold ($\mathbf A^a=(A^a_1,A^a_2,A^a_3)$).

Taking $A_0=0$ (temporal gauge), $E^a_i=-\partial_tA^a_i$. The second term of the magnetic field is the commutator part, and a magnetic field arises even if the potential is spatially uniform.

By (8), $\mathbf E^a$ and $\mathbf B^a$ turn in the colour index at each point. So "component 3 of the colour electric field" changes value with the gauge and is not a measurable quantity. What does not change are squared lengths such as $\sum_a|\mathbf E^a|^2=2\operatorname{tr}(\mathbf E\cdot\mathbf E)$. Following electromagnetism, in this chapter we take the following as the energy density (deriving it from the action is in 3-03).

Example. Typical quantities that do not change with the gauge are $\operatorname{tr}(F_{\mu\nu}F^{\mu\nu})$ and the trace of parallel transport around a closed path (the Wilson loop, 1-07 §06).

Both the energy (10) and the $F^a_{\mu\nu}F^{a\,\mu\nu}$ put in the action in 3-03 are squares of $F$. Putting in (7), they split into terms quadratic, cubic and quartic in $A$.

The first line is Maxwell's $F_{\mu\nu}F^{\mu\nu}$ repeated for the number of colours $N^2-1$, and on its own it is a collection of Maxwell fields that do not act on each other. The second and third lines contain $f^{abc}$ and vanish for U(1). Put in the action (3-03), they are interactions among gauge fields, and in perturbation theory (4-03) they become vertices where three or four gauge fields meet. In the classical Maxwell equations light does not scatter off light, but the Yang–Mills field scatters off itself.

The consequence is the non-linearity of the equations. Jaffe–Witten write “The Yang–Mills equations are nonlinear—in contrast to the Maxwell equations.” Known. Adding two solutions does not in general give a solution. ComputedFor random $A^a_\mu$, $\partial_\mu A^a_\nu$ the two sides of (11) agreed to 10 digits. The non-zero $f^{abc}$, counted as ordered, number 6 for SU(2) and 54 for SU(3).

We look at self-interaction in its smallest form. For SU(2) with $A_0=0$, and $A^a_i$ functions of time only, the magnetic field of (9) consists of the commutator term alone. Further putting $A^1_1=x(t)$, $A^2_2=y(t)$ and the rest 0, the non-zero fields are $E^1_1=-\dot x$, $E^2_2=-\dot y$, $B^3_3=g\,xy$, and (10) becomes the energy of a particle in the plane.

For uniform fields, the Yang–Mills equations of motion (3-03) become the equations of a particle with 9 components $A^a_i$ moving in $\tfrac12\sum(B^a_i)^2$, and this assignment is consistent with them Computed (by numerical differentiation, the force on the other 7 components is 0, and the Gauss constraint $\varepsilon^{abc}A^b_iE^c_i=0$ also holds). The Gauss constraint is the condition left by the equation for $A_0$ lost in the temporal gauge, the non-abelian version of $\nabla\cdot\mathbf E=0$ in electromagnetism. For uniform fields the derivative terms vanish, giving this form (4-01). For comparison, putting the two components in the same colour direction ($A^1_1=x$, $A^1_2=y$), the commutator is 0 and $V=0$, and the particle moves in a straight line, as in electromagnetism.

The non-abelian $V=\tfrac12g^2x^2y^2$ is 0 on the coordinate axes, so the valleys go on for ever, narrowing the further in (the width available at energy $E$ is $|y|\le\sqrt{2E}/(g|x|)$). Figure 2 runs two particles at the same time with velocities differing by $\delta$.

ComputedThe numbers in this figure were computed on this machine (the same integration rule as Table 3: fourth-order Runge–Kutta, step $\min(0.01,\,0.02/(g\max(|x|,|y|)))$, distance checked every 0.05). If JavaScript is disabled, Table 3 is the content of the figure.

In the non-abelian case, each time $\delta$ is made 1/10, the separation time grows by only about 8 (a straight-line fit to the 9 points $10^{-2}$–$10^{-10}$ gives slope 8.04; the increments are uneven because separation slows while the orbit is deep in a valley). This means the distance grows exponentially, and on the right it rises almost as a straight line. In the abelian case the distance is $\delta\,t$. The drift of the energy was at most $5\times10^{-9}$ at $t\le120$ Computed. This system has been studied as classical Yang–Mills mechanics. Savvidy (1983) argued that dynamics of this kind is a K-system (a dynamical system with strong mixing), but Dahlqvist–Russberg (1990) found a family of stable periodic orbits in this $x^2y^2$ system, refuting earlier claims that the motion is completely ergodic (according to the abstract). What the computation on this machine shows goes only as far as the separation visible in finite time for the initial values above.

The classical theory has no scale

(12) has no constant that could serve as a unit of length or time ($g$ only changes the scale of time). If $x(t),y(t)$ is a solution, then for any $\lambda\gt0$

is also a solution (substituting, $\lambda^3$ appears on both sides). Example. Making the energy 1/16, the motion of the same shape happens twice as slowly. Bringing the energy towards 0, the motion becomes as slow as we like, and the frequencies have no lower bound. ComputedSolving the initial values $\lambda=0.5,\,0.25,\,0.125$ separately with the same rule and dividing the position at time $5/\lambda$ by $\lambda$, all agreed with $(-0.401421,\,0.616689)$ to 6 digits.

The same holds for the whole field: keeping only the quadratic terms of (11) around $A=0$ gives a collection of Maxwell fields, and the frequencies $\omega=|\mathbf k|$ continue down to 0. Jaffe–Witten write that the classical equations describe “massless waves that travel at the speed of light” Known. Quantising, deep in the valley the lowest energy of the transverse oscillation $g|x|/2$ acts as a wall, and the spectrum of the operator obtained by quantising (12) is discrete, with a positive lowest eigenvalue Known (Simon 1983, the detour of 2-05). This is a theorem about a model with finitely many degrees of freedom keeping only uniform fields.

That quantum Yang–Mills as a field theory has a gap is widely accepted in physics, but has not become a theorem Physics.

We answer Pauli's question in the detour of §01 with formulas. For a vector field of mass $m$, the usual thing is to add $-\tfrac12m^2A^a_\mu A^{a\,\mu}$ to the Lagrangian density (for the electromagnetic field, the Proca field). But computing the change by (4),

The second term vanishes because $f^{abc}$ is antisymmetric in $a,b$ while $A^{a\,\mu}A^b_\mu$ is symmetric. The remaining first term comes from the inhomogeneous term of (3), and is not 0 if $\alpha$ depends on the point. The mass term breaks the demand that turning at each point be allowed Known. ComputedFor random $A$ and small $\alpha$, the two sides of (14) agreed to 4 digits or more for both SU(2) and SU(3).

So as long as gauge invariance is kept, the waves of the classical Yang–Mills field have no mass. Jaffe–Witten write “The massless nature of classical Yang–Mills waves was a serious obstacle to applying Yang–Mills theory to the other forces”. That the way past it differed for the weak force (the Higgs field, 3-07) and the strong force (asymptotic freedom, 4-06) we saw in 3-01 §06.

As the first of the properties a theory of the strong force must have at the quantum level, Jaffe–Witten list the mass gap: “there must be some constant $\Delta\gt0$ such that every excitation of the vacuum has energy at least $\Delta$” (0-03, 4-11). The classical field has no scale (§07), and the gap is thought to arise only on quantisation Physics. Showing existence and the mass gap as theorems for pure Yang–Mills theory is what the problem statement asks, and this chapter shows nothing about it.

What this chapter can say

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Next chapter: 3-03 The action and the equations ($\operatorname{tr}F^2$, the Bianchi identity, the equations of motion, self-duality). The geometry continues in 3-04.

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