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

The action and the equations — from one action come the equations, an identity, the energy and self-duality

The conventions are the same as in 3-02 §02: $\hbar=c=1$, the metric $(-,+,+,+)$, Hermitian $T^a$ ($\operatorname{tr}T^aT^b=\tfrac12\delta^{ab}$), and $D_\mu=\partial_\mu-igA_\mu$. On a field $X$ with values in the Lie algebra, $D_\mu X=\partial_\mu X-ig[A_\mu,X]$ acts (in components, $\partial_\mu X^a+gf^{abc}A^b_\mu X^c$), and the colour electric and magnetic fields are $E^a_i=F^a_{i0}$ and $B^a_i=\tfrac12\varepsilon_{ijk}F^a_{jk}$ (3-02 §05). The action replaces the $F$ of the Maxwell action ((8) of 2-04) with the non-abelian field strength and takes the trace.

The form is almost fixed by four requirements: gauge invariance (the rotation by $F'=\Omega F\Omega^{-1}$ disappears under the trace), Lorentz invariance, equations that close at second order in time derivatives, and no length scale in the coefficients of the action (only the terms with mass dimension $[F^2]=4$; counting dimensions in 2-04 §08). The last condition is needed because adding a higher-order invariant in $F$ such as $(\operatorname{tr}F^2)^2$ still leaves the equations second order. Its coefficient needs the fourth power of a length, which brings in a scale. Imposing all four, and if the group is simple, the quadratic invariants in $F$ are only linear combinations of (1) and the $\theta$ term (3-05), which does not change the equations (2-04 §07) Known. Example. For the uniform field $A^1_1=x(t)$, $A^2_2=y(t)$ (3-02 §07) the action per volume is $L=\tfrac12(\dot x^2+\dot y^2)-\tfrac12g^2x^2y^2$, and its Euler–Lagrange equations (2-01) are (12) of 3-02.

Rescaling as $\hat A=gA$, (1) becomes $-\frac{1}{2g^2}\int\operatorname{tr}\hat F_{\mu\nu}\hat F^{\mu\nu}$ (the same form as the $\frac{1}{4g^2}\int\operatorname{Tr}F\wedge *F$ of Jaffe–Witten, up to conventions of normalisation and sign Known). $g$ has no effect on the classical equations; in the quantum theory, as the combination $g^2\hbar$, it becomes the variable of the perturbative expansion (4-03).

Shifting $A$ by $\delta A$, (6) of 3-02 gives $\delta F_{\mu\nu}=D_\mu\delta A_\nu-D_\nu\delta A_\mu$. By the antisymmetry of $F^{\mu\nu}$, and because integration by parts moves the commutator term onto the other factor by $\operatorname{tr}([A,X]Y)=-\operatorname{tr}(X[A,Y])$ ($\delta A$ is 0 far away),

follows. The stationarity condition, that this is 0 for every $\delta A$, is the Yang–Mills equation Known.

Unlike Maxwell, the product of $A$ and $F$ left on the right-hand side is the colour current carried by the gauge field itself (as announced in 3-02 §06). (2) is a non-linear equation containing terms up to third order in $A$, and Jaffe–Witten write “The Yang–Mills equations are nonlinear—in contrast to the Maxwell equations.” Known. In differential forms it is $d_A{*F}=0$ (1-06).

The current is conserved only covariantly

When there is a matter field, a colour current appears on the right-hand side, with the same sign as in Maxwell: $D_\mu F^{\mu\nu}=-j^\nu$. On the left-hand side there is an identity that holds for any $A$ (each term is a commutator of a matrix with itself, so it is 0).

So $D_\nu j^\nu=0$ is required, and what is conserved in the ordinary sense is the sum of the matter current and the field current, $j^{a\nu}+gf^{abc}A^b_\mu F^{c\,\mu\nu}$ (the difference from the conservation of charge in 2-03).

Putting $F^{i0}=-E_i$ and $F^{ij}=\varepsilon_{ijk}B_k$ into $D_\mu F^{\mu\nu}=-j^\nu$ gives the laws of Gauss and Ampère–Maxwell ((M1) of 2-03) with $\nabla$ replaced by the covariant derivative $(\mathbf D_iX)^a=\partial_iX^a+gf^{abc}A^b_iX^c$ ($\rho=j^0$).

Gauss's law contains no second time derivative; it is a constraint at each time. With $A_0=0$, if it holds at the start, it keeps holding by the identity of §02. The constraint $\varepsilon^{abc}A^b_iE^c_i=0$ checked in 3-02 §07 is this equation for a uniform field.

This constraint tells us how many components can be chosen freely. For each colour, $A_\mu$ has 4 components. Since no time derivative of $A_0$ appears in (1) ($F_{i0}$ contains $\partial_0A_i$ but not $\partial_0A_0$), $A_0$ is not a dynamical variable and can be made $A_0=0$ by a gauge transformation. Of the 3 remaining components of $A_i$, the longitudinal one along the direction of propagation, seen in small waves, is removed by Gauss's law and by time-independent gauge transformations that preserve $A_0=0$. So $4-1-1=2$ remain for each colour, the same number as the two polarisations of light. For $SU(N)$ there are $N^2-1$ colours, so $2(N^2-1)$; for $SU(3)$ it is 16 Known (the quantum version is in 4-01).

The source-free half of the Maxwell equations came from building the field out of a potential (the $dF=0$ of 1-06). In the non-abelian case too, the following holds just because $F$ is the curvature of $A$ Known.

The reason is the Jacobi identity for operators, $[D_\lambda,[D_\mu,D_\nu]]+[D_\mu,[D_\nu,D_\lambda]]+[D_\nu,[D_\lambda,D_\mu]]=0$. Putting in $[D_\mu,D_\nu]=-igF_{\mu\nu}$ ((6) of 3-02) and $[D_\lambda,F_{\mu\nu}]=D_\lambda F_{\mu\nu}$, which comes from the product rule, the left-hand side is $-ig$ times (4). In the geometric notation it is (12) of 1-07, and in Jaffe–Witten $0=d_AF$. In electric and magnetic fields it is two equations, paired with (3).

Unlike electromagnetism, the ordinary divergence $\nabla\cdot\mathbf B^a=-gf^{abc}\mathbf A^b\cdot\mathbf B^c$ is not necessarily 0; looking at one colour alone, magnetic field lines may end partway. More precisely, $\nabla\cdot\mathbf B^a$ is a quantity that changes under gauge transformations, and the gauge-independent statement is the $\mathbf D\cdot\mathbf B=0$ of (5). The flux through the six faces of a small cube is measured in a different colour frame on each face; when they are parallel transported (1-07) to one common point before being added, the term of the order of the volume vanishes (the form of the first equation of (5) integrated over the cube). For a random field, the left-hand side of (4) was 0 within the error of the differences ($4\times10^{-7}$), and the cyclic sum with ordinary derivatives was 2.1–2.6 Computed.

The Noether conserved quantity for translations (2-01), tidied into a gauge-invariant form, is the energy–momentum tensor Known.

$T^{00}=\tfrac12\sum_a(|\mathbf E^a|^2+|\mathbf B^a|^2)$ is the energy density ((10) of 3-02) and $T^{0i}=\sum_a(\mathbf E^a\times\mathbf B^a)_i$ the momentum density; since they are summed over colours they do not change under gauge transformations. The energy density is at least 0, and it is 0 only where $F=0$ (pure gauge, 3-02 §04). Taking the divergence and using (4), two derivative terms of $F$ cancel, and

remains. It is 0 on solutions, and energy and momentum are conserved. If there is matter the right-hand side is $-j^{a\alpha}F^{a\,\nu}{}_{\alpha}$, and the field loses as much as the colour Lorentz force does on the matter. From $\eta_{\mu\nu}\eta^{\mu\nu}=d$ (the dimension of spacetime), the trace is

and it vanishes identically only for $d=4$. This is the stress side of saying that the classical action has no length scale (2-04, 1-12) Known. In the quantum theory, perturbative calculation gives the trace a $F^2$ term proportional to the $\beta$ function (the rate at which the coupling constant moves with the scale; 4-06) (the trace anomaly), and this is taken to be the same phenomenon as the dimensional transmutation of 4-07 Physics. A figure that places the coefficient of the trace, $1-d/4$, next to other quantities that switch at four is the dimension knob of 1-12 §10.

For random SU(2) and SU(3) fields, the difference between $D_\nu D_\mu F^{\mu\nu}$ and the two sides of (7) was within the error of the differences (within $2.2\times10^{-5}$), $T^{00}$, $T^{0i}$, the trace and the form of (3) were within rounding error, and the coefficient in (8) was $d=3,5,6$ also for $1-d/4$ Computed.

If particles were to be represented by classical fields, the candidates would be solutions whose energy stays in one place (static lumps). For pure Yang–Mills fields in $3+1$ dimensions, the only time-independent solution that is smooth everywhere and has finite energy is the vacuum Known. We follow the argument. If $T^{\mu\nu}$ is time-independent, (7) gives $\partial_kT^{kj}=0$ ($T^{\mu\nu}$ is symmetric by (6), so the divergence is the same whichever index it is taken on). Together with $\partial_kx^i=\delta_{ik}$ this gives $\partial_k(x^iT^{kj})=T^{ij}$, so integrating over a ball of radius $R$ and using the divergence theorem moves the volume integral to a surface integral (usable because the field is smooth inside the ball).

The right-hand side is of the order of the product of the area of the sphere $4\pi R^2$, the length of the arm $R$, and $|T|$ on the sphere. For example, if $T$ decays faster than $r^{-3}$, it goes to 0 as $R\to\infty$ and $\int T^{ij}d^3x=0$. In fact finite energy alone is enough (the detour below). Together with $-T^{00}+\sum_iT^{ii}=0$ from (8),

follows. The energy density is a continuous function that is at least 0, so it is 0 everywhere, and $F=0$ everywhere. In this form the outward pressure of the lump, $\sum_iT^{ii}$, equals the energy, and there is nothing to push back. Papers that put related conclusions in their titles are Deser (1976) “Absence of static solutions in source-free Yang–Mills theory” and Coleman (1977) “There are no classical glueballs” (bibliographic data only; the methods of proof have not been checked in the text). Adding a constant with the dimension of mass (a Higgs field and so on) makes the trace non-zero, and this argument no longer applies.

Detour: no assumption on the rate of decay is needed

Seen in a spherically symmetric field

In SU(2), taking the form $A_0=0$, $A^a_i=\varepsilon_{aij}x_j\,(1-w(r,t))/(gr^2)$, which is unchanged when space and colour are rotated together, the problem becomes one in a single variable $r=|\mathbf x|$.

$w=\pm1$ is a pure gauge with energy 0. On a three-dimensional field with differences, we checked that the energy density (to accuracy $10^{-8}$), Gauss's law and the rest of (3) reduce to (10) Computed.

ComputedThe numbers in this figure were computed on this machine. It solves the same differences as Table 1 ($h=0.05$, leapfrog with $\Delta t=h/2$) in the same order, and agrees with python3 to 12 digits. If JavaScript is disabled, Table 1 is the content of the figure.

The dip splits in two and travels at the speed of light, 1; the incoming peak reflects at the origin at $t\approx10$, and by $t=30$ nothing is left in $r\lt15$. The same held for $a=0.3,\ 1.8$ Computed. What is shown covers only these initial values and a finite range of time.

Detour: there is a static solution, but its energy is infinite

We move to imaginary time $t=-i\tau$, $x_4=\tau$ (2-04 §09). From $A_0\,dt=A_4\,d\tau$ we get $A_0=iA_4$, and also $\partial_t=i\partial_4$, so $F_{i0}=iF_{i4}$. Rewriting as $E^a_i=F^a_{i4}$, the sign of $|\mathbf E^a|^2$ in (1) flips, and $iS$ becomes $-S_E$. The Euclidean action $S_E$ is a sum of positive quantities, and in the quantum theory it enters the weight $e^{-S_E}$ (4-02).

The Hodge star $(*F)_{\mu\nu}=\tfrac12\varepsilon_{\mu\nu\rho\sigma}F_{\rho\sigma}$ ($\varepsilon_{1234}=1$, 1-12) exchanges $\mathbf E$ and $\mathbf B$ in this notation, so self-dual $F=*F$ means $\mathbf E^a=\mathbf B^a$, and anti-self-dual means $\mathbf E^a=-\mathbf B^a$. At each point $\tfrac12(|\mathbf E|^2+|\mathbf B|^2)=\tfrac12|\mathbf E\mp\mathbf B|^2\pm\mathbf E\cdot\mathbf B$, so

with equality only when self-dual ($Q\gt0$) or anti-self-dual ($Q\lt0$) Known. For a solution of the Yang–Mills equations with finite action, it is said that it extends to a field on $S^4$ and $Q$ is an integer (Uhlenbeck 1982, bibliographic data only). General fields that are not solutions are treated in 3-05 §03. In the geometric notation of 3-05 (anti-Hermitian $A$ and $c_2$) it is $Q=-c_2$ (the detour of 3-05).

A self-dual field is automatically a solution, because $D_\mu F_{\mu\nu}=\pm\tfrac12\varepsilon_{\mu\nu\rho\sigma}D_\mu F_{\rho\sigma}$ is 0 by the cyclic sum (4). Instead of the second-order (2), it is enough to solve the first-order $F=\pm*F$, and its solutions have the least action within the same $Q$. Jaffe–Witten §2 write that classical non-abelian gauge theory has played a large role in the study of three- and four-dimensional manifolds Known. The representative of this in four dimensions is the first-order (anti-)self-dual equation, and Donaldson (1983) used the space of its solutions for four-dimensional topology (3-06) Known. Real self-dual fields exist only in imaginary time (1-06).

As an example we borrow the BPST solution (Belavin–Polyakov–Schwartz–Tyupkin 1975), constructed in 3-05 Known. In 't Hooft's symbols ($\eta_{aij}=\varepsilon_{aij}$, $\eta_{ai4}=-\eta_{a4i}=\delta_{ai}$), the solution of size $\rho$ has the following form.

Since $\eta_a$ is self-dual, $\mathbf E^a=\mathbf B^a$, $S_E=8\pi^2/g^2$, $Q=1$ (Table 1 of 1-12). With differences we checked the right-hand formula of (13), $F=*F$ and $D_\mu F_{\mu\nu}=0$ to within $6\times10^{-10}$ Computed. We deform this in the direction of imaginary time. Pulling back by $M=\operatorname{diag}(1,1,1,s)$ as $A^{(s)}_\mu(x)=M_{\nu\mu}A_\nu(Mx)$, for $s\gt1$ the shape of the field shrinks by a factor $1/s$ in the $x_4$ direction, and for $s\lt1$ it stretches by a factor $1/s$. $\mathbf E^{(s)}=s\,\mathbf E(Mx)$, $\mathbf B^{(s)}=\mathbf B(Mx)$, and only the components $F_{i4}$ of $\mathbf E$ that have $x_4$ as an index are multiplied by $s$ (because $s$ multiplies the derivative in $x_4$ and $A_4$). For $s\ne1$ it is not a solution. Substituting $y=Mx$ ($d^4x=d^4y/s$) gives the following.

The action is least at $s=1$ and grows whether stretched or shrunk (the inequality of arithmetic and geometric means). $Q$ does not change because the integrand $\operatorname{tr}F\wedge F$ is a four-form that does not use the metric, and its integral does not change under orientation-preserving changes of coordinates (1-06). The action uses the Hodge star, that is, the metric, so it changes.

ComputedThe numbers in this figure were computed on this machine. The figure computes (14) and the closed formula for the action density on the spot (agreeing with Table 2).

The classical Yang–Mills field has no static lump of finite energy (§06), no lower bound on the frequency of waves (3-02 §07), and in imaginary time the minimum of the action in each $Q$ is attained by self-dual fields. None of these is a mass gap. Jaffe–Witten §2 call the mass gap and confinement “quantum properties that do not have classical analogs” Known. Since the classical theory has no scale, the scale of the gap is thought to enter in the process of quantisation (4-07) Physics. This chapter shows nothing about the quantum theory or about the mass gap.

A bridge to the chapters ahead. In 3-04 we study the shape of the space in which fields related by gauge transformations are regarded as the same (the gauge orbit space), and the reason why the gauge cannot be fixed in one way globally. That the $Q$ of §07 is an integer and the construction of the BPST solution are in 3-05, and the story of how the space of solutions of self-dual fields distinguishes four-dimensional topology is the detour of 3-06. The trace and scale of §05 continue on the quantum side in 4-06 (asymptotic freedom) and 4-07 (dimensional transmutation).

What this chapter can say

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Next chapter to read: 3-04 Gauge theory as geometry. For instantons, 3-05; for the lattice action, 5-01.

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