2026-10-01 · chapter Yang–Mills and the mass gapPart 2 Physical preparationfields and the wave equation

The idea of a field — when the number of beads becomes infinite, the coordinates of the system become a function

The shortest way into fields is to start from finitely many beads. Place $N$ beads at spacing $a=\ell/(N+1)$ on an interval of length $\ell$, join neighbours with taut threads, and fix both ends to walls. Let $q_j(t)$ be the sideways displacement of bead $j$ (the walls are $q_0=q_{N+1}=0$), $m=\rho a$ the mass of a bead ($\rho$ is the linear density), and $\sigma$ the tension.

If the difference $\Delta q$ from a neighbour is small, the thread between them stretches by $\sqrt{a^2+\Delta q^2}-a\approx\Delta q^2/(2a)$. We let the beads move only sideways, and discard the next term $-\Delta q^4/(8a^3)$. This makes the following equations linear (where linearity matters is section 08). Tension × stretch is the potential energy, so the Lagrangian (2-01) and the Euler–Lagrange equations take the following form.

Only neighbouring beads appear on the right-hand side. This locality survives all the way to gauge theory. Example. If $N=1$, it is a simple oscillation with $m\ddot q=-(2\sigma/a)q$, and when $\ell=\rho=\sigma=1$, $\omega=\sqrt8\approx2.828$.

Writing everything together as $\ddot{\mathbf q}=-K\mathbf q$, $K$ is the symmetric matrix with diagonal entries $2\sigma/(ma)$, $-\sigma/(ma)$ on either side, and 0 elsewhere. Being symmetric, it has $N$ orthogonal eigenvectors, and its eigenvalues are real (1-01). The eigenvalues are not negative either: $\mathbf q\cdot K\mathbf q$ is $2/m$ times the potential energy, a sum of squares. For an eigenvector $\mathbf v$, $\mathbf v\cdot K\mathbf v=\lambda|\mathbf v|^2$, so $\lambda\ge0$. Both ends stay at the walls, so the potential energy is 0 only when $\mathbf q=0$, and $\lambda=0$ does not occur either. So we write the eigenvalues as $\omega_n^2$ ($\omega_n\gt0$).

Taking the eigenvectors as new axes, each component obeys $\ddot Q_n=-\omega_n^2Q_n$. $N$ tangled beads split into $N$ mutually unrelated simple oscillations. These are the normal modes; each way of swinging is called a "mode", and $n$ the number of the mode. For this matrix a closed formula can be written for $n=1,\dots,N$.

It can be checked by substitution using $\sin(A+B)+\sin(A-B)=2\sin A\cos B$. Example. If $\ell=\rho=\sigma=1$ then $\omega_n=2(N+1)\sin\frac{n\pi}{2(N+1)}$, and when $n$ is much smaller than $N$, $\omega_n\approx n\pi$: the frequencies of the string in the next section.

Increase the number of beads and bring the spacing $a$ to 0. $\ell$, $\rho$, $\sigma$ are fixed, and the mass of a bead $m=\rho a$ goes to 0. What to fix when taking the limit returns as a question of the same form in lattice gauge theory (5-09). Reading the number $j$ as the position $x=ja$ and writing $q_j(t)=u(t,x)$, we get $(q_{j+1}-q_j)/a\to\partial_xu$, $\sum_ja(\cdots)\to\int dx$, $(q_{j+1}-2q_j+q_{j-1})/a^2\to\partial_x^2u$, and

$\mathcal L$ is called the Lagrangian density, and the formula below the wave equation. The same formula also comes from the Euler–Lagrange equation for fields $\partial_t\frac{\partial\mathcal L}{\partial(\partial_tu)}+\partial_x\frac{\partial\mathcal L}{\partial(\partial_xu)}-\frac{\partial\mathcal L}{\partial u}=0$, and this form is used both for the electromagnetic field (2-03) and for the Yang–Mills field (3-03).

A field is a system in which the index $j$ has been replaced by a continuous $x$. The "coordinates" of the system are not a set of numbers but a single function $u(t,\cdot)$. The normal modes of the string are $\sin(n\pi x/\ell)\cos\omega_nt$ with frequencies $\omega_n=n\pi c/\ell$ ($n=1,2,\dots$), with no upper limit.

ComputedThe numbers in the figure are computed on the spot from the closed formulas in the text.

ComputedTable 1 Ratio $\omega_n/(n\pi)$ of the frequencies of the chain and the string ($\ell=\rho=\sigma=1$). Values found numerically as eigenvalues of the matrix $K$; the relative difference from the closed formula was at most $3\times10^{-12}$.

With $n$ fixed the ratio approaches 1, but the column $n=N/2$ only approaches $\sin(\pi/4)/(\pi/4)\approx0.900$, and the column $n=N$ only $2/\pi\approx0.637$. The continuum limit is an approximation for waves whose wavelength is much longer than the spacing.

The solutions can be read in two ways. As travelling waves: for any twice differentiable $f,g$, $u=f(x-ct)+g(x+ct)$ is a solution. $f$ travels to the right and $g$ to the left, keeping their shape, at speed $c$. With initial shape $u_0$ and initial velocity 0, $u=\frac12[u_0(x-ct)+u_0(x+ct)]$, and the hump splits in two (with both ends fixed, extend $u_0$ to an odd function of period $2\ell$). Solutions of this form are said to go back to d'Alembert's paper of 1747.

As superpositions of standing waves: the equation is linear, so sums of normal modes are also solutions.

The formula for the coefficients comes from the orthogonality of the $\sin(n\pi x/\ell)$ (the integral of the product is 0 for $n\ne n'$ and $\ell/2$ for $n=n'$). It is a Fourier sine series (1-11). Example. For the triangle obtained by pulling the position $p$ to height $h$ and releasing, $b_n=\frac{2h\ell^2}{n^2\pi^2p(\ell-p)}\sin\frac{n\pi p}{\ell}$. The coefficients decrease only as $1/n^2$, and representing the corner needs modes up to high numbers.

ComputedThe numbers in the figure are computed on the spot from the partial sums of $b_n$ and d'Alembert's formula ($\ell=c=h=1$).

The sum $E$ of kinetic and potential energy does not depend on time. Using the wave equation, the time derivative of the integrand becomes $\sigma\,\partial_x(\partial_tu\,\partial_xu)$, and since both ends do not move the boundary terms vanish.

This is the field version of Noether's theorem (2-01). Splitting into normal modes, the cross terms vanish by orthogonality, $E=\sum_nE_n$, $E_n=\frac{\rho\ell}{4}(\dot b_n^{\,2}+\omega_n^2b_n^2)$. Each $E_n$ is conserved separately. A string is a collection of infinitely many mutually unrelated simple oscillations.

Example. For a plucked string ($s=p/\ell$), the share of mode $n$ is $f_n=\frac{2\sin^2(n\pi s)}{n^2\pi^2s(1-s)}$.

ComputedTable 2 The fraction of the energy held by the first $K$ modes. We also checked that the sum of all terms is 1 (to 5 digits or more with 2 million terms).

Plucking in the middle gives $f_1=8/\pi^2\approx0.81$. Near the end ($s=0.02$) the first 10 do not reach 40%, and the energy escapes to the short-wavelength side.

If each bead is also tied to the floor by a spring (constant $\kappa$) pulling it back to its original position, $\sum_j\frac{\kappa}{2}q_j^2$ is added to the potential energy. The continuum limit with $\Omega^2=\kappa/m$ fixed is the Klein–Gordon equation on the left, and substituting a plane wave $\cos(kx-\omega t)$ gives the dispersion relation on the right. Here $k$ is the wave number (the advance of phase per unit length; for wavelength $\lambda$, $k=2\pi/\lambda$), a different quantity from the mode number $n$.

If $\Omega=0$ then $\omega=c|k|$, and lengthening the wavelength brings the frequency towards 0. If $\Omega\gt0$ then always $\omega\ge\Omega$: the frequencies have a lower bound. In a box of length $\ell$ the wave numbers are restricted to $k_n=n\pi/\ell$, and the lowest frequency is $\omega_1=\sqrt{(c\pi/\ell)^2+\Omega^2}$. In a finite box it is positive even for $\Omega=0$, and what is asked is whether the lower bound remains when the box is enlarged. This is why the definition of the mass gap is written in infinite volume (4-11).

ComputedThe numbers in the figure are computed on the spot from $\omega=\sqrt{k^2+\Omega^2}$, $k_n=n\pi/\ell$ ($c=1$).

ComputedThe lowest frequency ($c=1$) for $\ell=1,10,100,1000$ is 3.142, 0.314, 0.0314, 0.00314 at $\Omega=0$, and 3.297, 1.048, 1.0005, 1.000005 at $\Omega=1$ (values of the closed formula). For $\ell\le100$ it agreed with the smallest eigenvalue of the matrix of a chain with spacing 0.05 to within a relative difference of 0.11% (largest at $\ell=1$).

A string has countably infinitely many degrees of freedom. The trouble comes when all modes are shaken with the same weight, the typical case being thermal equilibrium. A minimal definition (details in 2-08): in thermal equilibrium at temperature $T$, the probability density of a configuration is proportional to $\exp(-V/k_BT)$ ($k_B$ is Boltzmann's constant). If $V=\sum_n\frac{\kappa_n}{2}Q_n^2$, each $Q_n$ follows an independent Gaussian distribution (1-08) with variance $k_BT/\kappa_n$, and the mean potential energy is $k_BT/2$ for each (equipartition). For the string, $\kappa_n=\sigma n^2\pi^2/(2\ell)$ for $u=\sum_nq_n\sin(n\pi x/\ell)$, and three things happen.

1. The total energy is infinite, because $k_BT/2$ is given to each of infinitely many modes.
2. The displacement at a point is finite. $\operatorname{Var}u(x)=\sum_n\frac{2\ell k_BT}{\sigma n^2\pi^2}\sin^2\frac{n\pi x}{\ell}=\frac{k_BT}{\sigma}\cdot\frac{x(\ell-x)}{\ell}$. This random curve is called the Brownian bridge (KnownSheffield 2007).
3. The slope is not determined. $\operatorname{Var}\partial_xu=\frac{2k_BT}{\sigma\ell}\sum_n\cos^2\frac{n\pi x}{\ell}$ diverges. The curve does not break off, yet it is all corners everywhere. To be exact, it means that at no point can the slope be defined as a random variable.

ComputedThe figure computes $u=\sum_{n\le K}g_n\sqrt2\sin(n\pi x)/(n\pi)$ on the spot ($g_n$ are normal random numbers with a fixed seed, $\ell=k_BT/\sigma=1$).

ComputedRaising the dimension of space $d$ (a membrane, a field in a box), we count the variance at the centre of the box $[0,1]^d$ (in units of $k_BT/\sigma$) as a sum over modes with the number in each direction up to $K$ (doubling from $K=10$ to 160). The contribution of mode $n=(n_1,\dots,n_d)$ is proportional to $1/|n|^2$. The table and figure of the same quantity discretised on a lattice are in Table 3 and Figure 3 of 1-09, which this book takes as authoritative. A figure setting it beside quantities that switch at four is the dimension slider of 1-12 §10.

$d=1$ converges to $1/4$ (the value at the centre of the Brownian bridge). $d=2$ increases by $\ln2/(2\pi)\approx0.110$ each time $K$ is doubled (diverging as $\log K$); $d=3$ doubles, and $d=4$ quadruples. The number of modes in the shell of radius $r$ is proportional to $r^{d-1}$, with $1/r^2$ for each, so the sum has the form $\sum r^{d-3}$. The divergence comes from the short-wavelength side.

KnownWrite a field in a box of dimension $d$ as $\varphi(x)$ (the multidimensional version of the $u$ of the string). For $d\ge2$, the value $\varphi(x)$ of a hot field at a point has no meaning, and only the value $\int\varphi f\,dx$ obtained by multiplying by a smooth weight function $f$ and integrating has meaning as a random variable. In this sense, the field takes values not in functions but in distributions (Sheffield 2007, 1-09, 1-11). Writing a quantum field in imaginary time treats time like one more direction of space, and a free field in four-dimensional spacetime has a sum of the same form as the $d=4$ row (4-02).

The rougher the field, the harder it is to give meaning to products of fields (non-linear terms). To be exact, products of distributions cannot in general be defined (1-11). Divergences (4-04) are their naive appearance. Stating exactly how rough a field can be while non-linear terms can still be handled is the "subcritical / critical" of stochastic quantisation (6-09).

Uses ahead: the potentials of the electromagnetic field and their redundancy (2-03), the Lorentz invariance of $\partial_t^2/c^2-\partial_x^2$ (2-04, the article on spacetime; whether a wave leaves a tail is decided by the dimension of space: same article), the Fock space in which the modes become quantum oscillators (2-06), probability on fields (2-07).

In 3-02 one assumption breaks down. The splitting into normal modes worked because the equation was linear. The Yang–Mills field takes values in a Lie algebra at each point and interacts with itself, so the modes are no longer independent. Classical waves have no lower bound either. PhysicsEven so, in quantum theory a gap is thought to arise.

Detour: the string of confinement also vibrates by this equation

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Next chapter: 2-03 Electromagnetism is a gauge theory. To go ahead to quantisation, 2-06.

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