2026-10-01 · chapter Yang–Mills and the mass gapPart 1 Mathematical preparationFourier analysis and distributions

Fourier analysis and distributions — handling things without values at points by waves and averages

The free field in two or more dimensions has no value at a single point, because the variance at a point is infinite (1-09 §08). What has meaning is the value averaged with a smooth weight $f$, and its variance can be written as a double average of the two-point correlation $G$.

$G$ is called the propagator, and formally $E[\varphi(x)\varphi(y)]=G(x-y)$. $G$ is infinite at $x=y$, but the right-hand side of (1) is finite. Example: the variance of the value averaged over a box of side $\varepsilon$ is, for $d\ge3$, proportional to $\varepsilon^{2-d}$ by $G\approx c\,r^{2-d}$ (§07); in four dimensions it is 4 times larger each time the box is halved. Infinite at a point, finite when averaged.

Split a function of period $2\pi$ into a superposition of waves $e^{ikx}$. $e^{ikx}/\sqrt{2\pi}\ (k\in\mathbb Z)$ is an orthonormal basis of $L^2(-\pi,\pi)$ (the complex version of 1-10 §02), and the coefficients are extracted by inner products Known.

The last is Parseval's identity. Integrating by parts, differentiation is replaced by multiplying the coefficients by $ik$.

So smoothness shows in the decay of the coefficients. If the function is still square-integrable after $n$ derivatives, $\sum_kk^{2n}|c_k|^2\lt\infty$. With a jump, the coefficients decrease only as $1/k$.

Example: the partial sums of the square wave $f(x)=\operatorname{sign}x$ are $S_N(x)=\frac4\pi\sum_{k\ \text{odd}\le N}\frac{\sin kx}{k}$. The mean-square error goes to 0 as $\frac8{\pi^2}\sum_{k\ \text{odd}\gt N}k^{-2}\approx\frac4{\pi^2N}$ by Parseval, yet the maximum overshoots to about 1.179 beside the jump, and the ratio $\frac1\pi\int_0^\pi\frac{\sin t}{t}dt-\frac12=0.08949\cdots$ to the jump size 2 remains regardless of $N$. This is the Gibbs phenomenon; Wilbraham (1848) is said to have studied it first, Gibbs's letter of 1899 to have made it widely known, and Bôcher (1906) to have named it Known.

ComputedThe maximum of $S_N$ (the value at $x=\pi/(N+1)$) is 1.27324, 1.18233, 1.17898 at $N=1,9,999$ (limit $\frac2\pi\operatorname{Si}(\pi)$), and the mean-square error is 0.18943, 0.04040, 0.000405 (Parseval's formula and numerical integration agree; for $N=999$, 0.0004052846 agrees to 10 digits). It converges in mean square, and not uniformly. "Convergence" has meanings of different strength. The Fejér mean $\sigma_N=\frac1{N+1}(S_0+S_1+\cdots+S_N)$, which averages the partial sums, smooths with a positive kernel, so no overshoot occurs ($|\sigma_N|\le1$) Known. "Smooth first, then take the limit" is the basic move of approximation in §04.

ComputedThe numbers in this figure were computed on this machine (the partial sums of the series are added on the spot; the readout agrees with the computation above to 5 digits at $N=1,3,5,9,19,49,99,199$).

For functions without a period the frequencies are arranged continuously, and sums are replaced by integrals. The conventions of this book are as follows. Taking the measure as $d^dk/(2\pi)^d$ matches (10) of 1-09.

The same rules as for series hold Known. Length is preserved (Plancherel), differentiation becomes multiplication, and convolution $(f*g)(x)=\int f(y)g(x-y)\,d^dy$ becomes a product.

Then $\widehat{-\Delta f}=|k|^2\hat f$, and a differential equation with constant coefficients becomes division frequency by frequency. If $m\gt0$ the denominator is never 0, and the solution is determined uniquely.

Example: the transform of a Gaussian of width $\sigma$ is a Gaussian of width $1/\sigma$ (the uncertainty relation of 2-05). The transform of $e^{-m|x|}$ is $\frac{2m}{k^2+m^2}$: the corner at $x=0$ shows up as the slow decrease $1/k^2$ (the one-dimensional propagator of §07).

Objects without values at points are defined by their values when paired with smooth functions. The partners are called test functions, taken to be functions that can be differentiated any number of times and are 0 outside a bounded set (the whole set is $\mathcal D$. Example: $e^{-1/(1-|x|^2)}$ on $|x|\lt1$, 0 outside).

A distribution is a linear and continuous rule assigning a number $\langle T,\phi\rangle$ to a test function $\phi$ (continuous meaning that $\langle T,\phi_n\rangle\to0$ for sequences $\phi_n$ whose supports (the closure of the set of points where the function is non-zero) lie in one bounded set and all of whose derivatives approach 0 uniformly) Known. "Measuring by testing" in the title is a metaphor; to be exact, it means regarding this rule $\phi\mapsto\langle T,\phi\rangle$ itself as the object instead of values at points. The English word is distribution, and this is the "distributions" of the chapter title (not the "distribution" of probability). The idea of distributions is said to begin with Sobolev (1936) and to have been systematised by Schwartz Known.

Examples: (a) a function $f$ integrable on bounded sets becomes a distribution by $\langle f,\phi\rangle=\int f\phi\,d^dx$. (b) The $\delta$ of $\langle\delta,\phi\rangle=\phi(0)$ cannot be written as the integral of any function. Differentiation is defined by promoting integration by parts to a definition.

The right-hand side always makes sense, so every distribution can be differentiated any number of times. For the step function $H$ (1 for $x\gt0$, 0 for $x\lt0$), $\langle H',\phi\rangle=-\int_0^\infty\phi'\,dx=\phi(0)$, that is, $H'=\delta$.

Approximation: if $\rho\ge0$ is smooth with $\int\rho=1$ and $\rho_\varepsilon(x)=\varepsilon^{-d}\rho(x/\varepsilon)$, then $\langle\rho_\varepsilon,\phi\rangle\to\phi(0)$, that is, $\rho_\varepsilon\to\delta$. $\sin nx$ does not converge pointwise, but integration by parts gives $\frac1n$, so as a distribution it converges to 0.

For transforms we use partners wider than $\mathcal D$. On the set $\mathcal S$ of functions that can be differentiated any number of times and whose values and derivatives decrease faster than any power of $|x|$ (the Schwartz space), the transform maps $\mathcal S$ one-to-one onto $\mathcal S$ Known. The transform of a continuous linear rule on $\mathcal S$ (a tempered distribution) is defined to match $\int\hat f\phi=\int f\hat\phi$.

Example: since $\langle\hat\delta,\phi\rangle=\hat\phi(0)=\int\phi\,d^dx$, $\hat\delta=1$. Something concentrated at one point contains all frequencies with the same weight (the origin of the "white" in white noise).

Conversely, the transform of the constant 1 is $(2\pi)^d\delta$: something of the same size everywhere contains only the frequency 0. The rules of (5) also extend as they are, giving $\widehat{\partial_jT}=ik_j\hat T$ Known. This is why in §07 the propagator can be written $\hat G=1/(|k|^2+m^2)$.

Smoothness shows in the decay of the coefficients or of $\hat f$ ((3)(5)). We set up two rulers to measure this with numbers. The first is a length that puts weights on frequencies, and the set of objects for which it is finite is called the Sobolev space $H^s$. The larger $s$, the smoother the things admitted; the more negative $s$, the rougher the things admitted.

The second is behaviour under rescaling. The $T(\lambda\,\cdot)$ of a distribution is defined by $\langle T(\lambda\,\cdot),\phi\rangle=\lambda^{-d}\langle T,\phi(\cdot/\lambda)\rangle$, promoting the change-of-variables formula to a definition.

"Ruler" is a metaphor; to be exact, it means the two numbers $s$ of (9a) and $\alpha$ of (9b). Since $\hat\delta=1$, $\|\delta\|_{H^s}$ is finite only when $s\lt-d/2$. The scaling exponent is $-d$ for $\delta$ and $-a$ for $|x|^{-a}$.

For a random object $T$, when $T(\lambda\,\cdot)$ and $\lambda^\alpha T$ have the same probability distribution, we take the exponent to be $\alpha$. For white noise $\xi$ ($E[\xi(f)\xi(g)]=\int fg\,d^dx$, the $\mathbb R^d$ version of 1-09 §07), the covariance of the values of $\xi(\lambda\,\cdot)$ paired with test functions $\phi,\psi$ is $\lambda^{-2d}\int\phi(x/\lambda)\psi(x/\lambda)\,d^dx=\lambda^{-d}\int\phi\psi\,d^dx$, and a Gaussian with mean 0 is fixed by its covariance, so $\xi(\lambda\,\cdot)$ has the same probability distribution as $\lambda^{-d/2}\xi$. The exponent is $-d/2$. The massless free field ($m=0$, $d\ge3$) has covariance $G(\lambda x)=\lambda^{2-d}G(x)$ (the near-side formula of §07 is exact for $m=0$), so its exponent is $1-d/2$. For $d=2$, $G$ is logarithmic, so the same holds up to constants Known. Measuring white noise in spacetime, including time, by the parabolic scaling that multiplies time by $\lambda^2$ and space by $\lambda$, the exponent is $-(d+2)/2$ (Exercise 2.3 of Hairer's lectures) Known. This exponent decides in §10 whether products can be made.

The solution of the equation with $\delta$ on the right-hand side of (6) is the propagator. "Responding to δ" is a metaphor; to be exact, it means satisfying the first formula of (10) as an identity of distributions, and such a solution is called a fundamental solution. From $\frac1{|k|^2+m^2}=\int_0^\infty e^{-t(|k|^2+m^2)}dt$ and the transform of the Gaussian, there also comes a representation as the time integral of the fundamental solution $(4\pi t)^{-d/2}e^{-|x|^2/4t}$ (the heat kernel) of the heat equation $\partial_tu=\Delta u$.

$G$ is the covariance of the free field, and in physics the Euclidean propagator of a particle of mass $m$ (4-02, 4-03). With $r=|x|$, it is $e^{-mr}/2m$ for $d=1$, $e^{-mr}/4\pi r$ for $d=3$, and in general $(2\pi)^{-d/2}(m/r)^{d/2-1}K_{d/2-1}(mr)$ using the modified Bessel function $K_\nu(s)=\int_0^\infty e^{-s\cosh t}\cosh\nu t\,dt$ Known. What matters are its two faces.

On the near face $m$ disappears and it diverges as $d\ge2$. On the far face it decreases as $e^{-mr}$. "Face" is a metaphor; to be exact, it means that as $r\to0$ the ratio of $G(r)$ to the left formula of (11) approaches 1, and as $r\to\infty$ the ratio to the right formula approaches 1. Calling the $\xi$ for which correlations decrease as $e^{-r/\xi}$ the correlation length (2-08), $\xi=1/m$, and "rate of decrease of correlations = mass" is the free-field version of the definition of the mass gap (4-11).

ComputedThe numbers in this figure were computed on this machine (the integral representation of $K_\nu$ by the trapezoidal rule; agreeing with Table 1 to 7 digits).

ComputedTable 1 $G$ for $m=1$ and its ratio to the formulas of (11) (the near side for $d=2$ adds the constant $\ln2-\gamma$). The Bessel function formula and numerical integration of (10) agree to 15 digits.

The lattice propagator

The propagator on a lattice of spacing $a$ is the integrand of (10) of 1-09 multiplied by $e^{ik\cdot x}$ and integrated ((10) of 1-09 is its value at $x=0$). For $d=3$, $m=1$ at distance 1, the difference from the continuum $G$ shrinks in proportion to $a^2$: 0.9% at $a=1/8$ and 0.2% at $a=1/16$ Computed (5-01).

Even if it diverges at the origin, it is a distribution by (a) of §04 if it can be integrated near by. For powers it is a comparison with the dimension ($|S^{d-1}|$ is the area of the unit sphere).

$G\sim r^{2-d}$ is $a=d-2\lt d$, so it is a distribution in every dimension. Its $n$-th power is $a=n(d-2)$, integrable when $n(d-2)\lt d$; for the square when $d\lt4$, for the cube when $d\lt3$.

Powers of the propagator come from products of fields. The fourth moment of a Gaussian is a sum over ways of pairing (1-08 §06). In ${:}\varphi^2{:}=\varphi^2-E[\varphi^2]$, with the infinite $E[\varphi^2]=G(0)$ subtracted (the Wick product), only the pairings joining the $\varphi$ of $x$ with the $\varphi$ of $y$ remain (the pairings among the $x$ and among the $y$ cancel with what was subtracted) Known.

The variance of ${:}\varphi^2{:}(f)$ is finite when $G^2$ is integrable, $d\lt4$; for the cube, $6G^3$, when $d\lt3$. A figure setting this beside other computations that switch at four is the dimension slider of 1-12 §10.

ComputedTable 2 $I_n(\varepsilon)=\int_{\varepsilon\lt|x|\lt1}G(x)^n\,d^dx$ ($m=1$, numerical integration in the radius). Pressing a row draws it in Figure 3.

ComputedThe numbers in this figure were computed on this machine (a table embedding to 4 digits the same numerical integration in the radius as Table 2, done at $\log_{10}(1/\varepsilon)=1,1.25,\dots,6$).

$d=4,n=2$ grows by 0.02916 each time $\varepsilon$ is divided by 10, agreeing with $\ln10/8\pi^2$ from (11)(12) Computed. Four dimensions is the borderline where $G^2$ just stops being integrable, and the coupling constant being dimensionless in four dimensions (4-04) is the same counting of scales Known (1-12).

The product of a smooth $g$ and a distribution $T$ is determined by $\langle gT,\phi\rangle=\langle T,g\phi\rangle$. "Smoothing, multiplying and removing the smoothing" for two distributions makes three things happen. (i) It diverges: the area of $\rho_\varepsilon^2$ goes to infinity as $\varepsilon^{-d}\int\rho^2$. (ii) The way of smoothing remains: smooth $H$ with $\rho$ and $\delta$ with $\psi$ (both probability densities), and replace by $x=\varepsilon X$.

The limit is $c\,\delta$, but $c$ is determined by the way of smoothing. Taking $\rho$ to be the standard Gaussian and $\psi$ a Gaussian centred at $s$ gives $c=\Phi(s/\sqrt2)$ ($\Phi$ is the distribution function of the standard normal distribution), which can be any value between 0 and 1.

ComputedThe numbers in this figure were computed on this machine (closed formulas; checked against the numerical integration below).

ComputedIn numerical integration, the area of the product of $H$ and $\delta$ is 0.078650, 0.500000, 0.760250, 0.921350 at $s=-2,0,1,2$ (the same at $\varepsilon=0.1$ and 0.01, agreeing with $\Phi(s/\sqrt2)$ to 6 digits), and that of $\delta$ with itself is 1.038, 2.821, 2.197, 1.038 at $\varepsilon=0.1$, and 10 times that at $\varepsilon=0.01$.

(iii) Associativity breaks. For the principal value $\langle\mathrm{p.v.}\tfrac1x,\phi\rangle=\lim_{\varepsilon\to0}\int_{|x|\gt\varepsilon}\frac{\phi(x)}{x}dx$, as products "smooth function × distribution", $x\cdot\delta=0$ and $x\cdot\mathrm{p.v.}\frac1x=1$.

A product keeping all the properties of the product of functions cannot be put on the whole space of distributions. To be exact, it is said that there is no associative product on distributions (containing at least the continuous functions and $\delta$) that agrees with the ordinary product on continuous functions and for which differentiation satisfies the Leibniz rule $(fg)'=f'g+fg'$ (Schwartz 1954) Known.

The condition for products to be made can be written with the scaling exponent of §06. Let $C^\alpha$ be the following space. For $0\lt\alpha\lt1$, the set of Hölder continuous functions of order $\alpha$ ($|f(x)-f(y)|\le C|x-y|^\alpha$, $|x-y|\le1$); for $\alpha\gt1$, the same form of condition imposed on derivatives; for $\alpha\lt0$, the set of distributions whose scaling exponent in the sense of §06 is locally uniformly at least $\alpha$. The larger $\alpha$, the smoother. If the total roughness is positive, products can be made (Proposition 4.14 of Hairer 2014) Known.

"Defined continuously" means that extending the product of smooth functions by limits gives a single result independent of how the limit is approached. The same proposition states that, if $\alpha$ is not a natural number, $\alpha+\beta\gt0$ is also necessary.

Example: the step function $H$ is bounded but has a jump, and its roughness exponent is 0 ($H(\lambda x)=H(x)$; it lies in $\alpha\lt0$ for every $C^\alpha$). The one-dimensional $\delta$ is $-1$. The total is not positive, so (16) cannot be used, consistent with the phenomenon of (ii) in §09. The free field is slightly rougher than $1-d/2$ (it lies in $C^\alpha$ for $\alpha\lt1-d/2$), so its square can be made as the determined product of (16) only for $d\lt2$ Known. In §08 the variance of ${:}\varphi^2{:}(f)$ became finite for $d=2,3$ because the divergence of (i) in §09 was subtracted (renormalised) first, and the fact that it takes a form independent of the way of smoothing depends on probabilistic cancellation (the counting of Gaussian pairings).

The Yang–Mills connection $A$ is expected to be as rough as the free field at short distances Physics, and the $A\wedge A$ of the field strength $F=dA+A\wedge A$ (1-07) is a product that does not satisfy (16) in two or more dimensions. In two and three dimensions the analysis of products of this kind is being advanced by stochastic quantisation (6-09), and in four dimensions the continuum theory itself has not yet been constructed (6-10).

What this chapter can say

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Next chapter: 1-12 Detour: what happens only in four dimensions. For uses of the propagator see 4-03; for the axioms of fields with values in distributions, 4-09.

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