2026-10-01 · chapter Yang–Mills and the mass gapPart 1 Mathematical preparationHilbert spaces and spectra

Hilbert spaces and self-adjoint operators — measuring "the lowest" and "the next" in infinite dimensions

In finite dimensions the smallest eigenvalue and the gap to the next can always be computed (1-01). What is asked is whether the gap stays positive in the limit of more and more degrees of freedom. As an example, discretise on a lattice of $N$ equal parts the rule assigning $-u''$ to a function $u$ on $[0,1]$ vanishing at both ends (the same form as the Hessian matrix of 2-01 §05).

The eigenvectors are $u_j=\sin(k\pi j/N)$ (this can be checked by substitution). Fixing $k$ and letting $N\to\infty$ gives $\lambda_k(N)\to k^2\pi^2$. This is an eigenvalue of $-u''=\lambda u,\ u(0)=u(1)=0$. The top eigenvalue grows to near $4N^2$ and has no limit.

ComputedTable 1 Eigenvalues of the matrix of (1) (the difference between a tridiagonal eigenvalue computation and (1) is at most $10^{-15}$ relative).

ComputedThe numbers in the figure are computed on the spot from (1) (checked against Table 1).

The ratio is fixed by $t=k/N$ alone: $\lambda_k(N)/k^2\pi^2=\bigl(\sin(\pi t/2)/(\pi t/2)\bigr)^2$ (the dashed line on the left of Figure 1). At $k=N/2$, $8/\pi^2\approx0.81$; at the right end $t\to1$, $4/\pi^2\approx0.41$. Fixing $k$ and increasing $N$ gives $t\to0$, and the ratio approaches 1. An eigenvector with $k$ near $N$ is, at the lattice points, exactly the values of $\sin k\pi x$, but its oscillation is as fine as the spacing of the points, so joining the points makes it look like a different shape. In fact $\sin(k\pi j/N)=-(-1)^j\sin((N-k)\pi j/N)$: a slow oscillation multiplied by a sign that alternates point by point (visible on the right of Figure 1 by bringing $k$ near $N-1$). What remains in the limit is the lower eigenvalues and the gap, and the limiting object is an operator unbounded above. In the continuum limit of lattice gauge theory (5-05) too, what is read from the lattice is the lower end of the spectrum (the lightest state) Physics. That this limit exists in four dimensions has not been proved (6-10).

To treat functions as "vectors with infinitely many components" we need an inner product (1-01 §05) and completeness. Complete means that sequences whose terms come arbitrarily close to one another (Cauchy sequences) have limits inside the space. A complex vector space with these two is called a Hilbert space.

In $L^2$, functions that differ only on a set of length 0 are regarded as the same (1-08). Continuous functions alone are not complete: $\tanh(n(x-\frac12))$ come close to one another, yet their destination is a step function. To take the destination of "a sequence making the ratio smaller and smaller" in section 06, the complete $L^2$ is needed Known.

$e_k=\sqrt2\sin k\pi x\ (k\ge1)$ is an orthonormal basis of $L^2(0,1)$, and for $c_k=\langle e_k,f\rangle$, $\lVert f\rVert^2=\sum_k|c_k|^2$ (Parseval's identity) Known (1-11). Example. For $f=x(1-x)$, $c_k=4\sqrt2/(k\pi)^3$ for odd $k$ and 0 for even. $\sum_{k\,\text{odd}}32/(k\pi)^6=1/30$ agrees with $\int_0^1x^2(1-x)^2dx=1/30$. The $\ell^2$ of 1-09 connects to spaces of functions in this way.

An operator whose supremum of stretching factors $\lVert A\rVert=\sup_{\psi\ne0}\lVert A\psi\rVert/\lVert\psi\rVert$ is finite is called bounded. $-d^2/dx^2$ multiplies $e_k$ by $k^2\pi^2$, so it is not bounded, and there are functions whose second derivatives are not in $L^2$. So the subspace of functions it may be applied to, the domain $D(A)$, is specified as part of the operator. If $\langle\phi,A\psi\rangle=\langle A\phi,\psi\rangle$ on $D(A)$, it is symmetric. Self-adjointness is one step stronger: it says that the partners $\phi$ of this identity do not increase even when sought outside $D(A)$. To be exact, the set of $\phi$ for which there is a $\eta$ with $\langle\phi,A\psi\rangle=\langle\eta,\psi\rangle$ for all $\psi\in D(A)$ is taken as the domain $D(A^\dagger)$ of the adjoint $A^\dagger$, and $A^\dagger\phi=\eta$ is defined. If symmetric, $D(A)\subset D(A^\dagger)$, and when $D(A^\dagger)=D(A)$ it is self-adjoint. The spectral theorem and time evolution $e^{-itA}$ (2-05) presuppose self-adjointness Known. For the momentum $P=-i\,d/dx$, integration by parts gives the following.

(a) If $D=\{\psi(0)=\psi(1)=0\}$, it is symmetric, but the partner $\phi$ needs no condition, and $P^\dagger e^{izx}=ze^{izx}$ for every complex number $z$, so it is not self-adjoint. (b) If $D_\theta=\{\psi(1)=e^{i\theta}\psi(0)\}$, the partner satisfies the same condition and it is self-adjoint, with eigenvalues $2\pi n+\theta\ (n\in\mathbb Z)$. The phase difference $q\Phi/\hbar$ from a magnetic flux $\Phi$ threading the ring plays the role of $\theta$ (the Aharonov–Bohm effect of 2-03). For $-d^2/dx^2$ too, with $u=0$ at the ends (Dirichlet) the spectrum is $\{k^2\pi^2\}_{k\ge1}$, and with $u'=0$ (Neumann) it is $\{k^2\pi^2\}_{k\ge0}$. Boundary conditions are part of the operator, and change the lowest value.

In infinite dimensions there are cases where a map is one-to-one but its inverse is not bounded. So the spectrum is defined as follows.

The identity on the right holds when $A$ is self-adjoint and $\lambda\notin\sigma(A)$, and then $\sigma(A)$ is a closed subset of the real line Known. Example. The operator $M$ multiplying by $x$ on $L^2(0,1)$ has no eigenvalues, since $xf=\lambda f$ implies $f=0$. Even so, for $\lambda\in[0,1]$, $1/(x-\lambda)$ is not bounded, and $\sigma(M)=[0,1]$. A function of length 1 that is constant only on $[\lambda,\lambda+\varepsilon]$ has $\lVert(M-\lambda)f\rVert\le\varepsilon$: there are things arbitrarily close to eigenvectors (Weyl's criterion of 2-05 §03). Discretising makes the eigenvalues lattice points, which fill up $[0,1]$ (the lower eigenvalues of section 01 stay apart).

Isolated eigenvalues of finite multiplicity are called the discrete spectrum, and the rest the essential spectrum. The mass gap is the claim that the vacuum eigenvalue 0 is isolated and the rest of the spectrum lies at $\Delta\gt0$ or above (4-11).

A self-adjoint operator can be carried by a suitable unitary to "an operator multiplying by a real-valued function", like $M$ (the spectral theorem) Known. It is the infinite-dimensional version of the diagonalisation of matrices (1-01), and to be exact it includes parts that cannot be written with eigenvalues alone. Functions $g$ can be substituted, and if there is a basis of eigenvectors it takes the following form.

The right-hand side is for the case $A\ge0$, with the eigenspace of 0 spanned by $e_0$ of length 1 and the rest of the spectrum in $[\Delta,\infty)$; it follows by using $e^{-t\lambda}\le e^{-\Delta t}$ on each component. The gap is the slowest rate at which the components other than the lowest disappear (for the reading in imaginary time see 2-05 §05). Example 1. For the heat equation $\partial_tu=\partial_x^2u$ on a rod insulated at both ends, $A$ is the Neumann $-d^2/dx^2$, $\Delta=\pi^2$, and the deviation of the temperature from its mean is at most $e^{-\pi^2/2}\approx0.0072$ times at $t=0.5$. Example 2. The inverse of the Dirichlet $-d^2/dx^2$ is $(Gf)(x)=\int_0^1G(x,y)f(y)\,dy$, $G(x,y)=\min(x,y)-xy$ ($-(Gf)''=f$ and the zeros at both ends can be checked). Making $X(t)=B(t)-tB(1)$ from the Brownian motion $B$ of 1-09 gives zeros at both ends (the Brownian bridge), and its covariance $E[X(s)X(t)]$ is exactly $G(s,t)$. The eigenvalues of $G$ are $(k\pi)^{-2}$, and the finiteness of the sum $\sum_k(k\pi)^{-2}=1/6$ is the condition of 1-09 (4) for samples of the Brownian bridge to lie in $L^2$.

The lower end of the spectrum of a self-adjoint operator bounded below is the infimum of the Rayleigh quotient. The $n$-th too can be written in the form "take the maximum over a $n$-dimensional subspace, and the minimum over subspaces" (the min-max principle; for the eigenvalues below the essential spectrum, in increasing order counted with multiplicity) Known.

Expanded in eigenvectors, the Rayleigh quotient is a weighted average of eigenvalues $\sum\lambda_k|c_k|^2/\sum|c_k|^2$, and does not fall below $\lambda_1$. The eigenvalues of the matrix restricted to a subspace $V$ are upper bounds for $\lambda_1,\lambda_2,\dots$, and decrease as $V$ is enlarged (the Rayleigh–Ritz method). Example. Putting $x(1-x)$ into the Dirichlet $-d^2/dx^2$ gives the quotient $(1/3)/(1/30)=10\ge\pi^2$. For estimates from below there is Temple's inequality. If $R$ is the Rayleigh quotient of $\psi\in D(A)$, and we know a lower bound $\mu$ for the next eigenvalue ($R\lt\mu\le\lambda_2$), the following holds Known.

If $\psi=x(1-x)$, then $A\psi=2$, $\lVert A\psi\rVert^2/\lVert\psi\rVert^2=120$. Here $\lambda_2=4\pi^2$ is known exactly (section 03), so we can put $\mu=39$, giving $\lambda_1\ge10-20/29=9.310$. The Ritz value for $39.50\dots$ appearing below Table 2 is an upper bound for $\lambda_2$, and is no basis for $\mu$. The variational principle alone gives only upper bounds; holding it down from below needs other information. In Temple's inequality it is the lower bound $\mu$ for the next level (the convexity $V''\ge\kappa$ of section 08 is another example). This is, with finitely many degrees of freedom, the shape of why the proof of the mass gap is hard (2-05 §04, 4-11).

ComputedTable 2 Rayleigh–Ritz values on $V_n=\{x(1-x)p(x):\ p\ \text{of degree at most}\ n-1\ \text{}\}$ and the lower bound from (7) ($\mu=39$). The matrices were built exactly with fractions.

The antisymmetric functions added for even $n$ improve only $\lambda_2$, and the upper bound comes down to $42,\ 39.50155,\ 39.47847$ ($4\pi^2=39.47842$). At $n=5$, $\pi^2$ is caught in an interval of width $2\times10^{-6}$ (the lower end depends on $\mu=39\le4\pi^2$).

The Poincaré inequality is "a function with small slope does not stray far from its mean" put in numbers. On the interval $[0,1]$ it takes the following form, and the constant $1/\pi^2$ cannot be made smaller Known.

Proof. Writing $f-\bar f=\sum_{k\ge1}a_k\sqrt2\cos k\pi x$ with the Neumann eigenfunctions $1,\ \sqrt2\cos k\pi x$, the left-hand side is $\sum|a_k|^2$ and the integral on the right is $\sum k^2\pi^2|a_k|^2\ge\pi^2\sum|a_k|^2$. Equality holds only when $f=a+b\cos\pi x$. On an interval of length $L$ the constant is $(L/\pi)^2$, proportional to the square of the size (the version on a circle is (4) of 7-06).

This is the variational principle of section 06 used for "the next after 0" of the Neumann operator. In general, if for a self-adjoint $A\ge0$ only constants satisfy $Af=0$ and the gap is $\Delta$, then the best constant of $\operatorname{Var}(f)\le C_P\langle f,Af\rangle$ is $C_P=1/\Delta$, and by (5) the smoothing motion settles at rate $1/C_P$. The Dirichlet version $\int|f|^2\le\pi^{-2}\int|f'|^2$ ($f(0)=f(1)=0$) has the same number, but equality is at $\sin\pi x$, and the content is different. Example. For $f=x$ the left-hand side is $1/12$ and the right-hand side $1/\pi^2$, with ratio 0.82.

What field theory needs is the form for a probability measure $\mu(dx)=e^{-V(x)}dx/Z$ ($Z$ is the constant making the total 1). Write the inner product as $\langle f,g\rangle_\mu=\int\overline f\,g\,d\mu$, the space of functions of finite length in this inner product as $L^2(\mu)$, and the variance as $\operatorname{Var}_\mu(f)=\int\bigl|f-\textstyle\int f\,d\mu\bigr|^2d\mu$.

The best $C_P$ is the inverse of the gap of $Lf=-f''+V'f'$ after 0. $L$ is self-adjoint on $L^2(\mu)$, integration by parts gives $\langle f,Lf\rangle_\mu=\int|f'|^2d\mu$, and only constants satisfy $Lf=0$ (the same form as section 07; from the side of the ground state, 2-05 §08). The inequality on the left follows by putting in $f=x$. The one on the right is said to be a corollary of the variance inequality of Brascamp–Lieb (1976), and also follows from the logarithmic Sobolev inequality of Bakry–Émery (1985) (an inequality stronger than the Poincaré inequality) Known. In one dimension the following line suffices. Differentiating $Lf=\lambda f$, $g=f'$ gives $Lg+V''g=\lambda g$; the inner product with $g$ gives $\int|g'|^2d\mu+\int V''g^2d\mu=\lambda\int g^2d\mu$, hence $\lambda\ge\kappa$ (assuming there is an eigenfunction and the boundary terms vanish). For the normal distribution $V=x^2/(2\sigma^2)$ all three are $\sigma^2$ (Chernoff 1981).

ComputedThe numbers in the figure were computed on this machine (the best $f$ and $C_P$ were found with python3 and embedded; the weight $e^{-V_c}$ is computed on the spot).

ComputedTable 3 $C_P$ for $V_c=x^4/4-cx^2/2$ (found by turning $L$ into an operator of Schrödinger type and using three-point differences; the method is in the table of sources; halving the step agrees to 4 digits). At $c\lt0$, $\kappa=-c$.

With two valleys ($c\gt0$, height of the hill between the valleys $c^2/4$), $C_P$ grows much faster than $\operatorname{Var}(x)$, and at $c=4$ it is 9 times as large. The best $f$ becomes a step close to $-1$ in the left valley and $+1$ in the right, and its slope matters only on the hill, where there is almost no weight. A narrowing of the weight makes $C_P$ large.

When there are many variables $x=(x_1,\dots,x_n)$, replace the $|f'|^2$ of (9) by the squared length of the gradient $|\nabla f|^2=\sum_j|\partial_jf|^2$. The basics are the following two Known.

(i) The $C_P$ of a product measure of independent variables is the maximum of the $C_P$ of the factors, and does not depend on the number of variables. $L$ becomes the sum of the $L$ for each variable, the eigenfunctions are products $f_1(x_1)\cdots f_n(x_n)$ and the eigenvalues sums, so the value after 0 is the minimum of the gaps of the factors (assuming a basis of eigenfunctions).

(ii) The $C_P$ of the Gaussian measure with mean 0 and covariance matrix $C$ is the largest eigenvalue of $C$. Taking the eigenvectors of $C$ as orthogonal coordinates, the measure splits into a product of independent one-dimensional Gaussians whose variances are the eigenvalues of $C$ (1-08), and $|\nabla f|^2$ does not change under a change of orthogonal coordinates. The rest uses (i) and the one-dimensional $C_P=\sigma^2$ (section 08).

For the free field on the ring of 1-08 (10), $C=(-\Delta_{\mathrm{lat}}+m^2)^{-1}$. The eigenvalues of $-\Delta_{\mathrm{lat}}$ are at least 0, and 0 is the constant direction (the direction shifting every $x_j$ by the same amount), so the smallest eigenvalue of $C^{-1}$ is $m^2$, hence $C_P=1/m^2$. Combined with the correlation length $\xi\approx1/m$ (1-08 (11)) this gives $C_P\approx\xi^2$ (this $C_P$ is measured with the gradient in the field variables $x_j$. In lattice gauge theory, taking the angles of the links as variables changes the constant factor, so we always state which is used. 10-01).

Here we distinguish two gaps. For the free field, the inverse $m^2$ of $C_P$ is the gap of $L$ (the rate at which $e^{-tL}$ removes the deviation from the mean in (5)). On the other hand, the gap of the Hamiltonian when 1-08 (10) is read as an imaginary-time path integral is $1/\xi$ in units of the lattice spacing (1-08 (11); $\approx m$ when $m$ is small). The squared relation $C_P\approx\xi^2$ is a free-field computation; whether it holds in general in interacting theories is the question of 10-01, and has not been proved (the same form on a group is in 5-11).

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