2026-10-01 · chapter Yang–Mills and the mass gapPart 2 Physical preparationquantum mechanics and spectra

The bones of quantum mechanics — states are vectors, energies are a spectrum, and the gap is the space at the very bottom

In quantum mechanics a state is a vector $\psi$ of length 1 in a complex Hilbert space $\mathcal H$ (1-10). A sum $\alpha\psi+\beta\phi$ rescaled to length 1 is also a state (superposition), and $z\psi$, multiplied by $z$ with $|z|=1$, is the same state (the phase of 2-03). The inner product is taken to be conjugate-linear in the left argument.

On the left is a system with two choices (spin, or "the left valley or the right valley"); on the right is a particle on a line, and we read $|\psi(x)|^2$ as the probability density of position (this is said to go back to Born 1926). Measurable quantities correspond to self-adjoint operators $A$ (Hermitian matrices in finite dimensions, 1-01); the value of a measurement is one of the eigenvalues $a_k$, and its probability, using orthonormal eigenvectors $e_k$, is $|\langle e_k,\psi\rangle|^2$.

Example. Measuring $\sigma_z=\operatorname{diag}(1,-1)$ in $\psi=(1,1)/\sqrt2$ gives $\pm1$ half and half. For $\sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix}$, $\sigma_x\psi=\psi$, and $\psi$ is an eigenvector with eigenvalue $+1$, so a measurement certainly gives $+1$ ($\sigma_x,\sigma_z$ are the Pauli matrices, 1-04). In $L^2(\mathbb R)$ the position $X$ is the operator of multiplication by $x$, the momentum is $P=-i\hbar\,d/dx$, and from $XP-PX=i\hbar$ comes the uncertainty relation $\Delta X\,\Delta P\ge\hbar/2$ Known.

We call the operator of energy the Hamiltonian $H$. For a particle on a line it is $H\psi=-\frac{\hbar^2}{2m}\psi''+V\psi$. States obey the Schrödinger equation.

The right-hand end is the expansion in eigenvectors $H\phi_n=E_n\phi_n$, and each component only has its phase turning. $e^{-itH/\hbar}$ is unitary and preserves total probability (1-01; the converse is Stone's theorem Known). A state with a single eigenvector is a stationary state whose distribution does not change, and change arises only from the phase differences $(E_n-E_m)t/\hbar$.

Example (two levels). A system with the left valley $e_L$, the right valley $e_R$ and the tunnelling amplitude $g\gt0$.

Starting from $e_L$, the probability $P_R$ of being on the right goes back and forth with angular frequency $2g/\hbar=(E_+-E_-)/\hbar$. The gap between levels sets the speed of the back-and-forth (the $x=0$ of Figure 2 of 1-01). Section 07 gives an example where $g$ is exponentially small.

In infinite dimensions, values without eigenvectors also appear, so we use the spectrum $\sigma(H)$ (the set of $\lambda$ for which $H-\lambda$ has no bounded inverse, 1-10). For a self-adjoint operator, $\lambda\in\sigma(H)$ is the same as "there is a sequence with $\lVert\psi_k\rVert=1$, $\lVert(H-\lambda)\psi_k\rVert\to0$" (Weyl's criterion) Known.

The spectrum of a box with walls where $\psi=0$ consists only of eigenvalues (discrete spectrum). On a line the plane wave $e^{ipx/\hbar}$ is not a state, since $\int|\psi|^2=\infty$. Example. For a wave packet $\ell^{-1/2}\chi(x/\ell)e^{ipx/\hbar}$ of width $\ell$ ($\chi$ a smooth bump), $\lVert(H-p^2/2m)\psi\rVert$ approaches 0 like $1/\ell$, so every $E\ge0$ is in the spectrum without being an eigenvalue (continuous spectrum). Widening the box makes the spacing on the left tighter, approaching the right (section 06).

Detour: a well whose valleys go on for ever

We call $E_0=\inf\sigma(H)$ the ground-state energy, and, if it is an eigenvalue, its eigenvector $\phi_0$ the ground state. The variational principle (1-10) says the following.

Every trial function gives an upper bound, but a lower bound requires control of every $\psi$ (this asymmetry appears throughout the book, so it is summarised in the box in 1-10 §06). Three remarks. (i) The bottom need not be an eigenvalue (the free particle on a line). (ii) It does not fall to the bottom of the potential. For $V=\tfrac12m\omega^2x^2$, $\langle X^2\rangle\langle P^2\rangle\ge\hbar^2/4$ and the inequality of arithmetic and geometric means give $\langle H\rangle\ge\hbar\omega/2$ (2-06). (iii) In one dimension, if $V$ grows far away, the ground state is non-degenerate and can be taken positive, and the $n$-th eigenfunction has $n$ zeros (Sturm's oscillation theorem) Known.

Example. Putting a Gaussian $e^{-x^2/4s^2}$ into $\hbar=m=1$, $H=-\tfrac12d^2/dx^2+x^4$ gives $\langle H\rangle_s=1/8s^2+3s^4$, and its minimum $\tfrac3{16}48^{1/3}=0.68142$ is an upper bound for $E_0=0.667986$. The odd trial function $xe^{-x^2/4s^2}$ is orthogonal to the even ground state, so it gives an upper bound for $E_1$.

ComputedTable 1 The best upper bounds from Gaussian trial functions, and the eigenvalues of three-point differences.

The difference between two upper bounds is neither an upper nor a lower bound on the gap (depending on the trial functions it can be off in either direction). To bound it from below, we need estimates for every state orthogonal to the ground state. Showing the mass gap in quantum field theory is likewise, by definition, bounding from below the energy of the states orthogonal to the vacuum (4-11).

When the ground state is non-degenerate and $E_0$ is an isolated point of the spectrum, we call $\Delta=\inf(\sigma(H)\setminus\{E_0\})-E_0\gt0$ the gap. The exposition of the problem sets the energy of the vacuum to 0 and defines the mass gap as "the spectrum of $H$ has no points in $(0,\Delta)$" (Jaffe–Witten §4) Known. There are three readings. (i) Price: precisely, for states orthogonal to $\phi_0$, $\langle H\rangle\ge E_0+\Delta$. (ii) Oscillation: a superposition of $\phi_0$ and $\phi_1$ oscillates with angular frequency $\Delta/\hbar$ ((4)). (iii) Decay: formally putting $t=-i\tau$ into (3) (2-04), oscillation turns into decay.

For $c_0\ne0$ it approaches the ground state, and the rest becomes at most $e^{-\Delta\tau/\hbar}$ times as large. Write the left side of (7) as $\psi(\tau)$, and the weight of the ground state as $w_0(\tau)=|\langle\phi_0,\psi(\tau)\rangle|^2/\lVert\psi(\tau)\rVert^2$. $1-w_0$ is the probability of being outside the ground state, and Figure 1 and Table 2 follow it. For the imaginary-time correlation of a self-adjoint $A$ ($\langle\phi_0,A\phi_0\rangle=0$) it takes the following form.

From (8), the rough time for the correlation to vanish in the imaginary-time direction is at most $\hbar/\Delta$. This is a quantity with the dimension of time; in relativity it is multiplied by $c$ to give a length $\hbar c/\Delta$. Decay in the spatial direction (the cluster property) and the correlation length (2-08) are treated in 4-11, and the view as a sum over paths in 2-07.

ComputedThe numbers in the figure were computed on this machine (box $[0,1]$, $H=-d^2/dx^2$ ($\hbar=1$, $\hbar^2/2m=1$), $\Delta=3\pi^2$; the coefficients were found with python3, and the partial sums of (7) are computed on the spot).

ComputedTable 2 The weight outside the ground state $1-w_0$, and the prediction keeping only the next level $(c_1/c_0)^2e^{-2\Delta\tau}$.

The weight is the square of an amplitude, so the exponent is $2\Delta$. For a state symmetric about the centre, $c_1=0$, and the decrease is set by $E_2-E_0=8\pi^2$. What the gap guarantees is the slowest decrease, valid for every state.

The mass gap is asked about in infinitely wide space. A gap that is positive in a finite box can close when the box is widened. The exposition of the problem also says that a uniform gap, independent of the volume, could play a role in proving the infinite-volume limit, and that the proof would need new ideas (Jaffe–Witten §5, §6.5) Known. With $\hbar=m=1$ and box $[-L/2,L/2]$, we compare three $V$.

ComputedTable 3 The length of the box $L$ and the gap $E_1-E_0$ (three-point differences; $V=0$ agrees with the exact value $3\pi^2/2L^2$).

The gap of the free particle closes like $1/L^2$. $-\operatorname{sech}^2x$ has exactly one bound state $\operatorname{sech}x/\sqrt2$ ($E_0=-1/2$, which can be checked by substitution), and the next level drops to the bottom 0 of the continuous spectrum, so the gap tends to $1/2$ Known. Above the gap there may be continuous spectrum. Even in a field theory with a mass gap, above the vacuum is the continuous spectrum of particles, starting from $\Delta$ (2-04).

Example. For $N$ independent oscillators, the ground energy $N\hbar\omega/2$ grows with $N$ and diverges as $N\to\infty$. Even so, the gap stays $\hbar\omega$ (2-06). In quantum field theory the energy of the vacuum is set to 0 and the difference is what is asked about. Jaffe–Witten §4 states $H\Omega=0$ from the fact that the vacuum vector $\Omega$ is invariant under the Poincaré group (2-04).

Put two identical wells side by side and make the hill between them higher ($\hbar=m=1$).

With a single valley the lowest level is about $1/2$, and with two valleys it is doubly degenerate. Tunnelling between the valleys plays the role of the $g$ of (4), the levels split into an even $\phi_0$ and an odd $\phi_1$, and when the hill is high the splitting is as follows Known.

This is equation (2.8) of Zinn-Justin–Jentschura (2004), $\frac{2}{\sqrt{\pi g}}e^{-1/6g}$, with $g=1/4a^2$ (shifting the potential of that paper by $a$ gives (9); the height of the hill is $a^2/8=1/32g$). Perturbation theory splits $V_a$ around one of the valleys into an oscillator term and a remainder that is small when $g$ is small, expands the levels in powers of $g$, and finds the coefficients of each order in turn (4-03). That perturbation series does not depend on parity (evenness or oddness under $x\to-x$) and is the same for the two levels (the explanation of (2.13)–(2.14) in that paper) Known. The $e^{-1/6g}$ of the gap, expanded as a power series in $g$, has all coefficients 0. However high the order to which perturbation theory is computed, this gap does not appear. $S_0$ is the action of the classical solution (instanton) that rolls from hilltop to hilltop of $-V_a$ in imaginary time. Its counterpart in Yang–Mills is in 3-05, and quantities of the form $e^{-1/g^2}$ appear in 4-07 and 4-12.

ComputedThe numbers in the figure were computed on this machine (levels and wave functions by three-point differences in python3; the shading is $\tfrac12(\phi_0^2+\phi_1^2-2\phi_0\phi_1\cos\Delta t)$).

ComputedTable 4 Comparison of the gap of (9) (three-point differences) with (10). The rightmost column is the correction factor $1-\frac{71}{12}g-\frac{6299}{288}g^2$ from the coefficient (equation (2.19a)) of the expansion (2.13) that the same paper states as a conjecture Physics.

The ratio approaches 1, and also the correction factor (difference $2\times10^{-4}$ at $a=5$). The gap at $a=5$ is 6 orders of magnitude smaller than the level spacing (about 1), and the time $\pi/\Delta$ to move to the right is about $5\times10^6$.

In one dimension when $\phi_0\gt0$ (section 04), writing $\psi=f\phi_0$ with a real-valued $f$ and integrating by parts, the $V-E_0$ term vanishes by $H\phi_0=E_0\phi_0$.

$\psi\perp\phi_0$ means $\int f\,d\mu=0$, and then $\lVert\psi\rVert^2=\operatorname{Var}_\mu(f)$. Applying the variational principle to $\psi$ orthogonal to $\phi_0$, the following holds with the best constant.

The gap is inversely proportional to the Poincaré constant of the ground-state distribution $\mu$ Known (the Poincaré inequality is in 1-10 §07, the weighted form in §08). Here $\mu$ is the distribution of the ground state at a single time. Matching with $\psi=f\phi_0$, $H-E_0$ corresponds to $\tfrac{\hbar^2}{2m}L$ ($L$ is the operator of 1-10 §08 for this $\mu$), and the gap of $L$ is $2m\Delta/\hbar^2$. The $C_P\approx\xi^2$ of 1-10 §09 is a statement about the measure on the whole of the paths in imaginary time (1-08 (10)), and is a different quantity. Example. For the oscillator ($\hbar=m=\omega=1$), $\phi_0=\pi^{-1/4}e^{-x^2/2}$ (which can be checked by substitution), so $\mu$ is the normal distribution with variance $1/2$, and the gap 1 (the right column of Table 3, 2-06) gives $C_P=1/2$. "The Poincaré constant of a normal distribution equals its variance" has the same content as the gap $\hbar\omega$ of the oscillator (with equality at $f=x$). For the double well $a=3$, $C_P=1/2\Delta=36.9$. Taking $f$ close to "$-1$ on the left, $+1$ on the right", the variance is about 1, but the slope acts only on the hill between, where $\mu$ is almost absent. A waist in the distribution makes the gap small. The same form is also used in one-link integration (5-11, this site's papers).

What this chapter can say

Sources and reproduction

Next chapter: 2-06 The harmonic oscillator and creation and annihilation. Imaginary time continues in 2-07, and the main treatment of the definition of the gap is in 4-11.

← 2-04 Special relativity and spacetimeContents2-06 The harmonic oscillator and creation and annihilation →