2026-10-01 · chapter Yang–Mills and the mass gapPart 4 Quantisationthe definition of the mass gap

The definition of the mass gap — the lowest energy above the vacuum, and the uniform decay of correlations

The definition in §4 of the problem statement is 'A quantum field theory has a mass gap ∆ if H has no spectrum in the interval (0, ∆) for some ∆ > 0. The supremum of such ∆ is the mass m, and we require m < ∞.' (for a word-by-word reading, 0-03 §06). This sentence presupposes (a) a Hilbert space $\mathcal H$ and a self-adjoint $H\ge0$ (what the reconstruction of 4-10 §05 gives) and (b) a vacuum $\Omega$ unique up to a phase, with $H\Omega=0$ ((E4) in Table 1 of 4-10), and (c) asks about states orthogonal to the vacuum. In the words of §1, 'every excitation of the vacuum has energy at least ∆'. Under (b), the definition is the same as the following Known.

The second equality is the variational principle (1-10 §06). That (b) is needed can also be read from (1). If there were a state $\Omega'$ of energy 0 orthogonal to the vacuum, (1) would be 0 with $\psi=\Omega'$, but the first sentence of the problem statement alone would miss it, since 0 lies outside $(0,\Delta)$. Example. For the free field of mass $m$, (1) is also $m$ (we use the same letter), and for mass 0 it is 0 (4-10 §07).

In the notation of 4-10 §05, a bounded quantity $F$ at positive times gives a vector $[F]$, and the shift $U_\tau$ by $\tau$ in the time direction becomes $e^{-\tau H}$. Putting $\psi_F=[F]-\mathbb E[F]\,\Omega$, for the connected correlation (the correlation with the vacuum component subtracted),

The inequality follows from the spectral theorem (1-10 §05) Known. The converse also holds. If $\langle\psi_F,e^{-\tau H}\psi_F\rangle\le e^{-\Delta\tau}\|\psi_F\|^2$ for every bounded $F$, then $m\ge\Delta$.

To see this we use the spectral measure $\mu_F$ of $\psi_F$: the distribution of weight along the energy axis $E$ when $\psi_F$ is split into eigencomponents of $H$. If there is an eigenbasis $e_k$ ($He_k=\lambda_ke_k$), then $\mu_F=\sum_k|\langle e_k,\psi_F\rangle|^2\,\delta_{\lambda_k}$ ($\delta_\lambda$ is a weight 1 placed at the point $\lambda$), and even for continuous spectrum a distribution with the same meaning is fixed by the spectral theorem. Then $\langle\psi_F,e^{-\tau H}\psi_F\rangle=\int e^{-E\tau}d\mu_F(E)$. The left-hand side satisfies $\int e^{-E\tau}d\mu_F\ge e^{-\Delta'\tau}\mu_F([0,\Delta'))$ ($\Delta'\lt\Delta$), so as $\tau\to\infty$, $\mu_F([0,\Delta'))=0$. The set of all $\psi_F$ is dense in $\Omega^\perp$ (every vector of $\Omega^\perp$ can be approached arbitrarily closely), so $H\ge\Delta$ on $\Omega^\perp$. Since this includes $E=0$, the uniqueness of the vacuum comes out as well Known. The gap is equivalent to all connected correlations decaying at least at one rate that does not depend on the observable.

A single correlation $C_F(\tau)=\int e^{-E\tau}d\mu_F(E)$ is log-convex, that is, $\log C_F$ is convex ($C(\tau_1+\tau_2)^2\le C(2\tau_1)\,C(2\tau_2)$ in (5) of 4-10 corresponds to this). So the effective mass decreases monotonically.

$\operatorname{supp}\mu_F$ (the support) is the smallest closed set carrying the weight of $\mu_F$, and $E_F$ is the lowest energy that $\psi_F$ touches. $m_{\mathrm{eff}}$ at any time is an upper bound on $E_F$: from above it is bounded by a single $F$, but from below every $F$ is needed (the same shape as the box "The asymmetry of upper and lower bounds" in 1-10). On the lattice $U_a$ is the transfer matrix $T=e^{-aH}$, $m_{\mathrm{lat}}=\log(\lambda_0/\lambda_1)$ and $\Delta=m_{\mathrm{lat}}/a$. $\lambda_0$ and $\lambda_1$ are the largest and second-largest eigenvalues of $T$ (in gauge theory, on gauge-invariant states; section 08) ((1) of 2-09 is its smallest example).

We see that $E_F$ changes with the observable in the four-dimensional free field ($m=1$). The correlation of $O_k$, the Wick products of 4-10 §07 integrated over space to give momentum 0, is

($S$ is the two-point function of the four-dimensional free field, $K_1$ a modified Bessel function). $O_k$ creates exactly $k$ particles with total momentum 0, so $E_{O_k}=km$. At large $\tau$ the energy of each particle can be approximated as $m+|\vec p|^2/(2m)$, and the correlation of $k$ particles with total momentum 0 is $e^{-km\tau}$ multiplied by $\tau^{-3(k-1)/2}$ from the Gaussian integral over the three-dimensional momenta of the remaining $k-1$. So near the lower end $m_{\mathrm{eff}}\approx km+\frac{3(k-1)}{2\tau}$ Computed. Only $k=1$ sees the gap 1; however precisely ${:}\phi^2{:}$ is measured, it only approaches 2. This is because the number of particles is conserved for the free field; even with interactions, symmetries split the space into parts (for Yang–Mills, the channels of section 09). For the unprojected point-to-point $S(\tau,\vec0)$, even with the same $\phi$, it decreases slowly, like $1+\frac{3}{2\tau}$. The energies of moving particles form a continuum, and the integral over one three-dimensional momentum multiplies by $\tau^{-3/2}$ for the same reason; this is why lattice measurements sum over space (5-08).

ComputedThe numbers in this figure were computed on this machine (a table of (4) at steps of 0.5 in τ is embedded).

ComputedTable 1 $m_{\mathrm{eff}}(\tau)$ of the free field ($m=1$). In brackets, the lower end $E_F$. The momentum-0 $\phi$ is 1.00000 at every $\tau$, agreeing with the closed form $e^{-\tau}/2$ within double-precision rounding.

Every column decreases monotonically (checked at all points at steps of 0.25 for $\tau\le12$), agrees with the asymptotic form to within 0.13% at $\tau=16$, and stays above the lower end to the last. What can be read is an upper bound on $E_F$ Computed.

We call the range of $\tau$ that a measurement can see a window (precisely, a finite interval $[\tau_{\min},\tau_{\max}]$). The infimum (3) involves infinitely many $F$ and $\tau\to\infty$, so a measurement within a window sees only part of it. We look at a theory with no gap, made by placing side by side, independently, a free field $\phi$ of mass 1 and a free field $\chi$ of mass 0. The cross terms vanish, and the correlation of $O=\int d^3x\,[\phi+\varepsilon\,{:}\chi^2{:}]$ is

(the coefficient of the second term checked to 12 digits by numerical integration Computed). Up to the $\tau^*$ at which the two terms are equal, $m_{\mathrm{eff}}\approx1$, and it looks like "a theory with gap 1".

ComputedThe numbers in this figure were computed on this machine ((5) computed on the spot; τ* by bisection).

ComputedTable 2 $\tau^*$ and $m_{\mathrm{eff}}$ of (5).

Setting the two terms equal gives $\tau^*=\ln(1/\varepsilon^2)+\ln(4\pi^2\tau^*)$, so each time $\varepsilon$ becomes an order of magnitude smaller, $\tau^*$ moves out by about 5, $2\ln10\approx4.6$ plus a logarithmic correction, and for any finite window there is an $\varepsilon$ that looks like "gap 1" (far enough out, $m_{\mathrm{eff}}\approx1/\tau$). Measurements in a finite window give no lower bound on the gap Known. For how to read lattice numbers, 9-05 §07.

The gap of the mass operator $M=\sqrt{H^2-\vec P^{\,2}}$ had the same width as the gap of $H$ (0-03 §06). A one-particle state of mass $\mu$ is a state belonging to the eigenvalue $\mu$ of $M$ (Wigner's 1939 idea of seeing particles as irreducible representations of the Poincaré group), and it is isolated if there are no other points of $\sigma(M)$ near $\mu$. §5 of the problem statement places this outside the requirements, writing 'one would like to prove the existence of an isolated one-particle state (an upper gap, in addition to the mass gap)' Known.

The difference shows in the two-point function. The Euclidean two-point function of a scalar field $\phi$ ($\langle\Omega,\phi\,\Omega\rangle=0$) can be written with a positive measure $\rho$ as

which is called the Källén–Lehmann representation Known (the right-hand side is (3) of 4-10 superposed over $s$). If there is a gap $m$, the support of $\rho$ lies in $[m^2,\infty)$, and if $\phi$ touches a particle of mass $\mu$, $\rho$ has a point weight at $s=\mu^2$.

Example 1: a gap but no particle. Take $\rho$ to be the uniform weight on $[m^2,4m^2]$, and take the Gaussian measure with covariance $\int d\rho(s)(-\Delta+s)^{-1}$. Reflection positivity is a condition preserved by positive superpositions of covariances, so if it holds for each $s$ (4-10 §07) it also holds after integrating with the positive measure $\rho$. On reconstruction, the "one-particle" $M$ fills $[m,2m]$ continuously, and $\sigma(H)=\sigma(M)=\{0\}\cup[m,\infty)$. The gap is $m$, and $M$ has no eigenvalue other than 0. Fields of this kind are called generalised free fields (Greenberg 1961) Known. Example 2: a particle but no gap. In the theory of section 04, the one-particle state of $\phi$ has $M=1$, but two-particle states of $\chi$ fill $[0,\infty)$, and the eigenvalue 1 is buried in the continuous spectrum.

What physics expects for Yang–Mills is an isolated glueball at the lower end of the gap Physics. For the difficulty of telling an isolated point from the edge of a continuum using correlations, 9-05 §05.

Clustering without regard to rate (two quantities moved apart approach independence) is (E4), paired with the uniqueness of the vacuum (Table 1 of 4-10) Known. The massless free field also satisfies it, but its correlation decreases only as a power, $1/(4\pi^2|x|^2)$. Formula (2) of §5 of the problem statement says, for a local field $\mathcal O(\vec x)=e^{-i\vec P\cdot\vec x}\mathcal Oe^{i\vec P\cdot\vec x}$ with $\langle\Omega,\mathcal O\Omega\rangle=0$,

The outline on the Euclidean side is (2) and rotation invariance (E1). If two quantities localised within radius $\delta$ are a distance $R$ apart, rotating the direction joining them onto the $\tau$ axis, (2) gives a connected correlation at most (a constant independent of $R$) $\times e^{-m(R-2\delta)}$. At equal times and spacelike-separated points, the Wightman function (the vacuum expectation value of a product of fields; 4-09 §04) equals the value of the Schwinger function at time 0 (the analytic continuation of 4-10 §05), so this decay of the Euclidean correlation becomes (7) directly. The point field $\mathcal O(\vec x)$ is not a bounded quantity, so precisely one considers the field slightly smeared over radius $\delta$. The "$C\lt\Delta$" and "large enough" in (7) account for the constant in front and the width $2\delta$. The theorem on the Minkowski side is introduced in the introduction of Nachtergaele–Sims 2006: 'in [1] Fredenhagen proves a general theorem applicable to arbitrary strictly local theories' ([1] is Fredenhagen 1985) Known.

The converse does not hold in general; the same introduction writes that there are quantum lattice systems with a unique ground state and exponentially decaying correlations but no gap, and gives Example 2 of Nachtergaele 1996 as an example Known. In a rotation-invariant Euclidean theory, time is one of the directions. If "it decays at a uniform rate in every direction", it decays at the same rate in the time direction too, so the converse argument of section 02 (where the constant in front may depend on $F$) applies directly, and this is equivalent to the gap. A state mixing two phases (2-09 §08) breaks clustering itself, and assumption (b) of (1) fails.

(6) of 2-08 defined a correlation length from a single correlation. The correlation length of the theory is the supremum over observables.

Both are restatements of (3) Known (restoring units, $\xi=\hbar c/\Delta$; on the lattice $\xi_{\mathrm{phys}}=a\,\xi_{\mathrm{lat}}=1/\Delta$). Problem 5.1 of Chatterjee 2018 defines the lattice correlation length by plaquette correlations, and writes 'The number ξ represents the reciprocal of the mass of the lightest glueball' Known.

$\xi_{\mathrm{2nd}}^2=\sum_x|x|^2G(x)\big/\bigl(2d\sum_xG(x)\bigr)$ from second moments is also often used, and on the lattice the rate of decay also depends on the direction (4-02). What equals the inverse of the gap is the length along the direction of the transfer matrix (an axis). For the free field on the four-dimensional lattice (covariance $(-\Delta_{\mathrm{lat}}+m_0^2)^{-1}$) there are closed forms.

The $e$ in the third is a vector of length 1: on an axis $e=(1,0,0,0)$, on the diagonal $e=(1,1,1,1)/2$.

ComputedTable 3 The free field on the four-dimensional lattice. The two columns on the right are ratios to $\xi_{\text{axis}}$ (diagonal means the direction $e=(1,1,1,1)/2$). The third formula of (9) was checked by computing the two-point function on a two-dimensional periodic lattice with the fast Fourier transform, agreeing to within 0.11% on the axis and the diagonal.

When the correlation length is of the order of the lattice spacing ($m_0=1,2$), the three lengths disagree by 3 to 12%; at $\xi_{\text{axis}}=20$ the differences are of order $10^{-4}$, and rotational symmetry returns (the continuum limit; 2-08 §08). Outside free fields there is no guarantee that the lengths agree, and what is used for the gap is $\xi_{\text{axis}}$.

On a finite lattice, a positive gap is almost trivial. In the temporal gauge, the transfer matrix $T$ (5-03 §05) is an integral operator with a positive continuous kernel acting on the space $L^2(G^n)$ of values of the links at one time ($G$ a compact group, $n$ the number of links). A continuous kernel on a compact space makes $T$ a compact operator (precisely, a norm limit of finite-dimensional matrices), and its eigenvalues form a discrete sequence accumulating only at 0. Further, since the kernel is positive, Jentzsch's theorem (the infinite-dimensional version of the Perron–Frobenius theorem) makes the largest eigenvalue $\lambda_0$ simple, and $\lambda_1\lt\lambda_0$ Known. If the group is finite, $T$ is a finite matrix with positive entries, and it is literally the Perron–Frobenius theorem. Simplicity of the largest eigenvalue alone leaves open that the other eigenvalues crowd up to $\lambda_0$, so discreteness is needed. What is asked is whether $m_{\mathrm{lat}}(a,L)/a$ stays positive in the limits of volume $L\to\infty$ and lattice spacing $a\to0$. §6.5 of the problem statement writes 'New ideas are needed to prove the existence of a mass gap that is uniform in the volume of space-time.', and §5 also says that a uniform gap can play a role in the proof of the infinite-volume limit Known. (2) is an inequality between correlations (the $\|\psi_F\|^2=\mathbb E[\overline{\Theta F}F]-|\mathbb E[F]|^2$ on the right-hand side is also a correlation). So if it holds along a sequence of approximations with the same $\Delta$ and the correlations converge, it holds in the limit too. Using the converse of section 02 in the limiting theory gives $m\ge\Delta$ (that the set of all $\psi_F$ is dense in $\Omega^\perp$ comes from the way the reconstruction is built; 6-01).

The ways a finite-box gap closes as $L\to\infty$ are massless particles ($2\pi/L$ in a periodic box; 2-05 §06) and symmetry breaking (the tunnelling splitting between phases; 2-09 §08). In the latter each phase in infinite volume has a gap, and §6.3 of the problem statement writes that each phase of two-dimensional $\phi^4$ has 'a non-zero mass gap and a unique, Poincaré-invariant vacuum state' Known. As $a\to0$, if $\Delta$ is held fixed then $m_{\mathrm{lat}}=a\Delta\to0$, and a gap in lattice units that is uniformly positive means $\Delta=\infty$ in the continuum limit (5-09).

For Yang–Mills, the $\mathcal H$ of (1) is the space of gauge-invariant states (4-01), and the observables of (2) and (3) are local quantities built from the field strength such as $\mathrm{Tr}\,F_{ij}F_{kl}(x)$ (problem statement §4) and Wilson loops (4-10 §09). States split by the combination $J^{PC}$ of spin, parity and charge conjugation (channels) ($J$ is the spin, and $P$ and $C$ are the signs $\pm$ under spatial reflection and under the operation that replaces the field by its complex conjugate [the antiparticle]), and, as with $k$ in section 03, the minimum over channels of the lowest energy in each is $m$. In lattice numbers $0^{++}$ is the lowest (about 0.7 times $2^{++}$ for pure $SU(3)$; Chen et al. 2006; 5-08), and physics reads it as the lower end of the gap Physics.

Correlations of gauge-fixed fields are in general not guaranteed to be reflection positive (4-10 §04, §09), and for the gluon propagator it is read as violated Physics. The decay of a correlation that is not reflection positive does not give points of $\sigma(H)$. The string tension is fixed by the energy of a pair of static charges, and is a different quantity from excitations above the vacuum (9-05 §06). In mathematics there are long-standing results on the gap in lattice units at strong lattice coupling (small $\beta$) (5-04). Shen–Zhu–Zhu 2023, in 't Hooft's convention (writing the coupling of the action in the form $N\beta$; for conversion to other conventions, Table 4 of 5-01), showed that for $SU(N)$ with $|\beta|\lt1/(16(d-1))$ the correlations of a certain family of observables decay exponentially. The abstract calls this 'strictly positive mass gap', and also writes 'Our method improves earlier results' Known. To use it in the equivalence of section 02, that family must be dense. This is a result short of the limits of section 08, and for the four-dimensional continuum theory the $\mathcal H$ and $H$ of (1) have not yet been constructed (6-10).

More than ten quantities called "gap" appear in this book. Some are equivalent to the $\Delta$ of the problem statement, some are only upper bounds on $\Delta$, and some are different (the gap of another operator, or an additional requirement); confusing them changes the direction of a claim. The table below lists, one row each, the definition, the main chapters and the relation to $\Delta$ (the assignment of relations is this book's organisation; for the direction of upper and lower bounds, the box "The asymmetry of upper and lower bounds" in 1-10).

Appendix table The word "gap". "Equivalent" in the relation column means the same number under the stated assumptions.

11 and 12 are both Poincaré constants, but 11 concerns the measure on whole paths in imaginary time and 12 the measure at a single time, and only 12 is equivalent (2-05 §08). 5 and 9 point in opposite directions: 5 is an upper bound obtained from a single observable, 9 a lower bound over all observables and volumes.

What this chapter can say

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