2026-10-09 · chapter Yang–Mills and the mass gapPart 11 Further considerationsthe vacuum measure, the ground-state transformation, Kogut–Susskind · Kiichi, Shiori and Rin

Making time continuous — the Hamiltonian lattice and the vacuum measure

The setting is the same as for Proposition B of Part 10. Space is a finite lattice, the space of configurations at one time is $Q$ ($\mathbb R^n$, or a compact connected Riemannian manifold), the metric of the kinetic term is $M$, the Hamiltonian is $H=-\frac{\hbar^2}2\Delta_M+V$, and we place assumption (A1) (the spectrum is discrete, the ground state $\psi_0$ is non-degenerate, smooth and positive, $\Delta=E_1-E_0\gt0$). We set the vacuum distribution at one time, and the Dirichlet form and Poincaré constant on it, as

($F$ is a smooth, non-constant function on $Q$). In gauge theory we restrict $F$ to gauge-invariant functions, and $\Delta$ is the gap on the gauge-invariant subspace.

Proved here Proof. The map $U:F\mapsto F\psi_0$ is an isometry from $L^2(\nu)$ to $L^2(d\mathrm{vol})$, and takes the constant 1 to $\psi_0$. Hence $\langle F\rangle_\nu=0\iff F\psi_0\perp\psi_0$, and then $\|F\psi_0\|^2=\operatorname{Var}_\nu(F)$. For smooth $F$, the identity of the ground-state transformation ((18) of 10-01)

holds. Since $H-E_0$ is at least $\Delta$ on the orthogonal complement of $\psi_0$ (min-max), for smooth $F$ with $\langle F\rangle_\nu=0$ we have

Adding a constant changes neither side, so (4) holds for every smooth $F$. Equality is attained at $F=\psi_1/\psi_0$ (smooth by (A1), with $F\psi_0=\psi_1$, and the left-hand side of (3) is $\Delta\|\psi_1\|^2$). For $Q=\mathbb R^n$, the finiteness of the Dirichlet integral of $\psi_1/\psi_0$ is checked by the truncation lemma (Lemma 3.3) of the paper in preparation. ∎

In other words, the generator of Langevin dynamics on the vacuum distribution is unitarily equivalent, by the ground-state transformation, to $-(2/\hbar^2)\,\psi_0^{-1}(H-E_0)\,\psi_0$. Restatements of this type are well known (the restatements of stochastic quantisation and of supersymmetric quantum mechanics). Proposition V writes the gap part of it in this book's convention (1).

Calibration (one harmonic oscillator). For $Q=\mathbb R$, $M=1$, $V=\tfrac12\omega^2x^2$, $\hbar=1$, the gap is $\Delta=\omega$, so (2) gives $C_P(\nu)=1/(2\omega)$. $\psi_0^2$ is the normal distribution with variance $1/(2\omega)$, and this agrees with the fact that the Poincaré constant of a one-dimensional normal distribution equals its variance (with equality for linear functions). The row (α) of Table 3 of 10-01 ($1/(2m)$ for the free field) is this value Proved here.

Example 1 (checked by hand on two sites). On the two-site lattice of Example 6 of 10-01 ($\hbar=M=1$, $V=\tfrac12x^{\sf T}Kx$, the normal mode $u=(x_1+x_2)/\sqrt2$ with angular frequency $m_0$, $v=(x_1-x_2)/\sqrt2$ with angular frequency $\sqrt{m_0^2+2}$), $\nu$ is the independent normal distribution with variance $1/(2m_0)$ for $u$ and variance $1/(2\sqrt{m_0^2+2})$ for $v$. The Poincaré constant is the larger of the two variances, $1/(2m_0)$, and (2) gives $\Delta=m_0$, which agrees with the gap of Example 6 of 10-01 Proved here. If $m_0=\tfrac12$, then $C_P(\nu)=1$, the path measure has $C_P=\xi^2=4$, and $4=4\,C_P(\nu)^2$ ((5) in the next section).

Three quantities are called by the same symbol $C_P$. With Proposition V, the relations among the three line up in one table.

The equality in the second row

makes exact, on the family local in time and with interaction, the statement of 10-01 §04 that "for the free field $C_P^{(\beta)}=4(C_P^{(\alpha)})^2$" Proved here. In 10-01 this rereading was written as an analogy from the form of the free field. Under the assumptions of Proposition B, restricted to the family local in time, it is an equality, not an analogy. The amount by which the path $C_P$ stated by Assumption H exceeds $(2C_P(\nu)/\hbar)^2$ comes from functions that are not local in time (winding numbers and the topological charge).

Restated in physical units. Setting $\hbar=1$ in lattice units and converting to physical units by the bookkeeping length $\xi_{\rm RG}$, (2) becomes

Proved here. The mass gap (that the gap in physical units of the lattice stays positive) is the same as the Poincaré sentence "the Poincaré constant of the vacuum measure is bounded by the first power of the bookkeeping length". It is a Poincaré sentence that does not pass through Assumption H, and the growth is the first power, not the square (in terms of the dynamical exponent, the type $z=1$). But this is a restatement, and there is no new information on either side. An upper bound on $C_P(\nu)$ needs every $F$, and the difficulty is exactly that of a lower bound on the gap.

The topological problem does not arise for the vacuum measure. On $\nu$ we vary only single-valued functions at one time. For a free particle on the circle, $\nu$ is the uniform distribution on the circle and $C_P(\nu)=1$ is finite (Wirtinger's inequality for the circle of circumference $2\pi$), and (2) gives $\Delta=\hbar^2/2$ of Proposition C of 10-01. And the winding-number functional that made $C_P=\infty$ for the path measure (the angle of the lift, integrated in time) is not a function on $\nu$ Proved here. The four-dimensional topological charge is also a quantity fixed by an integral in time, so it is not among the functions on the vacuum distribution at one time Estimate. The boundary is the same as the inside and outside of $\mathcal F_1$ in 11-04.

For the 0+1-dimensional double well of 11-04 §07, the local Poincaré constant of the single-time window of the chain with time discretised at width $\varepsilon$ was exactly $C_P^{\rm loc}(1)=\varepsilon/(2m)$ when $\psi_0$ is the Schrödinger ground state ((17) of 11-04). The Dirichlet form of the window $(\hbar/\varepsilon)\int\psi_0^2|F'|^2$ is $\hbar/\varepsilon$ times $\mathcal E_\nu$ of (1), so this is exactly the value $C_P(\nu)=\hbar^2/(2\Delta)$ of Proposition V multiplied by $\varepsilon/\hbar$ Proved here. The ratio $\omega_b/m$ for the shortest window is this identity for the target (the double well) divided by that for the reference (the harmonic oscillator).

When time is discretised, the single-time distribution of the chain is the square of the top eigenvector of the transfer kernel (the symmetric Trotter product), and differs slightly from the Schrödinger $\psi_0^2$. On lattices with $V=(x^2-1)^2/8$, $\varepsilon=\tfrac12$, $\hbar=0.3,\ 0.15,\ 0.08$, the ratio of the direct value of the single-time window to $\varepsilon/(2m_{\rm tr})$ ($m_{\rm tr}$ from the second eigenvalue of the transfer) is $1.006,\ 1.013,\ 1.019$, and the ratio with the reference discretised too, $K_{\rm rel}(1)\cdot m_{\rm tr}$, is $1.037,\ 1.044,\ 1.050$ Reading. The identity is a statement in continuous time, and on a lattice with discretised time it is an approximation. This difference is the smallest form of the reason why assumption (D) of Part 10 ((20) of 10-01) remains.

The Hamiltonian of gauge theory with time continuous and only space made a lattice is the Kogut–Susskind lattice. The configuration space is the product of copies of the group, one per spatial link, $Q=G^{\{\text{spatial links}\}}$ (a compact connected manifold), and setting the metric of the kinetic term to $M=g^{-2}\cdot$ (the bi-invariant metric) to match the form of Proposition B, we have

$E_\ell^2$ is the Laplace–Beltrami operator on the group of the link $\ell$ with its sign changed (the square of the electric field), $U_p$ is the product around a plaquette, and the second term is the magnetic term. The conventions for the coefficients differ in the literature; here they are matched to the form of Proposition B. Physical states are gauge-invariant functions (the Gauss law), and $H_{\rm KS}$ commutes with gauge transformations, so $\psi_0$ is automatically gauge invariant. On this lattice, the quantity in the problem statement is exactly the gap of $H_{\rm KS}$ on the gauge-invariant subspace, and by Proposition V it can be written as the Poincaré constant of the vacuum distribution at one time $\nu=\psi_0^2\cdot$Haar.

The road that treats the gap directly on this lattice is one of the rows of the "table of other roads" in the table of materials of 11-00. Table 2 sets out how far it has reached and where it stopped.

Proved here Proof. $h_p$ is a multiplication operator on $L^2(Q,\text{Haar})$, and the multiplying function $2-\operatorname{tr}U_p$ is a non-constant real-analytic function on $Q$. If $(h_p-c_p)\psi=0$, then the support of $\psi$ lies in the level set $\{2-\operatorname{tr}U_p=c_pg^2\}$, but a level set of a non-constant real-analytic function has Haar measure 0, so $\psi=0$. ∎ That the electric term and the magnetic term do not commute is equally elementary ($E_\ell^2$ is a differential operator and does not commute with multiplication by $\operatorname{tr}U_p$). Whether a splitting that groups the electric and magnetic terms by sector also fails to be frustration-free is an expectation from their not commuting, and has not been shown Estimate. Nor have we checked the fact that the local Hilbert space $L^2(SU(2))$ is infinite-dimensional against the setting of the theorems.

Tools on the side of the vacuum measure. Moving the gap to $C_P(\nu)$ by Proposition V does not increase the tools available. We list three reasons.

• $\nu$ cannot be written explicitly. $-2\ln\psi_0$ is the "action" of $\nu$, and it is not local either. A criterion of Bakry–Émery type, which bounds the Poincaré constant by convexity, needs a lower bound on $\operatorname{Hess}(-\ln\psi_0)$. The result of Brascamp–Lieb type, that the ground state of a convex potential is log-concave, is from memory, and its bibliographic data have not been checked Assumption. Moreover, the magnetic term of (7) is a non-constant smooth function on a compact group, so its Hessian is at most 0 at a maximum point, and it cannot be uniformly convex Proved here. Whether a criterion with the Ricci curvature of the group added works uniformly as $g\to0$ is expected to meet an obstacle of the same type as the convexity-ceiling lemma of 11-06 Estimate.
• To set up on $\nu$ an argument of the local-to-global type for Glauber dynamics (from the Poincaré constants of local boxes and strong spatial mixing to a gap uniform in the volume, of the type in Martinelli's lectures), spatial mixing with respect to the boundary values of $\nu$ is needed. Since $\nu$ is not explicit, this is expected to be harder than for the Wilson measure Estimate (the statements of theorems of that type are from memory, not checked in the originals Assumption).
• Taking the temporal lattice spacing to 0 first (the Hamiltonian limit) is a different order of limits from Wilson's isotropic lattice. That both orders lead to the same continuum limit is an assumption Assumption (the background in 10-02).

In 11-04 §06 we saw, for the reference on a four-dimensional spacetime box, that "the $SU(2)$ colour singlet has no linear term, and the reference is exactly halved". We state the same thing on the side of the vacuum distribution at one time.

At leading order ($g\to0$, the three colour components independent Gaussians), the vacuum distribution $\nu$ is a Gaussian measure on the transverse fields of the spatial links; writing $\omega_k$ for the one-particle angular frequencies of the free Hamiltonian, the Langevin generator of the Dirichlet form (1) is unitarily equivalent to $-2(H-E_0)$ by the ground-state transformation of Proposition V ($\hbar=1$). The eigenvalues of $H-E_0$ are sums of one-particle angular frequencies $\sum_kn_k\omega_k$ ($n_k$ is the number of particles), and the eigenspaces correspond to the Wiener chaos ($n=\sum n_k$ gives the $n$-th chaos). On a finite spatial lattice, let the smallest angular frequency be $\omega_{\min}\gt0$ (modes with $\omega=0$, such as the constant mode of a periodic box, are treated separately).

A gauge-invariant function must be invariant under constant gauge transformations (global colour rotations $SO(3)$). A function in the first chaos $\sum c_k^aA_k^a$ is a vector in the colour index $a$, and $SO(3)$ has no fixed vector in $\mathbb R^3$, so the only invariant one is 0 Proved here. Hence the lowest test space of gauge-invariant functions is the second chaos (two particles), the smallest eigenvalue on it is $2\omega_{\min}$, and equality is attained at $F_w=\sum_a(\langle w,A^a\rangle^2-\mathbb E)$, with the two particles placed in the same lowest mode $w$ and contracted in colour Proved here. Reread through Proposition V, this is

Proved here (at leading order, three independent Gaussian colours, on a finite spatial lattice with $\omega_{\min}\gt0$). The lowest gauge-invariant excitation is at the two-particle threshold, and the Poincaré constant is half the one-particle constant. The reference $\Gamma_Q/2$ for the spacetime box in 11-04 is the same mechanism seen through the local Poincaré constant of the spacetime box. LeanLatticeGauge.WilsonLoopReference.no_linear_singlet (assumption: a linear form invariant under two rotations; conclusion: the coefficients are 0), chaos2_gamma (assumption: $0\lt n,g$ and the Wick moments as formulas; conclusion: the algebra by which the ratio is halved).

In physics it is said that "the lightest glueball is a bound state of two gluons". One can read (8) as the local Poincaré version of its leading order Estimate. However, a physical mechanism is not a proof (the table of physical mechanisms in 11-07), and (8) is a statement in the form of the leading order. In infinite volume $\omega_{\min}\to0$, so the right-hand side of (8) also goes to 0. The mass gap does not appear at leading order; what the leading order fixes is only the form "the gauge-invariant test space begins at two particles". How (8) changes for $SU(2)$ at finite $g$ has not been computed at all.

By Proposition V, measuring the Poincaré constant of the vacuum distribution is the same as measuring the gap. We set out the roads to measuring it by how far the machinery of the first step of Part 10 can be used.

• 0+1 dimensions. The machinery of the first step (the lattice eigenvalue problem of the transfer kernel, and the generalised eigenvalues of the weighted lattice Laplacian) computes the local Poincaré constant of the single-time window directly, and agreed with $\varepsilon/(2m_{\rm tr})$ within a factor of 1.006–1.019 (§03) Reading. In 0+1, the machinery measures $C_P(\nu)$ as it is.
• Samples. $\nu$ is not explicit, so samples are needed. The marginal distribution on a single-time slice of an anisotropic Wilson lattice with small temporal lattice spacing is expected to approach $\nu$ in the Hamiltonian limit Estimate. If the time step is finite, that slice is $\psi_0^2$ of the transfer kernel, and the discrepancy of §03 (1–2% at $\varepsilon=\tfrac12$ in 0+1) enters.
• Test functions. Since we restrict to gauge-invariant functions, we take them from polynomials in Wilson loops. As in 11-04 §06, loops of the size of the span of the box are needed.
• Graduation. $\mathcal E_\nu$ of (1) is measured in the metric of the kinetic term $M^{-1}=g^2\cdot$ (the bi-invariant metric). Unless the Langevin graduation is matched to this, the coefficient in (2) is off by a factor $\kappa$.
• Direction. The Rayleigh quotient on samples gives only a lower bound on $C_P(\nu)$, that is, an upper bound on the gap (the side obtained by placing a single test function). A lower bound on the gap needs every $F$, and the asymmetry between H↑ and H↓ of Part 10 ((15) of 10-01) appears as it is. The road to measuring is a road that measures the easy side of the gap.

This chapter places no computation. A computation that bounds the Kogut–Susskind gap of a small periodic lattice in intervals, truncating the representations of the electric field, stays at the numbers of a finite box, and does not become a bridge to a gap uniform in the volume Estimate.

• Proposition V is an identity, and we do not claim it has made the problem easier. There is no tool that bounds $C_P(\nu)$ uniformly in $\beta$ (or $g$).
• Local-to-global gap theorems do not apply to the splitting of the Kogut–Susskind lattice into electric and magnetic parts (Proposition F). That they do not apply to a splitting grouped by sector is an expectation and has not been shown.
• We have not checked the application of strong-coupling perturbative results to the Kogut–Susskind lattice. Results on the weak-coupling side were not found within the range searched.
• That the order taking the Hamiltonian limit first gives the same continuum limit as Wilson's lattice is an assumption.
• (8) is a statement at leading order (three independent Gaussian colours), and there is no value at finite $g$. The correspondence with glueballs is an estimate.
• This chapter does not show the mass gap. Nor does it show anything about Assumption H.

The machine-checking column covers only algebra. The identity (3), Proposition V, Proposition F and the measure steps of (8) are outside Lean (10-13).

Dependency graph (the numbers of premises are smaller than the number of the conclusion):

[01] ASM (A1)          discrete spectrum, ψ₀ > 0, Δ > 0               finite spatial lattice
[02] PRV identity      ground-state transformation ((18) of 10-01)
[03] PRV Prop. V       Δ = ħ²/(2 C_P(ν)) ⟸ [01][02]
[04] PRV three consts  sup over F₁ = (2C_P(ν)/ħ)² ⟸ [03] (+ Proposition B of 10-01, (2) of 11-04)
[05] PRV phys. units   m_phys ≥ c ⟺ C_P(ν) ≤ ξ_RG/(2c) ⟸ [03]
[06] PRV 0+1 window    the single-time window is ε/ħ times Prop. V ⟸ [03]     Lean one_slice_ratio
[07] RDG lattice disc. 1.006–1.019 (ψ₀ of the transfer with discretised time)
[08] PRV Prop. F       the electric/magnetic splitting is not frustration-free
[09] CIT local→global  gap theorems assuming frustration-freeness
[10] CIT perturbation  gap under relatively bounded quantum perturbations (abstract)
[11] ASM log-concave   ground state of a convex potential (from memory)
[12] ASM mixing        uniform gap from local boxes and spatial mixing (from memory)
[13] ASM limit order   the Hamiltonian limit and Wilson's lattice
[14] PRV no linear     the colour singlet has no linear term          Lean no_linear_singlet
[15] PRV two-particle  (8) ⟸ [03][14] (leading order)                Lean chaos2_gamma
[16] EST no fit        local→global does not apply to KS ⟸ [08][09]
[17] EST strong coup.  fits the perturbative type ⟸ [10]
[18] EST glueball      the reading of (8) ⟸ [15]
[19] EST no tool       no tool bounds C_P(ν) uniformly ⟸ [11][12][13][16]

The check for circles is only to see that on every "⟸" the numbers on the right are smaller than the number on the left. The conclusions of this chapter are the identities of [03]–[06] and the leading-order statement of [15], and there is no node for the mass gap. None of the assumption rows [01][11][12][13] contains a lower bound on the gap.

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Terms used in this chapter — the vacuum measure $\nu$, the ground-state transformation, Proposition V, the three Poincaré constants, the Hamiltonian lattice (Kogut–Susskind), frustration-free, local-to-global gap theorems, the colour singlet and the second chaos, the Hamiltonian limit. Their meanings are in A-02 Glossary and 11-A1.

Next chapter to read: 11-06 Paying with rarity instead of convexity — that convexity is not enough on groups, as a story of tools that pay with rarity.

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