2026-10-09 · chapter Yang–Mills and the mass gapPart 10 Towards a proof of the mass gap — building the first step of the staircaseline of construction, Assumption H, the one-sided bridge · Kiichi, Shiori and Rin

The line of construction and Assumption H — eight statements, the direction of the lattice version of the mass gap, and Assumption H in six stages

This chapter is about the side of the four-dimensional lattice. The group is $SU(2)$, the Wilson action is $S_W=\beta\sum_P(1-\tfrac12\operatorname{tr}U_P)$ (5-01), and $\beta=4/g^2$ is the inverse of the coupling, not an inverse temperature. All quantities are counted in lattice units (lattice spacing 1).

$m_{\rm lat}(\beta)$ is the gap of the transfer matrix on gauge-invariant states (the mass in lattice units), $\xi=1/m_{\rm lat}$ the correlation length, and $a(\beta)$ the lattice spacing measured in physical units. The mass in physical units is

There is more than one way to fix $a(\beta)$. If the string tension $\sigma$ is taken as the physical unit, $a\sqrt\sigma$ is the lattice spacing; in the four-dimensional record of this part, $a$ is set to the reciprocal of a length $\xi_{\rm RG}$ fixed by counting the steps that coarsen the lattice (§07). Which one is taken goes into the assumptions of the three chains (ii)(iii) (§05).

Example: if $m_{\rm lat}=0.514$ at $\beta=2.60$ (a Monte Carlo value of Lucini–Teper), then $\xi=1.95$ lattice spacings Reading. $\Delta$ is a quantity fixed only when this is divided by $a$; the numbers on the lattice alone do not fix it (5-09).

The problem statement (0-03) asks that a non-trivial quantum Yang–Mills theory exist on $\mathbb R^4$, and that the theory have a mass gap $\Delta>0$. On the road entering from the lattice, we line this up as a single line.

The order is to make the volume infinite first for the measure at each $\beta$, and then take $\beta\to\infty$ (the order of Chatterjee 2018; there is also the order of taking $a\to0$ first in a finite box, 9-04). Table 1 makes each stage of the line a statement, and the demand of the problem statement becomes the following overall picture.

(2) is an overall picture, not a theorem. (S5) is not in (2); it is a restatement of (S4) (§04).

The eight are not independent. Written with arrows "the source is a premise of the target", they are as follows (this book's arrangement; arrows marked $\mathrm H_{\rm band}$ hold only under the band statement of §05).

What is in hand in four dimensions is (S2) and (S7). One is tempted to say "once the limit can be taken, the rest is automatic", but a footnote of the problem statement does not accept weak existence by compactness alone as a solution unless the properties of the limit ('such as the existence of a mass gap and the axioms') are shown separately (Jaffe–Witten) Cited.

Example: three arrows enter (S4), and only the one from (S5) is an arrow under the band statement. (S3) is entered only from (S1) because of the order of making the volume infinite first.

If the "mass gap" on the lattice is read as "the gap in lattice units $m_{\rm lat}(\beta)$ is at least a fixed value for every $\beta$", in the continuum limit it means the opposite. As in (1), the mass in physical units is the mass in lattice units divided by the lattice spacing.

Proved here Just divide (1) by $a(\beta)$. As a metaphor, if the graduations of a ruler are made finer and finer but the thing measured is always a few graduations long, the actual length of the thing shrinks to 0. Precisely, under (5), $\xi$ is bounded, and the correlation length in physical units $a\xi$ goes to 0. What (S4) asks for is the opposite: that $m_{\rm lat}$ goes to 0 at the same rate as $a$.

This direction is exactly Problem 5.1 of Chatterjee (2018): 'Show that for any β > 0, there exists some ξ(β) ∈ (0, ∞) such that … Moreover, prove that lim_β→∞ ξ(β) = ∞.' Cited How to fix the scale is in 5-09.

Example: Table 2 takes the string tension $\sigma$ as the physical unit and regards $a\sqrt\sigma$ as the lattice spacing. Over $\beta=2.25\to2.60$, $a$ becomes 1/3.16 and $m_{\rm lat}$ 1/2.70, while $\Delta/\sqrt\sigma$ stays at 3.3–3.8. The last column is the case where $m_{\rm lat}$ stops at its value at $\beta=2.25$; it increases in inverse proportion to $a$.

$\Delta/\sqrt\sigma$ mostly increases and approaches the extrapolated value 3.844(61) of the original from below (it drops once, within errors, over $\beta=2.475\to2.55$) Reading. The window is $\xi\lesssim2$ lattice spacings, and it is not evidence that the limit exists (9-05).

We restate (S4) with the Poincaré constant $C_P$ of the lattice measure $\mu_\beta$. It is the supremum of the ratio of the variance of an observable to its sensitivity to the motion of the links (the Dirichlet form $\mathcal E$), and it is the time over which the slowest fluctuation is smoothed out in Langevin dynamics, which shakes the links while moving them towards lower action (1-10, 10-00 §04).

$\nabla_\ell$ is the derivative in the direction of link $\ell$ (measured in the angle of the link), and $A\asymp B$ means that the ratio is bounded above and below by positive constants independent of $\beta$. Langevin time has the dimension of length squared (diffusion), so $a^2C_P$ is the slowest relaxation measured in physical units, and (S5) says it goes neither to 0 nor to infinity.

Three quantities are called by the same symbol $C_P$, so we distinguish them (Table 3). The $C_P$ of this part is (β), the Poincaré constant of the path measure (the distribution of all links in spacetime).

For the free field $C_P^{(\beta)}=4\,(C_P^{(\alpha)})^2$ Proved here (just square the free-field column of Table 3; on a time lattice of spacing 1 the right-hand side is multiplied by $1+m^2/4$). Assumption H is a sentence about (β), so it can be reread as "the $C_P$ of the paths agrees with the square of the $C_P$ of the vacuum". In a gauge theory, however, the measure at a single time lives on the space of gauge orbits, and this rereading is an analogy from the form of the free field.

For the free field (with the graduation measured in field variables), $C_P$ is the largest eigenvalue of the covariance $(-\Delta_{\rm lat}+m_{\rm lat}^2)^{-1}$ (1-10)Cited.

For the free field, (S4) and (S5) say the same thing, and a uniform lattice gap $m_{\rm lat}\ge c$ is $C_P\le1/c^2$, that is, the side of $a^2C_P\to0$. Table 4 is an example with $\Delta=1$ held fixed and $a$ made smaller.

Each halving of $a$ makes $C_P$ four times and $\xi$ twice as large, with $a^2C_P=1$ and $a\xi\to1$. If the mass in lattice units is held at 1, $\Delta=2^k$ diverges while $C_P=1$ stays. Whether $C_P$ and $\xi^2$ grow with the same power in a theory with interaction is the Assumption H of the next section.

Detour: two times

In one sentence, Assumption H is the sentence "the time over which fluctuations are smoothed out grows with the same power as the square of the correlation length". In this section and the next we explain this sentence in six stages: (1) what we want to measure, (2) why we restate it as a quantity of a dynamics, (3) the statement, step by step, (4) what H gives, (5) what H does not give, (6) analogous cases. Each stage has a small example that can be checked by hand.

Its place in this part. Assumption H is one restatement chosen in order to get a handle on the problem of the mass gap. It is neither the only road to the mass gap nor a road settled to be right. Viewpoints other than H, and new ways of thinking that take H as a premise, are dealt with in Part 11.

Why this part puts H at the axis. The quantity the problem statement asks for is the gap in the direction of Euclidean time (the gap of the transfer matrix measured in physical units). H replaces it with a quantity — the relaxation time $C_P$ of Langevin dynamics and a power of the correlation length — that is fixed by the lattice measure alone, for which tools exist to assemble inequalities "for every $f$" from local parts, and whose algebra of assembly can be checked by machine. As we see in §06, this replacement can return to the side of the mass gap through the one-sided bridge; moreover, under the assumption of the one-sided bridge, the upper half H↑ of H is a stronger sentence than the gap in physical units on the lattice staying positive. Saying a stronger sentence in a language with more tools: this trade is the standpoint of this part, and all the chapters of Part 10 are built with H at the axis. That is what the title of this part, a study taking H as a handle, means.

(1) What we want to measure — the ruler and the smearing time

The part of the demand of the problem statement that can be stated on the lattice is (6): "the gap in physical units $\Delta=m_{\rm lat}/a$ stays positive as $\beta\to\infty$". Seen on the lattice, two things must happen at once: the ruler $\xi=1/m_{\rm lat}$ grows without bound compared with the graduations of the lattice, and the way it grows matches the inverse of the lattice spacing ($a\xi$ goes neither to 0 nor to infinity). In terms of the five quantities of 10-00 §05, the problem statement is a sentence asking about the relation between ① and ③.

Assumption H brings ② into this question. The time $C_P$ over which the slowest fluctuation in the measure is smoothed out is a diffusion time, so evening out a length $\xi$ should take on the order of $\xi^2$ (the detour of 10-00). So we make "② balances the square of ①" into one sentence, and state the relation between ① and ③ through ②. That is the idea of H.

Example 1 (counting the random walk on a single road by hand). On the single road of 10-00 §04 ($L$ cells, reflecting at both ends, uniform distribution, $\mathcal E(f)=\frac1L\sum_i(f_{i+1}-f_i)^2$), what corresponds to the ruler is the length of the road $L$. For $L=2$, $f=(1,-1)$ is the slowest fluctuation, with $\operatorname{Var}(f)=1$ and $\mathcal E(f)=\tfrac12\cdot2^2=2$, and the ratio is $\tfrac12$. For $L=3$, with $f=(1,0,-1)$, $\operatorname{Var}(f)=\tfrac23$ and $\mathcal E(f)=\tfrac13(1+1)=\tfrac23$, and the ratio is $1$. Both agree with the values of $C_P(L)=1/(2(1-\cos(\pi/L)))$ from (3) of 10-00, and for $L=4$ it is $1/(2-\sqrt2)\approx1.707$ Proved here. Doubling $L$ makes $C_P$ roughly four times as large, and for large $L$, $C_P\approx L^2/\pi^2$ (Table 3 of 10-00). This checks "the smearing time is the square of the ruler" on the smallest road.

Precisely, the $L$ of the single road is a length given from outside, not a ruler the measure carries itself (a correlation length). The simplest example where the two agree is the free field: from $C_P=1/m_0^2$ and $m_0^2=2(\cosh(1/\xi)-1)$, $C_P/\xi^2\to1$ (10-00 §04, Table 4 of this chapter). How ② and ① balance in a measure with interaction is the question of H.

(2) Why restate it as a quantity of a dynamics — measurable, with tools, within reach of checking

The quantity on the side of the ruler, $m_{\rm lat}$, is the gap of the transfer matrix, a quantity of an operator on the Hilbert space of gauge-invariant states. Placing one trial state gives an upper bound on $m_{\rm lat}$ (a lower bound on $\xi$), but a lower bound needs all states. An upper bound on $C_P$ also needs "every $f$", so the difficulty remains in the same direction. There are still three reasons to restate it.

(a) It can be measured from the lattice measure alone. As in (7), $C_P$ is fixed by expectation values under $\mu_\beta$ alone: the variance and the Dirichlet form. It can be defined directly from the measure, without going through the Hilbert space or reconstruction; one test function gives a lower bound, and readings of lower bounds can also be taken from the autocorrelations of Langevin dynamics or Markov chains (10-02 §04).

(b) There are tools to assemble "every $f$". For Poincaré inequalities, tools that assemble an upper bound on the whole from local parts are available in a form independent of dimension. With the conventions measure $\mu\propto e^{-S}$ and $\mathcal E(f)=\mathbb E|\nabla f|^2$,

The two on the left are Cited (the Bakry–Émery convexity criterion and the Holley–Stroock bounded perturbation; $\operatorname{osc}W=\sup W-\inf W$. Bibliographic data in the table of sources of 10-06), and the one on the right is Proved here (use the Poincaré inequality of each component on each term of the Efron–Stein inequality $\operatorname{Var}(f)\le\mathbb E\operatorname{Var}_1(f)+\mathbb E\operatorname{Var}_2(f)$ to get $\le\max$; equality for a function of one coordinate only). Adding to these the decomposition of the variance (the sum of the conditional variance and the variance of the conditional expectation, 10-05 §01), these four are the parts from which the first step is assembled. The logarithmic Sobolev inequality (LSI) has tools of the same type, among them convexity criteria split by scale (10-02 §05).

(c) Machine checking can reach it. The stage of combining the inequalities of the parts is algebra of real inequalities. The algebra can be checked in Lean, and numerical inequalities inside a finite window can be confirmed by interval arithmetic (computer-assisted). What is checked is the assembly from assumptions to conclusions, not the assumptions themselves (10-13).

Example 2 (where the tools work, for a Gaussian measure on two sites). Consider the measure $\mu\propto e^{-S}$ on $\mathbb R^2$, with $S(x_1,x_2)=\tfrac12(x_1-x_2)^2+\tfrac12m_0^2(x_1^2+x_2^2)$. The Hessian is $\begin{pmatrix}1+m_0^2&-1\\-1&1+m_0^2\end{pmatrix}$, with eigenvalues $m_0^2$ (direction $(1,1)$) and $m_0^2+2$ (direction $(1,-1)$). The criterion on the left of (9) gives $C_P\le1/m_0^2$. On the other hand, for $f=x_1+x_2$, since the covariance is the inverse of the Hessian, $\operatorname{Var}(f)=2/m_0^2$ and $\mathcal E(f)=|\nabla f|^2=2$, and the ratio is $1/m_0^2$. Upper and lower bounds agree, and $C_P=1/m_0^2$ Proved here. For the single road (Example 1) we found $C_P$ by diagonalising a matrix; in this example (9) gives the answer without diagonalisation. For the double well $V(x)=(x^2-1)^2/8$, at the top $V''(0)=-\tfrac12\lt0$, and the convexity criterion of (9) cannot be used on the whole. So one combines the decomposition of the variance, splitting valley by valley, with convexity and perturbation after the split (10-04 §06, 10-05 §04). That an upper bound can be assembled from parts for a measure with interaction that cannot be diagonalised — this is the first gain of the restatement.

(3) The statement of H, step by step — the band statement, the power statement, the upper and lower halves

First step: the band statement. Written plainly, it becomes the sentence "$C_P$ lies within the width of a constant factor of the square of the ruler". Comparing with the physical unit $a$ and a width of constant factors independent of $\beta$ (a band),

In numbers, it is the sentence that when $\xi_{\rm RG}=1/a=10$, $C_P$ lies between $100c$ and $100C$, and if $\xi_{\rm RG}=1000$, between $10^6c$ and $10^6C$. Langevin time has the dimension of length squared (diffusion), so $a^2C_P$ is the slowest relaxation measured in physical units, and (10) says it goes neither to 0 nor to infinity.

Example 3 (the single road satisfies the band statement). Taking $L$ as the ruler for the single road of Example 1 and setting $x=\pi/L$, $C_P(L)/L^2=\frac1{\pi^2}\cdot\frac{(x/2)^2}{\sin^2(x/2)}$. Since $t/\sin t$ increases on $(0,\pi/2]$, for $L\ge2$

(the equality on the right is $C_P=\tfrac12$ at $L=2$), and the band statement holds with $c=1/\pi^2\approx0.101$, $C=1/8$ Proved here.

Inserting the correlation length, the band statement splits into three factors.

The first factor is determined by the mass in physical units $m_{\rm phys}=m_{\rm lat}/a$, and the second is the factor of chain (i), "the Langevin gap $\lambda_1$ agrees with the square of the mass". Further splitting the first factor into "the square of the mass agrees with the string tension" (chain (ii)) and "the string tension agrees with the square of the lattice spacing" (chain (iii)), the three ratios are the three chains (7-12). All three are assumptions Assumption. What the upper and lower halves of each chain need is seen in 10-02 §05.

Second step: the power statement. The band statement is moved by the logarithmic factors brought in by the graduation of $\mathcal E$, the order of the bookkeeping and the renormalisation of stochastic time (the detour of §07). Lowering it to a level that ignores not only constant factors but also factors of powers of $\ln\xi$ gives Assumption H.

This sentence has the same form as (2) of the statement of the first step, 10-03 §05, $\lim_{\hbar\to0}\ln C_P(\hbar)/\ln\xi(\hbar)=2$. Only the family of measures differs: here the Wilson measure $\mu_\beta$ with $\beta\to\infty$, on the first step the path measure of the double well with $\hbar\to0$. So in this part (13) is called by the same name on both steps, as the dimension-independent "power form of H". Being a power statement, it does not change when $\mathcal E$ is multiplied by a power of $\beta$ ($\ln\beta/\ln\xi\to0$) Proved here Leanexponent_invariant_of_slowly_varying (assumptions: $X_n,\ell_n\gt0$, $\ln X_n/\ln a_n\to p$, $\ln\ell_n/\ln a_n\to0$. Conclusion: $\ln(X_n\ell_n)/\ln a_n\to p$. Algebra of sums of limits).

Example 4 (the difference between band and power, in numbers). Consider the family $C_P=\xi^2\ln\xi$. The ratio $C_P/\xi^2=\ln\xi$ keeps increasing, $2.30,\ 6.91,\ 13.8$ at $\xi=10,10^3,10^6$, so the band statement breaks. On the other hand $\ln C_P/\ln\xi=2+\ln\ln\xi/\ln\xi$ approaches 2 slowly, $2.362,\ 2.280,\ 2.190$, and the power statement holds Proved here (divide the values $0.834,\ 1.933,\ 2.626$ of $\ln\ln\xi$ by $\ln\xi=2.303,\ 6.908,\ 13.82$). Conversely the single road of Example 3 satisfies the band statement, but $\ln C_P/\ln L=2-(\ln\pi^2+o(1))/\ln L$ is $\ln6640/\ln256\approx1.587$ at $L=256$, far from 2. The power statement says only "2 in the limit", and allows a slow approach by the amount of the constants. The band statement implies the power statement ($\ln c/\ln\xi\to0$), and the converse fails, as $C_P=\xi^2\ln\xi$ shows Proved here.

(10) is a sentence that combines H with chains (ii)(iii), and in this book it is not called H. The names are fixed as in Table 5.

The two rows at the power level are moved neither by the choice of graduation nor by the logarithmic factors of §07. That is why this part places Assumption H as a power statement.

Third step: upper and lower halves. Separating the right and left of the band statement (10) gives H↑ and H↓.

$C_P$ is the supremum of (7), and $\xi_{\rm RG}=1/a$ is a length fixed separately from the measure, so the two have opposite quantifiers over the test function $f$.

Proved here This just writes out the definition of the supremum. H↓ needs only one test function to be shown; H↑ is a genuine Poincaré inequality over all functions. "Easy from above, hard from below" for eigenvalues (1-10) becomes, for their reciprocal $C_P$, "easy from below, hard from above". This asymmetry determines the whole of the roads and walls of 10-02.

Example 5 (families where only one holds). In the family $C_P=\xi_{\rm RG}^2\ln\xi_{\rm RG}$, for $\xi_{\rm RG}\ge e$ we have $a^2C_P\ge1$, so H↓ holds with $c=1$, and since $a^2C_P=\ln\xi_{\rm RG}\to\infty$, H↑ breaks (the power form of H holds, as in Example 4). Conversely, in a family where $C_P$ is bounded independently of $\beta$ (the form of the theorems on the strong-coupling side, Table 7 in §06), H↑ holds and H↓ breaks Proved here. The band statement is the sentence that neither of these two ways of breaking happens.

The remaining three of the six stages: (4) what H gives, (5) what H does not give, (6) analogous cases. (4) and (5) start from the fact that the quantity of the problem statement (the gap in Euclidean time) and the quantity of H (the Langevin relaxation time) are different quantities, and fix precisely the relation between H and the mass gap through the one-sided bridge joining the two and an example in which the converse of the bridge fails.

(4) What H gives — rereading as a mass, and the one-sided bridge

The version with equality. If chain (i) is put as the equality $C_P=1/m_{\rm lat}^2$, the two halves of (14) are reread as statements about the mass in physical units $m_{\rm phys}=m_{\rm lat}\,\xi_{\rm RG}$.

Proved here Proof: multiply both sides of $1/m_{\rm lat}^2\le C\xi_{\rm RG}^2$ in H↑ by $m_{\rm lat}^2/C$ to get $1/C\le(m_{\rm lat}\xi_{\rm RG})^2$. For H↓, the same with the inequality reversed. ∎ The level of proof is algebra, and two Lean theorems check this algebra.

theorem hup_is_gap_positivity {CP xi mlat K : ℝ}
    (hmlat : 0 < mlat) (hCP : CP = 1 / mlat ^ 2) (h : CP ≤ K * xi ^ 2) :
    1 ≤ K * (mlat * xi) ^ 2

theorem hdown_is_mass_finiteness {CP xi mlat c : ℝ}
    (hmlat : 0 < mlat) (hCP : CP = 1 / mlat ^ 2) (h : c * xi ^ 2 ≤ CP) :
    c * (mlat * xi) ^ 2 ≤ 1

LeanLatticeGauge.hup_is_gap_positivity (assumptions: $m_{\rm lat}\gt0$, $C_P=1/m_{\rm lat}^2$, $C_P\le K\xi^2$. Conclusion: $1\le K(m_{\rm lat}\xi)^2$) and LatticeGauge.hdown_is_mass_finiteness (assumptions: $m_{\rm lat}\gt0$, $C_P=1/m_{\rm lat}^2$, $c\,\xi^2\le C_P$. Conclusion: $c\,(m_{\rm lat}\xi)^2\le1$). Inside Lean all four letters are named real numbers; that it is the Poincaré constant of the Wilson measure, and that $C_P$ is a supremum (running over every $f$), are outside Lean (10-13). H↑ is "the mass does not collapse to 0", that is, a lower bound on the side of the mass gap; H↓ is the side "the mass does not escape to infinity".

The equality $C_P=1/m_{\rm lat}^2$ is the value for the free field, and is too strong an assumption to put down for a theory with interaction. However, the proof that derives a lower bound on the mass from H↑ uses only one side of the equality, $C_P\ge\kappa/m_{\rm lat}^2$. And on a lattice with continuous time, this one side can be shown as a proposition with its assumptions named.

Proved here Proof [Y3]. Set $m=\Delta/\hbar=1/\xi$ and $\varphi=(F-\langle F\rangle)\psi_0$. Since $\langle F\rangle=\int\psi_0\psi_1=0$, $\varphi=\psi_1$ and $A\varphi=m\varphi$.

(i) The variance. The correlations of the measure of (A2) are $\operatorname{Cov}(F(q(s)),F(q(t)))=\langle\varphi,e^{-|s-t|A}\varphi\rangle=\|\varphi\|^2e^{-m|s-t|}$ Cited (the formula writing correlations with the transfer operator, of the type of (12) in 2-07 §08 and 10-04 §04). Integrating over the square $[0,T]^2$ gives $\operatorname{Var}(f_T)=\|\varphi\|^2\bigl(2T/m-2(1-e^{-mT})/m^2\bigr)$ (close $\iint e^{-m|s-t|}=2\int_0^T(T-u)e^{-mu}du$ with one integration by parts).

(ii) The Dirichlet form. Differentiating $f_T$ with respect to the path $q(\tau)$ gives $dF(q(\tau))$ for $\tau\in[0,T]$ and 0 otherwise. From stationarity and (A3), $\mathcal E_G(f_T)=\hbar T\int\psi_0^2\langle dF,G\,dF\rangle\le\hbar T\lambda_G\int\psi_0^2|dF|_{M^{-1}}^2$.

(iii) The ground-state transformation identity. For a smooth real function $F$,

Proved here. Using $(V-E_0)\psi_0=\frac{\hbar^2}2\Delta_M\psi_0$ gives $(H-E_0)(F\psi_0)=-\frac{\hbar^2}2\bigl(\psi_0\Delta_MF+2\langle dF,d\psi_0\rangle_{M^{-1}}\bigr)=-\frac{\hbar^2}{2\psi_0}\operatorname{div}_M(\psi_0^2\,\nabla_MF)$, and multiplying by $F\psi_0$ and integrating by parts once gives the right-hand side (boundary terms vanish because $Q$ is compact without boundary, or the functions decay at infinity). ∎ It has the same form as the one-dimensional calculation of 1-10 §08 and 2-05 §08. Since $F\psi_0=\psi_1$, the left-hand side is $\Delta\|\varphi\|^2$, and $\int\psi_0^2|dF|_{M^{-1}}^2=(2m/\hbar)\|\varphi\|^2$; hence $\mathcal E_G(f_T)\le2\lambda_GmT\|\varphi\|^2$.

(iv) The ratio. Dividing (i) by (ii)(iii) gives $\operatorname{Var}/\mathcal E_G\ge\kappa\bigl(1/m^2-(1-e^{-mT})/(Tm^3)\bigr)$, which is the left of (17). $C_P$ is a supremum, so it is at least the right-hand side, and as $T\to\infty$, $C_P\ge\kappa\xi^2$. For the supremum in the canonical case, any $f$ local in time gives $\operatorname{Var}/T\to2\langle\varphi,A^{-1}\varphi\rangle$ and $\mathcal E/T=2\langle\varphi,A\varphi\rangle$, and using $1/a\le a/m^2$ for the eigenvalues $a\ge m$ of $A$ it is at most $\xi^2$, with equality when $\varphi$ lies in the eigenspace of eigenvalue $m$ (the same as (13) of 10-04 §04). ∎

LeanLatticeGauge.HupBridge.cont_ratio_closed (assumptions: $0\lt m$, $0\lt T$. Conclusion: the ratio of (i) divided by (ii)(iii) equals $1/m^2-(1-e^{-mT})/(Tm^3)$) and one_sided_bridge (assumptions: a lower bound on the variance, an upper bound on the Dirichlet form, $\operatorname{Var}/\mathcal E\le C_P$, positive numbers. Conclusion: $\kappa/m^2\le C_P$) check only the algebra of numbers in (i)–(iv). The path measure, the formula for correlations, (18), and that $C_P$ is a supremum are outside Lean.

$\kappa$ does not depend on the length of time $T$. $T$ enters only the factor $1-(\xi/T)(1-e^{-T/\xi})\in(0,1)$ of (17), which approaches 1 as $T/\xi\to\infty$. Setting $G=M^{-1}=1$ for a one-dimensional well gives Proposition L (10-04 §04), and Proposition B is its multidimensional version with a metric. In the language of the chains, Proposition B shows one side of chain (i), $\lambda_1\le m_{\rm lat}^2/\kappa$ (the half on the side of H↓), on a lattice with continuous time. Whether a claim of the same form exists in the literature has not been checked (it may be known).

Why the power is two. The variance is the time integral of correlations, $T\cdot(2/m)\|\varphi\|^2$, that is, it grows to first order in $1/\Delta$. The Dirichlet form, for the function $F=\psi_1/\psi_0$ of the slowest excitation, whose gradient is small, is $T\cdot2m\|\varphi\|^2$ by (18), that is, it becomes small to first order in $\Delta$. The ratio is $1/\Delta^2$. If one overlooks the smallness of the gradient and estimates the Dirichlet form as a quantity independent of $\Delta$, one drops to a lower bound of first order, $c/\Delta$, and the next (19) does not follow.

Note (two meanings of the symbol $\Delta$). In this chapter $\Delta$ is used with two meanings. The $\Delta(\beta)=m_{\rm lat}/a$ of (1) is the mass in physical units. The $\Delta=E_1-E_0$ of Proposition B and the examples of this section is the gap of the Hamiltonian with space made a finite lattice, tied to the correlation length in lattice units by $\xi=\hbar/\Delta$ (with $\hbar=1$ it corresponds to $m_{\rm lat}$, and the $\Delta(\beta)$ of (1) is that divided by the lattice spacing $a$). In Part 11, $\Delta$ is restricted to the meaning of Proposition B, $E_1-E_0$, and the mass in physical units is written $m_{\rm phys}$ (11-A1 §02).

Example 6 (checking the bridge by hand on a two-site lattice). Take the same two sites as in Example 2 as the spatial lattice, with continuous time ($\hbar=1$, $M=G=1$, $V=\tfrac12x^{\sf T}Kx$, where $K$ is the Hessian of Example 2). The normal modes are $u=(x_1+x_2)/\sqrt2$ (angular frequency $m_0$) and $v=(x_1-x_2)/\sqrt2$ (angular frequency $\sqrt{m_0^2+2}$); the first excitation raises the oscillation of $u$ by one, so $\Delta=m_0$, $\xi=1/m_0$ and $F=\psi_1/\psi_0\propto u$. On the paths, $u(\tau)$ is a stationary Gaussian process with $\mathbb E[u(s)u(t)]=e^{-m_0|s-t|}/(2m_0)$. For $f_T=\int_0^Tu\,d\tau$, $\operatorname{Var}(f_T)=\frac1{2m_0}\bigl(\frac{2T}{m_0}-\frac{2(1-e^{-m_0T})}{m_0^2}\bigr)$; since $|\nabla u|^2=\tfrac12+\tfrac12=1$, $\mathcal E(f_T)=T$, and the ratio is $1/m_0^2-(1-e^{-m_0T})/(Tm_0^3)$, exactly the right-hand side of (17). In numbers, with $m_0=\tfrac12$ ($\xi=2$) and $T=20$ the ratio is $4-0.99995/2.5\approx3.600$, and with $T=200$ it is $4-1/25=3.96$, approaching $\xi^2=4$ from below. (18) can also be checked: the left-hand side is $\Delta\,\mathbb E[u^2]=m_0\cdot\frac1{2m_0}=\tfrac12$ and the right-hand side is $\tfrac12\,\mathbb E|\nabla u|^2=\tfrac12$ Proved here. In this example the path measure is Gaussian and the largest eigenvalue of the covariance $(-\partial_\tau^2+K)^{-1}$ is $1/m_0^2$, so $C_P=\xi^2$, and the bridge holds with equality.

Combining H↑ with the bridge. Setting the $C_P\le C\xi_{\rm RG}^2$ of H↑ beside the $C_P\ge\kappa/m_{\rm lat}^2$ of Proposition B gives

Proved here. For example, with $\kappa=1$ and $C=4$, the gap in physical units stays at least $\tfrac12$ for every $\beta\ge\beta_0$. It is the form with the equality assumption of (16) weakened to one side, and in Lean

theorem hup_is_gap_positivity_of_lower {CP xi mlat K κ : ℝ}
    (hmlat : 0 < mlat) (hCP : κ / mlat ^ 2 ≤ CP) (h : CP ≤ K * xi ^ 2) :
    κ ≤ K * (mlat * xi) ^ 2

LeanLatticeGauge.HupBridge.hup_is_gap_positivity_of_lower (assumptions: $0\lt m_{\rm lat}$, $\kappa/m_{\rm lat}^2\le C_P$, $C_P\le K\xi^2$. Conclusion: $\kappa\le K(m_{\rm lat}\xi)^2$). Setting $\kappa=1$ recovers the theorem on the left of (16) (hup_is_gap_positivity_recovered). The right of (16), on the side of H↓, uses $C_P\le1/m_{\rm lat}^2$ (the hard side), so it cannot be weakened with the one-sided bridge.

We list the conditions for (19) to become a sentence on the side of the mass gap. First, $\kappa$ must be uniform in $\beta$. In the field graduation $\mathcal E_A$, $\kappa$ is a number independent of the coupling (its value depends on the convention for normalising the metric of the group), and the condition is met. In the link-angle graduation $\mathcal E_\vartheta$, $\kappa\propto g^2\to0$, and (19) gives $m_{\rm phys}^2\gtrsim g^2/C$, that is, it loses by a logarithm ((γ) of Table 3). Second, the treatment of time. Proposition B is a sentence on a lattice with continuous time (a Hamiltonian lattice with only space made a lattice); for the Wilson measure, which also discretises time, there is no counterpart of the ground-state transformation identity (18). On a time lattice of step 1, with $q=e^{-\Delta}$ the second eigenvalue of the transfer operator $\hat T$ (divided by its largest eigenvalue), putting down an assumption comparing the increment between neighbouring times with the gradient,