2026-10-09 · chapter Yang–Mills and the mass gapPart 10 Towards a proof of the mass gap — building the first step of the staircasedouble well, kinks, lower bound · Kiichi, Shiori and Rin

The path measure of the double well, and kinks

A particle of mass 1 sits in a potential with two valleys.

The valleys are at $x=\pm1$, the curvature at the bottom of a valley is $V''(\pm1)=1$ (the angular frequency of oscillation inside a valley is 1), and the top of the hill is at $x=0$ with $V''(0)=-\tfrac12$. This is (9) of 2-05 §07 with $a=1$.

To a path $x(\tau)$ in imaginary time $\tau$ ($0\le\tau\le T$) we attach the Euclidean action

and make a probability measure on paths with weight $e^{-S[x]/\hbar}$. This is the measure of (7) of 2-07 §05 (mass $m=1$); closing the path into a loop gives the partition function of (9) of 2-07 §06, that is, a classical chain in which the length $T$ of the loop plays the role of inverse temperature. In this part we take $T\to\infty$ (temperature 0, the side of the ground state) first, and then $\hbar\to0$. The quantity with the order reversed (holding $T$ fixed and taking $\hbar\to0$) is a different quantity. For short paths ($T\lt2\pi\sqrt2$), from the known theorem that the mean time to move between valleys grows at the rate $e^{T/(8\hbar)}$ Cited (Theorem 2.6 of Berglund–Gentz, of Eyring–Kramers type with the interval held fixed), $C_P/\xi^2$ is read as falling exponentially to 0 Reading (the rereading from mean times to eigenvalues is by reduction to two states).

The lattice. The numbers in the later chapters of this part are numbers on a lattice that discretises imaginary time with step $\varepsilon=\tfrac12$. With $x_i$ the value at site $i\in\mathbb Z$,

and the measure at $T=\infty$ is a one-dimensional nearest-neighbour Gibbs measure. The integral kernel of the transfer operator ((13) of 2-07 §08) is positive, so the measure is expected to be uniquely determined Assumption (using the integral-operator version of the Perron–Frobenius theorem of 2-07 §08 on $\mathbb R$. A statement of this form in an original source has not been checked in this part, so it is put as an assumption row). Some chapters also cite numbers with step $\varepsilon=0.25$, and say so each time.

The Dirichlet form and $C_P$. For a function $f$ of the path, we set the "size of the change when it is moved" to be

(the canonical Dirichlet form), and the Poincaré constant $C_P=\sup_f\operatorname{Var}(f)/\mathcal E(f)$ ((2) of 10-00 §04). The $\hbar$ attached to $\mathcal E$ is a normalisation matching the form differentiated with respect to the canonical field $x/\sqrt\hbar$ rather than the position $x$ (④ of 10-00 §05). The Langevin dynamics with this $\mathcal E$ is, on the lattice,

($W_i$ are independent Brownian motions); it has $e^{-S_{\rm lat}/\hbar}$ as equilibrium, and its gap is $1/C_P$ Cited (1-10 §07). Along an artificial time $t$ the whole path sways, and the slowest fluctuation decreases by a factor $1/e$ every time $C_P$.

Example (free field). With $V=\tfrac12x^2$ the measure is Gaussian, $C_P=1$, the correlation length is also $\xi=1$, and the ratio $C_P/\xi^2$ is exactly 1 Proved here LeanratioR_free_eq_one (assumptions: $C_P=1/m^2$, $m\ne0$). On the lattice it is the same in the zero-momentum direction.

Precisely, the measures of (2)–(5) are path measures of a 0+1-dimensional quantum-mechanical model, and no gauge field enters anywhere. The relation to four-dimensional gauge theory is only the analogy of 10-03 §04.

This model has two "times" of different character.

The first is a length in imaginary time, the correlation length $\xi$. With $\Delta=E_1-E_0$ (the tunnelling splitting) the difference between the lowest two levels of the quantum-mechanical Hamiltonian $H=-\frac{\hbar^2}{2}\frac{d^2}{dx^2}+V$, the correlation of two points on the path decreases as

(the ratio of eigenvalues of the transfer operator; the same line of argument as 2-07 §07) Cited. $\xi$ is a length in the imaginary-time direction; counted in lattice sites it is $\xi/\varepsilon=2\xi$.

The second is an artificial time, the smearing time $C_P$. It is the time for the whole path to approach equilibrium under the dynamics (5), a time in a direction different from imaginary time.

The quantity comparing the two is

For the free field $R=1$, and Assumption H (the power statement) is the same as "$\ln R$ is small compared with $\ln\xi$" (10-03 §05).

Precisely, the $\xi$ of (6) is the definition for the model with continuous time; for the lattice model it is fixed by $\xi=\varepsilon/\ln(\lambda_0/\lambda_1)$ from the top two eigenvalues $\lambda_0\gt\lambda_1$ of the transfer operator. As $\hbar\to0$ the difference between the lattice value and the continuum value enters the exponent as the difference of actions $S_0^{\rm lat}\ne S_0$, so lattice values are used when writing numbers (§03, §07).

When $\hbar$ is small, the weight $e^{-S/\hbar}$ concentrates on paths with small action. What makes the action 0 is the path staying in the right valley all along, $x\equiv1$, and the path staying in the left valley all along, $x\equiv-1$. The path with the next-smallest action, "moving from valley to valley", is the kink.

The stationarity equation of (2) is $\ddot x=V'(x)$ (motion in the potential $-V$, 2-07 §05), and solutions that settle into valleys as $\tau\to\pm\infty$ satisfy conservation of energy $\tfrac12\dot x^2=V(x)$. From (1), $\dot x=\sqrt{2V(x)}=(1-x^2)/2$, and solving this gives

Proved here (putting $\tfrac12\dot x^2=V$ into (2) makes the integrand $\dot x^2$, and $dx=\dot x\,d\tau$ turns it into an integral over $x$). The width of the kink is about 2 in $\tau$, and its action is the number $2/3$, independent of $\hbar$. This is the $S_0=2a^2/3$ ($a=1$) of 2-05 §07, in the language of physics the action of an instanton (3-05).

The position degree of freedom. The $\tau_0$ of (8) is arbitrary, and shifting the kink in the imaginary-time direction does not change the action. The second derivative of the action (the fluctuation operator) $-\partial_\tau^2+V''(x_{\rm cl})$ has $\dot x_{\rm cl}$ as an eigenfunction with eigenvalue 0 Proved here (differentiating the stationarity equation $\ddot x_{\rm cl}=V'(x_{\rm cl})$ with respect to $\tau$ gives $\dddot x_{\rm cl}=V''(x_{\rm cl})\dot x_{\rm cl}$). This flat direction (the zero mode) is one slow degree of freedom, the position of the kink. The next eigenvalue is $3/4$, and above it is the continuous part $[1,\infty)$, as is known Assumption (an operator of Pöschl–Teller type; the statement in the original source has not been checked in this part).

The diffusion coefficient. Projecting the noise of (5) onto the direction of the position of the kink, the position $X$ diffuses. The direction that shifts the position by one is $\dot x_{\rm cl}$, and the square of its length is $\int\dot x_{\rm cl}^2d\tau=S_0$, so the noise strength $\hbar$ is divided by $S_0$, giving

Cited The same projection count reproduces the coefficient of the theorem that a single interface in the stochastic Allen–Cahn equation on an interval converges to Brownian motion with diffusion coefficient $3/4$ (Theorem 2.1 of Bertini–Brassesco–Buttà). That is, at the level of "the motion of a single kink", (9) is a restatement of a known theorem. Reading In Langevin numerics following one kink on a ring with step $\varepsilon=0.25$, $D/(\hbar/S_0)$ was $1.00\pm0.03$ ($\hbar=0.02$), $1.04$ ($0.07$) and $1.14\pm0.05$ ($0.1$).

Precisely, (9) is a sentence about the process of the position of one kink, not about the gap $1/C_P$ of whole paths. Whether, on a long path with many kinks, the gap is determined by the diffusion of kinks is exactly the question of the upper side (Proposition 4.5).

How the ruler grows

The tunnelling splitting $\Delta$ is determined, in terms of paths in imaginary time, by "the weight of inserting one kink", $e^{-S_0/\hbar}$. The leading semiclassical term is ((10) of 2-05 §07 written with $\hbar$)

Cited (the formula of the leading term is a copy from the source of 2-05. When $\Delta$ is found by two independent numerical methods, the ratio to the leading term approaches 1, from 0.47 at $\hbar=0.5$ to 0.92 at $\hbar=0.05$ Reading), and the correlation length grows exponentially as $\xi\propto\hbar^{1/2}e^{S_0/\hbar}$. In the lattice model, instead of $S_0$ the action of the lattice kink $S_0^{\rm lat}=0.665266$ ($\varepsilon=\tfrac12$; it differs from $2/3$ by 0.2%) enters the exponent Reading. At $\hbar=10^{-3}$ this difference moves $\ln\xi$ by 1.4, so lattice values are used when writing numbers.

$C_P$ is a supremum, so one test function bounds it from below (10-00 §06). The plainest test function is a function $h(x)$ of a single time, summed over time,

(a test function local in time). In this family the lower bound reaches exactly the square of the ruler.

Proved here Proof [Y3]. Set $\varphi:=(h-\langle h\rangle)\psi_0$ and $A:=(H-E_0)/\hbar$. $A$ is the generator of imaginary time, and $A\ge m$ on functions orthogonal to $\psi_0$. Writing the correlations with the transfer operator (2-07 §08),

On the other hand $\delta f/\delta x(\tau)=h'(x(\tau))$, so $\mathcal E(f)/T=\hbar\int\psi_0^2h'^2\,dx$, and from the ground-state transformation identity $\langle\varphi,(H-E_0)\varphi\rangle=\frac{\hbar^2}{2}\int\psi_0^2h'^2\,dx$ (2-05 §08, 1-10 §08), $\mathcal E(f)/T=2\langle\varphi,A\varphi\rangle$. With the eigendecomposition of $A$, $\varphi=\sum_kc_ke_k$ (eigenvalues $a_k\ge m$), $1/a\le a/m^2$ gives

with equality when $\varphi$ lies in the eigenspace of eigenvalue $m$, that is, when $h-\langle h\rangle\propto\psi_1/\psi_0$. ∎ The transfer-operator formula (12) and the ground-state transformation identity are cited as known facts Cited. Leanlocal_rayleigh_ratio_le_one (assumptions: $0\lt m$, $a_k\ge m$, $c_k\ge0$, $\sum c_ka_k\gt0$) checks only the skeleton of finite sums of (13); the eigendecomposition, (12) and the ground-state transformation are outside Lean.

Proposition L holds not only for the double well but for every one-dimensional well. Whether a claim of the same form exists in the literature has not been checked (it may be known).

Readings beyond it. Proposition L also says that "for test functions local in time, $R=1$ is the ceiling". Seeing $R\gt1$ needs test functions that span times (for example, blurring the path in time and then looking at the sign, or counting the collective position of kinks). Evaluating the Rayleigh quotients of such test functions with samples drawn directly from the transfer operator (not samples approached by running a Markov chain), the lower bound on $R$ grew from 1.06 at $\hbar=0.4$ to 6.5 at $\hbar=0.07$ Reading (lower bound over a family, step $\varepsilon=0.25$, a window of the measure at $T=\infty$, variation between seeds ±10–15%). It also agreed, for $\hbar=0.3$ to $0.15$, with values from measuring the decay of the slowest observable in Langevin dynamics Reading (both are lower bounds, not upper bounds).

The type of growth, in the count at the level of physics that counts kinks (a dilute gas of kinks, the diffusion coefficient of (9)), is $R\approx S_0/\hbar\approx\ln\xi$, that is,

Assumption (the count is heuristic, not a proof). The numbers in this range alone cannot distinguish whether $R$ grows as $1/\hbar$ or as $e^{c/\hbar}$. In the former case the power form of H holds; in the latter it breaks. This is why, even for the double well, the sentence stated within the width of a constant factor, $C_P\le K\xi^2$, is read as being on the side that breaks.

Matching the form of (14), we put the target for the upper side as

Together with Proposition L this gives $1\le R\le KS_0/\hbar$, and the power form of H follows ((4) of 10-03 §05). (15) is the conclusion of Proposition 4.5 on top of conditions (H1)–(H4), and is not proved (10-05).

The naive road to showing (11) is to choose one slow variable (for example the mean position of the path, the magnetisation) and show that "functions orthogonal to it relax fast". For short paths ($T\ll\xi$, at most one pair of kinks) this form gives the right count. For long paths ($T\gg\xi$), however, this road cannot be used.

The reason can be seen in the Glauber dynamics of the one-dimensional Ising chain. The space spanned by functions of a single spin is closed under the generator, and the Fourier component of wave number $k$, $S_k=\sum_je^{ikj}\sigma_j$, is an eigenfunction with eigenvalue $\alpha(1-\gamma\cos k)$ ($\alpha$ is the rate, $\gamma=\tanh2K$). The magnetisation is $k=0$, and its ratio to the slowest component orthogonal to the magnetisation, $k=2\pi/N$, is

which approaches 1 when the chain is much longer than the correlation length ($N/\xi\to\infty$) Proved here (rewriting the ratio) Leanglauber_band_ratio (assumption: $\gamma\ne1$). For the formula of the eigenvalues itself, we saw to 8 digits that the two eigenvalues appear as they are in the diagonalisation of all states of chains with $N=8,10$ Reading (the statement in Glauber's original paper has not been checked in this part). The slowest fluctuation is not an isolated single mode but the lower edge of a continuous band. On long paths of the double well too, long-wave fluctuations of the density of kinks make the same band, and the Poincaré constant of the conditional measure with the magnetisation fixed stays of the same size as $\xi^2/D$.

So on long paths one has to take a number of slow variables proportional to the volume, or cut the problem into boxes of length $C\xi$. The latter is condition (H3) (reduction to the box dynamics) and (H4) (the gap inside the box) (10-11).

If the measure $e^{-S/\hbar}$ had a convex action (a log-concave density), the Bakry–Émery criterion would bound $C_P$ by the reciprocal of the curvature of the action, independently of length and of $\hbar$ Cited (1-10). The path measure of the double well is not convex. We look on the lattice (3) at where convexity breaks.

The second derivative of the action (3) (the Hessian) is a matrix with $2/\varepsilon+\varepsilon V''(x_i)$ on the diagonal and $-1/\varepsilon$ between neighbours. For $\varepsilon=\tfrac12$ the diagonal is $4+\tfrac12V''(x_i)$ and the neighbours $-2$. For uniform convexity over a long interval it suffices that the diagonal be at least the size $4$ of the sum of the two neighbours (diagonal dominance). Inside a valley ($|x_i|$ close to 1), $V''\approx1$ and the diagonal is $4.5$, which is enough, but at the top of the hill $x_i=0$, $V''(0)=-\tfrac12$, so

which falls short by $\tfrac14$ Proved here. For the path with every site at the top, $x\equiv0$, the smallest eigenvalue of the Hessian (in the uniform direction) is exactly $\varepsilon V''(0)=-\tfrac14$.

Between a pair of kinks, the interval where the path stays near the top makes a flat direction (a plateau) along which the action hardly changes. Conversely, confining the path to the convex core deep in a valley, $\{x_i\ge a\}$ ($a\gt1/\sqrt3$), Bakry–Émery gives $C_P\le2/(3a^2-1)$ uniformly in length and $\hbar$ Proved here (lower bound on curvature) Cited (of Bakry–Émery type) Leanconvex_core_curvature (assumptions: $1/3\lt a^2$, $0\le a\le x$). But the action of an excursion that leaves the core and comes back, $2/3-a+a^3/3$, is smaller than the action of a kink $2/3$ ($0.1535$ at $a=1/\sqrt3$) Leanexcursion_action_lt_kink (assumption: $0\lt a\lt1$), and an interval of length $\xi$ contains exponentially many excursions. The convex core alone does not give local Poincaré within a phase (condition (H1)).

This shortfall of $\tfrac14$ comes back, in changed form, to the same place on every road that builds the upper side. In later chapters it is called the common wall (10-09).

The later chapters of this part use three values of $\hbar$ as representatives. The order is always from large to small, $\hbar=3\cdot10^{-3}／10^{-3}／3\cdot10^{-4}$. The reason for the choice is that $\ln\xi$ changes widely, from about 218 to about 2213, making them representatives of the regime where the correlation length grows exponentially in $\hbar$. Table 2 gives values on the lattice $\varepsilon=\tfrac12$.

$\ln\xi$ is the value obtained for the difference of the top two eigenvalues of the transfer operator by Herring's formula, which involves no subtraction (the next detour). At intermediate $\hbar$ (0.2, 0.05, 0.04), the road of finding the eigenvalues of the matrix directly and the road of Herring's formula agreed to six digits Reading. That the remainder in the third row approaches a number independent of $\hbar$ is consistent with the form $\ln\xi=S_0/\hbar+\tfrac12\ln\hbar+O(1)$ of (10), and the difference from the continuum semiclassical value $\tfrac12\ln(\pi/16)\approx-0.814$ is read as a lattice correction.

The fourth row is the scale of the right-hand side of (15), and its difference from $2\ln\xi$ is $\ln(S_0/\hbar)$. The ratio of the two columns is close to 2, and approaches 2 as $\hbar$ is lowered. The "2" of the power form of H is a restatement of how small this difference is.

Precisely, Table 2 gives floating-point values, not values enclosed by interval arithmetic. $\xi$ is of size $e^{218}$ to $e^{2213}$, not a length that can be measured directly by Monte Carlo but a quantity read from the eigenvalues of the transfer operator. The three values of $\hbar$ are representative points, with a role different from the "nodes" and "windows" of later chapters (the distinction is in 10-12).

Detour: the tunnelling splitting by Herring's formula

Classification of the statements of this chapter

We put the statements of this chapter into one table in the format of the five grades of this part (Table 1 of 10-00 §01). The columns are the same as in the tables of propositions in later chapters.

What this chapter can say

Sources and reproduction

Words used in this chapter — double well, path measure, Dirichlet form, kink, zero mode, diffusion coefficient of kinks, tunnelling splitting, correlation length (of the double well), Herring's formula, plateau, convex core, phase (phase without kinks). Their meanings are in A-02 Glossary.

Next chapter to read: 10-05 The statement of the first step — four conditions and Proposition 4.5 — splits the upper side into four conditions by the algebra of the splitting of the variance, and states Proposition 4.5 divided into the parts shown and the assumption rows.

← 10-03 The four steps of the staircase — why we start from 0+1 dimensionsContents10-05 The statement of the first step — four conditions and Proposition 4.5 →