2026-10-01 · chapter Yang–Mills and the mass gapAppendicesglossary

Glossary — looking up the words of this book in one line, and jumping to the chapter of definition

Each entry consists of "the headword", "a meaning in 1–3 lines" and "→ the chapter where it is defined". The order is not alphabetical but follows the parts of the book, so the words needed to read an entry are usually a little above it. To search by word, you can filter with the box below (without JavaScript, use in-page search).

The backbone of this book is one formula. Set the energy of the vacuum of the Hamiltonian $H$ to 0; the distance from it to the next part of the spectrum is called the mass gap (4-11 §01).

To read this formula you need "spectrum", "Hamiltonian" and "vacuum", and to give it meaning for four-dimensional Yang–Mills theory you need everything from groups to lattices and reflection positivity. Figure 1 is that single road.

Without JavaScript, the twelve boxes of the figure and their chapter numbers can still be read as they are.

Table 1 counts how many chapters the main words appear in. "First chapter" is the chapter where the name first appears, not necessarily the chapter of definition.

ComputedThe numbers in this table were counted on this machine (a small script that counts words in the body text of the drafts; the numbers move as chapters are revised). Of the 26 words counted, we list the top 10 by number of chapters (excluding "connection", which overlaps with ordinary Japanese) and the 3 words central to Part 10 of this book (Assumption H, Poincaré constant, dynamical exponent).

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Yang–Mills theory is the theory of a mechanism for comparing the orientation of a group from place to place (the connection) and of the mismatch in that comparison (the curvature). The first half of Part 1 prepares the words for this. The key formula is the one line that makes the curvature $F$ from the connection $A$ (in the geometric way of writing; for conversion to the physical way, see A-01).

Example: if the group is $U(1)$ (complex numbers of absolute value 1), then $A\wedge A=0$ and $F=dA$, and we are back to the field strength of electromagnetism (2-03). For $SU(2)$, $A\wedge A\neq0$, and this is where the interaction of the field with itself comes from.

Group

A set with a product, an identity and inverses, whose product is associative. An abstraction of a collection of symmetries (operations that preserve a shape). → 1-02

SU(2), double cover

The group of 2×2 unitary matrices with determinant 1. The same as the quaternions of length 1; its shape is the three-dimensional sphere. It maps two-to-one onto the rotation group SO(3). → 1-02

Compact simple group

A bounded, closed, connected matrix group (a compact connected Lie group) whose Lie algebra is non-abelian and has no ideals other than 0 and the whole (simple in the sense of Lie groups; a finite centre, as in $SU(N)$, is allowed). The problem statement lets the gauge group be chosen arbitrarily from among these. → 0-03

Lie group, Lie algebra

A group whose product can be taken smoothly, and its tangent space at the identity equipped with the commutator $[X,Y]$. The Lie algebra is the "infinitesimal directions" of the group. → 1-03

Connection

A rule for comparing vectors at neighbouring points. The gauge field $A$ is a connection. → 1-07

Parallel transport, holonomy

Carrying a vector along a connection. The change (an element of the group) after going once round a closed path and returning is the holonomy. The prototype of the link variables of the lattice and of Wilson loops. → 1-07

Curvature

The leading part, proportional to the area, of the holonomy of a small loop. Formula (2). The field strength of gauge theory. → 1-07

Gauge transformation (geometry)

Changing the basis at each point by an element of the group. The connection changes as $A\mapsto gAg^{-1}-(dg)g^{-1}$ and the curvature as $F\mapsto gFg^{-1}$. → 1-07

Quantised Yang–Mills theory can be written, in imaginary time, as "a probability measure on configurations of the field". A configuration is infinitely many numbers, so we need probability in infinite dimensions and the language of the spectrum of operators. The inequality used most in this book is the Poincaré inequality, which bounds the variance by "how much the value changes when you shake things".

Example: if $\mu$ is the standard Gaussian distribution on the real line, then $C_P=1$, with equality for $f(x)=x$ (variance 1, expected squared derivative 1). Known The reciprocal of the best $C_P$ is the gap of the diffusion that has $\mu$ as its equilibrium (1-10).

Gaussian measure

A measure determined by its mean and covariance alone. The most basic measure that can be built even in infinite dimensions; constructive field theory often starts from here. → 1-08

There is no Lebesgue measure in infinite dimensions

On an infinite-dimensional separable Hilbert (Banach) space such as $\ell^2$, the only measure that is invariant under translations and gives finite values to bounded sets is 0. The reason why the "$\mathcal DA$" of the path integral has no literal meaning. → 1-09

Self-adjoint operator, spectrum

The infinite-dimensional version of a Hermitian matrix, and the set of its "eigenvalues" (including continuous parts). The Hamiltonian is self-adjoint and its spectrum is real. → 1-10

Variational principle (Rayleigh–Ritz)

The bottom of the spectrum (the smallest eigenvalue, if there is one) is the infimum of $\langle\psi,H\psi\rangle/\langle\psi,\psi\rangle$. However many trial functions you put in, you only get upper bounds. → 1-10

Poincaré inequality, Poincaré constant $C_P$

Formula (3), and the best constant that goes in it. Its value depends on the normalisation of $\mathcal E$ (what you differentiate by), so each chapter states its normalisation. → 1-10

The words of Part 2 have two faces, quantum mechanics and statistical mechanics. What quantum mechanics calls "the energy gap" and statistical mechanics calls "the length over which correlations decrease" become the same quantity through imaginary time. This is the rereading that runs through the whole book; as a formula, it is the following expansion using the vacuum $|0\rangle$ and the excitations $|n\rangle$ (4-11 §02).

Example: if $m=0.5$ then $\xi=2$, and the correlation at a separation of $t=10$ decreases by a factor of $e^{-5}\approx0.0067$. Moving the gap $m$ with the knob in Figure 2 changes the spacing of the levels on the left and the rate of decrease on the right together.

ComputedThe numbers in this figure and table were computed on this machine (values of the exponential function; the figure draws the same formula on the spot). The smaller the gap, the farther the correlation persists, and as $m\to0$ the correlation length becomes infinite. This is what "critical" looks like.

Action, Euler–Lagrange equation

A functional $S$ that assigns one number to a whole path. The classical motion is the path that makes $S$ stationary, and that condition is the Euler–Lagrange equation. → 2-01

State, Hamiltonian

A state is a unit vector in a Hilbert space; the Hamiltonian $H$ is the self-adjoint operator that generates the time evolution $e^{-itH}$ and represents the energy. → 2-05

Path integral

The way of writing quantum theory in which all paths are summed with weight $e^{iS/\hbar}$ (in imaginary time, $e^{-S}$). → 2-07

Imaginary time, Wick rotation

Substituting $t=-i\tau$. The oscillation $e^{iS}$ becomes the damping $e^{-S}$, and quantum theory becomes the probability of statistical mechanics. → 2-07

Correlation function, correlation length

The tie between the values at two distant points, and the length $\xi$ when it decreases as $e^{-r/\xi}$. In formula (4), the reciprocal of the gap. → 2-08

Phase transition, critical point, universality

When a parameter is moved, the free energy and other quantities change in a non-smooth (non-analytic) way. If a quantity jumps, it is a first-order transition; if nothing jumps but the correlation length becomes infinite, it is a continuous transition, and that point is a critical point. At a critical point, behaviour that does not depend on the details (universality) appears. → 2-08

Transfer matrix

The weight of one step of imaginary time made into a matrix (an operator) $T=e^{-H}$. The ratio of the largest eigenvalue to the next determines how correlations decrease. → 2-07

Langevin dynamics, Glauber dynamics

A noisy gradient flow that has a given measure as its equilibrium (the continuous case), and a stochastic dynamics that rewrites one point at a time (the discrete case). Not the time of the theory but a provisional time for producing samples. → 2-10

Dynamical exponent z, critical slowing down

The exponent when the relaxation time grows as a power of the correlation length, $\tau\sim\xi^{z}$. The phenomenon that the dynamics becomes slower the closer one is to criticality is critical slowing down. → 2-10

These are the words of Parts 3 and 4. The classical theory is the theory of "connections that minimise the square of the curvature", and when it is quantised, one length scale that the classical theory does not have is born. That scale is expressed by $\Lambda$, and at one loop it has the following form (4-07).

Example: for $SU(3)$, $11N/3=11$. As $g^2$ is made small, $\Lambda$ approaches 0 in the form $e^{-1/g^2}$, and does not show up in any power series in $g^2$. Known

Classical

Field strength

$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu-ig[A_\mu,A_\nu]$. The last commutator is the mark of non-commutativity: the field interacts with itself. → 3-02

Yang–Mills action

$S=\frac1{2g^2}\int\operatorname{tr}F_{\mu\nu}F_{\mu\nu}\,d^4x$ (Euclidean, written with $g$ absorbed into $A$). In four dimensions its value does not change when the scale is changed. → 3-03

Self-dual, instanton

A connection satisfying $F=\pm*F$ ($+$ is self-dual and $-$ anti-self-dual; the signs of the topological charge are opposite). A finite-action solution that brings the action all the way down to a constant times the absolute value of the topological charge. The representative is the BPST solution (a solution of topological charge 1 with a size around one point). → 3-03

Second Chern number, topological charge

$\frac1{8\pi^2}\int\operatorname{tr}F\wedge F$ (in the anti-Hermitian way of writing). An integer that does not change when the connection is changed continuously. → 3-05

Gauge orbit space, Singer's theorem

The space in which connections related by gauge transformations are identified. In the non-abelian case, there is no global way to choose one connection from each orbit continuously. → 3-04

Quantisation

Divergence, regularisation, renormalisation

That loop integrals become infinite at short distances; the cutoff that makes them temporarily finite; and readjusting the couplings so that observables remain finite when the cutoff is removed. → 4-04

Renormalisation group, β function

The rule for how the effective coupling flows when the scale at which one looks is changed. The speed of the flow is the β function $\mu\,dg/d\mu$. → 4-04

Asymptotic freedom

That the β function is negative and the coupling becomes weaker at shorter distances. For pure $SU(N)$, the one-loop coefficient $11N/3$ is what matters. → 4-06

Dimensional transmutation, Λ

That in a theory with only a dimensionless coupling, one length scale $\Lambda$ appears through quantisation. Formula (5). → 4-07

Gauge fixing, Faddeev–Popov, ghost

A condition that narrows the connections representing the same physics down to one, and the determinant that then appears in the integral. The determinant written in terms of fields is the ghost. → 4-08

BRST symmetry

A symmetry, involving the ghosts, that remains in the action after gauge fixing. It is used to check that physical quantities do not depend on the choice of gauge. → 4-08

Gribov copy, Gribov region

That one orbit contains several connections satisfying the gauge condition (a manifestation of Singer's theorem), and the region where the Faddeev–Popov operator is positive. → 4-08

Axioms and definitions

The Wightman axioms, Haag's theorem

The set of conditions that a quantum field theory on Minkowski spacetime should satisfy, and the theorem showing that an interacting field cannot be placed in the same representation as the free field (if the fields at time 0 are unitarily related to a free field and the vacuum is unique and invariant under spatial translations and rotations, then the field is a free field). → 4-09 axioms, Haag

Reflection positivity

The property that the correlation with the time-reflected function is non-negative. With it, a state space with a positive inner product can be built from the Euclidean measure. → 4-10

Schwinger functions

The correlation functions of the field in Euclidean (imaginary) time, $\langle\phi(x_1)\cdots\phi(x_n)\rangle$. The Wightman functions continued to imaginary time. → 4-10

The Osterwalder–Schrader axioms, reconstruction

The set of conditions that the Schwinger functions (just above) should satisfy, and the theorem that if they are satisfied, a Hilbert space and a Hamiltonian can be built. → 4-10

Mass gap

Formula (1). That the spectrum above the vacuum is separated by $\Delta>0$. It can be rephrased as the mass of the lightest particle, but this is a separate condition from the existence of particles. → 4-11

Non-trivial

That the constructed theory is not a free field (Gaussian). One of the requirements of the problem statement. → 0-03

These are the words of Part 5. A lattice replaces spacetime by a mesh of spacing $a$, and represents the gauge field by elements of the group placed on the edges (links) of the mesh. That gauge invariance is kept exactly is the strength of the lattice. The action is a sum over the small squares (plaquettes), one by one.

Example: for $SU(2)$ ($N=2$) with $\beta=2.4$, $g^2=4/\beta\approx1.67$. Making $\beta$ larger makes $g$ smaller, and by asymptotic freedom the lattice spacing $a$ measured in physical units is expected to approach 0 (the one-loop running, 4-07) Physics. Some references use a different normalisation; for conversion, see 5-01.

Link variable, lattice spacing

The element of the group $U_\ell$ placed on the edge joining neighbouring lattice points (the lattice version of the parallel transport of a connection), and the width of the mesh $a$. → 5-01

Plaquette, staple

The product $U_P$ of the four edges of a unit square (the holonomy of a small loop), and the product of the three edges with one edge removed. → 5-01

Wilson action

Formula (6). The lattice action that adds $1-\frac1N\operatorname{Re}\operatorname{tr}U_P$ for each plaquette; in the naive limit $a\to0$ it returns to the Yang–Mills action. → 5-01

Elitzur's theorem

On the lattice, the expectation value of a quantity that changes under gauge transformations (such as a component of a single link) is 0 in every volume and every limit. Local gauge symmetry does not break spontaneously. → 5-04

Wilson loop

The trace of the product of links along a closed path. Gauge invariant; it represents the weight for carrying a pair of heavy charges. → 5-02

Area law, perimeter law, string tension

The area law is that the expectation value of a Wilson loop decreases as the exponential of the enclosed area, $e^{-\sigma\cdot\text{area}}$; the perimeter law is that it decreases with the length of the perimeter. The coefficient $\sigma$ is the string tension. → 5-02

Confinement

That a colour charge cannot be taken out on its own. On the lattice it is often judged by the area law (the static potential growing in proportion to the distance), but there are several formulations. → 5-02

Strong-coupling expansion, character expansion

A method of expanding the weight in characters of the group when $\beta$ is small. Here the area law and a positive gap can be shown Known (Osterwalder–Seiler 1978). → 5-02

Continuum limit, scaling

The limit $\beta\to\infty$ ($a\to0$) that keeps quantities in physical units fixed. Scaling is when dimensionless ratios stop depending on $a$. → 5-05

Glueball

The lightest excitation of pure Yang–Mills regarded as a particle. On the lattice its mass is measured from how correlations decrease. → 5-08

Problem 5.1, the direction of the lattice version of the mass gap

Chatterjee's (2018) problem. For four-dimensional lattice gauge theory, show that for each $\beta$ the correlation length $\xi(\beta)$ exists and is finite, and that $\xi(\beta)\to\infty$ as $\beta\to\infty$. If the gap in lattice units stayed uniformly positive, the mass would become infinite in the continuum limit, so the question is posed in this direction. That the gap in physical units is kept is the step beyond ((S4)). → 5-09

These are the words of Part 6. Constructive field theory is the field that builds interacting field theories as probability measures and proves that they satisfy the axioms. Which dimensions are easy, and where things stop, can be estimated from the "dimension" of the coupling. The Yang–Mills coupling has a power of length.

Example: for $d=2$, $g^2$ has the dimension of mass squared, and for $d=3$ of mass to the first power, and it stops mattering at short distances (super-renormalisable). For $d=4$ it is dimensionless; it does not weaken by a power of the scale, and moves only logarithmically under the renormalisation group (marginal). In Yang–Mills, by asymptotic freedom, it weakens logarithmically at short distances (4-06). Known Where things stand in each dimension is in 9-01.

Constructive field theory

The field that builds field theories as mathematical objects (measures, operators) and checks the axioms. This book reads it divided into six stages. → 6-01

Triviality

That only a Gaussian (free field) remains when the continuum limit is taken. A result of the type shown for φ⁴ in four dimensions. → 6-08

Cluster expansion

A technique that splits the measure into a sum of contributions from small lumps far from each other, and shows convergence. Effective at strong coupling and large mass. → 6-04

Blocking, renormalisation group step

One step that coarsens the lattice by a factor $b$ and integrates out the fine fluctuations to get a new action. Repeating it controls the continuum limit. → 6-05

Ultraviolet stability

That, in finite volume, the upper and lower bounds of the final action after repeated blocking do not depend on the lattice spacing. A theorem in three dimensions, a theorem with a hypothesis in four. There are three readings of different strength. → 6-06, 6-07 three readings

Stochastic quantisation, regularity structures

The method of building a measure as the equilibrium of a Langevin equation, and the theory that gives meaning to singular stochastic partial differential equations. → 6-09

The words of Part 7 are the pictures by which physics, assuming that "the theory exists", explains the mass gap and confinement; most are at the level of Physics. The representative one is the picture in which the colour electric flux is squeezed into a thin tube and behaves as a vibrating string; the potential between heavy charges then takes the following form at long distances $R$ (7-03).

Example: for $d=4$ the second term is $-\pi/(12R)$. The first term is the area law (the string tension), and the second is the zero-point energy of the transverse vibrations of the tube. Physics That the coefficient of the second term does not depend on the details is a result under the assumption that a long tube can be regarded as an effective string.

Dual superconductivity, abelian projection

The picture of confinement in which, when magnetic monopoles condense, electric flux is squeezed into a tube, dually to the way a superconductor squeezes magnetic flux into a tube. Abelian projection is the procedure that partially fixes the gauge, shrinks the group to its abelian part (the diagonal components), and sees monopoles at the singularities of the fixing. → 7-02

Centre vortex

A sheet-like defect across which the phase shifts by an element of the centre of the group. The picture is that if there are many of them in disorder, Wilson loops show an area law. → 7-02

Effective string, Nambu–Goto, Lüscher term

The theory that regards the flux tube as a string weighted by its area, and the $1/R$ term of formula (8) produced by its fluctuations. → 7-03

Large N, 't Hooft coupling, planar diagrams

The limit $N\to\infty$ for $SU(N)$ with $\lambda=g^2N$ held fixed. Only the Feynman diagrams that can be drawn on a plane remain. → 7-04

Holography, AdS/CFT

The conjecture that the gravitational theory of a certain spacetime is the same thing as a gauge theory on its boundary. The radial direction corresponds to scale. → 7-08

One scale ("a single scale")

The physical premise that every dimensionful quantity of pure Yang–Mills is a constant times a power of $\Lambda$. On the lattice it can be written in the form $m^2\asymp\sigma$. → 7-12

From Part 8 on, words named within this book appear. The exhaustive list of assumptions is in A-06; here we list only the central set. The quantity $a^2C_P$ that the record kept on this machine sets up splits into a product of three ratios (7-12; $a=1/\xi_{\rm RG}$, $\lambda_1=1/C_P$).

Example: with the lattice values for SU(2), $\beta=2.60$ (the table in 7-12), $m/\sqrt\sigma=3.837$, so factor (ii) is $1/3.837^2\approx0.068$. Computed Whether (ii) can be bounded above and below by positive constants independent of $\beta$ is the question of one scale; the lattice values show the tendency, but are not a proof.

The dictionary and where things stand (Parts 8 and 9)

The two columns of settledness

A way of reading that attaches two grades separately to the correspondence between the physical and the mathematical vocabulary: (a) whether they correspond as definitions, and (b) whether the quantity has been shown to exist in four-dimensional Yang–Mills. → 8-01

Seam

A place where the physical way of digging and the mathematical way of digging do not connect. This book lines up seven of them by type. → 9-02

The study with Assumption H at its axis (Part 10)

The side of the four-dimensional lattice

(S1)–(S8)

The eight statements into which the problem statement is divided along the road that enters from the lattice (infinite volume, reflection positivity, Wilson loops, physical gap, slowness of fluctuations, recovery of rotations, OS reconstruction, non-triviality). (S5) is not among the requirements of the problem statement; it is a restatement of (S4). → 10-01

The bookkeeping length $\xi_{\rm RG}$

A length fixed by keeping the books on how the coupling decreases at each step that coarsens the lattice by a factor $b$, and taking the number of steps it takes to fall to a fixed value. We set $a:=1/\xi_{\rm RG}$ (the definition of the four-dimensional record in this part). → 10-01

Assumption H

The power statement that the Poincaré constant $C_P$ (the relaxation time of Langevin dynamics) grows with the same power as the square of the correlation length: $\ln C_P/\ln\xi\to2$. In four dimensions, for the Wilson measure as $\beta\to\infty$. There is no proof. → 10-01

The band statement

The statement "$C_P$ lies within a band of constant multiples of the square of the ruler": $a^2C_P\asymp1$. It is moved by logarithmic factors, so in this part it is not called H. → 10-01

H↑, H↓

The upper and lower halves of the band statement: H↑ is $a^2C_P\le C$ (an upper bound over all test functions), and H↓ is $a^2C_P\ge c$ (a lower bound for which one test function suffices). The hard one is H↑. Names distinct from Assumption H. → 10-01

$H_{\rm band}[\mathcal E]$

An assumption that compares within a band of constant multiples rather than by powers. Whether the band statement holds depends on the graduation of $\mathcal E$ (link angle or field), so it is written with the graduation named. An assumption distinct from H. → 10-01

Chains (i)(ii)(iii)

That each of the three factors into which $a^2C_P$ splits is $\asymp1$: relaxation and correlation length ((i)), correlation length and string length ((ii)), string length and bookkeeping length ((iii)). If all three hold, $a^2C_P\asymp1$ follows (the converse does not). → 10-01, 7-12

The one-sided bridge (Proposition B)

A proposition that, for the path measure with continuous time, a finite spatial lattice and $T=\infty$, under assumptions such as on the ground state, bounds $C_P$ from below by a constant times the square of the correlation length, using test functions local in time. It spans only the H↓ side. → 10-01

Topological sectors and the order of volume

In a periodic box, the topological charge $Q$, which moves between sectors with difficulty, also enters the test functions. Take the reading of the original sources for $SU(3)$ and above that the autocorrelation of $Q$ grows faster than $\xi^2$ (for $SU(2)$, a single primary source, close to $a^{-2}$), and apply to it, extended, the conjecture attached to the observation with open boundaries that HMC and Langevin are in the same universality class: then H↑ reads as false in a periodic box. In this book's order, where the volume is made infinite first, open boundaries are not expected to be a rescue (an assumption row). → the note to 10-01 §05

The lattice directions G0–G15′

The seventeen roads tried on the four-dimensional lattice. Each is a pair of one question and a set of tools applied to it. G9′ and G15′ are restatements of the original questions (the numbers are those of the previous edition's "directions 0–15′" with G prefixed). → 10-02

The lattice walls GA1–GE4

The reason, in one sentence, why a road stopped with a tool short of an arrow. The twenty-one walls are divided into five types: principle (GA), structure (GB), tool (GC), assumption (GD), measurement and literature (GE). → 10-02

Complete family

A set of links that removes exactly one link from every plaquette. If the remaining links are held fixed, the weights of the links of the family separate one by one and can be integrated independently of each other. → 10-02

The shortfall δ(b), the slope κ

The amount by which the window of one lattice step falls short of the continuum $\ln b$ (a number measured in the model of the one-loop kernel), and the slope of the one-loop running of the coupling $\kappa=11/(3\pi^2)$ (SU(2), $\beta=4/g^2$; a rewriting of known coefficients). → 10-02

The form of the answer

The statement "under Assumption H ($\eta=0$), (S4) ⟺ $\sum_k\delta(b_k)\lt\infty$". Besides H, it stands on the power form, on a model in which the errors of the steps are additive and bounded, and on δ being the value of the model of the one-loop kernel (assumption rows). → 10-02

Lower bound on the overlap

The input that, in the representation of the correlation by non-negative weights, the weight of the lightest state is at least $q$ times C(0). With it, a lower bound on the gap follows from correlations at finite $k$. Without it, spectroscopy ends with an upper bound on the gap (wall GA2). → 10-02

(P)

The assumption that the measure can be projected onto a one-dimensional chain in the scale $s=\ln\ell$. Under it, the two-weight Hardy inequality (a weighted inequality in one dimension whose best constant is bounded above and below by integrals of the two weights) can be used. It cannot be said to be weaker than H. → 10-02

T(c,1)

An inequality for the transfer operator saying that the weight of a tube of perimeter $\ell$ is at least $\sigma\ell+c/\ell$. If it holds, $m^2\ge4c\sigma$. There is no proof. → 10-02

The model of the first step

The staircase (four steps)

Four steps placed from an easy place to a hard place: the path measure of the double well in 0+1 dimensions ($\hbar\to0$), the Wilson measure of the four-dimensional lattice ($\beta\to\infty$), the continuum limit and the OS axioms, and the mass gap of the limiting theory. This part deals with the first and second steps. → 10-03

The power form of H

The statement of the first step, $\ln C_P/\ln\xi\to2$ ($\hbar\to0$). It is a statement of the same form as the four-dimensional Assumption H, differing only in the family of measures, so we call it by the same name on both steps. Distinct from the band statement, which speaks of a band of constant multiples. → 10-03, 10-03 §01

The five grades (proved here / cited / computer-assisted / reading / assumption)

The labels for the strength of claims in this part: elementary (the proof is placed in the text), a known theorem cited from the original source, computer-assisted, reading (floating point, Monte Carlo, fitting), assumption. One label per sentence. → 10-00

The double well

The potential energy with two valleys, $V(x)=(x^2-1)^2/8$. The valleys are at $x=\pm1$ (curvature 1 at the bottom), and the top at $x=0$ (curvature −1/2). → 10-04

The path measure

The probability measure that gives paths in imaginary time the weight $e^{-S/\hbar}$. $T\to\infty$ is taken first, then $\hbar\to0$. The numbers are taken on the lattice with step $\varepsilon=\tfrac12$. There is no gauge field in it. → 10-04

Tunnelling splitting

The difference between the two lowest levels of the Hamiltonian, $\Delta=E_1-E_0$. In terms of paths, it is determined by the weight $e^{-S_0/\hbar}$ of inserting one kink. → 10-04

Correlation length (of the double well)

The length in imaginary time over which the correlation between two points of a path decreases, $\xi=\hbar/\Delta$. At the leading semiclassical order it grows exponentially, $\xi\propto\hbar^{1/2}e^{S_0/\hbar}$. → 10-04, §03

Kink

A path that moves from one valley to the other; after the paths that stay in a valley, the path with the smallest action. Its action is $S_0=2/3$ (the action of the instanton), and its width is about 2 in imaginary time. → 10-04

Zero mode

The direction $\dot x_{\rm cl}$ with eigenvalue 0 of the fluctuation operator of the kink. It comes from the fact that shifting the kink in the direction of imaginary time does not change the action: one slow degree of freedom, the position of the kink. → 10-04

Diffusion coefficient of the kink

$D=\hbar/S_0$, obtained by projecting the Langevin noise onto the direction of the position of the kink. A statement about the position of one kink, not about the gap $1/C_P$ of the whole path. → 10-04

Herring's formula

A formula that writes the difference between the even and odd eigenvalues of the transfer operator directly as a ratio of inner products of eigenfunctions, without finding the two separately and subtracting. It avoids the loss of digits when $\hbar$ is small. → 10-04

Plateau

A flat direction, along which the action hardly changes, made by the stretch between a pair of kinks where the path stays near the top. The curvature of one site at the top falls short by $\tfrac14$ of the value needed for convexity ($\varepsilon=\tfrac12$). → 10-04

The convex core

The set deep in a valley, $\{x_i\ge a\}$ ($a\gt1/\sqrt3$). If the path is confined here, $C_P\le2/(3a^2-1)$ follows, of Bakry–Émery type, uniformly in the length and in $\hbar$. → 10-04

Phase (the phase without kinks)

The measure restricted to configurations in which the path stays on the side of one valley. For example $\mu^{(+)}$, restricted to $x_i\ge0$ at all sites (reflected at the wall $x_i=0$). The measure of condition (H1). → 10-05

The assembly of the first step

The splitting of the variance

An identity that splits the configuration space into the two valleys and divides the variance into "the variance within each valley" and "the difference of the means of the two valleys". It is the first tool for assembling an upper bound on $C_P$ from the constants of local Poincaré within the valleys and of the transport of the mean difference, but for long paths a single coarse variable is not enough. → 10-05

Local Poincaré

The Poincaré inequality for the measure restricted to one of the parts (a valley, or the phase without kinks). Its constant $C_{\rm loc}$ becomes condition (H1). → 10-05

Transport of the mean difference

An inequality that bounds the square of the difference of the means over the two valleys by the Dirichlet form. Its constant $C_{\rm md}$ gives the type of condition (H2). → 10-05

Conditions (H1)–(H4)

The four conditions into which the upper side of the first step is divided: (H1) local Poincaré in the phase without kinks, (H2) transport along the family of flat saddle points (inside the box, the rows of the tube and the core), (H3) reduction to the box dynamics, (H4) the gap inside the box. Each leaves assumption rows. They are distinct from Assumption H, which is a statement on the side of the conclusion. → 10-05

Proposition 4.5

The conditional proposition that, for the path measure of the double well, assuming conditions (H1)–(H4) and the premise rows, $C_P\le K\xi^2S_0/\hbar$ holds, and through $1\le R\le K'/\hbar$ the power form of H follows. It is algebra on top of the four conditions, and does not say that the first step is closed. → 10-05, 10-03

Scrap

A stretch of the phase without kinks, sandwiched between kink and kink inside the box (of length at most $C\xi$, with values at both ends). The version of (H1) on it is (H1^seg), which requires uniformity also in the end values. → 10-05

Tube

The collection of configurations around a kink, or a kink–antikink pair. The Poincaré inequality of the tube left after the configuration of kinks is held fixed is the row (H2^tube). The one on a stretch with both ends held in the two valleys is called "the bridge tube". → 10-05, 10-07

Core

The stretch inside the tube where the path stays near the top ($Q_k=(-x_c,0)^k$, $x_c=0.89$). The Hessian of the action has negative directions there; bounding its Poincaré constant uniformly in the length $k$ of the core is the row (T0′_P). → 10-07, 10-11

The box dynamics (reduction to the box)

The step that bounds the Poincaré constant of an infinitely long path by the Poincaré constant of the conditional measure on a box of length $C\xi$ with the values on both neighbours held fixed (condition (H3)). Proposition H3 has the form $C_P\le(1+1/n)\,C_{\rm box}/(1-c_{\rm alt})$ and does not use the logarithmic Sobolev inequality. → 10-05, 10-11

Even and odd boxes

The line covered by boxes of length $L$, neighbours overlapping by $l$ sites, divided into two families by the parity of their numbers. Boxes of the same family are apart from each other, and become independent of each other when the values outside the family are held fixed. → 10-11

Averaging over shifts

Shifting the arrangement of boxes by $l$ sites at a time in $n$ ways, and averaging the inequality for each shift. Since the overlap sites are disjoint across the shifts, the coefficient is only 1+1/n. → 10-05

Maximal correlation

For two σ-algebras, the supremum of the correlation between functions measurable with respect to each. For variables at two points, it can be expressed as the second singular value of the normalised matrix of the joint distribution. → 10-05, 10-11

Alternating maximal correlation

The maximal correlation $c_{\rm alt}$ of the two σ-algebras generated by the values outside the even family and the values outside the odd family. That this is below 1 uniformly in $\hbar$ is the assumption row (MIXalt) that remains in Proposition H3. → 10-05

The lemma of two projections

Lemma TP. If the maximal correlation $c$ of two σ-algebras is below 1, the variance is at most 1/(1−$c$) times the sum of the expectations of the two conditional variances. An elementary proof is placed in the text. → 10-05

The Efron–Stein inequality

The inequality that the variance of a function on a product measure is at most the sum of the expected variances when the coordinates are moved one at a time. Used for the inequality on the product of the even boxes. → 10-11

Reduced kink gas

A model that keeps only the slow variables inside the box (the number and positions of the kinks). If it is put into the same type as the Glauber dynamics of the one-dimensional Ising model (row (KG)), the gap takes the form of a sum of the creation and annihilation of pairs and the diffusion of kinks. → 10-05

Cross term

The row (X) that bounds the Dirichlet form of the slow side inside the box (the configuration of kinks $Y$) by (1+$\chi$) times the original Dirichlet form. That $\chi$ is uniform in $\hbar$ and in the boundary values is an assumption. → 10-05

Kept site

In the two-scale assembly of (H1), the site kept every 12 sites (the coarse side). When these are held fixed, the stretches between them become independent of each other, and the marginal on the kept sites becomes a nearest-neighbour chain. → 10-06

Bridge (a stretch with fixed ends)

A stretch with the values at both ends fixed. In (H1), the 11 sites between kept sites (the fine side); in (H2), a stretch with both ends held in the two valleys and one kink. → 10-06, 10-07

Nesting

The assembly that keeps sites 4 and 8 inside the non-convex 11-site bridge as one more level, splitting it into three small bridges (log-concave) and two sites. → 10-06

Helffer's criterion

The Poincaré version of a tool that judges the gap of a chain from the gap of one site and the strength of the coupling between neighbours. Used in Gershgorin form on the ring of kept sites. → 10-06

Margin

The key quantity of the criterion on the coarse side, $\hbar\rho$ − 2$\hbar\kappa$ (the gap of one site minus twice the coupling to a neighbour). That it is positive is confirmed by interval arithmetic for every pair of end values. → 10-06

Node (of $\hbar$)

A point of $\hbar$ confirmed with computer assistance. Proposition H1 is a statement at each of the six nodes $\hbar\in\{0.2,0.1,0.05,0.02,0.01,0.005\}$, with a margin of at least 0.067. Distinct from the nodes of the dependency graph. → 10-06

Subinterval

A small interval into which the range of $\hbar$ is cut in order to fill in the $\hbar$ between nodes. Within each subinterval, the movement of the expectation values with $\hbar$ is enclosed in an interval. → 10-06, 10-08

Allowance

In a subinterval, the amount of swelling permitted above the pointwise upper bound (the remainder after subtracting the pointwise upper bound from the upper limit fixed by the margin). If the same amount is allowed for every box, the worst box decides the allowance. → 10-06

Route

The seven lines of attack set up to close the row of the core of (H2) (the chain outside the window, the remaining numbers, the sector framework, small assumptions, the one remaining proposition, the seam of the ring, the range of $\hbar$). The numbers are names and do not express order or weight. → 10-07

Site window

The range $W$ of 17 sites around one site inside the core. The logarithmic Sobolev constant $A_W$ of the marginal of its mark is central to routes 2, 3 and 5. Distinct from the window of $\hbar$. → 10-07

The window of $\hbar$

The range $[3\cdot10^{-4},3\cdot10^{-3}]$ of $\hbar$ handled with computer assistance. The "window" in "the whole window" and "the lower end of the window" is this one; where they could be confused, "site" is added to the site window. → 10-07

The chain outside the window

The chain extending outside the site window. Its contraction ratio $q\lt1$ is the condition of routes 1 and 6; for the core of the bridge, $q_{\rm free}\le0.998854904$ became a computer-assisted inequality over the whole window of $\hbar$. Nothing is said outside the window or about $\hbar\to0$. → 10-07, 10-08

Ring

A short periodic path of length $T\ll\xi$ with a configuration containing one kink–antikink pair. → 10-07

Seam

The place on the ring where the separation $\ell$ of the pair shrinks and the pair disappears. The configuration there is the plateau at the top itself, and the same difficulty as in the core of the bridge tube appears. → 10-07

Lump

A run of sites close to the top ($|z|\lt\theta$, $\theta=\tfrac12$), taken as the unit of a sector that does not depend on the sign at the seam. With the rest held fixed, the action of a lump of $k$ sites is convex for $k\le7$. → 10-07

Outer chain and inner chain

The two kinds of chain that enter the logarithmic Sobolev constant of the site window of the ring. The outer chain is the chain on the side of fixed sign extending outside the window; its contraction is a computer-assisted inequality under conditions such as inside the window and pin $\le0$. The inner chain is the stage inside the window; its ratio reads 1.24–1.28 and exceeds 1 (a finite coefficient). → 10-07

Boundary stage

A finite stage that appears only for configurations in which the lump touches the edge of the window (pin in $(0,\tfrac12)$). Its ratio reads 1.004–1.045, and the computer-assisted row of the outer chain does not cover this pin. → 10-07

Sector framework

The procedure for bounding the gap of the core uniformly in $k$ by paying with the rarity of the places of negative curvature, divided into five stages (Σ.1)–(Σ.5) (sum over sectors, Gibbs in marginal form, localisation by component, locality of volume, composition). What remains as an assumption is not the structure but the size of the constants. → 10-09

Sector

The direct product of boxes for each type of the marks attached to the sites (on the side of the top or not). The variance, the Dirichlet form and the entropy split into sums over sectors, and the number of sectors does not enter the constants. → 10-09

Defect set

The set $N_j$ of configurations in which site $j$ or one of its neighbours is on the side of the top beyond the threshold. Negative curvature exists only inside it. → 10-09

Rarity exponent

The minimum $\ell_{\min}$ of the logarithm of the measure of sets containing a defect. Not a length; it does not depend on the length $k$ of the core. → 10-09

Closing condition

The condition $b(A_Wc_H+D_W+\ln2)\lt\kappa\ell_{\min}$ under which the right-hand side of the inequality assembled from the five stages of the sector framework is positive. If it is satisfied, the gap of the core is bounded below by a positive number independent of $k$. → 10-09

Needed value

The upper limit $A_{\max}$ obtained by rewriting the closing condition for $A_W$. That $A_W$ is smaller than this is the one remaining proposition (LS_W♭) (route 5). The present upper bound lies outside the needed value, and the lower bound from test functions does not exclude the needed value. → 10-09

The three strands

Attacks on the one remaining proposition in three forms that differ in the functional form of the condition: the componentwise form, the scalar form (Proposition P5), and the Feynman–Kac form. None of them is closed. → 10-09

Proposition P5

The proposition of the scalar strand, with a table of assumptions. With settings such as core length $k=20$, it bounds the remainder quantities at each of 8 nodes in $\hbar\in[3\cdot10^{-4},10^{-3}]$. The interval-arithmetic rows have not yet been run. → 10-09

Lifting assumption

Row Proxy of Proposition P5. The only assumption, which lifts the proxy ratio linearly outside the measured range (the values for lumps of two sites). → 10-09

The common wall

The same place where the three strands stop: at the top, the diagonal of one site, $4-\tfrac14=15/4$, does not reach the sum 4 of the couplings to its neighbours. A tool that has convexity as its entrance must pay for this ¼ with the rarity of the plateau, and $\hbar$ enters into the way it is paid. → 10-09

Diagonal dominance

That in each row of a matrix, the diagonal is at least the sum of the absolute values of the off-diagonal entries. For the Hessian of the core it is broken by just ¼ at the top, and on the face where every site is at the top, the smallest eigenvalue of a lump with $k\ge8$ is negative. → 10-09

Detour plan

A plan that moves one of the two sides of the closing condition, or the form of the inequality itself. R1–R24 were judged Alive, Conditional, On hold or Dead. → 10-10

The line that killed it

The fact that stopped a detour plan, shrunk to one line. The numbers are readings from a floating-point grid or a transfer matrix. → 10-10

Needed side, form, insufficient side (floor)

The three sides into which the detour plans are divided by direction. The needed side holds plans that make the needed value $A_{\max}$ larger; form, plans that change the functional form of the condition; the insufficient side (floor), plans that reduce the structural loss in the upper bound on $A_W$. → 10-10

Choice of structure and of tools

The two sources of the constants in the closing condition. The structural quantities ($\kappa$, $b$, $\ell_{\min}$) are decided by the model and by how the defect set is defined; the tool quantities ($c_H$, ln 2) depend on the choice of the steps of the proof and can be changed with the same set. What moved the needed value a great deal was the tool side. → 10-09, 10-10

Types of breakage

The reasons that appeared repeatedly in the lines that killed the dead detour plans, gathered into seven (one source, ceiling, paying twice, mixing up the currency, the address of $1/\hbar$, misdirection, transfer). Each type comes with a question to test against a plan. → 10-10

Paying twice

Type 3. Paying the same amount again on a set that an earlier step has already paid for. The original Bochner step paid for the negative curvature over the whole core, but that was inside the defect set. A type where fixing it has a large effect. → 10-10

Currency (speed of exit and entropy)

A word of type 4. What the Poincaré inequality measures is the speed of exit; what can pay for the rarity $\mu(B)\le e^{-\ell}$ is the currency of entropy. With only the ratio of the variance to the Dirichlet form, the rarity cancels between numerator and denominator. → 10-10

The address of $1/\hbar$

Type 5. If one misjudges which quantity $1/\hbar$ multiplies, a plan merely moves $1/\hbar$ into sectors, marks, comparison constants or the cost of a perturbation. On the plateau at the top, $1/\hbar$ lives not in the direction of the sum but in the radius from the corner of the top. → 10-10

Threshold of convexity

The boundary: the lower bound $\kappa(k)=8\sin^2\!\bigl(\pi/(2(k+1))\bigr)-\tfrac14$ on the Hessian of the action of the core is positive only up to $k\le7$. A core with $k\le7$ is bounded by convexity (Lemma BV), independently of the end values and of $\hbar$. → 10-11

Proposition CORE3

The proposition that bounds the Poincaré constant of the core in three stages by length $k$: $k\le7$ by convexity, $8\le k\le14$ by the lemma of two blocks and the maximal correlation of two points, and $k\ge15$ by applying the reduction to the box to the core. The inputs of stages 2 and 3 are measured values. → 10-11

The wall of conditioning

A wall standing in the direction in which the zero wants to move, because the stretch of the core was conditioned to be "exactly this stretch". In the core on the side of the box whose ends cross the top, this creates bistability. Not physical metastability, but something made by the choice of conditioning. → 10-11

The action barrier

Row (ΔS_core). The assumption that the Poincaré constant of the core on the box side is bounded in the form $e^{\Delta S_*/\hbar}$ (with a prefactor a power of $\hbar$), and that the barrier is smaller than $2S_0$ ($\Delta S_*\lt2S_0$). Under it, the local term of Proposition H4 vanishes as $\hbar\to0$. → 10-11

Computation and checking

Computer-assisted

What has been confirmed as an inequality by interval arithmetic (the grade "computer-assisted"). Premises such as the rounding convention are written on the same row. Nothing is said to hold outside the computed range. → 10-08, 10-00

Interval arithmetic

Computation that holds every number as "an interval that is sure to contain the true value". If the upper end of the final interval is smaller than 1 ($q^+\lt1$), then $q\lt1$ follows as an inequality. → 10-08

Outward rounding

When computing the ends of an interval, rounding the lower end down and the upper end up. In this part, the result of each operation is moved outward by one to the neighbouring floating-point number. Correct rounding in IEEE 754 is placed in a premise row. → 10-08

Loss of dependence

That intervals forget the ties between quantities coming from the same variable, and widen (evaluating $x-x^2$ naively on [0,1] gives [−1,1]). Fixed by subdividing finely or expanding about the centre. → 10-08, §04

Positive control, negative control

A positive control asks whether the machine reproduces the answer in a case where it is known in closed form. A negative control asks whether the machine rejects a claim that should not hold. → 10-08, 10-14

Pre-registration

Writing down, before running a computation, the expected range of values, the rate-limiting place and the pass line for the controls, and recording after the result whether they were right. It becomes a record that the pass line was not moved after seeing the result. → 10-08

The integrated table

A table that puts the rows of the five grades for the four conditions of the first step on one sheet, row by row. A map of where there are proofs and where assumptions remain; the proportion of rows does not express progress. → 10-12

The dependency graph

A list that turns the rows of the table into numbered nodes and lines up the premises of each row to the right of "⟸". Which rows the conclusion of Proposition 4.5 depends on is read by following the premises. → 10-12, 10-05

Node, leaf

A node is one row of the dependency graph. A leaf is a node with no "⟸": a row of assumption, known, computer-assisted or reading. → 10-12

The rule of numbering

The rule that the numbers of the premises to the right of "⟸" are always smaller than the number of the conclusion on the left. If it is kept on every row, the number goes down each time a premise is followed until a leaf is reached, and there is no circle. → 10-12

The required set of leaves

All the leaves that can be picked up by following the premises from the conclusion [111]. The only fork is the single "or" at [103], so there are two sets. → 10-12

The two roads (the core on the box side, the core on the sector side)

The two ways of supporting the fork [103] (the row that the Poincaré constant of the core is uniform). Road A supports it with Proposition CORE3 and the vanishing of the local term as $\hbar\to0$; road B with a uniform lower bound on the gap of the core in the sector framework and the regularity of the seam. → 10-15

Kinds of work

Six ways of classifying the work needed to close rows of assumption or reading: interval arithmetic, paper (writing a proof), bibliography (checking in the original source and raising it to "cited"), measurement, another method (out of reach of the present tools), method undecided. → 10-15, §03

A $\hbar$ that fits all

A $\hbar$ at which every row of the four conditions holds at the same time. Overlaying the ranges of the rows, there is none at present. The power form of H ultimately needs every row stated in a form uniform as $\hbar\to0$. → 10-12

The criterion of cleanness

That the output of #print axioms for a Lean theorem is entirely a subset of the three axioms propext, Classical.choice and Quot.sound. It shows that the proof has no holes and that no arbitrary axioms have been added. → 10-13

Putting the conclusion into a definition

The form in which the desired property is written into a definition and that definition is taken as an assumption. The conclusion becomes a copy of the assumption, and the proof closes in one line. It is invisible to the criterion of cleanness. → 10-13

Circle of dependence

The form in which theorem A takes the conclusion of B as an assumption and B takes the conclusion of A. Theorems are checked one at a time, so it is invisible to machine checking; it is checked with the rule of numbering of the dependency graph. → 10-13, 10-12

The four verdicts of reconciliation

The verdicts when the numbers of the record are compared with the known ones: Agreement, Conditional agreement (how they agree depends on the choice of step, box, convention or form of extrapolation), Disagreement, Not comparable. → 10-14

The words of Part 11

11-00 Entrance

Further considerations

The title of Part 11. A part that considers, in order, the shape of what hinders H, two candidates for restating while keeping H as a premise, two views that do not go through H, and what is at work in four dimensions. Many of its sentences are estimates. → 11-00

The trade of the restatement

Restating the quantity of the problem statement (the gap in Euclidean time) as the quantity of H (the relaxation $C_P$ in Langevin time: a quantity defined by a supremum, for which one test function gives a lower bound, and whose upper bound can be assembled from parts). The two are different quantities. → 11-00, 10-01

The estimate label Estimate

A sixth label added to the five labels of Part 10 (proved here, cited, computer-assisted, reading, assumption). Attached to sentences that are opinions offered as material for judgement, with neither proof nor numbers. → 11-00

11-01 A nearly conserved global quantity

A nearly conserved global quantity (one type)

A global quantity $Q$ whose variance is large and whose Dirichlet form is small (it is passed along through fluxes, or changes only in rare configurations), and which does not show up in the gap. Its relaxation time is decided not by the correlation length but by the scale of the events that break $Q$, and it hinders H↑. That the five examples fall into the same type is a classification (an estimate). → 11-01

The coefficient of breaking of conservation h

The density of the expectation of the square of the sum of the gradients of the charges of subregions (the gradient of the total charge). In the continuum it is 0 by topological invariance; only on the lattice is it non-zero, on configurations with dislocations. What H↑ as a whole asks about is this coefficient. This book has no value for the four-dimensional lattice. → 11-01

The transport coefficient d

The density of the expectation of the square of the first moment (the dipole) of the field of gradients of the charge. In the regime where $h$ is small, what the local H asks about is this coefficient. There is no value for the four-dimensional lattice. → 11-01, 11-03

The boundary-layer coefficient g

The quantity per unit area obtained by subtracting the volume term from the Dirichlet form of the charge cut by a half-space. It enters the boundary correction of the ratio for a sharp box. → 11-01

The crossing length $R_*$

A length determined by the ratio of the transport coefficient to the coefficient of breaking of conservation (the square of $R_*$ is dλ_W/h). The boundary at which the ratio of the charge of a smooth window is of transport type, proportional to $R^2$, in boxes shorter than this, and of barrier type, saturating at $\chi_t/h$, in longer boxes (an estimate resting on the assumption of cluster decay). → 11-01

Flux through the box

A field running uniformly through the box, which enters the space of tests if the gauge transformations at the boundary sites of the box are left without dividing out. It moves the reference value by a factor of 2.2–2.5 (a reading of the free-field reference numbers). An example where the same type arises not from the dynamics but from the choice of family. → 11-01, 11-03

11-02 The length at which conservation breaks

Three rulers

The candidates to compare $C_P$ against, divided into three: the static correlation length $\xi$, the length at which conservation breaks $\ell_*$, and the time for the global quantity to move by one $\tau_{\rm glob}$. They can take different values within the same model. → 11-02

The length at which conservation breaks $\ell_*$

The length at which the charge of a subregion begins to relax not by transport through the boundary but by breaking of conservation inside the region. The candidate definition is (2): the smallest $R$ at which the ratio Var/𝓔 reaches half its saturation value (an estimate). For two-dimensional $U(1)$ on a strip of width 2 with the outer boundary frozen, β = 2–6 and strip length $M\le256$, the length read from the reading of $\tau_{\rm int}$ (an upper bound on the gap) using the fit form $A/M^2+B$ is not proportional to the correlation length. → 11-02, 11-02 §03

Tree coordinates and link coordinates

In the Gaussian approximation on the hierarchical lattice, taking the values of the plaquettes as variables gives $C_P=1/\beta$ independent of the number of steps, while taking the link variables makes the gap shrink in inverse proportion to the square of the side. An example where the type of growth of $C_P$ changes with the choice of coordinates (the four-dimensional hierarchical lattice is an extension of this book). → 11-02

Conservation of flux

The property that when a link is moved, the values of the plaquettes (the flux) are only passed between neighbours. The source of the slowness of link coordinates. → 11-02

Flux dipole

The quantity $X$ obtained by summing the angles of the plaquettes of the strip with cosine weights. The slowest mode of the Gaussian approximation; the length at which conservation breaks is fixed from the bend in its relaxation. → 11-02, §04

Wrapping

The event in which the angle of one plaquette crosses $\pm\pi$. A rare event that crosses a barrier of height $2\beta$; it breaks conservation because the group is compact. → 11-02

Topological charge of the strip

The integer obtained by reducing the angles of the plaquettes of the strip to $(-\pi,\pi]$, summing and dividing by $2\pi$. It changes by one at each wrapping. We do not claim it is the same thing as the four-dimensional topological charge. → 11-02

Raw rate and static fluctuation

The rate of relaxation of the topological charge of the strip, split into the ratio of the rate of wrapping events (a dynamical quantity) and the static fluctuation $E$ (a quantity of the measure), which measures how much the charge moves in one event (an estimate of the chain). → 11-02

$\tau_{\rm glob}$

The time for the global quantity to move by one. The form that compares $C_P$ with $\max(\xi^2,\tau_{\rm glob})$ is a candidate estimate set up from readings of the numbers of the two-dimensional strip, and is not shown in any model. → 11-02

Two-dimensional test bed

For the two-dimensional $CP^{N-1}$ model, a comparison of freezing in a periodic box with the local ratio of a box as $N$ is varied. The ratio is a ratio of autocorrelations with the lattice side fixed at $L=64$, a different quantity from the static ratio of 11-03. The two-dimensional results imply nothing about four dimensions. → 11-02

11-03 The sector H and the local H

The algebra of time-averaged functionals $\mathcal D$

The collection of test functions built with smooth functions from functions at one time averaged over time with weights of bounded support. The $C_P$ of Part 11 is defined as a supremum over this. → 11-03

Window

The time interval $[0,c\ell]$ of $c$ times the ruler of length $\ell$ (in the version local also in space, a spacetime box of side $c\ell$). $\ell$ is not fixed by the gap but given from outside (in four dimensions, $\xi_{\rm RG}$). → 11-03

The local Poincaré constant $C_P^{\rm loc}(c)$

The supremum of $\operatorname{Var}/\mathcal E$ taken only over test functions whose weights have support inside the window. It increases in $c$, and returns to $C_P$ without assumption as $c\to\infty$. → 11-03

The local H (H^loc_c, $\mathrm H{\uparrow}^{\rm loc}_c$)

The statement that $C_P^{\rm loc}(c)$ grows with the same power as $\ell^2$, and its upper half $\mathrm H{\uparrow}^{\rm loc}_c(K)$ ($C_P^{\rm loc}(c)$ is at most $K$ times $\ell^2$). It has content different from H as a whole only when stated with one $c$ held fixed. The definition stands on continuous time, a finite spatial lattice, $T=\infty$ and the assumptions of Proposition B. The version for the four-dimensional Wilson measure is a copy of it, with the existence of the infinite-volume measure placed as a premise. → 11-03

The ratio form $K_{\rm rel}$

The ratio of the local Poincaré constant of a box for the target measure to the value of the massless reference on the same lattice, the same box and the same graduation. The form "bounded by some $K(c)$" is satisfied by the free-field computation for boxes in four spacetime dimensions that are local also in space, and so is empty; hence it is stated with this ratio. → 11-03

The massless reference and Γ

The reference is the local Poincaré constant $C_P^{\rm loc,0}(R)$ of a box for the infinite-volume massless free field, and $\Gamma$ is its coefficient $C_P^{\rm loc,0}(R)/R^2$. The numbers are readings of the free-field reference, not numbers for the target measure. → 11-03

Gauge quotient reference

The reference for the gauge field measured on functions invariant under gauge transformations at all sites of the box (for a contractible box, functions of the angles of the plaquettes). It does not depend on the parameter of gauge fixing, so this one is used. → 11-03

The sector H (H↑^sec)

The statement that, in a box with its physical side held fixed, the Poincaré constant within a sector of topological charge is bounded by the square of the bookkeeping length. It only pushes the obstacle to H into the quantity between sectors; it does not remove it. → 11-03

Sectors and the reflecting form

The sectors cut from the smoothed topological charge at regular values, and the Dirichlet form on them with no boundary condition imposed (the form of Langevin reflecting at the walls). The Poincaré constant of a sector is finite only when the sector is connected. → 11-03

The quantity between sectors $C_{\rm trans}$

The supremum of the ratio of the variance of the conditional expectation onto the sector number to the Dirichlet form. It gives the sandwich $C_{\rm trans}\le C_P\le\sup C_P^{\rm sec}+C_{\rm trans}$. Its physical reading is the rate of dislocations. → 11-03

Open box and closed box

A box of the conditional measure with the boundary values held fixed (the box of Proposition H3 in Part 10) is a closed box; a box of the marginal with the boundary not held fixed (the box of the local H) is an open box. → 11-03

11-04 Which H the one-sided bridge needs

The family local in time $\mathcal F_1$

The family of functionals obtained by integrating functions at one time over time with bounded weights. Under the assumptions of Proposition B and the canonical graduation, the supremum of $\operatorname{Var}/\mathcal E$ over this family is exactly $\xi^2$. The bridge uses only this part. → 11-04

The window factor $\varphi(x)$

$1-(1-e^{-x})/x$. The bracket in writing the ratio for the window $T$ as $\kappa\xi^2\varphi(T/\xi)=\kappa T^2g(mT)$; $\varphi(x)/x^2$ is strictly decreasing (Proposition G). Distinct from the angle of a plaquette $\varphi_p$. → 11-04

Proposition W

The proposition that writes the supremum over the family local in time with the support of the weights restricted to the window $[0,T]$ in closed form, via the root of a transcendental equation (under the assumptions of Proposition B and the canonical graduation). → 11-04

The two forms of the local H↑

The absolute form $\mathrm H{\uparrow}^{\rm loc}_c$ ($C_P^{\rm loc}(c)$ is at most $K(c)\ell^2$) and the ratio form $\mathrm H{\uparrow}^{\rm rel}_c$ ($K_{\rm rel}(c)$ is at most $K$). The constants correspond by $K(c)=K_{\rm rel}\,\Gamma\,c^2$. → 11-04

Threshold $K_*$

$K_*=\kappa/(\lambda_W\Gamma)$. Under the assumption $\text{(Loc-}\Delta_c)'$ and the identity of the ground-state transformation in infinite volume, if the ratio $K_{\rm rel}$ is smaller than this, a lower bound on the gap follows (algebra on top of assumptions). Its value changes with the choice of reference. → 11-04

Assumption $\text{(Loc-}\Delta_c)'$

The assumption that there is a local observable supported in the box such that the moment ratio of the spectral measure of the excitation it creates is at most the square of the infinite-volume gap $m_\infty$ plus a box-size correction γ/(cℓ)². It holds for the free field, but is not shown for any interacting model. → 11-04

The band $[K_*,1)$

For fields whose ratio lies in this range, the local bridge says nothing. The massive free field also lies in this band. Whether there is another argument that sees the mass inside the band is undecided. → 11-04

Colour singlet and the second chaos

Gauge invariants of $SU(2)$ (colour singlets) have no first-order term around the identity, and at the leading order ($\beta\to\infty$), with three independent Gaussian colours, they lie in the second chaos or higher. For this reason the colour-singlet reference is exactly half the value of the gauge quotient. → 11-04

Functionals of winding-number type

Functionals outside the family local in time, such as the functional that integrates with weights the increment of the lifted angle of a free particle on a circle. All of the amount by which H↑ is stronger than the gap lies outside this family. → 11-04

11-05 The Hamiltonian lattice and the vacuum measure

The vacuum measure ν

The distribution of the vacuum at one time (the distribution with density the square of the ground state, $\psi_0^2$). In gauge theory the functions are restricted to gauge-invariant ones. → 11-05

Ground-state transformation

The isometry that sends a function $F$ to $F\psi_0$. With it, the quadratic form of $(H-E_0)$ is carried to ħ²/2 times the Dirichlet form of $\nu$. → 11-05

Proposition V

Under assumption (A1), the gap can be written with the Poincaré constant of the vacuum measure as $\Delta=\hbar^2/(2C_P(\nu))$, and the supremum is attained by $\psi_1/\psi_0$. → 11-05

Three Poincaré constants

The $C_P$ of the path measure (a one-sided relation to the gap), the supremum over the family local in time ($\xi^2=\hbar^2/\Delta^2$), and the constant of the vacuum measure ($C_P(\nu)=\hbar^2/(2\Delta)$). → 11-05

The Kogut–Susskind lattice

The Hamiltonian of a gauge theory with time taken continuous and only space made a lattice (the sum of an electric term and a magnetic term). The quantity of the problem statement is exactly the gap on its gauge-invariant subspace. → 11-05

frustration-free

That, when the Hamiltonian is viewed as a sum of local terms, there is a state that realises the ground states of all the local terms at once. Local-to-global gap theorems need this. → 11-05

Proposition F

The sum into which the Kogut–Susskind lattice is split as an electric term and a magnetic term is not frustration-free ($g\gt0$, finite spatial lattice). For this reason local-to-global gap theorems do not apply to this splitting. → 11-05

The Hamiltonian limit

The order of limits in which the temporal lattice spacing is sent to 0 first. That it leads to the same continuum limit as Wilson's isotropic lattice is an assumption. → 11-05

11-06 Paying with rarity instead of convexity

The ceiling of convexity

On a compact manifold without boundary, the oscillation of a potential satisfying $\operatorname{Hess}V\ge-K\,g$ is at most $KD^2/2$ (Lemma 1). On $SU(2)$ a convex measure cannot leave the range of constant multiples of Haar, and cannot concentrate at the identity at weak coupling. → 11-06

The Bakry–Émery lower bound on curvature $\kappa_{\rm BE}$

The constant in $\operatorname{Ric}+\operatorname{Hess}V\ge\kappa_{\rm BE}\gt0$. If this holds, log-Sobolev and C_P ≤ 1/κ_BE follow (a copy of the statement in the original source). A different quantity from the comparison constant $\kappa$ of Proposition B. → 11-06

Fibre

Where the conditional measure lives on the fine fluctuations (products of many links) when the coarse field is held fixed. Reading the inductive convexity assumption of hierarchical models as Bakry–Émery on the fibres over the group is an estimate. → 11-06

The reversal of convex and non-convex places

For flat $\phi^4$, what is non-convex is small fields; on the group, it is the hemisphere far from the identity (large fields). On a closed manifold there is always a non-convex place, but the probability of being there is small. → 11-06

The rate of rarity s

The exponent when a bad set $B$ satisfies $\mu(B)\le e^{-s}$. The quantity written $\ell$ in 10-09 and 10-10 of Part 10; in Part 11 the letter was changed because $\ell$ is used for the ruler. → 11-06

Currency (entropy)

The distinction that the Poincaré inequality is a currency that measures the speed of exit, and that what can pay for rarity is the currency of entropy. A step that pays with rarity goes through an inequality of log-Sobolev type. → 11-06

The Holley–Stroock perturbation

The tool that says that if a measure is changed by a factor $e^{-U}$ with bounded $U$, the Poincaré constant changes by at most a factor $e^{\operatorname{osc}U}$. The cost is not rarity but the exponential of the oscillation (in this book, a standard form from memory, not checked against the original source). → 11-06

Lyapunov condition

The type that bounds the global Poincaré constant from a drift $LW\le-aW+b\mathbf 1_G$ returning from the bad set to the good set and a local Poincaré inequality on the good set. Convexity is used only on the good set. → 11-06

Field-dependent constants

The direction of letting the constants of the convexity assumption at each scale depend on the coarse field (but uniformly below 1). As long as uniformly below 1 is required, it does not get past the ceiling of convexity. → 11-06

The Polchinski flow on the group

The multiscale Bakry–Émery condition along the flow of covariances, carried over to convolution with the heat kernel on the group. A formulation on the group was not found within the range searched. → 11-06

The class of measures that pay with rarity

A candidate class of measures with the rules: locally convex on the good set, the bad set rare, a drift back from the bad set, and rarity paid only through entropy inequalities (an estimate). → 11-06

Subcriticality

The key condition of the solution theory of stochastic quantisation (regularity structures); it holds only when $d\lt4$. Four dimensions is exactly critical. → 11-06

11-07 Because it is four dimensions

Critical dimension

The dimension in which the coupling $g^2$ is exactly dimensionless. For Yang–Mills it is four, and the correction at each change of scale does not shrink by a power but remains. → 11-07

marginal

A term whose coupling is dimensionless and which, by power counting, neither strengthens nor weakens. Four-dimensional $\phi^4$ and Yang–Mills are both marginal, but the directions of their flows are opposite. → 11-07

Infrared asymptotic freedom

That the coupling approaches 0 at long distances (four-dimensional $\phi^4$). The asymptotic freedom of Yang–Mills is at short distances, the opposite direction. → 11-07

Dislocation

A small instanton, the size of the lattice spacing, passing through the lattice. The smoothed topological charge on the lattice changes only at such moments. → 11-07, 11-01

The four hurdles

A table that lines up, for each of the four steps of Part 10 (the 0+1 double well, the four-dimensional lattice, the continuum limit and the OS axioms, the mass gap of the limit), the hurdle, the known point reached, whether it is in reach, and the kind of cost. Whether it is in reach and the cost are estimates. → 11-07

The three roots (R1)–(R3)

The roots of the difficulty of the third step: (R1) the coupling is dimensionless and all scales act with the same weight, (R2) there is no expansion linking the ultraviolet and the infrared, (R3) non-abelian gauge invariance (there is no global gauge fixing, and large fields cannot be controlled by convexity). → 11-07

Ultraviolet stability (as used in Part 11)

That on a finite-volume lattice, the effective action on the unit lattice stays bounded after arbitrarily many repetitions of the block-spin transformation. Cited as a known point reached in the third step. → 11-07, the entry above

The window of coupling

The range that remains between the currency of rarity on the ultraviolet side (the rate of rarity is proportional to the effective coupling and runs out at the confinement scale) and the currency of curvature on the strong-coupling side, where neither is enough. A restatement of the second root (R2) (an estimate). → 11-07, 11-06

Dual superconductivity (as treated in Part 11)

The mechanism that explains confinement by the condensation of monopoles. The condensation has not been shown and the projection depends on the choice of gauge, so it is not used as a tool of proof. → 11-07, the entry above

Centre vortex (as treated in Part 11)

The mechanism by which vortices taking values in the centre carry the string tension. The evidence is lattice numerics, and there are neither estimate inequalities nor statements about the limit, so it is not used as a tool of proof. → 11-07, the entry above

11-08 Questions, and what is not claimed

Candidate rules (N1–N6 and three combinations)

Candidate systems that remove one assumption of an existing framework and put an alternative rule in its place: the axiomatisation of scale, the algebra of loops, conserved quantities and sectors, the class of measures that pay with rarity, the axiom of asymptotic freedom, the rule of curvature, and N1+N4, N3+N4, N2+N3. All are candidates, and not even their consistency has been checked. → 11-08

Status of the questions (done, partly done, not yet, new)

The status column of the table of questions in Part 11. "Done" means that the text has answered the question conditionally; "new" is a question added in the writing of this part. → 11-08

What does not move when settled

A column attached to each row of questions and candidate rules, stating what does not move even if that question is settled. Whichever question is settled, the mass gap problem does not move. → 11-08

The same letter or the same word can refer to different quantities in different chapters. This is because the conventions of each field are carried over; the meaning is declared at the start of each chapter, but it is missed when skimming. The heavy one is "gap": the gaps of three operators are called by the same word. The conversion between physical units and lattice units is the following line.

Example: if the gap in lattice units $m_{\rm lat}$ is held at 0.5 while $a$ goes from 1 to 1/16, $\Delta$ increases from 0.5 to 8. If it is made proportional to $a$, as $m_{\rm lat}=0.5a$, then $\Delta=0.5$ is kept. Computed What the continuum limit asks for is the latter (10-01).

Of the three gaps, the problem statement asks only for the first. The second passes to the first if the continuum limit exists and the ratio (10) converges ((S4), 10-01), and the third is a property of the dynamics that produces samples. Assumption H is a statement comparing the third with the second (10-01).

Detour: four uses of the word "dimension"

What this chapter can say

Sources and reproduction

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