2026-10-01 · chapter Yang–Mills and the mass gapPart 9 Where things stand, and the tasksa map of the tasks

The list of open tasks — joining the open questions by arrows of "if this were known, what would move"

A task in this chapter is a statement that has not yet become a theorem. What moves when a task becomes known can be written as an arrow "if A is known, B can be said", and we divide its strength into three. Theorem (a proved implication, or a definition), roadmap (a prospect that the problem statement or a survey writes as "once this is known, the next step should be possible"), and expectation (an implication at the level of physics). As a metaphor it is a table matching keys to doors; precisely, it is a list of rules of the form "if all the premises are in place, the conclusion follows".

The destination is the problem statement (Jaffe–Witten 2000, below JW; 0-03). Fix one group $G$, and divide the demand into four requirementsKnown (a summary of JW §4; the division into four is this book's arrangement).

$\mathsf E$ is the existence of the theory on $\mathbb R^4$, $\mathsf A$ the axioms (at least as strong as those of the book of Streater–Wightman [45] and the papers of Osterwalder–Schrader [35] cited by JW), $\mathsf N$ non-triviality, local fields and agreement with asymptotic freedom at short distances (JW §4), and $\mathsf M$ the gap $\Delta\gt0$ and $m\lt\infty$.

We add one more requirement $\mathsf C$, outside the problem statement. $\mathsf C$ is that the string tension $d$ of the continuum theory is not 0. $d$ is the constant of Problem 5.2 of Chatterjee 2018, written $\sigma$ in 6-10 §04. The physical expectation is that $d$ is positive.

Example: Theorem 2.4 of Chatterjee (2021) is a theorem arrow. If a lattice gauge theory has uniform exponential decay under arbitrary boundary conditions (Definition 2.3; the constants depend only on the group, $\beta$ (the coefficient of the coupling in the lattice action; not an inverse temperature) and the dimension of the lattice, not on the box or the boundary conditions), then centre symmetry is not broken. Further, rectangular Wilson loops in an irreducible representation that acts non-trivially on the centre obey the area law in every Gibbs measure (a distribution on the infinite lattice that follows the lattice weights inside any box once the outside is fixed)Known (9-05 §06). An example of a roadmap is JW §6.5's 'Such a result presumably would enable the study of the limit as T4 → R4' (from the uniform gap to the infinite-volume limit).

K1–K6 are tasks for the continuum theory, corresponding to stages 3–6 of the six stages (6-01) and the additional conditions of the problem statement. K7–K12 are lattice tasks, and tasks that turn lower dimensions and physical premises into mathematics. In this chapter limits are required along the whole sequence (this book's arrangement). Footnote 2 of JW (0-03 §08) writes that weak existence by compactness alone is not accepted as a solution of the existence part unless the properties of the limit (the gap and the axioms) are shown by other methodsKnown. The first three have the following form (notation as in 6-10).

$W_{\gamma_a}$ is the Wilson loop of a lattice path, $\langle\cdot\rangle^{\mathrm{ren}}$ is the expectation with a term proportional to the perimeter removed from its logarithm (formula (5.2) of Chatterjee 2018; 6-10 §04), $\mu_L$ is the measure on the box $T^4$ of side $L$, and $\Delta_L$ its gap. Requiring the limit along the whole sequence in K1 and K2 is this book's arrangement.

KnownTable 1 Twelve tasks

For details: K1 in 6-07, K5 in 6-08, K7 and K10 in 9-01 §07, K12 in 6-06. Example: what moves the four-dimensional cells of the board (9-01 Table 3) is K1–K4 and K7, and the three-dimensional row is K12. K8–K11 have no cell; they are tasks on the side of the seams.

Not many arrows have a proof. We look at three in turn.

Reflection positivity survives in the limit (R2)

The first is reflection positivity. It holds on Wilson's lattice (Osterwalder–Seiler 1978), and since it is an inequality, if the sequence of measures $\mu_n$ converges weakly it survives in the limitKnown (4-10). Converging weakly means that the expectation values of all bounded continuous quantities converge (0-03 §08).

$\Theta$ is time reflection, and $F$ a bounded continuous quantity determined by the side of positive times alone. Together with K4, the Osterwalder–Schrader reconstruction sets up a Hilbert space and $H\ge0$ (rule R2; JW also writes 'Establishing a quantum mechanical Hilbert space is part of the solution to this Millennium problem').

A uniform gap survives in the limit (R3)

The second is that a uniform gap survives in the limit (rule R3). Let $\tau_t$ be the operation of shifting by $t$ in the time direction, and write $\nu_F$ for the spectral measure of the vector made by $F$ with its vacuum component subtracted. $\nu_F$ is the same thing as the $\mu_F$ of 4-11 §02. In this chapter we use $\mu$ for the sequence of measures, so we change the symbol.

That the left-hand side is at most a constant times $e^{-\Delta t}$ for every bounded $F$ is the same as $H\ge\Delta$ in the directions orthogonal to the vacuumKnown (4-11 §02). The integral also includes the point $E=0$, so uniqueness of the vacuum comes out at the same time.

In the estimate box by box (6-10 formula (7)), if the rate of decay $\Delta_L\ge\Delta_*$ does not depend on $L$, the estimate survives in the limit and becomes a gap (the standard argument). The prefactor $\|A^*\Omega_L\|\,\|B\,\Omega_L\|$ in (7) is at most 1 if $|F|\le1$. This argument needs three premises. (i) The expectation values of the $F$ used converge along the sequence $L\to\infty$ (part of $\mathsf E$). (ii) The vectors of those $F$ are dense in the reconstructed Hilbert space. (iii) Since $T^4$ is also periodic in the time direction, the term $e^{-\Delta_L(L-t)}$ added to the correlations in the box vanishes as $L\to\infty$ (it does if $\Delta_L\ge\Delta_*$). What R3 gives is only $\Delta\gt0$; $m\lt\infty$ comes separately from the theory not being the vacuum alone (6-10 §06).

Example: the decay of a single $F$ tells us only the lower end of the support of $\nu_F$. The $\xi(\beta)$ of Problem 5.1 (K7) is the decay of plaquette correlations. Chatterjee defines $\xi$ by the limit $|x|\to\infty$ taken in every direction, but what connects to the spectrum of the transfer matrix (5-03) is the decay in the time direction. Read through the decay in the time direction, what is learned is the lower end of $H$ on the subspace made by the plaquette (this chapter's reading).

From centre symmetry to the area law (R6)

The third is Theorem 2.4 of Chatterjee (2021) from §01 (rule R6). Its introduction also cites Durhuus–Fröhlich (1980), which derives confinement in a gauge theory from exponential decay in a non-linear σ model one dimension lower, and Fröhlich (1979), which carries confinement from a $\mathbb Z_n$ theory to an $SU(n)$ theoryKnown (bibliographic data only). By the title of Fröhlich's paper the destination is the lattice Higgs theory of $SU(n)$; the introduction of Chatterjee 2021 presents it as "the $SU(n)$ theory".

JW writes prospects for the order of the tasks. For the construction on $T^4$ (K1) it says there are ample signs that extending known methods would reach it, writing 'One presumably needs to revisit known results at a deep level, simplify the methods, and extend them'. For the uniform gap (K3) it writes 'New ideas are needed', and goes on that this would make the infinite-volume limit (K2) possible. §5 also writes that a uniform gap 'may play a fundamental role' in the proof of the infinite-volume limitKnown. This is rule R8.

Precisely, the right-hand side of (6) is not the existence of the limit but the material of a uniform estimate for taking the limit, and how to assemble it is not written ('presumably').

Example: the two things R8 joins (uniform decay and the infinite-volume limit) were obtained together from the cluster expansion in models with a small parameter. Weak-coupling $\phi^4_2$ and $\phi^4_3$ are examplesKnown (6-04 §08). They did not pass in the order of deriving the infinite-volume limit from a uniform gap. Four-dimensional Yang–Mills has no such parameter at long distances (6-11 Obstacle 1). On the lattice side, Chatterjee (2018) places Problem 5.2 (K8) as 'the most basic question that needs to be solved before making any further progress' on the road through Wilson loopsKnown. This is a sentence in the direction of necessity, "needed first", and not an arrow of sufficiency from K8 to the requirements.

We put the rules of Table 2 into one figure. Pressing the box of a task makes that task "assumed known", and the conclusions of rules whose premises are all in place light up. What lights up is the smallest set that contains the conclusion whenever it contains all the premises (the closure).

ComputedHow this figure lights up is the closure (7) of the rules of Table 2, computed on the spot (the choice of arrows is this book's arrangement).

KnownTable 2 Rules (if all the premises are in place, the conclusion). R1–R7 are theorems or definitions, R8 a roadmap, R9 and R10 physical expectationsPhysics. The choice of rules is this book's arrangement. $\mathsf C$ is that of §01: the continuum string tension $d$ is not 0 (outside the problem statement)

Example: even if the "Lattice tasks" button makes K7–K10 and K12 assumed known, only $\mathsf C$ lights up, and not a single requirement. This restates in the language of arrows the "showing it on the lattice is enough" of 9-05 §08.

Among the sets of tasks that reach $\mathsf P_G$, we call those that no longer reach it when any one task is removed minimal sets. We examined the 4096 subsets at three levels and counted all the minimal setsComputed.

With theorems only there is one set, {K1,…,K6}; adding roadmaps, one set, {K1,K3,K4,K5,K6}; adding physical expectations as well, {K1,K2,K4,K5,K6,K8,K11} joins, making two. The same held for groups with trivial centre.

ComputedTable 3 For each task, whether it is in every minimal set for $\mathsf P_G$ (required), in some, or in none

Example: (a) K1, K4, K5 and K6 are needed on every road. (b) At the level of physical expectation, instead of the uniform gap K3, "the continuum string tension K8 and a single scale K11" reach the mass gap. The physical sentence that sees mass and string tension as two faces of one scale (formula (3) of 7-12) takes over the role of K3. (c) K7, K9, K10 and K12 are in no minimal set. The count uses only arrows checked in primary sources, so if an arrow from the lattice to a continuum requirement becomes a theorem, it must be redone.

The problem statement asks for $\mathsf P_G$ for every compact simple group $G$ (formula (2)). The candidate Lie algebras are four series and five exceptions, and with each Lie algebra come groups in forms divided by the centreKnown (1-04 Table 6). No general theorem carrying a result for one group to another was found in what this chapter read. Some arrows vanish depending on the group. Rule R6 uses that a finite-dimensional irreducible unitary representation $\pi$ acts non-trivially on the centre, that is, that there is an element $z$ of the centre with $\pi(z)\ne I$. An element of the centre commutes with every $\pi(g)$, so in an irreducible $\pi$ it acts as a scalar by Schur's lemma (1-04 §06).

Example: in the fundamental representation of $SU(N)$, $z=e^{2\pi i/N}I$ and $\omega=e^{2\pi i/N}$ (1-04 §07). For a group with trivial centre there is no representation satisfying (9). Besides the simply connected $G_2$, $F_4$ and $E_8$, this includes the forms divided by the centre $SO(3)=SU(2)/\mathbb Z_2$ and $PSU(N)=SU(N)/\mathbb Z_N$; every group divided by its centre has trivial centreKnown. The reason in one line: for a connected Lie group, the centre is the kernel of the adjoint representation (1-04 §06). $G/Z(G)$ is the same group as the image $\mathrm{Ad}(G)\subset GL(\mathfrak g)$ of the adjoint representation, and its adjoint representation is $\mathrm{Ad}(g)$ itself on $\mathfrak g$ (for a simple Lie algebra $\mathrm{ad}$ is injective). The kernel is the elements with $\mathrm{Ad}(g)=I$, that is, the identity alone. The "group with trivial centre" of Figure 1 is this case, and the arrow from K9 to K10 vanishes. Chatterjee notes that for groups with trivial centre (for example $SO(3)$) it is believed they may not confine at weak couplingPhysics. The problem statement asks for a gap for such groups too, so a road through confinement cannot be used for every group. On the other hand, the one-loop coefficient $16\pi^2b_0=\tfrac{11}{3}C_A$ is positive for every group (4-06 §02), and perturbative asymptotic freedom does not depend on the groupKnown (0-03).

JW §5 lists 'natural extensions': isolated one-particle states (an upper gap in addition to the gap), confinement, other four-dimensional gauge theories with fields added that keep asymptotic freedom and their dynamics, and the extension from $\mathbb R^4$ to general four-dimensional manifoldsKnown. None of these is a requirement, and in Figure 1 only $\mathsf C$ was placed outside. Isolated one-particle states can be written through the spectrum of the mass operator $\sqrt{H^2-\mathbf P^2}$ ($\mathbf P$ is the momentum) (9-02 seam 6).

Example: for a free field of mass $m$, the one-particle states of Fock space (2-06) sit at $m$ and two or more particles fill $[2m,\infty)$; the spectrum is $\{0\}\cup\{m\}\cup[2m,\infty)$, and the upper gap is $m$Known. JW §5 also writes that the clustering that follows from the gap (6-10 §08) may carry results on $\mathbb R^4$ to general four-dimensional manifolds ('may make it possible').

We use the two-column grades of the dictionary (8-01 §05) as common coordinates, and plot four lists on nine cells. (a) is "if both exist, are they the same", (b) is "does it exist in four dimensions". The grades of the dictionary rows are as in 8-01; the grades of seams, tasks and walls are this book's arrangement. The grounds are as follows.

Seams: take over the grade of the dictionary row corresponding to the pair of sentences in 9-02 Table 1 (1→1, 2→13, 3→9, 4→8, 5→2, 6→5, 7→14). Tasks: (a) is the strength of the rules leaving the task in Table 2 (theorem or definition = Settled, conditional on the group = Conditional, expectation = Unsettled; K7, K10 and K12, which have no rule, are unrated), (b) is the "Closest result" of Table 1 (Conditional if there is a version with limited scope). Walls: the grade of the dictionary row named by the column of 10-02 Roads tried on the four-dimensional lattice, and walls (H↑, H↓, η, chain → 13; (S4) → 8; limit → 7. One step and Σδ have no row in the dictionary and are unrated).

Example. The thirteen walls that bear on Assumption H gather in the same "Unsettled, None" cell as dictionary row 13 ($\Lambda$, "a single scale"), seam 2 and task K11. Although the types of wall (10-02 Roads tried on the four-dimensional lattice, and walls) divide into principle, structure, tool, assumption and measurement, on the dictionary they all bear on the same single pair. Only tasks enter "Settled, Conditional": those with a version of limited scope whose arrows to the requirements are theorems or definitions (K1, K2, K4).

ComputedThe assignment of grades was made on this machine by the rules above from 8-01 Table 2, 9-02 Table 1 and Tables 1 and 2 of this chapter, and 10-02 Table 3, and the counts (14, 7, 12, 21) were checked.

Table 4 The rows that fall in each of the nine cells and in unrated (numbers are the rows of each table; in the figure, dictionary rows carry # and seams carry S).

What this chapter can say

Sources and reproduction

Next chapter to read: 9-05 Common misconceptions, which takes ways of speaking that mistake the tasks back to the words of the problem statement. The tasks on the side of the record kept on this machine are in 10-15.

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