2026-10-01 · chapter Yang–Mills and the mass gapPart 8 A dictionary of two vocabularieshow to read the dictionary

How to read the dictionary — grading in two columns, and naming who judges "the same"

The "path integral", "vacuum" and "glueball" of physics are assigned in mathematics to "a probability measure", "a unit vector in the reconstructed Hilbert space (1-10)" and "an isolated eigenvalue of the mass operator $M=\sqrt{H^2-P^2}$ (the operator that reads the rest mass from the energy $H$ and the momentum $P$)". Whether words of different origins point at the same thing, and how certain the correspondence is, differs from row to row, and can only be checked row by row. That is the subject of this chapter.

There are three answers: it can be said by a theorem to be the same, it can be said only under assumptions, or it cannot be said. Moreover, in four dimensions the object on the side of mathematics has often not yet been made (0-03 §04). So we give a row the following form.

$\longleftrightarrow$ only places the two side by side as a pair; whether they are the same is stated by the two grades. Example: "instanton $\longleftrightarrow$ anti-self-dual connection" is (a) Settled, (b) Constructed (classical). For classical fields there is the precedent of Wu–Yang (1975) (abstract: 'identification with the mathematical concept of connections'; 1-07). An attempt to read the assumptions of this site's record in the vocabulary of physics is in 8-05.

There are nine entries: concept, the physical word and its origin, the mathematical word and its origin, column (a), column (b), column (c), judge, conditions (the assumptions needed for the identification and the gap that remains), and chapter.

The sample is the Feynman–Kac formula. The imaginary-time evolution of $H=-\tfrac12\Delta+V$ ($\Delta$ is the Laplacian, not the $\Delta$ of the mass gap) can be written as an expectation over Brownian motion $B_s$ started from $x$ (a random path made by piling up independent small Gaussian displacements; 1-08) Known.

Filling in the entries: the concept is "writing $e^{-tH}$ as an integral over paths", the physical word is the path integral (Feynman 1948), the mathematical word is the Feynman–Kac formula (Kac 1949), (a) Settled, (b) Not applicable (a row of quantum mechanics with finitely many degrees of freedom, so not asked; §04), the judge is a theorem, the conditions are "$V$ bounded below, etc.; the real-time $e^{iS}$ does not become a measure", and the chapter is 2-07. Two kinds of thing go into the Conditions entry. "$V$ bounded below, etc." is a hypothesis of the theorem, and "the real-time $e^{iS}$ does not become a measure" is a gap that remains even after the identification. Among the arrows, $\Longleftrightarrow$ is a theorem in both directions and $\Longrightarrow$ a theorem in one direction, while $\longleftrightarrow$, as in (1), only places the two side by side as a pair.

Column (c): the direction reachable by finite measurement. Besides (a) and (b) we attach one more grade: for a quantity on the side of mathematics, which direction follows as a rigorous inequality from numbers measured with finitely many observables, finite time $t$ and finitely many lattices. There are four values: Upper bound only, Lower bound only, Both and Out of reach (statements about limits, uniformity or all observables, which finite measurement settles in neither direction). Rows that name no quantity (rows that are only a correspondence of definitions or theorems) get "—". The grounds for the judgement are (8) and (9) of §08. A sum of exponentials with weights $w_n\ge0$ is log-convex, so the effective mass $m_{\mathrm{eff}}(t)$ decreases in $t$ and approaches $\mu_O$ from above, and at every finite $t$, $m\le\mu_O\le m_{\mathrm{eff}}(t)$ Known ((3) in 4-11). So the gap and its relatives (positions of eigenvalues, decay rates) fall under "Upper bound only", and the lengths and times that are their inverses (the correlation length $\xi$, the Poincaré constant $C_P$) under "Lower bound only". This is the box "The asymmetry of upper and lower bounds" in 1-10 made into an entry of the dictionary. The row of the sample, the Feynman–Kac formula, gets "—".

Column (a) asks: "if both objects existed, would they be the same?" Whether they exist is the job of column (b).

Even when Settled, the arrow has a direction. The sample is the Osterwalder–Schrader (OS) reconstruction theorem (4-10). We call the axioms of real-time quantum field theory the Wightman axioms (4-09), and the five conditions imposed on imaginary-time correlations (Schwinger functions) (E0)–(E4): (E0) regularity, (E1) Euclidean covariance, (E2) reflection positivity (correlations with the quantity reflected in the mirror of time are not negative), (E3) symmetry, (E4) clustering. (E0′) is the linear growth condition, a strengthening of (E0), requiring the estimates of the $n$-point functions to stay within roughly factorial growth in $n$ (4-10 §03) Known.

On the way there, analytically continuing the correlations of a Wightman theory to imaginary time, using that the energy is positive, gives (E0)–(E4). On the way back (E0′) is needed, and whether (E0′) follows from a Wightman theory is not known (Summers §3, 4-10 §03). Since the conditions differ on the way there and the way back, even though (a) is Settled we write the arrow as two $\Longrightarrow$, not as $\Longleftrightarrow$.

The sample of Conditional is the mass gap and the decay of correlations. §5 of the problem statement writes in one direction: 'An important consequence of the existence of a mass gap is clustering'. The converse needs a dense family of observables to decay at a uniform rate (§08).

Unsettled comes in three types. (i) No definition on the side of mathematics: the four-dimensional $\Lambda$. (ii) Definitions on both sides, but within the range we searched no theorem saying they are the same: centre symmetry (Chatterjee writes that the mathematical physics literature has no rigorous definition, and then that his definition on the lattice was inspired by the physical definition through the Polyakov loop). (iii) Correspondences inside conjectures or analogies: the radial direction of holography and the scale of renormalisation (7-08).

Read as two groups, "Settled" are pairs that are settled to point at the same thing, and "Conditional" and "Unsettled" are pairs that mathematical rigour has not been able to assert. Dictionary B (8-03 §01) adds a fourth grade, "Does not correspond", for pairs known to be different quantities. Row 9 of Table 2 (a false friend, the detour in §06) is graded at this level as B3 in 8-03.

Column (b) writes, with a level, whether the object the mathematical word points at has been made in four dimensions. There are four levels: classical (smooth connections), formal power series (each order of perturbation), lattice (spacing $a$ fixed), and continuum quantum theory (a measure and a Hilbert space on $\mathbb R^4$); the problem statement asks for the last (0-03 §03).

These are the three grades. For rows that are not objects of four-dimensional Yang–Mills (quantum mechanics with finitely many degrees of freedom, extra dimensions, and so on) (b) is not asked, and we write "Not applicable" (the sample in §02).

The last rule is the key. "None" as the name of a grade means "not yet constructed"; it has not been shown not to exist. The problem asks exactly about that existence. The measure on a finite lattice exists for every $\beta$ (the coefficient of the coupling in the lattice action) (5-01), but the continuum measure is its limit, and for four-dimensional non-abelian groups it has not yet been constructed Known.

$U_\ell$ is the link variable and $S_W$ the Wilson action. Example: instantons are Constructed classically (ADHM 1978), and the transfer matrix (the operator that advances imaginary time by one step on the lattice, §08) is Constructed on the lattice (Lüscher 1977). Rows 10 and 11 of Table 2 point at the same geometry, but 10 names the classical level and 11 the lattice level. The second Chern number of a smooth connection exists classically, so the (b) of 11 is Conditional because assigning an integer to a lattice field needs an admissibility condition ($U_p$ close to 1 on each plaquette). On the strongly coupled lattice an isolated one-particle state has also been shown (Schor 1984), but at the lattice level, and on the opposite side from the continuum limit (5-09).

With a one-column grade, "are they the same as definitions?" and "do they exist in four dimensions?" are mixed into one grade, and which of the two decided the grade changes from row to row.

Example: rows 4 and 5 of Table 2 are the same pair, yet asking about definitions gives Conditional, and asking about a four-dimensional instance gives Unsettled. The same holds for 6 and 7, and 1 (the path integral) is Settled as a definition but Conditional in one column. In two columns these fall into the same cell. With a one-column grade that "takes the weaker", the rows of continuum quantum theory almost all line up at "None", and the correspondences of definitions become invisible.

The positions are the grades of Table 2 (this book's reading). With JavaScript disabled, the still figure and the table can be read.

Table 2 Fourteen samples (details in 8-02–8-04). "One column" is the grade given without separating the two questions. (c) is column (c) of §02 (the direction reachable by finite measurement).

Notes on the words of Table 2. Row 11 names the lattice level (§04). The "theorem" in the judge entry of 12 is Hepp 1966 for scalar fields; for gauge theories Becchi–Rouet–Stora 1976 (for semisimple groups, with a choice of gauge in which every field has a mass, showing the identities of gauge symmetry to all orders of perturbation); and for massless $SU(2)$ pure Yang–Mills Efremov–Guida–Kopper 2017 (by flow equations of the renormalisation group). The source for pure Yang–Mills with a general compact group has not been checked. In 13, $\Lambda$ is the mass scale the theory generates by itself (4-07), $C_P$ is the Poincaré constant (how slowly configurations return to equilibrium under a stochastic gradient flow, i.e. Langevin dynamics; 1-10, 8-03), $\xi$ is the correlation length (the length over which correlations decrease as $e^{-r/\xi}$, the inverse of the gap in lattice units), and Assumption H is the assumption $C_P\asymp\xi^2$ made by the record kept on this machine (that the two are proportional up to constant factors; 10-01). The Fefferman–Graham expansion in 14 expands the metric of the interior in powers of the depth from the boundary (8-04).

The dictionary is ordered so that it can be looked up from the physical words that many readers know first, and the grades are judged after translating into mathematical statements (7-13 §01). When a physical word has two meanings, the row is split. The criterion for splitting is whether the grade in column (a) differs between the two meanings. If it is the same, they stay in one row; if it differs, they are split. The sample is "unitarity" Known.

Positive norm means that the inner product of states is positive definite (there are no states of negative norm). Asymptotic completeness means that the Hilbert space is exhausted by states of free particles in the far past and far future (asymptotic states) (8-02 §05). The upper row is a statement within the correspondence between OS and Wightman: from reflection positivity come a positive-definite Hilbert space and $H\ge0$ (reconstruction). The converse is the way-there direction of (3): the positivity of a Wightman theory becomes the reflection positivity of the Schwinger functions (4-10). The lower row is a statement under the Haag–Ruelle scattering theory, which builds asymptotic states by assuming an isolated one-particle mass shell.

Put into one row, the Settled grade would read as if it extended to the S-matrix. "Confinement" is another example of a split row. The three criteria are $V(R)\to\infty$ ($V(R)$ is the energy of a heavy quark pair placed at distance $R$, read from how Wilson loops decrease), the area law $\sigma\gt0$ ($\sigma=\lim_{R\to\infty}V(R)/R$ is the string tension), and unbroken centre symmetry (invariance under transformations by the centre of the group; in physics, the expectation of the Polyakov loop at finite temperature is 0). In physics they mean almost the same thing, but the mathematical theorems are one-directional and on the lattice. Chatterjee's Theorem 2.2 shows, on the lattice $\mathbb Z^d$ at fixed coupling, "centre symmetry unbroken $\Rightarrow V(R)\to\infty$" (for Wilson loops in representations acting non-trivially on the centre; 8-03 §03).

Detour: false friends

Theorem: there is a theorem in both directions or in one ((2), (3), (6)). Definition: the side of mathematics has fixed an object as the meaning of the physical word. Fixing the path integral as a countably additive probability measure is such a case. §4 of Jaffe's review lists as a "problem" of the time that the path integral had no definition as 'a true integral, given by a countably-additive Borel measure'. This way of fixing it does not go as far as agreement with other calculations. Convention: the habit of calling things by the same name. Calling the lowest state measured in each channel on the lattice a "glueball", and calling uniform exponential decay in lattice units a 'strictly positive mass gap' (Shen–Zhu–Zhu). Conjecture: for the correspondence between the radial direction and scale, the side of physics too writes that 'what cutoff on the field theory corresponds to a radial cutoff in the bulk?' is 'left unanswered' (Heemskerk–Polchinski). Analogy: pointing out a similarity of form. "A single scale" and Assumption H (7-12).

Example: row 5 of Table 2 is "Theorem + conditions, convention". The correspondence with an isolated eigenvalue is a conditional theorem (1-10), and calling the lowest state a "particle" is a convention. When they coexist we write both, and grade column (a) by the stronger.

The Conditions entry is where the hypotheses of theorems are kept. We read row 8 of Table 2 with a small model. The transfer matrix is the operator $T=e^{-H}$ that advances imaginary time by one step on the lattice (taking the step as 1; 4-10, 5-03). Let its levels be $0=E_0\lt E_1\lt E_2\lt\cdots$; the state $\Omega$ with $E_0=0$ is the vacuum. The correlation at imaginary-time separation $t$ can be written with the levels and the overlaps $w_n\ge0$ of the observable $O$ with each level (if the levels are discrete) Known.

The gap is $m=E_1$. The rate at which the correlation decreases is fixed by the first level with $w_n\gt0$.

What a single correlation can say is only the upper bound $m\le\mu_O$. One observable suffices for the upper bound, and equality needs $w_1\gt0$. The lower bound $m\ge c$ needs every member of a family of observables dense in the part orthogonal to the vacuum to decrease at rate at least $c$ (uniformity). Which observables overlap with $E_1$ is not known in advance, so a family is needed. This is the content of the conditions of row 8 (4-11 §02, 8-03. Easy from above and hard from below is the same shape as the variational principle: the box "The asymmetry of upper and lower bounds" in 1-10). Column (c) of the dictionary (§02) attaches this asymmetry to each row.

Example: if $E=(0.5,0.9,1.4)$, $w_2=1$, $w_3=0.5$ and $w_1=10^{-3}$, then $m_{\mathrm{eff}}(8)=0.892$, 1.78 times the gap, and it comes within 5% only from $t=24$ Computed. For $t\le8$ the difference between $w_1=10^{-3}$ and $w_1=0$ is at most 1.3% Computed; at finite $t$ a small overlap cannot be told apart from no overlap.

ComputedThe numbers in this figure were computed on this machine ((8) and (9) computed on the spot).

ComputedTable 3 $m_{\mathrm{eff}}(t)$ (with the same $E$, $w_2$ and $w_3$ as above). The right end is the first integer $t$ with $m_{\mathrm{eff}}\le1.05E_1$.

What the problem asks is to move column (b) from "None" to "Constructed" at the level of continuum quantum theory. "None" here means, as defined in §04, not that it has been shown not to exist but that it has not yet been made. What comes along then is shown by column (a).

In Table 2, column (b) is "None" in 9 rows, and 3, 10 and 12, which are "Constructed", are also at levels lower than continuum quantum theory. If a four-dimensional measure satisfying the way-back conditions $(\mathrm{E0}')$–$(\mathrm{E4})$ of (3) were made, the Settled 1 and 2 could be used as they are, and the reconstruction theorem would give a Hilbert space, a Hamiltonian and a vacuum. For the Conditional 4 and 5 the upper gap (§5 of the problem statement counts 'an upper gap, in addition to the mass gap' among the extensions), and for 8 a uniform decay rate, that is, the gap itself, have to be checked separately. The Unsettled 13 and 14 and the false friend 9 are not settled even when the measure is made. On this book's reading, this is also why existence and the gap stand side by side in one sentence in the problem statement (0-03 §01).

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