2026-10-01 · chapter Yang–Mills and the mass gapPart 8 A dictionary of two vocabulariesthe theory itself

Dictionary A: the theory itself — the Settled pairs line up in front of a single measure

The grades are the two columns of 8-01. (a) is "if both existed, would they be the same?", and (b) is whether the mathematical object exists in four-dimensional Yang–Mills theory (level in parentheses). The judge is what supports "the same" (theorem, definition, convention, and so on; 8-01 §07), and the chapter is the chapter that treats it in detail.

Notes on the words of Table 0. (c) is column (c) of 8-01 §02 (the direction reachable by finite measurement), given only for rows that name a quantity. A6 and A9 are rows where a single correlation gives an upper bound on the lowest point (8-01 §08), and A7 and A13 are statements about limits. "None" in (b) means not yet constructed at the continuum level, and "None (Constructed at the lattice level)" means rows made only at the lattice level. Only "ruled out" in A17 has the opposite sense: it is a theorem that a global section does not exist. ^* "Not applicable" in A1 means that, being a pair from quantum mechanics with finitely many degrees of freedom, it has no four-dimensional entry (8-01 §04).

Many of the Settled pairs lie inside a single premise. If on $\mathbb R^4$ there is a measure $\mu$ satisfying the Osterwalder–Schrader (OS) conditions, the reconstruction theorem gives the rest (4-10 §05) Known.

On the left are the OS conditions (growth, Euclidean covariance, reflection positivity, symmetry, clustering; 4-10 §03), and on the right the Hilbert space, Hamiltonian and vacuum, the passage from the Schwinger functions $S_n$ to the Wightman functions $W_n$, and the positivity of the spectral measure. In four-dimensional Yang–Mills theory the left is missing, so at the continuum level column (b) lines up almost entirely at "None". The exceptions are rows for which the premise is not enough even if it holds (A9, A10, A12), and rows decided outside the premise (A17 by Singer's theorem, A19 by Elitzur's theorem). Example: for the four-dimensional free field the left of (1) exists, and the objects of every row except the gauge rows are present.

"Container" is a metaphor. More precisely, it means the space on which the theory rests (the totality of field configurations), the measure on it, and the Hilbert space made from them.

A1 Path integral (quantum mechanics) ⟷ Feynman–Kac formula

The physical word is "a weighted sum over all the paths a particle can take", and the mathematical word is an expectation over Brownian motion. They can be said to be the same after moving to imaginary time and putting conditions such as the potential being bounded below.

Writing the $e^{-tH}$ of quantum mechanics in this form is the Feynman–Kac formula (Kac 1949; equation (2) of 8-01, 2-07 §05) Known. Feynman's (1948) sum has the real-time weight $e^{iS/\hbar}$, and moving it to imaginary time gives this formula. The real-time weight does not become a measure.

A2 Functional integral ⟷ Euclidean probability measure

The physical word is a sum over all field configurations with weight $e^{-S}$, and the mathematical word is a probability measure on the space of configurations. They can be said to be the same because the side of mathematics defined $\int\mathcal DA\,e^{-S}$ as a countably additive probability measure (the judge is a definition; §4 of Jaffe's review).

Example: the free field is the Gaussian measure with covariance $(-\Delta+m^2)^{-1}$ ($\Delta$ is the Laplacian, not the $\Delta$ of the mass gap) (4-10 §02). Two-dimensional Yang–Mills theory has also been constructed (Driver 1989, 6-02). On the lattice, equation (9) of 4-02 is a genuine measure. The measure of four-dimensional Yang–Mills has not been constructed, and the space it rests on (a completion of the space of connections divided by gauge) has not yet been fixed either (3-04).

A3 Space of states ⟷ reconstructed Hilbert space

The physical word is Fock space (the space that lines up states by number of particles, 2-06), and the mathematical word is the Hilbert space (3) made from the measure. They can be said to be the same for free fields, and, with interactions, when asymptotic completeness (§05) holds.

Let $\mathcal E_+$ be the totality of functions determined only by fields at positive times, and $\Theta$ time reversal (4-10 §05).

$\mathcal N$ is the totality of functions of norm 0, and dividing by it and completing gives $\mathcal H$. (i) For free fields it agrees with Fock space (4-10 §07). (ii) With interactions, the representation of the canonical commutation relations at time 0 becomes inequivalent to the Fock representation ($\phi^4_3$, Jaffe §6). The Fock space of physics is the space of asymptotic particles, so agreement needs asymptotic completeness.

"Three faces" is a metaphor. More precisely, it means the values of the same correlation read at different places (real time, imaginary time, time-ordered) of a single analytic function (A5).

A4 Wick rotation ⟷ OS reconstruction theorem

The physical word is the computational procedure of setting $t=-i\tau$ and rotating the contour, and the mathematical word is the theorem that builds a Wightman theory from Schwinger functions. They can be said to be the same under the conditions $(\mathrm{E0}')$–$(\mathrm{E4})$, in the direction from imaginary time to real time.

Wick rotation is Wick's (1954) procedure (4-02 §02), and its mathematical partner is the reconstruction theorem, which has a direction Known.

A condition on the ordering of times is needed because $S_n$ does not change when its arguments are exchanged, while $W_n$ changes with the order in which the operators stand (the example in A5). Care is needed in the reverse direction. The Schwinger functions obtained from a Wightman theory need not satisfy $(\mathrm{E0}')$. The first OS paper (1973) claimed an equivalence, but the proof had an error, which Glaser (1974) fixed. The second (1975) states it in the form of a sufficient condition with $(\mathrm{E0}')$ added (the abstract has '(necessary and) sufficient', with parentheses).

A5 Green functions ⟷ Wightman functions and Schwinger functions

The physical word "Green function" refers to any of the time-ordered product $G_T$, the Wightman function $W$ and the Schwinger function $S$, while mathematics defines these three as separate functions. They can be said to be the same in the sense of the theorem that the three are values of one analytic function. Example: for an oscillator of angular frequency $\omega$ ($\hbar=m=1$),

$W$ is the boundary value on the imaginary axis, $S$ is the restriction to the real axis folded by $|\tau|$, and $G_T$ is glued together by sign (Wightman 1956, Symanzik 1966). For $\tau\gt0$, $W(-i\tau)=F(\tau)=S(\tau)$, which is the two-point case of (4). For $\tau\lt0$, extending the formula for $W$ as it is grows as $e^{\omega|\tau|}$, disagreeing with $S(\tau)$. $W\to S$ follows from positive energy alone, but $S\to W$ needs $(\mathrm{E0}')$.

A6 Källén–Lehmann representation ⟷ spectral measure

The physical word is the formula that writes a two-point correlation as a superposition over energies (masses), and the mathematical word is the spectral measure of the Hamiltonian. They can be said to be the same when $H$ exists by reconstruction (under the premise of (1)). The representation of Källén (1952) and Lehmann (1954) can be written, in the form of equation (6) of 4-11, as

$P_H(E)$ is the projection onto the part of $H$ with energy at most $E$ (the spectral theorem of 1-10 §05). The positivity of the weights comes from reflection positivity (Table 2 of 4-10). Example: (5) is the case where $\rho$ has weight at the single point $E=\omega$, and on the lattice $\rho_O=\sum_n|\langle n|O|\Omega\rangle|^2\delta_{E_n}$ (equation (8) of 8-01). The breaking of positivity of the gauge-fixed gluon propagator is treated in 7-02.

"Contents" is also a metaphor. More precisely, it means the particular vector (the vacuum) and operators (the Hamiltonian, the mass operator) fixed on the Hilbert space of §02.

A7 Vacuum and uniqueness ⟷ $\Omega$ and clustering

The physical word is "the state with nothing in it", and the mathematical word is $\Omega=[1]$, obtained in the reconstruction as the class of the constant 1. They can be said to be the same when the vacuum of physics is read as "the state of lowest energy" (a theorem under the premise of (1)).

Uniqueness means that the eigenvalue 0 of $H$ is simple (Jaffe §2), and on the Euclidean side it is paired with clustering (E4) Known.

Non-uniqueness of the infinite-volume limit, degeneracy of the ground state and phase transitions are the same phenomenon (Jaffe §7.2). Example: on the low-temperature side of the two-dimensional Ising model, the difference between the lowest two levels closes with the width (2-09 §08). The identification with the procedure of physics that builds the vacuum by switching on interactions slowly from the vacuum without interactions (attributed to Gell-Mann and Low) is not rigorously justified, since Haag's theorem (attributed to Haag 1955) denies in general the existence of the interaction picture (the way of writing that evolves in time by the interaction alone on the space of free fields).

A8 Hamiltonian ⟷ logarithm of the transfer matrix

The physical word is the operator that generates the advance of time (the energy), and the mathematical word is the logarithm of the transfer matrix $T$ that advances imaginary time by one step on the lattice. They can be said to be the same at the lattice level; agreement with the continuum $H$ is conditional on the continuum limit.

$a$ is the lattice spacing in the time direction. A positive self-adjoint $T$ was made by Lüscher (1977) (5-03 §05), and it connects to the $H$ of Kogut–Susskind (1975) by sending the time step to 0 (Creutz 1977, 5-03 §07) Known. It is a different thing from the generator $L$ of stochastic quantisation (8-03 §05).

A9 One-particle state ⟷ isolated eigenvalue of the mass operator

The physical word is a particle classified as an irreducible representation of the Poincaré group (Wigner 1939), and the mathematical word is an isolated eigenvalue of the mass operator $M$. They can be said to be the same in the reconstructed theory, when that eigenvalue is isolated in the spectrum (4-11 §05).

$\vec P$ is the momentum, $\sigma(\cdot)$ the spectrum, and $\Omega^\perp$ the part orthogonal to the vacuum. Jaffe writes that scattering theory assumes this and 'It entails both a lower gap and an upper gap' (§7.1; §5 of the problem statement also counts the upper gap among the extensions) Known. It has been shown for weakly coupled $P(\phi)_2$ (Glimm–Jaffe–Spencer 1974), and there is also a one-particle state at strong coupling on the lattice (Schor 1984). Example: for the free field the continuous spectrum above $m$ starts at $2m$. The glueball row is in 8-03 §04.

"Unitarity" in physics is used in two senses, so we split it into A11 (no states of negative norm) and A12 (probability conserved in scattering) (8-01 §06). We put the tool of scattering (A10) first.

A10 S-matrix ⟷ Haag–Ruelle scattering theory

The physical word is the amplitude from a set of particles in the far past to a set of particles in the far future, and the mathematical word is an operator built from two isometries. They can be said to be the same when there is an isolated one-particle mass (A9).

The mass hyperboloid is the surface $p^2=m^2$ in momentum space (on the side $p^0=\sqrt{\vec p^{\,2}+m^2}$), and the one-particle states of mass $m$ lie on it. An isometry is a map that preserves inner products, and $\operatorname{Ran}$ is the range (the totality of values of the map). $\mathcal F$ is the Fock space of asymptotic particles, the space that lines up, by number of particles, the states of particles flying freely in the far past (or future). The scattering theory of Haag (1958) and Ruelle (attributed to 1962) builds the isometries $W^{\rm in},W^{\rm out}:\mathcal F\to\mathcal H$ from an isolated mass hyperboloid.

The rightward direction of (10) is elementary. For an isometric $W$, $W^*W=1$, and $WW^*$ is the projection onto the range; if the ranges are the same, $W^{\rm out}(W^{\rm out})^*$ does not change the values of $W^{\rm in}$, so $S^*S=(W^{\rm in})^*W^{\rm out}(W^{\rm out})^*W^{\rm in}=(W^{\rm in})^*W^{\rm in}=1$, and $SS^*=1$ likewise Paper. Example: for the free field $W^{\rm in}=W^{\rm out}$ and $S=1$. Asymptotic completeness is the claim that the two ranges are the whole of $\mathcal H$. Within the range we searched, we found no theorem deriving asymptotic completeness in the usual sense in general. Dybalski–Gérard (2014) showed a "generalised" form under lower and upper mass gaps Known. There are partial results for individual models (unitarity of the S-matrix for weakly coupled $P(\phi)_2$ on even states with energy below about $4m$: Spencer–Zirilli 1976).

A11 Positive norm ⟷ reflection positivity

The physical word is that there are no states of negative norm (Wightman positivity), and the mathematical word is the reflection positivity of the Schwinger functions. They can be said to be the same within the correspondence between OS and Wightman, and this is a theorem.

Reflection positivity gives the Hilbert space and $H\ge0$, and is preserved in limits (Jaffe §4), so if a limit exists it can be carried over from the lattice. It holds for Wilson's lattice (Osterwalder–Seiler 1978, Lüscher 1977) Known. Example: negative weights of Pauli–Villars type break it (4-10 §08).

A12 Unitarity of the S-matrix ⟷ positive norm + asymptotic completeness

The physical word is that probability is conserved in scattering ($S^*S=1$), and the side of mathematics is A11 with the agreement of the ranges in (10) added. They can be said to be the same when asymptotic completeness holds.

For gauge theories, the abstract of Kugo–Ojima (1979) writes that "unitarity of the physical S-matrix" is shown in general ('quite generally') on the basis of representations of the algebra of the BRS charge (the $Q_B$ of §08). This is a general theory assuming representations of the algebra and asymptotic fields, and its stage is a space with indefinite metric (an inner product that allows negative norms). Those assumptions have been checked within perturbation theory.

A13 Spacetime and lattice ⟷ the $a\to0$ limit of $a\mathbb Z^4$

The physical word is four-dimensional spacetime with Minkowski signature ($\mathbb R^4$ after Euclideanisation), and the side of mathematics is the $a\to0$ limit of measures on the lattice $a\mathbb Z^4$ of spacing $a$. They can be said to be the same when the limit exists and the rotational symmetry $O(4)$ comes back.

The rotational invariance of Euclideanised spacetime is the $O(4)$ invariance of (E1) (for the signature, the existing article One sign is different). What remains on the lattice is rotations by 90° and reflections, and the abstract of Wilson (1974) writes of the strong-coupling limit 'no Lorentz (or Euclidean) invariance' Known.

Example: for the free field on the lattice, the difference between correlations along an axis and along a diagonal vanishes as $a^2$ (Table 2 of 4-02). For pure gauge theories, the continuum limit of the lattice has been shown for non-abelian groups in two dimensions and for U(1) in three and four dimensions (Driver 1989, Gross 1983, Driver 1987, among others). The abstract for three dimensions writes that with the Villain action it converges to the free electromagnetic field. The free electromagnetic field is invariant under rotations, so rotational symmetry has come back too.

A14 Regularisation ⟷ family of measures with lattice spacing $a$

The physical word covers all devices that temporarily stop infinities, and what the side of mathematics can treat as a family of measures is mainly the lattice. They can be said to be the same when the limit with the device removed does not depend on the choice of device (universality).

§6.5 of the problem statement writes that there are few ways of recovering the reflection positivity lost in dimensional regularisation, Pauli–Villars and the like. What Reisz (1989) shows is that the continuum limit of the lattice is finite at each order of perturbation.

The non-perturbative (12) has been shown in two dimensions. Chevyrev–Shen (2023, arXiv:2302.12160) showed on the two-dimensional torus, for connected compact groups, that lattice gauge theories with the Wilson, Villain and Manton actions give the same limit. Convergence of the two-dimensional lattice approximation is also in Driver (1989).

A15 Renormalisability ⟷ BPHZ, Epstein–Glaser

The physical word is that infinities vanish at each order of perturbation after adjusting finitely many constants, and the mathematical word is the finiteness theorems of a subtraction rule (BPHZ) and of an inductive construction from causality (Epstein–Glaser). They can be said to be the same at the level of formal power series (4-05).

In physics, Dyson (1949), and for Yang–Mills 't Hooft (1971) and Becchi–Rouet–Stora (1976). In mathematics, Bogoliubov–Parasiuk (1957), Hepp (1966), Zimmermann (1969) and Epstein–Glaser (1973).

The abstract of Epstein–Glaser writes that the inductive construction gives 'at each order, finite terms' and is equivalent to BPHZ Known. It does not include convergence of the sum (the series is thought to diverge, 4-12).

A16 Field $\phi(x)$ ⟷ operator-valued distribution

The physical word is an operator $\phi(x)$ at each point $x$, and the mathematical word is an operator-valued distribution, in which only $\phi(f)$, smoothed with a test function $f$, is an operator. They can be said to be the same when $\phi(x)$ is read as "a symbol for the smoothed value, just before the smoothing width is sent to 0".

Distributions are in 1-11 §04 (the reading of fields is Jaffe §3) Known. For the free field of mass $m$ ($\omega_p=\sqrt{\vec p^{\,2}+m^2}$),

Example: taking $f$ to be a bump of integral 1 and width $\varepsilon$, $f_\varepsilon(x)=\varepsilon^{-4}g(x/\varepsilon)$, we have $\hat f_\varepsilon(p)=\hat g(\varepsilon p)$, so in four dimensions the right-hand side diverges as $\varepsilon^{-2}$ (1-09). The point value $\phi(x)$ is the limit as $f$ approaches $\delta$, so as an operator it does not exist. The point fields of Yang–Mills are $\mathrm{Tr}\,F_{ij}F_{kl}(x)$, and §4 of the problem statement requires those operators. Footnote 1 of the problem statement says that a natural one-to-one correspondence between classical differential polynomials and operators 'does not exist', and expects a correspondence between the spaces of dimension $\le d$ (0-03 §08). Dimension here is the power under a change of the length scale (2 for $F$, 4 for $\mathrm{Tr}\,F^2$; 1-11 §06). A Wilson loop is a quantity on a curve, outside the axioms of point fields (4-10 §09).

A17 Gauge fixing ⟷ section of the orbit space

The physical word is choosing one representative from each gauge orbit by a condition $F(A)=0$, and the mathematical word is a section of the projection to the orbit space $\mathcal A/\mathcal G$ (a map that assigns continuously one point to each orbit); as definitions they are the same. On the four-dimensional sphere, however, it is a theorem that there is no global section.

The geometry of slices is in 3-04 §05. The source of the Faddeev–Popov (1967) rule is

(15) holds when the condition surface $F=0$ meets each orbit exactly once. (15) evaluates the determinant at the single point $g=1$ and takes it outside the integral, so if it meets many times the right-hand side becomes a sum of ratios of the determinants at each intersection, and is neither 1 nor the number of intersections. Putting the determinant inside the integral gives the number of intersections $N(A)$.

The actual Faddeev–Popov rule uses $\det$ without absolute value, so it counts intersections with signs (the detour in 4-08 §05). Gribov (1978) pointed out that "exactly once" breaks down in non-abelian theories, and the abstract of Singer (1978) writes, on $S^4$, 'no continuous choice of exactly one connection on each orbit can be made' Known. The lattice integrates with Haar measure (the uniform probability measure on the group, unchanged by multiplication), so no gauge fixing is needed and it avoids this row (5-01, 4-08).

A18 Physical states ⟷ reconstruction from gauge invariants

The physical word is states satisfying the Gauss constraint, or BRST cohomology, and the mathematical word is the space made by carrying out (3) with gauge-invariant quantities only. They can be said to be the same up to the lattice level; in the continuum it is Unsettled.

The Gauss constraint is the non-abelian version of Gauss's law $\nabla\cdot\vec E=0$, and on the side of states it is the condition "the state does not change under gauge transformations at fixed time" (Kogut–Susskind 1975, 4-01). On the lattice, $\psi(U^g)=\psi(U)$ for gauge transformations $g$ at fixed time. The BRST $Q_B$ is an operator satisfying $Q_B^2=0$ (the BRST charge), and $\operatorname{Ker}Q_B/\operatorname{Im}Q_B$ is "the states annihilated by $Q_B$, identified up to the range of $Q_B$". An indefinite metric is an inner product that allows negative norms.

On the lattice, transition probabilities between gauge-invariant states are not negative (Lüscher 1977), and gauge fixing does not change the Green functions of gauge singlets (operators unchanged by gauge transformations) (Creutz 1977) Known. In the continuum it has not been shown that the two sides of (16) are the same, and non-perturbative BRST remains hard to define because of Gribov copies.

A19 Local symmetry does not break ⟷ Elitzur's theorem

The physical word is the claim "gauge symmetry (local symmetry) does not break spontaneously", and the mathematical word is a theorem stating it as a limit in an external field. They can be said to be the same when compared in the order "put in an external field, make the volume infinite, then remove the field".

The abstract of Elitzur (1975) argues, in the form 'It is argued that', that spontaneous breaking of local symmetry is impossible in a symmetric gauge theory without gauge fixing, and says it is shown in a simple system of abelian gauge fields on a lattice Known. Rigorous successors include a proof, for an abelian Higgs model on the lattice, that the magnetisation becomes 0 when the external field is removed (De Angelis–de Falco–Guerra 1978) Known. If the external field is 0, the Haar average makes the expectation 0 (equation (9) of 7-02). Spontaneous breaking is a question of the order in which the external field is removed (2-08).

Comparing in the order of removing the external field

We write Elitzur's argument out for the weight $\exp(\beta\sum_pU_p+h\sum_\ell U_\ell)$ of $\mathbb Z_2$ lattice gauge theory in $d$ dimensions (link variables $U_\ell=\pm1$, with $U_p$ the product of the four links round a plaquette). A gauge transformation at one point $x$ flips the $2d$ links touching $x$, and does not change $\sum_pU_p$. So for $\ell$ touching $x$,

Paper. The right-hand side depends neither on the volume nor on $\beta$. In the Ising model (global $\mathbb Z_2$) all spins must be flipped, and the upper bound from the same argument is $\tfrac12(e^{2h|\Lambda|}-1)$, which loses its meaning as the volume grows. The only difference is whether one symmetry transformation touches finitely many variables or all of them. The slope at $h=0$ is

On the left, the term with $\ell'=\ell$ is 1. The terms with $\ell'\ne\ell$ are 0 because, gauge-transforming at the endpoint of $\ell$ that does not touch $\ell'$, only $U_\ell$ of the two changes sign and $U_{\ell'}$ does not (in a box of side at least 3 such an endpoint always exists). The right is on the low-temperature side, where even distant spins keep the correlation $M_0^2$, so it is proportional to the volume.

ComputedThe numbers in this figure were computed on this machine (exact values by diagonalising the transfer matrix).

The Ising columns rise with $L$, and $\langle\sigma\rangle/(hL^2)$ at $h=10^{-4}$ is 0.9529, 0.9505, 0.9494, 0.9489, approaching 0.9479, the square of the formula $M_0=(1-\sinh^{-4}2\beta)^{1/8}$ cited for the spontaneous magnetisation (attributed to Yang 1952) Computed. In the gauge column, for $h\le10^{-2}$, $L=4$–10 agree to 6 decimal places, and the slope is the 1 of (18). The "breaking of gauge symmetry" of physics is said to be reread as, for example, the breaking of a global symmetry after gauge fixing Physics; taken literally it is a false friend (8-01).

Detour: gauge theory at β = ∞ is the Ising model

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