2026-10-01 · chapter Yang–Mills and the mass gapPart 0 EntranceEuclidean space and spacetime

Four dimensions in mathematics and four dimensions in physics — the same R⁴ has two rulers, and the problem asks for a theory that works with both

The problem statement (the Clay Mathematics Institute's description by Jaffe–Witten) asks to show that for any compact simple group $G$ a non-trivial quantum Yang–Mills theory exists "on $\mathbb R^4$" and has a mass gap $\Delta\gt0$ (a word-by-word reading is in 0-03). For this $\mathbb R^4$, the way of measuring length is not written.

The way of measuring appears in the surrounding sections. §3 introduces the axioms as 'Gårding and Wightman gave mathematically precise axioms for quantum field theories on R4 with a Minkowski signature', and §6 adds '(Equivalently in the Euclidean formulation one replaces Euclidean space-time R4 by R2 or R3.)'. Existence includes 'axiomatic properties at least as strong as those cited in [45, 35]', where [45] is Streater–Wightman (the axioms on the side of spacetime) and [35] is Osterwalder–Schrader (the axioms on the Euclidean side) Known. One can say that the same $\mathbb R^4$ has two rulers. To be exact, the set of points is the same, and what differs is the quadratic form that measures squared length ($c=1$).

Example. Reading the last coordinate of the point $(1,0,0,1)$ as time, $Q_{\mathrm E}=2$ but $Q_{\mathrm M}=0$, and it is joined to the origin exactly by light. $(0,0,0,1)$ has $Q_{\mathrm E}=1$, $Q_{\mathrm M}=-1$ (timelike). There is also the convention of writing the signature as $(+,-,-,-)$; the content is the same.

Fixing a ruler fixes the group of transformations that preserve it. $SO(4)$ is the group of $4\times4$ real matrices of determinant 1 that leave $Q_{\mathrm E}$ unchanged, and $SO(3,1)$ the one that leaves $Q_{\mathrm M}$ unchanged. The former is the rotation group of four dimensions and the latter the Lorentz group; together with translations they are the Euclidean group and the Poincaré group. Both have one direction for each of the $\binom42=6$ coordinate planes (What happens only in four dimensions). The difference is in the 3 planes that contain time, where rotations become boosts in spacetime (2-04).

This is because $\cos(i\varphi)=\cosh\varphi$ and $\sin(i\varphi)=i\sinh\varphi$ Known. This $\varphi$ is called the rapidity. A point with $x'=0$ in the lower formula of (2) moves along $x=t\tanh\varphi$, so the velocity of the boost is $v=\tanh\varphi$. Example. Putting $\varphi=0.7$, $(t,x)=(0.4,\,1.3)$ into the left gives $t'=-0.484091,\ x'=1.328286$, agreeing with the right Computed. The angle of a rotation goes once round in $2\pi$, but the rapidity has no upper bound (as $\varphi\to\infty$, $v$ only approaches 1). Restated in terms of quadratic forms, $SO(4)$ can move a point only on a sphere where $Q_{\mathrm E}$ is constant, but $SO(3,1)$ can carry it arbitrarily far on the hyperboloid $Q_{\mathrm M}=-1$. That $SO(4)$ is compact and $SO(3,1)$ non-compact is an expression of this difference (One sign is different §01).

The article What happens only in four dimensions looked at Euclidean $\mathbb R^4$ and its topology, and One sign is different at spacetime. Not every case matters for the problem. What matters most on the Euclidean side is that the 6 planes form 3 pairs of "perpendicular planes", so that the field strength $F$ splits in two (1-12). The field strength is the generalisation to Yang–Mills (3-02) of $F_{\mu\nu}$, which combines the electric and magnetic fields of electromagnetism (2-03), and it has one component for each coordinate plane. In electromagnetism, the 3 components of the electric field correspond to the 3 planes containing time, and the 3 components of the magnetic field to the 3 purely spatial planes. Writing the operation that moves the component of each plane to the perpendicular plane as the Hodge star $*$, the way the planes are divided becomes exactly the way $F$ is divided.

As examples, we list how the cases show up in the problem.

How to read the table. The top three rows (E) matter as tools and models in Part 3 (classical Yang–Mills) and Part 5 (the lattice). The fourth row (E, M) is used in sections 04–07 of this chapter to join the two four dimensions. Of the M rows, the light cone enters the axioms of Part 4 (4-09), and the gravity row is outside the problem.

The "four" that matters in the problem is not a phenomenon peculiar to four dimensions like exotic structures, but the four at which the dimension of the coupling constant vanishes (section 08). However, the proof of the existence of exotic $\mathbb R^4$ uses, together with Freedman's topological theorem, the same Yang–Mills equations (the space of their anti-self-dual solutions) (1-12, 3-06). Classical theory uses the solutions of the equations as a tool for seeing topology; quantum theory uses the action as a probability weight.

The first tool for going back and forth between the two four dimensions is to replace time $t$ by an imaginary number (Wick rotation; the name is said to come from Wick 1954; for the form in field actions, 4-02 §02 is the authoritative place). Putting $t=-i\tau$ into (1) turns $Q_{\mathrm M}$ into $Q_{\mathrm E}$. What changes can be seen in time evolution in quantum mechanics.

A state is a vector of length 1 in a Hilbert space (a complex vector space with an inner product, closed under limits; 1-10), and we assume it can be expanded in the eigenvectors $\psi_n$ (eigenvalues $E_n$) of the energy operator $H$ (the Hamiltonian). The component with energy $E_n$ turns with phase $e^{-iE_nt}$ ($\hbar=1$). Taking the lowest state as the vacuum $\Omega$ (energy 0) and writing $O\Omega=\sum_nc_n\psi_n$ for the vector obtained by applying an observable $O$, the correlation across a time $t$ is

$\vartheta$ is the angle by which time is rotated in the complex plane: $\vartheta=0$ is real time and $\vartheta=90^\circ$ imaginary time. If every $E_n$ is at least 0, then rotating $t$ into the lower half-plane keeps each term no greater than 1, and at $\vartheta=90^\circ$ oscillation turns into decay $e^{-E_n\tau}$. If there were a negative $E_n$, that term would grow and the sum would lose its meaning. §3 of the problem statement writes 'Wightman used positive energy to establish analytic continuation of expectations of Minkowski field theories to Euclidean space.' Known.

Example. We draw the connected part $C_{\mathrm{conn}}$, with the vacuum term removed, for a model with $E_n=1,\,1.7,\,2.6$ and weights $0.6,\,0.3,\,0.1$. The gap is $\Delta=1$.

ComputedThe numbers in this figure were computed on this machine (the sum (4) computed in the browser; Table 2 by python3). "Negative energy" is a term with $E=-0.3$ and weight 0.03.

In real time it is still 0.94 at $\tau=8$, and the correlation does not decrease. The more it is rotated, the faster it decreases; with negative energy added, the more it is rotated, the more it grows. The upper bound $|C_{\mathrm{conn}}|\le e^{-\Delta\tau\sin\vartheta}$ follows from the triangle inequality applied to each term of (4), since the weights sum to 1 and $E_n\ge\Delta$ Known. It was also checked at all points of a fine grid in $0\le\vartheta\le90^\circ$ and $0\le\tau\le12$ Computed.

Moving to imaginary time replaces the vocabulary of quantum theory with the vocabulary of probability. §3 of the problem statement sums up that Symanzik read Euclidean expectations as 'a statistical mechanical ensemble of classical Markov fields [46], with a probability density proportional to exp(−S)', and that Osterwalder and Schrader gave a general theory of the equivalence of the two axiom systems Known (the exact direction and conditions of the theorem are in 4-10).

The $e^{iS}$ on the left ($S$ is the action and $\phi$ the field; 2-01, 2-02) oscillates with size 1 and is not a probability. The $e^{-S_{\mathrm E}}$ on the right ($S_{\mathrm E}$ is the action moved to imaginary time) is positive, and when normalised can be a probability weight (2-07, 4-02). This is why on the mathematical side Yang–Mills becomes "the problem of constructing a probability measure on the space of connections" (a connection is the gauge field $A$; for details, 1-07). As examples, we list the main rows of the correspondence.

Every row finally passes through the single point "with reflection positivity, one can get back to a Hilbert space".

Detour: self-duality lives only in imaginary time

On the Euclidean side there is only a probability measure (expectation $\mathbb E$), with neither a Hilbert space nor a Hamiltonian. To recover them an inner product is needed, and the material is the mirror of time $\Theta:\tau\mapsto-\tau$. Let $X,Y$ be a quantity determined only by the fields on the side $\tau\gt0$, and $\Theta X$ the result of reversing the sign of every time in $X$ (a quantity on the side $\tau\lt0$), and set

The second half is reflection positivity (also called OS positivity). If it holds, $\langle\cdot,\cdot\rangle$ becomes an inner product (after identifying vectors of length 0), and the operation of shifting by $s$ in the direction of $\tau$ acts as $e^{-sH}$ ($H\ge0$) Known. §6.5 of the problem statement writes 'Establishing a quantum mechanical Hilbert space is part of the solution to this Millennium problem.'

Example. Put a particle rotating on a circle (moment of inertia $I$, angle $\theta$) on an imaginary-time lattice with step $a$. In this chapter $\theta$ is used for the angle of this particle (it is different from the angle of the $\theta$ term in 3-05 and elsewhere). Writing the angle at time $ka$ as $\theta_k$ and approximating one step of the action $\int\frac I2\dot\theta^2d\tau$ by $\frac Ia\bigl(1-\cos(\theta_{k+1}-\theta_k)\bigr)$, the weight is

$J$ is the strength with which the angles at neighbouring times are pulled into line, and in statistical mechanics this chain of weights is called the one-dimensional XY model (the vocabulary of partition functions and transfer matrices is in 2-08 and 2-09). $I_n$ is a modified Bessel function, and since the coefficients of its power series are all positive, $I_n(J)\gt0$ Known. The transfer matrix $T$ is the operator whose integral kernel is the weight of one step.

Substituting the expansion (8), $T$ maps $e^{in\theta}$ to itself with eigenvalue $2\pi I_n(J)$ (agreeing to 12 digits with a numerical computation that divides $\theta$ into 256 points Computed). If the plane of reflection is placed between lattice points, the left-hand side of (7) becomes a quadratic form sandwiching $T$, and since the eigenvalues are positive it is at least 0. In this way $H_a=-\frac1a\log\bigl(T/2\pi I_0(J)\bigr)\ge0$ is determined. Reflection positivity in Wilson's lattice gauge theory was shown by Osterwalder–Seiler 1978 (for the construction of the transfer matrix, also Lüscher 1977; 5-03).

The definition of the mass gap is on the side of spacetime; §4 of the problem statement writes 'A quantum field theory has a mass gap ∆ if H has no spectrum in the interval (0, ∆) for some ∆ > 0.' ($H$ is the Hamiltonian; the spectrum $\operatorname{spec}H$, in finite dimensions, is the set of eigenvalues). On the Euclidean side the same thing appears as decay of correlations. We call this the two faces of the mass gap. To be exact, they are a condition on the spectrum of the Hamiltonian and exponential decay of connected correlations (Schwinger functions with the vacuum contribution subtracted; Table 3). Setting $\vartheta=90^\circ$ in (4),

If $O$ overlaps with the lowest excitation, $-\frac{d}{d\tau}\log C_{\mathrm{conn}}$ approaches $\Delta$; in the model of Figure 1 it was 1.178, 1.086, 1.021, 1.001 at $\tau=1,2,4,8$ Computed. Measuring the masses of glueballs (candidates for excitations corresponding to Yang–Mills particles) on the lattice uses this face (5-08). In Euclidean space time and space are carried into each other by rotations, so decay in the direction of $\tau$ is also decay in spatial directions, which leads to the clustering of §5 of the problem statement Known. A local observable (a quantity determined only by the fields in a bounded region) rotated is again a local observable, so decay at the same rate holds in every direction. But this is a matter for after the reconstruction has given a Hilbert space and a Hamiltonian.

Example. Take $O=e^{i\theta}$ for the particle of section 06. The eigenvalues of $H=-\frac1{2I}\frac{d^2}{d\theta^2}$ are $\frac{n^2}{2I}$, the gap is $\Delta=\frac1{2I}$, and $O\Omega$ has only components with $n=1$, so $C_{\mathrm{conn}}(\tau)=e^{-\Delta\tau}$. On the lattice, an infinitely long chain gives $\langle e^{i(\theta_k-\theta_0)}\rangle=(I_1/I_0)^k$, and the rate of decay read as $\tau=ka$ is $\Delta(a)=-\frac1a\log\frac{I_1(J)}{I_0(J)}$. This is the gap on the lattice measured in physical units (the smallest positive eigenvalue of $H_a$ in section 06), and in lattice units it is $a\,\Delta(a)$ (the $m_{\mathrm{lat}}$ of Appendix A-01).

ComputedThe numbers in this figure were computed on this machine ($I_1/I_0$ and $\Delta(a)$ are the values of Table 4 computed with python3 and SciPy; the figure only joins the points $(ka,\,(I_1/I_0)^k)$).

On coarse lattices ($J\le2$) $\Delta(a)/\Delta$ is 1.4–1.6, a different number from the gap. Making the lattice finer brings it to 1 at almost the same rate as $1+\frac1{2J}$, and the second level approaches too. The correlation computed from products of transfer matrices on a chain of period 400 also agreed with $(I_1/I_0)^k$ to 10 digits Computed. In this example the existence of a gap is clear from the start. What is needed for four-dimensional Yang–Mills is that $\Delta(a)$ stays positive in the limit where the decay seen in lattice units $a\,\Delta(a)$ goes to 0, and, before that, to construct the Hilbert space of the limit (5-09).

The counting of dimensions is done on the Euclidean side, over $d$ equivalent coordinates. With $c=1$, time too is measured as a length, so 3+1-dimensional spacetime is $d=4$. Write the $d$-dimensional Euclidean action as $S_d[A]=\frac1{2g^2}\int_{\mathbb R^d}\mathrm{tr}\,F_{\mu\nu}F_{\mu\nu}\,d^dx$ (summed over $\mu,\nu$). This is the form of the action in Appendix A-01; here we take the convention that puts the coupling $g$ in front of the action and does not include it in $F$. With $\hbar=c=1$, writing the mass dimension as $[\,\cdot\,]$ (for the derivation, 2-04 and 1-12),

and only in four dimensions does the classical action have no scale of length. A mass cannot be built from classical constants, so the gap is expected to be a constant multiple $\Delta=C\Lambda$ ($C$ a dimensionless constant) of a scale $\Lambda$ born from quantum effects Physics. This is what lies behind 'the mass gap is not classically visible' in §6.6 of the problem statement (4-07). A figure that moves the dimension $d$ continuously and sets it beside other quantities that switch at four is the dimension slider in 1-12 §10.

Example. In three dimensions $[g^2]=1$, so $g^2$ itself becomes a mass scale, and by this counting it falls on the easy side. Chevyrev (arXiv:2202.13359 §1.1) also writes 'The cases d = 2, 3 are considered substantially simpler than d = 4', but in the same section the construction of the three-dimensional measure 'even on T3, remains open' Known. In two dimensions the continuum measure has been constructed; in four it has not (where things stand by dimension is in 6-10, misconceptions in 9-05).

For the particle on a circle the dictionary could be used as it is. In this chapter we call the way back the road that starts from a theory constructed on the Euclidean side and follows the dictionary in reverse to a theory on the side of spacetime. To be exact, the way back means obtaining, from a Euclidean probability measure by the OS reconstruction, a theory satisfying the Wightman axioms (one with a Hilbert space and $H\ge0$). For Yang–Mills there are three barriers along it: observables, regularisation, volume. The whole road, set out, is as follows.

The regularisation at the left end puts a minimal length into the theory (the lattice step $a$ or the like) so that infinities do not appear. $T^4$ is a four-dimensional box with each side joined periodically, to keep the volume finite. For four-dimensional non-abelian Yang–Mills, none of the arrows has yet been shown. §6 of the problem statement also writes of the construction on $T^4$ 'while the construction is not complete'. In two dimensions the road goes through from the lattice to the continuum measure on the plane $\mathbb R^2$ (6-02). The three barriers are as follows.

Observables. Only gauge-invariant quantities have meaning, that is, quantities whose values do not change under gauge transformations (the operation of choosing a group element at each point and replacing the field; 3-01). Local ones are composite quantities such as $\operatorname{Tr}F_{ij}F_{kl}(x)$ (§4 of the problem statement). Chatterjee (arXiv:1803.01950 §1), citing Seiler, writes 'the key observables in these theories do not take values at points, but at curves', and Driver 1989 says that Wilson loops are 'quite singular in d = 3 and especially d = 4' Known. This barrier bears on every stage of (11).

Regularisation. The barrier at the left end of (11). Wilson's lattice preserves reflection positivity but reduces rotations to steps of 90°. §6.5 of the problem statement writes that there are few methods for recovering reflection positivity lost through regularisation (quoted in 0-03 §03), and Magnen–Rivasseau–Sénéor 1993, which uses a continuum regularisation, too stops at considering that OS positivity (reflection positivity) can be shown with additional work Known.

Volume. The barrier at the second arrow of (11). Even on $T^4$, if there is reflection positivity, a finite-volume Hilbert space and transfer matrix are obtained, but a theory with Poincaré symmetry on $\mathbb R^4$ needs $T^4\to\mathbb R^4$. §6 of the problem statement writes 'Nor do present methods suggest how to obtain the existence of the infinite volume limit T4 → R4.' Known.

Example. For non-abelian Yang–Mills in three and four dimensions, a unique non-trivial continuum limit has not yet been shown (in three dimensions it is not constructed even on $T^3$; section 08). The final "?" of (11) is the body of the mass gap, but the gap is not a single arrow left over at the end. §6.5 of the problem statement writes 'New ideas are needed to prove the existence of a mass gap that is uniform in the volume of space-time', and goes on to expect that if a gap independent of the volume (uniform however large the box is made) were shown, it would become possible to study the $T^4\to\mathbb R^4$ limit Known. So the barriers of the gap and of volume are entangled (6-11, 9-02).

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Next chapter: 1-01 Vector spaces and linear maps. This chapter continues in 2-04, 4-02, 4-10 and 4-11.

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