2026-10-01 · chapter Yang–Mills and the mass gapPart 5 Lattice gauge theoryWilson's lattice

Wilson's lattice — put group elements on the edges, and gauge invariance survives without approximation

Euclideanised quantum Yang–Mills theory (4-02) formally integrates $e^{-S[A]}$ over all connections $A$. But there is no uniform measure on an infinite-dimensional space (1-09 §02), and the integral counts multiple times the $A$ related by gauge transformations (3-04, 4-08). Wilson replaced spacetime by a lattice. This is the abstract of the 1974 paper.

“It is shown how to quantize a gauge field theory on a discrete lattice in Euclidean space-time, preserving exact gauge invariance and treating the gauge fields as angular variables (which makes a gauge-fixing term unnecessary).”

"Angular variables" means holding the field as group elements rather than as elements of the Lie algebra. The problem statement (Jaffe–Witten §6.5) writes that, for compact groups and compactified spacetime, “the lattice approximation reduces the functional integration to a finite-dimensional integral”, and continues that one must check that the expectation values of gauge-invariant observables exist in the limit of lattice spacing 0 and infinite volume Known. We look at the limits in §08.

Conventions. Euclidean $\mathbb R^4$, indices 1 to 4, $\hbar=c=1$. As in 3-02, $A_\mu=A^a_\mu T^a$ (Hermitian, $\mathrm{tr}\,T^aT^b=\tfrac12\delta^{ab}$), $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu-ig[A_\mu,A_\nu]$, and the continuum action is $S=\tfrac12\int\mathrm{tr}\,F_{\mu\nu}F_{\mu\nu}\,d^4x$ (summed over all indices). $\beta$ is the inverse lattice coupling (a different quantity from the β function $\beta(g)$ of 4-06), and $\theta$ is just an angle (not the $\theta$ term).

Detour: the summer of 1973

On the lattice $a\mathbb Z^4$ of spacing $a$ (in finite volume, a periodic lattice with $L$ points per side), the edge joining $x$ and $x+a\hat\mu$ is called a link. In the continuum theory the gauge field was a tool for comparing frames at neighbouring points (3-01). On the lattice the tool itself becomes the variable: on each link we put $U_{x,\mu}\in G$ and read it as "carrying the value at $x+a\hat\mu$ into the frame of $x$". A gauge transformation, by $\Omega(x)\in G$ at each point, acts as follows.

If a matter field transforms as $\psi\mapsto\Omega\psi$, the difference $U_{x,\mu}\psi(x+a\hat\mu)-\psi(x)$ changes only by $\Omega(x)$ (the four-direction version of the chain of 3-02). From a smooth $A$, the link is the inverse of parallel transport (1-07) along the edge. Divide the edge into $n$ equal parts and multiply the small rotations from the left, starting from the initial end.

(1) is the transformation law $A'=\Omega A\Omega^{-1}-\tfrac ig(\partial\Omega)\Omega^{-1}$ of 3-02 accumulated along the edge Known. Example: for U(1), $U=e^{-i\theta}$, and if $\Omega(x)=e^{i\alpha(x)}$, (1) is $\theta\mapsto\theta+\alpha(x+a\hat\mu)-\alpha(x)$ (2-03 §03). For SU(2), $U=u_0+i\,\mathbf u\cdot\boldsymbol\sigma$ ($u_0^2+|\mathbf u|^2=1$), each link is a point of $S^3$, and $\tfrac12\mathrm{tr}\,U=u_0$.

A periodic lattice has $4L^4$ links, and the configuration space $G^{4L^4}$ is a compact manifold of real dimension $4(N^2-1)L^4$ (1-05). The gauge freedom $G^{L^4}$ is a quarter of it, the same proportion as removing 1 gauge component from the 4 components in the continuum.

A single link changes sandwiched between the $\Omega$ at its two ends, but when two links are joined end to end, the $\Omega^{-1}\Omega$ at the joining point cancels. Going round a closed path, only the $\Omega$ at the starting point remains, on both sides. The smallest closed path is a square of side $a$: the plaquette.

So $\mathrm{tr}\,U_P$ is invariant without approximation, whatever the lattice spacing and however rough the configuration. For a general closed path $C$, the trace of the product taken in the order of the path,

is called a Wilson loop. Made from a smooth field via (2), it approaches the continuum Wilson loop (the trace of the holonomy of 1-07) as the lattice path approaches a closed curve (Chatterjee 2018 §4) Known. Rectangular loops are used to test for confinement (5-02, 7-02). The order of multiplication matters. Multiplying the same four links in the order $U_{x,\mu}U_{x+a\hat\nu,\mu}^{-1}U_{x+a\hat\mu,\nu}U_{x,\nu}^{-1}$, factors such as $\Omega(x+a\hat\nu)^{-1}\Omega(x+a\hat\mu)$ get stuck between them and do not cancel. For U(1) the order does not matter (2-03).

ComputedThe numbers in this figure were computed on this machine. The browser computes the same configuration with the same formulas as Table 2 (if JavaScript is disabled, Table 2 gives the content).

ComputedApplying a random gauge transformation to a random configuration on a four-dimensional periodic lattice with 4 points per side, the change in the action of §04 was $2\times10^{-13}$ for SU(2) and 0 for SU(3). Building the sum of squares of the differences $\partial_\mu A_\nu-\partial_\nu A_\mu$ from the logarithms of the links (elements of the Lie algebra), for SU(2) 34,707 changes to 35,659. Even with the commutator added, for finite gauge transformations the transformation law of $A$ is inhomogeneous (the term $-\tfrac ig(\partial\Omega)\Omega^{-1}$ below (1)), so the differences are not exactly invariant. This is the meaning of "group elements rather than Lie algebra variables" in the recollection in the detour.

The sum takes every plaquette once ($6L^4$ of them on a periodic lattice). There are five properties Known.

1. Gauge invariant: each term is unchanged by (3).
2. Real: reversing the orientation leaves $\mathrm{Re}\,\mathrm{tr}\,U_P^\dagger$ the same.
3. Bounded: the eigenvalues have absolute value 1, so each term is between 0 and 2. The maximum is 2 for SU(2) ($U_P=-1$) and $3/2$ for SU(3) (all three eigenvalues $e^{2\pi i/3}$, or all three $e^{-2\pi i/3}$) Computed.
4. Minimum: $S_W=0$ only when all $U_P=1$. A pure gauge $\Omega(x)\Omega(x+a\hat\mu)^{-1}$ is an example. On a periodic lattice, so is a configuration with commuting elements $V_\mu$ placed uniformly in each direction. If its product once round, $V_\mu^L$, is not 1, it is not a pure gauge, because for a pure-gauge configuration the product once round is $\Omega(x)\Omega(x)^{-1}=1$.
5. Local: each term depends on four links only.

Chatterjee (2018) writes $\beta'\sum_P\mathrm{Re}\,\mathrm{tr}(I-U_P)$, so $\beta'=\beta/N$ (the same holds for the $\beta$ of SU(2) in 6-02).

Map a smooth $A$ to links via (2) and look at $a\to0$. Expanding (3) with the Baker–Campbell–Hausdorff formula $e^Xe^Y=e^{X+Y+\frac12[X,Y]+\cdots}$, which combines a product of exponentials into one, gives at the centre $c$ of the plaquette

The difference of $A$ on opposite edges gives $-iga^2(\partial_\mu A_\nu-\partial_\nu A_\mu)$, and the commutator gives $(-iga)^2[A_\mu,A_\nu]$; together they make the field strength (curvature = the shift after going once round a small loop; 1-07). Correspondence with the notation of 1-07: the anti-Hermitian connection and curvature of 1-07 correspond to $-igA$ and $-igF$ of this chapter, and parallel transport in 1-07 carries from the start to the end, so the links and plaquettes of this chapter are its inverses. This is why the sign in the exponent of (6) is opposite to 1-07's $U_p\approx\exp(-a^2F)$; the value of the trace does not change. For Hermitian $X$, $\mathrm{Re}\,\mathrm{tr}\,e^{-iX}=\mathrm{tr}\cos X=N-\tfrac12\mathrm{tr}\,X^2+\cdots$, so

Taking the centre of the expansion at the centre of the plaquette, the $a^5$ terms cancel by the symmetry of orientations, and the remaining error is $a^6$ per plaquette Known. Summing $a^{-4}$ plaquettes, the relative error is $a^2$, and this is the term that Symanzik improvement (5-05) removes. Counting plaquettes by a point $x$ and a pair $\mu\lt\nu$ and replacing $\sum_xa^4$ by an integral,

It agrees with the continuum action $S$ when $\beta=2N/g^2$, and this is where the $\beta$ in (5) comes from Known. Chatterjee (2018) §3 does the same calculation and writes that in four dimensions $\beta$ is classically independent of $a$, that in the quantum theory physicists believe $\beta$ should grow like $\log(1/a)$ though there are also doubts, and that “the question remains an open mathematical problem” Physics (5-05).

Converting conventions — five ways of writing one coupling

The same coupling $g^2$ is written with different letters and different normalisations in different references. This is why the same condition looks like different numbers in different chapters. The standard of this book is Wilson's $\beta=2N/g^2$ in (5) (the table of notation A-01 §06, where Table 7 lists six ways of writing it, adding $g^2$, with the same numerical examples). The other ways of writing are fixed by comparing term by term with the action (5).

$\beta_{\mathrm{Ch}}$ is the form writing the action as $\beta_{\mathrm{Ch}}\sum_P\mathrm{Re}\,\mathrm{tr}(I-U_P)$ (without $1/N$; (3) of 6-10), $\beta_{\mathrm{SZZ}}$ is 't Hooft's convention writing the inverse coupling in front of the action as $N\beta_{\mathrm{SZZ}}$ (Shen–Zhu–Zhu; 5-04), $\lambda$ is the 't Hooft coupling of large $N$ (7-04), and $\alpha$ is the coupling strength used in physics. Differences in the constant term of the action disappear in the normalisation $Z$ and do not change the measure.

Example: the Shen–Zhu–Zhu condition is $1/48$ independent of $N$ when written in $\beta_{\mathrm{SZZ}}$, $\lambda\gt96$ when written in $\lambda$, and $N^2/48$ when converted to Wilson's $\beta$. That $N$ disappears in $\beta_{\mathrm{SZZ}}$ and $\lambda$ is the advantage of 't Hooft's convention. The SU(2) value $\beta=2.5$ often used in numerical work is $\alpha\approx0.13$, and the edge of the condition is $\alpha\approx3.8$; the two are about 30 times apart. The conversion is this book's arithmetic, and the conditions themselves are to be read in the normalisation of each reference (5-04, 5-09, 6-10).

Looking at a single plaquette

For the constant SU(2) field $A_1=c\,T^1$, $A_2=c\,T^2$ there are no derivative terms, and $F_{12}=gc^2T^3$ comes only from the commutator (Example 2 in 3-02 §04). The links are $e^{-i\theta\sigma_1/2}$ and $e^{-i\theta\sigma_2/2}$ ($\theta=gac$), and the plaquette is the group commutator $U_1U_2U_1^{-1}U_2^{-1}$. We multiply in the form $u_0+i\,\mathbf u\cdot\boldsymbol\sigma$ of §02. From the product of Pauli matrices $\sigma_j\sigma_k=\delta_{jk}+i\epsilon_{jkl}\sigma_l$,

Putting $c=\cos\frac\theta2$ and $s=\sin\frac\theta2$, $U_1=(c;\,-s,0,0)$ and $U_2=(c;\,0,-s,0)$ (writing $u_0$ and $\mathbf u$ side by side), $U_1U_2=(c^2;\,-cs,-cs,-s^2)$ and $U_1^{-1}U_2^{-1}=(U_2U_1)^{-1}=(c^2;\,cs,cs,-s^2)$. The $u_0$ of the product of the two is $\tfrac12\mathrm{tr}\,U_P$, so

and the following closed form comes out.

For small $\theta$ the ratio is $1-\theta^2/6+\cdots$, and the relative error is proportional to $a^2$. For large $\theta$ the lattice value does not exceed 2, and returns to 0 at $\theta=2\pi$. The link sees the field only in the form $e^{-igaA}$, so it does not distinguish a field that is too strong from a weak one. This boundedness is one reason why the integral in §06 is finite.

ComputedThe numbers in this figure were computed on this machine. The curves are (9) (differing from the matrix calculation by at most $2\times10^{-16}$). If JavaScript is disabled: the ratio (lattice / continuum) is 0.998, 0.959, 0.845 and 0.501 at $\theta=0.1,\,0.5,\,1,\,2$, and the lattice value is at its maximum 2 at $\theta=\pi$ and 0 at $2\pi$.

Looking at a smooth four-dimensional field

We compare using a smooth SU(2) field on the torus $[0,1)^4$ (a sum of trigonometric functions with 8 wave numbers, $g=1$, $\beta=4$). The links truncate (2) at $n=16$, and since the continuum action is a trigonometric polynomial, it was integrated exactly as a sum over finitely many points ($S=184.2275$).

The difference is proportional to $a^2$, with coefficient near −5.06 for this field (the value depends on the field). Changing $n$ to 8, 16, 32 changes $S_W$ by 0.03 and 0.008 at $L=8$. The $\|U_P-e^{-iga^2F(c)}\|/a^2$ of (6) is 0.190, 0.119, 0.065 and 0.034 from $a=1/8$ to $1/64$, nearly halving each time $a$ is halved (ratios from 1.6 to 1.9), shrinking in proportion to $a$. With the sign of the commutator reversed it stayed at 5 to 6.5 Computed.

The quantum theory on a finite lattice is defined by the following probability measure (1-08 §01).

$dU_\ell$ is the Haar measure of $G$ (the measure of total weight 1 unchanged by multiplication from either left or right; for SU(2), the uniform distribution on $S^3$). It is finite: $0\lt e^{-S_W}\le1$ is continuous and the domain is compact, so $0\lt Z\le1$, and the boundedness of §04 also gives $Z\ge e^{-12\beta L^4}$ ((3) of 6-10) Known. For SU(3) with 4 points per side it is an ordinary multiple integral over 8,192 real dimensions.

No gauge fixing needed: Haar measure is unchanged by left and right multiplication, so moving a configuration by (1) changes neither $\prod dU_\ell$ nor $S_W$, and $\langle f\rangle=\langle f\circ\Omega\rangle$. In the continuum, gauge fixing was needed to choose representatives from orbits of infinite volume (4-08, 3-04), but on the lattice the whole of the gauge transformations $G^{L^4}$ is compact with weight 1, so the multiple counting is merely averaged. This is what "making the gauge-fixing term unnecessary" in the abstract means Known.

There are two consequences. (i) For general $f$, $\langle f\rangle$ equals the expectation value of $f$ averaged over gauge transformations. In particular, quantities that change non-trivially under gauge transformations (a single link $U_\ell$ or its components, $\mathrm{tr}\,U_\ell$, and so on) are cancelled by the Haar average and have expectation value 0 (the detour of 7-02; that this survives in infinite volume is Elitzur's 1975 theorem). Not every gauge-non-invariant quantity vanishes: the expectation value of $|\mathrm{tr}\,U_\ell|^2$ is positive. (ii) The links on a tree (a set of links containing no cycles) may be set to 1 (6-02 §02). Typical observables are Wilson loops and the correlation between two distant loops. The rate at which the correlation decreases is the mass on the lattice (4-11, 5-09), and Problem 5.1 in §5 of Chatterjee (2018) asks about it.

A device for avoiding infinities (a regularisation) changes part of the theory, and what it keeps decides the later work. On Wilson's lattice, gauge invariance survives exactly at each $a$, and of rotations and translations only rotations by 90° and shifts by the lattice spacing survive (the recovery of the continuum symmetries is to be shown in the limit; 6-10). Reflection positivity holds (Osterwalder–Seiler 1978; 5-03), and topological charge (3-05) is not defined as it stands.

The problem statement (JW §6.5) calls the fact that reflection positivity holds in Wilson's approximation “a major advantage” Known, because it is the key to building the Hamiltonian and the Hilbert space (4-10). By contrast, a continuum momentum cutoff breaks gauge invariance, which is restored with counterterms (6-07 §06). JW §6.5 also writes that when reflection positivity is lost with regularisations such as dimensional regularisation or Pauli–Villars, there are few ways of recovering it Known. On this basis, a regularisation that keeps all three (gauge invariance, reflection positivity, continuum rotations) has not been found as far as we have looked (this book's organisation; for the version that counts four by adding ultraviolet control, 6-10 §03).

The abstract of Lüscher (1982) states that non-abelian gauge fields on a four-dimensional hypercubic lattice with small action density have “an integer topological charge Q, which is invariant under continuous deformations of the field” Known. This $Q$ is said to return to the Chern number (3-05) in the classical continuum limit. (A note of this chapter: without the condition, $SU(N)^{\text{number of links}}$ is connected, so the only continuous integer-valued quantities are constants.) Nor is the action limited to Wilson's form: there are heat-kernel actions (Villain type; 6-02), and improved actions that add $1\times2$ rectangles to remove the $a^2$ difference of Table 3 (Symanzik improvement; 5-05). The view that all of them reach the same continuum limit is universality Physics.

The theory on a finite lattice was defined by (10). The theory of the problem statement needs two limits, $L\to\infty$ and $a\to0$ (in four dimensions thought to be paired with $\beta\to\infty$ Physics). The order is first to fix $\beta$ (that is, $a$) and take $L\to\infty$, then $a\to0$. Problem 5.1 in §5 of Chatterjee (2018) also takes the infinite-volume limit at fixed $\beta$ as its starting point Known. Even for the first, Chatterjee (2018) §3 writes that the uniqueness of the infinite-volume limit is not known in general for large $\beta$ above two dimensions Known. The rest of Part 5 shares the work. The expansion in small $\beta$ is 5-02, the Hamiltonian 5-03, infinite volume on the strong-coupling side 5-04, $\beta\to\infty$ and $a(\beta)$ 5-05, numerical integration 5-07 and 5-08, and what to hold fixed in the limit 5-09. This chapter shows nothing about the existence of the two limits or their mass gap.

What this chapter can say

Sources and reproduction

Next chapter to read: 5-02 The strong-coupling expansion (expanding (10) at small $\beta$ and seeing the area law).

← 4-12 Detour: the perturbation series divergesContents5-02 The strong-coupling expansion →