2026-10-01 · chapter Yang–Mills and the mass gapPart 4 Quantisationgauge fixing

Gauge fixing and Faddeev–Popov — correct locally, and globally the signs cancel

Conventions: Euclidean $\mathbb R^4$, $\hbar=c=1$, and we write the Hermitian $A_\mu=A^a_\mu T^a$ and the coupling $g$ separately (the style of 3-02; the connection of the geometry chapters is $\mathcal A=-igA$).

Perturbation theory (4-03) uses the inverse of the quadratic part of the action as the propagator. $S$ does not change under gauge transformations, so the quadratic part is flat in the direction $A=\partial\lambda$ and has no inverse.

Example. If $k=(0,0,0,\kappa)$, then $K=\kappa^2\,\mathrm{diag}(1,1,1,0)$, and the fourth component does not appear in the action Computed. On Wilson's lattice, where the volume of the orbits (4-02 §07) is finite, no fixing is needed (5-01 §06) Known. It is needed when one expands around $A=0$ while staying in the continuum.

Let us put into words why there is no inverse. The path integral $\int\mathcal DA\,e^{-S}$ (4-02) adds up all fields related by gauge transformations (one orbit). Along the orbit $e^{-S}$ does not change, so the Gaussian integral in that direction does not decay and diverges. Speaking in metaphor, the bottom of the valley is a single trench, flat without end along the trench. Precisely, the kernel $K$ of the quadratic form (2) has eigenvalue 0 in the direction of $k$. So we would like to take the integral along the orbit outside, choose one representative from each orbit and add up the rest. We first see how to take it out in a finite-dimensional model.

The weight $e^{-S(|x|)}$ on the plane does not change under the rotation $R_\theta$, and the orbits are circles centred at the origin. Here the angle of rotation $\theta$ plays the role of "the direction of the orbit". The aim is to bring the integral over $\theta$ to the surface without changing the value of $Z$, and for that we put a rewritten 1 into $Z$. We use the rule for the one-variable $\delta$ function.

If the condition $G(x)=0$ meets each circle exactly once, $\theta\mapsto G(R_\theta x)$ vanishes at a single point only, so the following "insertion of 1" holds.

Putting this into $Z=\int d^2x\,e^{-S}$ and changing variables $x\to R_\theta^{-1}x$, since $d^2x$, $S$ and $\Delta_G$ do not change under rotation, the integral over $\theta$ leaves the volume of the group $2\pi$.

This is the whole of the Faddeev–Popov procedure, and the weight $\Delta_G$ on the surface is the Faddeev–Popov determinant.

Example. Taking the condition to be the positive $x_1$ axis ($G=x_2$, $x_1\gt0$), $\Delta_G=r$, and (4) is $Z=2\pi\int_0^\infty r\,e^{-S}dr$: the Jacobian of polar coordinates. The determinant measures the size of the orbit (the same role as the Weyl factor in 3-04 §04).

We repeat (3)(4) with the group taken to be all pointwise gauge transformations and the condition $G^a=\partial_\mu A^a_\mu-\omega^a$. Since $\delta G^a=\frac1g\partial_\mu(D_\mu\lambda)^a$, the content of the determinant is the following $\mathcal M$. The constant factor $1/g$ and the sign become an overall constant of the determinant, absorbed into the overall constant of $Z$ (which cancels in ratios of expectation values), so we drop them.

Averaging $\omega$ with the weight $e^{-\int\omega^2/2\alpha}$ (many references write the parameter as $\xi$, but this book uses $\xi$ for the correlation length, so we write $\alpha$; the same letter as in 4-03), the $\delta$ function turns into a term of the action, and the quadratic form acquires an inverse.

$\alpha=1$ is the Feynman gauge and $\alpha\to0$ the Landau gauge (the inverse checked numerically Computed). To make $\det\mathcal M$ a term of the action as well, we use anticommuting numbers (Grassmann numbers: $\theta_i\theta_j=-\theta_j\theta_i$, $\theta_i^2=0$). The rules are $\int d\theta\,1=0$ and $\int d\theta\,\theta=1$. Since the expansion stops at $e^{-a\bar\theta\theta}=1+a\theta\bar\theta$,

Known. The determinant comes out with its sign (the source of the 0/0 of §08). In this way the determinant becomes an integral over anticommuting fields $c^a,\bar c^a$ (ghosts; a different meaning from the "ghost" in 4-10, which refers to the negative-weight states of Pauli–Villars).

($\simeq$ is integration by parts.) A ghost propagator $\delta^{ab}/k^2$ and a vertex proportional to $g f^{abc}p_\mu$ come out Known. Ghosts are not particles that come out, and appear only inside loops (they also enter the β function: 4-06 §03). Example. A real Gaussian integral in $n$ components gives $(\det A)^{-1/2}$ (2-07 §02). In the Feynman gauge ($\alpha=1$) with $g=0$ the kernel is $k^2\delta_{\mu\nu}$, so at each momentum the 4 components of the gauge field give $(k^2)^{-4/2}$ and the ghosts give $(k^2)^{+1}$ by (7), together $(k^2)^{-2/2}$: only the two transverse polarisations remain ($4-2=2$).

Detour: in the abelian case the ghosts disappear

The fixed action still has an anticommuting symmetry $s$ that mixes the gauge field and the ghosts (with an auxiliary field $b^a$ added).

$s\Psi=ib\,\partial A+\frac\alpha2b^2+\bar c\,\mathcal Mc$, and integrating out $b$ returns to (6)(8). From $sS=0$ and nilpotency $s^2=0$ (the Jacobi identity), two things follow formally Known.

• Independent of the choice of gauge. Changing $\Psi$ changes the action only by $s(\delta\Psi)$, and for $O$ annihilated by $s$, $\langle O\,s(\delta\Psi)\rangle=\langle s(O\delta\Psi)\rangle=0$. The abstract of Becchi–Rouet–Stora (1976) writes that they showed that, for semisimple groups, in linear gauges in which all fields are massive, the Slavnov identities (the $s$ symmetry written as relations between correlations) hold to all orders of renormalised perturbation theory, and that in the SU(2) Higgs–Kibble model examined as an example they showed, in the perturbative sense, that the physical S operator is unitary and independent of the gauge parameters (the 1975 paper is on the abelian Higgs–Kibble model). It is not a conclusion about massless pure Yang–Mills.
• Physical states. $s$ becomes an operator $Q_B$ acting on states (the BRST charge; $Q_B^2=0$). In covariant gauges the state space has an indefinite metric (the inner product $\langle\psi,\psi\rangle$ can be negative or 0), and as it stands it has no probabilistic interpretation. So we take the physical states to be $\operatorname{Ker}Q_B/\operatorname{Im}Q_B$ (BRST cohomology). This is "the states annihilated by $Q_B$, identified modulo the image of $Q_B$". Since $Q_B^2=0$, the image $\operatorname{Im}Q_B$ is contained in the kernel $\operatorname{Ker}Q_B$, and the quotient can be formed. The aim is that states in the image have inner product 0 with physical states, so dropping them by the identification does not change measurable quantities. States of negative or zero norm (the time and longitudinal polarisations, the ghosts) are dropped in this way, and the inner product on the remaining space is expected to be positive. The S operator is the operator that connects the states before and after scattering, and unitarity means preserving the total probability (the same property as $e^{-itH}$ in 2-05 §02). The abstract of Kugo–Ojima (1979) writes that, within a locally covariant operator formalism (a framework that assumes the theory and its asymptotic fields exist), they showed the unitarity of the physical S matrix from the algebra of the BRS charge Physics. The existence of the theory itself is assumed, and the identification in the continuum theory has not been shown (§09).

"BRST" is said to be the initials of Becchi, Rouet, Stora and Tyutin. Example. Computing $s^2$ symbolically on the part with constant fields, $s^2A^a$ and $s^2c^a$ had 0 terms for both SU(2) and SU(3). Changing the coefficient $-\tfrac12$ of $sc$ to $-1$ leaves 2 terms and 25 terms in $s^2A^a$ Computed.

We build a case in which the "exactly once" of (3) breaks down. Following the Landau gauge (3-04 §05), we take critical points of $F$ on the circle as representatives, $M=\partial_\varphi^2F$ as the determinant, and the set of local minima as the analogue of the Gribov region.

For large fields pairs of a local minimum and a local maximum are born, and at those points $(0,\pm r_*)$, $M=0$ (the analogue of the horizon). With the weight $e^{-r^2/2}$ we count $r^2$ (true expectation value 2) in four ways (for the difference between $|M|$ and $M$, see the detour of this section): (i) all copies with $|\det|$, (ii) only the Gribov region (local minima), (iii) only the global minimum of $F$ (the fundamental modular region), (iv) with signs (the BRST way of counting).

ComputedThe numbers in this figure were computed on this machine (the closed formulas of formula (10) and Table 1 evaluated on the spot; Table 1 agrees with a brute-force count of critical points to within 0.004).

When the number of copies varies from orbit to orbit, (i) is off, and (ii), which leaves copies inside the region, is not fixed either. (iii) is correct, but choosing the global minimum is a global comparison and cannot be written as a local condition. (iv) is 0/0, since local minima and local maxima come in equal numbers.

Example. The error of (i), $4ue^{-u}/(1+2e^{-u})$ ($2\times10^{-8}$ at $c=0.05$), is 0 at every order in powers of $c$, the same type as $e^{-1/x}$ in 4-07 §05 Computed.

Detour: counting intersections with signs

The abstract of Gribov (1978) writes that in the non-abelian case fixing $\partial_\mu A_\mu$ does not fix the gauge, and that restricting to the range where the FP determinant is positive seems to be enough Known. Reading the Landau gauge as the stationary points of the norm $\|A^g\|^2$,

$\Omega$ is the Gribov region, its boundary the horizon, and $\Lambda$ the fundamental modular region Known. $\bar\Omega$ is the closure of $\Omega$ (including its boundary). $\Omega$ is an open set given by $\mathcal M\gt0$, and $\Lambda$ a closed set given by $\le$; the boundary of $\Lambda$ has points touching the horizon (van Baal, §07), so we write it with the closure.

• Convex. $\mathcal M(A)$ is linear in $A$, so "$\langle\omega,\mathcal M(A)\omega\rangle\gt0$ for every $\omega$" is an intersection of half-spaces. The abstract of Zwanziger (1982) also writes 'convex region Ω which is bounded in every direction'. $\Lambda$ is also convex (the section "Complete gauge fixing" of van Baal 1997).
• Bounded in every direction (the abstract of Zwanziger 1982). The outline in finite dimensions: $\mathcal M_1$ in $\mathcal M(tA)=\mathcal M_0+t\mathcal M_1$ is a symmetric matrix with trace 0 ($f^{aba}=0$), so it has a negative eigenvalue, and for large $t$ the operator stops being positive.
• Every orbit passes inside the horizon (the abstract of Dell'Antonio–Zwanziger 1991: the norm attains its minimum on each orbit, and there the orbit meets the inside of the horizon).
• Copies remain even inside $\Omega$. Zwanziger (2004) argues that they do not affect expectation values Physics.

The solid line in Figure 1 is the analogue of $\Omega$, and the positive $x_1$ axis the analogue of $\Lambda$. For the effect on the propagator, see 7-02 §07.

Taking a constant SU(2) field $A^a_\mu=a\,\delta^{a3}\delta_{\mu1}$ in a periodic box of side $L$, $\mathcal M$ is diagonalised by plane waves $p=2\pi n/L$, and splits into colour component 3 (neutral) and $\omega^1\pm i\omega^2$ (charged).

The smallest non-trivial eigenvalue (excluding $n=0$) is $1-s$ at $n=(\pm1,0,0,0)$, and the horizon is at

van Baal (the section "Gauge fields on the three-torus" of 1997) does the same calculation in the Coulomb gauge on the three-torus with $C=gaL$, and shows that the Gribov region is the cube $|C_k|\le2\pi$ and that copies of $A=0$ under twisted (antiperiodic) gauge transformations lie on the horizon Known. The fundamental modular region is $|C_k|\le\pi$ ($s\le\frac12$) when antiperiodic transformations are also identified (pure Yang–Mills). When there are fields in the fundamental representation (quarks) and only periodic transformations are allowed, it is $|C_k|\le2\pi$ ($s\le1$), and its boundary touches the horizon Known.

ComputedThe numbers in this figure were computed on this machine (formula (12) evaluated on the spot). At $s=0$, 0.5, 1 and 2 the smallest non-trivial eigenvalue is 1, 0.5, 0 and −1 (in order: $A=0$, the boundary of the fundamental modular region [when antiperiodic transformations are also identified], the horizon, and the copy of $A=0$ under a periodic transformation).

We also checked on the lattice. On an $L^4$ lattice, setting the links in direction 1 to $e^{i\theta\sigma_3/2}$ and diagonalising the second derivative of the Landau gauge functional $\sum(1-\tfrac12\mathrm{Re\,tr}\,g_xU_{x,\mu}g^\dagger_{x+\mu})$ momentum by momentum, the charged eigenvalues agreed with $\tfrac12[\cos\tfrac\theta2-\cos(p_1\pm\tfrac\theta2)]+\tfrac12\sum_{\nu\ne1}(1-\cos p_\nu)$ (with lattice spacing 1, $\theta=ga$).

Computedpython3, NumPy. The difference from the closed formula is at most $3\times10^{-16}$.

The larger the box, the closer to the origin the horizon is in long-wavelength directions, and the abstract of Gribov writes that this restriction cancels the infrared singularity and leads to a linear interaction at long distances Physics. It has not become a theorem about the four-dimensional measure.

The ghosts gave $\det\mathcal M$ without absolute value (formula (7)). So the integral of BRST-symmetric gauge fixing counts the copies on an orbit with $\operatorname{sign}\det\mathcal M$.

We define the words used here. The Euler characteristic $\chi$: the sum over the critical points (those whose matrix of second derivatives is non-degenerate) of a smooth function on a compact manifold without boundary (1-05), each counted with the sign of the determinant of that matrix ($+1$ if the number of negative eigenvalues is even, $-1$ if odd). This number does not depend on the choice of function (the Poincaré–Hopf theorem). On a circle a minimum counts $+1$ and a maximum $-1$, giving $\chi(S^1)=0$; on the sphere $\chi(S^2)=2$. For a compact connected Lie group $G$ of positive dimension $\chi(G)=0$, and for product spaces $\chi(G^N)=\chi(G)^N$ Known. Morse theory is the name of the theory that reads the shape of a manifold by counting critical points in this way.

The last two equalities read the sum of signs as the Euler characteristic of $G^N$ ($N$ is the number of lattice points) (Schaden 1999's interpretation via Morse theory, formula (2) of Mehta et al. 2014; on orbits whose critical points are non-degenerate), and $\chi(G)=0$ Known (6-11 §05). Expectation values of gauge-invariant quantities are 0/0. Mehta et al. call this "Neuberger's 0/0 problem" and cite Neuberger's papers of 1986 and 1987, and the abstract of the 1987 paper says that this obstruction met non-perturbatively is formally present in the continuum formulation as well Known.

Example. The "sum of signs 0" in Figure 1 is its smallest version ($\chi(S^1)=0$). The same abstract also argues that at every order of the perturbative expansion the BRS invariance of the lattice does the job of local gauge invariance. This is because the cancellation of signs does not show up at any order of perturbation theory, the same type as $e^{-1/(18c^2)}$ in the example of §05. The ways around it change the way of counting Known.

• Do not fix (Wilson's lattice; Elitzur's theorem: 8-02 §09), or the maximal tree gauge, which has no Gribov problem (3-04 §08).
• Modified Landau gauge by stereographic projection: map the link variables to flat coordinates by stereographic projection before building the functional (von Smekal et al. 2007; abstract: for compact $U(1)$ it solves the Gribov problem, and used on the maximal abelian subgroup it may avoid the 0/0 of $SU(N)$; the continuum limit of the Landau gauge is unchanged).
• Orbifolding: replace the group manifold used for gauge fixing by a generalisation of manifolds that allows singular points (an orbifold), and count with Morse theory on it (Mehta et al. 2014; abstract: the partition function becomes independent of the orbit).
• Regularising with a mass: count with an action in which mass terms (the Curci–Ferrari mass) are added to the gauge field and the ghosts, regularising the 0/0 (Ghiotti–von Smekal–Williams 2006, abstract). Coset space: BRST counting not on the whole group but on $SU(2)/U(1)$ (the sphere $S^2$, with $\chi=2$) (Schaden).

For a comparison with the algebra of divergences, see 6-12.

What the problem statement asks for are correlations of gauge-invariant local fields (such as $\mathrm{tr}\,F_{\mu\nu}F_{\rho\sigma}$) and the gap of the reconstructed Hamiltonian (0-03). Gauge fixing is an indispensable tool in perturbation theory (§04). Elsewhere it has the following place Known.

• Relation to the axioms. The positivity needed for a probabilistic interpretation (reflection positivity, defined in 4-10) cannot be expected for fields fixed in a covariant gauge. That covariant gauges use a state space with an indefinite metric (§04) is a known framework, and Magnen–Rivasseau–Sénéor 1993 write that covariant gauges such as the Feynman gauge lack positivity and so cannot be used from the start of a construction (4-10 §09). So the reconstruction is done from gauge-invariant quantities. The identification of BRST cohomology with that positive-definite space has not been shown in the continuum (8-02 A18).
• Attempts at construction use it locally. On Bałaban, the exposition of the problem statement writes 'The choices vary locally in space-time, as well as on different length scales.' (Jaffe–Witten §6.5, 6-05, 6-07). Chevyrev–Shen in two dimensions also use lattice gauge fixing (abstract, 6-09).

What Singer's theorem forbids is only continuous global slices, and the 0/0 too is a phenomenon of fixing a covariant gauge BRST-symmetrically over the whole lattice. The remaining question is still "does the measure of gauge-invariant correlations exist in the four-dimensional continuum limit, and does the spectrum have a gap".

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