2026-10-15 · chapter Yang–Mills and the mass gapPart 12 The second roadthe principle of direction, reshaping fluctuations, the one-loop table · Kiichi, Shiori and Rin

Why fixed maps fail

1. Confinement needs the sector with a vortex not to be too expensive compared with the sector without, that is, a bound from above on the sector free energy $\Delta F=\ln(Z_+/Z_-)$ (to make $\theta=\tanh(\Delta F/2)$ small).
2. What gives a bound in that direction is the Gibbs variational principle with a map $T$ that carries the measure of sector $+$ to sector $-$: $\Delta F=\mathbb E_+[S\circ T-S-\ln J_T]-\mathrm{KL}$. Discarding the KL (relative entropy) gives an upper bound Proved here.
3. The discarded KL is the loss of the bound. Computing in a Gaussian model, the loss of maps that do not change the shape of the fluctuations (the identity map, group translations, gauge transformations, decimation) does not decrease as degrees of freedom are added. A map that stretches and shrinks the fluctuations while keeping volume brings the loss down to $\Delta F^2/n$ ($n$ is the number of degrees of freedom) Proved here.
4. The coefficient of asymptotic freedom $\tfrac{11}3$ splits at the one-loop level into an orbital part $-\tfrac13$ and a spin term $+4$. The spin term is traceless in the polarisations, so it does not appear in bounds from maps that do not change the shape of the fluctuations (first-order traces) Proved here (at the one-loop level).
5. As a result, at the one-loop level, a renormalisation comparison built with isometric maps can give only $\beta'\ge\beta+0.023$ per doubling (the same direction as $U(1)$), while a map that reshapes the fluctuations reproduces the $\beta-\beta'=0.258$ of asymptotic freedom Proved here (at the one-loop level).
6. In a two-dimensional model, fixed pointwise maps paid an extra cost growing faster than linearly in the lattice size $N_T$ in the twisted direction. In trials that redraw the fluctuations around the new background, this growth disappears Reading.
7. If the residual KL of a deterministic map grows with the area of the sheet, no area law follows from the variational bound Proved here (under that assumption). The role of the transport road lies in a qualitative target weaker than the form of the band itself.

Let $X_+$, $X_-$ be the sets of configurations of sectors $+$ and $-$, $\mu_\pm=e^{-S}/Z_\pm$ the Gibbs measures on each, and set the sector free energy to be $\Delta F=\ln(Z_+/Z_-)$ (the same orientation as $\Delta F_D$ of 12-06). The bias of the sector is $\theta=\tanh(|\Delta F|/2)$, so what is needed in the direction of confinement is an upper bound on $\Delta F$, in other words a lower bound on $Z_-/Z_+$.

Proved here A standard computation. Proof. The density of $\nu=T_\#\mu_+$ is $\nu(y)=\mu_+(x)/J_T(x)$ at $y=Tx$. Hence

Rearranging gives (1). The inequality follows from KL $\ge0$. ∎ On a finite configuration space ((F) of 12-05), if $T$ is a bijection then $J_T=1$, and the first term of the right-hand side of (1) is the $\mathbb E_+[\Delta S]$ of 12-05.

The other direction. The structural tools used in this book do not bound $\Delta F$ from above. Reflection positivity gives $Z_-\le Z_+$ ($\theta\ge0$, $\Delta F\ge0$), but that is a lower bound on $\Delta F$, the direction opposite to confinement (13-04) Proved here. Chessboard estimates replace $\mathbb E\prod\theta$ by a single density $\kappa_{cb}$, but saying $\kappa_{cb}\lt1$ still needs an upper bound on $\Delta F$ (13-05). The Tomboulis–Yaffe inequality carries a bound on $\theta$ to the Wilson loop, and does not bound $\theta$ itself (13-01). Within the range of this book's tools, upper bounds on $\Delta F$ have come only from variation of the type (1) (trial maps, trial measures) and from explicit constructions Reading (a judgement arranging the list of tools, not a theorem). We call this the principle of direction. A candidate tool for crossing the band must first of all give a bound in this direction.

We look at how large the KL of (1) is in a Gaussian model where everything can be computed. Take two Gaussian measures on $\mathbb R^n$, $\mu_0=N(0,\Delta_0^{-1})$ and $\mu_B=N(0,\Delta_B^{-1})$ (actions $S_0=\tfrac12x^{\mathsf T}\Delta_0x$, $S_B=\tfrac12x^{\mathsf T}\Delta_Bx$; $\Delta_0,\Delta_B$ positive definite). $\mu_0$ plays the role of "the fluctuations of sector $+$ (no background)", and $\mu_B$ that of "the fluctuations of sector $-$ (around the vortex background $B$)". The exact value is

$\lambda_k$ are the eigenvalues of $\Delta_0^{-1/2}\Delta_B\Delta_0^{-1/2}$. Putting a linear map $T$ into (1), the bound, using $M:=\Delta_0^{-1/2}T^{\mathsf T}\Delta_BT\Delta_0^{-1/2}$, is

(from $\mathbb E_0[x^{\mathsf T}Ax]=\mathrm{tr}(A\Delta_0^{-1})$).

A map isometric for the fluctuations is a linear map with $T^{\mathsf T}\Delta_0T=\Delta_0$ (it does not change the shape of the fluctuations of $\mu_0$, i.e. it preserves $\mu_0$). The identity map, linearisations of group translations and gauge transformations, and the linearisation of decimation (the operation of discarding some of the fine degrees of freedom and taking the rest as they are as degrees of freedom of the coarse lattice) belong to this class (for the treatment of decimation, see §08 (b)).

Proved here Proof. $O:=\Delta_0^{1/2}T\Delta_0^{-1/2}$ is an orthogonal matrix satisfying $O^{\mathsf T}O=1$, and $M=O^{\mathsf T}(\Delta_0^{-1/2}\Delta_B\Delta_0^{-1/2})O$. Hence $\mathrm{tr}\,M=\sum_k\lambda_k$ and $|\det T|=1$. The difference is the sum of $t-1-\ln t\ge0$. ∎ (4) is exactly the relative entropy $\mathrm{KL}(\mu_0\|\mu_B)$ of the two Gaussian measures. If the background changes the fluctuations in only a few directions (finitely many $\lambda_k$ different from 1), (4) does not decrease as $n$ is made large.

Proved here Proof. If $\det T=1$ then $\det M=r$, and conversely any positive definite $M$ with $\det M=r$ is realised by $T=\Delta_B^{-1/2}M^{1/2}\Delta_0^{1/2}$ (times an orthogonal matrix; $\det T=r^{-1/2}r^{1/2}=1$). By the arithmetic–geometric mean inequality $\mathrm{tr}\,M\ge n(\det M)^{1/n}=nr^{1/n}$, with equality at $M=r^{1/n}\cdot1$. The bound on the difference follows, with $u=2\Delta F/n$, from $e^u-1-u\le\tfrac{u^2}2e^{|u|}$. ∎

The map of Proposition 3 preserves volume ($\ln J_T=0$), stretches and shrinks the fluctuations in the few directions the background changes, and spreads that change of volume thinly over the many remaining directions, $\Delta F/n$ each. What is needed is not having a Jacobian (entropy), but changing the shape of the fluctuations.

Proved here (values of the closed formulae (2), (4), (5)). When small and large eigenvalues are mixed, the first-order trace $\tfrac12\sum(\lambda_k-1)$ and $\tfrac12\sum\ln\lambda_k$ can have opposite signs. In this example the change of sector is exactly cheap ($\Delta F\lt0$), yet the bound of the isometric map says it is expensive ($+1.75$). The spin term of the next section has exactly this form.

Place a constant abelian magnetic field $B$ in the (1,2) plane as background, and consider the fluctuations of the $SU(2)$ gluons around it in Feynman gauge. The operator of the quadratic action of the fluctuations is

where the first term is the orbital part (the square of the covariant derivative) and the second term is the spin term. The ghost operator is $-D^2$. The heat-kernel traces (per volume, omitting the common factor $1/(16\pi^2s^2)$ and the group factor $C_A=2$ = the number of charged components), as functions of the proper time $s$, are

With $x=Bs$, using $x/\sinh x=1-x^2/6+O(x^4)$ and $\cosh2x=1+2x^2+O(x^4)$ and picking out the coefficient of $x^2$ gives the following table.

Proved here (at the one-loop level. The expansion is a two-line computation: orbital $2(1-x^2/6)=2-x^2/3$, spin $(1-x^2/6)\cdot4x^2=4x^2+O(x^4)$.) In the conventions of this count, the coefficient $+\tfrac{11}3$ is the direction of asymptotic freedom and $-\tfrac13$ the opposite direction.

Proved here (at the one-loop level) Proof. $M_0^{-1}$ is proportional to $\delta_{\mu\nu}$ in the polarisation indices, and $F_{\mu\nu}$ is antisymmetric, so $\mathrm{tr}_{\mu\nu}(F_{\mu\nu})=0$. Hence the first-order trace of the spin term is 0, the bound equals $\tfrac12\mathrm{tr}((M_{\rm orb}-M_0)M_0^{-1})$, and by the inequality $t-1\ge\ln t$ of Proposition 2 it is at least $\tfrac12\ln\det(M_{\rm orb}M_0^{-1})$. ∎ The spin term takes effect at second order ($\cosh2Bs-1=O(B^2)$ in Table 2) and gives the $+4$ of antiscreening. A bound that looks only at first-order traces loses it completely. (Traces and determinants of infinite-dimensional operators are statements under the regularisation usually done in one-loop computations.)

The renormalisation comparison of Part 13 (13-06) aims to bound from above the vortex free energy of $(2N,\beta)$, with the whole lattice doubled, by that of the original size $(N,\beta')$, and to say $\beta'\lt\beta$ (the direction in which the coupling becomes stronger, asymptotic freedom). At the one-loop level, the coefficient $c$ around the vortex background fixes the shift of the coupling per doubling, and with $\beta=4/g^2$ and $b_0=c\,C_A/(16\pi^2)$, $\beta-\beta'=8b_0\ln2$.

Proved here (at the one-loop level) Proof. (a) follows from Proposition 4 and the fact that the bound is an upper bound (Proposition 1). The number is $8\cdot\tfrac{-1/3\cdot2}{16\pi^2}\ln2=-\tfrac{0.2575}{11}\approx-0.0234$. (b) applies Proposition 3 to the fluctuations shell by shell, and the number is $8\cdot\tfrac{22}{48\pi^2}\ln2\approx0.2575$. ∎

The attempt to build a renormalisation comparison with Migdal–Kadanoff-type decimation (Tomboulis 2007) was criticised on the point that the same argument also flows to strong coupling for $U(1)$ Cited (Ito–Seiler, Kanazawa; source table). Decimation belongs on the side that does not change the shape of the fluctuations, so Proposition 5 (a) can be read as giving a one-loop-level explanation of this criticism Reading (physics; an interpretation in which a one-loop computation pins down the gap in a non-perturbative argument, not a proof of the criticism itself).

A screening criterion. A candidate method for crossing the band must have a linearisation of the trial map that is non-isometric for the fluctuations — this is a necessary condition at the one-loop level Proved here (a restatement of Proposition 5 (a); at the one-loop level). The view that the same condition is needed at the non-perturbative level is Reading (physics). The driven insertion of 12-06 is cheap because the heat bath relaxes the shape of the fluctuations to fit the new coupling, which is consistent with this criterion Reading (physics).

Proposition 2 said that "the loss of isometric maps is not diluted by degrees of freedom". Numerics show that with fixed pointwise maps on the lattice, the loss moreover grows as the lattice spacing is made finer.

A two-dimensional model. Place the two-dimensional $O(3)$ model (spins $s_x\in S^2$, action $-\beta\sum s_x\cdot s_y$) on the strip $\mathbb Z\times(\mathbb Z/N_T)$, with a twist of a $\pi$ rotation about the $z$ axis in the periodic direction. The classical vacua are the north and south poles, and the action of the kink joining them (the two-dimensional version of the fractional instanton) approaches $2\pi\beta$ from below. It is a light test bed for the small twisted $T^2$ in four dimensions.

Reading The extra cost hardly depends on $\beta$ and grows faster than linearly in $N_T$. This is the opposite of the expected "decrease by $\ln N_T$ due to the running coupling".

On the other hand, in trials that do not carry the fluctuations over on the old background but redraw them from the Gaussian measure around the new background (the kink), the difference of the bound becomes the ratio of one-loop determinants $\tfrac12\ln[\det' H_{\rm kink}/\det H_{\rm vac}]$. Its value (with the two collective coordinates removed) changes only logarithmically, $-3.68,-2.94,-2.43,-2.11,-1.88$ at $N_T=3,4,6,8,10$, and the quantity including the measure of the rotational and translational collective coordinates was almost constant, $-4.91$ to $-4.85$ for $N_T=4$ to $12$ Reading (numerical computation of eigenvalues; no error estimate). The faster-than-linear growth has disappeared.

A four-dimensional slab. For $SU(2)$, $\beta=2.3$, the slab $2\times2\times8\times8$, an insertion that spreads the flux smoothly in a fixed colour direction ($\sigma_3$) raised the cost instead of lowering it: the mean of $\Delta S$ was 21.9 at spreading radius 1 and 36.0 at radius 2 Reading. This can be read as being because, in configurations without gauge fixing, the field in a fixed direction does not line up with the colour direction at each point Reading (physics). On the same slab, driven insertion (12-06) brought the mean work down to 2.9 with 16 steps.

How to read it. Fixed pointwise maps move the fluctuations "on a fixed background", so they fail to capture the difference of entropy of the fluctuations on the new background. The picture is that the loss of Proposition 2 piles up for each fine degree of freedom of the lattice Reading (physics; whether it is proportional to $N_T$ is a reading from three numerical points, and there is no general bound). The average of insertions (12-05), driven insertion (12-06) and redrawing the fluctuations are all tools for avoiding this cost.

The correct form of the band was $\theta(L,L)\le e^{-cL^2}$ (13-01). Since $\theta=\tanh(\Delta F/2)\le\Delta F/2$, showing this variationally requires the upper bound on $\Delta F(L,L)$ to be as small as order $2e^{-cL^2}$.

Proved here Proof. From (1), $\mathbb E_+[\cdots]=\Delta F+\mathrm{KL}\ge\mathrm{KL}\ge c'A$ (taking the orientation of the sectors so that $\Delta F\ge0$). $\tanh$ is monotone. ∎ This is a statement under an assumption, and whether the assumption holds must be checked map by map.

Proved here The standard fact that the relative entropy of a product measure is the sum of the relative entropies of the factors. ∎

That for fixed maps (not adapted to the fluctuations) the residual of each container is bounded below by $c_0\gt0$, so that the assumption of Proposition 6 holds, is the shared judgement of the work from two viewpoints, constructive field theory and probability Reading (physics, an estimate; the same goes for the judgement that the ceiling of triangular maps is of perimeter type). Under this judgement the transport road is not a tool for showing the form of the band itself, but a tool for a weaker qualitative target — $\Delta F(L,L)\to0$ (at any speed). $\Delta F(L,L)\to0$ is enough to rule out the Coulomb phase of $U(1)$ (where $\Delta F(L,L)$ approaches a positive constant) and the deconfined phase of $\mathbb Z_2$, but it yields neither confinement nor a mass gap Reading (physics; numerics in 13-06).

(a) The non-perturbative level. Propositions 4 and 5 and the top three rows of Table 3 are all statements at the one-loop (Gaussian) level. Bounding the error of maps that reshape the fluctuations at intermediate coupling and large fields (local relative entropy near the vortex sheet, and higher loops) is an open problem, and this book does not show it.

(b) The classification of isometric maps. The identity map and the linearisations of group translations and gauge transformations are isometric by definition. Putting decimation on the isometric side is a classification in the sense that it carries the fluctuations of the kept degrees of freedom over as they are; decimation is not a bijection, so this is not a literal application of Proposition 2.

(c) The principle of direction. "Upper bounds on $\Delta F$ come only from variation and explicit constructions" is a judgement about the list of this book's tools, not a theorem.

(d) The extra ultraviolet cost. Table 4 is three numerical points in a two-dimensional model, and does not show that four-dimensional gauge theory grows in the same way. In four dimensions cancellations through Ward identities are said to be possible, and whether they occur depends on gauge fixing and the covariance of the map.

(e) The assumption of Proposition 6. That the residual KL of fixed maps is proportional to the area can be said only in the product case of Proposition 7, assuming that the residual of each container is bounded below; for general maps it is an estimate.

(f) Novelty. The Gibbs variational principle, the relative entropy of Gaussian measures and the one-loop computation of the background field are all standard. No novelty is claimed for the view that lines them up as a screening criterion of "whether the shape of the fluctuations is changed". For scalar fields, Polchinski-type transport maps are known (Bauerschmidt–Bodineau–Dagallier); no example for gauge fields was found in the range searched (not yet checked).

This chapter has no machine-checked bundles.

Dependency graph (the numbers of premises are smaller than the number of the conclusion):

[01] PRV GIBBS   ΔF = E₊[S∘T − S − ln J] − KL                    bijection, Jacobian
[02] PRV RP      reflection positivity gives only ΔF ≥ 0           13-04
[03] RDG DIR     principle of direction (judgement) ⟸ [01][02]
[04] PRV ISO     loss of isometric maps not diluted in n ⟸ [01]   Gaussian
[05] PRV RESHAPE reshaping with det 1, loss ~ ΔF²/n ⟸ [01]        Gaussian
[06] PRV EX      example of Table 1 ⟸ [04][05]
[07] PRV 1LOOP   11/3 = −1/3 + 4                                   one-loop level
[08] PRV TRACE   spin term absent from first-order trace ⟸ [04][07]  one-loop level
[09] PRV ISO-RG  isometric comparison β′ ≥ β + 0.023 ⟸ [01][08]  one-loop level
[10] PRV SH-RG   reshaping comparison β − β′ = 0.258 ⟸ [05][07]   one-loop level
[11] RDG NUM     sign of doubling test (SU(2) +, U(1) −)           small lattices
[12] CIT MK      criticism of Migdal–Kadanoff interpolation
[13] RDG WHY-MK  one-loop explanation of [12] (physics) ⟸ [09][12]
[14] PRV CRIT    screening criterion: non-isometric ⟸ [09]        one-loop level
[15] RDG UV2D    extra cost of fixed maps (2D)
[16] RDG RESAMP  redrawing removes the growth (2D)
[17] RDG UV4D    fixed-direction insertion is expensive (4D slab)
[18] PRV NOAREA  residual KL ≥ c′A gives no area law ⟸ [01]      with assumption
[19] PRV ADD     KL adds up over independent containers
[20] RDG KLAREA  residual of fixed maps bounded below per container (estimate) ⟸ [19]
[21] RDG WEAKGOAL role of ΔF(L,L) → 0 (physics) ⟸ [18][20]

The way to check is to see by eye that every number to the right of "⟸" is smaller than the number on the left. There are no circles. There are no assumption rows. The conclusion of [18] applies to fixed maps only when the estimate of [20] is correct, and that distinction is expressed by the grade (Reading) of row [21]. This chapter has no node of confinement, no node of the mass gap, and no non-perturbative node.

What this chapter can say

Sources and reproduction

Words used in this chapter — the transport identity, the Gibbs variational principle, the principle of direction, maps isometric for the fluctuations, reshaping the fluctuations, the spin term, antiscreening, the residual relative entropy, the screening criterion. Their meanings are in A-02 Glossary (the words of Part 12 are to be added).

Next chapter to read: 12-08 The measured record — the numerics of Part 12 (the work of driven insertion, the vortex free energy when the $T^2$ is widened, the string tension of slabs) gathered in one place, with conditions.

← 12-06 Driven insertion and the flow lemmaContents12-08 The measured record →