2026-10-15 · chapter Yang–Mills and the mass gapPart 13 The crossover bandthe negative lemma, the electric flux, reflection positivity in z · Kiichi, Shiori and Rin

Structure alone cannot cross

In lattice gauge theory with the Wilson action on the lattice $\Lambda=(\mathbb Z/L_1)\times(\mathbb Z/L_2)\times(\mathbb Z/M)^2$, we place the background twist $n_{12}=1$ in the two transverse directions 1 and 2 (12-01). Directions 3 and 4 are the plane. What this chapter calls the structural inputs is the following set $\mathcal I$ of five.

1. Positivity of the transfer matrix: the transfer matrix $T$ with a planar direction (3 or 4) as time is a positive operator (5-03, Osterwalder–Seiler, Lüscher Cited).
2. The centre and duality identities: inserting twist flux is inserting a centre transformation, and it can be decomposed by the projections onto electric flux ('t Hooft Cited). Lemma 1 of §02 is one of this form.
3. Reflection positivity in the planar directions: the positivity of reflections through lattice points in directions 3 and 4 for the z ensemble with the twist fixed (Theorem 3 of §03).
4. Symmetries of the lattice: translations, rotations, reflections, and the fact that they preserve the class of the twist.
5. The finite lemmas: the container identity and nesting (12-03), the slab estimate $\sup_\Phi\theta_D\le\tanh(\beta L^2)^{|D|}$ (12-04), the flow lemma $\theta\le1-a$ (12-06). All of them use only properties of measures and finite sums.

The question is whether the lateral-widening inequality ((5) of 13-02) follows from $\mathcal I$. The answer, in §04, is "it does not". Before getting there, we set out two things that can certainly be said about the sector bias within $\mathcal I$.

Take the lattice to be $T^2_{L_1\times L_2}\times T^2_{M\times M}$, and write $Z(1,n_{34})$ for the partition function with, in addition to $n_{12}=1$, the twist flux $n_{34}\in\{0,1\}$ through the plane (3,4) inserted. The global bias of (8) of 13-01 was

We choose direction 3 as time. The space at one time is $T^2_{L_1L_2}\times T^1_M$ (directions 1, 2, 4), and in it there is the spatial magnetic flux $m_4=n_{12}=1$. The transfer matrix $T_3\ge0$ commutes with the centre transformation $\hat C_4$ of direction 4 (the operation that multiplies the links going once round direction 4 by the centre element $-1$), and $\hat C_4^2=1$. When space is split by the eigenvalues $(-1)^{e_4}$ of $\hat C_4$, $e_4\in\{0,1\}$ is the electric flux in direction 4 ('t Hooft Cited). Let the projections be $P_{e}$ and put

Lemma 1 (the electric-flux form) Proved here In the setting above, with $Y(0),Y(1)\ge0$,

Hence when $Z(1,0)+Z(1,1)\gt0$, $\theta=Y(1)/Y(0)\ge0$. If moreover $Z(1,1)\ge0$, then $\theta\le1$; that is, with $F_v\ge0$, $0\le\theta\le1$.

Proof The partition function is the trace of the transfer matrix, $Z(1,0)=\mathrm{Tr}\,T_3^{\,M}$, and the twist $n_{34}=1$ through the plane (3,4) is the same as inserting $\hat C_4$ somewhere around the time circle, so $Z(1,1)=\mathrm{Tr}(\hat C_4T_3^{\,M})$ ('t Hooft's construction Cited). Substituting $\hat C_4=P_0-P_1$ and $1=P_0+P_1$ gives (3). $P_e$ is an orthogonal projection commuting with $T_3$, so $P_eT_3^{\,M}P_e=P_eT_3^{\,M}$ is a positive operator and its trace is at least 0. When $Z(1,0)+Z(1,1)=2Y(0)\gt0$, putting (3) into (1) gives $\theta=2Y(1)/2Y(0)$. From $Z(1,1)=Y(0)-Y(1)\ge0$, $Y(1)\le Y(0)$. ∎

Two remarks. First, $\theta\ge0$ and $|Z(1,1)|\le Z(1,0)$ are spectral facts that follow from (3) and $Y(e)\ge0$ alone, but the $Z(1,1)\ge0$ needed for $\theta\le1$ is not. $Z(1,1)$ is at least 0 because it is the integral of a positive function containing the twisted plaquette weight $e^{-\frac\beta2\mathrm{tr}U_p}$, but this is the positivity of the Boltzmann weight and does not follow from the spectrum of the transfer matrix. Second, Lemma 1 is a lattice version of 't Hooft's electric–magnetic duality: "the vortex free energy (magnetic flux) is small ⟺ the state with electric flux $e_4=1$ is heavy compared with $e_4=0$". In the confining regime $Y(1)/Y(0)$ is expected to be small because of the electric string tension Reading (physics, 13-05 §04).

We add one elementary lemma, used when writing lateral widening in the global form.

Lemma 2 Proved here Let $\theta=\tanh(F/2)$, $\theta'=\tanh(F'/2)$, $F,F'\ge0$. If $F'\le KF$ ($K\ge1$), then $\theta'\le K\theta$.

Proof $\tanh$ is monotonically increasing and concave on $[0,\infty)$ with $\tanh0=0$, so for $K\ge1$, $\tanh(Kx)\le K\tanh x$ (for a concave function $g$ with $g(0)=0$, $g(x)=g(\tfrac1K\cdot Kx+(1-\tfrac1K)\cdot0)\ge\tfrac1Kg(Kx)$). Hence $\theta'=\tanh(F'/2)\le\tanh(KF/2)\le K\tanh(F/2)=K\theta$. ∎

That is, lateral widening in the form of the vortex free energy, $F_v(L')\le C(L'/L)^2F_v(L)$, implies lateral widening in the form of the bias.

Setting. Take the torus $\Lambda=(\mathbb Z/L)^2\times(\mathbb Z/M)^2$ with $M$ even. The variables are the links $U_\ell\in SU(2)$ and the plaquette variables $z_p\in\{\pm1\}$, and we impose the following constraints.

• For every three-dimensional cube $c$, $\prod_{p\in\partial c}z_p=1$ ($z$ is a closed $\mathbb Z_2$ 2-cochain).
• Fixing the global class, $n_{12}(z):=\prod_{p\in\Pi}z_p=-1$. $\Pi$ is a (1,2) plane with $(x_3,x_4)$ fixed, and since $z$ is closed this does not depend on the position of $\Pi$.

The measure is

($dU$ is the Haar measure). This is the ensemble of 13-01 §01 with the twist summed over, with the transverse twist fixed.

Reflection. Let $\Theta$ be the reflection through lattice points in direction 3, $x_3\mapsto-x_3$ (mod $M$). The fixed planes are $P_0=\{x_3=0\}$ and $P_1=\{x_3=M/2\}$. It acts on links by $U_{(x,\mu)}\mapsto U_{(\Theta x,\mu)}$ for directions $\mu\ne3$ and by $U_{(x,3)}\mapsto U_{(\Theta x-e_3,3)}^{-1}$ for direction 3, on plaquette variables by $z_p\mapsto z_{\Theta p}$, and on functions by $(\Theta G)(U,z):=\overline{G(\Theta U,\Theta z)}$.

Theorem 3 (reflection positivity in z) Proved here For any bounded function $G$ depending only on the variables of the right half $\Lambda_+=\{0\le x_3\le M/2\}$ (including the variables on the planes $P_0\cup P_1$), $\langle G\cdot\Theta G\rangle_\mu\ge0$. The same holds for direction 4.

Proof We show it in five steps.

Step 1 (splitting the variables). Split the variables into three. $\varphi$: the variables on the planes $P_0\cup P_1$ (the links in directions 1, 2, 4 within the planes, and the $z$ of the plaquettes of types (1,2), (1,4), (2,4) within the planes). $\psi_+$: the right variables (the direction-3 links leaving $0\le x_3\lt M/2$, the links in other directions with $0\lt x_3\lt M/2$, the plaquettes of type $(3,\nu)$ with $[x_3,x_3+1]\subset[0,M/2]$, and the $z$ of in-plane-type plaquettes with $0\lt x_3\lt M/2$). $\psi_-:=\Theta\psi_+$: the left variables. $\Theta$ does not move $\varphi$.

Step 2 (decomposing the weight). The plaquette weight $e^{\frac\beta2z_p\mathrm{tr}U_p}$ belongs to exactly one of right, left and the planes, since $p$ does, and goes into that one. The weights of plaquettes on the planes go into $A(\varphi)$. The cube constraints split in the same way: a cube containing direction 3 spans $[x_3,x_3+1]$ and so depends only on the variables of one side and of the planes; a cube not containing direction 3 depends only on the right variables if $0\lt x_3\lt M/2$, and only on the plane variables if it lies on a plane. The global constraint $n_{12}$, measured on $\Pi\subset P_0$, is a function of $\varphi$. So the weight has the form $W=A(\varphi)\,F(\psi_+,\varphi)\,F'(\psi_-,\varphi)$, with $A\ge0$, $F\ge0$.

Step 3 (the relation between the left and right factors). The action and the constraints are invariant under $\Theta$. Even when the orientation of a plaquette is reversed, in $SU(2)$ we have $\mathrm{tr}\,U_{\Theta p}=\mathrm{tr}\,U_p^{-1}=\mathrm{tr}\,U_p$. Cubes go to cubes, and the plane on which $n_{12}$ is measured is fixed. Hence $F'(\psi_-,\varphi)=F(\Theta\psi_-,\varphi)$.

Step 4 (invariance of the measure). The Haar measure is invariant under taking inverses, and the counting measure of $z$ is invariant under permutations.

Step 5 (positivity). From the above,

∎

The algebra from the last equality to the inequality (in the version with finite sums) is checked by ReflPos in Lean (ReflPos.refl_pos) (§06). The algebra, after splitting into factors, of the Cauchy–Schwarz form that follows from positivity, $\langle G\cdot\Theta H\rangle^2\le\langle G\cdot\Theta G\rangle\langle H\cdot\Theta H\rangle$ (for real $G,H$), is ReflPos (ReflPos.refl_cs). This is an input to the chessboard estimate (13-05).

Theorem 3 is the Osterwalder–Seiler argument for reflections through lattice points Cited (5-03) with the $\mathbb Z_2$ plaquette variables and the closedness constraint added. The key points are that each cube constraint fits on one side plus the planes, and that the global class $n_{12}$ lies in a plane orthogonal to the direction of reflection (3) and can be measured on the plane $P_0$.

Transverse reflections. If we try the same argument with reflections in directions 1 and 2, the twist gets in the way.

Lemma 4 Proved here In $SU(2)$, write $e^{\frac\beta2\mathrm{tr}U}=\sum_jc_j\chi_j(U)$ ($\chi_j$ is the character of spin $j$, $c_j=\frac{2(2j+1)}\beta I_{2j+1}(\beta)\gt0$). Then the twisted plaquette weight is

and the coefficients for half-integer $j$ are negative. Hence, with the twist fixed, the weights of plaquettes straddling the plane of a reflection in the transverse directions 1 and 2 are not of positive type, and the standard argument for reflection positivity does not hold there.

Proof $-1$ is a centre element of $SU(2)$, and $-\mathrm{tr}\,U=\mathrm{tr}(-U)$, $\chi_j(-U)=(-1)^{2j}\chi_j(U)$. Hence (5) follows from $e^{-\frac\beta2\mathrm{tr}U}=e^{\frac\beta2\mathrm{tr}(-U)}=\sum_jc_j\chi_j(-U)$. The formula for $c_j$ and its positivity are the standard expansion via the modified Bessel functions $I_n(\beta)\gt0$ ($\beta\gt0$) Cited. Being a weight of positive type (coefficients of the character expansion at least 0) is the property the standard argument needs for the plaquettes straddling the plane of reflection, and (5) does not satisfy it. ∎

The consequence of Lemma 4 is that reflection positivity gives no convexity or monotonicity in the transverse size $L$. The lateral-widening inequality is a comparison in $L$, so the road to obtaining it from reflections is closed from the start. What Lemma 4 says is that the standard argument cannot be used; it does not show that positivity of transverse reflections fails for all functions.

Lemma 5 (the negative lemma) Proved here The set of structural inputs $\mathcal I$ of §01 (positivity of the transfer matrix, the centre and duality identities (including Lemma 1), reflection positivity in the planar directions (Theorem 3), the symmetries of the lattice, the finite lemmas) holds, with the same proofs, in the four-dimensional Wilson theory in which the group is changed from $SU(2)$ to $\mathbb Z_2$ or U(1) and the twist variables are kept as $\pm1$. Hence, if there were an argument deriving the lateral-widening inequality using $\mathcal I$ alone, the same argument would derive the lateral-widening inequality for $\mathbb Z_2$ and U(1) too. Contrapositively, if the lateral-widening inequality is false for either $\mathbb Z_2$ or U(1), it cannot be derived from this set of inputs alone.

Proof For each input, we check the properties of the group it uses.

• Positivity of the transfer matrix: the Osterwalder–Seiler argument holds for the Wilson action of a compact group, for actions whose character-expansion coefficients are positive. $e^{\beta\,\mathrm{Re}\,U_p}$ for $\mathbb Z_2$ and U(1) satisfies this Cited.
• The centre and duality identities: all that is used is that $-1$ is in the centre of the group (the centre of $\mathbb Z_2$ and of U(1) is the group itself and contains $-1$), and that twist flux is the insertion of a centre transformation. The proof of Lemma 1 goes through as it stands.
• Reflection positivity: the only properties of the group used in the proof of Theorem 3 are $\mathrm{tr}\,U_{\Theta p}=\mathrm{tr}\,U_p^{-1}=\mathrm{tr}\,U_p$ in step 3 and the invariance of the Haar measure under inverses in step 4. For U(1), writing the action with $\mathrm{Re}\,U_p$, $\mathrm{Re}\,U^{-1}=\mathrm{Re}\,\overline U=\mathrm{Re}\,U$; for $\mathbb Z_2$, $U^{-1}=U$; both hold.
• Symmetries of the lattice: the definitions of the action and of the class of the twist have a form independent of the group, so they are the same.
• The finite lemmas: the container identity uses independence conditioned on the frame (the Markov property), the slab estimate uses conditioning on the orbits of flips of the plaquette variables, and the flow lemma uses only stationarity; none uses the non-commutativity of the group.

The first half is shown, so the second half is logic: if an argument uses only the properties of $\mathcal I$, it gives the same conclusion in every theory that satisfies $\mathcal I$. ∎

The premise of Lemma 5, "the lateral-widening inequality is false for $\mathbb Z_2$ or U(1)", is the estimate of 13-03 §02 Reading (physics; the count for the Coulomb phase of U(1) and the count for the wall of $\mathbb Z_2$ at weak coupling). Granting it, the conclusion is that the lateral-widening inequality does not follow from the structural inputs alone. This last step inherits the label of its premise, [Reading] (physics).

The same argument applies as it stands to the weak form ((C′) of 13-02) and to the uniform slab tension. For U(1) and $\mathbb Z_2$ the slab tension is estimated to go to 0 as $L$ grows, so neither follows from the same set of inputs alone Reading (physics).

We also write what Lemma 5 does not say. First, it does not say that $\mathcal I$ is useless for the band. If $L$ is fixed, Theorem 3 replaces frame decorrelation by the chessboard density $\kappa_{\rm cb}$ (13-05). Second, what comes out when one dynamical input is added to $\mathcal I$ is outside Lemma 5. Third, at strong coupling (small $\beta$) U(1) also confines, so what Lemma 5 separates is not the groups themselves but whether there is a non-confining phase.

By Lemma 5, an argument that crosses the band must use at least one input outside $\mathcal I$. That input must be false for U(1) and $\mathbb Z_2$ at weak coupling — the non-confining phase. The candidates that the study of this book can name are all properties of non-abelian dynamics.

Each row of Table 1 is an estimate, [Reading] (physics), and none of the inputs is in a form usable in the band $L\sim1/\Lambda$. On the strong-coupling side, there is a line of argument by a polyhedral expansion in which the lateral-widening inequality holds with the constant $C\approx1$ (there is a report that the factor $(L'/L)^2$ is consistent with the literature on the leading strong-coupling term (Münster 1981), but checking is not yet done). However, an estimate uniform in the planar size $M$ has not been confirmed, and this book places it under [Assumption]. At strong coupling U(1) also confines, so this row does not contradict Lemma 5.

Lemma 5 makes explicit a result that was expected; there is no surprise in it. Its role is to give one touchstone for reading arguments about the band: if an argument proceeds using only the properties of $\mathcal I$, it gives the same conclusion for U(1), so there is an error somewhere. The record in 13-08 of a road that drew the criticism that it does not distinguish U(1) is a real example of this touchstone.

The spectral part of Lemma 1 and the algebra of step 5 of Theorem 3 are five theorems in the Lean 4 files FluxDuality.lean and ReflPos.lean. We read the statements in the words of the text. All of them are statements about sums of real numbers over a finite set $\iota$ (or $\Phi,\Psi$).

FluxDuality.split: for families of reals $t,c:\iota\to\mathbb R$, under the assumption "for every $i$, $c_i=1$ or $c_i=-1$", $\sum_it_i-\sum_ic_it_i=2\sum_{c_i=-1}t_i$ and $\sum_it_i+\sum_ic_it_i=2\sum_{c_i=1}t_i$. In the text $t_i$ are the eigenvalues of $T_3^{\,M}$ and $c_i$ the eigenvalues of $\hat C_4$, and the two formulas are $Z(1,0)\mp Z(1,1)=2Y(1),\,2Y(0)$ of (3).

FluxDuality.twisted_le: under the assumptions "for every $i$, $0\le t_i$" and "$c_i=\pm1$", $\bigl|\sum_ic_it_i\bigr|\le\sum_it_i$. This is $|Z(1,1)|\le Z(1,0)$ of the text.

FluxDuality.bias_mem: under the assumptions "$0\le t_i$", "$c_i=\pm1$", "$0\le\sum_ic_it_i$", "$0\lt\sum_it_i+\sum_ic_it_i$", $0\le\theta\le1$ for $\theta:=(\sum t-\sum ct)/(\sum t+\sum ct)$. The third assumption $Z_{11}:=\sum ct\ge0$ is used for $\theta\le1$; it is not a spectral fact, and corresponds to the positivity of the Boltzmann weight in the text.

ReflPos.refl_pos: for finite sets $\Phi,\Psi$ and real functions $A:\Phi\to\mathbb R$, $F,G:\Psi\to\Phi\to\mathbb R$, under the assumption "for every $\varphi$, $0\le A(\varphi)$", $\sum_\varphi\sum_{\psi_1}\sum_{\psi_2}A(\varphi)F(\psi_1,\varphi)F(\psi_2,\varphi)G(\psi_1,\varphi)G(\psi_2,\varphi)\ge0$. It is the finite-sum version of $\int A\,|\int FG|^2\ge0$ in step 5 of Theorem 3, and the sign of $F$ does not matter.

ReflPos.refl_cs: under the same assumption $0\le A$, for $g(\varphi):=\sum_\psi F(\psi,\varphi)G(\psi,\varphi)$ and $h(\varphi):=\sum_\psi F(\psi,\varphi)H(\psi,\varphi)$, $\bigl(\sum_\varphi A\,g\,h\bigr)^2\le\bigl(\sum_\varphi A\,g^2\bigr)\bigl(\sum_\varphi A\,h^2\bigr)$. It writes the Cauchy–Schwarz form for reflections, $\langle G\cdot\Theta H\rangle^2\le\langle G\cdot\Theta G\rangle\langle H\cdot\Theta H\rangle$, as algebra after the weight has been split into factors.

What machine checking guarantees is only this algebra of finite sums and real numbers. That the transfer matrix is positive, that it commutes with $\hat C_4$, that the twist is the insertion of $\hat C_4$, that the weight splits into the form $A\cdot F\cdot F$ (steps 1–4 of Theorem 3), and the Haar measure lie outside Lean; the proofs in the text and the literature carry them. The axioms of the five theorems are, in the record, the three standard ones (propext, Classical.choice, Quot.sound), and they pass this book's re-check (A-04).

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

[01] CIT TM     positivity of the transfer matrix                     compact group, Wilson action
[02] CIT CEN    twist is a centre insertion, e₄ an eigenvalue         't Hooft
[03] PRV EFLUX  Lemma 1: θ = Y(1)/Y(0) ≥ 0 ⟸ [01][02]                 Lean split, twisted_le
[04] PRV LE1    θ ≤ 1 ⟸ [03]                                          uses Z(1,1) ≥ 0; Lean bias_mem
[05] PRV TANH   Lemma 2: F′ ≤ KF ⇒ θ′ ≤ Kθ                            K ≥ 1
[06] PRV RP     Theorem 3: reflection positivity in z, directions 3, 4  Lean refl_pos, refl_cs (algebra)
[07] CIT CHAR   character-expansion coefficients c_j > 0              SU(2)
[08] PRV TWX    Lemma 4: transverse reflections not of positive type ⟸ [07]  twist fixed
[09] PRV NOGO   Lemma 5: I holds for Z₂, U(1) too ⟸ [01][03][06]      from this set of inputs alone
[10] RDG FAIL   lateral widening false for U(1), Z₂                   13-03, estimate
[11] RDG NEED   does not follow from structural inputs alone ⟸ [09][10]
[12] PRV NOCVX  no convexity or monotonicity in L from reflections ⟸ [08]
[13] ASM SC     C ≈ 1 at strong coupling                              line of argument; uniformity in M unconfirmed
[14] RDG EL     Y(1)/Y(0) small in the confining regime               estimate

The check is to see by eye that every number to the right of a "⟸" is smaller than the number on the left. There are no cycles. The nodes that stand on paper are [03]–[06], [08], [09] and [12], and they rest on the literature rows [01], [02], [07]. Only the claim of the chapter title, [11], has the estimate row [10] as a premise, and it is "Reading (physics)". The assumption row [13] is not a premise of any conclusion.

What this chapter can say

Sources and reproduction

Terms used in this chapter — structural inputs, the transfer matrix, the centre transformation, the electric flux $e_4$, the sector bias $\theta$, the z ensemble, reflection through lattice points, positive type, the negative lemma, the lateral-widening inequality, touchstone. Their meanings are in A-02 Glossary and A-06 Index of assumptions.

Next chapter: 13-05 The chessboard and simultaneous flips — using Theorem 3 to replace frame decorrelation by a single density.

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