Variational lattice-gap bounds: plaquettes, electric trials and holonomies

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For a specified smooth gauge-invariant link function, the Feynman–Bijl identity bounds the physical Kogut–Susskind gap above by its electric gradient norm over its variance. Plaquettes give the strong-coupling order; an averaged spatial Polyakov loop gives a complementary exact susceptibility bound. Replacing its expectations by constant-mode values is a separate unproved small-volume step.

An abelian electric trial gives a structure-factor gaplessness criterion. For a semisimple gauge Lie algebra, a nonzero electric trial with constant colour coefficients is excluded by Gauss’s law. This exclusion neither proves a positive gap nor a universal string energy for all dressed trials. Continuum smearing and positive-measure moments have their own canonical treatment.

The Feynman–Bijl inequality

Let \(H\) be self-adjoint and bounded below with simple ground state \(\psi_0\) and next eigenvalue \(E_1\), and let \(O\) be symmetric with \(\psi_0\in D(O)\), \(O\psi_0\in D(H)\) and \(\langle\psi_0,O\psi_0\rangle=0\). Then \(O\psi_0\perp\psi_0\), so by the min-max principle \[E_1-E_0\ \le\ \frac{\langle O\psi_0,(H-E_0)O\psi_0\rangle}{\|O\psi_0\|^2} =\frac{\tfrac12\langle\psi_0,[O,[H,O]]\psi_0\rangle}{\langle\psi_0,O^2\psi_0\rangle},\] using \([O,[H,O]]=2OHO-O^2H-HO^2\) and \((H-E_0)\psi_0=0\). If \(O\) commutes with the Gauss generators, \(O\psi_0\) lies in the physical sector and the same bound holds for \(\Delta^{\rm phys}\). (Feynman, Phys. Rev. 94 (1954) 262, metadata; the argument is the two lines above.)

Link functions and strong-coupling order

Take \(H=\frac{\hbar c}{a}[\frac{g^2}{2}\sum_\ell(-\Delta_\ell)+\frac2{g^2}\sum_p(N-\operatorname{Re}\operatorname{tr}U_p)]\) as in the obligations map §1 and a real function \(O=f(U)\) with \(\langle f\rangle=0\). Then \([V,f]=0\), and for the Laplacian \(\Delta_\ell=\sum_aX_a^2\) in the left-invariant fields \(X_a\) of link \(\ell\), \[[f,[-\Delta_\ell,f]]=2\sum_a(X_af)^2=:2|\nabla_\ell f|^2,\] because \([\Delta,f]h=(\Delta f)h+2\sum_a(X_af)(X_ah)\) and the second commutator with \(f\) removes the \((\Delta f)\) terms and leaves \(-2\sum_a(X_af)^2h\). Hence

Proposition 1. For every real gauge-invariant \(f(U)\) with \(\langle f\rangle_0=0\), \[\Delta^{\rm phys}_{a,L}\ \le\ \frac{\hbar cg^2}{2a}\; \frac{\sum_\ell\langle|\nabla_\ell f|^2\rangle_0}{\langle f^2\rangle_0}.\]

Corollary 2 (plaquette). With \(f=\operatorname{Re}\operatorname{tr}U_p-\langle\operatorname{Re}\operatorname{tr}U_p\rangle_0\): for a link \(\ell\) of \(p\), \(X_a\operatorname{tr}U_p=\operatorname{tr}(T^aM)\) up to a phase, with \(M\) a cyclic rearrangement of \(U_p\), so by the \(SU(N)\) completeness relation \(\sum_a\operatorname{tr}(T^aM)\operatorname{tr}(T^aM^\dagger)=\tfrac12[\operatorname{tr}MM^\dagger-\tfrac1N|\operatorname{tr}M|^2]\le\tfrac N2\), and \(|\nabla_\ell\operatorname{Re}\operatorname{tr}U_p|^2\le|\nabla_\ell\operatorname{tr}U_p|^2\le N/2\). Four links give \(\sum_\ell|\nabla_\ell f|^2\le2N\) and \[\Delta^{\rm phys}_{a,L}\ \le\ \frac{\hbar c\,g^2N}{a\,\operatorname{Var}_0(\operatorname{Re}\operatorname{tr}U_p)} .\]

Corollary 3 (two-sided strong-coupling order). At \(g=\infty\) the ground state is the Haar measure and \(\operatorname{Var}(\operatorname{Re}\operatorname{tr}U_p)=\tfrac12\langle|\operatorname{tr}U|^2\rangle+\tfrac12\operatorname{Re}\langle(\operatorname{tr}U)^2\rangle\) equals \(1\) for \(SU(2)\) and \(\tfrac12\) for \(SU(N\ge3)\), by Peter–Weyl orthogonality of the fundamental character and the vanishing of \(\int(\operatorname{tr}U)^2dU\) for \(N\ge3\). The ground-state expectation of a local function is continuous in \(g^{-4}\) at \(g=\infty\) uniformly in the volume (Yarotsky’s Theorem 1, conclusion 4, analyticity of \(\omega(A)\) in the perturbation parameter; the finite-volume statement is analytic perturbation theory of a simple isolated eigenvalue), so there is \(g_1(N)\) with \(\operatorname{Var}_0\ge\tfrac14\) for \(g\ge g_1\). Combined with the strong-coupling note, \[\gamma\,\frac{g^2}{2}\,\frac{N^2-1}{2N}\,\frac{\hbar c}{a}\ \le\ \Delta^{\rm phys}_{a,L}\ \le\ 4N\,g^2\,\frac{\hbar c}{a} \qquad(g\ge\max(g_0,g_1)),\] uniformly in the lattice size. The strong-coupling gap is \(g^2\hbar c/a\) times a number bounded on both sides.

For \(SU(3)\) this reads \[\frac{2\gamma}{3}g^2\frac{\hbar c}{a}\le \Delta^{\rm phys}_{a,L}\le12g^2\frac{\hbar c}{a},\] with ratio \(18/\gamma\) between the sufficient bounds and both thresholds still required. The defining representation is complex, so charge conjugation also distinguishes \(C\)-even and \(C\)-odd channels. A \(C\)-even plaquette insertion need not see the global lowest excitation.

Why the plaquette says nothing about the continuum. As \(a\to0\) along the scaling curve, a weak-field diagnostic predicts a vanishing local plaquette variance (of order \(g^4\) when the plaquette angle is of order \(g\)). In that approximation the right side of Corollary 2 grows like \(\hbar c/(ag^2)\) and gives no finite physical upper bound. This is a limitation of the trial bound, not a determination of the actual gap. Writing \(R_O=\sum_\ell\langle|\nabla_\ell O|^2\rangle/\operatorname{Var}O\), the exact prefactor in Theorem 1 requires \(g^2R_O=O(a/\ell)\) to obtain an upper bound of order \(\hbar c/\ell\) at a fixed physical length \(\ell\). Thus \(R_O=O(a/(g^2\ell))\) suffices; omitting \(g^2\) gives the wrong scaling requirement. The smeared-susceptibility treatment states the additional continuum hypotheses.

Abelian electric trials and the structure factor

For compact \(U(1)\) write \(U_\ell=e^{i\theta_\ell}\), \(E_\ell=-i\partial/\partial\theta_\ell\), \(\theta_p=\sum_{\ell\in\partial p}\epsilon_{p\ell}\theta_\ell\), and \[H=\frac{\hbar c}{a}\Big[\frac{g^2}{2}\sum_\ell E_\ell^2+\frac1{g^2}\sum_p(1-\cos\theta_p)\Big]\] (other conventions rescale \(g^2\)). Gauss’s law generators \(G_x=\sum_{\ell\ni x}\pm E_\ell\) commute with every \(E_\ell\), so

Lemma 4. \(O_\varphi=\sum_\ell\varphi_\ell E_\ell\) with real \(\varphi\) is gauge invariant, and \(\langle O_\varphi\rangle_0=0\).

Proof. The \(E_\ell\) commute among themselves. Charge conjugation \(\theta\mapsto-\theta\) is a unitary symmetry of \(H\) under which the unique ground state is invariant and \(E_\ell\mapsto-E_\ell\). \(\square\)

With \([E_\ell,h(\theta)]=-i\partial_\ell h\) and the lattice curl \((d\varphi)_p=\sum_{\ell\in\partial p}\epsilon_{p\ell}\varphi_\ell\): \([O_\varphi,\cos\theta_p]=i(d\varphi)_p\sin\theta_p\), \([O_\varphi,\sin\theta_p]=-i(d\varphi)_p\cos\theta_p\), hence \[[O_\varphi,[H,O_\varphi]]=\frac{\hbar c}{ag^2}\sum_p(d\varphi)_p^2\cos\theta_p,\] and the variational identity above gives \[\Delta^{\rm phys}\le\frac{\hbar c}{2ag^2}\,\frac{\sum_p(d\varphi)_p^2\,c_0}{\langle O_\varphi^2\rangle_0}, \qquad c_0=\langle\cos\theta_p\rangle_0,\] where \(c_0\) is independent of \(p\) by translation invariance of the unique ground state. To bound \(\langle O_\varphi^2\rangle\) below use the uncertainty relation with the gauge-invariant \(\Theta=\sum_p\psi_p\sin\theta_p\): \(\langle O^2\rangle\langle\Theta^2\rangle\ge\tfrac14|\langle[O,\Theta]\rangle|^2\) with \(\langle[O_\varphi,\Theta]\rangle=-ic_0\sum_p\psi_p(d\varphi)_p\). Choosing \(\psi=d\varphi\):

Theorem 5. If \(c_0>0\), then for every real \(\varphi\) on the links with \(d\varphi\ne0\), \[\Delta^{\rm phys}_{a,L}\ \le\ \frac{2\hbar c}{a\,g^2\,c_0}\; \frac{\langle\Theta_\varphi^2\rangle_0}{\sum_p(d\varphi)_p^2}, \qquad\Theta_\varphi=\sum_p(d\varphi)_p\sin\theta_p .\] For a plane wave \(\varphi_\ell=\cos(k\cdot x_\ell)\,u_{i(\ell)}\) the ratio is the structure factor \(\hat S(k)\) of the plaquette sine at momentum \(k\), and \(\Delta^{\rm phys}\le2\hbar c\,\hat S(k)/(ag^2c_0)\) for every \(k\ne0\) allowed by the box.

Proof. Insert the uncertainty bound into the previous display: \(\langle O^2\rangle\ge c_0^2(\sum_p(d\varphi)_p^2)^2/(4\langle\Theta^2\rangle)\). \(\square\)

Reading. In a Coulomb phase \(\sin\theta_p\) is a curl of a nearly Gaussian field and \(\hat S(k)\to0\) as \(k\to0\); the theorem then forces \(\Delta\to0\) with the box, which is the gaplessness Guth and Fröhlich–Spencer prove at weak coupling (abstract, B78), obtained here as a consequence of two ground-state quantities, \(c_0\) and \(\hat S\). In a confining phase \(\hat S(0)>0\) and the bound is finite, consistent with the strong-coupling gap. The theorem does not decide which phase occurs; it shows that vanishing of this structure factor is a sufficient criterion for gaplessness in the specified channel. A nonvanishing upper bound gives no converse lower-gap criterion.

Semisimple Gauss law and the limits of the dressing argument

Proposition 6. Let \(G\) have semisimple Lie algebra and \(O=\sum_\ell\sum_a\varphi^a_\ell E^a_\ell\) with c-number coefficients. If \(O\) commutes with all Gauss generators, then \(\varphi=0\).

Proof. Write \(E^a_\ell=L^a_\ell\) for the left generators of link \(\ell\), which satisfy \([L^b_\ell,L^a_\ell]=if^{bac}L^c_\ell\), and let \(R^b_\ell\) be the right generators, which commute with all \(L^a_\ell\). The Gauss generator at \(x\) is \(G^b_x=\sum_{s(\ell)=x}L^b_\ell+\sum_{t(\ell)=x}R^b_\ell\) up to the sign convention. Then \([G^b_x,O]=i\sum_{s(\ell)=x}f^{bac}\varphi^a_\ell L^c_\ell\), with no contribution from the entering links. The \(L^c_\ell\) are linearly independent operators, so \([G^b_x,O]=0\) for all \(b,x\) forces \(f^{bac}\varphi^a_\ell=0\) for all \(b,c\) and every link, that is \([\varphi_\ell,T^b]=0\) for all \(b\): \(\varphi_\ell\) lies in the centre of the Lie algebra, which is zero under the semisimplicity hypothesis. \(\square\)

The proof identifies the exact boundary: coefficients must lie in the Lie algebra centre. A nonabelian group with a \(U(1)\) factor still admits central abelian trials; the former unrestricted “nonabelian” statement was too broad.

Field-dependent gauge-covariant coefficients can produce invariant electric trials, with ordering/domain obligations. A Wilson-loop dressing is one example; local curvature coefficients are another. For such an operator, \([K,O]\) generally contributes to the double commutator, but \(O\) is a differential operator rather than a multiplication function. Proposition 1’s gradient formula cannot be substituted for that extra term. The former universal positive cost proportional to \(g^2|C|\) was not established. Gauge invariance removes the specified c-number semisimple trial; it proves neither a mandatory long string nor a nonvanishing dressing energy or lower gap.

Spatial Polyakov-loop trials

The averaged holonomy and its exact bound

Fix a direction \(i\). The lattice decomposes into \(N_s^2\) lines parallel to \(i\), each of \(N_s\) links closing around the torus. For a line \(\lambda\) let \(P_\lambda=\operatorname{Re}\operatorname{tr}\prod_{\ell\in\lambda}U_\ell\), the ordered product along the line; it is gauge invariant (a closed loop), and \(\bar P_i=N_s^{-2}\sum_\lambda P_\lambda\). Let \(O=\bar P_i-\langle\bar P_i\rangle_0\).

Polyakov-loop bound. For every \(g>0\), \(N_s\ge2\) and \(SU(N)\) with the normalization of the obligations map, \[\Delta^{\rm phys}_{a,L}\ \le\ \frac{\hbar cg^2}{2a}\, \frac{\big\langle\sum_\ell|\nabla_\ell\bar P_i|^2\big\rangle_0}{\operatorname{Var}_0\bar P_i} \ \le\ \frac{\hbar c\,g^2N}{4L\,\operatorname{Var}_0\bar P_i}.\]

Proof. \(O\) is a real gauge-invariant function of the links with zero mean, so Proposition 1 of the upper-bound note applies and gives the first inequality. Each link in direction \(i\) belongs to exactly one line and links in other directions do not enter \(\bar P_i\), so \(\nabla_\ell\bar P_i=N_s^{-2}\nabla_\ell P_{\lambda(\ell)}\). For a link of \(\lambda\), \(X_a\operatorname{tr}\prod U=\operatorname{tr}(T^aM)\) up to a phase with \(M\) a cyclic rearrangement of the product, and the completeness relation gives \(\sum_a|\operatorname{tr}(T^aM)|^2=\tfrac12(N-|\operatorname{tr}M|^2/N)\le N/2\); the real part has smaller gradient. Hence \(\sum_\ell|\nabla_\ell\bar P_i|^2\le N_s^{-4}\cdot N_s^2\cdot N_s\cdot N/2=N/(2N_s)\), and \(\frac{\hbar cg^2}{2a}\cdot\frac{N}{2N_s}=\frac{\hbar cg^2N}{4L}\). \(\square\)

Conditional small-volume comparison

For the formal weak-coupling constant-mode comparison in one central holonomy patch, assume the gauge-invariant content of a Polyakov line is the holonomy of the constant mode: \(\prod_{\ell\in\lambda}U_\ell\approx\exp(iLa_i)\) with \(a_i=a_i^bT^b\) the constant mode of G07 Proposition 7, whose ground-state width is \(|a_i|\sim g^{2/3}/L\) in natural units. Since \(\operatorname{tr}T^b=0\), \[P_\lambda\approx\operatorname{Re}\operatorname{tr}e^{iLa_i}=N-\tfrac14L^2|\vec a_i|^2+O(L^4|a|^4),\qquad \nabla_\ell P_\lambda\approx\operatorname{Im}\operatorname{tr}(T^aLa_i)=\tfrac12La_i^a ,\] so the variance of \(\bar P_i\) is second order and the gradient first order in the constant mode: \[\operatorname{Var}_0\bar P_i\approx\tfrac1{16}L^4\operatorname{Var}(|\vec a_i|^2),\qquad \Big\langle\sum_\ell|\nabla_\ell\bar P_i|^2\Big\rangle_0\approx N_s^{-4}\cdot N_s^3\cdot\tfrac14L^2\langle|\vec a_i|^2\rangle =\frac{L^2\langle|\vec a_i|^2\rangle}{4N_s}.\] For the \(SU(2)\) constant-mode model of C133, \(La_i=g^{2/3}\xi_i\) with \(\xi_i\) distributed by the unit ground state \(\psi_0^{(3)}\) of \(\tfrac12(-\Delta_{\mathbb R^9}+\sum_{i<j}|\vec\xi_i\times\vec\xi_j|^2)\): \[\frac{\hbar cg^2}{2a}\cdot\frac{g^{4/3}\langle|\vec\xi_i|^2\rangle/(4N_s)}{g^{8/3}\operatorname{Var}(|\vec\xi_i|^2)/16} =\frac{\hbar c}{L}\,g^{2/3}\cdot\frac{2\langle|\vec\xi_i|^2\rangle}{\operatorname{Var}(|\vec\xi_i|^2)} .\] The bound is therefore \((\hbar c/L)g^{2/3}\) times the pure number \(2\langle|\vec\xi_i|^2\rangle/\operatorname{Var}(|\vec\xi_i|^2)\), finite because the unit ground state decays faster than any power along the valleys (the confining bound of C133 gives exponential decay of \(\psi_0\) in \(\sum_i|\vec\xi_i|\)). Consistency with the exact zero-mode gap \(\delta_1^{(3)}g^{2/3}\hbar c/L\) requires this number to be at least \(\delta_1^{(3)}\), which is the Feynman–Bijl inequality applied inside the zero-mode model with the trial operator \(|\vec\xi_i|^2\); the model version of the Polyakov-loop bound is exact and says exactly that.

The fundamental-loop calculation above is a one-patch diagnostic. On the full periodic \(SU(2)\) lattice, a centre transformation across a dual plane leaves the Hamiltonian invariant and changes every winding \(P_\lambda\) in that direction to \(-P_\lambda\). The unique positive ground state is centre invariant, hence \(\langle\bar P_i\rangle_0=0\), not the one-patch value near \(2\). Therefore the one-patch variance cannot be substituted into the global ground-state formula. The exact fundamental-loop inequality remains valid and can probe a charged sector.

For a centre-neutral trial in \(SU(2)\) use instead \[\bar A_i=N_s^{-2}\sum_\lambda(P_\lambda^2-1),\] the averaged adjoint character. Its centered multiplication preserves the trivial centre sector. Since each link enters only one winding line, \(\nabla_\ell\bar A_i=2P_{\lambda(\ell)}\nabla_\ell\bar P_i\) and \(|P_\lambda|\le2\), so \[\sum_\ell|\nabla_\ell\bar A_i|^2\le16/N_s,\qquad \Delta^{(0)}_{a,L}\le\frac{8\hbar c\,g^2} {L\operatorname{Var}_0\bar A_i}.\] Here \(\Delta^{(0)}\) is the excitation gap in the trivial centre sector; Rayleigh–Ritz applies there because the trial stays in that space. Near either central holonomy \(P_\lambda=\pm2\), \(A_\lambda-3\simeq\pm4(P_\lambda\mp2)\) in the corresponding patch. The gradient and variance both gain the same factor \(16\), leaving the displayed conditional \(g^{2/3}\) ratio unchanged. Matching that local neutral distribution and the other modes to the field theory remains unproved. A neutral-sector upper bound is distinct from the full gap and from a lower bound; the periodic benchmark makes the smaller inter-sector scale explicit.

Infinite-volume susceptibility condition

Suppose the lattice theory at fixed \(a\) and \(g\) has infinite-volume gap \(\Delta_a=\liminf_{N_s\to\infty}\Delta_{a,L}>0\) (T2\('\)), and write \(mc^2=\Delta_a\). the Polyakov-loop bound gives, for every \(L\), \[\operatorname{Var}_0\bar P_i\ \le\ \frac{\hbar cg^2N}{4L\,\Delta_{a,L}} .\] By translation invariance in the transverse plane, \(\operatorname{Var}_0\bar P_i=N_s^{-2}\sum_{y}C_i(y)\) with \(C_i(y)=\langle\operatorname{Re}P_\lambda(x)\operatorname{Re}P_\lambda(x+y)\rangle_c\) the connected two-point function of the spatial Polyakov loop at transverse separation \(y\). Hence

Susceptibility consequence. If the infinite-volume gap is \(mc^2>0\), then along any sequence of boxes on which \(\Delta_{a,L}\ge mc^2/2\), \[\sum_{y\in\text{transverse plane}}a^2\,C_i(y)\ \le\ \frac{\hbar cg^2N}{2mc^2}\,L ,\] and the averaged spatial Polyakov loop self-averages at least like \(1/L\).

Reading: in a confining phase the spatial Polyakov loop of length \(L\) has connected correlations decaying on a transverse scale \(\xi\), and the left side is of order \(\xi^2\langle P^2\rangle_c\), so the corollary is satisfied with room; in a phase where the loop orders or its correlations do not decay, the left side grows like \(L^2\) and the corollary forbids a gap. It is a necessary condition only, but it is one written entirely in terms of a single gauge-invariant correlation and the coupling, with no reference to the spectrum.

The Haar strong-coupling comparison

At \(g=\infty\) the links are independent Haar variables, distinct lines are independent, and \(\langle|\operatorname{tr}U_1\cdots U_{N_s}|^2\rangle=1\) for a product of independent Haar elements of \(SU(N)\) (the product is again Haar distributed), so \(\operatorname{Var}P_\lambda=\tfrac12\) for \(N\ge3\) and \(\operatorname{Var}\bar P_i=1/(2N_s^2)\). the Polyakov-loop bound then reads \(\Delta\le\hbar cg^2NN_s^2/(2L)\), which grows with the lattice and says nothing. The averaged loop is the wrong trial operator in the disordered phase, as the plaquette is in the ordered one; the two bounds are complementary, and neither is uniform in \(a\) along the scaling curve.

Consequence for STATE

The exact upper bounds retain their specified trial states, groups, variances, units and finite-volume domains. Strong-coupling order, a vanishing abelian structure factor and a Polyakov susceptibility constraint answer different questions. Neither a nonzero upper bound nor the absence of one constant-colour trial establishes a lower gap. The small-volume spectral transport and the continuum equal-time smearing hypotheses remain open.