The weak-coupling gap in a small box reduces to three inequalities through the Schur complement

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A lower bound \(\Delta\ge\delta_1^{(3)}g^{2/3}(1-Cg^{2/3})\,\hbar c/L\) for the torus Hamiltonian with a momentum cutoff follows from a proved gap-transfer lemma and three inequalities about polynomial operators on finitely many oscillators. The lemma: if \(P\) is an orthogonal projection, \(D=\bar PH\bar P\ge\mu\) on \(\operatorname{ran}\bar P\), and the Schur-complement error \(\eta=\sup_{E\le a_1}\|(D-E)^{-1/2}\bar PHP\|^2\) satisfies \(a_1-\eta<\mu\), then the two lowest eigenvalues of \(H\) obey \(a_0-\eta\le E_0\le a_0\) and \(E_1\ge a_1-\eta\), so \(\operatorname{gap}(H)\ge\operatorname{gap}(PHP)-\eta\). Applied with \(P\) the projection onto the ground state of the nonzero momentum modes, the three inequalities are: (H1) exciting any nonzero mode costs at least \((2\pi\hbar c/L)(1-Cg^{2/3})\) in the presence of the constant modes and the couplings; (H2) the coupling between the constant-mode sector and the excited nonzero-mode sector has Schur error \(\eta\le Cg^{4/3}\hbar c/L\), which requires the projection to follow the constant mode (Section 3a) and a gauge-invariant cutoff; (H3) the effective constant-mode Hamiltonian \(PHP\), which is C133’s operator plus the nonzero-mode zero-point energy as a function of the constant modes, has gap \(\ge\delta_1^{(3)}g^{2/3}(1-Cg^{2/3})\hbar c/L\). The scaling analysis of the cubic and quartic couplings shows why each inequality is expected with the stated powers, and identifies the smallness parameter as \(g^{2/3}\), Lüscher’s expansion parameter. The lemma is proved; the three inequalities are the open content of the lower side of the small-volume corner. Constants are kept explicit with the conventions of G07 Proposition 7: \(g\) dimensionless, fields of dimension inverse length, every energy a multiple of \(\hbar c/L\). Nothing here is promoted.

1. The gap-transfer lemma

Let \(H\) be self-adjoint, bounded below, with compact resolvent on a Hilbert space \(\mathcal H\), and \(P\) an orthogonal projection with \(P\,D(H)\subset D(H)\); write \(\bar P=1-P\), \(A=PHP|_{\operatorname{ran}P}\), \(D=\bar PH\bar P|_{\operatorname{ran}\bar P}\), \(B=PH\bar P\). Let \(a_0\le a_1\le\cdots\) be the eigenvalues of \(A\) and \(E_0\le E_1\le\cdots\) those of \(H\).

Lemma 1. Suppose \(D\ge\mu\) and, for some \(\eta\ge0\), \(\|(D-E)^{-1/2}B^*\|^2\le\eta\) for all \(E\le a_1\), with \(a_1<\mu\). Then \[a_0-\eta\le E_0\le a_0,\qquad E_1\ge a_1-\eta,\qquad E_1-E_0\ \ge\ (a_1-a_0)-\eta .\]

Proof. For \(E<\mu\), \(D-E\) is invertible and the Schur complement of \(H-E\) with respect to the decomposition \(\operatorname{ran}P\oplus\operatorname{ran}\bar P\) is \(F(E)=A-E-B(D-E)^{-1}B^*\) on \(\operatorname{ran}P\). The block factorization \[H-E=\begin{pmatrix}1&B(D-E)^{-1}\\0&1\end{pmatrix} \begin{pmatrix}F(E)&0\\0&D-E\end{pmatrix} \begin{pmatrix}1&0\\(D-E)^{-1}B^*&1\end{pmatrix}\] is a congruence, so by Sylvester’s law of inertia (the number of negative eigenvalues, finite here because \(H\) has compact resolvent and \(E<\mu\le\inf D\)) the number of eigenvalues of \(H\) below \(E\) equals the number of negative eigenvalues of \(F(E)\), since \(D-E>0\) contributes none. The hypothesis gives \(0\le B(D-E)^{-1}B^*\le\eta\), hence \(A-E-\eta\le F(E)\le A-E\) and, by min-max, the number of negative eigenvalues of \(F(E)\) lies between \(\#\{k:a_k<E\}\) and \(\#\{k:a_k<E+\eta\}\). Taking \(E\) slightly above \(a_0\) shows \(E_0\le a_0\); taking \(E=a_0-\eta\) shows no eigenvalue of \(H\) lies below \(a_0-\eta\); taking \(E=a_1-\eta\) shows at most one eigenvalue of \(H\) lies below \(a_1-\eta\), so \(E_1\ge a_1-\eta\). \(\square\)

The lemma transfers a gap from the compressed operator \(PHP\) to \(H\) at the price \(\eta\), which is the size of the coupling between \(\operatorname{ran}P\) and its complement measured against the energy \(D-E\) needed to leave \(\operatorname{ran}P\). Everything below is the choice of \(P\) and the estimation of \(a_1-a_0\), \(\mu\) and \(\eta\).

2. The cutoff torus Hamiltonian and the projection

On \(T^3_L\) in temporal gauge, with \(SU(2)\) for definiteness, momentum cutoff \(|k|\le\Lambda\) and the convention of G07 Proposition 7 (\(S=-\frac1{4g^2}\int F^aF^a\)), decompose \(A_i(x)=a_i+\tilde A_i(x)\) into the constant mode and the nonzero modes \(\tilde A_i(x)=L^{-3}\sum_{0<|k|\le\Lambda}\hat A_i(k)e^{ik\cdot x}\), with conjugate momenta \(p_i\) and \(\tilde\Pi_i\). Then \[H=H_0+H_\perp+W,\] \[H_0=\frac{g^2c}{2\hbar L^3}\sum_i|\vec p_i|^2+\frac{\hbar cL^3}{2g^2}\sum_{i<j}|\vec a_i\times\vec a_j|^2,\qquad H_\perp=\sum_{0<|k|\le\Lambda}\Big[\frac{g^2c}{2\hbar}|\hat\Pi(k)|^2+\frac{\hbar c}{2g^2}|k\times\hat A(k)|^2\Big]\] (with \([a^a_i,p^b_j]=i\hbar\delta_{ij}\delta^{ab}\) and the corresponding normalization of \(\hat\Pi\), so that the free part has energies \(\hbar c|k|\)), and \(W\) collects every term of \(\frac{\hbar c}{2g^2}\int|F_{ij}|^2\) that is neither purely constant-mode nor quadratic in \(\tilde A\) at \(a=0\): \[W=W_2+W_3+W_4,\quad W_2=\frac{\hbar c}{g^2}\int\big(\partial_i\tilde A_j-\partial_j\tilde A_i\big)\cdot\big([a_i,\tilde A_j]-[a_j,\tilde A_i]\big)+\frac{\hbar c}{2g^2}\int\big|[a_i,\tilde A_j]-[a_j,\tilde A_i]\big|^2,\] \[W_3=\frac{\hbar c}{g^2}\int\bar F_{ij}\cdot[\tilde A_i,\tilde A_j],\qquad \bar F_{ij}=\partial_i\tilde A_j-\partial_j\tilde A_i+[a_i,\tilde A_j]-[a_j,\tilde A_i]+[a_i,a_j],\] \[W_4=\frac{\hbar c}{2g^2}\int\big|[\tilde A_i,\tilde A_j]\big|^2 ,\] where the purely constant part \(\frac{\hbar cL^3}{2g^2}|[a_i,a_j]|^2\) of \(\frac{\hbar c}{2g^2}\int|F|^2\) is in \(H_0\) and \(W_4\) is quartic in the nonzero modes. The Gauss constraint is imposed on states; the constant gauge transformations act on \(a_i\) by the adjoint action and on the nonzero modes covariantly, and the nonconstant ones are fixed by keeping the transverse nonzero modes, the standard small-volume gauge choice.

Let \(\Omega_\perp\) be the ground state of \(H_\perp\), the Gaussian vacuum of the free transverse modes, with energy \(E_\perp=\hbar c\sum_k|k|\) (a constant that cancels from gaps) and first excitation energy \(\mu_\perp=2\pi\hbar c/L\). Set \[P=1_{\text{const}}\otimes|\Omega_\perp\rangle\langle\Omega_\perp| .\] \(P\) commutes with the constant gauge transformations because \(\Omega_\perp\) is invariant under them, so Lemma 1 can be applied inside the physical sector. The compression is \[A=PHP=H_0+\langle\Omega_\perp,W\Omega_\perp\rangle(a)+E_\perp,\] where \(\langle\Omega_\perp,W_2\Omega_\perp\rangle(a)\) is a function of the constant modes only (the Gaussian expectation of a quadratic form in \(\tilde A\) with \(a\)-dependent coefficients), \(\langle W_3\rangle\) vanishes (odd in \(\tilde A\)), and \(\langle W_4\rangle\) is a constant. Thus \(A\) is C133’s operator plus a potential \(U(a)=\langle W_2\rangle(a)\), which is the leading change of the nonzero-mode zero-point energy with the constant background.

3. The three inequalities and their expected sizes

Rescale \(\vec a_i=g^{2/3}L^{-1}\vec\xi_i\) and \(\hat A(k)=g\,\hat A'(k)\) so that \(H_0=g^{2/3}(\hbar c/L)\,h_3(\xi)\) with \(h_3\) the unit operator of C133 and \(H_\perp\) is \(g\)-independent in the primed variables. Then, counting powers of \(g\) at fixed \(L\) and at typical values \(|\xi|=O(1)\), \(|\hat A'|=O(1)\):

With these sizes the three inequalities that Lemma 1 needs are:

(H1) \(D=\bar PH\bar P\ \ge\ E_\perp+\frac{2\pi\hbar c}{L}(1-C_1g^{2/3})\) on > \(\operatorname{ran}\bar P\): any state with an excited nonzero mode costs > at least one quantum \(2\pi\hbar c/L\) up to the shift produced by \(W\) in a background \(|a|\sim g^{2/3}/L\).

(H2) \(\eta=\sup_{E\le a_1}\|(D-E)^{-1/2}\bar PWP\|^2\ \le\ C_2\,g^{4/3}\hbar c/L\). > The leading contribution is the \(W_2\) term linear in \(a\) (order > \(g^{2/3}\hbar c/L\), creating one or two nonzero quanta) squared and > divided by \(2\pi\hbar c/L\); the \(W_3\) term of order \(g\hbar c/L\) > contributes \(g^2\hbar c/L\).

(H3) \(\operatorname{gap}(A)=\operatorname{gap}\big(H_0+U(a)\big)\ \ge\ \delta_1^{(3)}g^{2/3}(1-C_3g^{2/3})\,\hbar c/L\). > \(U(a)\) is even in \(a\) (charge conjugation), its linear term vanishes, > and its quadratic term is of order \(g^{4/3}\hbar c/L\), a relatively bounded perturbation of \(h_3\) in the rescaled variables.

Given (H1)–(H3), Lemma 1 with \(\mu=E_\perp+\frac{2\pi\hbar c}{L}(1-C_1g^{2/3})\) and \(a_1-\eta<\mu\) (true for small \(g\) since \(a_1-a_0=O(g^{2/3}\hbar c/L)\) and \(a_0-E_\perp=O(g^{2/3}\hbar c/L)\)) yields \[\Delta\ \ge\ \delta_1^{(3)}\,g^{2/3}\,\frac{\hbar c}{L}\,\big(1-Cg^{2/3}\big).\] This is the lower side of the small-volume corner S, matching the upper side obtained in the Polyakov-average note up to pure numbers, and it is the leading term of Lüscher’s expansion (abstract, B78) as a two-sided bound.

3a. Correction: the projection must follow the constant mode, and the cutoff must be gauge invariant

The counting in the table is right for the sizes of the operators but (H2) fails for the projection \(P\) of Section 2 when the cutoff is a sharp momentum cutoff. The term of \(W_2\) linear in \(a\) creates pairs of nonzero quanta \((k,-k)\) from \(\Omega_\perp\) with amplitude of order \(g^{2/3}|\xi|/L\) for every \(k\), so \(\|\bar PW_2P\psi\|^2\sim(g^{4/3}/L^2)(\Lambda L)^3|\xi|^2\) and, after dividing by the pair energy \(2|k|\), \[\eta\ \sim\ \frac{g^{4/3}\hbar c}{L^2}\sum_{0<|k|\le\Lambda}\frac{|\xi|^2}{2|k|} \ \sim\ \frac{g^{4/3}\hbar c}{L}\,(\Lambda L)^2\,|\xi|^2 ,\] quadratically divergent in the cutoff. This is the second-order energy shift \(-c\,|a|^2\Lambda^2L^3\) of the constant mode. A quadratic term in \(a\) is allowed by the constant gauge transformations, which act by conjugation and leave \(\operatorname{tr}a_i^2\) invariant, and the exact one-loop potential of the constant modes on the torus does contain one: the sum \(\tfrac12\sum_k[\omega_k(a)-\omega_k(0)]\) is a periodic function of the holonomy angles \(La_i\) (shift of the summation variable), so its small-\(a\) expansion is \(\hbar c\,[c_2(La)^2/L+O((La)^4/L)]\) with a finite pure number \(c_2\), the Casimir-type Weiss potential, of order \(g^{4/3}\hbar c/L\) in the rescaled variables and cutoff independent. The quadratic \(\Lambda^2\) divergence above is an artefact of the sharp momentum cutoff, which breaks the periodicity in \(La\); a periodic (lattice) cutoff, or any regularization compatible with the large gauge transformations, gives the finite coefficient. Two changes make the reduction usable.

Fibered projection. \(H_\perp+W_2\) is quadratic in \(\tilde A\) at fixed \(a\), so it has, for \(|a|\) small against \(2\pi/L\), a Gaussian ground state \(\Omega_\perp(a)\) with frequencies \(\omega_k(a)\), the spectrum of the covariant transverse curl in the constant background (for non-commuting \(a_i\) the background carries a constant field strength and the lowest shifted frequency is the Nielsen–Olesen mode, \(\omega^2=(2\pi/L)^2-c\,|[a,a]|\), still positive when \(g^{4/3}\ll(2\pi)^2\)). Take \[P=\int^\oplus da\;|\Omega_\perp(a)\rangle\langle\Omega_\perp(a)| ,\] the Born–Oppenheimer projection. Then \(W_2\) is absorbed exactly, the compression is \[A=PHP=H_0+U(a)+\text{(non-adiabatic terms)},\qquad U(a)=\tfrac12\sum_k\big[\omega_k(a)-\omega_k(0)\big]+\text{const},\] and \(U(a)\) is the one-loop effective potential of the constant modes on the torus: even in \(a\), with the finite quadratic Casimir term \(c_2(La)^2/L\), a finite quartic term, and a quartic term \(\propto|[a,a]|^2\log(\Lambda L)\) that renormalizes \(1/g^2\). In rescaled variables the finite part of \(U\) is \(O(g^{4/3}/L)\) and the logarithm is \(O(g^{4/3}\cdot g^{2}\log\Lambda L\,/L)\) relative to \(H_0\)’s \(g^{2/3}/L\): (H3) holds with \(g\) understood as the renormalized coupling at the scale \(L\), which is Lüscher’s statement. With a periodic cutoff the \(a\)-independent projection of Section 2 also works at the level of sizes, since its second-order Schur term is the same finite Casimir sum; the fibered projection is cleaner because it removes \(W_2\) from the error altogether. The non-adiabatic terms come from \(p_i\) acting on \(\Omega_\perp(a)\) (the Berry connection of the fibered vacuum) and are of relative order \(g^{4/3}\).

Schur error. With the fibered \(P\), \(\bar PWP\) contains only \(W_3\), \(W_4\) and the non-adiabatic terms. \(W_3\) creates three quanta with amplitude of order \(g/L\) per mode triple, and its second-order contribution is the \(a\)-dependent part of a two-loop vacuum energy, whose divergent part is \(a\)-independent and cancels from the gap; the finite \(a\)-dependent part is of order \(g^2\hbar c/L\). (H2) is therefore expected in the form \(\eta\le C_2g^{4/3}(1+g^{2/3}\log\Lambda L)\,\hbar c/L\), with a gauge-invariant cutoff, and the constants \(C_i\) grow with \(\log(L/a)\) on a lattice: the reduction holds at fixed cutoff, and the cutoff dependence is exactly the running of the coupling.

Relative form of the lemma. The constant-mode coordinate \(\xi\) is unbounded and the pair amplitudes grow with \(|\xi|\), so \(\eta\) cannot be a uniform operator bound; the usable hypothesis is relative to \(A\): \(B(D-E)^{-1}B^*\le\epsilon(A-a_0)+\eta\) with \(0\le\epsilon<1\). Then \(F(E)\ge(1-\epsilon)(A-a_0)+a_0-E-\eta\), the count of negative eigenvalues of \(F(E)\) is at most \(\#\{k:a_0+(1-\epsilon)(a_k-a_0)-\eta<E\}\), and the same argument gives \(E_0\ge a_0-\eta\), \(E_1\ge a_0+(1-\epsilon)(a_1-a_0)-\eta\), hence \(\operatorname{gap}(H)\ge(1-\epsilon)(a_1-a_0)-\eta\). The confining potential of \(h_3\) controls \(\langle|\xi|^2\rangle\) in terms of \(A-a_0\), which is what makes \(\epsilon\) small, of order \(g^{2/3}\).

The corrected list is: (H1) as before for \(H_\perp(a)\); (H2) in the relative form for the fibered Schur error; (H3) for \(h_3+U\) with \(U\) the finite one-loop torus potential including its Casimir quadratic term.

4. What the reduction shows

The smallness parameter is \(g^{2/3}\) in every entry of the table, which is the reason the small-volume expansion is a series in \(g^{2/3}\). The mechanism transferred is the zero-point confinement of C133: \(A\) is that operator plus a small even potential, and the nonzero modes enter only through the price \(\eta\) of leaving their vacuum. The reduction also shows where the small-volume regime ends: (H1) fails when \(C_1g(L)^{2/3}\) reaches order one, that is when \(\Delta_0(L)\sim2\pi\hbar c/L\), which is the crossover \(z\simeq2\) of Lüscher–Münster in the variable \(z=\Delta L\). Beyond it, leaving the nonzero-mode vacuum is no longer expensive compared with the constant-mode gap, no projection of the form \(P\) separates scales, and the argument gives nothing, which is the statement of T2\('\)’s difficulty in this language.

5. Consequence for STATE

The weak-coupling lower bound is now three inequalities on explicit polynomial operators, with a proved transfer lemma. The next step on the lower side is the fibered (H2): the non-adiabatic terms and the \(W_3\) Schur error against the nonzero-mode gap, on a lattice cutoff, with the C133 ground state as weight; (H1) and (H3) are relatively bounded perturbation statements, (H3) requiring the finite one-loop torus potential. Their proof would close the small-volume corner S from both sides and make the crossover statement the whole remaining content of T2\('\) at fixed cutoff.