In three dimensions the mass gap is the positivity of one function at infinity

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For pure Yang–Mills in \(d=3\) (two space dimensions) the coupling \(g^2\) is itself a mass, so the finite-volume gap of the continuum theory on a periodic square torus of side \(L\) has the exact form \[\Delta(L)=g^2\hbar^2c\,f(x),\qquad x=g^2\hbar L\ \text{(dimensionless)},\] by dimensional analysis alone, and the Jaffe–Witten conjecture for \(d=3\) is the single statement \[\lim_{x\to\infty}f(x)=C\in(0,\infty),\qquad m=C\,g^2\hbar^2/c .\] The small-\(x\) end of \(f\) is known from the constant-mode sector: with two spatial components the zero-mode Hamiltonian is C133 with \(D=2\), \(m_{\rm eff}=L^2/(g^2c)\) and \(g_B^2=L^2c/g^2\), giving \(\Delta_0(L)=\delta_1^{(2)}\,g^2\hbar^2c\,x^{-2/3}\), that is \(f(x)=\delta_1^{(2)}x^{-2/3}[1+o(1)]\) as \(x\to0\). On the lattice, the strong-coupling theorem of the previous note with \(\nu=2\) gives a gap \(\ge\gamma_2(g_{\rm lat}^2/2)C_2\hbar c/a\) uniform in the lattice size, and the continuum limit is the statement that \(\delta_\infty(g_{\rm lat})/g_{\rm lat}^2\) stays bounded below as \(g_{\rm lat}^2=g^2\hbar a\to0\): the gap-to-coupling ratio is bounded below at both ends of the coupling axis, and the conjecture is its positivity in between. Contrast \(d=4\), where the ratio must pass from a power law \(\propto g^2\) at strong coupling to the essential singularity \(e^{-1/(2b_0g^2)}\) at weak coupling. The three-dimensional problem is therefore the natural first target: it has no transmutation, a single scaling variable, and both ends of \(f\) in hand. The expected value \(C=N/(2\pi)\) (Karabali–Nair, physics argument; metadata, B78) is a prediction to be proved, not an input. Nothing here is promoted.

1. Units and the one-variable form

With \(S=\frac1{4g^2}\int F^a_{\mu\nu}F^{a\mu\nu}\,d^3x\) and the quantization \(e^{iS/\hbar}\), the dimensional census of G07 §4 gives \([1/g^2]=\) action\(\cdot\)length in \(d=3\), so \(g^2\hbar=1/\ell_g\) is an inverse length, \(\mu:=g^2\hbar^2/c=\hbar/(c\ell_g)\) is a mass, and \(\varepsilon_3:=\mu c^2=g^2\hbar^2c\) is the unique energy formed from \((\hbar,c,g)\). The continuum theory on the periodic torus \(T^2_L\times\mathbb R\) has the fixed constants \(\hbar\), \(c\), \(g\) and the box size \(L\), and a gap \(\Delta(L)\), when the finite-volume theory exists, depends on them only. By the floor/unit theorem C131 applied to the observable “energy”, \(\Delta(L)\) is \(\varepsilon_3\) times a function of the single dimensionless combination \(x=L/\ell_g=g^2\hbar L\), because \((\hbar,c,g)\) span exactly one energy and \(L\) adds exactly one dimensionless ratio. Setting \(f(x)=\Delta(L)/\varepsilon_3\): \[\Delta(L)=\varepsilon_3\,f(x),\qquad\varepsilon_3=g^2\hbar^2c,\qquad x=g^2\hbar L .\] The infinite-volume gap, if it exists, is \(m c^2=\varepsilon_3\lim_{x\to\infty}f(x)\), and Jaffe–Witten’s \(0<m<\infty\) is \(0<\lim f<\infty\). Vacuum-energy renormalization, the only ultraviolet divergence of the superrenormalizable theory relevant here, shifts \(E_0\) and \(E_1\) equally and leaves \(f\) untouched.

2. The small-\(x\) end from the constant modes

Temporal gauge on \(T^2_L\) leaves two constant spatial components \(a_1,a_2\) in the Lie algebra. With \(S=\frac1{4g^2}\int F^a_{\mu\nu}F^{a\mu\nu}d^2x\,c\,dt\), \(F_{0i}=c^{-1}\dot a_i\) and \(F_{12}=\vec a_1\times\vec a_2\), the constant-mode Lagrangian is \(\frac{L^2}{2g^2c}|\dot{\vec a}|^2-\frac{L^2c}{2g^2}|\vec a_1\times\vec a_2|^2\) and \[H_0=\frac{g^2c}{2L^2}\sum_{i=1}^2|\vec p_i|^2+\frac{L^2c}{2g^2}\,|\vec a_1\times\vec a_2|^2, \qquad[a_i^a,p_j^b]=i\hbar\delta_{ij}\delta^{ab},\] which is C133 with \(D=2\), \(m_{\rm eff}=L^2/(g^2c)\) and \(g_B^2=L^2c/g^2\); \(\hbar\) enters only through \(g^2\) and the commutator. Its energy unit is \[\varepsilon=\hbar^{4/3}g_B^{2/3}m_{\rm eff}^{-2/3} =\hbar^{4/3}\Big(\frac{L^2c}{g^2}\Big)^{1/3}\Big(\frac{L^2}{g^2c}\Big)^{-2/3} =\hbar^{4/3}c\,g^{2/3}L^{-2/3} =\frac{\hbar c}{L}\Big(\frac{L}{\ell_g}\Big)^{1/3} =\varepsilon_3\,x^{-2/3},\] using \(g^{2/3}\hbar^{1/3}=\ell_g^{-1/3}\) and \(\hbar c/L=\varepsilon_3/x\). So the constant-mode gap is \(\Delta_0(L)=\delta_1^{(2)}\varepsilon_3x^{-2/3}\) with \(\delta_1^{(2)}>0\) the pure number of C133 for \(D=2\), and \[f(x)=\delta_1^{(2)}\,x^{-2/3}\,[1+o(1)],\qquad x\to0,\] where \(o(1)\) stands for the coupling to the nonzero momentum modes, controlled perturbatively by the small-volume expansion in the same way as in \(d=4\) (Lüscher’s method; the \(d=3\) expansion parameter is \(x^{1/3}\) rather than \(g^{2/3}\), since \(x\) is the only parameter). The claim \(f\to\infty\) as \(x\to0\) is the statement that the small box is gapped with a gap growing like \(L^{-2/3}\), slower than the \(1/L\) of a free particle in a box: the zero-point mechanism of G08 gives a softer confinement than a hard wall.

3. The lattice route in \(d=3\) and the strong-coupling end

On the periodic square lattice \((a\mathbb Z/N_sa\mathbb Z)^2\) the Kogut–Susskind Hamiltonian is \[H=\frac{\hbar c}{a}\Big[\frac{g_{\rm lat}^2}{2}\sum_\ell(-\Delta_\ell) +\frac{2}{g_{\rm lat}^2}\sum_p(N-\operatorname{Re}\operatorname{tr}U_p)\Big], \qquad g_{\rm lat}^2=g^2\hbar a=\frac a{\ell_g},\] with \(g_{\rm lat}\) dimensionless. Theorem 1 of the obligations map holds verbatim in two dimensions: \(\Delta_{a,L}=(\hbar c/a)\,\delta(g_{\rm lat};N_s,G)>0\) with a unique physical ground state. The strong-coupling theorem holds with \(\nu=2\), \(\Lambda_0=\{0,1\}^2\), two links and one plaquette per site: \[\beta=\frac{4}{g_{\rm lat}^4C_2}\cdot2N=\frac{16N^2}{g_{\rm lat}^4(N^2-1)},\qquad \Delta_{a,L}\ge\gamma_2\,\frac{g_{\rm lat}^2}{2}\,C_2\,\frac{\hbar c}{a},\] for \(g_{\rm lat}\ge g_0^{(2)}(N)=\big(16N^2/[(N^2-1)\beta_*(2,\{0,1\}^2)]\big)^{1/4}\), uniformly in \(N_s\), by the same hypothesis check.

The continuum limit is \(a\to0\) at fixed \(g^2\), \(L\), that is \(g_{\rm lat}^2\to0\) with \(N_s=L/a\) and \(x=g^2\hbar L=g_{\rm lat}^2N_s\) fixed. The lattice gap in units of \(\varepsilon_3\) is \[\frac{\Delta_{a,L}}{\varepsilon_3}=\frac{\hbar c}{a\,g^2\hbar^2c}\,\delta =\frac{\delta(g_{\rm lat};N_s,G)}{g_{\rm lat}^2},\] so the continuum finite-volume gap exists iff \(\delta(g_{\rm lat};x/g_{\rm lat}^2,G)/g_{\rm lat}^2\to f(x)\) as \(g_{\rm lat}\to0\), and the infinite-volume mass is \[\frac{mc^2}{\varepsilon_3}=\lim_{g_{\rm lat}\to0}\ \frac{\delta_\infty(g_{\rm lat})}{g_{\rm lat}^2}, \qquad\delta_\infty(g_{\rm lat})=\liminf_{N_s\to\infty}\delta(g_{\rm lat};N_s,G).\] At strong coupling the same ratio is bounded below by \(\gamma_2C_2/2\). The \(d=3\) conjecture on the lattice route is therefore:

T3\(_{(3)}\). The function \(c(g_{\rm lat})=\delta_\infty(g_{\rm lat})/g_{\rm lat}^2\), which satisfies \(c\ge\gamma_2C_2/2\) for \(g_{\rm lat}\ge g_0^{(2)}\), stays bounded below by a positive constant for all \(g_{\rm lat}>0\) and has a finite limit \(C\) as \(g_{\rm lat}\to0\).

The abelian contrast in \(d=3\). For compact \(U(1)\) in three dimensions Göpfert and Mack (abstract, B78) prove a gap at every coupling, so T2\('\) holds for the abelian group here, and their Debye mass \(m_D^2=(2\beta/a^3)e^{-\beta v(0)/2}\), \(\beta=4\pi^2/e^2_{\rm lat}\), makes \(c(g_{\rm lat})=\delta_\infty/g_{\rm lat}^2\) vanish with an essential singularity as \(g_{\rm lat}\to0\): the continuum photon is massless. In \(d=3\) the abelian/non-abelian distinction is therefore the limit of the ratio, zero against a positive \(C\), with positivity at each coupling shared by both. Everything in this note up to here is a collection of known facts in one normalization; the only statement of this programme’s own is the reduction to \(f\) and its two ends.

No exponential enters. In \(d=4\) the corresponding ratio \(\delta_\infty(g)/g^2\) is bounded below at strong coupling by the same theorem and must vanish like \(g^{-2}e^{-1/(2b_0g^2)}\) at weak coupling; the two ends of the \(d=4\) problem have different functional forms and the lattice gap must interpolate between them, which is what makes T3 in \(d=4\) a transmutation statement. In \(d=3\) both ends are linear in \(g_{\rm lat}^2\).

4. What is in hand and what is not

Open in \(d=3\): (i) the infinite-volume limit of the finite-volume construction; (ii) any lower bound on \(f\) away from the two ends; (iii) the identification of \(f\)’s small-\(x\) correction with the stochastic construction’s estimates. Item (ii) is the conjecture itself, and the one-variable form means that a single inequality \(f(x)\ge f_->0\) for \(x\ge x_0\), proved for the continuum finite-volume theory, together with the existence of \(\lim f\), would settle the \(d=3\) mass gap.

5. Consequence for STATE

The \(d=3\) target is now a statement about one function of one variable with both ends known: \(f(x)\sim\delta_1^{(2)}x^{-2/3}\) at \(x\to0\) and \(f\to C\) conjectured at \(x\to\infty\), with the lattice ratio \(\delta_\infty/g_{\rm lat}^2\) bounded below at strong coupling. It replaces the \(d=3\) line of STATE. The next theorem-sized step in \(d=3\) is a lower bound \(f(x)\ge f_->0\) on an interval \(x\in[x_0,x_1]\) beyond the small-volume expansion, for which the candidate tool is a variational or operator-inequality argument on the torus Hamiltonian that keeps the zero-point mechanism of the constant modes while controlling the nonzero modes by their Gaussian part, which is what the weak-coupling lattice bound of STATE step 3 is meant to prepare.