An action floor on transverse phase-space area produces the Yang–Mills quantum-mechanical gap, and a gap forces an action unit

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In the Yang–Mills quantum-mechanical model of G07 the two directions of the analogy between a positive action floor \(h>0\) and a mass gap can both be stated and proved inside one solved system. Forward: classical mechanics plus the single postulate that no transverse oscillation has phase-space area below \(h/2\) confines the escaping valley motion to \(|x|\le2\sqrt m\,E/(\hbar g)\) at energy \(E\), makes the accessible phase-space volume finite with the growth \(E^{3/2}\ln E\) that Simon records for the true eigenvalue counting function, and reproduces through Bohr–Sommerfeld quantization the energy unit \(\varepsilon=\hbar^{4/3}g^{2/3}m^{-2/3}\) of the exact spectrum; the operator inequality behind G07’s Theorem 5 is this postulate made rigorous by the uncertainty inequality. The abelian model has no transverse oscillation, so the same floor constrains nothing and its spectrum stays continuous: with \(h>0\) assumed, a commutative field still has no gap. Backward: in the classical model with constants \(m\) and \(g\), any positive finite gap \(\Delta\), whatever extra constant produces it, supplies the action unit \(\Delta^{3/4}m^{1/2}g^{-1/2}\), so within the model “gap” and “action unit” are dimensionally equivalent. The same equivalence holds for Yang–Mills in \(d=2\) and \(d=3\), where the classical coupling is dimensionful, and fails exactly in \(d=4\), the action-critical dimension, where a mass gap together with \(c\) admits no action unit. The floor postulate is a phase-space area statement; the necessity theorem sought in STATE item 1 is the derivation of such a floor. This is G08, exploratory, with no ledger promotion.

1. The transverse phase-space area and the floor postulate

Take the scalar model of G07, Theorem 5, with classical Hamiltonian \[H_{\rm cl}=\frac{p_x^2+p_y^2}{2m}+\frac{g^2}{2}x^2y^2 .\] At fixed \(x\neq0\) the transverse motion is the oscillator \(E_y=p_y^2/2m+\tfrac12m\omega_x^2y^2\) with \(\omega_x=g|x|/\sqrt m\).

Proposition 1. The transverse orbit at fixed \(x\) and transverse energy \(E_y\) is the ellipse \(\{p_y^2/2m+\tfrac12m\omega_x^2y^2=E_y\}\), and its enclosed phase-space area is \[J_y=\oint p_y\,dy=\frac{2\pi E_y}{\omega_x}=\frac{2\pi\sqrt m\,E_y}{g|x|}.\]

Proof. The ellipse has semi-axes \(\sqrt{2mE_y}\) in \(p_y\) and \(\sqrt{2E_y/(m\omega_x^2)}\) in \(y\); the product times \(\pi\) gives \(2\pi E_y/\omega_x\). \(\square\)

\(J_y\) is the adiabatic invariant of the transverse oscillation: along the valley it is conserved to leading order when \(\omega_x\) changes slowly on the oscillation’s time scale, \(|\dot x/x|\ll\omega_x\), that is \(|x|\gg x_{\rm ad}(E)=(2E)^{1/4}g^{-1/2}\) at energy \(E\) (Landau–Lifshitz, Mechanics §49; the note uses only Proposition 1 and the inequality below, which need no adiabatic limit).

Floor postulate \(\mathsf F(h)\). No admitted state has a transverse oscillation of phase-space area below \(h/2=\pi\hbar\): \(J_y\ge h/2\) whenever \(x\ne0\), and \(J_x\ge h/2\) whenever \(y\ne0\).

The value \(h/2\) is the Bohr–Sommerfeld ground-state area \((n+\tfrac12)h\) at \(n=0\); any positive multiple of \(h\) gives the same conclusions with a different pure number.

2. Forward: the floor confines the valley and produces the gap

Theorem 2. Under \(\mathsf F(h)\), every admitted classical state of energy \(E\) satisfies \[|x|\le X(E)=\frac{2\sqrt m\,E}{\hbar g},\qquad |y|\le X(E),\] and the transverse energy obeys \(E_y\ge\hbar\omega_x/2=\hbar g|x|/(2\sqrt m)\). In particular the admitted region of the energy shell is bounded, whereas without the floor the shell contains the unbounded valley motions \(y=p_y=0\), \(x(t)=x_0+p_xt/m\) of every energy \(p_x^2/2m>0\).

Proof. For \(x\ne0\), Proposition 1 and \(E_y\le E\) give \(J_y=2\pi\sqrt m\,E_y/(g|x|)\le2\pi\sqrt m\,E/(g|x|)\); the floor \(J_y\ge\pi\hbar\) then forces \(|x|\le2\sqrt mE/(\hbar g)\). Reading the same inequality as a bound on \(E_y\) gives \(E_y\ge\hbar g|x|/(2\sqrt m)\). The \(y\) bound is symmetric. \(\square\)

Remarks.

1. The floor is the quantum bound. The inequality \(E_y\ge\hbar g|x|/(2\sqrt m)\) is word for word the oscillator bound of G07, Theorem 5 (Lemma 0 there): \(-\frac{\hbar^2}{2m}\partial_y^2 +\frac{g^2}{2}x^2y^2\ge\hbar g|x|/(2\sqrt m)\). The quantum theorem is the floor postulate enforced by the uncertainty inequality \(\int(\mu|\phi'|^2+\nu y^2|\phi|^2)\ge\sqrt{\mu\nu}\int|\phi|^2\), which is the statement that no normalizable state occupies a transverse phase-space area below \(h/2\). So in this model the implication “\(h>0\) gives the gap” is exact, with \(h\) entering only through the area floor, and the rest of quantum kinematics is not used.
2. Self-consistency of the adiabatic reading. The cutoff \(X(E)\) lies in the adiabatic region: \(X(E)/x_{\rm ad}(E)=2^{3/4}(E/\varepsilon)^{3/4}\) with \(\varepsilon=\hbar^{4/3}g^{2/3}m^{-2/3}\), so for \(E\gg\varepsilon\) the floor acts where \(J_y\) is conserved, and near the ground state, \(E\sim\varepsilon\), the classical reading is only an estimate.
3. Counting. Without the floor the phase-space volume \(|\{H_{\rm cl}\le E\}|\) is infinite for every \(E>0\) (G07, Theorem 5, proof of (iv)). With the floor, the transverse area at fixed \(x\) and fixed \(p_x\) is \(2\pi(E-p_x^2/2m)/\omega_x\); integrating over \(|p_x|\le\sqrt{2mE}\) gives \((8\pi/3)\sqrt{2m}\,E^{3/2}\sqrt m/(g|x|)\) per unit \(x\), and integrating \(|x|\) from a fixed inner scale to \(X(E)\) gives \[|\{H_{\rm cl}\le E\}\cap\mathsf F(h)| =\frac{16\sqrt2\,\pi m}{3g}\,E^{3/2}\ln\frac{X(E)}{x_0}+O(E^{3/2}),\] which grows like \(E^{3/2}\ln E\). Simon states (Ann. Phys. 146 (1983) 209, p. 3, passage read) that the true counting function \(N(E)\) of \(-\Delta+x^2y^2\) grows like \(E^{3/2}\ln E\), while the classical volume is infinite. The floor therefore reproduces the growth law of the exact spectrum; the identification \(N(E)\approx|\{\cdot\}|/h^2\) is the Weyl heuristic, stated here as a consistency check and not as a theorem.
4. Semiclassical levels. With \(J_y=(n+\tfrac12)h\) conserved, the valley motion has effective Hamiltonian \(p_x^2/2m+\kappa_n|x|\), \(\kappa_n=(n+\tfrac12)\hbar g/\sqrt m\), and Bohr–Sommerfeld quantization \(\oint p_x\,dx=(N+\tfrac12)h\) with \(\oint p_x\,dx=\tfrac{8\sqrt{2m}}{3}E^{3/2}/\kappa_n\) gives \[E^{\rm BS}_{n,N}=\varepsilon\Big[\frac{3\pi(2n+1)(2N+1)}{16\sqrt2}\Big]^{2/3}, \qquad\varepsilon=\hbar^{4/3}g^{2/3}m^{-2/3}.\] The unit \(\varepsilon\) is the exact one of G07, Theorem 5(4), as Theorem 1 of G07 requires; the pure numbers are semiclassical estimates of \(e_n\) and are labelled as such. The lowest level and the first spacing are positive: the old quantum theory of this model already has a gap.

Corollary 3 (the commutative model). For the abelian matrix model, and for \(D=1\), the potential vanishes identically: there is no transverse oscillation, \(\mathsf F(h)\) imposes no condition, the classical energy shells stay unbounded and the quantum spectrum is \([0,\infty)\) (G07, Theorem 6(5)). An action floor produces a gap only in the presence of the commutator potential, which is what supplies the transverse ellipses whose area the floor bounds.

For the SU(2) model with \(D\ge2\) the same argument applies to each of the two transverse directions of \(\vec x_j\) relative to \(\vec x_i\): the floor gives \(E_\perp\ge\hbar g|\vec x_i|/\sqrt m\), the bound of G07, Theorem 6(1), and \(|\vec x_i|\le\sqrt m\,E/(\hbar g)\) on the energy shell.

3. Backward: a gap forces an action unit, except in the action-critical case

Proposition 4. Let the classical model with fixed constants \(m\), \(g\) (dimension vectors \(d_m=(1,0,0)\), \(d_g=(\tfrac12,-1,-1)\)) be enlarged by any fixed constants, and suppose the enlarged theory has a positive finite gap \(\Delta\) depending on the fixed constants only. Then the theory has the action unit \[A=\Delta^{3/4}\,m^{1/2}\,g^{-1/2},\] and conversely an action unit \(A\) gives the energy unit \(A^{4/3}g^{2/3}m^{-2/3}\).

Proof. Energy has dimension vector \((1,2,-2)\) and action \((1,2,-1)\). The system \(a\,d_m+b\,d_g+c\,(1,2,-2)=(1,2,-1)\) has the unique solution \(a=\tfrac12\), \(b=-\tfrac12\), \(c=\tfrac34\): the length and time components give \(-b+2c=2\), \(-b-2c=-1\), hence \(c=\tfrac34\), \(b=-\tfrac12\), and the mass component gives \(a=\tfrac12\). So \(m^{1/2}g^{-1/2}\Delta^{3/4}\) has action dimension whatever the extra constants are. The converse is G07, Theorem 3 with \(k=4\). \(\square\)

Within the model, therefore, “the gap is positive” and “an action unit exists” are the same dimensional statement, and G07’s Theorem 5 supplies the dynamical content, \(0<\delta_1<\infty\), that turns the unit into the number \(\Delta=\delta_1\varepsilon\).

Proposition 5 (which field theories share this equivalence). With the classical coupling \(1/g^2\) of dimension action\(\cdot\)length\(^{4-d}\), the speed \(c\) and a gap \(\Delta\), an action unit exists in \(d=2\) and \(d=3\) and does not exist in \(d=4\):

Proof. Solve \(a\,d_{1/g^2}+b\,d_c+c'\,(1,2,-2)=(1,2,-1)\): for \(d=3\), adding the length and time equations gives \(2a=1\), then \(c'=\tfrac12\), \(b=-\tfrac12\); for \(d=2\), \(3a=1\), \(c'=\tfrac23\), \(b=-\tfrac23\); for \(d=4\) the mass component forces \(b=1\), the length component then forces \(a=0\), and the time component reads \(-2=-1\). The \(k=-2\) row is G07, Section 6. \(\square\)

The equivalence “gap if and only if action unit” thus holds whenever the classical theory carries a dimensionful coupling, and fails exactly at the action-critical dimension, where the classical theory carries no constant at all and a mass gap with \(c\) yields a length only after \(\hbar\) is supplied. This is the dimensional reason the four-dimensional problem is one-directional: the gap needs a scale generated by transmutation, and that scale does not by itself return an action unit.

4. What this decides for the positive-action question

• Direction. In the solved model the analogy is an equivalence at the level of units and an implication at the level of dynamics: the floor postulate, used only through the transverse area, produces the gap (Theorem 2 and Remark 1), and the gap returns the unit (Proposition 4). The mass-gap problem is the forward direction with \(\hbar\) and the gauge group given; the necessity problem of STATE item 1 is the derivation of the floor itself. Inside this model those two problems have the same dimensional content, and the necessity problem is exactly as hard as proving a gap from a consistency premise that does not name \(\hbar\).
• The second ingredient. Corollary 3 states the commutative case with \(h>0\) assumed: an action floor alone gives no gap; the commutator potential is what creates transverse oscillations for the floor to act on. Any transported statement “\(h>0\) gives a gap” must name the structure that plays this role.
• Areas. The floor is a bound on a phase-space area, the ellipse of Proposition 1. The programme’s anchor, Galileo’s area \(vF\tau^3/(6m)\) between the inertial line and the parabola, is a configuration-space area converted to action by the factor \(3F/v\) (N01 §1). The model shows what a successful floor would have to look like on the phase-space side: a lower bound on the area of a closed transverse orbit, invariant under the adiabatic deformation of the orbit. A constant-force motion has no closed transverse orbit, which is one more statement of why the Galileo class is similarity-covariant (G07, Proposition 14).
• Gate. For gap-track work the supplied/generated gate of G07 can be applied in phase-space terms: a generated gap needs a family of closed orbits whose area can fall below \(h\) along a non-compact direction.

5. Strategic consequence

G08 closes the loop inside the Yang–Mills quantum-mechanical model: floor gives gap, gap gives unit, and the commutative model is inert under the floor. Two dimensional facts are retained for the programme: the equivalence of gap and action unit holds exactly when the classical theory has a dimensionful coupling, and fails at \(d=4\); and the phase-space area of a closed transverse orbit is the object a floor must bound. Neither changes STATE item 1’s target, which remains the derivation of the floor from consistency premises; N02 gains the phase-space form of the floor as the statement to aim at. No ledger claim is promoted; Theorem 2 and Propositions 1, 4, 5 are elementary and the semiclassical remarks are labelled estimates. Sources are in B79.