Fefferman, Fröhlich and Graf (PNAS 1996; Comm. Math. Phys. 1997) proved HH-stability -- stability of matter of the second kind -- for NN nonrelativistic spin-1/2 electrons and MM static nuclei of charges at most ZZ interacting through Coulomb forces and the quantized radiation field with an ultraviolet cutoff Λ\Lambda: the ground-state energy is bounded below by −ε(Z) Z2M-\varepsilon(Z)\,Z^2 M with ε(Z)≤const⋅Z3/2\varepsilon(Z)\le\mathrm{const}\cdot Z^{3/2} proven and ε(Z)≤const⋅Z\varepsilon(Z)\le\mathrm{const}\cdot Z conjectured, while the constant in the corresponding bound for arbitrary cutoff grows linearly in Λ\Lambda because the Hamiltonian is unrenormalized. The problem is the conjecture the authors attached to this theorem: prove or disprove the conjectured linear ZZ-dependence of ε(Z)\varepsilon(Z), which would match the physical −const⋅Z3-\mathrm{const}\cdot Z^3 per-atom energy scale, and determine the optimal ZZ-dependence of ε(Z)\varepsilon(Z).

Background

Stability of matter of the second kind (HH-stability) is the statement that the ground-state energy of a system of NN electrons and MM nuclei interacting through Coulomb forces is bounded below by a negative constant times the number of particles, so that macroscopic matter cannot release an energy growing faster than linearly with its size. Proving it requires the uncertainty principle, the Pauli exclusion principle (fermionic statistics of the electrons), and a Lieb--Thirring inequality, which bounds the sum of the negative eigenvalues, or the local kinetic energy, of a many-fermion Hamiltonian from below by a phase-space constant. When the electromagnetic field is quantized, the kinetic energy is modified by minimal coupling to the photon field and the field contributes its own Hamiltonian, and stability becomes a question about nonrelativistic quantum electrodynamics (QED).

The model is the Pauli--Fierz Hamiltonian: NN nonrelativistic spin-1/2 electrons with the Pauli kinetic-energy operator, MM static nuclei of atomic numbers at most ZZ (an atomic number is the nuclear charge in units of the electron charge), Coulomb interparticle potentials, and the transverse quantized radiation field in the Coulomb gauge, with an ultraviolet cutoff Λ\Lambda (a restriction of photon wave vectors to ∣k∣≤Λ|k|\le\Lambda, imposed because the model is otherwise ill-defined: the field's zero-point energy and the electron self-energy, the electron's energy of interaction with its own radiation field, both diverge as Λ→∞\Lambda\to\infty). The coupling strength is the fine-structure constant α=e2/ℏc≈1/137\alpha=e^2/\hbar c\approx 1/137. Following the source paper, energies are measured in units where 2mc2α2=12mc^2\alpha^2=1, so the hydrogen ground-state energy is −4-4 and a typical inner electron in an atom of atomic number ZZ has energy of order −Z2-Z^2.

Fefferman, Fröhlich and Graf proved (PNAS 1996; full proofs in Comm. Math. Phys. 190 (1997) and Adv. Math. 124 (1996)): (i) for arbitrary NN, MM, ZZ, α\alpha and cutoff Λ\Lambda, the ground-state energy is bounded below by −K M-K\,M with KK finite but growing (at most) linearly in Λ\Lambda -- the linear Λ\Lambda-growth reflects that the Hamiltonian is the unrenormalized one, in which each electron carries a self-energy diverging with the cutoff; (ii) under the cutoff condition α−1Λ≤Z2α−3/4\alpha^{-1}\Lambda\le Z^2\alpha^{-3/4} and sufficiently small αZ\alpha Z and α5/2Z3\alpha^{5/2}Z^3, the energy is bounded below by −ε(Z) Z2M-\varepsilon(Z)\,Z^2 M with ε(Z)≤const⋅Z3/2\varepsilon(Z)\le\mathrm{const}\cdot Z^{3/2} independent of α\alpha, NN and MM; equivalently H≥−const⋅Z7/2MH\ge-\mathrm{const}\cdot Z^{7/2}M. Since a nucleus binds only of order ZZ electrons at energies of order Z2Z^2, the ground-state energy of an atom or ion is of order −Z3-Z^3, and the authors conjectured that the estimate improves to ε(Z)≤const⋅Z\varepsilon(Z)\le\mathrm{const}\cdot Z, i.e. H≥−const⋅Z3MH\ge-\mathrm{const}\cdot Z^3 M; they attribute the gap to the ultraviolet divergence of the field's zero-point energy. The proof combines a magnetic Lieb--Thirring inequality for the Pauli operator in an arbitrary static magnetic field (a version of the Lieb--Thirring bound that remains uniform in the field strength), a Calderón--Zygmund decomposition of space (a multi-scale partition of R3\mathbb{R}^3 into cubes adapted to the local nucleus density and field strength), and Fock-space estimates (Fock space is the state space of the quantized field). The authors also note that treating nuclei as static, spinless point charges is not an innocent approximation, because real nuclei have magnetic moments coupling to the radiation field.

Later work sharpened the picture without closing the authors' question. Lieb's 2002 survey records that stability with the quantized field is established but that renormalization (constructing a well-defined limit as Λ→∞\Lambda\to\infty after subtracting divergent self-energies, e.g. by redefining the physical electron mass -- mass renormalization) and pair production remain to be included. Lieb and Loss (Comm. Math. Phys. 258 (2005)) proved the thermodynamic-limit lower bound for the same model with dynamical nuclei (of mass at least 1800 electron masses) but under small-coupling conditions (α<1/98\alpha<1/98, Z≤39Z\le 39 at physical α\alpha), with the hyperfine (nuclear magnetic moment) interaction excluded; they state that renormalization is yet to be resolved and that the true self-energy divergence lies between Λ3/2\Lambda^{3/2} and Λ12/7\Lambda^{12/7}, unsettled. A one-particle renormalization program (Hainzl; Hainzl--Seiringer; Lieb--Loss; Hiroshima--Ito) handles a single electron with renormalized mass, and a 2026 result of Bach, Ballesteros and Mlinarzik shows Λ3/2\Lambda^{3/2} growth of the minimal energy for the translation-invariant (no Coulomb potential) Pauli--Fierz model, but many-body Coulomb stability with a renormalized, cutoff-independent constant remains open, as does the conjectured linear ZZ-dependence of ε(Z)\varepsilon(Z).

Problem Statement

For the Pauli-Fierz Hamiltonian HH of NN nonrelativistic spin-1/2 electrons and MM nuclei of atomic numbers at most ZZ, with Coulomb interparticle forces, Coulomb gauge, and ultraviolet cutoff Λ\Lambda, in units where 2mc2α2=12mc^2\alpha^2=1, determine the sharp stability constant of matter: prove or disprove the conjecture of Fefferman, Fröhlich and Graf (PNAS 1996, Remark (ii) to Theorem 1) that under the hypotheses of their Theorem 1(ii) -- the cutoff condition α−1Λ≤Z2α−3/4\alpha^{-1}\Lambda\le Z^2\alpha^{-3/4} and sufficiently small αZ\alpha Z and α5/2Z3\alpha^{5/2}Z^3 -- the constant in the stability bound H≥−ε(Z) Z2MH\ge-\varepsilon(Z)\,Z^2 M (with ε(Z)\varepsilon(Z) independent of α\alpha, NN and MM) satisfies ε(Z)≤const⋅Z\varepsilon(Z)\le\mathrm{const}\cdot Z, improving the proven rate ε(Z)≤const⋅Z3/2\varepsilon(Z)\le\mathrm{const}\cdot Z^{3/2} (i.e. H≥−const⋅Z7/2MH\ge-\mathrm{const}\cdot Z^{7/2}M) to H≥−const⋅Z3MH\ge-\mathrm{const}\cdot Z^3 M, which would match the physical −const⋅Z3-\mathrm{const}\cdot Z^3 energy scale of an atom or ion. Determining the optimal ZZ-dependence of ε(Z)\varepsilon(Z) further requires a matching lower bound on ε(Z)\varepsilon(Z) establishing that the resulting rate is sharp.

References

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