Proc. Natl. Acad. Sci. USA Vol. 93, pp. 15009–15011, December 1996 Mathematics
Stability of nonrelativistic quantum mechanical matter coupled to the (ultraviolet cutoff) radiation field
CHARLES FEFFERMAN*†,JU¨RG FRO¨HLICH‡, AND GIAN MICHELE GRAF‡
*Department of Mathematics, Fine Hall, Princeton University, Princeton, NJ 08544; and ‡Theoretical Physics, Eidgeno¨ssiche Technische Hochschule Ho¨nggerberg, CH-8093 Zurich, Switzerland
Contributed by Charles Fefferman, October 14, 1996
ABSTRACT We announce a proof of H-stability for the where L2(ޅ3, d3x) R ރ2 is the Hilbert space for one electron of 1 3 quantized radiation field, with ultraviolet cutoff, coupled to spin ⁄2 in physical space ޅ , ⌳ denotes an antisymmetric tensor arbitrarily many non-relativistic quantized electrons and product, and Ᏺ is the photon Fock space—i.e., the symmetric static nuclei. Our result holds for arbitrary atomic numbers tensor algebra—over the one-photon Hilbert space L2(ޒ3, d3k) and fine structure constant. We also announce bounds for the R ރ2. The factors ރ2 account for the spin states of electrons energy of many electrons and nuclei in a classical vector and the helicities of photons, respectively. Electrons are fer- potential and for the eigenvalue sum of a one-electron Pauli mions—i.e., they satisfy the Pauli principle, giving rise to Hamiltonian with magnetic field. antisymmetric tensor products—while photons are bosons corresponding to symmetric tensor products. In this note, we describe a variety of mathematical results The Hamiltonian of the system, in the presence of M static concerning the quantum theory of systems of static nuclei and nuclei, is given by nonrelativistic electrons with spin coupled to a classical, static H ϭ HPauli ϩ Hfield, [2] magnetic field and͞or to the ultraviolet cutoff quantized radiation field. Our results and their proofs are based, in part, where on earlier work in refs. 1 through 7. Details will appear in refs. N 8–10. H ϭ D͞ 2 ϩ V , [3] A typical system we propose to study consists of an arbitrary Pauli i c iϭ1 number, N, of nonrelativistic electrons with electric charge Ϫe, 3 1 Ѩ bare mass m Ͼ 0, spin ⁄2 and a bare gyromagnetic factor g ϭ ͑⌳͒ ͑i͒ D͞ ϭ Ϫi Ϫ A␣ ͑x ͒ ⅐ ␣ , [4] 2, an arbitrary number, M, of nuclei of atomic number ՅZ, for i ␣ i ␣ϭ1 ͩ Ѩxi ͪ some arbitrary, but fixed integer Z Ͻϱ, and an arbitrary number of photons which describe the transverse degrees of ␣ 3 xi is the ␣th component of the position, xi ʦ ޅ ,oftheith freedom of the quantized electromagnetic field. The dynamics (⌳) electron; A␣ (x)isthe␣th component of the quantized of the system, generated by a self-adjoint Hamilton operator H, electromagnetic vector potential, A(⌳)(x), with an ultraviolet 3 (i) conserves the number of nuclei and electrons, but the number cutoff ⌳, at the point x ʦ ޅ (see below); ␣ stands for the of photons is arbitrary and changes in time. matrix Our concern is to show that, in the ground state of the system corresponding to the infimum of the spectrum of H, the energy I ⅐⅐⅐ ␣ ⅐⅐⅐ I, [5] per charged particle (electron or nucleus) is bounded uniformly 2 RN in N and M, for fixed Z Ͻϱ. This property of nonrelativistic, on spin space (ރ ) , where ␣, ␣ ϭ 1, 2, 3 are the three Pauli quantum-mechanical matter is called H-stability. In ref. 11 the matrices, and the 2ϫ2 matrix ␣ in Eq. 5 appears in the ith reader may find many important results and background factor of the tensor product; VC denotes the Coulomb poten- material on stability of matter, as well as applications thereof. tial—i.e., When the coupling of nuclei to the transverse degrees of 1 Zl freedom of the electromagnetic field is turned off H-stability VC ϭ Ϫ ͉xi Ϫ xj͉ iϭ1,⅐⅐⅐,N ͉xi Ϫ yl͉ of the systems described above is a mathematical consequence 1ՅiϽjՅN lϭ1,⅐⅐⅐,M of the H-stability of systems in which nuclei are treated as ZlZk static. In this note we consider static nuclei, in accordance with ϩ , [6] ͉yl Ϫ yk͉ the fact that their masses are much larger than the mass of an 1ՅlϽkՅM electron, but we emphasize that, because of their magnetic 3 moments, this is not an entirely innocent approximation for where yl ʦ ޅ denotes the position of the lth static nucleus and most realistic nuclei; see the discussion in refs. 7 and 8. Zl Յ Z its atomic number, l ϭ 1, ⅐⅐⅐,M; finally, Furthermore, interactions between electrons and photons with energies large compared with a typical energy of an electron Ϫ1 ଙ 3 H ϭ ␣ ␣ ͑k͉͒k͉a ͑k͒d k [7] in an atom of atomic number ՅZ are turned off with the help field ͵ ϭϮ of an ultraviolet cutoff. The Hilbert space of pure state vectors of a system of N is the Hamiltonian of the transverse degrees of freedom of the electrons and an arbitrary number of photons is given by quantized electromagnetic field. The constant ␣ ϭ e2͞បc (ប is Planck’s constant and c is the velocity of light) is the dimen- 2 3 3 2⌳N 1 Ᏼ ϭ ͑L ͑ޅ , d x͒ ރ͒ Ᏺ, [1] sionless fine-structure constant, with ␣ Ϸ ⁄137 in nature. We have chosen units for energy where 2mc2␣2 ϭ 1, i.e., the ground state energy of a hydrogen atom is Ϫ4. Furthermore, The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked ‘‘advertisement’’ in accordance with 18 U.S.C. §1734 solely to indicate this fact. †To whom reprint requests should be addressed.
15009 Downloaded by guest on September 29, 2021 15010 Mathematics: Fefferman et al. Proc. Natl. Acad. Sci. USA 93 (1996)
we impose the Coulomb gauge on our choice of electromag- denote the corresponding magnetic field. We set netic potentials; in particular, VC is purely electrostatic and [⅐A(⌳)(x) ϭ 0. The ultraviolet cutoff vector potential A(⌳)(x)is D͑x)ϭ min ͉x Ϫ yk͉. [13ٌ given by 1ՅkՅM 2 ޅ3 3 ރ2 ⌳N A͑⌳͒͑x͒ ϭ A͑⌳͒͑x͒ ϩ A͑⌳͒͑x͒, Let ʦ (L ( , d x) R ) . We consider the energy Ϫ ϩ functional where ͑, A͒ ϭ ͗, HPauli͘ ␣1͞2 A͑⌳͒͑x͒ ϭ ⌳͑k͉͒k͉Ϫ1͞2a ͑k͒ ͑k͒eik⅐xd3k, Ϫ 2 ͵ ϩ ͕⌫͉B͑x͉͒2 ϩ C͑⌫, Z͒L2ٌ͉͑ B͒͑x͉͒2͖ ϭϮ ͵ ޅ3 A͑⌳͒͑x͒ ϭ ͑A͑⌳͒͑x͒͒ଙ, [8] ϩ Ϫ ϫ eϪLϪ1D͑x͒d3x, [14] k denotes a wave vector, and the direction of propagation, where HPauli is the N-electron Hamiltonian defined in eqs. 3 k͉͞k͉, and the two polarization vectors (k), ϭϮ, form an and 4, but with A(⌳) replaced by the classical vector potential orthonormal basis of ޒ3 R ރ, for each k ʦ ޒ3; moreover, the A cutoff function ⌳(k) satisfies 0 Յ ⌳(k) Յ 1, and , and ⌫, C(⌫, Z), and L are constants yet to be chosen. The proof of Theorem 1 is based on a lower bound for the supp ⌳͑ ⅐ ͒ ʕ ͕k͉͉k͉Յ⌳͖, [9] energy functional Ᏹ which we describe next. THEOREM 2. Given ⌫Ͼ0and Z Ͻϱ, and for a suitable choice ૺ where ⌳ is a finite constant. The operators a(k) and a(k) are of the constant C(⌫,Z)Ͻϱin Eq. 14, there are finite, positive creation and annihilation operators on Ᏺ satisfying the canon- constants c(⌫,Z)and CЈ(⌫,Z)depending only on ⌫ and Z such ical commutation relations that, for arbitrary L Յ c(⌫, Z),
# # ଙ ͑3͒ Ϫ1 ͓a͑k͒ , aЈ͑k͒ ͔ ϭ 0, ͓a ͑k͒, aЈ͑kЈ͒ ͔ ϭ ␦Ј␦ ͑k Ϫ kЈ͒, [10] inf ͑, A͒ ՆϪCЈ͑⌫,Z͒L M, [15] ͉͉͉͉ϭ1,A where a# ϭ a or aૺ. where M is the number of nuclei. If ⌫ Ն Z2 there are finite, positive We are now prepared to describe our main results concern- Ϫ1 ing H-stability of the systems described above. constants c and C such that, for the choices c(⌫,Z)ϭcZ [i.e., HEOREM c(⌫,Z)is proportional to the Bohr radius of an electron bound T 1. (i) For an arbitrary number, N, of electrons, an 2 arbitrary number, M, of static nuclei of atomic numbers ՅZ, for to a nucleus of atomic number Z] and C(⌫,Z)ϭCZ , one has arbitrary values of Z Ͻϱand of the fine-structure constant ␣, and that for an arbitrary ultraviolet cutoff ⌳ (see expression 9), there exists CЈ͑⌫, Z͒ Յ CZ2. [16] a finite constant E(␣,Z) (depending only on ␣ and Z) such that Remarks:(i) The proof of Theorem 2 is outlined in ref. 8 and H Ն ϪE(␣,Z)⌳⅐M. [11] carried out in full detail in ref. 9. In ref. 9, bounds on the constants c(⌫, Z), C(⌫, Z), and CЈ(⌫, Z) are given in the three (ii) For arbitrary N, M, and Z, as above, for an ultraviolet cutoff 2 2 Ϫ1 2 Ϫ3/4 regimes ⌫ Ն Z , Z Ն ⌫ Ն 1, and ⌫ Յ 1. In using Theorem 2 ␣ ⌳ Յ Z ␣ , and for sufficiently small values of ␣Z and Ϫ2 5/2 3 3/2 to prove Theorem 1, one sets ⌫ϭconst. ␣ . Since, in nature, ␣ Z , there exists a finite constant (Z) Յ const.Z indepen- 1 2 ␣ Ϸ ⁄137 and Z Ͻ 100, the regime ⌫ Ն Z is the most important dent of ␣,N, and M such that one for physics. 2 H Ն Ϫ(Z)Z2⅐M. [12] (ii)If⌫ՆZ the bounds in Theorem 2 imply that A ՆϪ 3 [17] Remarks:(i) The linear dependence on M (and indepen- inf ͑ , ͒ CЈZM, ͉͉͉͉ϭ1,A dence of N) in bounds 11 and 12 implies H-stability of the systems considered here. The linear dependence on ⌳ of our for some finite constant CЈ independent of the number N of bound 11 is related to the fact that the Hamiltonian H is the electrons. In our units of energy, the ground state energy of an unrenormalized Hamiltonian of quantum electrodynamics with atom or ion consisting of a nucleus of atomic number Z and an nonrelativistic matter (see the discussion in ref. 8 and, espe- arbitrary number of electrons is ՆϪconst. Z7/3 (see refs. 12 and cially, in ref. 7). 13). The fact that, in part ii of Theorem 1, we only have that H Ն (ii) In our units, a typical energy of an inner electron bound Ϫconst. Z7/2 M is a consequence of the ultraviolet divergence of in an atom of atomic number Z is ϪZ2. Thus the ultraviolet the zero-point energy of the quantized electromagnetic field cutoff energy ␣Ϫ1⌳ϳZ2␣Ϫ3/4 for photons is much larger than (see ref. 8). a typical energy of such an electron, because, in nature, ␣ Ϸ (iii) The strategy developed in ref. 9 (see also ref. 6) to prove 1 ⁄137. A nucleus of atomic number Z can bind of the order of Theorem 2 is to reduce the stability bound 15 to local stability Z electrons (12). The ground state energy of the resulting of matter in small cubes of physical space ޅ3 whose sizes are bound state (an atom or ion) is therefore bounded from below chosen to depend on the configuration of nuclei and on the by Ϫconst. Z3, in our units. We therefore conjecture that our local behavior of the magnetic field B. These cubes form a estimates on (Z) in inequality 12 can be improved to (Z) Յ Calde´ron–Zygmunddecomposition of ޅ3. const. Z (see also Theorem 2 below). To show that Theorem 2 implies Theorem 1, we write the (iii) A proof of Theorem 1 can be found in ref. 8. It combines Hamiltonian H introduced in Eqs. 2-7 as the sum of two terms, arguments in ref. 7 with a new result on H-stability of nonrelativistic quantum-mechanical matter in an arbitrary H ϭ HI ϩ މHII, static, external magnetic field established in ref. 9, which is where stated below. ͑⌳͒ Let A(x) be an arbitrary, classical electromagnetic vector HI ϭ HPauli ϩfield͑A ͒, potential with square-integrable second derivatives in x and with ٌ⅐A(x) ϭ 0 (Coulomb gauge). Let B(x) ϭ curl A(x) with Downloaded by guest on September 29, 2021 Mathematics: Fefferman et al. Proc. Natl. Acad. Sci. USA 93 (1996) 15011
of an electron, in our units. We also define b(x) ϭ r(x)Ϫ2 as the 2 2 field͑A͒ ϭ ͕⌫͉B͑x͉͒ ϩ C͑⌫, Z͒L strength of an effective magnetic field. ͵ (1) ޅ3 Our main result on H , proven in ref. 10, is the following theorem. 2 ϪLϪ1D͑x͒ 3 ϫ ٌ͉͑ B͒͑x͉͒ }e d x, THEOREM 4. There are finite, positive constants C, CЈ,CЉsuch that, for an arbitrary vector potential A(x), and
͑⌳͒ HII ϭ Hfield Ϫfield͑A ͒. 5͞2 3 3͞2 3 Ϫei Յ CЈ ͵ v(x) d x ϩ CЉ ͵ b(x) v(x)d x, [20] It is clear that Theorem 2 proves an appropriate lower bound i ޅ3 ޅ3 on H [because, in the ‘‘Schro¨dinger representation’’ of Fock I and space Ᏺ, A(⌳) can be treated as a classical vector potential (8)], while a lower bound on HII follows from surprisingly simple Fock space estimates (7, 8). b(x)2d3x Յ C ͉B(x)͉2d3x. [21] In refs. 7 and 10 the following result related to Theorem 2 is ͵ ͵ ޅ3 ޅ3 proven: Let As a corollary one obtains a result of ref. 5: There is a finite, ⍀ ϭ ͕x ʦ ޅ3͉͉x Ϫ y ͉ Յ ͑Z ϩ 1͒Ϫ1, l ϭ 1, ⅐⅐⅐,M͖ l positive constant K such that
and define the classical field energy in the region ⍀ by 3͞4 5͞2 3 2 3 Ϫ ei Յ K ͵ v͑x͒ d x ϩ ͵ ͉B͑x͉͒b d x 2 3 i ͭ ޅ3 ͩ ޅ3 ͪ ,field͑A; ⍀͒ ϭ ⌫ ͵ ٌ͉͑ A͒͑x͉͒ d x ⍀ 1͞4 with ⌫ϭ(8␣2)Ϫ1. ϫ v͑x͒4d3x . [22] Ϫ1 ͵ THEOREM 3. There is a constant Ͼ 0 such that if ⌫ (Z ϩ ͩ ޅ3 ͪ ͮ 1) Ͻ , then A Ն 2 HPauli ϩ field( ; ⍀) ϪK(Z ϩ 1) (N ϩ M), The relevance of inequalities 20, 21, or 22 for proofs of for some finite constant K (depending only on ⌫). stability of matter in magnetic fields is explained in ref. 10, Theorem 3 can be used to prove a variant of part ii of respectively ref. 5. See also ref. 14. Theorem 1 (see ref. 7) in the (physically relevant) range ␣ Յ 1 Ϫ1 Ϫ5/4 1. Fro¨hlich, J., Lieb, E. H. & Loss, M. (1986) Commun. Math. Phys. ⁄132, Z Յ 6, and ␣ ⌳ Յ ␣ (Z ϩ 1). The proof of Theorem 3 is based on methods developed in 104, 251–270. ref. 5. Estimates extending the key estimates underlying the 2. Lieb, E. H. & Loss, M. (1986) Commun. Math. Phys. 104, results in refs. 5 and 7 are proven in ref. 10. They have the 271–282. following flavor: As in Eq. 4, define 3. Loss, M. & Yau, H. T. (1986) Commun. Math. Phys. 104, 283–290. 4. Fefferman, C. (1995) Proc. Natl. Acad. Sci. USA 92, 5006–5007. 3 Ѩ 5. Lieb, E. H., Loss, M. & Solovej, J. P. (1995) Phys. Rev. Lett. 75, D͞ ϭ Ϫi Ϫ A ͑x͒ ⅐ , 985–989. ͩ Ѩx␣ ␣ ͪ ␣ ␣ϭ1 6. Minicozzi, W. (1994) Notes to Lectures by C. Fefferman at Stanford University. where A(x) ϭ (A (x), A (x), A (x)) is an arbitrary vector 1 2 3 7. Bugliaro, L., Fro¨hlich, J. & Graf, G. M. (1996) Phys. Rev. Lett., potential, and let B ϭ curl A be the magnetic field correspond- in press. ing to A. Let v(x)beapositive (bounded, measurable) function 8. Fefferman, C., Fro¨hlich, J. & Graf, G. M. (1996) Commun. Math. ޅ3 on . We consider the one-electron Pauli Hamiltonian Phys., in press. H͑1͒ ϭ D͞ 2 Ϫ v. [18] 9. Fefferman, C., On electrons and nuclei in a magnetic field, Adv. Math., in press. (1) 10. Bugliaro, L., Fefferman, C., Fro¨hlich, J., Graf, G. M. & Stubbe, Let {ei}iϭ0,1,2,⅐⅐⅐ be the negative eigenvalues of H (ordered J. (1996) A Lieb–Thirring Bound for a Magnetic Pauli Hamilto- such that e0 Յ e1 Յ e2 Յ ⅐⅐⅐). We define a basic, B-dependent length scale r(x) as the solution of the equation nian, preprint. 11. Lieb, E. H. (1991) The Stability of Matter: From Atoms to Stars: Selecta of Elliott H. Lieb, ed. Thirring, W. (Springer, Heidelberg). r͑x͒Ϫ1 ϭ ͑r͑x͒Ϫ1͑y Ϫ x͉͒͒B͑y͉͒2d3y, [19] 12. Ruskai, M. B. (1982) Commun. Math. Phys. 85, 325–327. ͵ 13. Lieb, E. H., Sigal, I. M., Simon, B. & Thirring, W. (1984) Phys. Rev. Lett. 52, 994–996. 1 2 Ϫ2 with (x) ϭ (1 ϩ ⁄2͉x͉ ) . For a homogeneous magnetic field 14. Sobolev, A. V. (1996) Lieb–Thirring Inequalities for the Pauli B, r(x) is proportional to ͉B͉Ϫ1/2, which is the cyclotron radius Operator in Three Dimensions, preprint. Downloaded by guest on September 29, 2021