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(E3, d3x) R C2 is the Hilbert space for one electron of 1 3 quantized radiation field, with ultraviolet cutoff, coupled to spin ⁄2 in physical space E , L denotes an antisymmetric tensor arbitrarily many non-relativistic quantized electrons and product, and ^ 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(R3, d3k) and fine structure constant. We also announce bounds for the R C2. The factors C2 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 5 HPauli 1 Hfield, [2] magnetic field andyor 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 5 Dy 2 1 V , [3] A typical system we propose to study consists of an arbitrary Pauli O i c i51 number, N, of nonrelativistic electrons with electric charge 2e, 3 1 ­ bare mass m . 0, spin ⁄2 and a bare gyromagnetic factor g 5 ~L! ~i! Dy 5 2i 2 Aa ~x ! z sa , [4] 2, an arbitrary number, M, of nuclei of atomic number #Z, for i O a i a51 S ­xi D some arbitrary, but fixed integer Z ,`, and an arbitrary number of photons which describe the transverse degrees of a 3 xi is the ath component of the position, xi [ E ,oftheith freedom of the quantized electromagnetic field. The dynamics (L) electron; Aa (x)istheath component of the quantized of the system, generated by a self-adjoint Hamilton operator H, electromagnetic vector potential, A(L)(x), with an ultraviolet 3 (i) conserves the number of nuclei and electrons, but the number cutoff L, at the point x [ E (see below); sa 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 ^ zzz ^ sa ^zzz ^ I, [5] per charged particle (electron or nucleus) is bounded uniformly 2 RN in N and M, for fixed Z ,`. This property of nonrelativistic, on spin space (C ) , where sa, a 5 1, 2, 3 are the three Pauli quantum-mechanical matter is called H-stability. In ref. 11 the matrices, and the 232 matrix sa 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 5 O 2 O uxi 2 xju i51,zzz,N uxi 2 ylu of the systems described above is a mathematical consequence 1#i,j#N l51,zzz,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 1 O , [6] uyl 2 yku the fact that their masses are much larger than the mass of an 1#l,k#M electron, but we emphasize that, because of their magnetic 3 moments, this is not an entirely innocent approximation for where yl [ E denotes the position of the lth static nucleus and most realistic nuclei; see the discussion in refs. 7 and 8. Zl # Z its atomic number, l 5 1, zzz,M; finally, Furthermore, interactions between electrons and photons with energies large compared with a typical energy of an electron 21 w 3 H 5 a a ~k!ukua ~k!d k [7] in an atom of atomic number #Z are turned off with the help field O E l l l56 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 a 5 e2y\c (\ is Planck’s constant and c is the velocity of light) is the dimen- 2 3 3 2LN 1 * 5 ~L ~E , d x! ^C! ^^, [1] sionless fine-structure constant, with a ' ⁄137 in nature. We have chosen units for energy where 2mc2a2 5 1, i.e., the ground state energy of a hydrogen atom is 24. 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 ¹zA(L)(x) 5 0. The ultraviolet cutoff vector potential A(L)(x)is D~x)5 min ux 2 yku. [13] given by 1#k#M 2 E3 3 C2 LN A~L!~x! 5 A~L!~x! 1 A~L!~x!, Let c [ (L ( , d x) R ) . We consider the energy 2 1 functional where «~c, A! 5 ^c, HPaulic& a1y2 A~L!~x! 5 L~k!uku21y2a ~k!« ~k!eikzxd3k, 2 2p O E l l 1 $GuB~x!u2 1 C~G, Z!L2u~¹ ^ B!~x!u2% l56 E E3 A~L!~x! 5 ~A~L!~x!!w, [8] 1 2 3 e2L21D~x!d3x, [14] k denotes a wave vector, and the direction of propagation, where HPauli is the N-electron Hamiltonian defined in eqs. 3 kyuku, and the two polarization vectors «l(k), l 56, form an and 4, but with A(L) replaced by the classical vector potential orthonormal basis of R3 R C, for each k [ R3; moreover, the A cutoff function L(k) satisfies 0 # L(k) # 1, and , and G, C(G, Z), and L are constants yet to be chosen. The proof of Theorem 1 is based on a lower bound for the supp L~ z ! # $kuuku#L%, [9] energy functional % which we describe next. THEOREM 2. Given G.0and Z ,`, and for a suitable choice , where L is a finite constant. The operators al(k) and al(k) are of the constant C(G,Z),`in Eq. 14, there are finite, positive creation and annihilation operators on ^ satisfying the canon- constants c(G,Z)and C9(G,Z)depending only on G and Z such ical commutation relations that, for arbitrary L # c(G, Z), # # w ~3! 21 @al~k! , al9~k! # 5 0, @al ~k!, al9~k9! # 5 dll9d ~k 2 k9!, [10] inf «~c, A! $2C9~G,Z!L M, [15] uucuu51,A where a# 5 a or a,. where M is the number of nuclei. If G $ Z2 there are finite, positive We are now prepared to describe our main results concern- 21 ing H-stability of the systems described above. constants c and C such that, for the choices c(G,Z)5cZ [i.e., HEOREM c(G,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 #Z, for to a nucleus of atomic number Z] and C(G,Z)5CZ , one has arbitrary values of Z ,`and of the fine-structure constant a, and that for an arbitrary ultraviolet cutoff L (see expression 9), there exists C9~G, Z! # CZ2. [16] a finite constant E(a,Z) (depending only on a and Z) such that Remarks:(i) The proof of Theorem 2 is outlined in ref. 8 and H $ 2E(a,Z)LzM. [11] carried out in full detail in ref. 9. In ref. 9, bounds on the constants c(G, Z), C(G, Z), and C9(G, Z) are given in the three (ii) For arbitrary N, M, and Z, as above, for an ultraviolet cutoff 2 2 21 2 23/4 regimes G $ Z , Z $ G $ 1, and G # 1. In using Theorem 2 a L # Z a , and for sufficiently small values of aZ and 22 5/2 3 3/2 to prove Theorem 1, one sets G5const. a . Since, in nature, a Z , there exists a finite constant «(Z) # const.Z indepen- 1 2 a ' ⁄137 and Z , 100, the regime G $ Z is the most important dent of a,N, and M such that one for physics.