Stability of Matter for the Hartree-Fock Functional of the Relativistic Electron-Positron Field

We investigate stability of matter of the Hartree-Fock functional of the relativistic electron-positron eld { in suitable second quantization { interacting via a second quantized Coulomb eld and a classical magnetic eld. We are able to show that stability holds for a range of nuclear charges Z 1 ; ::; Z K Z and ne structure constants that include the physical value of and elements up to holmium (Z = 67).


Introduction
Electrons and positrons can be described just interacting with themselves and the electromagnetic eld.However, in many interesting applications these particles do not exist separated from the rest of the world but interact with nuclei, in fact very often with many nuclei.It is therefore of interest, to investigate the stability of quantum electrodynamics, the basic theory describing relativistic electrons and positrons, when coupled to many nuclei.A standard model to incorporate nuclei is to assume them as external sources of the electric eld and minimize the energy over all possible pairwise distinct nuclear positions.This is known as the Born-Oppenheimer approximation.
Stability in the context of eld theory means, that the energy is bounded from below by a multiple of the number operator of the electron-positron eld plus a constant times the number of nuclei involved.In fact, we would like to show positivity of the energy.
The purpose of this paper is to make a step towards this direction.Based on paper of Chaix et al. 4] we showed 2] that the Hartree-Fock functional of the vacuum and of atoms with su ciently small nuclear charge is nonnegative (with or without self-generated magnetic eld) provided the Sommerfeld ne structure constant = e 2 is also small where e is the elementary charge unit.These results included the physical value 1=137 and atoms with atomic number up to 67 (holmium).Here we show that positivity even holds when the number of nuclei is no longer restricted, in fact without any essential loss: it holds again up to holmium for the physical value of .
Our paper is organized as follows: For the readers convenience we x some notations in Section 2 and Appendix B. Some inequalities used in the proof are collected in Appendix A. Section 3 contains our positivity result for the Hartree-Fock functional disregarding the magnetic eld.Section 4 extends this to the case when the self-generated magnetic eld of the particle is taken into account on a classical level.
2 Definition of the Problem Before stating our problem precisely, we x our notations following 2].(See also Appendix B for additional notations.)Dirac Operator The operator for a particle of charge e, in magnetic eld r A, and interacting with K nuclei of same charge is acting in the four components vector space H = L 2 (R 3 ) C 4 .The 4 4 matrices and are the Dirac matrices in the standard representation 14].The vector potential A is assumed to be such that the magnetic induction B = r A is square integrable.The multiplication operator eV is the electric potential of K nuclei with charge eZ located at R 1 ; : : : ; R K , i.e., V (x) := K X k=1 Z jx R k j : (1) Note that D A;V is self-adjoint with form domain H 1=2 (R 3 ) C 4 under the assumption on e and Z stated in Theorems 1 and 2.
Energy of a State We de ne D to be the set of all states with nite kinetic energy, i.e., P i;j2Z (D 0;0 ) i;j (: i j : ) converges absolutely where colons denote normal ordering where we xed an orthonormal basis such that all basis vectors e i are in H 1=2 (R 3 ) C 4 .We denote by (D A;V ) i;j = (e i ; D A;V e j ), and by W i;j;k;l , the matrix elements of the two-body Coulomb potential W(x; y) = 1=jx yj, where dx denotes the product measure (Lebesgue measure in the rst factor and counting measure in the second factor) of G := R 3 f1; (3) where D(f; g) := (1=2) R R 6 dxdyf(x)g(y)jx yj 1 is the Coulomb scalar product, (x; y) := P i;j2Z (e i ; e j )e i (x)e j (y), (x; y) := P i;j2Z (e i ; e j )e i (x)e j (y) (note the di erence to ), and (x) := P 4 =1 (x; x).(We use the notation x := (x; ) 2 R3 f1; : : : ; 4g.)We also remind the reader that = e2 .
The main goal of this paper is to show positivity of E A;V; ( ) for quasi-free states.More notations can be found in Appendix B.

Stability of Relativistic Matter without Magnetic Field
We prove here, in the case A = 0, that the energy functional E A;V; de ned in (2) is positive on generalized Hartree-Fock states for suitable choice of the electron subspace and and Z small enough.More precisely, H + := 0;1) (D 0;V eff )](H) is the positive spectral subspace associated to D 0;0 + V e , where Here := fx 2 R 3 : jx R j jx R k j; 8k = 1; : : : ; Kg denotes the -th Voronoi cell and M is the characteristic function of the set M. Our rst result is Theorem 1. Pick H + := 0;1) (D 0;V eff )](H) as electron subspace.Let L 1=2;3 be the constant in the Lieb-Thirring inequality 2 for moments of order 1=2.If 2 (0; 1), 2 0; 4= ] and Z 2 0; 1) are such that Remark that we do not assume that 0 is not in the spectrum of D 0;V eff .This means in particular that H + includes the null space of D 0;V eff .Note also that is a free parameter that we can use to optimize the value of and Z. Instead of giving a cumbersome analytic formula, Figure 1 gives the result when picking suitably.The proof of the theorem consists of ve steps: Replace the Dirac operator D 0;V by D 0;V eff which is done by reducing the Coulomb potential V in each Voronoi cell to a one-nucleus/electron Coulomb potential V e .
Dominate the exchange energy W X by the kinetic energy.
Control the di erence of the kinetic energy and the energy of the modi ed Dirac operator D 0;V eff by applying the Birman-Koplienko-Solomyak inequality 3] to obtain a Schr odinger like operator.X s;t=1 Z j (x; y)j 2 jx yj dxdy tr (jrj 1) 2 ] tr(jD 0;0 j 2 ) tr(jD 0;0 j( ++ )): ( 6) So far we have not used the choice of the subspaces H + and H speci ed in the hypothesis.In order to control the trace in (6) with the trace on the right hand side of ( 5), we now use that H + is the positive spectral subspace of D 0;V eff , i.e., H + := 0;1) (D 0;V eff ) (H).This implies tr(D 0;V eff ) = tr(jD 0;V eff j( ++ )), and thus E 0;V; tr h jD 0;V eff j 4 jD 0;0 j ( ++ If we bound below the trace on the right hand side of ( 7) by using the Birman-Koplienko-Solomyak inequality 3] (see also Appendix A), and noting that 0 ++ 1, we obtain tr jD 0;V eff j jD 0;0 =4j ( ++ ) tr jD 0;V eff j jD 0;0 j=4 tr n (D 0;V eff ) 2 2 2 (D 0;0 where the subscript minus denotes the negative part (jAj A)=2 of the operator A.
To bound the trace on the right hand side of (8) from below, we use the localized Hardy inequality of Lieb and Yau 12, Formula (5.2)] (see also Appendix A), K times with k = 1; : : : ; K and B k : ) jf(x)j 2 dx : (9) Inequality (9) together with (8) gives E 0;V; tr (" Using the Lieb-Thirring inequality (see Appendix A) for the exponent 1=2 in ( 10) implies E 0;V; Note that the numerical value of the Lieb-Thirring constant L 1=2;3 does not exceed 0:06003.In (11), we have estimated the rst term in the parenthesis with Inequality (4.6) in 8].

Inclusion of the Magnetic Field
We now consider the whole energy functional E A;V; given in (3), i.e., we include also magnetic elds B := r A of nite eld energy.Theorem 2. Take H + := 0;1) (D A;V eff )](H).If 2 (0; 1), 0 2 (0; 1), 2 0; 4= ] and Z 2 0; 1) verify Again, note that and 0 are free parameters that can be picked arbitrarily within the given ranges.However, we refrain to give cumbersome optimal expressions.Instead we { once again { optimize numerically, insert, and show the result in Figure 2.

B Notations
We collect some additional notation that was already used in 2]: Fock Space and Field Operators For a given orthogonal decomposition L 2 (R 3 ) R 4 = H + H into the one-particle electron and positron subspace, one constructs, following 14] (see also 6] and 2]), the Fock space F. We denote the orthogonal projections onto H + and H are denoted by P H+ and P H respectively.For any f 2 H, we also denote the particle annihilation (respectively creation) operator by a(f) (respectively a (f)) and the antiparticle annihilation (respectively creation) operator by b(f) (respectively b (f)).(Note that { according to the convention used in 6] and also here { a(f) = a(P H+ f) and b(f) = b(P H f).) They ful ll the canonical anticommutation relations for all f and g in H fa(f); a(g)g = fa (f); a (g)g = fb(f); b(g)g = fb (f); b (g)g = 0; (17) fa(f); a (g)g = (f; P H+ g) ; fb (f); b(g)g = (f; P H g) where f ; g denotes the anticommutator.
For any f 2 H, the eld operator is the antilinear bounded operator (f) := a(f) + b (f) acting in F. Its adjoint is linear and equal to (f) = a (f) + b(f).Given an orthonormal basis f: : : ; e 2 ; e 1 ; e 0 ; e 1 ; : : : g of H, where vectors with negative indices are in H and vectors with nonnegative indices are in H + , we denote a i := a(e i ), a i := a (e i ), b i := b(e i ), b i := b (e i ), i := a i +b i and i := a i +b i .with ++ := P H+ P H+ , + := P H+ P H , + := P H P H+ = + , and := P H P H appropriately restricted.Similarly ++ := P H+ P H+ , + := P H+ P H , + := P H P H+ = t + , and := P H P H also appropriately restricted.
(x)e j (y)e k (x)e l (y) jx yj

Figure 1 :
Figure1: The plain curve gives an estimate from below of the critical value of the pair ( ; Z), for which the energy E 0;V; is positive.For the physical value 1=137:0359895 we obtain Z 0:489576, i.e., Z 67:089649.The dashed curve is the one obtained in 2] in the case of a single nucleus of atomic number Z

2 Z
Estimate the resulting expression by a localized Hardy inequality of Lieb and Yau 12] going back to Dyson and Lenard 5].Apply the Lieb-Thirring inequality 10] for moment 1=2 to estimate the trace.Proof.Set d k to be half the distance of the k-th nucleus to its nearest neighbor, then the electrostatic inequality of Lieb and Yau 12], p. 196, Formula (4.4), implies with Documenta Mathematica 3 (1998) 353{364 d (x) := (x)dx E 0;V; tr(D 0;V ) + U + D( ; ) s inequality (see Appendix A) and then Inequalities (22) and (23) we get 2 4

Figure 2 :
Figure 2: The plain curve gives an estimate from below of the critical value of the pair ( ; Z), for which the energy E A;V; is positive.For the physical value 1=137:0359895 we obtain Z 0:48899985, i.e., Z 67:0105779.The dashed curve

2 (
p 1 and consider two non-negative self-adjoint linear operators C and D such that C p D p ] 1=p is trace class.Then C D] is trace class tr C D] tr C p D p ] 1=p(Birman, Koplienko, and Solomyak 3], see also 9]).Diamagnetic Inequalities Let A 2 L 2 loc (R 3 ; R 3 ), then, for all u with juj 2 H 1 (Simon 13]) and for all u 2 D(jpj) (juj; jpj juj) (u; jp + Aju) (see 8, Formula (5.7)]).(Note that we allow for the right side to be in nite.)Documenta Mathematica 3 (1998) 353{364 Electrostatic Inequality Let be any bounded Borel measure on R3 , then with the notations of Theorem 1 we have 12, Lemma 1

4jx
and d any positive real number.If B d (R) denotes the ball in R3 with center R and radius d, then, for any f 2 L 2 (B d (R)) such that rf 2 L 2 (B d (R)) we have 12, Formula (5.Rj 2 (1 + jx Rj 2 4d 2 ) jf(x)j 2 dx:Lieb-Thirring Inequality (d = 3, = 1=2) Given a positive constant , a real vector eld A with square integrable gradients, and a real valued function V in L 2 (R 3 ), we have for V + := (jV j + V )andThirring 11]  for the case A = 0 and Avron, Herbst, andSimon   1]  for the general case).
One-Particle Density Matrix A trace class operator on H H is called a onewhere the superscript t refers to transposition, i.e., given our basis xed initially, the matrix elements of B t are (B t ) i;j := B j;i .Since the Hilbert space H is the orthogonal sum of H + and H , we can write