V. BACH, J.M. BARBAROU, B. HELFFER, AND H. SIEDENTOP Since quantum electrodynamics depends on the denition of the electron space, it is interesting to

Transcription

1 ON THE STABILITY OF THE RELATIVISTIC ELECTRON-POSITRON FIELD VOLKER BACH, JEAN-MARIE BARBAROU, BERNARD HELFFER, AND HEIN SIEDENTOP Abstract. We study the energy of relativistic electrons and positrons interacting through the second quantized Coulomb interaction and a self-generated magnetic eld. As states we allow generalized Hartree-Fock states in the Fock space. Our main result is the assertion of positivity of the energy, if the atomic numbers and the ne structure constant are not too big. We also discuss the dependence of the result on the dressing of the electrons (choice of subspaces dening the electrons). 1. Introduction Relativistic particles particles should be described by quantum electrodynamics. This theory dates back to the very early days of quantum mechanics. It can be traced back to Born and Jordan's [5] famous paper clarifying Heisenberg's matrix mechanics which was followed by many important works. But despite these eorts it is still unclear, whether the energy in quantum electrodynamics is bounded from below; the existence of a vacuum state, i.e., a state with lowest but nite energy (ground state) remains elusive. In view of this unsatisfactory situation various approximate models have been studied. In this context one can mention the work of Conlon [8], Feerman [9], Lieb et al. [15, 1, 13, 14]. All of these models assume that the Hamiltonian is particle number conserving. In fact the positron number is assumed to be zero and the electron number enters only as a parameter. There is, however, an interesting observation of Chaix, Iracane, and Lions [6, 7] that concerns an approximate quantum electrodynamics. They consider the second quantized relativistic electron-positron eld interacting via the second quantized electric eld disregarding any contribution from the magnetic eld and no external eld present. They argue that the energy of generalized Hartree-Fock states is nonnegative provided the ne structure constant is less than 1=(4). Here, we prove and extend this result in various directions. The paper is organized a follows Section gives an introduction into the basic terminology of the subject and allows us to formulate our result properly. Section 3 contains the above mentioned stability result of Chaix et al. [7]. To describe atoms and molecules one needs to include also the external potentials of the nuclei. Section 4 treats this for the atomic case. Besides a constraint on we also obtain a constraint on. In Section 5 we are able to include the classical magnetic eld generated by the particles. As usual it destabilizes the problem further. The result is condensed in Figure 1 where we obtain a critical curve which is the boundary of the region where we obtain positivity of the energy. Date April 17, Key words and phrases. Dirac operator, Stability, QED, Generalized Hartree-Fock states. Thanks go to Christian Tix for useful help with the numerical optimization of Formula (31) resulting in Figure 1. Financial support of the European Union through the TMR network FMR- CT is gratefully acknowledged. 1

2 V. BACH, J.M. BARBAROU, B. HELFFER, AND H. SIEDENTOP Since quantum electrodynamics depends on the denition of the electron space, it is interesting to study the consequences on the lowest possible energy. This is done in Section 6. The result is that taking the positive spectral subspace of the one-particle Dirac operator including the external eld is the optimal choice.. Definition of the Problem and Statement of the Result For the readers convenience and to state our results precisely, we introduce some notations following mainly Thaller [18] and partially [1]. Dirac Operator The Dirac operator of a particle of charge e in the electric eld erv and magnetic eld r A is D A;V = ( 1 i r + ea) + m + e V acting in H = L (R 3 )C 4. The 44 matrices and are dened as follows = ; = 0 i i 0 where i, i = 1; ; 3, are the Pauli matrices, i 1 = ; 1 0 = ; i 0 3 = ; i = 1; ; 3 ; Although the operator can be dened for more general vector potentials A, it suces for our purpose to assume B = r A to be square-integrable which we will do henceforth. (Physically B is the magnetic induction and erv the electric eld.) We are primarily interested in describing atoms, i.e., we pick V (x) = =jxj where is the atomic number of the atom under consideration. Again, we remark that our results transcribe easily to more general potentials. We also set = e, the Sommerfeld ne structure constant. We assume that D A;V is self-adjoint with form domain H 1= (R 3 ) C 4 and that zero is not in the spectrum of D A;V. Projections Spectral projections on the positive subspace and negative subspace of D A;V are respectively denoted by P+ A;V and P A;V. We will also use the following notation jd A;V j = P+ A;V DA;V P A;V DA;V. Fock Space Given a closed subspace H + of H, we dene the one-particle electron subspace as F (1) + = H +. The one-particle positron subspace is F (1) = C(H +), where C is the charge conjugation operator dened as (C )(x) = i (x). The projections onto H + and H are denoted by P H+ and P H. The n-particle electron subspace is F (n) + = n^ =1 and the m-particle positron subspace is F (m) = m^ =1 H + ; CH (1) We also set F (0) + = F (0) = C. The total Fock space is F = 1M n;m=0 F (n) + F(m) Field Operators We denote by x = (x; s) an element of G = R 3 f1; ; 3; 4g. Assume f; g H

3 STABILITY 3 First, we dene in F the electron annihilation operator a(f). On the subspace F (n+1;m) it acts as () (a(f) ) (n;m) (x 1 ; ; x n ; y 1 ; ; y m ) (3) = p n + 1 G dx (P H+ f)(x) (n+1;m) (x; x 1 ; ; x n ; y 1 ; ; y m ) where dx denotes the product measure (Lebesgue measure in the rst factor and counting measure in the second factor) of G. By linearity it extends to a bounded operator on F. Note that the map f 7! a(f) is anti-linear. Next, we introduce the electron creation operator by dening it rst on F (n1;m) (a (f) ) (n;m) = 1 p n n j=1(1) j+1 P H+ f(x j ) (n1;m) (x 1 ; ; ^x j ; ; x n ; y 1 ; ; y m ) (4) (5) (with the understanding that a negative superscript indicates the zero element). As before it extends by linearity to a bounded operator on F. The operator a (f) is the adjoint of a(f) and the map f 7! a (f) is linear. We also have the canonical anti-commutation relations fa(f); a(g)g = fa (f); a (g)g = 0 ; fa(f); a (g)g = (f; P H+ g) We dene the positron annihilation operator by (6) (b(g) ) (n;m) (x 1 ; ; x n ; y 1 ; ; y m ) = (1) np m + 1 and the positron creation operator (7) (b (g) ) (n;m) (x 1 ; ; x n ; y 1 ; ; y m ) G dy [CP H g](y) (n;m+1) (x 1 ; ; x n ; y; y 1 ; ; y m ); = (1)n p m m k=1(1) k+1 [CP H g](y k ) (n;m1) (x 1 ; ; x n ; y 1 ; ; ^y k ; ; y m ) We observe that, due to the anti-linearity of C, the map g 7! b(g) is linear and the map g 7! b (g) is anti-linear. Similarly to (4) and (5), we have for these operators the anti-commutation relations fb(f); b(g)g = fb(f) ; b (g)g = 0 ; fb(f) ; b(g)g = (f; P H g) Note that, for all f H, a(f) and b(f) annihilate the vacuum vector = 1. This property characterizes the vacuum vector uniquely up to phase factors. We then dene the eld operator (f) on the Fock space F by (f) = a(f) + b (f) This eld operator is bounded on F and depends anti-linearly on f. adjoint operator (f) = a (f) + b(f) is also bounded and depends linearly on f. Matrix Elements We x an orthonormal basis f ; e ; 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 also assume that all basis vectors e i are in H 1= (R 3 ) C 4. In addition we set a i = a(e i ), a i = a (e i ), b i = b(e i ), b i = b (e i ), The

4 4 V. BACH, J.M. BARBAROU, B. HELFFER, AND H. SIEDENTOP i = a i + b i and i = a i + b i. By (D A;V ) i;j we denote the matrix elements of the Dirac operator, i.e., (D A;V ) i;j = (e i ; D A;V e j ); and by W i;j;k;l, we denote the matrix elements of the two-body Coulomb potential W (x; y) = 1=jx yj, i.e, W i;j;k;l = (e i e j ; W e k e l ) = dx dy e i(x)e j (y)e k (x)e l (y) G G jx yj Note that W i;j;k;l C under our assumptions on the basis fe i g i. Energy We dene D to be set of all states on B(F), i.e., all positive continuous linear forms on the set of bounded operators B(F) with (1) = 1, which have also nite kinetic energy, i.e., P i;j (D0;0 ) i;j ( i j ) < 1, where colons denote normal ordering (Thaller [18], p. 85). Then, the energy E A;V; D! R in the state is (8) E A;V; () = (D A;V ) i;j ( i j )+ i;j i;j;k;l W i;j;k;l ( i j l k )+ 1 8 R 3 B ; where the Sommerfeld ne structure constant has the physical value of about 1= []. The main purpose of this paper is to estimate when E A;V; is bounded from below or not as a function of and. One-Particle Density Matrix A trace class operator on H H is called a one-particle density operator (1-pdm), if it has the following properties = and 1 1. (9) = (10) (11) with = and t = where 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. We note that the Hilbert space H is the orthogonal sum H + and H. This means that we can write = B + C A + + 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. To conclude this item we note that each state denes a 1-pdm in a natural way through the formula (h; g) = [ (g 1 ) + (~g )][ (h 1 ) + ( h ~ )] where h = (h 1 ; h ) H, g = (g 1 ; g ) H and given f = P k ke k, we dene ~ f = Pk ke k. Note that the conjugation dened this way traces back to the fact that the density matrix is given with respect to a xed basis (See also [1], beginning of Section.b).

5 STABILITY 5 (1) (13) To characterize all 1-pdm for which there exist also corresponding states (representability), we also introduce unrenormalized one-particle density matrices + PH ur = 1 ( + P H ) Then, is representable (according to (11)), if in addition to = we have 0 ur 1. (The representability can be shown by adapting a proof from [1].) The matrix elements are in this case i;j = ( j i ), ( ++ ) i;j = (a j a i), ( + ) i;j = (b j a i ), ( ) i;j = (b i b j) and i;j = ( j i ), ( ++ ) i;j = (a j a i ), ( + ) i;j = (b j a i), ( ) i;j = (b j b i ). For any given dene 0 ++ = + + or equivalently, for g; h H, (h; 0 g) = ( (g) (h)). The canonical anticommutation relations imply 0 ~ 1. Thus, noting that + + = + +, we get ; Hartree-Fock States The set of generalized Hartree-Fock states (or quasi-free states with nite particle number) is the set of states that fulll i) For all nite sequences of operators d 1 ; d ; ; d K, where d i stands for a i, a i, b i, or b i, we have (d 1 d d K1 ) = 0 and (d 1 d d K ) = S ; sgn()(d (1) d () ) (d (K1) d (K) ) (14) (15) where S is the set of permutations such that (1) < (3) < < (K 1) and (i 1) < (i) for all 1 i K 1. This implies in particular (d 1 d d 3 d 4 ) = (d 1 d )(d 3 d 4 ) (d 1 d 3 )(d d 4 ) + (d 1 d 4 )(d d 3 ) ii) The state has a nite particle number, i.e., if N = P i (a i a i + b i b i) denotes the particle number operator, we have (N) < 1, or equivalently, written in terms of the one-particle density matrix, tr( ++ ) < 1. We write D HF for the set of all generalized Hartree-Fock states with nite kinetic energy, i.e., P i;j (D0;0 ) i;j ( i j ) < 1. Hartree-Fock Functional By we denote the set of all one-particle density matrices with nite kinetic energy. The Hartree-Fock functional is dened by EA;V; HF! R ; A;V;() = tr(d A;V ) + j(x; y)j dxdy + D( jx ; ) yj j(x; y)j dxdy + 1 B jx yj 8 R 3 E HF where D(f; g) = (1=) R R 6 dxdyf(x)g(y)jxyj 1, (x; y) = P i;j (e i; e j )e i (x)e j (y), (x; y) = P i;j (e i; e j )e i (x)e j (y) (note the dierence to ), and (x) = P 4 =1 (x; x). In Section 3.1 we will be able to identify it as the energy of Hartree-Fock states written in terms of the one-particle density matrix.

6 6 V. BACH, J.M. BARBAROU, B. HELFFER, AND H. SIEDENTOP As an immediate consequence of the representability of any unrenormalized 1-pdm (see [1]), we note that its inmum over the one-particle density matrices with nite kinetic energy is equal to the inmum of E A;V; over D HF. 3. The Existence of the Hartree-Fock Vacuum without External Potentials We are now ready to state our rst result which has been claimed and argued for by Chaix et al. [7]. Theorem 1. Pick for the one-particle electron subspace H + = P+ 0;0 E 0;0; j DHF is nonnegative for [0; 4=]. H. Then, 3.1. Reduction to 1-pdm. First, we express the energy E 0;0; () in terms of the matrix elements of the 1-pdm. Expanding the quartic term in (8) yields the 48 summands ( (16) (17) i j l k ) =(a i a j )(a l a k ) (a i a l )(a j a k ) + (a i a k )(a j a l ) (a i a j )(b k a l) + (a i b k)(a j a l) (a i a l)(a j b k) + (a i a j )(b l a k) (a i b l )(a j a k) + (a i a k)(a j b l ) + (a i a j )(b l b k) (a i b l )(a j b k) + (a i b k)(a j b l ) + (a i b j)(a l a k ) (a i a l)(b j a k ) + (a i a k)(b j a l ) + (a i b k)(b j a l ) (a i b j )(b ka l ) + (a i a l )(b kb j ) (a i b l )(b j a k ) + (a i b j)(b l a k) (a i a k)(b l b j) + (a i b l )(b kb j ) (a i b k)(b l b j ) + (a i b j )(b l b k) (a j b i)(a l a k ) + (a j a l)(b i a k ) (a j a k)(b i a l ) (a j b k)(b i a l ) + (a j b i )(b ka l ) (a j a l )(b kb i ) + (a j b l )(b i a k ) (a j b i)(b l a k) + (a j a k)(b l b i) (a j b l )(b kb i ) + (a j b k)(b l b i ) (a j b i )(b l b k) + (b i b j )(a l a k ) (b i a l )(b j a k ) + (b i a k )(b j a l ) (b kb i )(b j a l ) + (b kb j )(b i a l ) (b ka l )(b i b j ) + (b l b i)(b j a k ) (b l b j)(b i a k ) + (b l a k)(b i b j ) + (b l b k)(b i b j ) (b l b i )(b kb j ) + (b l b j )(b kb i ) After factoring some of the terms in (16), we obtain where E 0;0; () = T () + (W P() + W D () W ()); T () = (D 0;0 ) i;j ( i;j = tr jd 0;0 j( ++ ) i j ) = tr D 0;0 is the kinetic energy in the state. (The terms of the form (a i b j ) and (b i a j ) disappeared in the sum because (D 0;0 ) i;j = (e i ; D 0;0 e j ) = 0, if i and j have opposite sign.) The pairing part of the potential is W P () = i;j;k;l W i;j;k;l (a i a j ) + (b i b j ) (a j b i) + (a i b j) ((a k a l ) + (b kb l ) (a l b k) + (a k b l)) = W i;j;k;l ( ) i;j ( ) k;l i;j;k;l

7 STABILITY 7 We note that none of the matrix elements of involved in this expression is charge-conserving. Furthermore W D () = i;j;k;l W i;j;k;l ((a i a k) + (a i b k) + (b i a k ) (b k b i)) ((a l a j) + (a l b j ) + (b la j ) (b j b l)) = W i;j;k;l ( ) k;i ( ) j;l i;j;k;l is the direct part, i.e, the classical electrostatic energy. It exclusively contains matrix elements of charge conserving operators. Finally, the exchange energy W () = i;j;k;l W i;j;l;k ((a i a k) + (a i b k) + (b i a k ) (b k b i)) (a l a j ) + (a l b j ) + (b la j ) (b j b l) = W i;j;l;k ( ) k;i ( ) j;l i;j;k;l also contains only matrix elements of charge conserving operators. (The dierence between the direct and the exchange part of the energy is the permutation of the indices k and l in the matrix elements of W.) The positivity of the operator W immediately implies that W P () is positive, for all states. Thus, the minimization of the energy E 0;V; () over the whole space D HF is equivalent to the minimization over the subspace D HFC of D HF, for which all non-charge conserving terms vanish, i.e., = 0. (We note that the resulting operator 0 is again a one-particle density matrix If D HF then 0, resulting 1 0 from by conjugation with the unitary matrix, i.e., substituting by its 0 1 negative, is a one-particle density matrix. Since D HF is convex, ( + 0 )= = 0, is again a one-particle density matrix. Thus (18) inf E 0;0; () = inf T () + D HF D HFC (W D() W ()) Note also that W D () is positive for all. 3.. Bound on the Interaction Potential. Now we prove that W () can be controlled by the kinetic energy T (), if is smaller than 4=. Assume k to be a measurable complex function on R 3 R 3 which is tempered in the second variable and whose Fourier transform in the second variable is also measurable. Then Kato's inequality (Kato [11], V, x5, Formula (5.33), see also Herbst [10]), ju(x)j R dx (u; jrju) 3 jxj gives, for any xed x, (19) k(x; ); 1 j xj k(x; ) (k(x; ); jrjk(x; )) Since has nite kinetic energy { and thus nite particle number { we have ++ S 1 (H + ) and S 1 (H ). Moreover, from inequality (1), one deduces that + and +, and thus, are Hilbert-Schmidt operators. Recall also that is self-adjoint. In abuse of notation we denote by (x; y) the integral kernel of.

8 8 V. BACH, J.M. BARBAROU, B. HELFFER, AND H. SIEDENTOP From (19) we obtain 4 j(x; y)j (0) dxdy jx yj tr[(jrj 1) ] = tr(jd0;0 j ) s;t=1 Now, on one hand, we have the relation tr(jd 0;0 j ) = tr jd 0;0 j( ) = tr jd 0;0 j ( ) + ( ) (1) tr jd 0;0 j( ++ ) = T () ; where in the last inequality we have used the relations (1) and (13). On the other hand we have j(x; y)j () W () = dx dy jx yj Thus, () together with (0) and (1) gives (3) W () T () This inequality and (18) prove Theorem 1. G 4. The Existence of the Hartree-Fock Vacuum with External Potentials We are interested in describing also systems with an external potential. To be denite, we will restrict to the most interesting case, namely V (x) = jxj, i.e., a nucleus that interacts via Coulomb forces. As the space H + dening the Fock space we take P 0;V + (H). The energy E 0;V; in a state then becomes E 0;V; () = ( (D 0;V (4) ) i;j i j + W i;j;k;l i j l k ) i;j where (D 0;V ) i;j are the matrix elements of D 0;V = D 0;0 + V and W i;j;k;l = (e i e j ; W e j e k ) is given as in Section. We begin with the Lemma 1. For c [0; 1=) we have jd 0;0 j D 0;0 c=jxj =(1 c) Proof. We abbreviate k = 1=(1c). Since the square root is an operator monotone function it suces to prove (5) which is equivalent to the requirement (6) (D 0;0 ) k (D 0;0 c=jxj) ; kd 0;0 k kk(d 0;0 c=jxj) k ; for all in S. If we estimate the right hand side from below by the triangle inequality, we get the stronger inequality (7) Rearranging turns (7) into (8) kd 0;0 k kkd 0;0 k kck =jxjk kc=(k 1)k =jxjk k( + m ) 1= k ; which is true for kc=(k 1) 1= because of Hardy's inequality 1 u(x) 4 R 3 jxj dx jru(x)j dx R 3 Solving for k gives the desired inequality. G

9 STABILITY 9 The main result of this section is Theorem. If the one-particle electron subspace is picked to be H + = P 0;V + (H), then the energy E 0;V; is nonnegative on D HF, if ; 0 fulll 4(1)=. Proof. We can imitate the proof of Theorem 1 up to (). We merely need to use Lemma 1 in addition. We have (9) jxj 4 jd0;0 j 4(1 ) D0;0 jxj ; under the condition that 1. We need to estimate the right hand side by D 0;0 ()=jxj, which is immediate under the stated hypothesis on. 5. The Inclusion of Magnetic Fields As charged particles electrons and positrons are also coupled to the magnetic eld B = r A, where A is a vector potential. Taking this into account, the energy of electrons and positrons interacting via a quantized electric eld - which has been integrated out in the Coulomb gauge { and a classical magnetic eld is given by (30) E A;V; () = i;j (D A;V ) i;j i + W i;j;k;l i j l k 1 A 1 + B ; 8 R 3 where (D A;V ) i;j are the matrix elements of D A;V = (ir+ea)+m+v. The potentials V and W are the same as in Section 4. We pick the electron subspace H + = P A;V + (H), i.e., explicitly depending on the vector potential which we require to have nite energy, i.e., R R 3 B < 1. Theorem 3. For [0; 1], [0; 4=] and [0; 1), and (31) (1 ) 4( 1 1)() 16 > 0 ; L 1 ;3 (1 ) 4( 1 1)() 3 (1 ) ; implies that E A;V; is nonnegative on D HF, where L 1 ;3 is the constant in the Lieb- Thirring inequality for moments of order 1=. Before proving the theorem we remark that the saturating the above inequality is uniquely determined through a third order equation. Since the resulting expression is somewhat lengthy, we will skip it here. Instead we show a plot of the curve (see Figure 1), dened by equality in (31), which estimates from below the critical curve separating stability and instability. Proof. Analogously to the proofs of Theorems 1 and and, in addition, using the diamagnetic inequality (Simon [17] and [1] [note that the last reference contains a typo in Formula (5.4) in the exponent t should be subtracted]), we estimate (3) dx dyj(x; y)j =jx yj 4 tr( j ir + p Aj) Note that e = p. To show positivity of (30) it hence suces to show h tr jd A;V j 4 j ir + p i (33) Aj + 1 B 0 ; 8 Insert Figure 1 here

10 10 V. BACH, J.M. BARBAROU, B. HELFFER, AND H. SIEDENTOP = Figure 1. The shown curve estimates from below the critical value of the pair (; ), for which the energy E A;V; is positive. It is obtained from Theorem 3 and Formula (31). On and below the line, non-negativity of the energy in Hartree-Fock states is guaranteed. Note that we assert stability from to = 4=, if = 0. However, we do not assert stability if > 4= even if = 0. For the physical value 1= we obtain , i.e., since 0 1. At this point we are in a position to follow the argument in [13] By the Birman-Koplienko-Solomyak inequality [3] the rst summand in (33) is estimated from below by (34) tr ( (D A;V ) 4 (ir + p 1 A) Separating the mixed terms in the rst summand and observing that m = 0 gives a lower bound, we obtain the following lower bound for the rst summand in (34), (35) tr ( (1 ) (ir + p 1 A) ( 1)() jxj ) 16 (ir + p 1 A) The rst summand is essentially the Pauli operator. Thus, using Hardy's inequality, (35) is estimated from below by (36) tr ( (1 ) 4( 1 1)() 16 (ir + p A) (1 ") p jbj Assuming now that the parameters ", and are such that (1 ) 4( 1 1)() 16 > 0 ; we apply the Lieb-Thirring inequality for the moment 1= and see that (36) is estimated from below by L 1 ;3 (1 ) 4( 1 1)() 3 (37) (1 ) B ; 64 where L 1 ; Combining (33) and (37) shows that it suces to have L 1 ;3 (1 ) 4( 1 1)() 3 (38) (1 ) ) )

11 STABILITY 11 Optimizing with respect to yields the desired result. 6. Optimality of the Quantization The second quantization is not unique, since { as is obvious from the denition of the Fock space (1) and the energy (8) { it depends on what is considered the oneparticle electron space H +. Since we are working with functional that turns out to be unbounded from below, we would like to make use of the freedom of choice with respect to H + and investigate the maximal stability range (see also [16]). Thus, we consider (39) E(H + ) = inf DHF E A;V;() where H + is any closed subspace of H such that, if D(D A;V ) is the domain of D A;V, we have D(D A;V ) \ H + is dense in H + and D(D A;V ) \ (H + ) is dense in H = (H + ). We denote by S the set of all such subspaces. The optimal H + in S can indeed be found and is unique Theorem 4. Consider D A;V with values of and as in Theorem 3 and such that 0 6 (D A;V ). For any H + in S n fp+ A;V Hg we get (40) E(H + ) < E(P A;V + H) = 0 Before we turn to the proof, we mention that the use of the Coulomb potential V in the energy functional is not essential our proof easily extends to other external potentials where zero is not in the spectrum of the one-particle operator and the energy is bounded from below. Proof. First note that as a direct consequence of Theorem 3, E(P+ A;V (H)) = 0. We now prove that if the orthogonal projection P H+ onto H + does not commute with P A;V + and P A;V, then the corresponding E(H + ) is strictly negative. Assume to the contrary that there exist two normalized vectors p H + \ D(D A;V ) and n H \D(D A;V ) such that the real part of (p; D A;V n) is non-zero. Without loss of generality we may pick the orthonormal system ; e 1 ; e 0 ; of H such that e 1 = n and e 0 = p. We consider the vector ' H dened as ' = (1 ) 1= + a 0 b 1, where (0; 1), and is the vacuum. The corresponding one particle density matrix is 0 = ; 0 with = 0;i 0;j (1 ) 1 j;1 i;0 (1 ) 1 i;1 j;0 ; j;0 i;0 where k;l is the Kronecker symbol. Computing the Hartree-Fock energy E HF A;V; yields (41) EA;V;() HF = (D 0;1 + D 1;0 ) + O( ) ; i.e., we have a non-vanishing term linear in plus a term of higher order in, which can be always made negative by appropriately choosing the sign of and picking suciently close to zero. Thus, an element H + in S can give nonnegative E(H + ), only if for all p H + \ D(D A;V ) and n H \ D(D A;V ) we have (p; D A;V n) = 0. Under our assumptions on S, this implies that the projection P H+ commutes with D A;V and therefore commutes with all spectral projections of D A;V (see e.g. Birman and Solomyak [4], x6.3, Theorem ), and in particular with P A;V + and P A;V. Now we consider H + in S such that the projection P H+ commutes with P A;V + and P A;V. Suppose that there exists u H + such that u 6 P+ A;V (H). Then, under our

12 1 V. BACH, J.M. BARBAROU, B. HELFFER, AND H. SIEDENTOP assumptions on H +, there exists a non-zero vector ~u in D(D A;V ) \ H + \ P A;V (H). Without loss of generality, we can take an orthonormal basis ; e 1 ; e 0 ; of H such that e 0 = ~u. We now pick the following one-particle density matrix, = 0 1 i;0 j; C 0 0 i;0 j;0 0A The corresponding Hartree-Fock state is the pure state associated to ' = a 0. The Hartree-Fock energy is easily computed and we get EA;V; HF = D 0;0 < 0. Similarly, if H A;V + \ (P (H)) c 6= ;, then the corresponding Hartree-Fock energy E(H + ) is also strictly negative. Thus, a Hilbert space H + in S is a maximizer only if H + = P A;V + (H). References [1] V. Bach, E. H. Lieb, and J. P. Solovej. Generalized Hartree-Fock theory and the Hubbard model. J. Stat. Phys., 76(1&)3{89, [] R.M. Barnett, C.D. Carone, D.E. Groom, T.G. Trippe,, C.G. Wohl, B. Armstrong, P.S. Gee,, G.S. Wagman, F. James, M. Mangano, K. Monig,, L. Montanet, J.L. Feng, H. Murayama, J.J. Hernandez, A. Manohar, M. Aguilar-Benitez, C. Caso, R.L. Crawford, M. Roos, N.A. Tornqvist, K.G. Hayes, K. Hagiwara, K. Nakamura, M. Tanabashi, K. Olive, K. Honscheid, P.R. Burchat, R.E. Shrock, S. Eidelman, R.H. Schindler, A. Gurtu, K. Hikasa, G. Conforto, R.L. Workman, C. Grab, and C. Amsler. Review of particle physics. Phys. Rev. D, 54(1)1{ 70, July [3] M. Sh. Birman, L. S. Koplienko, and M.. Solomyak. Estimates for the spectrum of the dierence between fractional powers of two self-adjoint operators. Soviet Mathematics, 19(3)1{6, Translation of Izvestija vyssich. [4] M. Sh. Birman and M.. Solomjak. Spectral theory of selfadjoint operators in Hilbert space. Mathematics and its Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht, Translated from the 1980 Russian original by S. Khrushchev and V. Peller. [5] M. Born and P. Jordan. ur Quantenmechanik. eitschrift fur Physik, 34858{888, 195. [6] P. Chaix and D. Iracane. From quantum electrodynamics to mean-eld theory I. the Bogoliubov-Dirac-Fock formalism. J. Phys. B., (3)3791{3814, December [7] P. Chaix, D. Iracane, and P. L. Lions. From quantum electrodynamics to mean-eld theory II. Variational stability of the vacuum of quantum electrodynamics in the mean-eld approximation. J. Phys. B., (3)3815{388, December [8] Joseph G. Conlon. The ground state energy of a classical gas. Commun. Math. Phys, 94(4)439{458, August [9] C. L. Feerman and R. de la Llave. Relativistic stability of matter { I. Revista Matematica Iberoamericana, (1, )119{161, [10] Ira W. Herbst. Spectral theory of the operator (p + m ) 1= e =r. Commun. Math. Phys., 5385{94, [11] Tosio Kato. Perturbation Theory for Linear Operators, volume 13 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, 1 edition, [1] Elliott H. Lieb, Michael Loss, and Heinz Siedentop. Stability of relativistic matter via Thomas-Fermi theory. Helv. Phys. Acta, 69(5/6)974{984, December [13] Elliott H. Lieb, Heinz Siedentop, and Jan Philip Solovej. Stability and instability of relativistic electrons in classical electromagnetic elds. J. Statist. Phys., 89(1-)37{59, Dedicated to Bernard Jancovici. [14] Elliott H. Lieb, Heinz Siedentop, and Jan Philip Solovej. Stability of relativistic matter with magnetic elds. Phys. Rev. Lett., 79(10)1785{1788, September [15] Elliott H. Lieb and Horng-Tzer Yau. The stability and instability of relativistic matter. Commun. Math. Phys., {13, [16] Marvin H. Mittleman. Theory of relativistic eects on atoms Conguration-space Hamiltonian. Phys. Rev. A, 4(3)1167{1175, September [17] Barry Simon. Kato's inequality and the comparison of semigroups. J. Funct. Anal., 3(1)97{ 101, [18] Bernd Thaller. The Dirac Equation. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1 edition, 199.

13 STABILITY 13 Fachbereich Mathematik, Technische Universitat Berlin, D-1063 Berlin, Germany address Lehrstuhl I fur Mathematik, Universitat Regensburg, D Regensburg, Germany address jean-marie.barbaroux@mathematik.uni-regensburg.de Departement de mathematiques, B^atiment 45, F Orsay Cedex, France address bernard.helffer@math.u-psud.fr Lehrstuhl I fur Mathematik, Universitat Regensburg, D Regensburg, Germany address heinz.siedentop@mathematik.uni-regensburg.de