Stability of Matter for the Hartree-Fock Functional

of the Relativistic Electron-Positron Field^{1}

Volker Bach, Jean-Marie Barbaroux, Bernard Helffer, Heinz Siedentop

Received: September 11, 1998

Communicated by Bernold Fiedler

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

Keywords and Phrases: Dirac operator, stability of matter, QED, generalized Hartree-Fock states

1 Introduction

Electrons and positrons can be described just interacting with themselves and the electromagnetic field. 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 field and minimize the energy over all possible pairwise distinct nuclear positions. This is known as the Born-Oppenheimer approximation.

Stability in the context of field theory means, that the energy is bounded from below by a multiple of the number operator of the electron-positron field 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 sufficiently small nuclear charge is nonnegative (with or without self-generated magnetic field) provided the Sommerfeld fine structure constant α = e^{2} is also small where e is the elementary charge unit. These results included the physical value α ≈ 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 α.

1 Financial support of the European Union through the TMR network FMRX-CT 96-0001 is gratefully acknowledged.

Documenta Mathematica 3 (1998) 353–364

Our paper is organized as follows: For the readers convenience we fix 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 field. Section 4 extends this to the case when the self-generated magnetic field of the particle is taken into account on a classical level.

2 Definition of the Problem

Before stating our problem precisely, we fix our notations following [2]. (See also Appendix B for additional notations.)

Dirac Operator The operator for a particle of charge −e, in magnetic field ∇×A, and interacting with K nuclei of same charge is

D^{A,V} := α · (\frac{1}{i}∇ + eA) + mβ + e^{2}V,

acting in the four components vector space ℌ = L^{2}(ℝ^{3})⊗ℂ^{4}. The 4×4 matrices α and β are the Dirac matrices in the standard representation [14]. The vector potential A is assumed to be such that the magnetic induction B = ∇ × 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=1}^{K} \frac{Z}{|x − R_{k}|}.  (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 define 𝔇 to be the set of all states ρ with finite kinetic energy, i.e., ∑_{i,j∈ℤ}(D^{0,0})_{i,j} ρ(: Ψ_{i}^{∗}Ψ_{j}: ) converges absolutely where colons denote normal ordering where we fixed 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/|x − y|, i.e.,

W_{i,j;k,l} = (e_{i} ⊗ e_{j}, W e_{k} ⊗ e_{l}) = ∫_{G} dx ∫_{G} dy \frac{\overline{e_{i}(x)e_{j}(y)}e_{k}(x)e_{l}(y)}{|x − y|}

where dx denotes the product measure (Lebesgue measure in the first factor and counting measure in the second factor) of G := ℝ^{3} × {1, 2, 3, 4}. The energy of a state ρ ∈ 𝔇 is thus

E_{A,V,α}(ρ) = ∑_{i,j∈ℤ}(D^{A,V})_{i,j} ρ(: Ψ_{i}^{∗}Ψ_{j}: ) + αU

+\frac{α}{2} ∑_{i,j,k,l∈ℤ} W_{i,j;k,l} ρ(: Ψ_{i}^{∗}Ψ_{j}^{∗}Ψ_{l}Ψ_{k}: ) + \frac{1}{8π} ∫_{ℝ^{3}} B^{2}, (2)

with U := ∑_{1≤κ<k≤K} Z^{2}/|R_{κ} − R_{k}| describing the energy of the nuclei.

Energy of Generalized Hartree-Fock States Following the proof of Theorem 1 in [2], we can show that for all generalized Hartree-Fock states ρ ∈ 𝔇_{HF} (see Appendix B), the energy (2) can be rewritten as a functional of Γ_{ρ}, the 1-pdm of ρ:

E_{A,V,α}(ρ) = E_{A,V,α}^{HF}(Γ_{ρ}) := tr(D^{A,V} γ) + αU + \frac{α}{2} ∫ dxdy \frac{|υ(x, y)|^{2}}{|x − y|} + αD(ρ_{γ}, ρ_{γ})

− \frac{α}{2} ∫ dxdy \frac{|γ(x, y)|^{2}}{|x − y|} + \frac{1}{8π} ∫_{ℝ^{3}} B^{2},

(3)

where D(f, g) := (1/2) ∫_{ℝ^{6}} dxdy\overline{f(x)}g(y)|x − y|^{−1} is the Coulomb scalar product, υ(x, y) := ∑_{i,j∈ℤ}(e_{i}, υe_{j})e_{i}(x)e_{j}(y), γ(x, y) := ∑_{i,j∈ℤ}(e_{i}, γe_{j})e_{i}(x)\overline{e_{j}(y)} (note the difference to υ), and ρ_{γ}(x) := ∑_{σ=1}^{4} γ(x, x). (We use the notation x := (x, σ) ∈ ℝ^{3} × {1, …, 4}.) We also remind the reader that α = e^{2}.

The main goal of this paper is to show positivity of E_{A,V,α}(ρ) for quasi-free states.

More notations can be found in Appendix B.

3 Stability of Relativistic Matter without Magnetic Field

We prove here, in the case A = 0, that the energy functional E_{A,V,α} defined in (2) is positive on generalized Hartree-Fock states for suitable choice of the electron subspace and α and Z small enough. More precisely, ℌ_{+} := [χ_{[0,∞)}(D^{0,V_{eff}} )](ℌ) is the positive spectral subspace associated to D^{0,0} + αV_{eff}, where

V_{eff} := −Z ∑_{k=1}^{K} \frac{χ_{Υ_{k}}(x)}{|x − R_{k}|}.  (4)

Here Υ_{κ} := {x ∈ ℝ^{3} : |x − R_{κ}| ≤ |x − R_{k}|, ∀k = 1, …, K} denotes the κ-th Voronoi cell and χ_{M} is the characteristic function of the set M. Our first result is

Theorem 1. Pick ℌ_{+} := [χ_{[0,∞)}(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 ϵ ∈ (0, 1), α ∈ [0, 4/π] and Z ∈ [0, ∞) are such that

1 − ϵ − π^{2}α^{2}/16 − 4(1/ϵ − 1)α^{2}Z^{2} > 0,

2See Appendix A.

Documenta Mathematica 3 (1998) 353–364

0 0.2 0.4 _{0  0.4  0.8  1.2}  α 4/π α_{0} αZ 0.5

Figure 1: The plain curve gives an estimate from below of the critical value of the pair (α, αZ), for which the energy E_{0,V,α} is positive. For the physical value α ≈ 1/137.0359895 we obtain αZ ≈ 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

and

\frac{26296πL_{1/2,3}(1/ϵ − 1)^{2}}{105(1 − ϵ − π^{2}α^{2}/16 − 4(1/ϵ − 1)α^{2}Z^{2})^{3/2}} α^{3}Z^{2} ≤ 1,

then E_{0,V,α} is nonnegative on 𝔇_{HF}.

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 ϵ is a free parameter that we can use to optimize the value of α and Z. Instead of giving a cumbersome analytic formula, Figure 1 gives the result when picking ϵ suitably.

The proof of the theorem consists of five 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_{eff}.

• Dominate the exchange energy W_{X} by the kinetic energy.

• Control the difference of the kinetic energy and the energy of the modified Dirac operator D^{0,V_{eff}} by applying the Birman-Koplienko-Solomyak inequality [3] to obtain a Schrödinger like operator.

• 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 dν(x) := ρ(x)dx

E_{0,V,α} ≥ tr(D^{0,V} γ) + αU + αD(ρ_{γ}, ρ_{γ}) − \frac{α}{2} ∫ dxdy \frac{|γ(x, y)|^{2}}{|x − y|}

≥ tr(D^{0,V_{eff}} γ) + \frac{αZ^{2}}{8} ∑_{k=1}^{K} d_{k}^{−1} − \frac{α}{2} ∫ dxdy \frac{|γ(x, y)|^{2}}{|x − y|} .  (5)

Using Kato’s inequality (see Appendix A) and then Inequalities (22) and (23) we get

\frac{2}{π} ∑_{s,t=1}^{4} ∫ \frac{|γ(x, y)|^{2}}{|x − y|} dxdy ≤ tr[(|∇| ⊗ 1)γ^{2}] ≤ tr(|D^{0,0}|γ^{2})

≤ tr(|D^{0,0}|(γ_{++} − γ_{−−})). (6)

So far we have not used the choice of the subspaces ℌ_{+} and ℌ_{−} specified 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_{+} := [χ_{[0,∞)}(D^{0,V_{eff}} )] (ℌ). This implies tr(D^{0,V_{eff}} γ) = tr(|D^{0,V_{eff}} |(γ_{++} − γ_{−−})), and thus

E_{0,V,α} ≥ tr [(|D^{0,V_{eff}} | − \frac{πα}{4}|D^{0,0}|) (γ_{++} − γ_{−−})] + \frac{αZ^{2}}{8} ∑_{k=1}^{K} d_{k}^{−1}.  (7)

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 ≤ γ_{++} − γ_{−−} ≤ 1, we obtain

tr [(|D^{0,V_{eff}} | − απ|D^{0,0}/4|) (γ_{++} − γ_{−−})] ≥ − tr (|D^{0,V_{eff}} | − απ|D^{0,0}|/4)_{−}

≥ − tr {[(D^{0,V_{eff}} )^{2} − π^{2}α^{2}(D^{0,0})^{2}/16]_{−}^{1/2}}

≥ − tr {[(1 − ϵ − π^{2}α^{2}/16)(D^{0,0})^{2} − (1/ϵ − 1)α^{2}V_{eff}^{2}]_{−}^{1/2}} (8)

where the subscript minus denotes the negative part (|A| − 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} := B_{d_{k}}(R_{k}), we have

∫_{ℝ^{3}} |∇f(x)|^{2}dx ≥ ∑_{k=1}^{K} (∫_{B_{k}} (\frac{1}{4|x − R_{k}|^{2}} − \frac{1}{d_{k}^{2}} (1 + \frac{|x − R_{k}|^{2}}{4d_{k}^{2}} )) |f(x)|^{2}dx) . (9)

Inequality (9) together with (8) gives

E_{0,V,α} ≥ − tr {[ (1 − ϵ − α^{2}π^{2}/16 − 4(1/ϵ − 1)α^{2}Z^{2}) (D^{0,0})^{2}

− (1/ϵ − 1)α^{2}Z^{2} ∑_{k=1}^{K} (\frac{χ_{Υ_{k}\B_{k}}(x)}{|x − R_{k}|^{2}} − \frac{4}{d_{k}^{2}} (1 + \frac{|x − R_{k}|^{2}}{4d_{k}^{2}} )χ_{B_{k}(x)})]−^{\frac{1}{2}}\} + \frac{αZ^{2}}{8} ∑_{k=1}^{K} d_{k}^{−1}. (10)

Using the Lieb-Thirring inequality (see Appendix A) for the exponent 1/2 in (10) implies

E_{0,V,α} ≥ \frac{−L_{1/2,3}(\frac{1}{ϵ} − 1)^{2}α^{4}Z^{4}}{(1 − ϵ − π^{2}α^{2}/16 − 4(1/ϵ − 1)α^{2}Z^{2})^{3/2}} \{ ∑_{k=1}^{K} ∫_{Υ_{k}\B_{k}} \frac{1}{|x − R_{k}|^{4}} dx

+16 ∑_{k=1}^{K} ∫_{B_{k}} \frac{1}{d_{k}^{4}} (1 + \frac{|x − R_{k}|^{2}}{4d_{k}^{2}} )^{2} dx\} + \frac{αZ^{2}}{8} ∑_{k=1}^{K} d_{k}^{−1}  (11)

≥ (\frac{αZ^{2}}{8} − \frac{π(3 + 64(1/3 + 1/10 + 1/112)L_{1/2,3}(1/ϵ − 1)^{2}α^{4}Z^{4}}{(1 − ϵ − \frac{π^{2}α^{2}}{16} − 4(1/ϵ − 1)α^{2}Z^{2})^{\frac{3}{2}}} ) ∑_{k=1}^{K} d_{k}^{−1}.

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 first term in the parenthesis with Inequality (4.6) in [8].

4 Inclusion of the Magnetic Field

We now consider the whole energy functional E_{A,V,α} given in (3), i.e., we include also magnetic fields B := ∇ × A of finite field energy.

Theorem 2. Take ℌ_{+} := [χ_{[0,∞)}(D^{A,V_{eff}} )](ℌ). If ϵ ∈ (0, 1), ϵ′ ∈ (0, ∞), α ∈ [0, 4/π] and Z ∈ [0, ∞) verify

1 − ϵ − π^{2}α^{2}/16 − 4(1/ϵ − 1)α^{2}Z^{2} > 0,  (12)

\frac{26296πL_{1/2,3}(1/ϵ − 1)^{2}(1 + ϵ′)}{105(1 − ϵ − π^{2}α^{2}/16 − 4(1/ϵ − 1)α^{2}Z^{2})^{3/2}} α^{3}Z^{2} ≤ 1,  (13)

and

\frac{8πL_{1/2,3}(1 − ϵ)^{2}(1 + 1/ϵ′)}{(1 − ϵ − π^{2}α^{2}/16 − 4(1/ϵ − 1)α^{2}Z^{2})^{3/2}} α ≤ 1  (14)

then E_{A,V,α} is nonnegative on 𝔇_{HF}.

Again, note that ϵ and ϵ′ 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.

Proof. By the (relativistic) diamagnetic inequality (see, e.g., the appendix of [8], see also Appendix A)

\frac{α}{2} ∫ dx ∫ dy|γ(x, y)|^{2}/|x − y| ≤ \frac{πα}{4} tr(γ^{∗}| − i∇ + \sqrt{α}A|γ).  (15)

0 0.2 0.4 _{0  0.4  0.8  1.2}  α 4/π α_{0} αZ 0.5

Figure 2: The plain curve gives an estimate from below of the critical value of the pair (α, αZ), for which the energy E_{A,V,α} is positive. For the physical value α ≈ 1/137.0359895 we obtain αZ ≈ 0.48899985, i.e., Z ≈ 67.0105779. The dashed curve shows the critical curve obtained in [2] in the case of a single nucleus. The numerical value where both curves cut the abscissa is α_{0} ≈ 0.5235.

Now, following the proof of Theorem 1 using (5) to (8) and (15), we obtain for ℌ_{+} := [χ_{[0,∞)}(D^{A,V_{eff}} )](ℌ) and for any ϵ ∈ (0, 1)

T := tr((D^{A,0} + αV_{eff})γ) − \frac{α}{2} ∫ \frac{|γ(x, y)|^{2}}{|x − y|} dxdy

≥ − tr \{[(1 − ϵ)(D^{A,0})^{2} − (\frac{1}{ϵ} − 1)α^{2}V_{eff}^{2} − \frac{π^{2}α^{2}}{16} | − i∇ + \sqrt{α}A|^{2}]−^{\frac{1}{2}}\}

≥ − tr \{[(1 − ϵ − \frac{π^{2}α^{2}}{16} )| − i∇ + \sqrt{α}A|^{2} − (\frac{1}{ϵ} − 1)α^{2}V_{eff}^{2} − (1 − ϵ)\sqrt{α}|B|]−^{\frac{1}{2}}\} .

Combining first (9) with the nonrelativistic diamagnetic inequality for Schrödinger operators (Simon [13], see also Appendix A) gives

∫_{ℝ^{3}} |(−i∇ + \sqrt{α}A)f(x)|^{2}dx

≥ ∑_{k=1}^{K} (∫_{B_{k}} (\frac{1}{4|x − R_{k}|^{2}} − \frac{1}{d_{k}^{2}} (1 + \frac{|x − R_{k}|^{2}}{4d_{k}^{2}} )) |f(x)|^{2}dx) . (16)

Using this inequality we are able to control the |x − R_{k}|^{−2} singularities for V_{eff}^{2} in balls of radius d_{k} around R_{k} by a piece of (−i∇ + \sqrt{α}A)^{2}. This gives

T ≥ − tr \{[ (1 − ϵ − \frac{π^{2}α^{2}}{16} − 4(\frac{1}{ϵ} − 1)α^{2}Z^{2}) | − i∇ + \sqrt{α}A|^{2} − (1 − ϵ)\sqrt{α}|B|

−(\frac{1}{ϵ} − 1)α^{2}Z^{2} ∑_{k=1}^{K} (\frac{χ_{Υ_{k}\B_{k}}(x)}{|x − R_{k}|^{2}} − \frac{4}{d_{k}^{2}} (1 + \frac{|x − R_{k}|^{2}}{4d_{k}^{2}} )χ_{B_{k}}(x))]−^{\frac{1}{2}}\}

The Lieb-Thirring inequality for the moment 1/2 implies

T ≥ \frac{−L_{1/2,3}}{(1 − ϵ − \frac{π^{2}α^{2}}{16} − 4(\frac{1}{ϵ} − 1)α^{2}Z^{2})^{3/2}} ∫_{ℝ^{3}} \{(\frac{1}{ϵ} − 1)α^{2}Z^{2}[ ∑_{k=1}^{K} \frac{χ_{Υ_{k}\B_{k}}(x)}{|x − R_{k}|^{2}}

+4 ∑_{k=1}^{K} \frac{1}{d_{k}^{2}} (1 + \frac{|x − R_{k}|^{2}}{4d_{k}^{2}} ) χ_{B_{k}(x)}] + \sqrt{α}(1 − ϵ)|B|\}^{2}dx

≥ \frac{−L_{1/2,3}}{(1 − ϵ − \frac{π^{2}α^{2}}{16} − 4(\frac{1}{ϵ} − 1)α^{2}Z^{2})^{3/2}} \{(1 + ϵ′)(\frac{1}{ϵ} − 1)^{2}α^{4}Z^{4}

×[ ∑_{k=1}^{K} ∫_{ℝ^{3}} \frac{χ_{Υ_{k}\B_{k}}(x)}{|x − R_{k}|^{4}}dx + 16 ∑_{k=1}^{K} \frac{1}{d_{k}^{4}} ∫_{ℝ^{3}} (1 + \frac{|x − R_{k}|^{2}}{4d_{k}^{2}} )^{2} χ_{B_{k}(x)}dx]

+(1 + \frac{1}{ϵ′})(1 − ϵ)^{2}α ∫_{ℝ^{3}} B^{2}dx\}.

Collecting all terms and using the previous inequality gives with δ := 3 + 64(1/3 + 1/10 + 1/112) – for any ϵ′ ∈ (0, ∞) and under assumptions (12)-(14) –

E_{A,V,α} ≥ tr [(D^{A,0} + αV_{eff})γ] − \frac{α}{2} ∫ \frac{|γ(x, y)|^{2}}{|x − y|} dxdy + \frac{αZ^{2}}{8} ∑_{k=1}^{K} d_{k}^{−1} + ∫_{ℝ^{3}} B^{2}

≥ \frac{αZ^{2}}{8} [1 − \frac{8πδL_{1/2,3}(\frac{1}{ϵ} − 1)^{2}(1 + ϵ′)α^{3}Z^{2}}{(1 − ϵ − \frac{π^{2}α^{2}}{16} − 4(\frac{1}{ϵ} − 1)α^{2}Z^{2})^{3/2}} ] ∑_{k=1}^{K} d_{k}^{−1}

+\frac{1}{8π} [1 − \frac{8πL_{1/2,3}(1 − ϵ)^{2}(1 + \frac{1}{ϵ′})α}{(1 − ϵ − \frac{π^{2}α^{2}}{16} − 4(\frac{1}{ϵ} − 1)α^{2}Z^{2})^{3/2}} ] ∫_{ℝ^{3}} B^{2}.

A Inequalities

BKS Inequality Let 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 ∈ L_{loc}^{2}(ℝ^{3}, ℝ^{3}), then, for all u with |u| ∈ H^{1}(ℝ^{3})

∫_{ℝ^{3}}(∇|u|)^{2} ≤ ∫_{ℝ^{3}} |(−i∇ − A)u|^{2}

(Simon [13]) and for all u ∈ D(|p|)

(|u|, |p| |u|) ≤ (u, |p + A| u)

(see [8, Formula (5.7)]). (Note that we allow for the right side to be infinite.)

Electrostatic Inequality Let ν be any bounded Borel measure on ℝ^{3}, then with the notations of Theorem 1 we have [12, Lemma 1]

\frac{α}{2} ∫_{ℝ^{3}} ∫_{ℝ^{3}} \frac{dν(x)dν(y)}{|x − y|} − α ∫_{ℝ^{3}}(V (x) − V_{eff}(x))dν(x) + αU ≥ \frac{αZ^{2}}{8} ∑_{k=1}^{K} \frac{1}{d_{k}} .

Kato’s Inequality Let H_{0} be the closure of the essentially self-adjoint operator −Δ on C_{0}^{∞}(ℝ^{3}). Then for u ∈ D(H_{0}^{1/2}) and a ∈ ℝ^{3}, ([7, chap. V, §5, Formula (5.33)])

∫_{ℝ^{3}} |x − a|^{−1}| u(x)|^{2}dx ≤ \frac{π}{2} ∫_{ℝ^{3}} |k| |û(k)|^{2}dk ≤ \frac{π}{2} (|H_{0}|u, u).

Localized Hardy Inequality Let R be any point in ℝ^{3} and d any positive real number. If B_{d}(R) denotes the ball in ℝ^{3} with center R and radius d, then, for any f ∈ L^{2}(B_{d}(R)) such that ∇f ∈ L^{2}(B_{d}(R)) we have [12, Formula (5.2)]

∫_{B_{d}(R)} |∇f(x)|^{2}dx ≥ \frac{1}{d^{2}} ∫_{B_{d}(R)} (\frac{d^{2}}{4|x − R|^{2}} − (1 + \frac{|x − R|^{2}}{4d^{2}} )) |f(x)|^{2}dx.

Lieb-Thirring Inequality (d = 3, γ = 1/2) Given a positive constant μ, a real vector field A with square integrable gradients, and a real valued function V in L^{2}(ℝ^{3}), we have for V_{+} := (|V | + V)/2

tr \{[(−iμ∇ − A)^{2} − V ]_{−}^{1/2}\} ≤ \frac{L_{1/2,3}}{μ^{3}} ∫_{ℝ^{3}} V_{+}^{2}

(see Lieb and Thirring [11] for the case A = 0 and Avron, Herbst, and Simon [1] for the general case).

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 ∈ 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 fulfill the canonical anticommutation relations for all f and g in ℌ

{a(f), a(g)} = {a^{∗}(f), a^{∗}(g)} = {b(f), b(g)} = {b^{∗}(f), b^{∗}(g)} = 0,  (17)
{a(f), a^{∗}(g)} = (f, P_{ℌ_{+}}g) , {b^{∗}(f), b(g)} = (f, P_{ℌ_{−}}g)  (18)

where { , } denotes the anticommutator.

For any f ∈ ℌ, the field operator is the antilinear bounded operator

Ψ(f) := a(f) + b^{∗}(f)

acting in 𝔉. Its adjoint is linear and equal to Ψ^{∗}(f) = a^{∗}(f) + b(f). Given an orthonormal basis {…, e_{−2}, e_{−1}, e_{0}, e_{1}, … } 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}), Ψ_{i} := a_{i}+b_{i}^{∗} and Ψ_{i}^{∗} := a_{i}^{∗}+b_{i}.

One-Particle Density Matrix A trace class operator Γ on ℌ × ℌ is called a oneparticle density operator (1-pdm), if

• Γ = Γ^{∗} and −1 ≤ Γ ≤ 1.

•

Γ = (\begin{matrix}γ & υ \\ υ^{∗} & −\overline{γ} \end{matrix})  (19)

with

γ^{∗} = γ and υ^{t} = −υ  (20)

where the superscript t refers to transposition, i.e., given our basis fixed 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

Γ = (\begin{matrix}γ_{++} & γ_{+−} & υ_{++} & υ_{+−} \\ γ_{−+} & γ_{−−} & υ_{−+} & υ_{−−} \\ υ_{++}^{∗} & υ_{−+}^{∗} & −γ̄_{++} & −γ̄_{+−} \\ υ_{+−}^{∗} & υ_{−−}^{∗} & −γ̄_{−+} & −γ̄_{−−} \end{matrix})

with γ_{++} := P_{ℌ_{+}}γP_{ℌ_{+}}, γ_{+−} := P_{ℌ_{+}}γP_{ℌ_{−}}, γ_{−+} := P_{ℌ_{−}}γP_{ℌ_{+}} = γ_{+−}^{∗}, and γ_{−−} := P_{ℌ_{−}}γP_{ℌ_{−}} appropriately restricted. Similarly υ_{++} := P_{ℌ_{+}}υP_{ℌ_{+}}, υ_{+−} := P_{ℌ_{+}}υP_{ℌ_{−}}, υ_{−+} := P_{ℌ_{−}}υP_{ℌ_{+}} = −υ_{+−}^{t}, and υ_{−−} := P_{ℌ_{−}}υP_{ℌ_{−}} also appropriately restricted.

For each state ρ ∈ 𝔇, we define the associated 1-pdm Γ_{ρ} by its matrix elements as

(h, Γ_{ρ}g) = ρ (: [Ψ^{∗}(g_{1}) + Ψ(g̃_{2})][Ψ(h_{1}) + Ψ^{∗}(h̃_{2})]: )  (21)

where h := (h_{1}, h_{2}) ∈ ℌ^{2}, g := (g_{1}, g_{2}) ∈ ℌ^{2} and given f = ∑_{k∈ℤ} λ_{k}e_{k}, we define f̃ = ∑_{k∈ℤ} \overline{λ_{k}}e_{k}. The colons denote normal ordering, i.e., anticommuting all stared operators to the left ignoring the anticommutators. Note that for a fixed basis, Γ_{ρ} is uniquely defined. The matrix elements of Γ_{ρ} are thus γ_{i,j} = ρ(: Ψ_{j}^{∗}Ψ_{i}: ), (γ_{++})_{i,j} = ρ(a_{j}^{∗}a_{i}), (γ_{+−})_{i,j} = ρ(b_{j}a_{i}), (γ_{−−})_{i,j} = −ρ(b_{i}^{∗}b_{j}) and υ_{i,j} = ρ(: Ψ_{j}Ψ_{i}: ), (υ_{++})_{i,j} = ρ(a_{j}a_{i}), (υ_{+−})_{i,j} = ρ(b_{j}^{∗}a_{i}), (υ_{−−})_{i,j} = ρ(b_{j}^{∗}b_{i}^{∗}).

We also recall that

γ_{++}^{2} + γ_{+−}γ_{−+} ≤ γ_{++},  (22)
γ_{−+}γ_{+−} + γ_{−−}^{2} ≤ −γ_{−−}.  (23)

holds [2].

States – Generalized Hartree-Fock States A state is a bounded positive linear form ρ on the space of bounded operators on 𝔉 with ρ(1) = 1. The set of generalized Hartree-Fock states (or quasi-free states with finite particle number) is the set of states ρ that fulfill

i) For all finite sequences of operators d_{1}, d_{2}, ⋯, d_{2K}, where d_{i} stands for a(f), a^{∗}(f), b(f), or b^{∗}(f), we have ρ(d_{1}d_{2} ⋯ d_{2K−1}) = 0 and

ρ(d_{1}d_{2} ⋯ d_{2K}) = ∑_{σ∈S} sgn(σ)ρ(d_{σ(1)}d_{σ(2)}) ⋯ ρ(d_{σ(2K−1)}d_{σ(2K)})

where S is the set of permutations σ such that σ(1) < σ(3) < ⋯ < σ(2K − 1) and σ(2i − 1) < σ(2i) for all 1 ≤ i ≤ K. This implies in particular

ρ(d_{1}d_{2}d_{3}d_{4}) = ρ(d_{1}d_{2})ρ(d_{3}d_{4}) − ρ(d_{1}d_{3})ρ(d_{2}d_{4}) + ρ(d_{1}d_{4})ρ(d_{2}d_{3}). (24)

ii) The state ρ has a finite particle number, i.e., if N := ∑_{i∈ℤ}(a_{i}^{∗}a_{i} + b_{i}^{∗}b_{i}) denotes the particle number operator, we have ρ(N) < ∞, or equivalently, written in terms of the one-particle density matrix, tr(γ_{++} − γ_{−−}) < ∞.

We write 𝔇_{HF} for the set of all generalized Hartree-Fock states ρ with finite kinetic energy, i.e., ∑_{i,j∈ℤ}(D^{0,0})_{i,j}ρ(: Ψ_{i}^{∗}Ψ_{j}: ) is absolutely convergent.

References

[1] J. Avron, I. Herbst, and B. Simon. Schrödinger operators with magnetic fields. I. General interactions. Duke Math. J., 45(4):847–833, December 1978.

[2] Volker Bach, Jean-Marie Barbaroux, Bernard Helffer, and Heinz Siedentop. On the stability of the relativistic electron-positron field. Commun. Math. Phys., 1999.

[3] M. Sh. Birman, L. S. Koplienko, and M. Z. Solomyak. Estimates for the spectrum of the difference between fractional powers of two self-adjoint operators. Soviet Mathematics, 19(3):1–6, 1975. Translation of Izv. Vysš. Učebn. Zaved. Matematika.

[4] P. Chaix, D. Iracane, and P. L. Lions. From quantum electrodynamics to meanfield theory: II. Variational stability of the vacuum of quantum electrodynamics in the mean-field approximation. J. Phys. B., 22(23):3815–3828, December 1989.

[5] Freeman J. Dyson and Andrew Lenard. Stability of matter I. J. Math. Phys., 8:423–434, 1967.

[6] Bernard Helffer and Heinz Siedentop. Form perturbations of the second quantized Dirac field. Mathematical Physics Electronic Journal, Accepted for publication.

[7] Tosio Kato. Perturbation Theory for Linear Operators, volume 132 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, 1 edition, 1966.

[8] 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 1996.

[9] Elliott H. Lieb, Heinz Siedentop, and Jan Philip Solovej. Stability and Instability of Relativistic Electrons in Classical Electromagnetic Fields. J. Stat. Phys., 89(1/2):37–59, 1997.

[10] Elliott H. Lieb and Walter E. Thirring. Bound for the kinetic energy of Fermions which proves the stability of matter. Phys. Rev. Lett., 35(11):687–689, September 1975. Erratum: Phys. Rev. Lett., 35(16):1116, October 1975.

[11] Elliott H. Lieb and Walter E. Thirring. Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In Elliott H. Lieb, Barry Simon, and Arthur S. Wightman, editors, Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann. Princeton University Press, Princeton, 1976.

[12] Elliott H. Lieb and Horng-Tzer Yau. The stability and instability of relativistic matter. Commun. Math. Phys., 118:177–213, 1988.

[13] Barry Simon. Kato’s inequality and the comparison of semigroups. J. Funct. Anal., 32(1):97–101, 1979.

[14] Bernd Thaller. The Dirac Equation. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1 edition, 1992.

Volker Bach

Fachbereich Mathematik

Technische Universität Berlin D-10623 Berlin Germany Jean-Marie Barbaroux Lehrstuhl für Mathematik I Universität Regensburg D-93040 Regensburg Germany

Bernard Helffer

Département de mathématiques Bâtiment 425 Université Paris-Sud F-91405 Orsay Cédex France Heinz Siedentop Lehrstuhl für Mathematik I Universität Regensburg D-93040 Regensburg Germany