A Nonlinear Model for Relativistic Electrons at Positive Temperature
aa r X i v : . [ m a t h - ph ] F e b A Nonlinear Model for Relativistic Electrons atPositive Temperature
Christian HAINZL a , Mathieu LEWIN b & Robert SEIRINGER c a Department of Mathematics, UAB, Birmingham, AL 35294-1170, USA. [email protected] b CNRS & Department of Mathematics (CNRS UMR8088), University ofCergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex,FRANCE.
[email protected] c Department of Physics, Jadwin Hall, Princeton University, P.O. Box 708,Princeton, New Jersey 08544, USA. [email protected]
February 26, 2008
Abstract
We study the relativistic electron-positron field at positive tem-perature in the Hartree-Fock-approximation. We consider both thecase with and without exchange term, and investigate the existenceand properties of minimizers. Our approach is non-perturbative inthe sense that the relevant electron subspace is determined in a self-consistent way. The present work is an extension of previous work byHainzl, Lewin, S´er´e, and Solovej where the case of zero temperaturewas considered.
Contents
Introduction 21 The reduced Bogoliubov-Dirac-Fock free energy 7
Proofs 15
A Appendix: Integral Representation of Relative Entropy 24References 25
Introduction
In Coulomb gauge and when photons are neglected, the Hamiltonian ofQuantum Electrodynamics (QED) reads formally [16, 2, 14, 13] H ϕ = Z Ψ ∗ ( x ) D Ψ( x ) dx − Z ϕ ( x ) ρ ( x ) dx + α Z Z ρ ( x ) ρ ( y ) | x − y | dx dy . (1)Here Ψ( x ) is the second-quantized field operator satisfying the usual anti-commutation relations, and ρ ( x ) is the density operator ρ ( x ) = P σ =1 [Ψ ∗ ( x ) σ , Ψ( x ) σ ]2 = P σ =1 { Ψ ∗ ( x ) σ Ψ( x ) σ − Ψ( x ) σ Ψ ∗ ( x ) σ } , (2)where σ is the spin variable. In (1), D = − i α · ∇ + β is the usual freeDirac operator, α is the bare Sommerfeld fine structure constant and ϕ isthe external potential. The matrices α = ( α , α , α ) and β are the usual4 × ~ = c = m = 1. In QED, one main issue is the minimizationof the Hamiltonian (1). However, even if we implement an UV-cutoff, theHamiltonian is unbounded from below, since the particle number can bearbitrary.In a formal sense this problem was first overcome by Dirac, who sug-gested that the vacuum is filled with infinitely many particles occupyingthe negative energy states of the free Dirac operator D . With this axiomDirac was able to conjecture the existence of holes in the Dirac sea whichhe interpreted as anti-electrons or positrons . His prediction was verified byAnderson in 1932. Dirac also predicted [6, 7] the phenomenon of vacuumpolarization: in the presence of an electric field, the virtual electrons aredisplaced and the vacuum acquires a non-uniform charge density.In Quantum Electrodynamics Dirac’s assumption is sometimes imple-mented via normal ordering which essentially consists of subtracting thekinetic energy of the negative free Dirac sea, in such a way that the kineticenergy of electrons as well as positrons (holes) becomes positive. With thisprocedure the distinction between electrons and positrons is put in by hand.2t was pointed out in [14] (see also the review [13]), however, that normalordering is probably not well suited to the case α = 0 of interacting particles(the interaction is the last term of (1)). Instead a procedure was presentedwhere the distinction between electrons and positrons is not an input butrather a consequence of the theory. The approach of [14] is rigorous and fullynon-perturbative, but so far it was only applied to the mean-field (Hartree-Fock) approximation, with the photon field neglected. It allowed to justifythe use of the Bogoliubov-Dirac-Fock model (BDF) [4], studied previouslyin [10, 11, 12]. The purpose of the present paper is to extend these resultsto the nonzero temperature case.The methodology of [14] is a two steps procedure. First, the free vacuumis constructed by minimizing the Hamiltonian (1) over Hartree-Fock states ina box with an ultraviolet cut-off, and then taking the thermodynamic limitwhen the size of the box goes to infinity. The limit is a Hartree-Fock state P − describing the (Hartree-Fock) free vacuum [14, 13]. It has an infinite energy,since it contains infinitely many virtual particles forming the (self-consistent)Dirac sea. We remark that this state is not the usual sea of negative electronsof the free Dirac operator because all interactions between particles are takeninto account, but it corresponds to filling negative energies of an effectivemean-field translation invariant operator.The second step of [14] consists of constructing an energy functional thatis bounded from below in the presence of an external field, by subtracting the(infinite) energy of the free self-consistent Dirac sea. The key observationis that the difference of the energy of a general state P minus the (infinite)energy of the free vacuum P − can be represented by an effective functional(called Bogoliubov-Dirac-Fock (BDF) [4]) which only depends on Q = P −P − , describing the variations with respect to the free Dirac sea. The BDFenergy was studied in [10, 11, 12]. The existence of ground states was shownfor the vacuum case in [10, 11] and in charge sectors in [12]. For a detailedreview of all these results, we refer to [13]. An associated time-dependentevolution equation, which is in the spirit of Dirac’s original paper [6], wasstudied in [15].Let us now turn to the case of a non zero temperature T = 1 /β >
0. Weconsider a Hartree-Fock state with one-particle density matrix 0 ≤ P ≤ renormalized densitymatrix γ = P − /
2. We remark that the anticommutator in (2) is a kind ofrenormalization which does not depend on any reference as normal orderingdoes (it just corresponds to subtracting the identity divided by 2). Theanticommutator of (2) is due to Heisenberg [16] (see also [18, Eq. (96)]) andit is necessary for a covariant formulation of QED, see [22, Eq. (1 . F QED T ( γ ) = tr( D γ ) − α Z ϕ ( x ) ρ γ ( x ) + α Z Z ρ γ ( x ) ρ γ ( y ) | x − y |− α Z Z tr C | γ ( x, y ) | | x − y | − T S ( γ ) (3)where the entropy is given by the formula S ( γ ) = − tr (cid:0) ( + γ ) ln( + γ ) (cid:1) − tr (cid:0) ( − γ ) ln( − γ ) (cid:1) . (4)The (matrix-valued) function γ ( x, y ) is the formal integral kernel of theoperator γ and ρ γ ( x ) := tr C γ ( x, x ) is the associated charge density. Theabove formulas are purely formal; they only make sense in a finite box withan ultraviolet cut-off, in general.As in [14] the first step is to define the free vacuum at temperature T ,which is the formal minimizer of (3) when ϕ = 0. Following [14], one canfirst confine the system to a box, then study the limit as the size of the boxgoes to infinity and identify the free vacuum as the limit of the sequence ofground states. Alternatively, it was proved in [14] that the free vacuum canalso be obtained as the unique minimizer of the free energy per unit volume.In the nonzero temperature case, this energy reads T T ( γ ) =1(2 π ) Z B (0 , Λ) tr C [ D ( p ) γ ( p )] dp − α (2 π ) Z Z B (0 , Λ) tr C [ γ ( p ) γ ( q )] | p − q | dp dq + T (2 π ) Z B (0 , Λ) tr C (cid:2) (cid:0) + γ ( p ) (cid:1) ln (cid:0) + γ ( p ) (cid:1) + (cid:0) − γ ( p ) (cid:1) ln (cid:0) − γ ( p ) (cid:1)(cid:3) dp and it is defined for translation-invariant states γ = γ ( p ) only, under theconstraint − / ≤ γ ≤ /
2. Here, B (0 , Λ) denotes the ball of radius Λcentered at the origin. The real number Λ > γ ,and prove several interesting properties of it. In particular, we shall see thatit satisfies a nonlinear equation of the form˜ γ = 12 (cid:18)
11 + e βD ˜ γ −
11 + e − βD ˜ γ (cid:19) (5)i.e. it is the Fermi-Dirac distribution of a (self-consistent) free Dirac opera-tor, defined as D ˜ γ = D − α ˜ γ ( x, y ) | x − y | . (The last term stands for the operator having this integral kernel.) Thisextends results of [14] to the T > γ from theenergy of any state γ . In this way one obtains a Bogoliubov-Dirac-Fock freeenergy at temperature T = 1 /β which can be formally written as F T ( γ ) = “ F QED T ( γ ) − F QED T (˜ γ )”= T H ( γ, ˜ γ ) − α Z ϕ ( x ) ρ [ γ − ˜ γ ] ( x ) + α Z Z ρ [ γ − ˜ γ ] ( x ) ρ [ γ − ˜ γ ] ( y ) | x − y |− α Z Z tr C | ( γ − ˜ γ )( x, y ) | | x − y | (6)where H is the relative entropy formally defined as T H ( γ, ˜ γ ) = “ tr( D ˜ γ ( γ − ˜ γ )) − T S ( γ ) + T S (˜ γ )” . We shall consider external field of the form ϕ = ν ∗ | x | , where ν representsthe density distribution of the external particles, like nuclei, or molecules.In Section 2.2, we show how to give a correct mathematical meaning tothe previous formulas and we prove that the BDF free energy is boundedfrom below. An important tool is the following inequality T H ( γ, ˜ γ ) ≥ tr (cid:2) | D ˜ γ | ( γ − ˜ γ ) (cid:3) ≥ tr (cid:2) | D | ( γ − ˜ γ ) (cid:3) . (7)This implies that the relative entropy can control the exchange term andenables us to show that F T is bounded from below.Unfortunately, like for the T = 0 case, the free BDF energy is not con-vex, which makes it a difficult task to prove the existence of a minimizer.Although we leave this question open, we derive some properties for a po-tential minimizer in Section 2.2. In particular we prove that any minimizer γ satisfies the following nonlinear equation γ = 12 (cid:18)
11 + e βD γ −
11 + e − βD γ (cid:19) (8)where the (self-consistent) Dirac operator reads D γ = D + αρ γ ∗ | · | − − αϕ − α γ ( x, y ) | x − y | . Compared with the zero temperature case, the main difficulty in provingthe existence of a minimizer comes from localization issues of the relativeentropy which are more involved than in the zero temperature case.As a slight simplification, we thoroughly study the reduced
Hartree Fockcase for
T >
0, where the exchange term (the first term of the second lineof (3)) is neglected. In the zero-temperature case, this model was alreadystudied in detail in [11] and [9]. The corresponding free vacuum is now5imple: it is the Fermi-Dirac distribution corresponding to the usual freeDirac operator D , γ = 12 (cid:18)
11 + e βD −
11 + e − βD (cid:19) . The reduced Bogoliubov-Dirac-Fock free energy is obtained in the same wayas before by subtracting the infinite energy of the free Dirac see γ to the(reduced) Hartree-Fock energy. It is given by F red T ( γ ) = T H ( γ, γ ) − α Z ϕρ [ γ − γ ] + α Z Z ρ [ γ − γ ] ( x ) ρ [ γ − γ ] ( y ) | x − y | dx dy,H ( γ, γ ) being defined similarly as before. As this functional is now convex,we can prove in Theorem 2 that it has a unique minimizer ¯ γ , which satisfiesthe self-consistent equation¯ γ = 12 (cid:18)
11 + e βD ¯ γ −
11 + e − βD ¯ γ (cid:19) where D ¯ γ := D + αρ ¯ γ − γ ∗ | · | − − αϕ in this case.Additionally we show in Theorem 3 that this minimizer has two interest-ing properties. First, ¯ γ − γ is a trace-class operator. In the zero temperaturecase, on the other hand, it was proved in [11] that the minimizer is nevertrace-class for α >
0. This was indeed the source of complications concern-ing the definition of the trace (and hence of the charge) of Hartree-Fockstates [10] when T = 0. This is related to the issue of renormalization[11, 14, 9]. Although we do not minimize in the trace-class in the case T = 0 but rather in the Hilbert-Schmidt class because the free energy isonly coercive for the Hilbert-Schmidt norm, it turns out that the minimizeris trace-class nevertheless.The second (and related) interesting property shown in Theorem 3 belowis that the total electrostatic potential created by the density ν and thepolarized Dirac sea decays very fast. More precisely we prove that ρ ¯ γ − ν ∈ L ( R ) and ( ρ ¯ γ − ν ) ∗ | x | ∈ L ( R ) . Necessarily, the charge of ρ ¯ γ and the charge of the external sources haveto be equal. More precisely the effective potential has a much faster decayat infinity than 1 / | x | , which shows that the effective potential is screened.In other words due to the positive temperature, the particles occupying theDirac-sea have enough freedom to rearrange in such a way that the exter-nal sources are totally shielded. Within non-relativistic fermionic plasma6his effect is known as Debye-screening . Let us emphasize that in orderto recover such a screening, it is essential to calculate the Gibbs-state in aself-consistent way.These two properties of the minimizer of the reduced theory probablyalso hold for the full BDF model with exchange term. However, like for thecase T = 0, the generalization does not seem to be straightforward.The paper is organized as follows. The first section is devoted to thepresentation of our results for the reduced model which is simpler and forwhich we can prove much more than for the general case. In the secondsection, we consider the original Hartree-Fock model with exchange term.We prove the existence and uniqueness of the free Hartree-Fock vacuum,define the BDF free energy in the presence of an external field and providesome interesting properties of potential minimizers. In the last section weprovide some details of proofs which are a too lengthy to be put in the maintext. Acknowledgments.
M.L. acknowledges support from the ANR project“ACCQUAREL” of the French ministry of research. R.S. was partially sup-ported by U.S. NSF grant PHY-0652356. and by an A.P. Sloan fellowship.
Throughout this paper, we shall denote by S p ( H ) the usual Schatten classof operators Q acting on a Hilbert space H and such that tr( | Q | p ) < ∞ .The UV cut-off is implemented like in [10, 11, 12, 14] in Fourier space byconsidering the Hilbert space H Λ := n ψ ∈ L ( R , C ) | supp ˆ ψ ⊂ B (0 , Λ) o , (9)with B (0 , Λ) denoting the ball of radius Λ centered at the origin. We denoteby ‘tr’ the usual trace functional on S ( H Λ ). Within the reduced theory,the free vacuum at temperature T = β − > H Λ defined by γ = 12 (cid:18)
11 + e βD −
11 + e − βD (cid:19) . (10)Notice when T → β → ∞ ), we recover the usual formula [10, 11, 9] γ = − D / | D | .We assume that T > D , the spectrum of γ does7ot include 0 or ± /
2. In fact, it is given by σ ( γ ) = " −
12 + e − βE (Λ) e − βE (Λ) , −
12 + e − β e − β ∪ " − e − β e − β , − e − βE (Λ) e − βE (Λ) (11)where E (Λ) = √ . Also the charge density of the free vacuum γ attemperature T vanishes: ρ γ = 12(2 π ) Z B (0 , Λ) tr C (cid:18)
11 + e βD ( k ) −
11 + e − βD ( k ) (cid:19) dk = 0 . (12)We shall denote the class of Hilbert-Schmidt perturbations of γ by K : K := (cid:26) γ ∈ B ( H Λ ) | γ ∗ = γ, − ≤ γ ≤ , γ − γ ∈ S ( H Λ ) (cid:27) . (13)The relative entropy reads H ( γ, γ ) = tr (cid:20) (cid:0) + γ (cid:1) (cid:0) ln (cid:0) + γ (cid:1) − ln (cid:0) + γ (cid:1)(cid:1) + (cid:0) − γ (cid:1) (cid:0) ln (cid:0) − γ (cid:1) − ln (cid:0) − γ (cid:1)(cid:1) (cid:21) . (14)Note that since γ ∈ K is a compact perturbation of γ , we always have σ ess ( γ ) = σ ess ( γ ). Hence σ ( γ ) only contains eigenvalues of finite multiplicityin the neighborhood of ± /
2. Using the integral formulaln a − ln b = − Z ∞ (cid:20) a + t − b + t (cid:21) dt = Z ∞ a + t ( a − b ) 1 b + t dt, (15)we easily see that Eq. (14) is well defined as soon as γ ∈ K , γ − γ ∈ S ( H Λ ),since the spectrum of γ does not contain ± / γ − γ ∈ K is merely Hilbert-Schmidt, we may define the relativeentropy by the integral formula H ( γ, γ ) = tr (cid:18)Z −
21 + 2 uγ ( γ − γ ) 1 − | u | uγ ( γ − γ ) 11 + 2 uγ du (cid:19) (16)It is clear that this provides a well defined object in K as one has ∀ γ ∈ K , ∀ u ∈ [ − , , ≤ − | u | uγ ≤ ≤
11 + 2 uγ ≤ ǫ ǫ >
0, by (11). It is not difficult to see that (16) and (14) coincidewhen γ − γ ∈ S ( H Λ ). We shall discuss this in the appendix. But (16) hasthe advantage of being well-defined for all γ ∈ K , and hence we use (16) fora definition of H henceforth.Our first result is the Theorem 1 ( Properties of Relative Entropy).
The functional γ H ( γ, γ ) defined in (16) is strongly continuous on K for the topology of S ( H Λ ) . It is convex, hence weakly lower semi-continuous (wlsc). Moreover,it is coercive on K for the Hilbert-Schmidt norm: ∀ γ ∈ K , T H ( γ, γ ) ≥ tr (cid:0) | D | ( γ − γ ) (cid:1) (17) where we recall that T = β − is the temperature. Coercive in this context means that H ( γ, γ ) → ∞ if (cid:12)(cid:12)(cid:12)(cid:12) γ − γ (cid:12)(cid:12)(cid:12)(cid:12) S ( H Λ ) →∞ . This follows from (17) since | D | ≥ Proof of Theorem 1.
First we prove that H ( · , γ ) is strongly continuous forthe S ( H Λ ) topology. This is indeed a consequence of the following Lemma 1.
Let γ, γ ′ ∈ K . Then we have for some constant C (dependingon Λ ) and all ≤ η ≤ , (cid:12)(cid:12) H ( γ, γ ) − H ( γ ′ , γ ) (cid:12)(cid:12) ≤ Cη (cid:12)(cid:12)(cid:12)(cid:12) γ − γ ′ (cid:12)(cid:12)(cid:12)(cid:12) S ( H Λ ) + Cη (cid:16)(cid:12)(cid:12)(cid:12)(cid:12) γ − γ (cid:12)(cid:12)(cid:12)(cid:12) S ( H Λ ) + (cid:12)(cid:12)(cid:12)(cid:12) γ ′ − γ (cid:12)(cid:12)(cid:12)(cid:12) S ( H Λ ) (cid:17) . (18) Proof.
We use Formula (16) and split the integrals as follows: Z − = Z − η − + Z − η − η + Z − η . We estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) tr Z − η − η du (cid:18)
11 + 2 uγ ( γ − γ ) 1 − | u | uγ ( γ − γ ) 11 + 2 uγ −
11 + 2 uγ ( γ ′ − γ ) 1 − | u | uγ ′ ( γ ′ − γ ) 11 + 2 uγ !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cη (cid:12)(cid:12)(cid:12)(cid:12) γ − γ ′ (cid:12)(cid:12)(cid:12)(cid:12) S ( H Λ ) using in particular1 − | u | uγ − − | u | uγ ′ = 2 u − | u | uγ ( γ ′ − γ ) 11 + 2 uγ ′ ≤ (1 + 2 uγ ′ ) − ≤ η − as γ ′ ∈ K and − η ≤ u ≤ − η . Similarly (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) tr Z − η du
11 + 2 uγ ( γ − γ ) 1 − | u | uγ ( γ − γ ) 11 + 2 uγ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cη (cid:12)(cid:12)(cid:12)(cid:12) γ − γ (cid:12)(cid:12)(cid:12)(cid:12) S ( H Λ ) . The other terms are treated in the same way.Convexity of γ H ( γ, γ ) is a simple consequence of the integral rep-resentation (16). In fact, the integrand is convex for any fixed u ∈ [ − , γ ( γ − γ ) 11 + 2 uγ ( γ − γ )= 1(2 u ) (cid:18) uγ − − uγ + (1 + 2 uγ ) 11 + 2 uγ (1 + 2 uγ ) (cid:19) is clearly convex.Finally, we prove Formula (17). Consider the following function f ( x, y ) = ( + x ) (cid:0) ln( + x ) − ln( + y ) (cid:1) + ( − x ) (cid:0) ln( − x ) − ln( − y ) (cid:1) defined on ( − / , / . Minimizing over x for fixed y , one finds that f ( x, y ) ≥ ( x − y ) C ( y ) where C ( y ) = ln (cid:16) / y / − y (cid:17) / (2 y ). If we write y as y = 12 (cid:18)
11 + e h −
11 + e − h (cid:19) , (19)we obtain C ( y ) = h tanh( h/ − ≥ max( | h | , y takes the form(19), we deduce f ( x, y ) ≥ max (cid:8) ( x − y ) | h | , x − y ) (cid:9) . Assume now that X and Y are self-adjoint operators acting on a Hilbertspace H , with − / ≤ X, Y ≤ / Y = 12 (cid:18)
11 + e H −
11 + e − H (cid:19) for some H . By Klein’s inequality [23, p. 330], one also has H ( X, Y ) = tr f ( X, Y ) ≥ max (cid:8) tr( X − Y ) | H | , X − Y ) (cid:9) . (20)This gives (17), taking X = γ and Y = γ .10 .2 Existence of a minimizer and Debye screening Now we are able to define the reduced Bogoliubov-Dirac-Fock energy at tem-perature T = β − . For this purpose, we introduce the Coulomb space C := { ρ ∈ S ′ ( R ) | D ( ρ, ρ ) < ∞} (21)where D ( f, g ) = 4 π Z R | k | − b f ( k ) g ( k ) dk. (22)We remark that the Fourier transform of Q = γ − γ in an L -functionwith support in B (0 , Λ) × B (0 , Λ). Hence Q ( x, y ) is a smooth kernel and ρ Q ( x ) = tr C ( Q ( x, x )) is a well defined function. Indeed, the map γ ∈ K 7→ ρ γ − γ ∈ L ( R ) is continuous for the topology of S ( H Λ ). It is easy to seethat the Fourier transform of ρ γ − γ is given by the formula \ ρ γ − γ ( k ) = 1(2 π ) / Z | p + k/ |≤ Λ | p − k/ |≤ Λ tr C h \ ( γ − γ )( p + k/ , p − k/ i dp. (23)We also define our variational set by K C := (cid:8) γ ∈ K | ρ γ − γ ∈ C (cid:9) . (24)The reduced Bogoliubov-Dirac-Fock energy reads F red T ( γ ) = T H ( γ, γ ) − αD ( ν, ρ γ − γ ) + α D ( ρ γ − γ , ρ γ − γ ) (25)and it is well-defined on K C by Theorem 1. In (25), ν ∈ C is an externaldensity creating an electrostatic potential − ν ∗ / | x | . The number α > fine structure constant . The following is an easy consequence onTheorem 1: Theorem 2 ( Existence of a minimizer).
Assume
T > , α ≥ and ν ∈ C . Then F red T satisfies ∀ γ ∈ K C , F red T ( γ ) ≥ − α D ( ν, ν ) (26) hence it is bounded below on K C . It has a unique minimizer ¯ γ on K C . Theoperator ¯ γ satisfies the self-consistent equation ¯ γ = 12 (cid:18)
11 + e βD ¯ γ −
11 + e − βD ¯ γ (cid:19) ,D ¯ γ := D + α ( ρ ¯ γ − γ − ν ) ∗ | · | − . (27) Remark 1.
When T = 0, a similar result was proved in [11, Thm 3], butthere might be no uniqueness in this case.11 emark 2. If there is no external field, ν = 0, we recover that the optimalstate is γ − γ = 0, and its energy is zero, by (26). Proof of Theorem 2.
Eq. (26) is an obvious consequence of positivity of therelative entropy H and positive definiteness of D ( · , · ). The existence of aminimizer is obtained by noticing that F red T is weakly lower semi-continuousfor the topology of S ( H Λ ) and C , by Theorem 1. As F red T is convex andstrictly convex with respect to ρ γ − γ , we deduce that all the minimizers sharethe same density. Next we notice that ± / / ∈ σ (¯ γ ) since the derivative ofthe relative entropy with respect to variations of an eigenvalue is infiniteat these two points. Hence ¯ γ does not saturate the constraint and it isa solution of Eq. (27). This a fortiori proves that ¯ γ is unique, since D ¯ γ depends only on the density ρ ¯ γ − γ .Now we provide some interesting properties of any solution of Eq. (27),thus in particular of our minimizer ¯ γ . Theorem 3 ( Debye Screening).
Assume
T > , α > and ν ∈ C ∩ L ( R ) . Any γ ∈ K that solves Eq. (27) is a trace-class perturbation of γ ,i.e., γ − γ ∈ S ( H Λ ) . Its charge density ρ γ − γ is an L ( R ) function whichsatisfies Z R ρ γ − γ = Z R ν and (cid:0) ρ γ − γ − ν (cid:1) ∗ | x | ∈ L ( R ) . (28)This result implies that the particles arrange themselves such that thetotal effective potential (cid:0) ρ γ − γ − ν (cid:1) ∗ / | x | has a decay much faster than 1 / | x | .This implies that the nuclear charge of the external sources is completelyscreened.The proof of Theorem 3 is lengthy and is given later in Section 3.1. When the exchange term is not neglected, the free vacuum is no longer de-scribed by the operator γ introduced in the previous section. Instead it isanother translation-invariant operator ˜ γ that solves a self-consistent equa-tion. Following ideas from [14], we define in this section ˜ γ as the (unique)minimizer of the free energy per unit volume. We consider translation-invariant operators γ = γ ( p ) acting on H Λ and such that − / ≤ γ ≤ / − / ≤ γ ( p ) ≤ /
2, for a.e. p ∈ B (0 , Λ), in It can indeed be proved that H ( · , γ ) is strictly convex but we do not need that here. C × C hermitian matrices. The free energy per unit volumeof such a translation-invariant operator γ at temperature T is given by [14] T T ( γ ) = 1(2 π ) (cid:18) Z B (0 , Λ) tr C [ D ( p ) γ ( p )] dp − α (2 π ) Z Z B (0 , Λ) tr C [ γ ( p ) γ ( q )] | p − q | dp dq − T S ( γ ) (cid:19) (29)where the entropy is defined as S ( γ ) = − Z B (0 , Λ) tr C (cid:2) (cid:0) + γ ( p ) (cid:1) ln (cid:0) + γ ( p ) (cid:1) + (cid:0) − γ ( p ) (cid:1) ln (cid:0) − γ ( p ) (cid:1) (cid:3) dp. The free energy is defined on the convex set of matrix-valued functions, suchthat, for all p ∈ B (0 , Λ), γ ( p ) is a hermitian 4 × A := (cid:8) γ : B (0 , Λ) → M | γ ( p ) ∗ = γ ( p ) , − / ≤ γ ( p ) ≤ / p ∈ B (0 , Λ) (cid:9) . (30) Theorem 4 ( The free vacuum at temperature T ). For all
T > andall ≤ α < /π , the free energy per unit volume T T in (29) has a uniqueminimizer ˜ γ on A . It is a solution of the self-consistent equation ˜ γ = 12 (cid:18)
11 + e βD ˜ γ −
11 + e − βD ˜ γ (cid:19) D ˜ γ = D − α ˜ γ ( x,y ) | x − y | . (31) Furthermore, ˜ γ has the form ˜ γ ( p ) = f ( | p | ) α · p + f ( | p | ) β (32) with f , f ≤ a.e. on B (0 , Λ) and D ˜ γ satisfies | D ˜ γ | ≥ | D | . (33)Here and in the following, we shall identify operators with their integralkernels for simplicity of the notation. That is, the last term in the secondline of (31) denotes the operator with integral kernel given by ( γ − ˜ γ )( x,y ) | x − y | ,where ( γ − ˜ γ )( x, y ) is the integral kernel of the translation-invariant operator γ − ˜ γ (it is a function of x − y ). Remark 3.
The assumption α < /π guarantees that the functional (29) isbounded from below, independently of the UV cutoff Λ, which is arbitraryin this paper. This is a consequence of Kato’s inequality. For α > /π thisis not the case [5, 17].For comparison, we note that in the non-interacting case α = 0, thefunctions f ( | p | ) and f ( | p | ) appearing in Theorem 4 are given by f ( | p | ) = f ( | p | ) = 12 E ( p ) (cid:18)
11 + e βE ( p ) −
11 + e − βE ( p ) (cid:19) .
13 similar result was proved in the zero temperature case in [14]. As in[14], it is possible to justify the introduction of T T by a thermodynamic limitprocedure. The proof of Theorem 4 is given in Section 3.3.Like for the reduced case, we have that σ (˜ γ ) ⊂ (cid:20) −
12 + ǫ, − ǫ (cid:21) ∪ (cid:20) ǫ, − ǫ (cid:21) for some ǫ >
0. This can be seen from (33) and the fact that D ˜ γ is a boundedoperator on H Λ due to the presence of the ultraviolet cut-off. Notice alsothat we have formally ρ ˜ γ ≡ As in Section 1.2, one can consider the Bogoliubov-Dirac-Fock energy withan external field. It is formally obtained by subtracting the infinite freeenergy of the free vacuum at temperature
T > γ . This procedure can be justified like in [14] by a thermodynamiclimit procedure. Using the same notation as in Section 1.2, the Bogoliubov-Dirac-Fock free energy reads F T ( γ ) = T H ( γ, ˜ γ ) − αD ( ν, ρ γ − ˜ γ ) + α D ( ρ γ − ˜ γ , ρ γ − ˜ γ ) − α Z Z tr C | ( γ − ˜ γ )( x, y ) | | x − y | dxdy, (34)where H is the relative entropy defined like in Section 1.1. Like for thereduced case, we see that the functional F T is well-defined on the followingconvex set˜ K C := (cid:26) γ ∈ B ( H Λ ) | γ ∗ = γ, − ≤ γ ≤ , γ − ˜ γ ∈ S ( H Λ ) , ρ γ − ˜ γ ∈ C (cid:27) . (35)Note that although the function γ H ( γ, ˜ γ ) is convex, F T is not a convexfunctional because of the presence of the exchange term. This is of course agreat obstacle in proving the existence of a minimizer, and we have to leavethis as an open problem. Following the method of Theorem 1, we shall showthat ∀ γ ∈ ˜ K C , T H ( γ, ˜ γ ) ≥ tr (cid:0) | D ˜ γ | ( γ − ˜ γ ) (cid:1) . (36)With the aid of this inequality we can prove the Theorem 5 ( Minimizer in External Field).
Assume that ≤ α < /π and that T > . We have ∀ γ ∈ ˜ K C , F T ( γ ) ≥ − α D ( ν, ν ) (37) and hence F T is bounded below on ˜ K C . ssume that γ ∈ ˜ K C is a minimizer of F T . Then it satisfies the self-consistent equation γ = 12 (cid:18)
11 + e βD γ −
11 + e − βD γ (cid:19) ,D γ := D ˜ γ + α ( ρ γ − γ − ν ) ∗ | · | − − α ( γ − ˜ γ )( x,y ) | x − y | (38) with D ˜ γ defined in (31). It is unique when ≤ α π ( − α π s α/ − απ/ π / / ! D ( ν, ν ) / ) − ≤ . (39)The proof of Theorem 5 is provided in Section 3.4. Let γ be a solution of γ = 12 (cid:18)
11 + e βD γ −
11 + e − βD γ (cid:19) ,D γ := D + α ( ρ γ − γ − ν ) ∗ | · | − . (40)For the sake of simplicity, we define ρ := ρ γ − γ − ν and V = α ( ρ γ − γ − ν ) ∗| · | − . Note that ∇ V ∈ L ( R ) as ρ ∈ C , hence V ∈ L ( R ). Following [11,p. 4495], we may use the Kato-Seiler-Simon inequality (see [20] and [21,Thm. 4.1]) ∀ p ≥ , || f ( − i ∇ ) g ( x ) || S p ( L ( R )) ≤ (2 π ) − /p || g || L p ( R ) || f || L p ( R ) (41)to obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V | D | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S ∞ ( H Λ ) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V | D | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S ( H Λ ) ≤ C ′ || V || L ( R ) ≤ C || ρ || C . This shows that | D γ | ≤ (1 + αC || ρ || C ) | D | . Thanks to the cut-off in Fourierspace, we deduce that D γ is a bounded operator or H Λ . Recall Duhamel’sformula e βD γ = e βD + β Z e tβD γ V e (1 − t ) βD dt. (42)Denoting K := β R e tβD γ V e (1 − t ) βD dt and using (42), we have K = K + K ′ := β Z dt e tβD V e (1 − t ) βD + β Z dt Z t ds e sβD γ V e ( t − s ) βD V e (1 − t ) βD .
15e obtain for the self-consistent solution γ − γ = 11 + e βD γ −
11 + e βD = −
11 + e βD K
11 + e βD −
11 + e βD K ′
11 + e βD + 11 + e βD K
11 + e βD γ K
11 + e βD (43)which we write as γ − γ = A + B where A = −
11 + e βD K
11 + e βD = − β Z e tβD e βD V e (1 − t ) βD e βD dt. As V ∈ L ( R ) and D γ is bounded, using the cut-off in Fourier space and theKato-Seiler-Simon inequality (41), we have K ∈ S ( H Λ ). Hence we obtainthat K ′ ∈ S ( H Λ ) and B ∈ S ( H Λ ).The next step is to compute the density of A . The kernel of A is givenby b A ( p, q ) = − β (2 π ) − / Z e tβD ( p ) e βD ( p ) b V ( p − q ) e (1 − t ) βD ( q ) e βD ( q ) dt. Using (23), we obtain c ρ A ( k ) = − αC ( | k | ) | k | b ρ ( k )where C ( | k | ) := β π Z | p + k/ |≤ Λ | p − k/ |≤ Λ dp Z dt tr C " e tβD ( p + k/ e βD ( p + k/ e (1 − t ) βD ( p − k/ e βD ( p − k/ . (44)Inserting this into the self-consistent equation (43) gives b ρ ( k ) = − b ν ( k ) − αC ( | k | ) | k | b ρ ( k ) + c ρ B ( k ) (45)or, equivalently, b ρ ( k ) = b b ( k )( − b ν ( k ) + c ρ B ( k )) , (46)and b V ( k ) = 4 π b b ( k )( − b ν ( k ) + c ρ B ( k )) , (47)where b := F − (cid:18) | k | | k | + αC ( | k | ) (cid:19) and b := F − (cid:18) | k | + αC ( | k | ) (cid:19) , (48)with F − denoting the inverse Fourier transform. Our main tool will be thefollowing 16 roposition 1 ( Properties of b , b ). The two functions b ( x ) and b ( x ) ,defined in (48) and (44) , belong to L ( R ) . We postpone the proof of Proposition 1 to Section 3.2 and first completethe proof of Theorem 3. First we claim that ρ B ∈ L ( R ). To see this, wetake a function ξ ∈ L / ( R ) ∩ C ∞ ( R ) and compute | tr( Bξ ) | = | tr( B B (0 , Λ) ( p ) ξ B (0 , Λ) ( p )) |≤ || B || S ( H Λ ) (cid:12)(cid:12)(cid:12)(cid:12) B (0 , Λ) ( p ) ξ B (0 , Λ) ( p ) (cid:12)(cid:12)(cid:12)(cid:12) S / ( H Λ ) . Writing ξ = | ξ | / sgn( ξ ) | ξ | / and using the Kato-Seiler-Simon inequality(41) twice in S ( H Λ ), we obtain | tr( Bξ ) | ≤ C || B || S ( H Λ ) || ξ || L / ( R ) where C depends on the cut-off Λ. This proves by duality that ρ B ∈ L ( R ).Next we use a boot-strap argument. As ν ∈ L ( R ) and ρ B ∈ L ( R ),we get from (47) and Proposition 1 that V ∈ L ( R ). Inserting in thedefinition of K ′ and using (41) once more, we obtain that K ′ ∈ S / ( H Λ ),hence B ∈ S ( R ) and ρ B ∈ L ( R ). Using again (47) and Proposition 1,we get that V ∈ L ( R ), hence B ∈ S ( H Λ ) and ρ B ∈ L ( R ). This finishesthe proof of Theorem 3, by (46), (47) and Proposition 1. The proof proceeds along the same lines as in the Appendix of [9]. In thefollowing we shall denote by P +0 and P − the projection onto the positive andnegative spectral subspace of D , respectively. As multiplication operatorsin momentum space, P +0 ( p ) = 12 (cid:18) α · p + βE ( p ) (cid:19) , P − ( p ) = 12 (cid:18) − α · p + βE ( p ) (cid:19) . The function C in (44) can be written as C ( | k | ) = βπ Z | p + k/ |≤ Λ | p − k/ |≤ Λ dp ×× Z dt tr C " e tβE ( p + k/ e βE ( p + k/ e (1 − t ) βE ( p − k/ e βE ( p − k/ P +0 ( p + k/ P +0 ( p − k/ + tr C " e tβE ( p + k/ e βE ( p + k/ e − (1 − t ) βE ( p − k/ e − βE ( p − k/ P +0 ( p + k/ P − ( p − k/ . (49)17ence C ( | k | ) =1 π Z | p + k/ |≤ Λ | p − k/ |≤ Λ
11 + e βE ( p + k/ e βE ( p + k/ − e βE ( p − k/ E ( p + k/ − E ( p − k/
2) 11 + e βE ( p − k/ ×× (cid:20) p + k/ · ( p − k/
2) + 1 E ( p + k/ E ( p − k/ (cid:21) dp + 1 π Z | p + k/ |≤ Λ | p − k/ |≤ Λ
11 + e βE ( p + k/ e βE ( p + k/ − e − βE ( p − k/ E ( p + k/
2) + E ( p − k/
2) 11 + e − βE ( p − k/ ×× (cid:20) − ( p + k/ · ( p − k/
2) + 1 E ( p + k/ E ( p − k/ (cid:21) dp. For the sake of clarity, we denote by C ( | k | ) (resp. C ( | k | )) the first (resp.second) integral of the previous formula. By the monotonicity of the expo-nential function, it is easily seen that C ( | k | ) ≥ C ( | k | ) ≥ C ( | k | ) = 8 π | k | Z Z Λ ( | k | )0 dz Z | k | z dv e βw ( k,z ) e β ( w ( k,z )+ v ) sinh( βv ) v ××
11 + e β ( w ( k,z ) − v ) z w ( k, z ) − (1 − z ) + 8 π | k | Z | k | Z Λ ( | k | )0 dz Z z dv e β ( E (Λ) − z ) e β ( E (Λ) − z + v ) sinh( βv ) v
11 + e β ( E (Λ) − z − v ) ×× (cid:18) ( E (Λ) − z ) − | k | (cid:19) , (50) C ( | k | ) = 8 π | k | Z Z Λ ( | k | )0 dz Z | k | z dv e βv e β ( w ( k,z )+ v ) sinh( βw )1 + | k | (1 − z ) ××
11 + e β ( v − w ( k,z )) (cid:18) | k | − v (cid:19) z − z + 8 π | k | Z | k | Z Λ ( | k | )0 dz Z z dv e βv e β ( E (Λ) − z + v ) sinh( β ( E (Λ) − z )) E (Λ) − z ××
11 + e β ( v + z − E (Λ)) (cid:18) | k | − v (cid:19) . (51)In the above formulas we have used the notation (as in [9]) Z Λ ( r ) = √ − p − r ) r . Z Λ is a decreasing C ∞ function on [0 , Z Λ (0) =Λ /E (Λ), Z Λ (2Λ) = 0. We have also used the shorthand notation w ( k, z ) = r | k | (1 − z ) / − z . All integrands of the above formulas are real analytic functions of r = | k | ona neighborhood of [0 , k = 0. We deducethat C and C are smooth functions on [0 , Z Λ (2Λ) = 0, one alsosees that C (2Λ) = C ′ (2Λ) = C (2Λ) = C ′ (2Λ) = 0. A Taylor expansion ofthe first integral of C yields C (0) = 4 π β Z E (Λ)1 t dt (1 + e − βt )(1 + e βt ) > . The end of the proof of Proposition 1 is then the same as in [9, Prop. 17].First we notice that as C ( r ) is bounded and has a compact support, b and b are in L ∞ ( R ). We now prove that they decay at least like | x | − at infinitymeaning that they also belong to L ( R ). To this end we write for b = b or = b the inverse Fourier transform in radial coordinates: ∀ x ∈ R \ { } , b ( x ) = 1 √ π | x | Z ( r b b ( r )) sin( r | x | ) dr. (52)Integrating by parts and using b b (2Λ) = b b ′ (2Λ) = 0 yields ∀ x ∈ R \ { } , b ( x ) = 1 √ π | x | b b ′′ (2Λ) cos(2Λ | x | ) − b b ′ (0) − Z ( r b b ) (3) ( r ) cos( r | x | ) dr ! . (53)This completes the proof of Proposition 1. The proof is inspired by ideas from [14]. We denote I := inf γ ∈A T T ( γ ). Westart by introducing the following auxiliary minimization problem J = inf γ ∈B T T ( γ ) (54)where B ⊂ A is given by B := { γ ∈ A , γ ( p ) = f ( | p | ) α · p + f ( | p | ) β, f , f ≤ } . (55) Lemma 2.
There exists a minimizer ˜ γ ∈ B for (54) . roof of Lemma 2. The functional T T is weakly lower semi-continuous forthe weak- ∗ topology of L ∞ ( B (0 , Λ)). This is because − S is convex and theexchange term is continuous for the weak topology of L ( B (0 , Λ)) as shownin [14]. Also B is a bounded closed convex subset of L ∞ ( B (0 , Λ)). Hencethere exists a minimum.
Lemma 3.
Let ˜ γ ∈ B be a minimizer of (54) . Then there exists an ǫ > such that | ˜ γ | ≤ / − ǫ .Proof. For x ∈ [1 / , / s ( x ) := (cid:0) + x (cid:1) ln (cid:0) + x (cid:1) + (cid:0) − x (cid:1) ln (cid:0) − x (cid:1) is an even function of x . Because of the special form of ˜ γ , we have ˜ γ ( p ) = k ˜ γ ( p ) k I C for all p ∈ B (0 , Λ), where k · k denotes the matrix norm. Hence ∀ γ ∈ B , S ( γ ) = − Z B (0 , Λ) s ( k γ ( p ) k ) dp. The derivative of s is infinite at x = 1 / { p ∈ B (0 , Λ) | / − ǫ ≤ k ˜ γ ( p ) k ≤ / } has zero measure for ǫ small enough.Let us now write the first order condition satisfied by ˜ γ . Since k ˜ γ ( p ) k ≤ / − ǫ for some ǫ small enough, we can consider a perturbation of the form γ ( p ) = ˜ γ ( p ) + t ( g ( | p | ) α · p + g ( | p | ) β )with g , g ≤ t > Z B (0 , Λ) tr C (cid:20)(cid:18) D ˜ γ ( p ) + T ln 1 / γ ( p )1 / − ˜ γ ( p ) (cid:19) (cid:0) g ( | p | ) α · p + g ( | p | ) β (cid:1)(cid:21) dp ≥ g , g ≤ x ln (cid:16) / x / − x (cid:17) is odd, hence ∀ γ ∈ B , ln (cid:18) / γ ( p )1 / − γ ( p ) (cid:19) = sgn( γ ) ln (cid:18) / k γ ( p ) k / − k γ ( p ) k (cid:19) , (57)with sgn( γ ) = γ/ | γ | . We obtain thatln (cid:18) / γ ( p )1 / − ˜ γ ( p ) (cid:19) = ˜ γ ( p ) F ( k ˜ γ ( p ) k )where F ( x ) = ln (cid:16) / x / − x (cid:17) /x . On the other hand, we can write D ˜ γ = d ( | p | ) α · p + d ( | p | ) β d and d are given by [14, Eq. (72)-(73)]. Using f , f ≤
0, weimmediately see that d ( | p | ) ≥ d ( | p | ) ≥ , (58)which in particular proves that k D ˜ γ ( p ) k ≥ k D ( p ) k ≥ | p | . (59)All this gives D ˜ γ + T ln 1 / γ / − ˜ γ = (cid:0) d ( | p | ) + T f ( | p | ) F ( k ˜ γ ( p ) k ) (cid:1) α · p + (cid:0) d ( | p | ) + T f ( | p | ) F ( k ˜ γ ( p ) k ) (cid:1) β. (60)Inserting this in (56), we obtain the first order conditions (cid:26) d ( | p | ) + T f ( | p | ) F ( k ˜ γ ( p ) k ) ≤ d ( | p | ) + T f ( | p | ) F ( k ˜ γ ( p ) k ) ≤ . (61)In particular, because of (58) we infer that (cid:26) f ( | p | ) F ( k ˜ γ ( p ) k ) ≤ − /Tf ( | p | ) F ( k ˜ γ ( p ) k ) ≤ − /T. (62)As F ( k ˜ γ ( p ) k ) ≥ f ≥ −k ˜ γ ( p ) k , we obtain from (62) the inequality | γ ( p ) | F ( k γ ( p ) k ) ≥ /T . Hence k γ ( p ) k ≥ e /T − e /T ) . (63)This inequality means that f and f cannot vanish simultaneously. But wecan indeed prove that each of them cannot vanish, as expressed in the Lemma 4.
Let ˜ γ ( p ) = f ( | p | ) α · p + f ( | p | ) β be a minimizer of (54) . Thenthere exists an ǫ > such that f ≤ − ǫ and f ≤ − ǫ. Proof.
By Lemma 3 we know that k ˜ γ ( p ) k ≤ / − ǫ for some ǫ >
0. By(62) and the monotonicity of F we obtain f k ≤ − T F (1 / − ǫ ) for k = 0 , Lemma 5.
Let ˜ γ ∈ B be a minimizer of (54) . Then it solves the self-consistent equation ˜ γ = 12 (cid:18)
11 + e βD ˜ γ −
11 + e − βD ˜ γ (cid:19) . (64)21 roof. As the constraints are not saturated by Lemmas 3 and 4, we obtainthat the derivative vanishes, i.e. (cid:26) d ( | p | ) + T f ( | p | ) F ( k ˜ γ ( p ) k ) = 0 d ( | p | ) + T f ( | p | ) F ( k ˜ γ ( p ) k ) = 0 . (65)which means that D ˜ γ + T ln 1 / γ / − ˜ γ = 0 . Hence ˜ γ solves (64).Now we prove that the operator ˜ γ defined in the previous step is theunique minimizer of T T on the full space A defined in (30), not merely onthe subset B in (55). We have T T ( γ ) −T T (˜ γ ) = T H ( γ, ˜ γ ) − α (2 π ) Z Z B (0 , Λ) tr C [( γ − ˜ γ )( p )( γ − ˜ γ )( q )] | p − q | dp dq where H is the relative entropy per unit volume H ( γ, ˜ γ ) = (2 π ) − Z B (0 , Λ) tr C (cid:2) (cid:0) + γ (cid:1) (cid:0) ln (cid:0) + γ (cid:1) − ln (cid:0) + ˜ γ (cid:1)(cid:1) + (cid:0) − γ (cid:1) (cid:0) ln (cid:0) − γ (cid:1) − ln (cid:0) − ˜ γ (cid:1)(cid:1) (cid:3) dp. (66)We shall use the important Lemma 6.
For H in (66) the inequality T H ( γ, ˜ γ ) ≥ (2 π ) − Z B (0 , Λ) tr C | D ˜ γ ( p ) | ( γ ( p ) − ˜ γ ( p )) dp (67) holds for all γ ∈ A .Proof. This is a simple application of (20), taking X = γ ( p ), Y = ˜ γ ( p ) andintegrating over the ball B (0 , Λ).Using Lemma 6 and the formula α (2 π ) Z Z B (0 , Λ) tr C [ γ ( p ) γ ( q )] | p − q | dp dq = α Z R tr C | ˇ γ ( x ) | | x | dx where ˇ γ ( x ) is the Fourier inverse of the function γ ( p ), we find T T ( γ ) − T T (˜ γ ) ≥ (2 π ) − (cid:18) Z B (0 , Λ) tr C | D ˜ γ ( p ) | ( γ ( p ) − ˜ γ ( p )) dp − α Z R tr C | (ˇ γ − ˇ γ )( x ) | | x | dx (cid:19) (68)22or all γ ∈ A . We now use ideas of [1, 10, 14]. Kato’s inequality | x | − ≤ π/ |∇| gives α Z R tr C | (ˇ γ − ˇ γ )( x ) | | x | dx ≤ απ Z B (0 , Λ) tr C | p | ( γ ( p ) − ˜ γ ( p )) dp. By (59) we deduce T T ( γ ) − T T (˜ γ ) ≥ (1 − πα/ π ) − Z B (0 , Λ) tr C | D ˜ γ ( p ) | ( γ ( p ) − ˜ γ ( p )) dp. Hence ˜ γ is the unique minimizer of T T on A when 0 ≤ α < /π . Thiscompletes the proof of Theorem 4. The lower bound (37) is obtained by following an argument of [1, 10]. By(36) we have for all γ ∈ ˜ K C F T ( γ ) ≥ tr (cid:0) | D ˜ γ | ( γ − ˜ γ ) (cid:1) − α Z Z tr C | ( γ − ˜ γ )( x, y ) | | x − y | dxdy + α D ( ρ γ − ˜ γ − ν, ρ γ − ˜ γ − ν ) − α D ( ν, ν ) . (69)By (33) together with Kato’s inequality | x | − ≤ ( π/ |∇| ≤ ( π/ | D | , Z Z tr C | ( γ − ˜ γ )( x, y ) | | x − y | dxdy ≤ π (cid:0) | D ˜ γ | ( γ − ˜ γ ) (cid:1) which yields (37) when 0 ≤ α < /π .Assume now that γ is a minimizer of F T on ˜ K C . The proof that it satisfiesthe self-consistent equation (38) is the same as in the case of the reducedBDF functional in Section 1.2. Note that because of (69) and inf ˜ K C F T ≤ (cid:18) π − α (cid:19) Z Z tr C | ( γ − ˜ γ )( x, y ) | | x − y | dxdy + α D ( ρ γ − ˜ γ , ρ γ − ˜ γ ) ≤ α D ( ν, ν ) . It was proved in [11, p. 4495] that this implies (under the condition (39))that | D γ | ≥ d − | D | (70)with d = ( − α π s α/ − απ/ π / / ! D ( ν, ν ) / ) − . γ ′ ∈ ˜ K C and use that H ( γ ′ , ˜ γ ) = H ( γ, ˜ γ ) + H ( γ ′ , γ )+ tr " ( γ ′ − γ ) ln + γ − γ ! − ln + ˜ γ − ˜ γ !! . (71)Inserting Eq. (38) for our minimizer γ and Eq. (31) for ˜ γ , we obtain forany γ ′ ∈ ˜ K C the formula F T ( γ ′ ) = F T ( γ )+ T H ( γ ′ , γ )+ α D ( ρ γ ′ − γ , ρ γ ′ − γ ) − α Z Z tr C | ( γ − ˜ γ )( x, y ) | | x − y | dxdy. (72)We may use one more time (15) and the self-consistent equation (38) toobtain T H ( γ ′ , γ ) ≥ tr (cid:0) | D γ | ( γ ′ − γ ) (cid:1) . By (70) and Kato’s inequality as before, we eventually get F T ( γ ′ ) ≥ F T ( γ )+ α D ( ρ γ ′ − γ , ρ γ ′ − γ )+ (cid:18) πd − α (cid:19) Z Z tr C | ( γ − ˜ γ )( x, y ) | | x − y | dxdy. Hence we obtain that any minimizer is unique when απd/ ≤
1, as stated.Let us remark that the expression in last term of (71) is indeed a trace-classoperator. It would, however, have been sufficient to choose γ ′ as trace classperturbation of γ and conclude the rest by a density argument. A Appendix: Integral Representation of RelativeEntropy
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