## Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation

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@MISC{Hainzl_existenceof,

author = {Christian Hainzl and Mathieu Lewin and Éric Séré},

title = {Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation },

year = {}

}

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985 |
Quantum field theory
- Itzykson, Zuber
- 1980
(Show Context)
Citation Context ...8, 18, 13, 31, 29] that the density ρ αϕ associated with Q αϕ = P αϕ −P 0 is never well-defined, since it diverges pointwise logarithmically. In physics literature [12, 28, 9, 10] (see also the books =-=[20, 23, 32]-=-), a procedure called charge renormalization aims at extracting the main information from ραϕ and a renormalized density ρ αϕ ren is used to replace the ill-defined density in the Hamiltonian D ren Qα... |

329 |
The Quantum theory of fields Vol
- Weinberg
(Show Context)
Citation Context ...8, 18, 13, 31, 29] that the density ρ αϕ associated with Q αϕ = P αϕ −P 0 is never well-defined, since it diverges pointwise logarithmically. In physics literature [12, 28, 9, 10] (see also the books =-=[20, 23, 32]-=-), a procedure called charge renormalization aims at extracting the main information from ραϕ and a renormalized density ρ αϕ ren is used to replace the ill-defined density in the Hamiltonian D ren Qα... |

224 |
The Dirac equation
- Thaller
- 1992
(Show Context)
Citation Context ...:= −iα · ∇ + β (1) where α = (α1,α2,α3) and k=1 ( ) I2 0 β = , αk = 0 −I2 ( ) 0 σk , σk 0 with σ1 = ( ) 0 1 , σ2 = 1 0 ( ) ( ) 0 −i 1 0 , σ3 = . i 0 0 −1 We follow here the notation of Thaller’s book =-=[30]-=-, and of [2]. We have chosen a system of units such that � = c = 1, and also such that the mass me of the electron is normalized to 1. The operator D 0 acts on 4-spinors, i.e. functions Ψ ∈ H := L 2 (... |

134 |
Quantum Electrodynamics of Strong Fields
- Greiner, Müller, et al.
- 1985
(Show Context)
Citation Context ...rticular normal ordering of the second-quantized Hamiltonian (see the Appendix for details). Remark also that the vacuum polarization terms play an essential role within the treatment of muonic atoms =-=[14]-=-. 2.1 Supertrace-class operators In order to define the BDF-functional properly, we introduce the concept of supertrace-class operators, in the spirit of [30, Section 5.7]. In this section only, we wo... |

108 |
The quantum vacuum: an introduction to quantum electrodynamics
- Milonni
- 1994
(Show Context)
Citation Context ...8, 18, 13, 31, 29] that the density ρ αϕ associated with Q αϕ = P αϕ −P 0 is never well-defined, since it diverges pointwise logarithmically. In physics literature [12, 28, 9, 10] (see also the books =-=[20, 23, 32]-=-), a procedure called charge renormalization aims at extracting the main information from ραϕ and a renormalized density ρ αϕ ren is used to replace the ill-defined density in the Hamiltonian D ren Qα... |

61 |
The Hartree-Fock theory for Coulomb systems
- Lieb, Simon
- 1977
(Show Context)
Citation Context ..., section 8.2]). In physics, self-consistent equations are usually derived as Euler-Lagrange equations of an energy functional. It is the case, for instance, in the nonrelativistic Hartree-Fock model =-=[40]-=-. Similarly, the self-consistent equation for the vacuum has a variational interpretation: it is satisfied by a minimizer of the Bogoliubov-Dirac-Fock (BDF) energy functional. This functional was firs... |

53 |
The index of a pair of projections
- Avron, Seiler, et al.
- 1994
(Show Context)
Citation Context ...P 0 )Q(1 − P 0 ) and −P 0 QP 0 are non-negative trace-class operators. We now use the proof of [1, Theorem 4.1]. Since Q ∈ S2, we infer Q 3 ∈ S1 and so (P,P 0 ) is a Fredholm pair, in the language of =-=[1]-=-. Therefore, tr(Q 3 ) is an integer and satisfies tr(Q 3 ) = tr(Q 2n+1 ) for all n ≥ 1. Now we have (P − P 0 ) 3 = (P − P 0 ) 2 P − (P − P 0 ) 2 P 0 = P(P − P 0 )P + P 0 (P − P 0 )P 0 Applying this re... |

44 |
Error bound for the Hartree-Fock energy of atoms and molecules
- Bach
- 1992
(Show Context)
Citation Context ... operators, which can be interpreted as one-particle density matrices of quasi-free states. This kind of extension is standard for mean-field models depending only on the one-body density matrix (see =-=[40, 3, 5]-=-). In the whole paper, we assume that the nuclear charge density n = −∆ϕ/4π belongs to the Hilbert space C = { f ∈ L 2 (R 3 , R), D(f,f) < ∞ } , where ∫ f(k)g(k) D(f,g) = 4π dk. We will choose the f... |

42 |
Variational Principle for Many-Fermion Systems
- Lieb
- 1981
(Show Context)
Citation Context ...t operators, which can be interpreted as one-particle density matrices of quasifree states. This kind of extension is standard for mean-field models depending only on the one-body density matrix (see =-=[38, 3, 5]-=-). In the whole paper, we assume that the nuclear charge density n = −∆ϕ/4π belongs to the Hilbert space where C = { f ∈ L 2 (R 3 , R), D(f,f) < ∞ } , ∫ f(k)g(k) D(f,g) = 4π dk. |k| 2 We will choose... |

41 |
The radiation theories of Tomonaga
- Dyson
- 1949
(Show Context)
Citation Context ...wn since the very beginning of QED [8, 18, 13, 31, 29] that the density ρ αϕ associated with Q αϕ = P αϕ −P 0 is never well-defined, since it diverges pointwise logarithmically. In physics literature =-=[12, 28, 9, 10]-=- (see also the books [20, 23, 32]), a procedure called charge renormalization aims at extracting the main information from ραϕ and a renormalized density ρ αϕ ren is used to replace the ill-defined de... |

30 | On the Stability of the relativistic electron-positron field
- Bach, Barbaroux, et al.
- 1999
(Show Context)
Citation Context ...β (1) where α = (α1,α2,α3) and k=1 ( ) I2 0 β = , αk = 0 −I2 ( ) 0 σk , σk 0 with σ1 = ( ) 0 1 , σ2 = 1 0 ( ) ( ) 0 −i 1 0 , σ3 = . i 0 0 −1 We follow here the notation of Thaller’s book [30], and of =-=[2]-=-. We have chosen a system of units such that � = c = 1, and also such that the mass me of the electron is normalized to 1. The operator D 0 acts on 4-spinors, i.e. functions Ψ ∈ H := L 2 (R 3 , C 4 ).... |

23 | Generalized Hartree-Fock theory and the Hubbard model
- Bach, Lieb, et al.
- 1994
(Show Context)
Citation Context ... operators, which can be interpreted as one-particle density matrices of quasi-free states. This kind of extension is standard for mean-field models depending only on the one-body density matrix (see =-=[40, 3, 5]-=-). In the whole paper, we assume that the nuclear charge density n = −∆ϕ/4π belongs to the Hilbert space C = { f ∈ L 2 (R 3 , R), D(f,f) < ∞ } , where ∫ f(k)g(k) D(f,g) = 4π dk. We will choose the f... |

23 | Solutions of the Dirac-Fock equations for atoms and molecules
- Esteban, Séré
- 1999
(Show Context)
Citation Context ...th the constraint −P ≤ γ ≤ 1−P. This energy is easily seen to be non intrinsic and unbounded from below: infP inf−P ≤γ≤1−P E [2] P (γ) = −∞ (see the properties of the Dirac-Fock functional defined in =-=[11]-=-). Therefore, a procedure which takes the form supP inf−P ≤γ≤1−P E [2] (γ) inspired by [24] was considered in [2], leading to the solution P = P αϕ . In fact, as explained in [5], the vacuum polarizat... |

23 |
Spinor representation of infinite orthogonal groups
- Shale, Stinespring
- 1965
(Show Context)
Citation Context ...ace F. This state has to be a solution to the analogue of (69) |ΩP | F = 1, aP(f)ΩP = 0 and bP(f)ΩP = 0 (71) for all f ∈ HΛ. The answer is given by the celebrated Theorem 4 (Shale-Stinespring Theorem =-=[52]-=-). There exists a dressed vacuum ΩP in the Fock space F satisfying (71) if and only if P − P 0 is a HilbertSchmidt operator. In this case, ΩP is unique up to a phase factor. There are many proofs of t... |

22 |
From quantum electrodynamics to mean field theory: II. Variational stability of the vacuum of quantum electrodynamics in the mean-field approximation
- Chaix, Iracane, et al.
- 1989
(Show Context)
Citation Context ... part of the spectrum of the free Dirac operator D 0 . In the presence of an external field, these virtual particles react and the vacuum becomes polarized. In this paper, following Chaix and Iracane =-=[5]-=-, we consider the Bogoliubov-Dirac-Fock model, which is derived from QED. The corresponding BDF-energy takes the polarization of the vacuum into account and is bounded from below. A BDF-stable vacuum ... |

21 |
Discussion of the infinite distribution of electrons in the theory of the positron
- Dirac
- 1934
(Show Context)
Citation Context ...en P is chosen to be P αϕ . Remark that P αϕ is the projector which is obtained after the first iteration of the fixed-point algorithm if we start at P 0 . It is known since the very beginning of QED =-=[8, 18, 13, 31, 29]-=- that the density ρ αϕ associated with Q αϕ = P αϕ −P 0 is never well-defined, since it diverges pointwise logarithmically. In physics literature [12, 28, 9, 10] (see also the books [20, 23, 32]), a p... |

21 |
On Bogoliubov transformations for systems of relativistic charged particles
- Ruijsenaars
- 1977
(Show Context)
Citation Context ... Fock space F satisfying (71) if and only if P − P 0 is a HilbertSchmidt operator. In this case, ΩP is unique up to a phase factor. There are many proofs of this Theorem in the literature, see, e.g., =-=[54, 36, 47]-=- and the references in [20]. This result explains why we assumed in the previous section that P − P 0 ∈ S2(HΛ). Notice that ΩP can be expressed as a rotation of the bare vacuum in the Fock space, ΩP =... |

20 |
The regular external field problem in quantum electrodynamics
- Klaus, Scharf
- 1977
(Show Context)
Citation Context ...Γ is trace-class, since |x−y| 1 −α(ρQ−Zn)∗ |x−y| 1/2 and Γ(x,y) |x−y| 1/2 are in S2 by (13). Let us now define D = D |·| = D0 + α(ρQ − Zn) ∗ 1 |·| and P ′ = χ (−∞;0)(D). By the result of Klaus-Scharf =-=[21]-=- (see also [17] and the proof of Theorem 4), it is known that P ′ − P 0 ∈ S2(HΛ). Thus P ′ DΓP ′ = DP ′ ΓP ′ ∈ S1(HΛ) since Γ ∈ S P0 ′ 1 = SP 1 by Theorem 1, and D is bounded by the proof of Lemma 3. ... |

20 |
The S-Matrix in Quantum Electrodynamics
- Dyson
- 1949
(Show Context)
Citation Context ...wn since the very beginning of QED [8, 18, 13, 31, 29] that the density ρ αϕ associated with Q αϕ = P αϕ −P 0 is never well-defined, since it diverges pointwise logarithmically. In physics literature =-=[12, 28, 9, 10]-=- (see also the books [20, 23, 32]), a procedure called charge renormalization aims at extracting the main information from ραϕ and a renormalized density ρ αϕ ren is used to replace the ill-defined de... |

17 |
Bemerkungen zur Diracschen Theorie des Positrons
- Heisenberg
- 1934
(Show Context)
Citation Context ...en P is chosen to be P αϕ . Remark that P αϕ is the projector which is obtained after the first iteration of the fixed-point algorithm if we start at P 0 . It is known since the very beginning of QED =-=[8, 18, 13, 31, 29]-=- that the density ρ αϕ associated with Q αϕ = P αϕ −P 0 is never well-defined, since it diverges pointwise logarithmically. In physics literature [12, 28, 9, 10] (see also the books [20, 23, 32]), a p... |

15 | Self-consistent solution for the polarized vacuum in a no-photon QED model, preprint, arXiv: math-ph/0404047
- Hainzl, Lewin, et al.
(Show Context)
Citation Context ...is possible to remove the smallness assumption on the potential, but for this purpose the constructive fixed-point approach must be replaced by a direct – and non-constructive – minimization argument =-=[28]-=-. The regularity assumption cannot be dropped: this is a well known phenomenon in QED when P 0 is chosen as reference for normal ordering (see e.g., [35]). But this regularity is not really a restrict... |

14 | Nonrelativistic limit of the Dirac-Fock equations
- Esteban, Séré
(Show Context)
Citation Context ...ctional has critical points which are solutions of the Dirac-Fock equations [15, 44], but these critical points have an infinite Morse index, and the rigorous definition of a ground state is delicate =-=[16, 17]-=-. 3The second problem with Dirac-Fock is its physical derivation: one would like to interpret this model as a variational approximation of Quantum Electrodynamics (QED), which is believed to be the e... |

14 |
Über die Elektrodynamik des Vakuums auf Grund der Quantentheorie des Elektrons
- Weisskopf
- 1936
(Show Context)
Citation Context ...ally referred to as mass renormalization. In fact in [16, Section 3] these subtractions were justified by the first two guiding principles (denoted as W1 and W2) formulated and justified by Weisskopf =-=[33]-=-. The definition of the normal ordering depends on the annihilation and creation operators and it thus depends on the projector P which is used. Now we want to express H in terms of : − :P for some P.... |

13 |
Théorie du positron. Solvay report
- Dirac
- 1934
(Show Context)
Citation Context ...spectrum of D 0 is not bounded from below is the source of many difficulties in Relativistic Quantum Mechanics. To explain why a free electron does not dissolve into the lower continuum, Dirac’s idea =-=[13, 14]-=- was to postulate that in the absence of external field, the vacuum contains infinitely many virtual electrons which completely fill up the negative part of the spectrum of D 0 . This Dirac Sea should... |

12 |
Quantum Electrodynamics II. Vacuum Polarization and SelfEnergy
- Schwinger
- 1949
(Show Context)
Citation Context ...wn since the very beginning of QED [8, 18, 13, 31, 29] that the density ρ αϕ associated with Q αϕ = P αϕ −P 0 is never well-defined, since it diverges pointwise logarithmically. In physics literature =-=[12, 28, 9, 10]-=- (see also the books [20, 23, 32]), a procedure called charge renormalization aims at extracting the main information from ραϕ and a renormalized density ρ αϕ ren is used to replace the ill-defined de... |

11 | Renormalization of the regularized relativistic electronpositron field
- Lieb, Siedentop
(Show Context)
Citation Context ...see the resolution of the Hartree equations in [34]. For the determination of projectors describing the vacuum, the fixed-point approach has been used for the first time by E.H. Lieb and H. Siedentop =-=[22]-=-. We use the Banach fixed-point theorem as in [22], but our model is different and the necessary estimates are much more delicate. A model also inspired by [5] was studied by V. Bach, J.-M. Barbaroux,... |

9 | The sharp bound on the stability of the relativistic electron-positron field in Hartree-Fock approximation - Hundertmark, Röhrl, et al. |

9 |
The relativistic self-consistent
- Swirles
- 1935
(Show Context)
Citation Context ... problems associated with standard relativistic quantum chemistry calculations. In these calculations, electrons near heavy nuclei are usually treated, in first approximation, by the Dirac-Fock model =-=[53]-=-, a variant of Hartree-Fock in which the kinetic energy operator −∆/2 is replaced by the free Dirac operator D 0 . This approach gives results that are in excellent agreement with experimental data [3... |

8 | On the Hartree-Fock equations of the electron/positron field
- Barbaroux, Farkas, et al.
(Show Context)
Citation Context ...ximizes, in the Hartree-Fock approximation, the ground-state energy of the normal-ordered Hamiltonian. From a mathematical viewpoint, Mittleman’s max-min principle has been investigated in the papers =-=[4, 19, 7, 6]-=-. In the case of zero or one electron [4, 19], it works very well and one shows that the projector P αϕ of the Furry picture is the optimal reference. But it seems, from the counterexample given in [6... |

8 |
Une Méthode de Champ Moyen Relativiste et Application à l’Etude du Vide de l’Electrodynamique Quantique
- Chaix
- 1990
(Show Context)
Citation Context ... particular case. This equation can be interpreted as the Euler-Lagrange equation associated with the minimization of the Bogoliubov-Dirac-Fock (BDF) energy defined by Chaix and Iracane [5] (see also =-=[7]-=-). The study of this functional was our original motivation for solving (4). In this paper, we first give a rigorous meaning to formula (4) and then show the existence of a solution by a fixed-point a... |

8 | Vacuum Polarization in Fock Space - Klaus, Scharf |

8 |
Theory of relativistic effects on atoms: Configuration-space hamiltonian. Phys
- Mittleman
- 1981
(Show Context)
Citation Context ...s of [2] have neglected the terms describing the polarization of the vacuum and, as this is explained in [5], the resulting energy is not bounded from below. Therefore a max-min procedure inspired by =-=[24]-=- was considered, leading to the solution P = P αϕ . Since we keep the vacuum polarization terms, the BDF energy is bounded from below and we can define the polarized vacuum as the minimum of this ener... |

8 |
Theory of positron production in heavyion collision
- Reinhardt, Müller, et al.
- 1981
(Show Context)
Citation Context ...is important for high-Z atoms [43] and even plays a crucial role in muonic atoms [18, 24]. It also explains the production of electron-positron pairs, observed experimentally in heavy ions collisions =-=[1, 46, 37, 50, 20]-=-. When the external field is not too strong, a good approximation (called the Furry picture [21]) is to define the polarized vacuum as the projector P αϕ := χ (−∞;0)(D αϕ ). Note that in reality, the ... |

7 | A max-min principle for the ground state of the DiracFock functional
- Esteban, Séré
(Show Context)
Citation Context ...ctional has critical points which are solutions of the Dirac-Fock equations [15, 44], but these critical points have an infinite Morse index, and the rigorous definition of a ground state is delicate =-=[16, 17]-=-. 3The second problem with Dirac-Fock is its physical derivation: one would like to interpret this model as a variational approximation of Quantum Electrodynamics (QED), which is believed to be the e... |

7 | Non-Perturbative Mass and Charge Renormalization in Relativistic no-photon Quantum
- Hainzl, Siedentop
(Show Context)
Citation Context ...n is used to replace the ill-defined density in the Hamiltonian D ren Qαϕ = Dαϕ + αρ αϕ ren ∗ 1 − αQ(x,y) | · | |x − y| . 3This procedure has been recently clarified by C. Hainzl and H. Siedentop in =-=[16]-=-, where it is in addition shown that Dren Qαϕ is then a well defined self-adjoint operator, under some reasonable assumption on ϕ. Some interesting features of ρ αϕ ren, in the case of strong external... |

7 |
Nonregularity of the Coulomb potential in quantum electrodynamics
- Klaus
- 1980
(Show Context)
Citation Context ...– and non-constructive – minimization argument [28]. The regularity assumption cannot be dropped: this is a well known phenomenon in QED when P 0 is chosen as reference for normal ordering (see e.g., =-=[35]-=-). But this regularity is not really a restriction from the point of view of physics: point-like nuclei do not exist in nature. In [4], the operator D0Q is assumed to be trace class, so that the expre... |

7 |
Polarization effects in the positron theory
- Uehling
- 1935
(Show Context)
Citation Context ...en P is chosen to be P αϕ . Remark that P αϕ is the projector which is obtained after the first iteration of the fixed-point algorithm if we start at P 0 . It is known since the very beginning of QED =-=[8, 18, 13, 31, 29]-=- that the density ρ αϕ associated with Q αϕ = P αϕ −P 0 is never well-defined, since it diverges pointwise logarithmically. In physics literature [12, 28, 9, 10] (see also the books [20, 23, 32]), a p... |

7 | Stability of the relativistic electron-positron field of atoms in Hartree-Fock approximation: Heavy elements
- Brummelhuis, Röhrl, et al.
- 2001
(Show Context)
Citation Context ...[22], but our model is different and the necessary estimates are much more delicate. A model also inspired by [5] was studied by V. Bach, J.-M. Barbaroux, B. Helffer and H. Siedentop in [2] (see also =-=[4]-=-) . Although our model coincide with [2] when there is no external potential (i.e. Z = 0), it is very different in the presence of an external electrostatic field (i.e. Z = 0). Indeed, the authors of... |

6 |
Some Physical Consequences of Vacuum Polarization
- Foldy, Eriksen
- 1954
(Show Context)
Citation Context ...the calculation of the Lamb shift for the ordinary hydrogen atom (comparing to other electrodynamic phenomena), but it is important for high-Z atoms [43] and even plays a crucial role in muonic atoms =-=[18, 24]-=-. It also explains the production of electron-positron pairs, observed experimentally in heavy ions collisions [1, 46, 37, 50, 20]. When the external field is not too strong, a good approximation (cal... |

6 |
Particle interpretation for external field problems
- Fierz, Scharf
- 1979
(Show Context)
Citation Context ...is important for high-Z atoms [43] and even plays a crucial role in muonic atoms [18, 24]. It also explains the production of electron-positron pairs, observed experimentally in heavy ions collisions =-=[1, 46, 37, 50, 20]-=-. When the external field is not too strong, a good approximation (called the Furry picture [21]) is to define the polarized vacuum as the projector P αϕ := χ (−∞;0)(D αϕ ). Note that in reality, the ... |

6 | Trace Ideals and their Applications. Vol 35 - Simon - 1979 |

5 |
Form perturbation of the second quantized Dirac field
- Helffer, Siedentop
- 1998
(Show Context)
Citation Context ... writing down the formal unregularized no-photon Hamiltonian ∫ Hur = dxΨ ∗ (x)D αϕ Ψ(x) + α ∫ 2 ∫ dx dy Ψ∗ (x)Ψ(x)Ψ∗(y)Ψ(y) , (62) |x − y| which acts on the Fock space F. As explained for instance in =-=[30, 17]-=-, the free vacuum may not belong to the domain of this formally defined operator. Therefore, the formal energy of the free vacuum is substracted from (62) by a procedure which is called “normal orderi... |

5 |
Linear modifications in the Maxwell field equations
- Serber
(Show Context)
Citation Context |

5 |
On the Theory of the Electron and
- Furry, Oppenheimer
- 1934
(Show Context)
Citation Context |

5 | On the Vacuum Polarization Density caused by an External Field, Ann. Henri Poincaré 5
- Hainzl
- 2004
(Show Context)
Citation Context ... Dren Qαϕ is then a well defined self-adjoint operator, under some reasonable assumption on ϕ. Some interesting features of ρ αϕ ren, in the case of strong external fields, were obtained by Hainzl in =-=[15]-=-. We do not want to give a precise definition of ρ αϕ ren here and we refer the reader to [16, 15, 25]. For the same reason as for ραϕ , it can be seen that if P is a solution to equation (4), then ρQ... |

5 | Trace Ideals and their Applications. Vol 35 of London Mathematical Society Lecture Notes Series - Simon - 1979 |

4 |
The electromagnetic shift of energy levels
- French, Weisskopf
- 1949
(Show Context)
Citation Context |

4 |
On Bound States and Scattering in Positron Theory
- Furry
(Show Context)
Citation Context ... production of electron-positron pairs, observed experimentally in heavy ions collisions [1, 46, 37, 50, 20]. When the external field is not too strong, a good approximation (called the Furry picture =-=[21]-=-) is to define the polarized vacuum as the projector P αϕ := χ (−∞;0)(D αϕ ). Note that in reality, the polarized vacuum modifies the electrostatic field, and the virtual electrons react to the correc... |

4 |
A symmetry theorem in the positron theory
- Furry
- 1937
(Show Context)
Citation Context ... matrices are traceless and the remaining terms are odd in η and vanish after integration. This can be easily generalized to ρ0,2k for all k, and is known as Furry’s Theorem in the physics literature =-=[22]-=-. • ρ2,0. We use here a method similar to what we have done above. We estimate for some ζ ∈ C ′ ∩ L 2 and Qζ := Q2,0ζ | Qζ(p,p)| ≤ (2π) −5/2 ≤ 4(2π) −5/2 ∫ +∞ −∞ ∫ +∞ −∞ ∫∫ dη ∫∫ dη E(η) | R(p,p1)... |

4 | On the theory of the electron and positÏve - Furry, Oppenheimer - 1934 |