根据狄拉克的想法，真空由无限多的虚电子组成，这些虚电子完全填满了自由狄拉克算符$D^0$的负谱部分。在存在外场的情况下，这些虚粒子发生反应，真空变得极化。本文在Chaix和Iracane（{ \it J. Phys. B}，22，3791- 3814，1989）的基础上，考虑了由无光子QED导出的Bogoliubov-Dirac-Fock模型。相应的BDF能量考虑了真空的极化，并且从下面有界。一个BDF稳定的真空被定义为这个能量的最小值。如果它存在，这样的极小是一个自洽方程的解。我们证明了存在一个唯一的极小的BDF-能量在存在一个外部静电场，通过一个固定点的方法。这种极小化被解释为极化真空。
英文摘要：
According to Dirac's ideas, the vacuum consists of infinitely many virtual electrons which completely fill up the negative 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 ({\it J. Phys. B}, 22, 3791--3814, 1989), we consider the Bogoliubov-Dirac-Fock model, which is derived from no-photon QED. The corresponding BDF-energy takes the polarization of the vacuum into account and is bounded from below. A BDF-stable vacuum is defined to be a minimizer of this energy. If it exists, such a minimizer is solution of a self-consistent equation.
We show the existence of a unique minimizer of the BDF-energy in the presence of an external electrostatic field, by means of a fixed-point approach. This minimizer is interpreted as the polarized vacuum.
We review some results of our paper arXiv:1602.05171v2 on the "nonlinear quasifree approximation" to the many-body Schrödinger dynamics of Bose gases. In that paper, we derive, with the help of this approximation, the time-dependent Hartree-Fock-Bogoliubov (HFB) equations, providing an approximate description of the dynamics of quantum fluctuations around a Bose-Einstein condensate and study properties of these equations.
We consider the gravitational collapse for neutron stars in the Hartree-Fock-Bogoliubov theory. We prove that when the number particle becomes large and the gravitational constant is small such that the attractive interaction strength approaches the Chandrasekhar limit mass slowly, the minimizers develop a universal blow-up profile. It is given by the Lane-Emden solution.
We prove the existence of minimizers for Hartree-Fock-Bogoliubov (HFB) energy functionals with attractive two-body interactions given by Newtonian gravity. This class of HFB functionals serves as model problem for self-gravitating relativistic Fermi systems, which are found in neutron stars and white dwarfs. Furthermore, we derive some fundamental properties of HFB minimizers such as a decay estimate for the minimizing density. A decisive feature of the HFB model in gravitational physics is its failure of weak lower semicontinuity. This fact essentially complicates the analysis compared to the…
In this article, we prove the global well-posedness of the time-dependent Hartree-Fock-Bogoliubov (TDHFB) equations in $\mathbb{R}^{1+1}$ with two-body interaction potentials of the form $N^{-1}v_N(x) = N^{β-1} v(N^βx)$ where $v$ is a sufficiently regular radial function $v \in L^1(\mathbb{R})\cap C^\infty(\mathbb{R})$. In particular, using methods of dispersive PDEs similar to the ones used in Grillakis and Machedon, Comm. PDEs., (2017), we are able to show for any scaling parameter $β>0$ the TDHFB equations are globally well-posed in some Strichartz-type spaces independent of $N$, cf. (Bach …
We formulate the Bogoliubov variational principle in a mathematical framework similar to the generalized Hartree-Fock theory. Then we analyze the Bogoliubov theory for bosonic atoms in details. We discuss heuristically why the Bogoliubov energy should give the first correction to the leading energy of large bosonic atoms.
In this article, we use quasifree reduction to derive the time-dependent Hartree-Fock-Bogoliubov (HFB) equations describing the dynamics of quantum fluctuations around a Bose-Einstein condensate in $\mathbb R^d$. We prove global well-posedness for the HFB equations for sufficiently regular pair interaction potentials, and establish key conservation laws. Moreover, we show that the solutions to the HFB equations exhibit a symplectic structure, and have a form reminiscent of a Hamiltonian system. In particular, this is used to relate the HFB equations to the HFB eigenvalue equations encountered …
We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree-Fock and Hartree-Fock-Bogoliubov type equations, which describe the evolution of attractive fermionic systems (e. g. white dwarfs). Our main results are twofold: First, we extend the recent blowup result of [Hainzl and Schlein, Comm. Math. Phys. \textbf{287} (2009), 705--714] to Hartree-Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree-Fock-Bogoliubov mode…
We prove local in time, uniform in $N$, estimates for the solutions $ϕ$, $Λ$ and $Γ$ of a coupled system of Hartree-Fock-Bogoliubov type with interaction potential $v_N(x-y) =N^{3 β} v(N^β(x-y))$, with $β<1$ and $v$ a Schwartz function (satisfying additional technical requirements). The initial conditions are general functions in a Sobolev-type space, and the expected correlations in $Λ$ develop dynamically in time. As shown in our previous work, as well as the work of J. Chong, (both in the case $β<2/3$), using the conserved quantities of the system of equations, this type of local in time es…
Canonical transformations (Bogoliubov transformations) for fermions with an infinite number of degrees of freedom are studied within a calculus of superanalysis. A continuous representation of the orthogonal group is constructed on a Grassmann module extension of the Fock space. The pull-back of these operators to the Fock space yields a unitary ray representation of the group that implements the Bogoliubov transformations.
We provide general conditions for which bosonic quadratic Hamiltonians on Fock spaces can be diagonalized by Bogoliubov transformations. Our results cover the case when quantum systems have infinite degrees of freedom and the associated one-body kinetic and paring operators are unbounded. Our sufficient conditions are optimal in the sense that they become necessary when the relevant one-body operators commute.
In this letter we present a new expression for the overlaps of wavefunctions in Hartree-Fock-Bogoliubov based theories. Starting from the Pfaffian formula by Bertsch et al (Phys. Rev. Lett. 108,042505 (2012)), an exact and computationally stable formula for overlaps is derived. We illustrate the convenience of this new formulation with a numerical application in the context of the particle-number projection method. This new formula allows for substantially increased precision and versatility in chemical, atomic, and nuclear physics applications, particularly for methods dealing with superfluid…
The derivation of effective evolution equations is central to the study of non-stationary quantum many-body sytems, and widely used in contexts such as superconductivity, nuclear physics, Bose-Einstein condensation and quantum chemistry. We reformulate the Dirac-Frenkel approximation principle in terms of reduced density matrices, and apply it to fermionic and bosonic many-body systems. We obtain the Bogoliubov-de-Gennes and Hartree-Fock-Bogoliubov equations, respectively. While we do not prove quantitative error estimates, our formulation does show that the approximation is optimal within the…
