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  • VARIATIONAL METHODS IN RELATIVISTIC QUANTUM

    MECHANICS

    MARIA J. ESTEBAN, MATHIEU LEWIN, AND ERIC SERE

    Abstract. This review is devoted to the study of stationary solutions of lin-ear and nonlinear equations from relativistic quantum mechanics, involving theDirac operator. The solutions are found as critical points of an energy func-tional. Contrary to the Laplacian appearing in the equations of nonrelativisticquantum mechanics, the Dirac operator has a negative continuous spectrumwhich is not bounded from below. This has two main consequences. First, theenergy functional is strongly indefinite. Second, the Euler-Lagrange equationsare linear or nonlinear eigenvalue problems with eigenvalues lying in a spectralgap (between the negative and positive continuous spectra). Moreover, sincewe work in the space domain R3, the Palais-Smale condition is not satisfied.For these reasons, the problems discussed in this review pose a challenge in theCalculus of Variations. The existence proofs involve sophisticated tools fromnonlinear analysis and have required new variational methods which are nowapplied to other problems.

    In the first part, we consider the fixed eigenvalue problem for models of afree self-interacting relativistic particle. They allow to describe the localizedstate of a spin-1/2 particle (a fermion) which propagates without changingits shape. This includes the Soler models, and the Maxwell-Dirac or Klein-Gordon-Dirac equations.

    The second part is devoted to the presentation of min-max principles al-lowing to characterize and compute the eigenvalues of linear Dirac operatorswith an external potential, in the gap of their essential spectrum. Many con-sequences of these min-max characterizations are presented, among them anew kind of Hardy-like inequalities and a stable algorithm to compute theeigenvalues.

    In the third part we look for normalized solutions of nonlinear eigenvalueproblems. The eigenvalues are Lagrange multipliers, lying in a spectral gap.We review the results that have been obtained on the Dirac-Fock model whichis a nonlinear theory describing the behavior of N interacting electrons in anexternal electrostatic field. In particular we focus on the problematic definitionof the ground state and its nonrelativistic limit.

    In the last part, we present a more involved relativistic model from Quan-tum Electrodynamics in which the behavior of the vacuum is taken into ac-count, it being coupled to the real particles. The main interesting feature of

    this model is that the energy functional is now bounded from below, providingus with a good definition of a ground state.

    Date: June 21, 2007.1991 Mathematics Subject Classification. 49S05, 35J60, 35P30, 35Q75, 81Q05, 81V70, 81V45,

    81V55.Key words and phrases. Relativistic quantum mechanics, Dirac operator, variational methods,

    critical points, strongly indefinite functionals, nonlinear eigenvalue problems, ground state, non-relativistic limit, Quantum Chemistry, mean-field approximation, Dirac-Fock equations, Hartree-Fock equations, Bogoliubov-Dirac-Fock method, Quantum Electrodynamics.

    1

  • 2 MARIA J. ESTEBAN, MATHIEU LEWIN, AND ERIC SERE

    Contents

    Introduction 2Notations and basic properties of the Dirac operator 71. Nonlinear Dirac equations for a free particle 81.1. Soler models: existence by O.D.E. techniques 91.2. Soler models: existence by variational techniques 101.3. Existence of solutions by perturbation theory 111.4. Nonlinear Dirac equations in the Schwarzschild metric 121.5. Solutions of the Maxwell-Dirac equations 121.6. Nonlinear Dirac evolution problems 172. Linear Dirac equations for an electron in an external field 192.1. A variational characterization of the eigenvalues of Dc + V 202.2. Numerical method based on the min-max formula 252.3. New Hardy-like inequalities 272.4. The nonrelativistic limit 272.5. Introduction of a constant external magnetic field 283. The Dirac-Fock equations for atoms and molecules 303.1. The (non-relativistic) Hartree-Fock equations 303.2. Existence of solutions to the Dirac-Fock equations 313.3. Nonrelativistic limit and definition of the Dirac-Fock ground state 344. The mean-field approximation in Quantum Electrodynamics 364.1. Definition of the free vacuum 404.2. The Bogoliubov-Dirac-Fock model 424.3. Global minimization of EBDF: the polarized vacuum. 454.4. Minimization of EBDF in charge sectors 474.5. Neglecting Vacuum Polarization: Mittlemans conjecture 50References 53

    Introduction

    In this paper, we present various recent results concerning some linear and non-linear variational problems in relativistic quantum mechanics, involving the Diracoperator.

    Dirac derived his operator in 1928 [44], starting from the usual classical expres-sion of the energy of a free relativistic particle of momentum p R3 and massm

    (1) E2 = c2|p|2 +m2c4

    (c is the speed of light), and imposing the necessary relativistic invariances. Bymeans of the usual identification

    p i~where ~ is Plancks constant, he found that an adequate observable for describ-ing the energy of the free particle should therefore be a self-adjoint operator Dcsatisfying the equation

    (2) (Dc)2 = c2~2 +m2c4.

  • VARIATIONAL METHODS IN RELATIVISTIC QUANTUM MECHANICS 3

    Taking the causality principle into account, Dirac proposed to look for a localoperator which is first order with respect to p = i~:

    (3) Dc = ic~ +mc2 = ic~3

    k=1

    kk + mc2,

    where 1, 2, 3 and are hermitian matrices which have to satisfy the followinganticommutation relations:

    (4)

    k + k = 2 k 1,k + k = 0,

    2 = 1.

    It can be proved [158] that the smallest dimension in which (4) can take place is 4(i.e. 1, 2, 3 and should be 4 4 hermitian matrices), meaning that Dc hasto act on L2(R3,C4). The usual representation in 2 2 blocks is given by

    =

    (I2 00 I2

    ), k =

    (0 kk 0

    )(k = 1, 2, 3) ,

    where the Pauli matrices are defined as

    1 =

    (0 11 0

    ), 2 =

    (0 ii 0

    ), 3 =

    (1 00 1

    ).

    By (1), the time-dependent Dirac equation describing the evolution of a freeparticle is

    (5) i~

    t = Dc.

    This equation has been successfully used in physics to describe relativistic particleshaving a spin 1/2.

    The main unusual feature of the Dirac equation is the spectrum of Dc which isnot bounded from below:

    (6) (Dc) = (,mc2] [mc2,).Compared with non-relativistic theories in which the Schrodinger operator/(2m)appears instead of Dc, property (6) leads to important physical, mathematical andnumerical difficulties. Indeed, if one simply replaces /(2m) by Dc in the ener-gies or operators which are commonly used in the non-relativistic case, one obtainsenergies which are not bounded from below.

    Although there is no observable electron of negative energy, the negative spec-trum plays an important role in physics. Dirac himself suspected that the negativespectrum of his operator could generate new interesting physical phenomena, andhe proposed in 1930 the following interpretation [45, 46, 47]:

    We make the assumption that, in the world as we know it, nearlyall the states of negative energy for the electrons are occupied, withjust one electron in each state, and that a uniform filling of all thenegative-energy states is completely unobservable to us. [47]

    Physically, one therefore has to imagine that the vacuum (called the Dirac sea) isfilled with infinitely many virtual particles occupying the negative energy states.With this conjecture, a real free electron cannot be in a negative state due to thePauli principle which forbids it to be in the same state as a virtual electron of theDirac sea.

  • 4 MARIA J. ESTEBAN, MATHIEU LEWIN, AND ERIC SERE

    With this interpretation, Dirac was able to conjecture the existence of holes inthe vacuum, interpreted as anti-electrons or positrons, having a positive chargeand a positive energy. The positron was discovered in 1932 by Anderson [3]. Diracalso predicted the phenomenon of vacuum polarization: in presence of an electricfield, the virtual electrons are displaced, and the vacuum acquires a nonconstantdensity of charge. All these phenomena are now well known and well establishedin physics. They are direct consequences of the existence of the negative spectrumof Dc, showing the crucial role played by Diracs discovery.

    Actually, in practical computations it is quite difficult to deal properly with theDirac sea. As a consequence the notion of ground state (state of lowest energywhich is supposed to be the most stable for the system under consideration)is problematic for many of the models found in the literature. Numerically, theunboundedness from below of the spectrum is also the source of important practicalissues concerning the convergence of the considered algorithms, or the existence ofspurious (unphysical) solutions.

    Diracs interpretation of the negative energies will be an implicit assumption inall this review in the sense that we shall (almost) always look for positive energysolutions for the electrons. In the last section, we present a model from QuantumElectrodynamics (QED) in which the nonlinear behavior of the Dirac sea will befully taken into account.

    Mathematically, most of the energy functionals that we shall consider are stronglyindefinite: they are unbounded from below and all their critical points have an in-finite Morse index. Note that the mathematical methods allowing to deal withstrongly indefinite functionals have their origin in the works of P. Rabinowitz con-cerning the study of nonlinear waves [133] and also the study of periodic solutionsof Hamiltonian systems [134]. Many results have followed these pioneering works,and powerful theories have been devised, in particular in the field of periodic orbitsof Hamiltonian