Relativistic model Hamiltonians for many-electron systems

2 Relativistic model Hamiltonians for many-electron systems 

To start with, consider systems consisting of N dynamical electrons and positrons and K fixed nuclei with Coulomb interactions between all pairs of particles. The clamped-nuclei approximation (the Bom-Oppenheimer approximation) may be legitimate because of the huge difference in mass between electrons and nuclei. Stability of matter means that the energy of such a model system is bounded from below by a negative constant times the number of particles E —C(N 4- K). Such a condition is necessary for some basic physical properties such as the existence of the thermodynamical limit. 

Recent investigations (Lieb et al. 1996) have concentrated on, as a first task, optimizing the stability result for the Chandrasekhar operator 

The mathematical problem associated with the Dirac Hamiltonian, i.e. the starting point of the relativistic theory of atoms, can be phrased in simple terms. The electron-positron field can have states of arbitrarily negative energy. As a general feature of the Dirac spectrum this instability occurs even in the case of extended nuclei and even in the absence of any nucleus (free Dirac spectrum), the energy is not bounded from below. This gives rise to the necessity of renormalization and well-established renormalization schemes have been around for many decades. Despite their successful applications in physics, we may ask instead whether there exist states that allow for positivity of the energy. 

A rigorous mathematical model for the relativistic electron-positron field in the Hartree-Fock approximation has been recently proposed (Bach et al. 1999). It describes electrons and positrons with the Coulomb interaction in second quantization in an external field using generalized Hartree-Fock states. It is based on the standard QED Hamiltonian neglecting the magnetic interaction A = 0 and is motivated by a physical treatment of this model (Chaix and Iracane 1989 Chaix et al. 1989). 

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