Relativistic quantum mechanics on Fock space

Egon Marx

Malcolm McMillan

Ophelliam Loiselet

Abstract Quantum mechanics describes multi-electronic systems by using probabilistic theories. All of these theories provide a probabilistic conception of quantum mechanics, with an astonishing accuracy (for instance the Helium atom ground state is computed with more than 20 digits of accuracy) but unfortunately seems to be a bit “disconnected” with macro-scale physics. Indeed, the real observable world is for us a 3D+time Minkowski space but quantum mechanics is expressed in multi-dimensional Hilbert space: which is not intuitively physical but seems to be purely mathematical. In this article we try to construct a relativistic equation that gives an approximately correct description of atomic physics in the spacetime (i.e the “real space”) for 2 electrons systems: i.e a 2-electrons Dirac equation. The starting point is to use the wave mechanics concept of De Broglie and Schrödinger: considering the electron as a matter-wave. For this we start from the Hartree equation, which gives already a spacetime description of each electron wave function. Then, we switch to Hartree-Fock representation, which gives birth to the exchange potential, that can be also expressed in spacetime. By extending the concept of exchange potential to the relativistic 4-exchange: we obtain a relativistic 2-electrons spacetime Dirac-Hartree-Fock (STDHF) equation (i.e in Minkowski space). That equation takes into account relativistic effects but not correlation effects, which is the difficulty of this study. Then we will show the equivalence of the 2-electrons STDHF equation with the Hartree-Fock equation in non-relativistic, stationary and scalar regime. Electron correlation issues will be discussed as well.

Walter Greiner

Sol Wieder

Ophelliam Loiselet

Quantum mechanics describes multi-electronic system by using non-relativistic probabilistic theories (non-relativistic many-body Hamiltonian, Bethe-Salpeter equation, Breit equation, density functional theory and so on).All of these theories are approximations and provide a probabilistic conception of quantum mechanics, which prevents it to be compatible with general relativity (or "macro scale" physics).Indeed, the real observable world (or Minkowski space) is for us a 3D+time space but quantum mechanics is expressed in multi-dimensional Hilbert space, which is not physical but purely mathematical.In this article we try to make a relativistic equation that gives a correct description of atomic physics in the real three-dimensional space for N ≥ 2 electrons systems: i.e a N-electrons Dirac equation.The starting point is to use the wave mechanics concept of De Broglie and Schrödinger: considering the electron as a mini oscillating cloud or matter-wave.For this we start from the Hartree equation, which gives already a 3D space description of each electron wave function.Then we jump into Hartree-Fock theory, that takes into account the electron exchange and can be also expressed in 3D, thanks to Slater-Condon rules.By extending the concept of exchange potential to a relativistic 4-exchange potential, and doing the same for the electron-electron repulsion potential, we have obtained a relativistic N electrons Dirac equation, in 3D+time space, including new relativistic interactions.The non-relativistic equivalence to Hartree-Fock equations will be demonstrated, which shows that the equation is already correct to describe light atoms.