N-electrons Dirac equation in Minkowski space
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.
•Jul 3, 2024•0 citations