  In the author's previous works, it is derived from the Dirac equation that
particles can have negative kinetic energy (NKE) solutions, and they should be
treated on an equal footing as the positive kinetic energy (PKE) solutions.
More than one NKE particles can make up a stable system by means of
interactions between them and such a system has necessarily negative
temperature. Thus, many-body theories for NKE systems are desirable. In this
work, the many-body theories for NKE systems are presented. They are
Thomas-Fermi method, Hohenberg-Kohn theorem, Khon-Sham self-consistent
equations, and Hartree-Fock self-consistent equations. They are established
imitating the theories for PKE systems. In each theory, the formalism of both
zero temperature and finite negative temperature are given. In order to verify
that tunneling electrons are of NKE and real momentum, an experiment scenario
is suggested that lets PKE electrons collide with tunneling electrons.
