  The solutions of a particle's Dirac equation contains a negative kinetic
energy (NKE) branch. Such an energy spectrum has an upper limit but no lower
limit, so that the system with this spectrum, called NKE system, is of negative
temperature. Fundamental formulas of statistical mechanics and thermodynamics
of NKE systems are presented. All the formulas have the same forms of those of
positive kinetic energy (PKE) systems. Almost all thermodynamic quantities,
except entropy and specific heat, have a contrary sign compared to those of PKE
systems. Specially, pressure is negative and its microscopic mechanism is
given. Entropy is always positive and Boltzmann entropy formula remains valid.
The three laws of thermodynamics remain valid, as long as the thermodynamic
quantities have a negative sign. Negative temperature Carnot engine can work
between two negative temperatures. Since the NKE levels need not be fully
filled, it is argued that the concept of Dirac's Fermion Sea can be totally
abandoned.
