Electrons and holes in planar systems and condensation by scattering of electrons-holes pairs in high-TC materials

Quantum field-theory is developed for treating electrons and holes in planar systems. Non-unitary representations of Dirac equation in the plane are developed. These equations can be used for treating holes-electron pairing in high TC materials where in a non-relativistic approximation we have a relatively small energy band gap. It is shown that while holes or electrons satisfy separately Pauli-exclusion principle, with anti-commutation relations, the holes-electron pairs follow boson commutation relations. The distribution for electrons-holes pairs in planar systems is calculated for the case in which scattering effects between different pairs can be neglected. It is shown that for high surface density in which many pairs are included in an area with wave length dimension, Bose-Einstein condensation can occur. The ground state of the Bose condensation is calculated and the calculations are compared with those of the BCS theory. The physical implications to high TC currents in planar systems are discussed. The present model is related to the chemistry of high-TC materials. PACS: 03.65.