0 Thermodynamics of Bose - Einstein condensation of relativistic gas

We show that the free relativistic wave equation which describes the particle without or with rest mass has more than one part of energy spectrum. One part of energy spectrum is beginning with rest energy and it is not limited by above. This part of spectrum is called by us as normal. Another part of energy spectrum is beginning with the zero energy. This part of spectrum is called by us as anomalous since the zero energy corresponds to the infinite group velocity.The presence of the zero in the energy spectrum permits to consider the Bose-Einstein condensation. We show that the heat capasity has the finite discontinuity at the condensation temperature. The last means that we have the phase transition at the condensation point. It is well known that the construction of the thermodynamics requires the knowledge of particle energy spectrum because the principal thermodynamic characteristic ,namely, equilibrium temperature distribution function depends on particle energy (frequency). We denote the last by f r , f r = (exp(¯ hω − µ/k B T) + r) −1 , r = 0, ±1. where the relativistic energy ¯ hω is equal to ¯ hω = ¯ hc k 2 + k 2 0 , (1) Here k 2 is a square of 3-D momentum and k 0 is the inverse Compton wave length. We have superhigh energy at k >> k 0 , and in this case energy is equal to ¯ hω ≈ ¯ hck (ultrarelativistic case). In the opposite case k << k 0 the energy spectrum of Eq(1) transforms into non relativistic energy spectrum. Energy spectrum of Eq(1) corresponds to eigenfunctions (plane waves) of Klein-Gordon-Fock (KGF) equation Ψ (+) = exp (i(−ωt + k z z)) exp i(k 1 x 1 + k 2 x 2). (2) Later we will explain the representation of the solution of Eq.(2) as a product of two exponential functions. The plane waves of Eq.(2) are the basis for Green functions (GF)