ON THE BOSE-EINSTEIN CONDENSATION OF FREE RELATIVISTIC BOSONS WITH OR WITHOUT MASS

The Bose-Einstein condensation of free relativistic particles [epsilon = (M2c4 + c2p2)1/2 - Mc2] is studied rigorously. For massless bosons epsilon = cp), the condensation transition of third (second) order occurs in 2 (3) dimensions (D). The molar heat capacity follows the T2 (T3) law below the condensation temperature T(c) {k(B)T(c) = (2-pi-h2c2n/1.645)1/2 [(pi-2h3c3n/1.202)1/3]}, reaches 4.38 (10.8) R at T = T(c), and approaches the high-temperature-limit value 2 (3) R with no jump (a jump equal to 6.75R) in 2 (3)D. For finite-mass (M) bosons, the phase transition occurs only in 3D with the condensation temperature T(c) always smaller than that of the corresponding nonrelativistic bosons [epsilon = (2M)-1p2]. If the mass M is reduced to zero, the condensation temperature T(c) grows monotonically and reaches eventually that of massless relativistic bosons. This mass-dependence of T(c) is therefore distinct from the case of nonrelativistic bosons, where T(c) grows to infinity as M --> 0. A brief discussion is given for a possible connection with the normal-to-super transition of the independently moving Cooper pairs (bosons).