On Bose-Einstein Condensation in Any Dimension

Arbitrarily large ground state population is a general property of any ideal bose gas when conditions of degeneracy are satisfied; it occurs at any dimension D. For D = 1, the condensation is diffuse, at D = 2 it is a sort of quasi-condensate. The discussion is made by following a microscopic approach and for finite systems. Some astrophysical consequences are discussed, as well as the temperature-dependent mass case. ∗Permanent address: Grupo de Fisica Teórica, ICIMAF, Academia de Ciencias de Cuba, Calle E No. 309,Vedado, La Habana 4, Cuba. 1 What is Bose-Einstein condensation? At present there is a renewed interest in Bose-Einstein condensation (BEC), particularly after the experimental realization of it [1]. Actually, BEC is one of the most interesting problems of quantum statistics. It occurs in a free particle Bose gas at a critical temperature Tc, and is a pure quantum phenomenon, in the sense that no interaction is needed to be assumed to exist among the particles. BEC is interesting for condensed matter (superfluidity, superconductivity) but is has also increasing interest in high energy physics (electroweak phase transition, superfluidity in neutron stars). The consequences of its occurrence at dimensions different from D = 3 may have interest equally in these two fields of physics. Bose -Einstein condensation is understood as the steady increase of particles in the state with zero energy [3], or as the macroscopically large number of particles accumulating in a single quantum state [2], and its connection with the theory of phase transitions is actually a property of BEC in dimensions D > 2, since it has a critical temperature at which the phenomenon of condensation starts. But as different from BEC, phase transitions theory assumes, in general, some interaction among the particles [2], and properties of non-analyticity of thermodynamic quantities appear in the thermodynamic limit N = N /VN ,V→∞, where N and V are respectively the number of particles and volume of the system. It has been also investigated the possible connection of BEC with spontaneous symmetry breaking (SSB) [4], [5]. Actually, there is a close analogy, but not a full correspondence among them. The SSB assumes also interaction among the fields, i.e., systems with infinite number of degrees of freedom. In systems of low dimensionality, it happens that no SSB of a continuous symmetry occurs in one or two spatial dimensions D according the the Mermin-Wagner theorem [6]; (see also [8], for a proof that there are no Goldstone bosons in one dimension). Thre is, however, a close correspondence between phase transitions theory and SSB. Concerning BEC, it is usually stated [7] that in the thermodynamic limit BEC is not possible in D = 2 and that it neither occurs