boson gas behaves as a one dimensional system [@ROJAS1996148]. Once the dimension is reduced, a further decrease in the temperature, or a increase in the magnetic field increases the population of the states around $p_3 \approx 0$ until it becomes a macroscopic quantity. But, due to the one dimensionality of the system, a critical temperature can not be defined, and the condensation is diffuse [@ROJAS1997].

For the neutral boson gas, the coupling between the field and the spin magnetic moment does not imply quantization for any momentum component. All of them are preserved as quantum observable and the three dimensionality of the neutral gas is kept for any temperature and magnetic field. For a Bose gas in three dimensions a critical temperature is well defined, and that is why it undergoes a phase transition to a usual BEC. However, it is still possible to find a formal analogy between the charged and the neutral spin-one gas in what respect to the condensation if we restrict the later to move in one dimension.

### Condensation in the one dimensional system

Let us consider a gas of neutral vector bosons with $p_{\perp}=0$. The statistical contribution to the thermodynamical potential has the form

