  Standard arguments state that Bose Einstein condensation (BEC) cannot occur
in dimensions lower than three in the thermodynamic limit as the expressions
for the number of bosons in the excited states are unbounded. These arguments
imply that the number density is infinite, which is an extraordinary condition.
As an alternative to this pedagogy, we explore the use of regularization
techniques to show that the number density of bosons in the excited state is
finite in two and one dimensions at low temperatures, making it possible to
have a BEC transition in the thermodynamic limit. We suggest creating a
two-dimensional optical traps of increasing sizes to test this hypothesis as
the thermodynamic limit is approached.
