The grand canonical partition function of a bosonic quantum gas can be
expressed as a functional integral of a complex-valued field. This
elegant formulation of quantum field theory, since the interatomic
potential is typically unknown, is usually implemented assuming a
zero-range two-body interaction between the bosons. However, due to
this assumption, the zero-point energy of the gas diverges and it is
necessary to apply proper regularization techniques. In this talk, I
will implement the functional integral approach to obtain the
regularized equation of state of a two-dimensional spherical Bose gas.
The key ingredient to obtain a finite and observable result is the
scattering theory, which allows to relate the zero-range interaction
strength to the experimental parameters. This new result for the
equation of state provides an insight into finite-size corrections to
the thermodynamics of a curved Bose gas and offers a benchmark for the
current experiments on shell-shaped Bose-Einstein condensates.
