  The Klein-Gordon-Einstein equations of classical real scalar fields have
time-dependent solutions (periodic in time). We show that quantum real scalar
fields can form non-oscillating (static) solitonic objects, which are quite
similar to the solutions describing boson stars formed with classical and
quantum complex scalar fields (the latter will be studied in this paper). We
numerically analyze the difference between them concerning the mass of boson
stars. On the other hand, we suggest an interesting test (a viable process that
the boson star may undergo in the early universe) for the formation of boson
stars. That is, it is questioned that after a second-order phase transition (a
simple toy model will be considered here), what is the fate of the boson star
composed of quantum real scalar field.
