  When families of quantum systems are equipped with a continuous family of
Hamiltonians such that there is a gap in the common spectrum one can define a
notion of a Berry connection. In this note we stress that, in general, since
the Hilbert bundle defining the family of quantum systems does not come with a
canonical trivialization there is in fact not a single Berry connection but
rather a family of Berry connections. Two examples illustrate that this remark
can have physical consequences.
