  In the standard electroweak theory that describes nature, the Chern-Simons
number associated with the vacua as well as the unstable sphaleron solutions
play a crucial role in the baryon number violating processes. We recall why the
Chern-Simons number should be generalized from a set of discrete values to a
dynamical (quantum) variable. Via the construction of an appropriate Hopf
invariant and the winding number, we discuss how the geometric information in
the gauge fields is also captured in the Higgs field. We then discuss the
choice of the Hopf variable in relation to the Chern-Simons variable.
