  Relations between critical exponents, or scaling laws, at both continuous and
discontinuous quantum phase transitions are derived and discussed. In general
there are multiple dynamical exponents at these transitions, which complicates
the scaling description. Some rigorous inequalities are derived, and the
conditions needed for these inequalities to be equalities are discussed. New
scaling laws involving the specific-heat exponents are derived and and
contrasted with their counterparts at classical phase transitions. We also
generalize the ideas of Fisher and Berker and others for applying (finite-size)
scaling theory near a classical first-order transition to the quantum case. We
then apply and illustrate all of these ideas by using the quantum ferromagnetic
phase transition in metals as an explicit example. This transition is known to
have multiple dynamical scaling exponents, and in general it is discontinuous
in clean systems, but continuous in disordered ones. Furthermore, it displays
many experimentally relevant crossover phenomena that can be described in terms
of fixed points, originally discussed by Hertz, that ultimately become unstable
asymptotically close to the transition and give way to the asymptotic fixed
points. These fixed points provide a rich environment for illustrating the
general scaling concepts and exponent relations. We also discuss the
quantum-wing critical point at the tips of the tricritical wings associated
with the discontinuous quantum ferromagnetic transition from a scaling point of
view.
