  We study the effect of coupling a spin bath environment to a system which, at
low energies, can be modeled as a quantum Ising system. A field theoretic
formalism incorporating both thermal and quantum fluctuations is developed to
derive results for the thermodynamic properties and response functions, both
for a toy model and for the $LiHoF_4$ system, in which spin-8 electronic spins
couple to a spin-$7/2$ nuclear spin bath: the phase transition then occurs in a
system of electronuclear degrees of freedom, coupled by long-range dipolar
interactions. The quantum Ising phase transition still exists, and one
hybridized mode of the Ising and bath spins always goes soft at the transition.
