  We examine the low-energy excitations in the vicinity of the quantum critical
point in LiHoF$_4$, a physical realization of the Transverse Field Ising Model,
focusing on the long-range fluctuations which soften to zero energy at the
ferromagnetic quantum phase transition. Microwave spectroscopy in tunable
loop-gap resonator structures identifies and characterizes the soft mode and
higher-energy electronuclear states. We study these modes as a function of
frequency and magnetic fields applied transverse and parallel to the Ising
axis. These are understood in the context of a theoretical model of a soft
electronuclear mode that interacts with soft photons as well as soft phonons.
We identify competing infrared divergences at the quantum critical point,
coming from the photons and the electronuclear soft mode. It is an incomplete
cancellation of these divergences that leads to the muted but distinct
signatures observed in the experiments. The application of a longitudinal
magnetic field gaps the soft mode. Measurements well away from the quantum
critical point reveal a set of ``Walker'' modes associated with ferromagnetic
domain dynamics.
