  In the first part of the thesis we consider the constraints of causality and
unitarity for particles interacting via strictly finite-range interactions. We
generalize Wigner's causality bound to the case of non-vanishing partial-wave
mixing. Specifically we analyze the system of the low-energy interactions
between protons and neutrons. We also analyze low-energy scattering for systems
with arbitrary short-range interactions plus an attractive $1/r^{\alpha}$ tail
for $\alpha\geq2$. In particular, we focus on the case of $\alpha=6$ and we
derive the constraints of causality and unitarity also for these systems and
find that the van der Waals length scale dominates over parameters
characterizing the short-distance physics of the interaction. This separation
of scales suggests a separate universality class for physics characterizing
interactions with an attractive $1/r^{6}$ tail. We argue that a similar
universality class exists for any attractive potential $1/r^{\alpha}$ for
$\alpha\geq2$.
  In the second part of the thesis we present lattice Monte Carlo calculations
of fermion-dimer scattering in the limit of zero-range interactions using the
adiabatic projection method. The adiabatic projection method uses a set of
initial cluster states and Euclidean time projection to give a systematically
improvable description of the low-lying scattering cluster states in a finite
volume. We use L\"uscher's finite-volume relations to determine the $s$-wave,
$p$-wave, and $d$-wave phase shifts. For comparison, we also compute exact
lattice results using Lanczos iteration and continuum results using the
Skorniakov-Ter-Martirosian equation. For our Monte Carlo calculations we use a
new lattice algorithm called impurity lattice Monte Carlo. This algorithm can
be viewed as a hybrid technique which incorporates elements of both worldline
and auxiliary-field Monte Carlo simulations.
