  We review recent developments in the theory of quantum dynamics in ultra-cold
atomic physics, including exact techniques, but focusing on methods based on
phase-space mappings that are appli- cable when the complexity becomes
exponentially large. These phase-space representations include the truncated
Wigner, positive-P and general Gaussian operator representations which can
treat both bosons and fermions. These phase-space methods include both
traditional approaches using a phase-space of classical dimension, and more
recent methods that use a non-classical phase-space of increased
dimensionality. Examples used include quantum EPR entanglement of a four-mode
BEC, time-reversal tests of dephasing in single-mode traps, BEC quantum
collisions with up to 106 modes and 105 interacting particles, quantum
interferometry in a multi-mode trap with nonlinear absorp- tion, and the theory
of quantum entropy in phase-space. We also treat the approach of variational
optimization of the sampling error, giving an elementary example of a nonlinear
oscillator.
