  Can high energy physics be simulated by low-energy, non-relativistic,
many-body systems, such as ultracold atoms? Such ultracold atomic systems lack
the type of symmetries and dynamical properties of high energy physics models:
in particular, they manifest neither local gauge invariance nor Lorentz
invariance, which are crucial properties of the quantum field theories which
are the building blocks of the standard model of elementary particles.
  However, it turns out, surprisingly, that there are ways to configure atomic
system to manifest both local gauge invariance and Lorentz invariance. In
particular, local gauge invariance can arise either as an effective, low
energy, symmetry, or as an "exact" symmetry, following from the conservation
laws in atomic interactions. Hence, one could hope that such quantum simulators
may lead to new type of (table-top) experiments, that shall be used to study
various QCD phenomena, as the confinement of dynamical quarks, phase
transitions, and other effects, which are inaccessible using the currently
known computational methods.
  In this report, we review the Hamiltonian formulation of lattice gauge
theories, and then describe our recent progress in constructing quantum
simulation of Abelian and non-Abelian lattice gauge theories in 1+1 and 2+1
dimensions using ultracold atoms in optical lattices.
