  Quantum simulations of High Energy Physics, and especially of gauge theories,
is an emerging and exciting direction in quantum simulations. However,
simulations of such theories, compared to simulations of condensed matter
physics, must satisfy extra restrictions, such as local gauge and Lorentz
invariance. In this paper we discuss these special requirements, and present a
new method for quantum simulation of lattice gauge theories using ultracold
atoms. This method allows to include local gauge invariance as a fundamental
symmetry of the atomic Hamiltonian, arising from natural atomic interactions
and conservation laws (and not as a property of a low energy sector). This
allows us to implement elementary gauge invariant interactions for three
lattice gauge theories: compact QED (U(1)), SU(N) and Z_N, which can be used to
build quantum simulators in 1+1 dimensions. We also present a new loop method,
which uses the elementary interactions as building blocks in the effective
construction of quantum simulations for d+1 dimensional lattice gauge theories
(d>1), without having to use Gauss's law as a constraint, as in previous
proposals. We discuss in detail the quantum simulation of 2+1 dimensional
compact QED and provide a numerical proof of principle. The simplicity of the
already gauge invariant elementary interactions of this model suggests it may
be useful for future experimental realizations.
