  We study space-time symmetries in Non-Commutative (NC) gauge theory in the
(constrained) Hamiltonian framework. The specific example of NC CP(1) model,
posited in \cite{sg}, has been considered. Subtle features of Lorentz
invariance violation in NC field theory were pointed out in \cite{har}. Out of
the two - Observer and Particle - distinct types of Lorentz transformations,
symmetry under the former, (due to the translation invariance), is reflected in
the conservation of energy and momentum in NC theory. The constant tensor
$\theta_{\mu\nu}$ (the noncommutativity parameter) destroys invariance under
the latter.
  In this paper we have constructed the Hamiltonian and momentum operators
which are the generators of time and space translations respectively. This is
related to the Observer Lorentz invariance. We have also shown that the
Schwinger condition and subsequently the Poincare algebra is not obeyed and
that one can not derive a Lorentz covariant dynamical field equation. These
features signal a loss of the Particle Lorentz symmetry. The basic observations
in the present work will be relevant in the Hamiltonian study of a generic
noncommutative field theory.
