  The fundamental issues of microcausality and energy positivity conditions are
important to be investigated in the context of spontaneous Lorentz violating
theories. We check the microcausality and energy positivity conditions for a
free rank-2 antisymmetric tensor field with spontaneous Lorentz violating term
and a classically equivalent vector theory from dispersion relations and
propagators, and find that the two theories lead to different conditions. While
the antisymmetric tensor theory satisfies energy positivity condition and can
satisfy microcausality condition for a particular choice of vacuum value, the
vector theory violates both energy positivity and microcausality conditions.
