  We analyze the implications of the microlocal spectrum/Hadamard condition for
states in a (linear) quantum field theory on a globally hyperbolic spacetime
$M$ in the context of a (distributional) initial value formulation. More
specifically, we work in a $3+1$-split $M\cong\mathbb{R}\times\Sigma$ and give
a bound, independent of the spacetime metric, on the wave front sets of the
initial data for a quasi-free Hadamard state in the quantum field theory
defined by a normally hyperbolic differential operator $P$ acting in a vector
bundle $E\stackrel{\pi}{\rightarrow}M$. This aims at a possible way to apply
the concept of Hadamard states within approaches to quantum field
theory/gravity relying on a Hamiltonian formulation, potentially without a
(classical) background metric $g$.
