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We analyze 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 3+1-split M ≅ ℝ × Σ 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 a quantum field theory defined by a normally hyperbolic differential operator P acting in a vector bundle E → π 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.

The microlocal spectrum condition, initial value formulations, and background independence