The Necessity of the Hadamard Condition

Hadamard states are generally considered as the physical states for linear quantized fields on curved spacetimes, for several good reasons. Here, we provide a new motivation for the Hadamard condition: for "ultrastatic slab spacetimes" with compact Cauchy surface, we show that the Wick squares of all time derivatives of the quantized Klein-Gordon field have finite fluctuations only if the Wick-ordering is defined with respect to a Hadamard state. This provides a converse to an important result of Brunetti and Fredenhagen. The recently proposed "S-J (Sorkin-Johnston) states" are shown, generically, to give infinite fluctuations for the Wick square of the time derivative of the field, further limiting their utility as reasonable states. Motivated by the S-J construction, we also study the general question of extending states that are pure (or given by density matrices relative to a pure state) on a double-cone region of Minkowski space. We prove a result for general quantum field theories showing that such states cannot be extended to any larger double-cone without encountering singular behaviour at the spacelike boundary of the inner region. In the context of the Klein-Gordon field this shows that even if an S-J state is Hadamard within the double cone, this must fail at the boundary.