The necessity of the Hadamard condition

Abstract

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.