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S10e. Hadamard states and their Hilbert spaces
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In his book on qunatum field theory in curved spacetime
Wald delineates a class of 2-point functions called Hadamard states
that have locally the same kind of singular behavior as the flat
free 2-point functions. This class of states is also natural from
several other points of view, though I cannot give details off-hand
since this is slightly outside my field of knowledge.
Associated to each Hadamard state is a Gaussian state |0>
of the quantum field which is constructed from the 2-point function
via Wick's theorem. This state is often called a 'vacuum state',
though this is not quite appropriate, unless one allows the vacuum
to carry gravitational and electromagnetic fields. A more appropriate
name would be a 'coherent state' since it is the generalization of
coherent states in the Fock spaces considered in optics.
Each Gaussian state produces a Hilbert space of wave functions
consisting of linear combinations of the a*_k1 a*_k2 ...|0>,
weighted by sufficiently smooth functions of the k's to render
their norm finite.
All states in this Hilbert space are also physically reasonable,
but they do not have the same basic (vacuum-like)
status as the Hadamard states since they are no longer Gaussian,
and hence are harder to work with.
But you can evaluate in such a state by
expanding everything in terms of vacuum expectations of expressions
in a's and a^*'s and applying Wick's theorem. Their leading singular
behavior is probably the same as for the Gaussian state itself,
though I haven't tried to check this.