# Local causal structures, Hadamard states and the principle of local covariance in quantum field theory

January 06, 2010

In the framework of the algebraic formulation, we discuss and analyse some
new features of the local structure of a real scalar quantum field theory in a
strongly causal spacetime. In particular we use the properties of the
exponential map to set up a local version of a bulk-to-boundary correspondence.
The bulk is a suitable subset of a geodesic neighbourhood of any but fixed
point p of the underlying background, while the boundary is a part of the
future light cone having p as its own tip. In this regime, we provide a novel
notion for the extended *-algebra of Wick polynomials on the said cone and, on
the one hand, we prove that it contains the information of the bulk counterpart
via an injective *-homomorphism while, on the other hand, we associate to it a
distinguished state whose pull-back in the bulk is of Hadamard form. The main
advantage of this point of view arises if one uses the universal properties of
the exponential map and of the light cone in order to show that, for any two
given backgrounds M and M' and for any two subsets of geodesic neighbourhoods
of two arbitrary points, it is possible to engineer the above procedure such
that the boundary extended algebras are related via a restriction homomorphism.
This allows for the pull-back of boundary states in both spacetimes and, thus,
to set up a machinery which permits the comparison of expectation values of
local field observables in M and M'.

Keywords:

*none*