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

TL;DR

This paper develops a local bulk-to-boundary correspondence in algebraic quantum field theory, enabling the comparison of local observables and states across different curved spacetimes using light cone structures.

Contribution

It introduces a new local framework for quantum fields on curved spacetimes, utilizing exponential maps and light cones to relate bulk and boundary algebras and states.

Findings

Constructed a local *-algebra of Wick polynomials on light cones.

Proved the injectivity of the algebra homomorphism from bulk to boundary.

Established a method to compare expectation values of local observables across different spacetimes.

Abstract

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'.