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Communications in Mathematical Physics

, Volume 304, Issue 2, pp 459–498 | Cite as

Local Causal Structures, Hadamard States and the Principle of Local Covariance in Quantum Field Theory

  • Claudio DappiaggiEmail author
  • Nicola Pinamonti
  • Martin Porrmann
Article

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 an arbitrary 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 aforesaid 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

Integral Kernel Double Cone Cauchy Surface Null Cone Extended Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Claudio Dappiaggi
    • 1
    • 2
    Email author
  • Nicola Pinamonti
    • 2
  • Martin Porrmann
    • 3
  1. 1.Erwin Schrödinger Institut für Mathematische PhysikWienAustria
  2. 2.II. Institut für Theoretische PhysikUniversität HamburgHamburgGermany
  3. 3.Quantum Research Group, School of PhysicsUniversity of KwaZulu-Natal and National Institute for Theoretical PhysicsDurbanSouth Africa

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