Abstract

We prove essential self-adjointness of the spatial part of the linear Klein-Gordon operator with external potential for a large class of globally hyperbolic manifolds. The proof is conducted by a fusion of new results concerning globally hyperbolic manifolds, the theory of weighted Hilbert spaces and related functional analytic advances.

Highlights

  • Quantum field theory (QFT) in curved spacetime studies the behavior of quantum fields that propagate in the presence of a classical gravitational field, where the quantum behavior of the gravitational field is neglected

  • We investigate the case of static globally hyperbolic spacetimes, i.e., spacetimes where N and all components of h are time independent

  • Our proof of essential self-adjointness of w2 holds for all static globally hyperbolic spacetimes

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Summary

Introduction

Quantum field theory (QFT) in curved spacetime studies the behavior of quantum fields that propagate in the presence of a classical gravitational field, where the quantum behavior of the gravitational field is neglected. It is seen as an intermediate (and mostly rigorous) step towards a complete theory of quantum gravity (see [1,2,3,4,5] for excellent reviews). One fruitful context arises from the restriction to globally hyperbolic spacetimes. The advantage of this class of spacetimes is the existence of a (noncanonical) choice of time, or equivalently the existence of a global Cauchy surface

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Klein-Gordon Theory on Globally Hyperbolic Spacetimes
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Weighted Manifolds and Essential Self-Adjointness
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Main Result on Self-Adjointness
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Complete Riemannian Manifolds
Static Globally Hyperbolic Spacetimes
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Globally Hyperbolic Spacetimes
Discussion
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Full Text
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