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
Summary
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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