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The Quantization of Linear Gravitational Perturbations and the Hadamard Condition

Hunt, David Stephen (2012) The Quantization of Linear Gravitational Perturbations and the Hadamard Condition. PhD thesis, University of York.

Available under License Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 UK: England & Wales.

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The quantum field theory describing linear gravitational perturbations is important from a cosmological viewpoint, in particular when formulated on de Sitter spacetime, which is used in inflationary models. There is currently an ongoing controversy pertaining to the existence of a de Sitter invariant vacuum state for free gravitons. This thesis is a mathematically rigorous study of the theory and all constructions are performed in as general a setting as is possible, which allows us to then specialise to a particular spacetime when required. In particular, to study the case of de Sitter spacetime with a view to resolving the aforementioned controversy. The main results include the full construction of the classical phase space of the linearized Einstein system on a background cosmological vacuum spacetime, which includes proving when various gauge choices can be made. In particular, we prove that within a normal neighbourhood of any Cauchy surface, in a globally hyperbolic spacetime, one may pass to the synchronous gauge. We also consider the transverse-traceless gauge but show that there is a topological obstruction to achieving this, which rules out its general use. In constructing the phase space it is necessary to obtain a weakly non-degenerate symplectic product. We prove that this can be achieved for the case that the background spacetime admits a compact Cauchy surface by using results from the Arnowitt-Deser-Misner (ADM) formalism, specifically the initial data splittings due to Moncrief. The system is quantized using Dirac's prescription, which permits the construction of an algebra of observables consisting of gauge-invariant smeared fields. It is shown that this algebra satisfies a time-slice condition. Finally, the states of the system are considered: we formulate the Hadamard condition and show that the Fock vacuum in Minkowski spacetime satisfies this definition.

Item Type: Thesis (PhD)
Academic Units: The University of York > Mathematics (York)
Identification Number/EthosID: uk.bl.ethos.564159
Depositing User: Mr David Stephen Hunt
Date Deposited: 10 Jan 2013 15:19
Last Modified: 08 Sep 2016 13:01
URI: http://etheses.whiterose.ac.uk/id/eprint/3156

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