The Quantization of Linear Gravitational Perturbations and the Hadamard Condition

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.