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The main goal of this thesis is to quantize the Einstein-Hilbert actionextended by the quadratic curvature terms within the canonical quantizationapproach, thus formulating quantum geometrodynamics of the higher derivativetheories. The motivation is to provide an alternative to the standard canonicalquantization based on the Einstein-Hilbert action alone, because the latterdoes not generate the quadratic curvature terms in the semiclassical limit. Aparticular formulation of a semiclassical approximation scheme is employedwhich ensures that the effects of the quadratic curvature terms becomeperturbative in the semiclassical limit. This leaves the classical GeneralRelativity intact, while naturally giving rise to its first semiclassicalcorrections. Another topic of interest is a classical theory where thequadratic Ricci scalar and the Einstein-Hilbert term are absent from theaction, which then enjoys the symmetry with respect to the conformaltransformation of fields (local Weyl rescaling). We pay a special attention tothis case, since it provides a natural setting for the absence of the notion ofa physical length scale. Certain useful model-independent tools are alsoconstructed in this thesis. Firstly, dimensionless coordinates and theunimodular decomposition of the metric are used to expose the only conformallyvariant degree of freedom, making the geometrical origin of the physical lengthscale apparent. With such an approach several earlier results become much moretransparent. Secondly, using unimodular-conformal variables a model-independentgenerator of conformal field transformations is constructed in terms of which areformulation of the definition of conformal invariance is given. Thirdly, itis argued that a canonical quantization scheme makes more sense to be based onthe quantization of generators of relevant transformations, than on first classconstraints.

Quantum Geometrodynamics of Higher Derivative Theories with and without Conformal Symmetry