In 1980, Christ and Lee wrote a paper titled 'Operator ordering and Feynman rules in gauge theories'. In this paper they showed that the Hamiltonian formulation of a non-abelian gauge theory in a general non-covariant gauge is complicated by problems of operator ordering, leading to new nonlocal interactions they call V<SUB>1</SUB> + V<SUB>2</SUB>. In particular this applies to the familiar Coulomb gauge. More recently, Cheng and Tsai have arrived at the same ordering by careful attention to the fact that the Gauss-law constraint is not an operator equation. On the other hand, it is known that the naive Coulomb gauge Feynman rules in non-abelian gauge theory give rise to ambiguous integrals, in addition to the usual ultraviolet divergences. This thesis shows how these ambiguities can be resolved to all orders in perturbation theory. This is done by defining a gauge that interpolates smoothly between the Feynman gauge and the Coulomb gauge, and then using a diagrammatic method to combine the different contributions that arise. The extra terms V<SUB>1</SUB> + V<SUB>2</SUB> of Christ and Lee are then identified with certain two loop ambiguous terms. The only remaining question is whether the method of resolving the ambiguities is compatible with renormalisation. An investigation into the extra complications that renormalisation introduces is carried out.

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Direct detection of gravitational radiation, predicted by Einstein’s general theory of relativity, remains one of the most exciting challenges in experimental physics. Due to their relatively weak interaction with matter, gravitational waves promise to allow exploration of hitherto inaccessible objects and epochs. Unfortunately, this weak coupling also hinders detection with strain amplitudes at the Earth estimated to be of order 10^−21. Due to their wide bandwidth and theoretical sensitivity, kilometre-scale Michelson style interferometers have become the preferred instrument with which to attempt ground based detection. A worldwide network of first generation instruments has been constructed and prodigious volumes of data recorded. Despite each instrument approaching or having reached its design sensitivity, a confirmed detection remains elusive. Planned upgrades to these instruments aim to increase strain sensitivity by an order of magnitude, commencing the era of second generation detectors. Entry into this regime will be accompanied by an entirely new set of challenges, two of which are addressed in this work. As advanced interferometers are commissioned, instrumental artifacts will give way to fundamental noise sources. In the region of peak sensitivity it is expected that thermal noise in the interferometers’ dielectric mirror coatings will be the principal source of displacement noise. Theory suggests that increasing the spot size of laser light incident on these mirrors will reduce the measured thermal noise. In the first part of this work we examine one method of realising larger spots. By adopting non-spherical mirrors in the interferometers’ arms it is possible to create resonators which support a wide, flat-topped field known as the mesa beam. This beam has been shown to theoretically reduce all forms of mirror thermal noise without being significantly more difficult to control. In this work we investigate these claims using a bespoke prototype mirror. The first results regarding a non-Gaussian beam created in a manner applicable to a gravitational wave interferometer are presented. A common theme among all second generation interferometer designs is the desire to maximise circulating power. This increased power is partnered by commensurately increased thermal perturbations. Since the attractive properties of the mesa beam are produced by the fine structure of its supporting mirrors, it is important that we understand the effects absorption of stored optical power could have on mesa fields. In the second part of this work we report on numerical evaluations of measured thermal noise and mesa beam intensity profile as a function of absorbed power. Increased optical power also has less obvious consequences. As a result of radiation pressure, there exists a pathway between optical energy stored in an interferometer’s arms and mechanical energy stored in the acoustic modes of its test masses. Under appropriate conditions, this coupling can excite one or more test masses to such a degree that interferometer operation becomes impossible. In the final part of this work we determine whether it is possible to mitigate these parametric instabilities using electrostatic actuators originally designed to control the position and orientation of the test masses.

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