The Casimir effect refers to the primarily attractive force between material bodies due to quantum fluctuations in the electromagnetic field. The Casimir effect is difficult to calculate in general, since it is sensitive to the exact shapes of the bodies and involves contributions from all frequencies. As a result, calculating the Casimir effect between general bodies usually requires a numerical approach. The worldline method computes Casimir energies by creating an ensemble of space-time paths corresponding to a virtual particle interacting with the bodies. This method was originally developed for a scalar fields coupled to an idealized background potential, rather than the vector electromagnetic field interacting with media. This thesis presents work on extending the worldline method to account for the material properties of the interacting bodies, and the polarizations of electromagnetism. This thesis starts by covering background material on path integrals, and quantizing the electromagnetic field in media. The electromagnetic field is decomposed in terms of two scalar fields for planar bodies, where these scalar fields correspond to the transverse-electric and transverse-magnetic polarizations of the electromagnetic field. The worldline path integrals are developed for both polarizations, and solved analytically. Next, numerical methods are developed and tested in the context of planar bodies. The starting positions, and scale of the paths, and shape of the paths are sampled via Monte Carlo methods. The transverse-magnetic path integral also requires specialized methods for estimating derivatives, and path construction. The analytical and numerical results for both worldline path integrals are in agreement with known solutions. Finally, specialized methods are developed for computing derivatives of the worldline Casimir-energy path integrals, allowing for efficient numerical computations of Casimir forces and torques.

Dominated by Heisenberg's uncertainty principle, vacuum is not quantum mechanically an empty void, i.e. virtual pairs of particles appear and disappear persistently. This nonlinearity subsequently provokes a number of phenomena which can only be practically observed by going to a high-intensity regime. Pair production beyond the so-called Sauter-Schwinger limit, which is roughly the field intensity threshold for pairs to show up copiously, is such a nonlinear vacuum phenomenon. From the viewpoint of Dirac's front form of Hamiltonian dynamics, however, vacuum turns out to be trivial. This triviality would suggest that Schwinger pair production is not possible. Of course, this is only up to zero modes. While the instant form of relativistic dynamics has already been at least theoretically well-played out, the way is still open for investigating the front form. The aim of this thesis is to explore the properties of such a contradictory aspect of quantum vacuum in two different forms of relativistic dynamics and hence to investigate the possibility of finding a way to resolve this ambiguity. This exercise is largely based on the application of field quantization to light-front dynamics. In this regard, some concepts within strong field theory and light-front quantization which are fundamental to our survey have been introduced, the order of magnitude of a few important quantum electrodynamical quantities have been fixed and the basic information on a small number of nonlinear vacuum phenomena has been identified. Light-front quantization of simple bosonic and fermionic systems, in particular, the light-front quantization of a fermion in a background electromagnetic field in (1+1) dimensions is given. The light-front vacuum appears to be trivial also in this particular case. Amongst all suggested methods to resolve the aforementioned ambiguity, the discrete light-cone quantization (DLCQ) method is applied to the Dirac equation in (1+1) dimensions. Furthermore, the Tomaras-Tsamis-Woodard (TTW) solution, which expresses a method to resolve the zero-mode issue, is also revisited. Finally, the path integral formulation of quantum mechanics is discussed and, as an alternative to TTW solution, it is proposed that the worldline approach in the light-front framework may shed light on different aspects of the TTW solution and give a clearer picture of the light-front vacuum and the pair production phenomenon on the light-front.

In dieser Arbeit wird der Ansatz der Weltlinienquantenfeldtheorie (WQFT) zur Berechnung von Observablen des klassischen allgemeinen relativistischen Zweikörpersystems vorgestellt. Kompakte Körper wie Schwarze Löcher oder Neutronensterne werden im Rahmen einer effektiven Feldtheorie mit Weltlinienfeldern beschrieben. Die WQFT behandelt alle Weltlinienfelder gleichberechtigt mit dem Gravitationsfeld und ist definiert als die tree-level-Beiträge eines Pfadintegrals auf diesen Feldern. Zuerst wird die effektive feldtheoretische Beschreibung von kompakten Körpern mit Weltlinien und die post-Minkowski'schen Approximation der Streuung dieser Körpern vorgestellt. Die Einbeziehung des Spins wird mit besonderem Augenmerk auf ihre supersymmetrische Beschreibung in Form von antikommutierenden Grassmann-Variablen analysiert. Anschließend wird die WQFT mit einer Diskussion ihrer in-in Schwinger-Keldysh-Formulierung, ihrer Feynman-Regeln und Graphengenerierung sowie ihrer on-shell Einpunktfunktionen vorgestellt. Die Berechnung von Streuobservablen erfordert im Allgemeinen die Auswertung von Multi-Loop-Integralen, und wir analysieren die Zwei-Loop-Integrale, die in der dritten post-Minkowski'schen Ordnung der Weltlinienobservablen auftreten. Schließlich wenden wir uns den Ergebnissen der WQFT zu und beginnen mit der gravitativen Bremsstrahlung bei der Streuung zweier rotierender Körper. Diese Wellenform wird zusammen mit der Strahlungsinformation der Linear- und Drehimpulsflüsse diskutiert. Der gesamte abgestrahlte Drehimpuls führender post-Minkowski'schen Ordnung wird abgeleitet. Wir präsentieren dann die Ergebnisse des konservativen und strahlenden Impulses und des Spin-Kicks bei dritter post-Minkowski'scher Ordnung und quadratischer Ordnung in Spins zusammen mit der Abbildung der ungebundenen Ergebnisse auf einen konservativen (gebundenen) Hamiltonian bei der entsprechenden perturbativen Ordnung. / In this work the worldline quantum field theory (WQFT) approach to computing observables of the classical general relativistic two-body system is presented. Compact bodies such as black holes or neutron stars are described in an effective field theory by worldline fields with spin degrees of freedom efficiently described by anti-commuting Grassmann variables. Novel results of the WQFT include the gravitational bremsstrahlung at second post-Minkowskian order and the impulse and spin kick at third post-Minkowskian order all at quadratic order in spins. Next, the WQFT is presented with a comprehensive discussion of its in-in Schwinger-Keldysh formulation, its Feynman rules and graph generation and its on-shell one-point functions which are directly related to the scattering observables of unbound motion. Here, we present the second post-Minkowskian quadratic-in-spin contributions to its free energy from which the impulse and spin kick may be derived to the corresponding order. The computation of scattering observables requires the evaluation of multi-loop integrals and for the computation of observables at the third post-Minkowskian order we analyze the required two-loop integrals. Our discussion uses retarded propagators which impose causal boundary conditions of the observables. Finally we turn to results of the WQFT starting with the gravitational bremsstrahlung of the scattering of two spinning bodies. This waveform is discussed together with its radiative information of linear and angular momentum fluxes. Lastly we present the conservative and radiative impulse and spin kick at third post-Minkowskian order and quadratic order in spins together with the a conservative Hamiltonian at the corresponding perturbative order. The results obey a generalized Bini-Damour radiation-reaction relation and their conservative parts can be parametrized in terms of a single scalar.