This study investigates finite-volume effects in physical processes that involve the combination of long-range hadronic matrix elements with electroweak loop integrals. We adopt the approach of implementing the electroweak part as the infinite-volume version, which is denoted as the EW$_\infty$ method in this work. A general approach is established for correcting finite-volume effects in cases where the hadronic intermediate states are dominated by either a single particle or two particles. For the single-particle case, this work derives the infinite volume reconstruction (IVR) method from a new perspective. For the two-particle case, we provide the correction formulas for power-law finite-volume effects and unphysical terms with exponentially divergent time dependence. The finite-volume formalism developed in this study has broad applications, including the QED corrections in various processes and the two-photon exchange contribution in $K_L\to\mu^+\mu^-$ or $\eta\to\mu^+\mu^-$ decays.

This PhD thesis gives a comprehensive treatment of ab initio lattice Monte Carlo simulations of ultracold Bose gases by means of the complex Langevin algorithm. Since the field-theoretic action of non-relativistic bosons is a complex quantity, the corresponding path integral features a complex weight and is not accessible to standard Monte Carlo techniques. The complex Langevin algorithm represents an approach to overcome this obstacle, thereby providing the intriguing possibility of numerically exact simulations of interacting Bose-Einstein condensates within the field-theoretic framework. After reviewing the coherent-state path integral description of ultracold Bose gases as well as the complex Langevin method, we present the results of simulations in several physical scenarios. While parts of the thesis are based on arXiv:2204.10661 and arXiv:2304.05699 that treat the 3D and 2D homogeneous gas with contact interactions, it contains additional material covering external trapping potentials as well as Bose gases with long-range dipolar interactions.

We investigate high and low energy implications of a gauge dual description of the Standard Model. The high energy electric theory features gauge dynamics involving only fermionic matter fields, while the low energy magnetic description features a quasi-supersymmetric spectrum testable at colliders. The flavour theory is constructed via operators generated at the Planck scale. We further show that duality opens novel avenues for theories of grand unification.

Given a base manifold $M$ and a Lie group $G$, we define $\widetilde{\cal A}_M$ a space of generalized $G$-connections on $M$ with the following properties: - The space of smooth connections ${\cal A}^\infty_M = \sqcup_\pi {\cal A}^\infty_\pi$ is densely embedded in $\widetilde{\cal A}_M = \sqcup_\pi \widetilde{\cal A}^\infty_\pi$; moreover, in contrast with the usual space of generalized connections, the embedding preserves topological sectors. - It is a homogeneous covering space for the standard space of generalized connections of loop quantization $\bar{\cal A}_M$. - It is a measurable space constructed as an inverse limit of of spaces of connections with a cutoff, much like $\bar{\cal A}_M$. At each level of the cutoff, a Haar measure, a BF measure and heat kernel measures can be defined. - The topological charge of generalized connections on closed manifolds $Q= \int Tr(F)$ in 2d, $Q= \int Tr(F \wedge F)$ in 4d, etc, is defined. - On a subdivided manifold, it can be calculated in terms of the spaces of generalized connections associated to its pieces. Thus, spaces of boundary connections can be computed from spaces associated to faces. - The soul of our generalized connections is a notion of higher homotopy parallel transport defined for smooth connections. We recover standard generalized connections by forgetting its higher levels. - Higher levels of our higher gauge fields are often trivial. Then $\widetilde{\cal A}_\Sigma = \bar{\cal A}_\Sigma$ for $\dim \Sigma = 3$ and $G=SU(2)$, but $\widetilde{\cal A}_M \neq \bar{\cal A}_M$ for $\dim M = 4$ and $G=SL(2, {\mathbb C})$ or $G=SU(2)$. Boundary data for loop quantum gravity is consistent with our space of generalized connections, but a path integral for quantum gravity with Lorentzian or euclidean signatures would be sensitive to homotopy data.