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We describe particles in a potential by a special diffusion process, the maximal entropy random walk (MERW) on a lattice. Since MERW originates in a variational problem, it shares the linear algebra of Hilbert spaces with quantum mechanics. The Born rule appears from measurements between equilibrium states in the past and the same equilibrium states in the future. Introducing potentials by the observation that time, in a gravitational field running in different heights with a different speed, MERW respects the rule that all trajectories of the same duration are counted with equal probability. In this way, MERW allows us to derive the Schrödinger equation for a particle in a potential and the Darwin term of the nonrelativistic expansion of the Dirac equation. Finally, we discuss why quantum mechanics cannot be simply a result of MERW, but, due to the many analogies, MERW may pave the way for further understanding.

Maximal center gauge (MCG) aims to detect some of the most important vacuum configurations, suggesting thick magnetic flux tubes quantized to non-trivial center elements of the gauge group being responsible for confinement. Due to the NP-hardness of a global maximization of the gauge functional only numeric procedures aiming for local maxima are possible. We observe a linear decrease in the string tension with increasing gauge functional value of the local maxima. This implies that the request to get as close as possible to the absolute maximum is unattainable. We compare global properties of the ensemble of local maxima with other methods to detect center vortices and with determinations of the string tension from full configurations. This comparison indicates that the information about the number and positions of center vortices is contained in the structure of the ensemble of local maxima. This may pave the way for a future more successful formulation of the gauge condition.

New analysis regarding the structure of center vortices is presented: Using data from gluonic SU(2) lattice simulation with Wilson action, a correlation of fluctuations in color space to the curvature of vortex fluxes was found. Finite size effects of the S2-homogeneity hint at color homogeneous regions on the vortex surface.

In search for an effective model of quark confinement we study the vacuum of SU(2) quantum chromodynamic with lattice simulations using Wilson action. Assuming that center vortices are the relevant excitations causing confinement, we analyzed their physical size and their color structure. We present confirmations for a vanishing thickness of center vortices in the continuum limit and hints at their color structure. This is the first time that algorithms for the detection of thick center vortices based on non-trivial center regions has been used.

In the model of topological particles we have four types of topologically stable dual Dirac monopoles with soft cores and finite mass. We discuss the steps for getting a Dirac equation for these particles. We show for the free and the interacting case that we arrive at the Dirac equation in the limit, where the soft solitons approach singular dual Dirac monopoles.

With the idea to find geometric formulations of particle physics we investigate the predictions of a three-dimensional generalization of the Sine-Gordon model, very close to the Skyrme model and to the Wu-Yang description of Dirac monopoles. With three rotational degrees of freedom of spatial Dreibeins, we formulate a Lagrangian and confront the predictions to electromagnetic phenomena. Stable solitonic excitations we compare with the lightest fundamental electric charges, electrons, and positrons. Two Goldstone bosons we relate to the properties of photons. These particles are characterized by three topological quantum numbers, which we compare to charge, spin, and photon numbers. Finally, we conjecture some ideas for further comparisons with experiments.

Gauge fields control the dynamics of fermions, and, in addition, a back reaction of fermions on the gauge field is expected. This back reaction is investigated within the vortex picture of the QCD vacuum. We show that the center vortex model reproduces the string tension of the full theory also in the presence of fermionic fields.

The center vortex model of quantum-chromodynamics can explain confinement and chiral symmetry breaking. We present a possible resolution for problems of the vortex detection in smooth configurations and discuss improvements for the detection of center vortices.