Lifshitz symmetries and nonrelativistic holography

In this dissertation we cover topics within the main themes of Lifshitz symmetries and nonrelativistic holography.
Nonrelativistic theories are typically less constrained than relativistic ones, which makes them often more cumbersome to work with. Via holography one can have acces to domains of a field theory which are typically computationally inaccessible using only field theoretical methods. To get a better understanding of what holography can do for us, however, one first requires a proper understanding of the field theory itself. Thus, as a side effect of studying nonrelativistic holography, nonrelativistic field theories have become a field of (renewed) interest in itself. Motivated by quantum critical systems we restrict ourselves to a specific set of nonrelativistic symmetries, namely Lifshitz symmetries.
In Chapters 2, 4, 5, and 6 we perform research in a bottom up holographic setup, where the canonical relativistic methods are generalized to a nonrelativistic setting. We study relaxation and isotropization of a strongly coupled nonrelativistic plasma via a specific holographic model, in which it turns out that the approach known from relativistic holography was well generalizable to nonrelativistic holography, at the cost of being more involved at the computational level. In both cases it turned out that z=d, when the dynamical critical exponent equals spatial dimensions, respectively, is an important point in parameter space.
In Chapter 7 we compare features from a specific field theoretic model and the same gravitational model as used in Chapters 2, 4, 5, and 7. We find that specific correlators, for z=d, which are obtainable analytically from both sides of the duality, coincide.
In order to compute quantities beyond relaxation times, e.g. viscosities, one has to study the hydrodynamics of a system. Although the computation of hydrodynamic quantities can be done via holography, for the boundary interpretation of these quantities one needs to study the hydrodynamics of the appropriate symmetry group. In Chapter 2 a framework is constructed in which this can be done. As a proof of principle, the speed of sound was computed for a gas of Lifshitz particles with various statistics.