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The development of quantum mechanics during the 20th century gave rise to a completely new way of describing physics. The interpretation of quantum theory is inherently difficult: for example, many-body systems are described by a so called density matrix which has no straightforward analogue in classical theory. However, in the 30’s Wigner proposed an alternative way of describing many-body systems, using a quasi-probability distribution function. This made the connection between classical and quantum kinetic theory clearer.

This thesis is concerned with modelling of quantum effects in plasmas. The focus lies on describing plasmas containing spin-1/2 particles. For this purpose, new models, based on quantum kinetic theory, are derived. This is achieved by starting from the evolution equation for the density matrix and applying a combination of the Wigner transformation for the position degree of freedom and the Q-transformation for the spin. The properties of the resulting kinetic theory are then investigated and it is shown to satisfy basic necessary criteria such as energy conservation. The kinetic equation is then used to derive a fluid theory for spin-1/2 particles.

In this thesis the kinetic and fluid models are applied to different problems in quantum plasma physics. For example it will be shown that the quantum electrodynamic correction to the electron g-factor can give rise to a wave mode which lacks classical analogue, and that spin may affect the damping rate of Alfvén waves. The models will also be applied to nonlinear problems and it will be shown that they give rise to modifications of the so called spin ponderomotive force.

The theory of quantum plasmas dates back to the first attempts of Wigner, Moyal and others to find a suitable formalism to describe many body quantum systems, and from this foundation a rich field has emerged. A plasma is a system consisting of many charged particles, and in a quantum plasma these particles have quantum mechanical properties, such as for example spin. Quantum properties of individual particles are often negligible on macroscopic scales, but due to collective interaction in the plasma certain phenomena arise that can only be explained by considering the fundamental quantum properties of the particles.

In this thesis kinetic descriptions, derived following the work of Wigner et al. and extended to also include particle spin, are employed to study various nonlinear phenomena related to the magnetic field generation and the ponderomotive force in quantum plasmas consisting of spin 1/2 particles. A specific model to study the special case of low temperature degenerate quantum plasmas by only considering dynamics on the surface of the velocity distribution sphere is also derived, and is applied to the problem of nonlinear Landau damping. Furthermore, by taking moments of the full kinetic theory a more nimble spin plasma fluid model is derived. This formalism is then applied to a variety of nonlinear problems involving the ponderomotive force and wave-wave interaction. Where possible the fluid model is shown to be in agreement with results derived from the considerably more complex full kinetic theory.