  First, the present work is concerned with generalising constructions and
results in quantum field theory on curved spacetimes from the well-known case
of the Klein-Gordon field to Dirac fields. To this end, the enlarged algebra of
observables of the Dirac field is constructed in the algebraic framework. This
algebra contains normal-ordered Wick polynomials in particular, and an extended
analysis of one of its elements, the stress-energy tensor, is performed. Based
on detailed calculations of the Hadamard coefficients of the Dirac field, it is
found that a construction of a stress-energy tensor fulfilling necessary
physical properties is possible. Additionally, the mathematically sound
Hadamard regularisation prescription of the stress-energy tensor is compared to
the mathematically less rigorous DeWitt-Schwinger regularisation and it is
found that both prescriptions are essentially equivalent in rigorous terms.
  While the aforementioned results hold in generic curved spacetimes,
particular attention is also devoted to a specific class of Robertson-Walker
spacetimes with a lightlike Big Bang hypersurface. Employing holographic
methods, Hadamard states for the Klein-Gordon and the Dirac field are
constructed. These states are preferred in the sense that they constitute
asymptotic equilibrium states in the limit to the Big Bang hypersurface.
Finally, solutions of the semiclassical Einstein equation for quantum fields of
arbitrary spin are analysed in the flat Robertson-Walker case. One finds that
these solutions explain the measured supernova Ia data as good as the $\La$CDM
model. Hence, one arrives at a natural explanation of dark energy and a simple
quantum model of cosmological dark matter.
  It is the hope of the author that the present thesis can serve as an
accessible introduction to the field of (algebraic) quantum field theory on
curved spacetimes and its recent developments.
