In quantum field theory the objects of interest are the n-point vacuum expectations which can be
calculated from the path integral. The path integral usually used in physics is not well-defined and
the main motivation for this thesis is to give axioms that a well-defined path integral candidate
has to at least satisfy for it to be physically relevant - we want the path integral to have properties
which allow us to reconstruct the physically interesting objects from it.
The axioms given in this thesis are called the Osterwalder-Schrader axioms and the reconstruction of
the physical objects from the path integral satisfying the axioms is called the Osterwalder-Schrader
reconstruction. The Osterwalder-Schrader axioms are special in the sense that they are stated in
terms of the Euclidean spacetime instead of the physically relevant Minkowski spacetime. As the
physical objects live in Minkowski spacetime this means that when reconstructing the physically
relevant objects we have to go back to Minkowski spacetime at some point.
This thesis has three parts (and an introduction). In the first part we give a brief introduction
to parts of functional analysis which we will need later - theory about distributions and about
generators of families of operators. The second part is about the Osterwalder-Schrader axioms
and about the reconstruction of the physically relevant objects from the path integral. In the last
part we check that the path integral for the free field of mass m satisfies the Osterwalder-Schrader
axioms.
