  This paper introduces the notion of exponential arcs in Hilbert space and of
exponential arcs connecting vector states on a sigma-finite von Neumann algebra
in its standard representation. Results from Tomita-Takesaki theory form an
essential ingredient. Starting point is a non-commutative Radon-Nikodym theorem
that involves positive operators affiliated with the commutant algebra. It is
shown that exponential arcs are differentiable and that parts of an exponential
arc are again exponential arcs. Special cases of probability theory and of
quantum probability are used to illustrate the approach.
