  We develop the axiom system proposed by Kontsevich and Segal to define
volume-dependent field theories (VFTs), a class of non-topological quantum
field theories whose dependence on the background metric factors through the
associated density. We construct a well-defined Lorentzian limit of a
Wick-rotated VFT defined on smooth, possibly degenerate Lorentzian bordisms
with incoming and outgoing boundary both nonempty. When the VFT is reflection
positive, this extends the main theorem given by Kontsevich and Segal.
