  Requiring causality on measurements in quantum field theory seems to impose
strong conditions on a self-adjoint operator to be really measurable. This may
seem limiting and artificial in the operator language of algebraic quantum
field theory (AQFT), but is essential for a truly relativistic theory. Recent
publications attempt to deal with this issue by including the apparatus into
the formalism, connecting AQFT with measurement theory, but other options have
been suggested. In this essay, I discuss the causality conditions on
self-adjoint operators both in the language of AQFT and in the language of
quantum information theory. I then present measurement theory in AQFT,
modelling the apparatus as a quantum field with coupling to the measured system
restricted to a region of spacetime. I highlight how this approach leads to a
causally well behaved theory. Finally, I attempt to formulate the causality
conditions on measurements in the Feynman path integral approach, using the
concept of decoherent histories. I claim that the path integral approach has
problems with causality similar to the operator based approaches and that even
here causality is an a posteriori condition.
