  Suppose the postulate of measurement in quantum mechanics can be extended to
quantum field theory, then a local projective measurement at some moment on an
object locally coupled with a relativistic quantum field will result in a
projection or collapse of the wavefunctional of the combined system defined on
the whole time-slice associated with the very moment of the measurement, if the
relevant degrees of freedom have nonzero correlations. This implies that the
wavefunctionals in the same Hamiltonian system but defined in different
reference frames would collapse on different time-slices passing through the
same local event where the measurement was done. Are these post-measurement
states consistent with each other? We illustrate that the quantum states of the
Raine-Sciama-Grove detector-field system started with the same initial Gaussian
state defined on the same initial time-slice, then collapsed by the
measurements on the pointlike detectors on different time-slices in different
frames, will evolve to the same state of the combined system up to a coordinate
transformation when compared on the same final time-slice. Such consistency is
guaranteed by the spatial locality of interactions and the general covariance
in a relativistic system, together with the spatial locality of measurements
and the linearity of quantum dynamics in its quantum theory.
