It seems to me that the whole fuss about switching to euclidean time to remove wildly oscillatory behaviour in many integrals found in QFT to be unnecessary. For instance if we calculate <0|0>j the result is a path integral / <0|0>[j=0]. However we cheat in our mathematics by demanding that <0|0>[j=0] is equal to 1. Mathematically <0|0>j=0 is actually some integral with wildly oscilatory behaviour because we choose to take |0> at t'=+inf and <0| at t=-inf. It is entirely reasonable to declare the amplitude to be 1 but there is no reason to say that the phase is zero. We could have simply recognized the phase to be a meaningless quantity undetectable to experiment. Now going back to <0|0>[j=0], we foolishly choose to integrate to infinity when in fact all the action happens between t1 and t2, when j~=0. Therefore, let us simply take the beginning time and the end time to be large (again quite foolish) but restricted to the condition E0*(t' - t) = 2*pi*N, for some large N. Now all our integrals end up with no phase contribution due to time evolution outside of [t1,t2]. If we did not artificially contrive t' and t as above, we end up with some meaningless phase, so be it. Sure its not elegant but its saves a few pages in your QFT textbook. Am i missing something?