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Hello! I am reading from Schwarz book on QFT the Path Integral chapter and I am confused about something. I attached a SS of that part. So we have $$<\Phi_{j+1}|e^{-i\delta H(t_j)}|\Phi_{j}>=N exp(i\delta t \int d^3x L[\Phi_j,\partial_t \Phi_j])$$ What happens when we have the left and right most terms i.e. $$<\Phi_{1}|e^{-i\delta H(t_0)}|0>$$ and $$<0|e^{-i\delta H(t_n)}|\Phi_{n}>$$ Another thing that I am confused about, where are we using the fact that the state in the beginning and the end is ##|0>##? I see we are using the boundaries on time, but I can't see where we use explicitly the fact that we start and end with vacuum i.e. if we had any ##<f|i>## where would we have a different term?