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The quantity $Z[j]=\int D\phi\exp[iS[\phi,j]]$ represents the vacuum-to-vacuum transition amplitude in presense of an external source $j(x)$. Shouldn't the quantity $Z[0]$ i.e., the vacuum-to-vacuum transition amplitude in absence of the external source $j(x)$ be unity? But the integral $\int D\phi\exp[iS[\phi]]$ is not unity. This is because if the system is initially prepared in the vacuum state then in the absence of the source it will continue to be in that state.

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Whenever you calculate a quantity like a transition amplitude, you have to normalize it. This is familiar from normal QM or stat mech. It's extra-true when you're using path integrals, since the definition of the path integral often drops irrelevant prefactors.

What should the normalization factor be? You've just discovered the normalization factor is $Z[0]$.

In other words: the vacuum-to-vacuum amplitude is really $Z[j]/Z[0]$. Plugging in $j=0$ then gives 1, as desired.

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