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.


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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