In some QFT books it is written that the generating functional(adsbygoogle = window.adsbygoogle || []).push({});

[tex]Z[J]=\int \mathcal{D}\phi e^{i\int d^{4}x(\mathcal{L}_{o} +V(\phi) +J\phi) }[/tex]

can be expressed in equivalent form:

[tex]Z[J]=e^{i\int d^{4}xV(\phi)} \int \mathcal{D}\phi e^{i\int d^{4}x(\mathcal{L}_{o} +J\phi )}[/tex].

The only argument supporting this statement I found is that [tex]V(\phi)[/tex] does not depend on J. But I'm still suspicious about it because we have still to integrate over all possible paths [tex]\mathcal{D}\phi[/tex], which is ommited in the second definition of the generating functional.

So...can anybody explain me why these two froms of [tex]Z[J][/tex] are equivalent?

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# Interaction term in path integral

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