- #1
QuantumDevil
- 29
- 0
In some QFT books it is written that the generating functional
[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?
[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?
Last edited: