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I need help carrying out the following calculation:

We have the generating functional of free theory:

$$Z[f] = \exp\left(\frac{i}{2} \int d^4xd^4y f(x)f(y)\Delta(x-y)\right) $$

where $f$ is an external field and $\Delta(x-y)$ the Green function of the Klein-Gordon equation.

The two-point function $\langle{0}| T\phi(x) \phi(y) |0\rangle$ is then calculated by the rule

$$\frac{1}{i^2} \frac{\delta Z[f]}{\delta f (x_1) \delta f (x_2)}\bigg|_{f=0}$$

The result should of course be $i\Delta(x-y)$.

Can someone show me the explicit steps? I just obtain gibberish.

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Omitting space-time indices (cf. DeWitt notation): $$ Z[f]=\mathrm e^{\frac12f\cdot\Delta\cdot f} $$

Therefore, $$ Z'[f]=\Delta\cdot f\ \mathrm e^{\frac12f\cdot\Delta\cdot f} $$ and $$ Z''[f]=\Delta\ \mathrm e^{\frac12f\cdot\Delta\cdot f}+(\Delta\cdot f)^2\ \mathrm e^{\frac12f\cdot\Delta\cdot f} $$

Therefore, setting $f=0$, we get $$ Z''[0]=\Delta $$ as required.

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  • $\begingroup$ Thank you. At least I was on the right track and apparently did some calculation errors. $\endgroup$
    – Jakob Elias
    Commented Apr 26, 2017 at 13:16

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