I am studying the Schwinger-Keldysh formalism. Basically, we double the number of degrees of freedom for the upper and lower branches.

Let´s consider the case where we have a certain field, coupled to an external current, given by the Lagrangian:

$$L \equiv (\partial_\mu \varphi)(\partial^\mu\varphi) - U(\varphi) + j\varphi$$

Now, in the Schwinger-Keldysh technique, we are going to have the fields and currents:

$$\varphi_+;\varphi_-;j_+;j_-$$

After we perform whatsoever the calculations we want to, typically something like:

$$ \langle 0_{in}|P \space φ_{−}(x_1) · · · φ_{−}(x_{n}) \space φ_{+}(y_1) · · · φ_{+}(y_{p})|0_{in}\rangle $$ ($P$ stands for the ordering in the Schwinger-Keldysh contour, that is to say normal ordering in the uppper branch and anti-time ordering in the lower branch) we take the external fields in both branches to be identical.

Does it have any physical meaning to keep the external fields different in our final result?