# Time ordering operator, interaction Lagrangian, QED

In summary, the problem statement involves calculating a quantity using the time-ordering operator and a Lagrangian function. The solution should include terms up to second order in the coupling constant. To simplify the expression, one can use properties of the time-ordering operator and the nature of the initial and final states.

## Homework Statement

I am trying to calculate the following quantity:
$$<0|T\{\phi^\dagger(x_1) \phi(x_2) exp[i\int{L_1(x)dx}]\}|0>$$

where:

$$L_1(x) = -ieA_{\mu}[\phi^* (\partial_\mu \phi ) - (\partial_\mu \phi^*)\phi]$$[/B]

I am trying to find an expression including the propagators, wick's theorem, and then calculate the Feynman diagrams in position space.
The solution should include terms of up to ## e^2 ## order.

## The Attempt at a Solution

I am not sure if the Lagrangian ## L_1(x) ## is the interaction Lagrangian, so that I should use it as it is in the integral, or if I should extract an interaction Lagrangian out of it.
I also have to tell you that I am not expected to calculate complex integrals in every detail, or be occupied with infinities.
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