Wick contraction in scalar QED

[Markus Kahn]

Homework Statement

Expand
$$\left\langle 0\left|\mathrm{T}\left[\exp \left(i \int_{-T}^{T} \mathrm{d} y^{4} \mathcal{L}_{\mathrm{int}}(y)\right)\right]\right| 0\right\rangle,$$
where we have
$$\mathcal{L}_{int}(x)= iqA_\mu (\phi\partial^\mu\phi^*-\phi^*\partial^\mu\phi)+i^2q^2 A_\mu A^\mu |\phi|^2$$
up to second order in $q$.

Relevant Equations

All given above

While writing out the Dyson series due to the time ordering above I encountered the two expressions
$$T(\mathcal{L}_{int}(x))\quad \text{and}\quad T(\mathcal{L}_{int}(x)\mathcal{L}_{int}(y))$$
I was able to write out the first term in terms of contractions using Wick's theorem and then finally in terms of propagators... But when I try to do the same for the second term I encountered some problems. If you expand the second term by inserting the definition of $\mathcal{L}_{int}(x)$ one of the terms this will result in is
$$\langle A_\mu(x)A_\nu(y)\phi(x)[\partial^\mu \phi^*(x)]\phi(y)[\partial^\nu \phi^*(y)] \rangle, $$
where the $\langle\dots \rangle$ is to be understood as the sum over all possible contractions of the fields inside the brackets. The relations I'm familiar with in this context are
$$\langle A_\mu (x)A_\nu(y)\rangle = G_V^{\mu\nu}(x-y)\quad \langle \phi^*(x)\phi(y) \rangle=G_F(x-y)\quad \langle \phi(x)\phi(y) \rangle=0,$$
where $G_V$ is the photon propagator and $G_F$ the Feynman propagator of the real scalar field.

My issue is now, that I don't know how one is supposed to evaluate expressions like
$$\langle A_\mu(x)\phi(y)\rangle\quad \text{and}\quad \langle \phi(x)\partial^\mu\phi^*(y)\rangle,$$
I'm not even sure if these make sense since $\partial^\mu \phi$ is a vector while $\phi$ is obviously a scalar (same for $A_\mu$ and $\phi$). But if you really can't evaluate the two examples, then how does one evaluate
$$\langle A_\mu(x)A_\nu(y)\phi(x)[\partial^\mu \phi^*(x)]\phi(y)[\partial^\nu \phi^*(y)] \rangle$$

Any hints are appreciated.

 

[mark726]

it is important to have a clear understanding of the mathematical expressions you are working with. In this case, it seems like you are working with a time-ordered product of operators, which can be expanded using Wick's theorem. However, when trying to expand the second term, you encountered some difficulties.

Firstly, it is important to note that the time-ordered product is a formal expression that is defined as a series in powers of the interaction Hamiltonian. Therefore, it is not necessary to evaluate the expressions in the way you would for a regular operator.

In the first term, you were able to write it in terms of contractions using Wick's theorem and then in terms of propagators. However, in the second term, it may not be possible to do the same. This is because Wick's theorem only applies to products of fields at the same spacetime point. In the second term, you have a product of fields at different points, making it more complicated.

In order to evaluate expressions like $\langle A_\mu(x)\phi(y)\rangle$ and $\langle \phi(x)\partial^\mu\phi^*(y)\rangle$, you can use the definition of the propagator and the commutation/anticommutation relations of the fields. For example, using the commutation relation $[\phi(x),\partial^\mu\phi^*(y)]=\delta^{(4)}(x-y)$, you can evaluate the second expression as $\langle \phi(x)\partial^\mu\phi^*(y)\rangle=\langle \phi(x)\phi^*(y)\partial^\mu\rangle-\partial^\mu\langle \phi(x)\phi^*(y) \rangle=0$. Similarly, you can evaluate the first expression as $\langle A_\mu(x)\phi(y)\rangle=\langle \phi(y)A_\mu(x)\rangle=0$.

As for the final expression, you can use the same techniques to evaluate it. However, it may be a complicated expression and you may need to use more advanced techniques to simplify it. It is also possible that it cannot be simplified and may require numerical calculations instead.

In summary, it is important to have a clear understanding of the definitions and commutation/anticommutation relations of the fields involved in order to evaluate expressions involving time-ordered products. You may also need to use more advanced techniques to simplify the expressions,