I'm currently trying to study a chapter in Altland & Simons, "Condensed Matter Field Theory" (2nd edition) and I'm stuck at the end of section 9.5.2, page 579.

Given the euclidean Chern-Simons action for a gauge field $a_µ$ that is coupled to a current $j_µ$

$$ S[a_µ,j_µ] = ∫d^3x (j_µ a_µ + \frac{iθ}4ε_{µνλ}a_µ ∂_ν a_λ) $$

the task is to integrate out the gauge field and obtain the effective action for the current.

Since this is a gauge field, we have to take care about the superfluous gauge degree of freedom. Altland & Simons note that one way to do it would be to introduce a gauge fixing term $α (∂_µ a_µ)^2$ and let $α\to ∞$ at the end.

However, this does not seem to work. In momentum space, the Chern-Simons action plus gauge fixing terms is proportional to

$$∫ d^3q\ a_µ(-q) \left(
\begin{array}{ccc}
\alpha q_0^2 & -i q_2 & i
q_1 \\
i q_2 & \alpha q_1^2 & -i
q_0 \\
-i q_1 & i q_0 & \alpha
q_2^2
\end{array}
\right)_{µν} a_ν(q) .$$

To get the effective action for the current, I just have to invert this matrix, which we call $A_{µν}$, and send $α\to ∞$. But this can't be. For instance, one entry of the inverse matrix reads

$$ A^{-1}_{01} = \frac{-q_0 q_1-i q_2^3 \alpha
}{q_1^2 q_2^2 q_0^2 \alpha
^3- α(q_0^4+q_1^4+q_2^4) } $$

and this vanishes in the limit $α\to∞$. Same for the other entries. This is bad.

My question, hence

How to properly perform the functional integral over a gauge field $a_µ$ with a gauge fixing contribution $α(∂_µ a_µ)^2$ where $α\to ∞$?

I am aware that there are other methods, for instance to integrate only over the transverse degrees of freedom, as Altland & Simons note. I don't mind learning about them as well, but I would like to understand the one presented here in particular. Not to mention that I may have made a simple mistake in the calculation above.

This post imported from StackExchange Physics at 2014-04-05 17:31 (UCT), posted by SE-user Greg Graviton