# Determining this vacuum expectation

I am trying to find the analytic expression for the result that follows from evaluating this vacuum expectation value:

$\langle0\vert;\prod_{i=1}^M \prod_{j=1}^N \hat{a}(y_{ij}) \hat{a}^\dagger(y'_{ij});\vert0\rangle$.

Where I denote the anti-normal ordering with the "$;$" symbol. I have already determined a similar, but simpler, expectation value which was:

$\langle0\vert;\prod_{i=1}^N \hat{a}(y_{i}) \hat{a}^\dagger(y'_{i});\vert0\rangle = \sum_\sigma \prod_{i=1}^N \delta(y_i - y'_{\sigma(i)}).$

How could I reach a similar expression for the former expression? The main difficulty is that there are different number of annihilation operators to the creation operators.

The usual commutator definitions hold for the field operators.

This expression arises from the normalisation of a wavefunction which generates a series of Guassian sources (i.e. creation operators operating on the vaccuum state, hence why you can see the creation operators on the right of the expectation value.) Thanks.

• A bit of context would be nice. Why is this expression interesting? What have you tried? – ACuriousMind Sep 17 '15 at 12:49
• @ACuriousMind, question edited. – Sid Sep 17 '15 at 15:18
• @ACuriousMind, I have made use of the Q-Representation since it is well suited to evaluate the inner product for anti-normally ordered products. However, I am not able to find an analytic expression still. – Sid Sep 19 '15 at 15:18