Can the product of two integrals be simplified into one using dummy variables?

[Hepth]

If I have an integral:
$$A = \int \frac{d^d p}{(2 \pi)^d} Z[p]$$

And I want $$A^* A$$

Is it
$$A^* A = \int \frac{d^d p}{(2 \pi)^d} Z^*[p] Z[p]$$ ?

Because the "p" is the same, and really it would be integral 1 times integral 2 times a delta, which should make it just one.

I don't think its true, and an example in mathematica doesn't work.

Is there a shortcut to get the product of two integrals to be one?:
$$(\int \frac{d^d p}{(2 \pi)^d} Z^*[p])( \int \frac{d^d p}{(2 \pi)^d} Z[p]) =$$

hmm, the product of two sums...

 

[jasonRF]

Recall that the variables you integrate over are dummy variables. So you should write
$$(\int \frac{d^d q}{(2 \pi)^d} Z^*[q])( \int \frac{d^d p}{(2 \pi)^d} Z[p]) = \int \frac{d^d q}{(2 \pi)^d} \int \frac{d^d p}{(2 \pi)^d} Z^*[q] Z[p]$$.

Whether or not you can do something clever with this to make it a single integral depends on the form of $$Z$$.