Understanding Vacuum Fluctuations Mathematically

[quantumfireball]

Vaccum fluctuations?

How to understand vacuum fluctuations mathematically without getting into the virtual particles that is so stereotypical of POP sci articles?
Am i right in saying that the vacuum expectation value of the square of electric field is inversely proportional to the fourth power of l.
where l is the length of the cube in where you are measuring the vev?

 

[olgranpappy]

here's a simple way to put it in terms of creation/annihilation operators (with all indices/sums suppresed).

In QED the electric field operator is given by (suppressing a bunch of indices and constants, etc):

$E\sim (a+a^\dagger)$

where 'a' annihilates and 'a^\dagger' creates.

then if <whatever> indicates the vacuum expectation value of 'whatever'

$$<E>=0$$
but

$$<E^2>\ne 0$$