A note on the renormalized square of free quantum fields in space–time dimension d⩾4

We consider the renormalized or Wick square of the free quantum field, which is well-defined as a Hida distribution. It is known that this is a random variable for space–time dimension d⩽3. In this report, by considering the characteristic function, we show that this Wick square is not a random variable in dimension d⩾4.