The fact that probability amplitudes are complex-valued means that under addition (“superposition”) they exhibit quantum interference (the addition of their complex phases) in stark contrast to the addition of probability densities, for which this cannot happen.

For instance on some probability space$(X,\mu)$ there are the probability amplitudes $\exp(i \pi/2) \mu$ and $\exp(-i \pi/2)\mu$ whose associated probability densities are both just $\mu$ itself again. But the sum of these two probability amplitudes vanishes, in contrast to the sum of their associated probability densities. This is known as “destructive” quantum interference.

Scattering amplitudes

In perturbative quantum field theory the key probability amplitudes considered are scattering amplitudes, which encode the probability for a given configuration of field quanta to come in from the far past, interact with each other and hence “scatter off” each other, and then re-emerge as some other set of field quanta in the far future.