  The Fourier transform is often used to connect the Lorentzian energy
distribution for resonance scattering to the exponential time dependence for
decaying states. However, to apply the Fourier transform, one has to bend the
rules of standard quantum mechanics; the Lorentzian energy distribution must be
extended to the full real axis $-\infty<E<\infty$ instead of being bounded from
below $0\leq E <\infty$ (``Fermi's approximation''). Then the Fourier transform
of the extended Lorentzian becomes the exponential, but only for times $t\geq
0$, a time asymmetry which is in conflict with the unitary group time evolution
of standard quantum mechanics. Extending the Fourier transform from
distributions to generalized vectors, we are led to Gamow kets, which possess a
Lorentzian energy distribution with $-\infty<E<\infty$ and have exponential
time evolution for $t\geq t_0 =0$ only. This leads to probability predictions
that do not violate causality.
