Fundamental impossibility of a superradiant neutrino laser

Here we address the fundamental question whether an idealized system of $N$ atoms will show collective behavior and superradiance when it emits fermions instead of photons. We show that the maximum emission is $\propto N$ and not $\propto N^2$ which proves the absence of superradiance and shows that the recent proposal to realize a superradiant neutrino laser is impossible. This can be understood as either destructive interference of fermionic transition amplitudes, or Pauli blockade by collective excitations with fermionic nature. On the other hand, states with low excitation can show collective behavior. We derive the exact solution of the fermionic Dicke problem and analyze the decay dynamics in various regimes.