Radiation from Dirac fermions caused by a projective measurement

The theory of radiation of photons from Dirac particles caused by a projective measurement is developed. The explicit expressions for the inclusive probability to record a chain of events that the Dirac fermion had been measured in a certain state and after that the photon was recorded are derived. Stimulated and spontaneous radiations are considered. It is shown that in both cases the properties of radiation due to measurement resembles the properties of edge or transition radiation. In the case of stimulated radiation from a single particle, its wave function creates photons coherently as a charged fluid, i.e., the amplitudes of radiation from the points of the particle wave packet are summed. In the case of spontaneous radiation, the radiation of photons is incoherent, i.e., the probabilities of radiation from the points of the particle wave packet are added up. It is shown that stimulated radiation due to measurement can be used to trace the dynamics and collapse of the wave function of the Dirac particle. A systematic procedure taking into account a finiteness of the measurement time is presented. It is established that radiation due to measurement can be used as a source of hard photons, but the finiteness of the measurement time imposes an upper bound on the energy of radiated photons: the measurement time must be smaller than the radiation formation time, the latter being in inverse proportion to the photon energy. In the ultrarelativistic limit, the radiation formation time can be rather large. Several examples of radiation due to measurement of the state of free Dirac particles are investigated in detail. Namely, we scrutinize the radiation due to measurement of the spin projection, of the momentum, and of the coordinate for a general initial state of Dirac particles. The particular cases of uncorrelated and entangled particles in the beam are considered.