Causality and time dependence in quantum tunneling

Quantal penetration through a (stationary) one-dimensional potential barrier is considered as a time evolution of an initially prepared wave packet. The large-time asymptotics of the process is concerned. Locality of the potential imposes certain analytical properties of the interaction amplitudes in the energy representation. The results are presented in terms of development of the phase-space (Wigner's) quasi-distribution. The phase-space evolution kernel is constructed, and it is shown that in the presence of a positive potential no part of the distribution is transported faster than the free particle. For the case of a small initial momentum uncertainty, the deformation of the coordinate density is considered, including a possible advance of its maximum, which would not mean any noncausal signal transport. Supported by G. I. F.