On Quantum Tunneling in Real Time

A detailed real time description of quantum tunneling in the semiclassical limit is given, using complex classical trajectories. This picture connects naturally with the ideas of post-selection and weak measurement introduced by Aharonov and collaborators. I show that one can precisely identify the {\it complex} classical trajectory which a post-selected tunneling particle has followed, and which dominates the path integral in the limit as Planck's constant $\hbar$ tends to zero. Detailed analytical calculations are presented for tunneling in cubic and quartic potentials. For a long post-selected tunneling time, the imaginary part of the tunneling coordinate is found to achieve very large values just before the particle tunnels. I discuss how the real and imaginary parts of the particle's coordinate may, in principle, be independently measured using weak measurements. It would be very interesting to observe this effect, which would demonstrate the essential role of complex numbers in our closest possible classical description of reality. Extensions to quantum field theory and general relativity are briefly discussed.