In the Feynman lectures on Physics, he talks about quantum tunneling and, basically, there's a non-zero amplitude to be found in the barrier, where the particle's energy is below the rest mass, i.e. kinetic energy is negative, and because $E_k = p^2 / (2m)$, that implies $p$ is imaginary. Can this actually be observed? Can you collapse the wave function by putting a detector in the barrier and get a count? What properties does matter have in this state?