Weak value and the wave–particle duality

Abstract

The weak value, introduced by Aharonov et al. (Phys Rev Lett 60:1351, 1988), extends the conventional scope of physical observables in quantum mechanics, allowing them to bear values outside the range of eigenvalues. On one hand, the weak value sheds new light on understanding quantum paradoxes, and also the method of obtaining it—the weak measurement—is expected to be useful for precision measurement. On the other hand, it poses serious questions on its physical meaning due to the unconventional features including the possible complexity of its value. In this paper, we point out that this complexity of the weak value has relevance to the wave–particle duality of quantum mechanics, at least in the nonrelativistic regime of a massive particle system. Namely, we show that the wave nature manifests itself in the imaginary part of the weak value while the particle nature appears in the real part. This is illustrated by the double slit experiment, where we find, without conflicting with complementarity, that the trajectory of the particle is inferable from the weak value even under the presence of interference.