Abstract

In a recent paper (Berry in Eur J Phys 42: 015401, 2020), the boundaries of superoscillatory regions (the regions where a function oscillates faster than its fastest Fourier component) of waves described by the Helmholtz equation in a uniform medium were related to zeros of the quantum potential, arising in the Madelung formulation of quantum mechanics. We generalize this result, showing that the relativistic counterpart, which is, essentially, a Klein-Gordon equation, exhibits the same behaviour, but in spacetime, giving rise to anomalous values for the local mass (instead of local momenta, in the classical case). This amounts to the generalization of the condition for superoscillations from the magnitude of a vector, to the norm of a four-vector. For a photon, boundaries of superoscillatory regions are surfaces on which the photon’s trajectory is locally light-like, separating time-like and space-like regions, which correspond to real and imaginary local mass, respectively.