Rindler's metric is an interesting way to incorporate a set of uniformly accelerated observers into space-time coordinates; this is consistent with special and general relativity. It is known [1] that such an acceleration gives rise to the famous Unruh effect, yet to be confirmed experimentally [2]. Interestingly, its Galilean limit already shows the appearance of quantized modes for massive particles in free space. This happens when a wall or a boundary condition is moving in an accelerated trajectory in free space and in the presence of a field.

Here we show that such a boundary condition, when viewed as a material point-like obstacle in motion, gives rise to quantized modes for the Klein-Gordon and Maxwell fields, despite the flatness of the original space. Technically, the description involves Hankel functions of imaginary arguments and indices, as well as the fall-to-the-origin potential $-1/x^2$, also known as quantum anomaly.

This counterintuitive result can always be compared with a reversed physical situation where a field representing a particle is subjected to a fictitious force against a rigid wall. Our view draws a parallel with the well-known experiment of a vertically bouncing neutron [3], where bound states appear and Galileo's universality of free fall is obviously violated.

With this result, the reverse image of the Unruh effect that describes radiation against a detector in motion, is indeed that of a bouncing particle for Schroedinger, Klein-Gordon and Maxwell equations. It is yet to be argued that this represents an experimental confirmation of particle creation in second quantization.

[1] L. C. B. Crispino, A. Higuchi, and G. E. A. Matsas, Rev Mod Phys 80, 787 (2008)

[2] G. Cozzella, A. G. S. Landulfo, G. E. A. Matsas, and D. A. T. Vanzella, Phys Rev Lett 118, 161102 (2017)

[3] R. Colella, A. Overhauser, and S. Werner, Phys Rev Lett 34, 1472 (1975)

Poster presentation.

1. The Schroedinger equation under a moving boundary condition is solved. Quantized modes are obtained.
2. The Klein-Gordon field shows spectral anomalous potentials, leading also to bound states upon specification of one boundary condition.
3. A component of the Maxwell field for vertical linear polarization also suffers quantization when bouncing against a single mirror in motion.
4. In first quantization, these problems show that free field modes overlap significantly with bound states produced by a moving wall.
5. This gives us a glimpse to nontrivial transition amplitudes, yet to be exploited in second quantization.