We investigate the properties of a quantum walk which can simulate the behavior of a spin 1/2 particle in a model with an ordinary spatial dimension, and one extra dimension with warped geometry between two branes. Such a setup constitutes a \(1+1\) dimensional version of the Randall–Sundrum model, which plays an important role in high energy physics. In the continuum spacetime limit, the quantum walk reproduces the Dirac equation corresponding to the model, which allows to anticipate some of the properties that can be reproduced by the quantum walk. In particular, we observe that the probability distribution becomes, at large time steps, concentrated near the “low energy” brane, and can be approximated as the lowest eigenstate of the continuum Hamiltonian that is compatible with the symmetries of the model. In this way, we obtain a localization effect whose strength is controlled by a warp coefficient. In other words, here localization arises from the geometry of the model, at variance with the usual effect that is originated from random irregularities, as in Anderson localization. In summary, we establish an interesting correspondence between a high energy physics model and localization in quantum walks.

Anglés-Castillo, A., Pérez, A. A quantum walk simulation of extra dimensions with warped geometry. Sci Rep 12, 1926 (2022). https://doi.org/10.1038/s41598-022-05673-2

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When quantum electrodynamics, the quantum field theory of electrons and photons, was being developed after World War II, one of the major challenges for theorists was calculating a value for the Lamb shift, the energy of a photon resulting from an electron transitioning from one hydrogen hyperfine energy level to another.

The effect was first detected by Willis Lamb and Robert Retherford in 1947, with the emitted photon having a frequency of 1,000 megahertz, corresponding to a photon wavelength of 30 cm and an energy of 4 millionths of an electronvolt—right on the lower edge of the microwave spectrum. It came when the one electron of the hydrogen atom transitioned from the 2P_1/2 energy level to the 2S_1/2 level. (The leftmost number is the principal quantum number, much like the discrete but increasing circular orbits of the Bohr atom.)

Conventional quantum mechanics didn’t have such transitions, and Dirac’s relativistic Schrödinger equation (naturally called the Dirac equation) did not have such a hyperfine transition either, because the shift is a consequence of interactions with the vacuum, and Dirac’s vacuum was a “sea” that did not interact with real particles.