A full theory harmonizing the description of the gravitational field, presently provided by classic general relativity, with the principles of quantum mechanics is still lacking. However, once a theory of quantum gravity is devised, spacetime is expected not to be viewed as a continuous entity but to manifest a granular or foam-like structure in experiments involving distances approaching a “minimal length.” Even if spacetime is continuous on all scales, it is expected that distances smaller than the minimal length cannot be accessed due to a modification of the Heisenberg Uncertainty Principle (HUP) into a Generalized Uncertainty Principle (GUP). This has consequences on many predictions associated to quantum mechanics so that the exploration of experiments that might, at least in principle, reveal this distortion has produced a large phenomenological research literature. In this paper, which was awarded an Honorable Mention in the 2020 Essay Competition of the Gravity Research Foundation, I consider the distortion of dispersion forces, such as the unretarded van der Waals force and the retarded Casimir-Polder force, due to the GUP. Interestingly, despite the explosively growing body of published research on minimal length scenarios, I am aware of only one paper, published by Panella in 2007, in which the effects on interatomic forces are analyzed within the rigorous framework of quantum field theory. This is in contrast with the several papers on modifications of the macroscopic Casimir effect, that is, the pressure between two macroscopic, parallel, perfectly conducting plates separated by an empty gap. With the consent of the Editor of the issue reproducing the winning essays, Professor Dharam Vir Ahluwalia, my winning paper was expanded to offer a review of previous work and to show my independent calculation of the unretarded van der Waals potential by means of a computer algebra system (CAS). I was able to confirm that, as Panella had originally found, in this regime, corresponding to an infinite speed of light, the GUP has no effect. In order to consider the retarded regime, instead of carrying out a rigorous calculation, I used heuristic arguments designed to shed light of the physical meaning of the result. My approach was founded on a calculation, published by Mania and Maziashvili in 2011, of the distortion of the black-body radiation spectrum, including the zero-point-field, due to the GUP. As shown by Spruch in 1986, the standard zero-point-field can be used to obtain an estimate of the Casimir-Polder potential. By following the path illustrated by Spruch, but replacing the standard expression for the zero-point field by its distorted counterpart, I was able to recover the result by Panella to within a numerical factor. Although a handful of papers have appeared justifying the expression of the Casimir potential in minimal length by means of elementary arguments, to the best of my knowledge, my discussion is the only one published to-date offering a heuristic derivation of the distorted Casimir-Polder potential. Once these tools are developed, I apply my results to the particular case of dispersion forces in neutron-neutron scattering, first considered by Arnold in 1973 and again very recently, and much more thoroughly, by Babb and collaborators. My calculations show that, although the effects of GUP distortions are extremely small, considering neutron-neutron collisions, at interparticle distances as small as one Fermi, yields a remarkable gain in the magnitude of the distortion term of over 18 orders of magnitude. The analysis of detailed predictions about this type of experiments from the GUP-distorted Schrödinger equation is presently underway and will appear in future publications.
