Monopoles, electric dipoles, and the inverse square potential

Research is a fulfilling enterprise, especially when it takes us on an unexpected path that leads to new insights even when others may have crossed the same path.

A couple years ago on Christmas eve, I was toying with a calculation based on my realization that a potential of the form V(x) = x^q captures all of the interesting toy models of quantum mechanics such as the harmonic oscillator (q=2), the particle in a box (q = ∞), and the hydrogen atom (q= -1). I wanted to derive a general analytical expression for the energy spectrum as a function of q so that I could analyze the nonlinear-optical properties of all such systems. To my delight, I was able to get such an expression using a mathematical trick. I was am certain that I was not the first to do so, but I nevertheless felt satisfied.

A plot of my results showed a removable singularity at q=0, which corresponds to a free particle that has no bound states (bound states are needed for my calculation of the nonlinear-optical properties), and a divergence at q = -2. I dutifully wrote up the results and left them sitting on my computer for a couple years. The work was given new life when I handed it over to new graduate student.

After working on the problem for a summer and part of a semester, he found a huge literature on the x^-2 potential, the location of the divergence in my plot. It turns out that this potential has all sorts of interesting mathematical problems. It has a continuum of bound states -- very unusual and in fact an impossibility; and, the wave functions and their duals do not span the same space -- a peculiar state of affairs. The known fix, as the student found in the literature, is to exclude a small region near x = 0 and then solve the problem in the limit as this small region tends to zero. Doing so fixes the problems by essentially removing the point-like properties of the dipole. 

The x^-2 potential describes the influence of an electric dipole moment on a point charge. Certainly nature must reconcile these funny mathematical difficulties given that dipoles and point charges exist. But how?

Nature is very clever by not allowing true point electric dipoles to exist.  Instead, all dipoles are extended dipoles where the potential deviates from x^-2 at close range, thus removing the pathological behavior. In fact, this may be the reason why an ideal point dipole is not observed in any particle. If it existed, serious paradoxes would ensue.

The electron has a very tiny electric dipole moment beyond the limits of measurability with today's technology, and arises from the CP-violating part of the CKM matrix in the standard model. The moment is tiny because CP violation involves quarks that are created as virtual particles, interact with the electron, and then are annihilated. As a result, the dipole moment is not a point dipole but is due to a sea of virtual quarks. All elementary particles have only small electric dipole moments with an extended charged cloud, so nature avoids potential pathologies by forbidding point dipoles.

Magnetic dipoles, which are a consequence of moving charges through spin and orbital angular momentum, are a different story. Spin angular momentum does not originate from spread out charge that spins in physical space.The spin anguar momentum lives in its own space so the associated magnetic dipole moment is pointlike, and will lead to pathologies when interacting with a magnetic monopole. Nature has apparently avoided such problems by forbidding the existence of magnetic monopoles. While there are theories that allow for magnetic monopoles, searches for them have come up short and I don't think they will be found.

In summary, the interaction of a point dipole that interacts with a point charge leads to all kinds of pathologies that can only be resolved by demanding that the dipole be an extended object that avoids being a true x^-2 potential. Nature seems to have solved the problem by allowing point electric charges but no point electric dipoles. Magnetic point dipoles, on the other hand exist; but, there are no magnetic monopoles with which they can interact.  The symmetry between the electric and magnetic parts of Maxwell's equations is broken to avoid unphysical behavior.

I cannot claim to be the first person to have this idea and wouldn't be surprised if this argument is common knowledge. Or, my argument may be faulty.  This does not make my musings any less exciting to me. I continue to be in awe of the power of physics, which allows us to ponder domains that are far removed from our daily experiences. My job not only allows me to think of such things, but encourages it; it is indeed a wonderful life

In studying the complex ramifications of the simple x^-2 potential when applied to nonlinear optics leads to insights into why point electric dipoles and magnetic monopoles can't exist.  How can I sleep tonight!