Renormalization, Symmetry Breaking, and Discrete Scale Invariance

One of the most basic applications of quantum mechanics is the study of a two body system with a spherically symmetric interaction. This problem is characterized simply by giving the mass of each particle as well as the inter-particle potential V (r) which is a function of the inter-particle separation only. This problem can be reduced in a straightforward way to a simpler one particle problem by introducing the reduced mass m = m1m2/(m1 +m2). All that remains is then to solve the familiar one particle Schrodinger equation for a particle of mass m in a force field given by V (r) with the appropriate boundary conditions. As we shall see, our cavalier attitude towards this final step is not entirely justified. We investigate a system interacting via a central potential V (r) = c/r, our basic goal being to determine the bound state spectrum and the scattering amplitudes of the system. Without giving too much away too soon, let us say that this naive goal is not physically realistic for our potential. It turns out that the r−2 potential is a special transition case leading to a large class of so called singular potentials. Singular potentials are so called for their singular behavior near the origin r = 0 and one finds that in general these potentials, if not supplemented with additional information, lead to ill defined physical systems. How does this singularity manifest itself in the physics of the potential V = g/r? When we put our potential into the Schrodinger equation and turn the mathematical crank, we find two linearly independent solutions as we would expect. However, we find that for a sufficiently attractive potential the usual boundary conditions (vanishing at infinity and regularity at zero) do not single out a unique physical solution. This means that without further information, the Schrodinger equation and the usual boundary conditions do not suffice to specify the physical properties of the r−2 interaction. In a true physical system, the r = 0 behavior of the Schrodinger equation will certainly not be valid. Since a particle must in general have momentum inversely proportional to a to sense features of the potential on the order of a (think de Broglie waves), at the very least relativistic effects will eventually be important as the particle begins to sense r = 0. This limit is actually much exaggerated, the r−2 behavior of the potential is modified by any number of other effect in a true physical system. In general we find different physical systems will probably have quite different physical mechanism for ameliorating the r = 0 behavior of the potential. Having seen that the r = 0 behavior of the potential is connected to high momentum (and thus high energy) modes, we might expect that low energy observables, low compared to the energy scale of new physics, might be independent of the specific cutoff mechanisms of different physical systems. In other words, we would expect physically a kind of universal behavior, at least among the low energy observables, for systems interacting via a r−2 tail. We may thus calculate low energy observables using any particular cutoff we like, probably a computationally convenient one, so long as it regulates the r = 0 behavior of the potential or equivalently the high momentum modes. The final piece of the puzzle amounts to erasing the cutoff dependence of low energy observables. Since naively imposing a momentum cutoff affects observables at all energy scales we turn to the machinery of renormalization theory to keep our low energy observables independent of the cutoff. In addition to the cutoff, renormalization theory demands that we also supply a counterterm within our governing equation that is a function of the cutoff. We choose the functional dependance of the counterterm such that the low energy observables of the theory remain fixed as we vary the cutoff. The resulting theory is mathematically well defined because of the high momentum cutoff. The low energy observables calculated using our modified Schrodinger equation will be, by construction, independent of the cutoff. These low energy observables will be universal to all systems interacting via an r−2 tail in the sense that they do not depend on the details of the cutoff mechanism but merely on the presence of such a mechanism.