David Miller with Tanmay Vachaspati

From Inverse Scattering To Field Theory

In a general quantum mechanical problem, one can reconstruct an analytic potential from the spectrum of eigenvalues for the system using symmetric reflectionless potentials. This technique works well for reconstructing potentials in simple 1-dimensional problems.

    We are interested in the field theory underlying an excitation spectrum of eigenvalues of a topological defect in the field. The potential detailed by this set of eigenvalues is inherently connected to the field theory underlying it. This means that one can determine the complete field theory from the excitation spectrum of the defect. The process for recovering the field theory is not guaranteed to be analytic however, so numerical processes are necessary in some cases. Cases where the field theory is analytic serve as useful examples for qualitative analysis. We can use this method to determine the field theory in problems that are one-dimensional and ones in which we have S-wave eigenfunctions. Specifically, we wish to consider a generally more complicated spectrum of eigenvalues, and see what we can determine about the field theory underlying it, and the transitions therein when the eigenvalues change.