The Solution of the Relativistic Schrodinger Equation for the $\delta'$-Function Potential in 1-dimension Using Cutoff Regularization

TL;DR

This paper solves the relativistic Schrödinger equation with a derivative delta potential in 1D using cutoff regularization, revealing behaviors similar to quantum field theories such as asymptotic freedom and conformal fixed points.

Contribution

It develops a new regularization procedure for the relativistic delta prime potential, analyzing its bound and scattering states, and compares it with the non-relativistic case.

Findings

The reciprocal of the bare coupling constant is UV divergent.

The relativistic system behaves like the delta function potential with quantum field theory features.

In the massless limit, the system exhibits dimensional transmutation and an infrared fixed point.

Abstract

We study the relativistic version of Schr\"odinger equation for a point particle in 1-d with potential of the first derivative of the delta function. The momentum cutoff regularization is used to study the bound state and scattering states. The initial calculations show that the reciprocal of the bare coupling constant is ultra-violet divergent, and the resultant expression cannot be renormalized in the usual sense. Therefore a general procedure has been developed to derive different physical properties of the system. The procedure is used first on the non-relativistic case for the purpose of clarification and comparisons. The results from the relativistic case show that this system behaves exactly like the delta function potential, which means it also shares the same features with quantum field theories, like being asymptotically free, and in the massless limit, it undergoes dimensional transmutation and it possesses an infrared conformal fixed point.