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We consider the Schr\"odinger equation for a relativistic point particle inan external 1-dimensional $\delta$-function potential. Using dimensionalregularization, we investigate both bound and scattering states, and we obtainresults that are consistent with the abstract mathematical theory ofself-adjoint extensions of the pseudo-differential operator $H = \sqrt{p^2 +m^2}$. Interestingly, this relatively simple system is asymptotically free. Inthe massless limit, it undergoes dimensional transmutation and it possesses aninfra-red conformal fixed point. Thus it can be used to illustrate non-trivialconcepts of quantum field theory in the simpler framework of relativisticquantum mechanics.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Asymptotic Freedom, Dimensional Transmutation, and an Infra-red  Conformal Fixed Point for the $\delta$-Function Potential in 1-dimensional  Relativistic Quantum Mechanics