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

We consider the Schrödinger equation for a relativistic point particle in an external 1-dimensional $δ$-function potential. Using dimensional regularization, we investigate both bound and scattering states, and we obtain results that are consistent with the abstract mathematical theory of self-adjoint extensions of the pseudo-differential operator $H = \sqrt{p^2 + m^2}$. Interestingly, this relatively simple system is asymptotically free. In the massless limit, it undergoes dimensional transmutation and it possesses an infra-red conformal fixed point. Thus it can be used to illustrate non-trivial concepts of quantum field theory in the simpler framework of relativistic quantum mechanics.