Asymptotic freedom, dimensional transmutation, and an infrared conformal fixed point for the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>δ</mml:mi></mml:mrow></mml:math>-function potential in one-dimensional relativistic quantum mechanics

We consider the Schr\"odinger equation for a relativistic point particle in an external one-dimensional $\ensuremath{\delta}$-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 pseudodifferential 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 infrared conformal fixed point. Thus it can be used to illustrate nontrivial concepts of quantum field theory in the simpler framework of relativistic quantum mechanics.

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