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

Abstract

We consider the Schrödinger equation for a relativistic point particle in an external 1-dimensional $\delta$-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 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.

References in corpus (1)

• Non locality and causal evolution in QFT

Cited by in corpus (4)

• One-Dimensional Semirelativistic Hamiltonian with Multiple Dirac Delta Potentials
• Majorana Fermions in a Box
• A Perturbative Approach to the Tunneling Phenomena
• A Solution of the Relativistic Schrödinger Equation for the $\delta$-Function Potential in 1-dimensiona with Cutoff Regularization