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 (12)

• Light Composite Higgs from Higher Representations versus Electroweak Precision Measurements -- Predictions for LHC
• Conformality Lost
• Self-adjoint extensions of operators and the teaching of quantum mechanics
• Chiral phase structure of QCD with many flavors
• Non-perturbative regularization and renormalization: simple examples from non-relativistic quantum mechanics
• Perturbative Renormalization in Quantum Mechanics
• From a Particle in a Box to the Uncertainty Relation in a Quantum Dot and to Reflecting Walls for Relativistic Fermions
• Localization and Causality for a free particle
• Walking near a Conformal Fixed Point: the 2-d O(3) Model at theta near pi as a Test Case
• Minimal Position-Velocity Uncertainty Wave Packets in Relativistic and Non-relativistic Quantum Mechanics
• Causality and Localization Operators
• Non locality and causal evolution in QFT

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• Self-adjoint extension approach for singular Hamiltonians in (2+1) dimensions
• On the Number of Bound States of Point Interactions on Hyperbolic Manifolds
• 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
• Tunneling dynamics of the relativistic Schrodinger/Salpeter equation