[Paper Review] Equivalence of renormalization with self-adjoint extension in Green's function formalism

D.K. Park, Sahng-Kyoon Yoo|ArXiv.org|Dec 15, 1997

Quantum Mechanics and Non-Hermitian Physics1 references4 citations

TL;DR

This paper establishes the equivalence between renormalization and self-adjoint extension in the Green's function formalism for two- and three-dimensional delta-function plus harmonic oscillator potentials. By deriving energy-dependent Green's functions through both methods and imposing a specific relation between the self-adjoint extension parameter and the renormalized coupling constant, the authors show that both approaches yield identical results, unifying two distinct mathematical frameworks in quantum field theory and quantum mechanics.

ABSTRACT

Energy-dependent Green's functions for the two and three dimensional $δ$-function plus harmonic oscillator potential systems are derived by incorporating the renormalization and the self-adjoint extension into the Green's function formalism, respectively. It is shown that both methods yield an identical Green's function if a certain relation between the self-adjoint extension parameter and the renormalized coupling constant is imposed.

Motivation & Objective

• To investigate the mathematical equivalence between renormalization and self-adjoint extension in quantum mechanical systems with singular potentials.
• To resolve the ambiguity in defining self-adjoint extensions for singular interactions like the delta function in higher dimensions.
• To unify two different approaches—renormalization and self-adjoint extension—within the Green's function formalism.
• To derive consistent energy-dependent Green's functions using both methods and compare their results.

Proposed method

• Derives energy-dependent Green's functions for 2D and 3D systems with a delta-function plus harmonic oscillator potential using the Green's function formalism.
• Applies renormalization techniques to handle divergences arising from the singular delta interaction.
• Implements self-adjoint extension by introducing a parameter that characterizes the boundary condition at the origin.
• Constructs the Green's function via the spectral representation, incorporating the self-adjoint extension parameter as a boundary condition parameter.
• Compares the resulting Green's functions from both methods under a specific mapping between the self-adjoint extension parameter and the renormalized coupling constant.
• Demonstrates that the two formalisms yield identical Green's functions when this parameter mapping is applied.

Experimental results

Research questions

• RQ1 Can the self-adjoint extension method reproduce the same Green's function as renormalization in singular quantum systems?
• RQ2 What is the precise mathematical relationship between the self-adjoint extension parameter and the renormalized coupling constant?
• RQ3 Do both approaches yield consistent physical predictions in two and three dimensions for delta-function plus harmonic oscillator potentials?
• RQ4 Is the equivalence between renormalization and self-adjoint extension robust across different spatial dimensions?
• RQ5 How does the Green's function formalism facilitate the comparison of these two foundational approaches in quantum theory?

Key findings

• The Green's functions derived via renormalization and self-adjoint extension are identical when a specific relation is imposed between the self-adjoint extension parameter and the renormalized coupling constant.
• The equivalence holds in both two and three spatial dimensions, demonstrating the robustness of the mapping across dimensions.
• The self-adjoint extension parameter effectively encodes the same physical information as the renormalized coupling constant in the renormalization approach.
• The energy-dependent Green's functions derived through both methods are mathematically consistent and physically equivalent under the parameter mapping.
• The study provides a rigorous bridge between two distinct mathematical frameworks—renormalization and self-adjoint extension—within a unified Green's function framework.
• The results confirm that self-adjoint extension is not just a mathematical tool but a physically equivalent alternative to renormalization in singular quantum systems.