[Paper Review] On Zero-Mass Bound States in Super-Membrane Models

Jens Hoppe|ArXiv.org|Sep 30, 1996

Quantum chaos and dynamical systems3 citations

TL;DR

This paper investigates zero-mass bound states in supermembrane matrix models by analyzing symmetry-reduced Schrödinger-type equations with fermionic contributions modeled as attractive δ-function potentials. It proves that real solutions to these equations are not square-integrable, ruling out physical real bound states, though complex solutions remain mathematically possible, suggesting a need for further analysis in complexified configuration spaces.

ABSTRACT

For the simplest case of a supermembrane matrix model, various symmetry reductions are given, with the fermionic contribution(s) (to an effective Schrödinger equation) corresponding to an attractive $δ$-function potential (towards zero-area configurations). The differential equations are real, and are shown not to admit square-integrable real solutions (even when allowing non-vanishing boundary conditions at infinity). Complex solutions, however, are not excluded by this argument.

Motivation & Objective

• To analyze the existence of zero-mass bound states in the simplest supermembrane matrix model.
• To investigate how fermionic contributions affect the effective potential in reduced symmetry models.
• To determine whether square-integrable real solutions exist for the resulting Schrödinger-type equations.
• To explore the implications of the absence of real solutions for the physical interpretation of zero-mass states.
• To assess the viability of complex solutions as candidates for physical bound states.

Proposed method

• Reduces the supermembrane model using various symmetry assumptions to simplify the dynamics.
• Derives an effective Schrödinger equation with a fermionic contribution modeled as an attractive δ-function potential at zero-area configurations.
• Analyzes the resulting differential equations, which are real-valued and time-independent.
• Applies standard quantum mechanical criteria for square-integrability to determine the existence of normalizable states.
• Considers boundary conditions at infinity and evaluates whether non-vanishing limits allow for integrable solutions.
• Explores the mathematical possibility of complex solutions despite the exclusion of real ones.

Experimental results

Research questions

• RQ1 Do zero-mass bound states exist in the reduced supermembrane matrix model with fermionic δ-function potentials?
• RQ2 Are there square-integrable real solutions to the effective Schrödinger equation derived from the symmetry-reduced model?
• RQ3 What is the role of the fermionic contribution in shaping the effective potential toward zero-area configurations?
• RQ4 Can non-vanishing boundary conditions at infinity allow for physical solutions in the absence of square-integrability?
• RQ5 Are complex solutions viable candidates for describing zero-mass bound states when real solutions are ruled out?

Key findings

• No square-integrable real solutions exist for the effective Schrödinger equation derived from the symmetry-reduced supermembrane model.
• The fermionic contribution generates an attractive δ-function potential centered at zero-area configurations, which is crucial for the analysis.
• Even with non-vanishing boundary conditions at infinity, the real differential equations do not admit normalizable solutions.
• The absence of real solutions does not rule out the existence of complex solutions, which remain mathematically possible.
• The results suggest that zero-mass bound states, if they exist, must be described by complex wave functions, requiring further investigation beyond real-valued quantum mechanics.