[Paper Review] Non-existence of ground states in the translation invariant Nelson model

Thomas Norman Dam|arXiv (Cornell University)|Jul 31, 2018

Nonlinear Waves and Solitons8 references4 citations

TL;DR

This paper proves the non-existence of ground states in the translation-invariant massless Nelson model with ultraviolet cutoff, using rotation invariance and non-degeneracy of the ground state instead of requiring differentiability or infrared regularity. The result extends prior work by removing restrictive assumptions on the spectral behavior of the fiber Hamiltonians.

ABSTRACT

In this paper we consider the massless translation invariant Nelson model with ultraviolet cutoff. It is proven that the fiber operators have no ground state if there is no infrared cutoff.

Motivation & Objective

• To establish the absence of ground states in the massless, translation-invariant Nelson model under minimal assumptions.
• To remove the need for strong regularity conditions on the mass shell, such as differentiability or infrared cutoffs, which were required in prior works.
• To demonstrate that ground states do not exist for any non-zero coupling strength μ and all ξ ∈ ℝ³.
• To use only rotation invariance and non-degeneracy of the ground state as structural assumptions, rather than spectral smoothness.
• To provide a rigorous foundation for understanding why the mass gap limit strategy fails in this model.

Proposed method

• Analyzes the fiber Hamiltonians Hμ(ξ) as a direct integral over ξ ∈ ℝ³, exploiting the translation invariance of the model.
• Applies the HVZ theorem to study the spectral structure and identify the absence of isolated ground state energies.
• Uses rotation invariance of the map ξ ↦ inf(σ(Hμ(ξ))) to constrain the possible behavior of the spectral infimum.
• Employs the pull-through formula for the annihilation operator A to relate the action of Hμ(ξ) on vectors to the resolvent of Hμ(ξ−k)+ω(k).
• Relies on the non-degeneracy of the ground state and the structure of the Fock space to rule out the existence of an L²-normalizable vector minimizing the energy.
• Constructs a contradiction by assuming a ground state exists and showing that the resulting vector cannot be square-integrable due to the lack of infrared regularization.

Experimental results

Research questions

• RQ1 Does the massless, translation-invariant Nelson model admit a ground state when no infrared cutoff is imposed?
• RQ2 Can the non-existence of ground states be proven without assuming differentiability or regularity of the mass shell function ξ ↦ inf(σ(Hμ(ξ)))?
• RQ3 What structural properties—such as rotation invariance and non-degeneracy—suffice to rule out ground states in this model?
• RQ4 Why does the standard strategy of taking the mass gap to zero fail in this context?
• RQ5 How does the absence of a ground state affect the physical interpretation of the model, particularly in relation to Fock representations?

Key findings

• The fiber Hamiltonians Hμ(ξ) have no ground state for any μ ≠ 0 and all ξ ∈ ℝ³ in the massless, translation-invariant Nelson model.
• The non-existence of ground states holds even without assuming infrared regularity or differentiability of the spectral infimum function.
• The result is established using only rotation invariance of the energy-momentum map and the non-degeneracy of the ground state, without requiring strong spectral smoothness.
• The pull-through formula for the annihilation operator is used to derive a resolvent identity that leads to a contradiction if a ground state exists.
• The absence of a ground state implies that the usual 'mass gap to zero' limit strategy fails to produce a ground state, supporting the physical intuition of instability in the massless case.
• The model does not support a ground state in the standard Fock representation, though Pizzo has shown existence in a non-equivalent Fock representation, indicating a fundamental structural issue in the standard formulation.