[Paper Review] Ground States for translationally invariant Pauli-Fierz Models at zero Momentum

David Hasler, Oliver Siebert|arXiv (Cornell University)|Jul 2, 2020

Spectral Theory in Mathematical Physics26 references4 citations

TL;DR

This paper proves the existence of a ground state for the translationally invariant Pauli-Fierz model at zero total momentum, using a non-perturbative compactness argument based on a natural energy inequality. The key result establishes that the fiber Hamiltonian $ H(0) $ has a ground state for all coupling constants, resolving infrared divergence issues critical at zero momentum.

ABSTRACT

We consider the translationally invariant Pauli-Fierz model describing a charged particle interacting with the electromagnetic field. We show under natural assumptions that the fiber Hamiltonian at zero momentum has a ground state.

Motivation & Objective

• To establish the existence of a ground state for the non-relativistic QED Hamiltonian restricted to zero total momentum.
• To resolve the critical infrared singularity that arises at zero momentum, where standard perturbative methods fail.
• To provide a non-perturbative, coupling-constant-independent proof that avoids reliance on ultraviolet cutoffs.
• To extend previous results—previously known only for small coupling or spinless cases—to the general case with spin.
• To lay a foundation for asymptotic expansions of binding energy and spectral properties in systems like the hydrogen atom.

Proposed method

• Utilizes a compactness argument based on weak convergence and spectral properties of the fiber Hamiltonian $ H(0) $ at zero momentum.
• Employs an energy inequality derived from the lower bound $ H_{m}( ilde{ heta}) riangleright E_m( ilde{ heta}) + ilde{ heta}^2/2 $, ensuring uniform control over the spectrum.
• Applies operator inequalities involving $ d ilde{ heta} $ and $ ilde{ heta}^2 $ to control the essential spectrum and prove gap existence.
• Uses the unitary transformation $ ilde{I} $ to relate $ H_m( ilde{ heta}) $ and $ H_m(- ilde{ heta}) $, preserving spectral structure under parity.
• Establishes the positivity of the spectral gap $ \Delta_m(\xi) > 0 $ for $ |\xi| \leq 1 $, crucial for excluding essential spectrum in the ground state sector.
• Relies on the compactness of $ (1 + H_{f,m})^{-1} \Gamma(j_1^2) $ on finite-particle subspaces to control weak limits in spectral analysis.

Experimental results

Research questions

• RQ1 Does the fiber Hamiltonian $ H(0) $ of the translationally invariant Pauli-Fierz model possess a ground state for arbitrary coupling strength?
• RQ2 Can the existence of a ground state be established without perturbative assumptions or ultraviolet cutoffs?
• RQ3 How does the infrared singularity at zero momentum affect the spectral structure of the system?
• RQ4 What conditions ensure the positivity of the spectral gap $ \Delta_m(\xi) $, and how does this relate to the existence of a ground state?
• RQ5 Can the non-perturbative method used here be generalized to other critical models with similar infrared divergences?

Key findings

• The fiber Hamiltonian $ H(0) $ has a ground state for all values of the coupling constant, under natural assumptions.
• The energy inequality $ H_{m}( ilde{ heta}) \triangleright E_m(\tilde{\theta}) + \tilde{\theta}^2/2 $ holds for all $ \tilde{\theta} $, ensuring spectral control.
• $ \Delta_m(\xi) > 0 $ for all $ |\xi| \leq 1 $, which implies a positive spectral gap and excludes the essential spectrum from the bottom of the spectrum.
• The proof is non-perturbative and independent of any ultraviolet cutoff, making it robust across coupling strengths.
• The method provides a framework that could be adapted to prove ground state existence in other critical models with infrared singularities.
• The existence of the ground state enables future asymptotic expansions of the binding energy and spectral data via other methods.