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Enhanced binding for a quantum particle coupled to scalar quantized field

Volker Betz, Tobias Schmidt, Mark Sellke

TL;DR

The paper proves enhanced binding for a single quantum particle linearly coupled to a scalar quantum field (the one-particle Nelson model) by employing a functional-integral approach and the Gaussian correlation inequality. By formulating a Nelson path measure $\hat{\mathbb P}_{\delta,\alpha,T}$ and performing a sequence of confinement, half-time recurrence, and spectral analyses, the authors show that for sufficiently large coupling $\alpha$ a ground state exists even when the external potential alone would not bind. The results connect localization of Brownian paths under attractive field-induced interactions to spectral binding, yielding a rigorous existence and uniqueness result for the ground state under the stated assumptions. This methodology provides a robust, non-perturbative route to enhanced binding beyond small-coupling perturbations and clarifies the role of pathwise localization in particle-field systems.

Abstract

Enhanced binding of a quantum particle coupled to a quantized field means that the Hamiltonian of the particle alone does not have a bound state, while the particle-field Hamiltonian does. For the Pauli--Fierz model, this is usually shown via the binding condition, which works less well in the case of a linear coupling to a scalar field. In particular, the case of a single particle linearly coupled to a scalar field has been open so far. Using a method relying on functional integrals and the Gaussian correlation inequality, we obtain enhanced binding for this case. From a statistical mechanics point of view, our result describes a localization phase transition (in the strength of the pair potential) for a Brownian motion subject to an external and an attractive pair potential.

Enhanced binding for a quantum particle coupled to scalar quantized field

TL;DR

The paper proves enhanced binding for a single quantum particle linearly coupled to a scalar quantum field (the one-particle Nelson model) by employing a functional-integral approach and the Gaussian correlation inequality. By formulating a Nelson path measure and performing a sequence of confinement, half-time recurrence, and spectral analyses, the authors show that for sufficiently large coupling a ground state exists even when the external potential alone would not bind. The results connect localization of Brownian paths under attractive field-induced interactions to spectral binding, yielding a rigorous existence and uniqueness result for the ground state under the stated assumptions. This methodology provides a robust, non-perturbative route to enhanced binding beyond small-coupling perturbations and clarifies the role of pathwise localization in particle-field systems.

Abstract

Enhanced binding of a quantum particle coupled to a quantized field means that the Hamiltonian of the particle alone does not have a bound state, while the particle-field Hamiltonian does. For the Pauli--Fierz model, this is usually shown via the binding condition, which works less well in the case of a linear coupling to a scalar field. In particular, the case of a single particle linearly coupled to a scalar field has been open so far. Using a method relying on functional integrals and the Gaussian correlation inequality, we obtain enhanced binding for this case. From a statistical mechanics point of view, our result describes a localization phase transition (in the strength of the pair potential) for a Brownian motion subject to an external and an attractive pair potential.

Paper Structure

This paper contains 13 sections, 20 theorems, 133 equations.

Table of Contents

  1. Introduction and Results
  2. Other approaches for showing the existence of bound states
  3. The binding condition
  4. The effective mass
  5. Persistence of binding
  6. Proof of Main Results
  7. Notation and the Gaussian correlation inequality
  8. A preliminary confinement result
  9. Proof of Theorem \ref{['recurrence_half_time_theorem']}
  10. Proof of Theorem \ref{['main_thm']} and Proposition \ref{['simple_gs']}
  11. Proof of Proposition \ref{['approx_markov']}
  12. Two basic results from quantum theory
  13. Criteria for the existence of ground states

Key Result

Theorem 1.2

Assume potential_assumption, pair_potential_assumption and local_decrease. Fix $\delta >0$ and let Then, there exists $\alpha^* > 0$ such that $H(\alpha)$ has a ground state for all $\alpha \;\geqslant\; \alpha^*$.

Theorems & Definitions (41)

  • Example 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 31 more