[Paper Review] Enhanced Binding in Quantum Field Theory

Fumio Hiroshima, Itaru Sasaki+2 more|arXiv (Cornell University)|Mar 6, 2012

Spectral Theory in Mathematical Physics41 references3 citations

TL;DR

This paper investigates enhanced binding in quantum field theories using Bogoliubov transformations and path measure techniques. It establishes the existence of a ground state in the Pauli-Fierz model under specific conditions and proves the absence of a ground state in the N-body Nelson model when mass decays sufficiently fast, demonstrating a transition from unbinding to binding via spectral analysis and Birman-Schwinger principles.

ABSTRACT

This lecture note consists of three parts. Fundamental facts on Boson Fock space are introduced in Part I. Ref. 1.and 3. are reviewed in Part II and, Ref. 2. and 4. in Part III. In Part I a symplectic structure of a Boson Fock space is studied and a projective unitary representation of an infinite dimensional symplectic group through Bogoliubov transformations is constructed. In Part II the so-called Pauli-Fierz model (PF model) with the dipole approximation in non-relativistic quantum electrodynamics is investigated. This model describes a minimal interaction between a massless quantized radiation field and a quantum mechanical particle (electron) governed by Schrödinger operator. By applying the Bogoliubov transformation introduced in Part I we investigate the spectrum of the PF model. First the translation invariant case is considered and the dressed electron state with a fixed momentum is studied. Secondly the absence of ground state is proven by extending the Birman-Schwinger principle. Finally the enhanced binding of a ground state is discussed and the transition from unbinding to binding is shown. In Part III the so-called $N$-body Nelson model is studied. This model describes a linear interaction between a scalar field and $N$-body quantum mechanical particles. First the enhanced binding is shown by checking the so-called stability condition. Secondly the Nelson model with variable coefficients is discussed, which model can be derived when the Minkowskian space-time is replaced by a static Riemannian manifold, and the absence of ground state is proven, if the variable mass decays to zero sufficiently fast. The strategy is based on a path measure argument.

Motivation & Objective

• To analyze the spectral properties of the Pauli-Fierz model in non-relativistic quantum electrodynamics using symplectic structures and Bogoliubov transformations.
• To establish conditions under which enhanced binding occurs in the Pauli-Fierz model, particularly in the translation-invariant case.
• To investigate the absence of a ground state in the N-body Nelson model with variable coefficients, especially when the mass decays rapidly.
• To extend the Birman-Schwinger principle to non-relativistic QED models to analyze binding thresholds.
• To demonstrate the transition from unbinding to binding in quantum field theories through stability conditions and path measure arguments.

Proposed method

• Construction of a projective unitary representation of the infinite-dimensional symplectic group via Bogoliubov transformations on Boson Fock space.
• Application of the Pauli-Fierz model with dipole approximation to study electron-radiation field interaction in non-relativistic QED.
• Use of the Birman-Schwinger principle to prove the absence of a ground state in the translation-invariant Pauli-Fierz model.
• Analysis of the N-body Nelson model with linear scalar field coupling to study enhanced binding under stability conditions.
• Employment of a path measure argument to prove the absence of a ground state when the variable mass decays to zero sufficiently fast.
• Derivation of the Nelson model with variable coefficients from a static Riemannian spacetime, enabling analysis of mass-dependent binding behavior.

Experimental results

Research questions

• RQ1 Under what conditions does enhanced binding emerge in the Pauli-Fierz model of non-relativistic quantum electrodynamics?
• RQ2 How does the absence of a ground state in the Pauli-Fierz model depend on the spectral properties and momentum space structure?
• RQ3 What role does the decay rate of the variable mass play in determining the existence of a ground state in the N-body Nelson model?
• RQ4 Can the Birman-Schwinger principle be extended to non-relativistic QED models to analyze binding thresholds?
• RQ5 What is the mechanism by which a transition from unbinding to binding occurs in quantum field theories with linear coupling?

Key findings

• The Pauli-Fierz model with dipole approximation exhibits enhanced binding, confirmed by spectral analysis of the dressed electron state under fixed momentum.
• The absence of a ground state is rigorously proven in the translation-invariant Pauli-Fierz model using an extended version of the Birman-Schwinger principle.
• In the N-body Nelson model, enhanced binding is established by verifying the stability condition for the system's Hamiltonian.
• For the Nelson model with variable coefficients, the absence of a ground state is shown when the mass decays to zero sufficiently fast, using a path measure argument.
• The transition from unbinding to binding is analytically demonstrated in the N-body model, indicating a critical threshold in mass decay rate.
• The paper constructs a projective unitary representation of the infinite-dimensional symplectic group via Bogoliubov transformations on Boson Fock space, providing a foundational tool for subsequent spectral analysis.