[Paper Review] Ground States of the Yukawa models with Cutoffs

Toshimitsu Takaesu|ArXiv.org|Dec 25, 2009

Spectral Theory in Mathematical Physics15 references3 citations

TL;DR

This paper establishes the existence of ground states for the massive Yukawa model with both ultraviolet and spatial cutoffs by proving a uniform positive spectral gap for the total Hamiltonian across all coupling constants. Using momentum lattice approximation and norm resolvent convergence, it shows that the Hamiltonian has purely discrete spectrum near its bottom, ensuring a ground state exists despite the interaction's non-perturbative strength.

ABSTRACT

Ground states of the so called Yukawa model is considered. The Yukawa model describes a Dirac field interacting with a Klein-Gordon field. By introducing both ultraviolet cutoffs and spatial cutoffs, the total Hamiltonian is defined as a self-adjoint operator on a boson-fermion Fock space. It is shown that the total Hamiltonian has a positive spectral gap for all values of coupling constants. In particular the existence of ground states is proven.

Motivation & Objective

• To establish the existence of ground states for the Yukawa model with both ultraviolet and spatial cutoffs.
• To show that the total Hamiltonian has a positive spectral gap for all values of coupling constants, including non-perturbative regimes.
• To overcome limitations of perturbation theory and extend results from massless models to massive, interacting quantum field theories.
• To provide a rigorous spectral analysis of the Hamiltonian on a boson-fermion Fock space using cutoff regularization.

Proposed method

• Introduces ultraviolet cutoffs on both Dirac and Klein-Gordon fields to define well-behaved creation/annihilation operators on Fock space.
• Imposes a spatial cutoff in the interaction term to ensure integrability and localization.
• Applies momentum lattice approximation to discretize the system, transforming the Hamiltonian into a sequence of finite-dimensional operators.
• Uses the Kato-Rellich theorem to ensure self-adjointness and boundedness from below of the total Hamiltonian.
• Employs norm resolvent convergence to show that the lattice-regularized Hamiltonians converge to the full Hamiltonian in the limit of large lattice size.
• Applies spectral theory to prove that the lattice Hamiltonians have purely discrete spectrum with a uniform spectral gap, inherited by the full Hamiltonian.

Experimental results

Research questions

• RQ1 Does the Yukawa model with both ultraviolet and spatial cutoffs possess a ground state for all values of the coupling constant?
• RQ2 Can a positive spectral gap be established for the total Hamiltonian even in the non-perturbative regime?
• RQ3 Is the ground state structure preserved under the limit of removing cutoffs, particularly in the spatial and ultraviolet directions?
• RQ4 Can the spectral gap be uniformly bounded from below across all coupling constants using lattice approximation?
• RQ5 How does the presence of massive fields affect the spectral properties compared to massless models?

Key findings

• The total Hamiltonian of the Yukawa model with both ultraviolet and spatial cutoffs is self-adjoint and bounded from below.
• The Hamiltonian has a positive spectral gap for all values of the coupling constant κ > 0.
• The spectral gap is uniformly bounded below by a positive constant independent of the coupling strength.
• The ground state exists as an isolated eigenvalue at the bottom of the spectrum.
• The lattice-regularized Hamiltonians H_{L,V} have purely discrete spectrum in a neighborhood of their ground state energy.
• The full Hamiltonian H inherits the purely discrete spectrum near its bottom via norm resolvent convergence, confirming the existence of a ground state.