On a Direct Description of Pseudorelativistic Nelson Hamiltonians

Abstract

Abstract interior-boundary conditions (IBC's) allow for the direct description of the domain and the action of Hamiltonians for a certain class of ultraviolet-divergent models in Quantum Field Theory. The method was recently applied to models where nonrelativistic scalar particles are linearly coupled to a quantised field, the best known of which is the Nelson model. Since this approach avoids the use of ultraviolet-cutoffs, there is no need for a renormalisation procedure. Here, we extend the IBC method to pseudorelativistic scalar particles that interact with a real bosonic field. We construct the Hamiltonians for such models via abstract boundary conditions, describing their action explicitly. In addition, we obtain a detailed characterisation of their domain and make the connection to renormalisation techniques. As an example, we apply the method to two relativistic variants of Nelson's model, which have been renormalised for the first time by J. P. Eckmann and A. D. Sloan in 1970 and 1974, respectively.

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Cited by in corpus (14)

• Hamiltonians Without Ultraviolet Divergence for Quantum Field Theories
• The Renormalised Bogoliubov-Fröhlich Hamiltonian
• A nonrelativistic quantum field theory with point interactions in three dimensions
• The resolvent of the Nelson Hamiltonian improves positivity
• Interior-Boundary Conditions for Schrodinger Operators on Codimension-1 Boundaries
• Interior-Boundary Conditions for the Dirac Equation at Point Sources in 3 Dimensions
• Hamiltonians for polaron models with subcritical ultraviolet singularities
• Non-Fock Ground States in the Translation-Invariant Nelson Model Revisited Non-Perturbatively
• Extended State Space for describing renormalized Fock spaces in QFT
• A Lower Bound on the Critical Momentum of an Impurity in a Bose-Einstein Condensate
• Feynman-Kac formula and asymptotic behavior of the minimal energy for the relativistic Nelson model in two spatial dimensions
• Boundary Conditions that Remove Certain Ultraviolet Divergences
• Implementing Bogoliubov Transformations Beyond the Shale-Stinespring Condition
• On the resolvent of $H+A^{\ast }+A$