Weak coupling limit and removing an ultraviolet cutoff for a Hamiltonian of particles interacting with a quantized scalar field

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12 Citations (Scopus)

Abstract

An interaction system consisting of particles and a quantized scalar field is considered. The Hamiltonian of the system is defined as a self-adjoint operator in a Hilbert space. An ultraviolet cutoff is imposed on the Hamiltonian. A renormalized Hamiltonian is defined by subtracting a renormalization term from the Hamiltonian. Our aim in this paper is to remove the ultraviolet cutoff and take the weak coupling limit simultaneously for the renormalized Hamiltonian. By using a functional integral that contains a vector-valued stochastic integral, a Schrödinger Hamiltonian with a many-body Coulomb potential (resp., Yukawa potential) is derived, if the mass of the quantized scalar field is zero (resp., positive).

Original languageEnglish
Pages (from-to)1215-1236
Number of pages22
JournalJournal of Mathematical Physics
Volume40
Issue number3
DOIs
Publication statusPublished - Jan 1 1999
Externally publishedYes

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Weak Coupling
Ultraviolet
Scalar Field
cut-off
scalars
Yukawa potential
Coulomb potential
Hilbert space
operators
Coulomb Potential
Functional Integral
Stochastic Integral
interactions
Self-adjoint Operator
Renormalization
Zero
Term
Interaction

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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author = "Fumio Hiroshima",
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AB - An interaction system consisting of particles and a quantized scalar field is considered. The Hamiltonian of the system is defined as a self-adjoint operator in a Hilbert space. An ultraviolet cutoff is imposed on the Hamiltonian. A renormalized Hamiltonian is defined by subtracting a renormalization term from the Hamiltonian. Our aim in this paper is to remove the ultraviolet cutoff and take the weak coupling limit simultaneously for the renormalized Hamiltonian. By using a functional integral that contains a vector-valued stochastic integral, a Schrödinger Hamiltonian with a many-body Coulomb potential (resp., Yukawa potential) is derived, if the mass of the quantized scalar field is zero (resp., positive).

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