### 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 language | English |
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Pages (from-to) | 1215-1236 |

Number of pages | 22 |

Journal | Journal of Mathematical Physics |

Volume | 40 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 1 1999 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Weak coupling limit and removing an ultraviolet cutoff for a Hamiltonian of particles interacting with a quantized scalar field.** / Hiroshima, Fumio.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Weak coupling limit and removing an ultraviolet cutoff for a Hamiltonian of particles interacting with a quantized scalar field

AU - Hiroshima, Fumio

PY - 1999/1/1

Y1 - 1999/1/1

N2 - 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).

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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UR - http://www.scopus.com/inward/citedby.url?scp=0033241358&partnerID=8YFLogxK

U2 - 10.1063/1.532796

DO - 10.1063/1.532796

M3 - Article

AN - SCOPUS:0033241358

VL - 40

SP - 1215

EP - 1236

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 3

ER -