Fumio Hiroshima | Last modified date：2020.06.16 |

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Reseacher Profiling Tool Kyushu University Pure

Academic Degree

Doctor of science

Country of degree conferring institution (Overseas)

Yes

Field of Specialization

Analysis

Total Priod of education and research career in the foreign country

02years06months

Outline Activities

I am studying the spectral analysis of operators on an infinite dimensional space. Especially, from the mathematical standpoint, we investigate the quantum field theory on pseudo-Riemannian manifolds by using operator theory, micro-local analysis, theory of one-parameter semigroup, stochastic analysis, functional integral.

The Hamiltonian that appears in quantum field theory can be mathematically regarded as an unbounded self-adjoint operator on Hilbert space. The spectrum of the self-adjoint operator is analyzed nonperturbatively and abstractly. The Hamiltonian of quantum field theory has an eigenvalue embedded in a continuous spectrum when the coupling constant is zero. It has been found that unlike discrete eigenvalues, the analysis of embedded eigenvalues is subtle, and is different from the behavior of discrete eigenvalues. Kato's regular theory exists in the perturbation theory of discrete eigenvalues, and very many things have been established. However, there is no general theory about our analysis of the perturbations of the eigenvalues.

The goal is to analyze the existence / absence of the Hamiltonian ground state and the degree of degeneracy, analysis of infrared divergence / ultraviolet divergence, spectral scattering theory, resonance, renormalization theory, Gibbs measure, etc. In recent years, we have also been studying CCR representation theory and lattice Schrodinger-operators.

Keywords:

Feynmann-Kac formula, Gibbs measure, renormalization, Boson-Fock space, Euclidean field, infinite-dimensional Ornstein-Uhlenbeck process, Levy process, subordinator, exponential decay, Gaussian decay, ultraviolet cut, infrared cut, ground state, self-adjoint operator, noncommutative Perron-Frobenius theorem, local convergence of measure, Kato potential, Bernstein function, Pauli-Fierz model, Nelson model, spin-boson model, non-commutative harmonic oscillator (NCHO), Rabi model, spectrum zeta function, time operator, CCR, weak Weyl relation, Laplacian on lattice, semi-classical analysis

The Hamiltonian that appears in quantum field theory can be mathematically regarded as an unbounded self-adjoint operator on Hilbert space. The spectrum of the self-adjoint operator is analyzed nonperturbatively and abstractly. The Hamiltonian of quantum field theory has an eigenvalue embedded in a continuous spectrum when the coupling constant is zero. It has been found that unlike discrete eigenvalues, the analysis of embedded eigenvalues is subtle, and is different from the behavior of discrete eigenvalues. Kato's regular theory exists in the perturbation theory of discrete eigenvalues, and very many things have been established. However, there is no general theory about our analysis of the perturbations of the eigenvalues.

The goal is to analyze the existence / absence of the Hamiltonian ground state and the degree of degeneracy, analysis of infrared divergence / ultraviolet divergence, spectral scattering theory, resonance, renormalization theory, Gibbs measure, etc. In recent years, we have also been studying CCR representation theory and lattice Schrodinger-operators.

Keywords:

Feynmann-Kac formula, Gibbs measure, renormalization, Boson-Fock space, Euclidean field, infinite-dimensional Ornstein-Uhlenbeck process, Levy process, subordinator, exponential decay, Gaussian decay, ultraviolet cut, infrared cut, ground state, self-adjoint operator, noncommutative Perron-Frobenius theorem, local convergence of measure, Kato potential, Bernstein function, Pauli-Fierz model, Nelson model, spin-boson model, non-commutative harmonic oscillator (NCHO), Rabi model, spectrum zeta function, time operator, CCR, weak Weyl relation, Laplacian on lattice, semi-classical analysis

Research

**Research Interests**

- Spectral analysis of QFT

keyword : quantum field theory, Fock space, embedded eigenvalues, ground states, scattering theory, pseudo Riemann manifold, resonances, renormalization group, functional integrations, Gibbs measures

1993.04. - Semi-classical analysis of QFT

keyword : Wigner measure, coherent state, Pauli-Fierz model, Maxwell equations

2019.04. - Feynman-Kac formula and related topics

keyword : Schroedinger operator, relativistic Schroedinger operator, Schroedinger operator with spin, relativistic Schroedinger operator with spin, PF model, Nelson model, spin-boson model

2006.04. - Gibbs measure and localizations

keyword : Gibbs measure, ground states, UV renormalization, exponential decay

2000.04. - Spectral zeta functions

keyword : Rabi model, non-commutative harmonic oscillator

2013.04. - Time operators and CCR representations

keyword : time operator, CCR

2009.04. - Schrodinger operator on lattice

keyword : lattice, spectrum

2012.04.

**Academic Activities**

**Books**

**Reports**

**Papers**

**Presentations**

1. | Fumio Hiroshima, Localizations of bound states of a renormalized Hamiltonian, Scattering, microlocal analysis and renormalisation,Mittag-Leffler Institute, Stockholm, 2020.06. |

2. | Fumio Hiroshima, Semi-classical analysis and Wigner measure for non-relativistic QED, NAKAMURA60, Gakushuin univ. , 2020.01. |

3. | Fumio Hiroshima, Stochastic renormalization, MSJ, Nippon univ. , special lecture, 2019.09. |

4. | Fumio Hiroshima, Hierarchy of CCR representations, Aarhus university, seminar, Denmark, 2018.10. |

5. | Fumio Hiroshima, Integral kernels of semigroup generated by a model in quantum field theory, 18th Workshop: Noncommutative Probability, Operator Algebras, Random Matrices and Related Topics, with Applications,Bedlewo, Poland, 2018.07. |

6. | Fumio Hiroshima, Ground state of renormalized Nelson model, Aarhus university, Math. Phys. seminar, Denmark, 2017.12. |

7. | Fumio Hiroshima, Schroedinger operators on lattice, Second Summer School: Various aspects of mathematical physics,St Petersburg,Russia, 2017.07, [URL]. |

8. | Fumio Hiroshima, Feynman-Kac formula and its application to quantum physics, Mathematical Analysis of Interacting Quantum Systems, Rennes univ., France, 2017.03. |

9. | Fumio Hiroshima, Analysis of time operators, La Sapienza univ. Math. Phys. seminar, Rome, 2017.02. |

10. | Fumio Hiroshima, Time operator associated with Schroedinger operators, QUTIS, Bilbao, Spain, 2016.09. |

11. | Fumio Hiroshima, Threshold ground state of the semi-relativistic Pauli-Fierz Hamiltonian,Aarhus university, Aarhus univ. Seminar, Denmark, 2016.02. |

12. | Fumio Hiroshima, Stochastic analysis of quantum field theory, MSJ, Kyoto Ind. univ. , special lecture, 2015.09. |

13. | Fumio Hiroshima, Analysis of ground state of quantum field theory by Gibbs measures, International Congress of Mathematical Physics(ICMP) 2015,Chile, 2015.08. |

14. | Fumio Hiroshima, Spectrum of semi-relativistic QED by a Gibbs measure, The 51 winter school of theoretical physics (Karpacz Winter Schools in Theoretical Physics), Poland, 2015.02. |

15. | Fumio Hiroshima, Functional integral approach to mathematically rigorous quantum field theory, TJASSST2013, Tunisia, 2013.11. |

16. | Fumio Hiroshima, Gibbs measure approach to spin-boson model, International conference on stochastic analysis and applications, Tunisia, 2013.10. |

17. | Fumio Hiroshima, Non-relativistic QFT and Gibbs measures, math and phys summer school lecturer, Univ. Tokyo, 2013.09. |

18. | Fumio Hiroshima, Spectrum of non-commutative harmonic oscillator and related models, Bologna univ. seminar, Italy, 2013.09. |

19. | Fumio Hiroshima, Enhanced binding for quantum field models, Universite de Rennes 1 seminar, France, 2013.03. |

20. | Fumio Hiroshima, Gibbs measure approach to properties of ground state in QFT, Laboratoire d’Analyse,Topologie,Probabilités seminar, Universite d'Aix Marseilles,Luminy, France, 2013.03. |

21. | Fumio Hiroshima, Removal of UV cutoff of the Nelson model by stochastic analysis, 量子場の数理とその周辺, 2012.11. |

22. | Fumio Hiroshima, Spectral analysis of QFT by functional integrals with jump processes, SPA,Osaka, 2010.09. |

23. | Fumio Hiroshima, Functional integrations in quantum field theory, The 24th Max Born symposium, Wroclaw Univ., 2008.09. |

24. | Fumio Hiroshima, Feynman-Kac formula, Mini-course on Feynman-Kac formulas and their applications, Wien University, WPI, 2006.03, [URL]. |

25. | Fumio Hiroshima, Spectral analysis of Schrodinger operators coupled to a quantum field, MSJ, Hokkaido univ., special lecture, 2004.09. |

26. | Fumio Hiroshima, Self-adjointness of the Pauli-Fierz model for arbitrary coupling constants, Relativistic quantum system and quantum electrodynamics, Mathematisches Forschungsinstitut Oberwolfach, 2001.08. |

27. | Fumio Hiroshima, Spectral analysis of atoms interacting with a quantized radiation field, math and phys summer school lecturer, Gakushuin univ., 2000.09. |

28. | Fumio Hiroshima, Ground states of a system interacting with a radiation field: existence, uniqueness and expression, UAB Mathematical Physics Congress 99, Alabama univ. USA, 1999.03. |

29. | Fumio Hiroshima, Analysis of the Pauli-Fierz model for arbitrary coupling constant, Workshop on Open Classical and Quantum Field Theory, Lille univ. ,France, 1999.06. |

**Awards**

- MSJ 2019 Analysis prize

Educational

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