arXiv subjects

F. Hiroshima

Publications and source records attributed to F. Hiroshima.

8 recordsLinked to original sources

arXiv2013

Ultraviolet Renormalization of the Nelson Hamiltonian through Functional Integration

Starting from the N-particle Nelson Hamiltonian defined by imposing an ultraviolet cutoff, we perform ultraviolet renormalization by showing that in the zero cutoff limit a self-adjoint operator exists after a logarithmically divergent term is subtracted from the original Hamiltonian. We obtain this term as the diagonal part of a pair interaction appearing in the density of a Gibbs measure derived from the Feynman-Kac representation of the Hamiltonian. Also, we show existence of a weak coupling limit of the renormalized Hamiltonian and derive an effective Yukawa interaction potential between the particles.

math-ph↗

arXiv2011

Removal of UV cutoff for the Nelson model with variable coefficients

We consider the Nelson model with variable coefficients. Nelson models with variable coefficients arise when one replaces in the usual Nelson model the flat Minkowski metric by a static metric, allowing also the boson mass to depend on position. We study the removal of the ultraviolet cutoff.

math-ph↗

arXiv2011

Infrared divergence of a scalar quantum field model on a pseudo Riemannian manifold

A scalar quantum field model defined on a pseudo Riemann manifold is considered. The model is unitarily transformed the one with a variable mass. By means of a Feynman-Kac-type formula, it is shown that when the variable mass is short range, the Hamiltonian has no ground state. Moreover the infrared divergence of the expectation values of the number of bosons in the ground state is discussed.

math-ph↗

arXiv2010

Pauli-Fierz model with Kato-class potentials and exponential decays

Generalized Pauli-Fierz Hamiltonian with Kato-class potential $\KPF$ in nonrelativistic quantum electrodynamics is defined and studied by a path measure. $\KPF$ is defined as the self-adjoint generator of a strongly continuous one-parameter symmetric semigroup and it is shown that its bound states spatially exponentially decay pointwise and the ground state is unique.

math-ph↗

arXiv2004

Mass Renormalization in Non-relativistic Quantum Electrodynamics with Spin 1/2

The effective mass $\mass$ of the the Pauli-Fierz Hamiltonain with ultraviolet cutoff $Λ$ and the bare mass $m$ in nonrelativistic QED with spin 1/2 is investigated. Analytic properties of $\mass$ in coupling constant $e$ are shown and explicit forms of constants $a_1(Λ/m)$ and $a_2(Λ/m)$ depending on $Λ/m$ such that $$\mass/m =1 + a_1(Λ/m) e^2+ a_2(Λ/m) e^4+ {\mathcal O}(e^6)$$ are given. It is shown that the spin interaction enhances the effective mass and that there exist strictly positive constants $b_1,b_2, c_1$ and $c_2$ such that $$\d b_1\leq \lim_{Λ\to\infty} \frac{a_1(\Lamdda/m)}{\log (Λ/m)}\leq b_2, c_1\leq \lim_{Λ\to\infty} \frac{a_2(Λ/m)}{(Λ/m)^2}\leq -c_2.$$ In particular $a_2(Λ/m)$ does not diverges as $\pm [\log(Λ/m)]^2$ but $-(Λ/m)^2$.

math-ph↗

arXiv2003

Local exponents and infinitesimal generators of canonical transformations on Boson Fock spaces

A one-parameter symplectic group $\{e^{t\dA}\}_{t\in\RR}$ derives proper canonical transformations on a Boson Fock space. It has been known that the unitary operator $U_t$ implementing such a proper canonical transformation gives a projective unitary representation of $\{e^{t\dA}\}_{t\in\RR}$ and that $U_t$ can be expressed as a normal-ordered form. We rigorously derive the self-adjoint operator $\D(\dA)$ and a phase factor $e^{i\int_0^t\TA(s)ds}$ with a real-valued function $\TA$ such that $U_t=e^{i\int_0^t\TA(s)ds}e^{it\D(\dA)}$. Key words: Canonical transformations(Bogoliubov transformations), symplectic groups, projective unitary representations, one-parameter unitary groups, infinitesimal self-adjoint generators, local factors, local exponents, normal-ordered quadratic expressions.

math-ph↗

arXiv2002

Ground state degeneracy of the Pauli-Fierz Hamiltonian inlcuding spin

We consider an electron, spin 1/2, minimally coupled to the quantized radiation field in the nonrelativistic approximation, a situation defined by the Pauli-Fierz Hamiltonian $H$. There is no external potential and $H$ fibers as $\int^\oplus H_p dp$ according to the total momentum $p$. We prove that the ground state subspace of $H_p$ is two-fold degenerate provided the charge $e$ and the total momentum $p$ are sufficiently small. We also establish that the total angular momentum of the ground state subspace is $\pm1/2$ and study the case of a confining external potential.

math-ph↗

arXiv2002

Ground state degeneracy of the Pauli-Fierz Hamiltonian with spin

We consider an electron, spin 1/2, minimally coupled to the quantized radiation field in the nonrelativistic approximation, a situation defined by the Pauli-Fierz Hamiltonian $H$. There is no external potential and $H$ fibers according to the total momentum $p$. We prove that the ground state subspace of $H$ with a total momentum $p$ is two-fold degenerate provided the charge $e$ and the total momentum $p$ are sufficiently small. We also establish that the total angular momentum of the ground state subspace is $\pm1/2$ and study the case of a confining external potential.

math-ph↗