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Fumio Hiroshima

Publications and source records attributed to Fumio Hiroshima.

54 records · Page 3Linked to original sources
arXiv2011

A Probabilistic Representation of the Ground State Expectation of Fractional Powers of the Boson Number Operator

We give a formula in terms of a joint Gibbs measure on Brownian paths and the measure of a random-time Poisson process of the ground state expectations of fractional (in fact, any real) powers of the boson number operator in the Nelson model. We use this representation to obtain tight two-sided bounds. As applications, we discuss the polaron and translation invariant Nelson models.

math-ph↗
arXiv2011

The No-Binding Regime of the Pauli-Fierz Model

The Pauli-Fierz model $H(α)$ in nonrelativistic quantum electrodynamics is considered. The external potential $V$ is sufficiently shallow and the dipole approximation is assumed. It is proven that there exist constants $0<α_-< α_+$ such that $H(α)$ has no ground state for $|α|<α_-$, which complements an earlier result stating that there is a ground state for $|α| > α_+$. We develop a suitable extension of the Birman-Schwinger argument. Moreover for any given $δ>0$ examples of potentials $V$ are provided such that $α_+-α_-<δ$.

math-ph↗
arXiv2011

Infrared problem for the Nelson model on static space-times

We consider the Nelson model with variable coefficients and investigate the problem of existence of a ground state and the removal of the ultraviolet cutoff. 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. A physical example is obtained by quantizing the Klein-Gordon equation on a static space-time coupled with a non-relativistic particle. We investigate the existence of a ground state of the Hamiltonian in the presence of the infrared problem, i.e. assuming that the boson mass tends to 0 at infinity.

math-ph↗
arXiv2010

Absence of ground state for the Nelson model on static space-times

We consider the Nelson model on some static space-times and investigate the problem of absence of a ground state. 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 investigate the absence of a ground state of the Hamiltonian in the presence of the infrared problem, i.e. assuming that the boson mass $m(x)$ tends to $0$ at spatial infinity. Using path space techniques, we show that if $m(x)\leq C |x|^{-μ}$ at infinity for some $C>0$ and $μ>1$ then the Nelson Hamiltonian has no ground state.

math-ph↗
arXiv2010

Path Integral Representation for Schroedinger Operators with Bernstein Functions of the Laplacian

Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard Feynman-Kac formula is taken here by subordinated Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which hypercontractivity of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.

math-ph↗
arXiv2010

On the ionization energy of the semi-relativistic Pauli-Fierz model for a single particle

A semi-relativistic Pauli-Fierz model is defined by the sum of the free Hamiltonian $H_{\rm f}$ of a Boson Fock space, an nuclear potential $V$ and a relativistic kinetic energy: $$ H=\sqrt{[σ\cdot(\mathbf{p}+e\mathbf{A})]^2+M^2} - M + V + H_{\rm f}. $$ Let $-e_0<0$ be the ground state energy of a semi-relativistic Schrödinger operator $$ H_{\rm p}=\sqrt{\mathbf{p}^2+M^2} - M + V. $$ It is shown that the ionization energy $E^{\rm ion}$ of $H$ satisfies $$E^{\rm ion}\geq e_0>0$$ for all values of both of the coupling constant $e\in\mathbb{R}$ and the rest mass $M\geq 0$. In particular our result includes the case of M=0.

math-ph↗
arXiv2008

Strong time operators associated with generalized Hamiltonians

Let the pair of operators, $(H, T)$, satisfy the weak Weyl relation: $Te^{-itH} = e^{-itH}(T + t)$, where $H$ is self-adjoint and $T$ is closed symmetric. Suppose that g is a realvalued Lebesgue measurable function on $\RR$ such that $g \in C^2(R K)$ for some closed subset $K \subset \RR$ with Lebesgue measure zero. Then we can construct a closed symmetric operator $D$ such that $(g(H), D)$ also obeys the weak Weyl relation.

math-ph↗
arXiv2008

Physical state for non-relativistic quantum electrodynamics

A physical subspace and physical Hilbert space associated with asymptotic fields of nonrelativistic quantum electrodynamics are constructed through the Gupta-Bleuler procedure. Asymptotic completeness is shown and a physical Hamiltonian is defined on the physical Hilbert space.

math-ph↗
arXiv2008

Gibbs measures with double stochastic integrals on a path space

We investigate Gibbs measures relative to Brownian motion in the case when the interaction energy is given by a double stochastic integral. In the case when the double stochastic integral is originating from the Pauli-Fierz model in nonrelativistic quantum electrodynamics, we prove the existence of its infinite volume limit.

math-ph↗
arXiv2008

Functional Integral Representation of the Pauli-Fierz Model with Spin 1/2

A Feynman-Kac-type formula for a Lévy and an infinite dimensional Gaussian random process associated with a quantized radiation field is derived. In particular, a functional integral representation of $e^{-t\PF}$ generated by the Pauli-Fierz Hamiltonian with spin $\han$ in non-relativistic quantum electrodynamics is constructed. When no external potential is applied $\PF$ turns translation invariant and it is decomposed as a direct integral $\PF = \int_\BR^\oplus \PF(P) dP$. The functional integral representation of $e^{-t\PF(P)}$ is also given. Although all these Hamiltonians include spin, nevertheless the kernels obtained for the path measures are scalar rather than matrix expressions. As an application of the functional integral representations energy comparison inequalities are derived.

math-ph↗
arXiv2006

Enhanced binding for N-particle system interacting with a scalar bose field I

An enhanced binding of an $N$-particle system interacting through a scalar bose field is investigated, where $N\geq 2$. It is not assumed that this system has a ground state for a zero coupling. It is shown, however, that there exists a ground state for a sufficiently large values of coupling constants. When the coupling constant is sufficiently large, $N$ particles are bound to each other by the scalar bose field, and are trapped by external potentials. Basic ideas of the proofs in this paper are applications of a weak coupling limit and a modified HVZ theorem.

math-ph↗
arXiv2006

Fiber Hamiltonians in the non-relativistic quantum electrodynamics

A translation invariant Hamiltonian $H$ in the nonrelativistic quantum electrodynamics is studied. This Hamiltonian is decomposed with respect to the total momentum $\tot$: $$H=\int_{\BR} ^\oplus \fri(P) dP,$$ where the self-adjoint fiber Hamiltonian $\fri(P)$ is defined for arbitrary values of coupling constants. It is discussed a relationship between rotation invariance of $H(P)$ and polarization vectors, and functional integral representations of $n$ point Euclidean Green functions of $H(P)$ is given. From these, some applications concerning with degeneracy of ground states, ground state energy and expectation values of suitable observables with respect to ground states are given.

math-ph↗
arXiv2004

Mass renormalization in nonrelativistic QED

In nonrelativistic QED the charge of an electron equals its bare value, whereas the self-energy and the mass have to be renormalized. In our contribution we study perturbative mass renormalization, including second order in the fine structure constant $α$, in the case of a single, spinless electron. As well known, if $m$ denotes the bare mass and $\mass$ the mass computed from the theory, to order $α$ one has $$\frac{\mass}{m} =1+\frac{8α}{3π} \log(1+\half (Λ/m))+O(α^2)$$ which suggests that $\mass/m=(Λ/m)^{8α/3π}$ for small $α$. If correct, in order $α^2$ the leading term should be $\displaystyle \half ((8α/3π)\log(Λ/m))^2$. To check this point we expand $\mass/m$ to order $α^2$. The result is $\sqrt{Λ/m}$ as leading term, suggesting a more complicated dependence of $m_{\mathrm{eff}}$ on $m$.

math-ph↗
arXiv2004

Regularities of ground states of quantum field models

Regularities and higher order regularities of ground states of quantum field models are investigated through the fact that asymptotic annihilation operators vanish ground states. Moreover a sufficient condition for the absence of a ground state is given.

math-ph↗
arXiv2004

Multiplicity of ground states in quantum field models: applications of asymptotic fields

The ground states of an abstract model in quantum field theory are investigated. By means of the asymptotic field theory, we give a necessary and sufficient condition for that the expectation value of the number operator of ground states is finite, from which we obtain a wide-usable method to estimate an upper bound of the multiplicity of ground states. Ground states of massless GSB models and the Pauli-Fierz model with spin 1/2 are investigated as examples.

math-ph↗
arXiv2003

Localization of the number of photons of ground states in nonrelativistic QED

One electron system minimally coupled to a quantized radiation field is considered. It is assumed that the quantized radiation field is {\it massless}, and {\it no} infrared cutoff is imposed. The Hamiltonian, $H$, of this system is defined as a self-adjoint operator acting on $\LR\otimes\fff\cong L^2(\BR;\fff)$, where $\fff$ is the Boson Fock space over $L^2(\BR\times\{1,2\})$. It is shown that the ground state, $\gr$, of $H$ belongs to $\cap_{k=1}^\infty D(1\otimes N^k)$, where $N$ denotes the number operator of $\fff$. Moreover it is shown that, for almost every electron position variable $x\in\BR$ and for arbitrary $k\geq 0$, $\|(1\otimes \N)\gr (x) \|_\fff \leq D_ke^{-δ|x|^{m+1}}$ with some constants $m\geq 0$, $D_k>0$, and $δ>0$ independent of $k$. In particular $\gr\in \cap_{k=1}^\infty D (e^{β|x|^{m+1}}\otimes N^k)$ for $0<β<δ/2$ is obtained.

math-ph↗