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

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

Ultra-Weak Time Operators of Schroedinger Operators

In an abstract framework, a new concept on time operator, ultra-weak time operator, is introduced, which is a concept weaker than that of weak time operator. Theorems on the existence of an ultra-weak time operator are established. As an application of the theorems, it is shown that Schroedinger operators H with potentials V obeying suitable conditions, including the Hamiltonian of the hydrogen atom, have ultra-weak time operators. Moreover, a class of Borel measurable functions $f$ such that $f(H)$ has an ultra-weak time operator is found.Comment: We add Sections 1.1,1.2 and 1.

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^{-\delta |x|^{m+1}} with some constants $m\geq 0$, $D_k>0$, and $\delta>0$ independent of $k$. In particular \gr\in \cap_{k=1}^\infty D (e^{\beta |x|^{m+1}}\otimes N^k) for $0<\beta<\delta/2$ is obtained.Comment: 43page

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

A Feynman-Kac-type formula for a L\'evy 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.Comment: This is a revised version. This paper will be published from J. Funct. Ana

Mass Renormlization in the Nelson Model

The asymptotic behavior of the effective mass $m_{\rm eff}(\Lambda)$ of the so-called Nelson model in quantum field theory is considered, where $\Lambda$ is an ultraviolet cutoff parameter of the model. Let $m$ be the bare mass of the model. It is shown that for sufficiently small coupling constant $|\alpha|$ of the model, $m_{{\rm eff}}(\Lambda)/m$ can be expanded as $m_{{\rm eff}}(\Lambda)/m= 1+\sum_{n=1}^\infty a_n(\Lambda) \alpha^{2n}$. A physical folklore is that $a_n(\Lambda)\sim [\log \Lambda]^{(n-1)}$ as $\Lambda\to \infty$. It is rigorously shown that $$0<\lim_{\Lambda\to\infty}a_1(\Lambda)<C,\quad C_1\leq \lim_{\Lambda\to\infty}a_2(\Lambda)/\log\Lambda\leq C_2$$ with some constants $C$, $C_1$ and $C_2$.Comment: It has been published in International Journal of Mathematics and Mathematical Sciences, vol. 2017, Article ID 476010

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.Comment: 17 page