scholarly journals LOCALIZATION OF THE NUMBER OF PHOTONS OF GROUND STATES IN NONRELATIVISTIC QED

2003 ◽  
Vol 15 (03) ◽  
pp. 271-312 ◽  
Author(s):  
FUMIO HIROSHIMA
Keyword(s):  
Radiation Field ◽  
Ground State ◽  
Fock Space ◽  
Adjoint Operator ◽  
Ground States ◽  
Electron System ◽  
Number Operator ◽  
Position Variable ◽  
Boson Fock Space ◽  
Quantized Radiation Field

One electron system minimally coupled to a quantized radiation field is considered. It is assumed that the quantized radiation field is massless, and no infrared cutoff is imposed. The Hamiltonian, H, of this system is defined as a self-adjoint operator acting on L2 (ℝ3) ⊗ ℱ ≅ L2 (ℝ3; ℱ), where ℱ is the Boson Fock space over L2 (ℝ3 × {1, 2}). It is shown that the ground state, ψg, of H belongs to [Formula: see text], where N denotes the number operator of ℱ. Moreover, it is shown that for almost every electron position variable x ∈ ℝ3 and for arbitrary k ≥ 0, ‖(1 ⊗ Nk/2) ψg (x)‖ℱ ≤ Dk e-δ|x|m+1 with some constants m ≥ 0, Dk > 0, and δ > 0 independent of k. In particular [Formula: see text] for 0 < β < δ/2 is obtained.

scholarly journals Self-adjointness of the semi-relativistic Pauli–Fierz Hamiltonian

2015 ◽  
Vol 27 (07) ◽  
pp. 1550015 ◽  
Author(s):  
Takeru Hidaka ◽  
Fumio Hiroshima
Keyword(s):  
Radiation Field ◽  
Quantum Electrodynamics ◽  
Fock Space ◽  
Total Momentum ◽  
Momentum Operator ◽  
The Self ◽  
External Potential ◽  
Boson Fock Space ◽  
Quantized Radiation Field

The spinless semi-relativistic Pauli–Fierz Hamiltonian [Formula: see text] in quantum electrodynamics is considered. Here p denotes a momentum operator, A a quantized radiation field, M ≥ 0, Hf the free Hamiltonian of a Boson Fock space and V an external potential. The self-adjointness and essential self-adjointness of H are shown. It is emphasized that it includes the case of M = 0. Furthermore, the self-adjointness and the essential self-adjointness of the semi-relativistic Pauli–Fierz model with a fixed total momentum P ∈ ℝd: [Formula: see text] is also proven for arbitrary P.

scholarly journals Spectral Theory for a Mathematical Model of the Weak Interaction—Part I: The Decay of the Intermediate Vector BosonsW±

2009 ◽  
Vol 2009 ◽  
pp. 1-52 ◽  
Author(s):  
J.-M. Barbaroux ◽  
J.-C. Guillot
Keyword(s):  
Ground State ◽  
State Energy ◽  
Fock Space ◽  
Adjoint Operator ◽  
Weak Decay ◽  
Spectral Interval ◽  
Coupling Constant ◽  
Small Coupling ◽  
Small Coupling Constant ◽  
Intermediate Vector

We consider a Hamiltonian with cutoffs describing the weak decay of spin 1 massive bosons into the full family of leptons. The Hamiltonian is a self-adjoint operator in an appropriate Fock space with a unique ground state. We prove a Mourre estimate and a limiting absorption principle above the ground state energy and below the first threshold for a sufficiently small coupling constant. As a corollary, we prove the absence of eigenvalues and absolute continuity of the energy spectrum in the same spectral interval.

scholarly journals LOCAL EXPONENTS AND INFINITESIMAL GENERATORS OF CANONICAL TRANSFORMATIONS ON BOSON FOCK SPACES

2004 ◽  
Vol 07 (04) ◽  
pp. 547-571 ◽  
Author(s):  
F. HIROSHIMA ◽  
K. R. ITO
Keyword(s):  
Canonical Transformation ◽  
Symplectic Group ◽  
Unitary Operator ◽  
Fock Space ◽  
Adjoint Operator ◽  
The Self ◽  
Canonical Transformations ◽  
Fock Spaces ◽  
Infinitesimal Generators ◽  
Boson Fock Space

A one-parameter symplectic group {etÂ}t∈ℝ derives proper canonical transformations indexed by t on a Boson–Fock space. It has been known that the unitary operator Ut implementing such a proper canonical transformation gives a projective unitary representation of {etÂ}t∈ℝ on the Boson–Fock space and that Ut can be expressed as a normal-ordered form. We rigorously derive the self-adjoint operator Δ(Â) and a local exponent [Formula: see text] with a real-valued function τÂ(·) such that [Formula: see text].

GROUND STATE OF THE YUKAWA MODEL WITH CUTOFFS

2011 ◽  
Vol 14 (02) ◽  
pp. 225-235 ◽  
Author(s):  
TOSHIMITSU TAKAESU
Keyword(s):  
Ground State ◽  
Spectral Gap ◽  
Fock Space ◽  
Adjoint Operator ◽  
Coupling Constants ◽  
Dirac Field ◽  
Yukawa Model ◽  
Klein Gordon

The ground state of the Yukawa model is considered. The Yukawa model describes the system of a Dirac field interacting with a Klein–Gordon field. By introducing ultraviolet cutoffs and spatial cutoffs, the total Hamiltonian is defined as a self-adjoint operator on a boson–fermion Fock space. It is shown that the total Hamiltonian has a positive spectral gap for all values of coupling constants. In particular, the existence of the ground state is proven.

scholarly journals STABILITY OF GROUND STATES IN SECTORS AND ITS APPLICATION TO THE WIGNER–WEISSKOPF MODEL

2001 ◽  
Vol 13 (04) ◽  
pp. 513-528 ◽  
Author(s):  
ASAO ARAI ◽  
MASAO HIROKAWA
Keyword(s):  
Excited State ◽  
Scalar Field ◽  
Ground State ◽  
Adjoint Operator ◽  
Ground States ◽  
The Other ◽  
Coupling Constant ◽  
A Value ◽  
First Excited State ◽  
The One

We consider two kinds of stability (under a perturbation) of the ground state of a self-adjoint operator: the one is concerned with the sector to which the ground state belongs and the other is about the uniqueness of the ground state. As an application to the Wigner–Weisskopf model which describes one mode fermion coupled to a quantum scalar field, we prove in the massive case the following: (a) For a value of the coupling constant, the Wigner–Weisskopf model has degenerate ground states; (b) for a value of the coupling constant, the Wigner–Weisskopf model has a first excited state with energy level below the bottom of the essential spectrum. These phenomena are nonperturbative.

scholarly journals Singular Bogoliubov transformations and inequivalent representations of canonical commutation relations

2019 ◽  
Vol 31 (08) ◽  
pp. 1950026 ◽  
Author(s):  
Asao Arai
Keyword(s):  
Fock Space ◽  
Sufficient Conditions ◽  
Bogoliubov Transformation ◽  
Fock Representation ◽  
Canonical Commutation Relations ◽  
Commutation Relations ◽  
Direct Sum ◽  
Boson Fock Space ◽  
Inequivalent Representations ◽  
Bogoliubov Transformations

We introduce a concept of singular Bogoliubov transformation on the abstract boson Fock space and construct a representation of canonical commutation relations (CCRs) which is inequivalent to any direct sum of the Fock representation. Sufficient conditions for the representation to be irreducible are formulated. Moreover, an example of such representations of CCRs is given.

INTERMEDIATE NUMBER-SQUEEZED STATE OF THE QUANTIZED RADIATION FIELD

1995 ◽  
Vol 09 (25) ◽  
pp. 1673-1683 ◽  
Author(s):  
B. BASEIA ◽  
A.F. DE LIMA ◽  
A.J. DA SILVA
Keyword(s):  
Radiation Field ◽  
Intermediate State ◽  
Phase State ◽  
Squeezed State ◽  
Number State ◽  
Binomial State ◽  
Quantized Radiation Field

Following previous strategies by Stoler et al. which introduced the binomial state and Baseia et al. which introduced the intermediate number phase state, we introduce a new intermediate state of the quantized radiation field, which reduces to the number state and squeezed state in two different limits. This interpolating state exhibits nonclassical effects as sub-Poissonian, antibunching and squeezing, obtained from the corresponding expressions as function of the interpolating parameters.

scholarly journals MATRIX THEORY, HILBERT SCHEME AND INTEGRABLE SYSTEM

1998 ◽  
Vol 13 (34) ◽  
pp. 2731-2742 ◽  
Author(s):  
YUTAKA MATSUO
Keyword(s):  
Hilbert Scheme ◽  
Matrix Theory ◽  
Fock Space ◽  
Homology Class ◽  
Orthogonal Basis ◽  
Inner Product ◽  
Virasoro Symmetry ◽  
Hilbert Scheme Of Points ◽  
The Matrix ◽  
Boson Fock Space

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relates the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These bases can be related to the eigenfunctions of Calogero–Sutherland (CS) equation and the deformation parameter of MNS is identified with coupling of CS system. We discuss the structure of Virasoro symmetry in this model.

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