Ground state properties of the Nelson Hamiltonian: A Gibbs measure-based approach

@article{Betz2001GroundSP,
  title={Ground state properties of the Nelson Hamiltonian: A Gibbs measure-based approach},
  author={Volker Betz and Fumio Hiroshima and J{\'o}zsef Lőrinczi and Robert Minlos and Herbert Spohn},
  journal={Reviews in Mathematical Physics},
  year={2001},
  volume={14},
  pages={173-198}
}
The Nelson model describes a quantum particle coupled to a scalar Bose field. We study properties of its ground state through functional integration techniques in case the particle is confined by an external potential. We obtain bounds on the average and the variance of the Bose field both in position and momentum space, on the distribution of the number of bosons, and on the position space distribution of the particle. 

Ground states and associated path measures in the renormalized Nelson model

We prove the existence, uniqueness, and strict positivity of ground states of the possibly massless renormalized Nelson operator under an infrared regularity condition and for Kato decomposable

FKN Formula and Ground State Energy for the Spin Boson Model with External Magnetic Field

We consider the spin boson model with external magnetic field. We prove a path integral formula for the heat kernel, known as Feynman–Kac–Nelson (FKN) formula. We use this path integral

A Functional Central Limit Theorem for Polaron Path Measures

The application of the Feynman‐Kac formula to Polaron models of quantum theory leads to the path measure of Brownian motion perturbed by a pair potential that is translation invariant both in space

Central limit theorem for Gibbs measures on path spaces including long range and singular interactions and homogenization of the stochastic heat equation

We consider a class of Gibbs measures defined with respect to increments of $d$-dimensional Wiener measure. The underlying Hamiltonian is defined by interactions that are invariant under uniform

Spectrum of the semi-relativistic Pauli–Fierz model II

69 INTEGRAL KERNELS OF THE RENORMALIZED NELSON HAMILTONIAN

In this article we consider the ground state of the renormalized Nelson Hamil‐ tonian in quantum field theory by using the integral kernel of the semigroup gen‐ erated by the Hamiltonian. By

Ground State Properties in the Quasi-Classical Regime

We study the ground state energy and ground states of systems coupling non-relativistic quantum particles and force-carrying Bose fields, such as radiation, in the quasi-classical approximation. The

Functional Central Limit Theorems and P(ϕ)1-Processes for the Relativistic and Non-Relativistic Nelson Models

We construct P(ϕ)1-processes indexed by the full time-line, separately derived from the functional integral representations of the relativistic and non-relativistic Nelson models in quantum field

Integral kernels of the renormalized Nelson Hamiltonian

  • Fumio
  • Physics, Mathematics
  • 2019
In this article we consider the ground state of the renormalized Nelson Hamiltonian in quantum field theory by using the integral kernel of the semigroup generated by the Hamiltonian. By introducing
...

References

SHOWING 1-10 OF 21 REFERENCES

Pointwise bounds for Schrödinger eigenstates

TLDR
Using probabilistic techniques the authors prove pointwise upper bounds for Lq-Schrödinger eigenstates and pointwise lower bounds for the corresponding groundstate and generalize Schnol's and Simon's ones.

Interaction of Nonrelativistic Particles with a Quantized Scalar Field

We demonstrate the mathematical existence of a meson theory with nonrelativistic nucleons. A system of Schrodinger particles is coupled to a quantized relativistic scalar field. If a cutoff is put on

Quantum Mechanics and Path Integrals

QUANTUM ELECTRODYNAMICS OF CONFINED NONRELATIVISTIC PARTICLES

Abstract We consider a system of finitely many nonrelativistic, quantum mechanical electrons bound to static nuclei. The electrons are minimally coupled to the quantized electromagnetic field; but we

Gibbs Measures and Phase Transitions

TLDR
This comprehensive monograph covers in depth a broad range of topics in the mathematical theory of phase transition in statistical mechanics and serves both as an introductory text and as a reference for the expert.

Functional integration and quantum physics

Introduction The basic processes Bound state problems Inequalities Magnetic fields and stochastic integrals Asymptotics Other topics References Index Bibliographic supplement Bibliography.

Existence of Dressed One Electron States in a Class of Persistent Models

In this paper the dynamics of a class of simple models (so called persistent models) including Nelson's model is studied. These models describe conserved, non relativistic, scalar electrons

On the Existence of Ground States for Massless Pauli-Fierz Hamiltonians

We consider in this paper the problem of the existence of a ground state for a class of Hamiltonians used in physics to describe a confined quantum system (”matter”) interacting with a massless

The free Markoff field