Narrow Results

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, ψ_g, of H belongs to , 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 ⊗ N^k/2) ψ_g (x)‖_ℱ ≤ D_k e^-δ|x|m+1 with some constants m ≥ 0, D_k > 0, and δ > 0 independent of k. In particular for 0 < β < δ/2 is obtained.

We consider, in an abstract form, a system of "quantum particles" coupled to a Bose field. It is shown that, under suitable hypotheses, the composed system can have a ground state even if the uncoupled particle system has no ground state.

The effective mass m_eff of the Pauli–Fierz Hamiltonian with ultraviolet cutoff Λ and the bare mass m in non-relativistic quantum electrodynamics (QED) with spin ½ is investigated. Analytic properties of m_eff in coupling constant e are shown. Let us set . The explicit form of constant a₂(Λ/m) depending on Λ/m is given. It is shown that the spin interaction enhances the effective mass and that there exist strictly positive constants c₁ and c₂ such that

Generalized Pauli–Fierz Hamiltonian with Kato-class potential K_PF in nonrelativistic quantum electrodynamics is defined and studied by a path measure. K_PF 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.

The spinless semi-relativistic Pauli–Fierz Hamiltonian

is also proven for arbitrary P.

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.

Recent developments in rigorous functional integration applied to the Nelson and Pauli-Fierz models are presented.