Reviews in Mathematical Physics

Framework for a general discussion of environmentally induced classical properties, like superselection rules, privileged basis and classical behavior, in quantum systems with both finite and infinite number of degrees of freedom is proposed. A number of examples showing that classical properties do not have to be postulated as an independent ingredient are given. In particular, it is shown that infinite open quantum systems in some cases may behave like simple classical dynamical systems.

The non-relativistic (scaling) limit of a particle-field Hamiltonian H, called a Dirac–Maxwell operator, in relativistic quantum electrodynamics is considered. It is proven that the non-relativistic limit of H yields a self-adjoint extension of the Pauli–Fierz Hamiltonian with spin 1/2 in non-relativistic quantum electrodynamics. This is done by establishing in an abstract framework a general limit theorem on a family of self-adjoint operators partially formed out of strongly anticommuting self-adjoint operators and then by applying it to H.

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.