Reviews in Mathematical Physics

We consider the massless Nelson model with two types of massive particles which we call atoms and electrons. The atoms interact with photons via an infrared regular form-factor and thus they are Wigner-type particles with sharp mass-shells. The electrons have an infrared singular form-factor and thus they are infraparticles accompanied by soft-photon clouds correlated with their velocities. In the weak coupling regime, we construct scattering states of one atom and one electron, and demonstrate their asymptotic clustering into individual particles. The proof relies on the Cook’s argument, clustering estimatesblack, and the non-stationary phase method. The latter technique requires sharp estimates on derivatives of the ground state wave functions of the fiber Hamiltonians of the model, which were proven in the earlier papers of this series. Although we rely on earlier black studies of the atom–atom and electron–photon scattering in the Nelson model, the paper is written in a self-contained manner. A perspective on the open problem of the electron–electron scattering in this model is also given.

We consider the quantum evolution $e^{-i\frac{t}{\hslash }{H}_{\beta }}{\psi }_{\xi }^{\hslash }$ of a Gaussian coherent state ${\psi }_{\xi }^{\hslash }\in L^2(ℝ)$ localized close to the classical state $\xi \equiv (q,p)\in {ℝ}^2$, where $H_{\beta }$ denotes a self-adjoint realization of the formal Hamiltonian $-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+\beta {\delta }_0^{\prime }$, with ${\delta }_0^{\prime }$ the derivative of Dirac’s delta distribution at $x=0$ and $\beta$ a real parameter. We show that in the semi-classical limit such a quantum evolution can be approximated (with respect to the $L^2(ℝ)$-norm, uniformly for any $t\in ℝ$ away from the collision time) by $e^{\frac{i}{\hslash }{A}_t}{e}^{it{L}_B}{\varphi }_x^{\hslash }$, where $A_t=\frac{p^{2}t}{2m}$, ${\varphi }_x^{\hslash }(\xi ):={\psi }_{\xi }^{\hslash }(x)$ and $L_B$ is a suitable self-adjoint extension of the restriction to ${{𝒞}}_c^{\infty }({\mathcal{ℳ}}_0)$, ${\mathcal{ℳ}}_0:=\{(q,p)\in {ℝ}^2|q\ne 0\}$, of ($-i$ times) the generator of the free classical dynamics. While the operator $L_B$ here utilized is similar to the one appearing in our previous work [C. Cacciapuoti, D. Fermi and A. Posilicano, The semi-classical limit with a delta potential, Ann. Mat. Pura Appl. 200 (2021) 453–489], in the present case the approximation gives a smaller error: it is of order ${\hslash }^{7/2-\lambda }$, $0<\lambda <1/2$, whereas it turns out to be of order ${\hslash }^{3/2-\lambda }$, $0<\lambda <3/2$, for the delta potential. We also provide similar approximation results for both the wave and scattering operators.

The purpose of this paper is to give different interpretations of the first non-vanishing term (quadratic) of the ground state asymptotic expansion for a spin system in quantum electrodynamics, as the spin magnetic moments go to 0. One of the interpretations makes a direct link with some classical physics laws. A central role is played by an operator $A_M$ acting only in the finite-dimensional spin state space and making the connections with the different interpretations, and also being in close relation with the multiplicity of the ground state.