We prove that in the Euclidean representation of the three dimensional massless Nelson model the t = 0 projection of the interacting measure is absolutely continuous with respect to a Gaussian measure with suitably adjusted mean. We also determine the Hamiltonian in the Fock space over this Gaussian measure space.
We provide a new construction of infraparticle states in the massless Nelson model. The approximating sequence of our infraparticle state does not involve any infrared cut-offs. Its derivative w.r.t. the time parameter $t$ is given by a simple explicit formula. The convergence of this sequence as $t\to \infty$ to a non-zero limit is then obtained by the Cook method combined with stationary phase estimates. To apply the latter technique we exploit recent results on regularity of ground states in the massless Nelson model, which hold in the low coupling regime.
For the massless Nelson model we provide detailed information about the dependence of the normalized ground states $\checkψ_{P,σ}$ of the fiber single-electron Hamiltonians $H_{P,σ}$ on the total momentum $P$ and the infrared cut-off $σ$. This information is obtained with the help of the iterative analytic perturbation theory. In particular, we derive bounds of the form \[ \|\partial_{P_i}\checkψ_{P,σ}\|,\ \| \partial_{P_i} \partial_{P_j}\checkψ_{P,σ} \|\leq \frac{c}{σ^{δ_{λ_0}}}, \] for some constant $c$ and a function of the maximal admissible coupling constant $λ_0\mapsto δ_{λ_0}$ s.t. $\li…
In the theory of point interactions, one is given a formal expression for a quantum mechanical Hamiltonian. The interaction terms of the Hamiltonian are singular: they can not be rigorously defined as a perturbation (in the operator or form sense) of an unperturbed free operator. A similar situation occurs in Quantum Field Theory, where it is known as the ultraviolet problem. Recently, it was shown that some of the tools used in the context of point interactions can be adapted to solve the problem of directly defining a Hamiltonian for the Nelson model. This model provides a well studied examp…
We consider a model for a massive uncharged non-relativistic particle interacting with a massless bosonic field, widely referred to as the Nelson model. It is well known, that an ultraviolet renormalized Hamilton operator exists in this case. Further, due to translation-invariance, it decomposes into fiber operators. In this paper, we treat the renormalized fiber operators. We give a description of the operator and form domains and prove that the fiber operators do not have a ground state. Our results hold for any non-zero coupling constant and arbitrary total momentum. Our proof for the absen…
In this paper we consider the massless translation invariant Nelson model with ultraviolet cutoff. It is proven that the fiber operators have no ground state if there is no infrared cutoff.
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, clust…
In the context of the massless Nelson model, we prove that two non-relativistic nucleons interacting with a massless meson field do not bind when a sufficiently strong Coulomb repulsion between the nucleons is added to the Hamiltonian. The result holds for both the renormalized and unrenormalized theories, and can also be applied to the so-called piezoelectric polaron model, which describes an electron interacting with the acoustical vibrational modes of a crystal through the piezoelectric interaction. The result can then be interpreted as well as a no-binding statement about piezoelectric bip…
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 electrostatic potentials fulfilling a binding condition. If the infrared regularity condition is violated, then we show non-existence of ground states of the massless renormalized Nelson operator with an arbitrary Kato decomposable potential. Furthermore, we prove the existence, uniqueness, and strict positivity of ground states of the massless renormalized Nelson operator in a non-Fock representation w…
Let $H_{P,σ}$ be the single-electron fiber Hamiltonians of the massless Nelson model at total momentum $P$ and infrared cut-off $σ>0$. We establish detailed regularity properties of the corresponding $n$-particle ground state wave functions $f^n_{P,σ}$ as functions of $P$ and $σ$. In particular, we show that \[ |\partial_{P^j}f^{n}_{P,σ}(k_1,\ldots, k_n)|, \ \ |\partial_{P^j} \partial_{P^{j'}} f^{n}_{P,σ}(k_1,\ldots, k_n)| \leq \frac{1}{\sqrt{n!}} \frac{(cλ_0)^n}{σ^{δ_{λ_0}}} \prod_{i=1}^n\frac{ χ_{[σ,κ)}(k_i)}{|k_i|^{3/2}}, \] where $c$ is a numerical constant, $λ_0\mapsto δ_{λ_0}$ is a posit…
We propose an abstract framework describing energy-renormalized Hamiltonians in terms of local algebras. Within the framework, we examine the positivity improvingness of the semigroup generated by the renormalized Hamiltonian. As examples, we discuss the renormalized Nelson Hamiltonian and the renormalized Nelson Hamiltonian at fixed total momentum. The characteristic features of our approach are as follows:(i) in contrast to the probabilistic approach in the Schrödinger representation, our method works well in the Fock representation; (ii) the method covers the massless case.
We consider solutions to the Einstein-massless-scalar field system with a positive cosmological constant, arising from sufficiently regular, near-FLRW, initial data. We establish global existence in the future direction and derive their precise asymptotic behavior towards infinity. As a corollary, we infer that, unlike the FLRW background, the perturbed solutions do not describe a regular irrotational stiff fluid with linear equation of state $p=ρ$, for general asymptotic data at infinity. The reason for the breakdown of this interpretation is that the gradient of the scalar field stops being …
