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On a Direct Description of Pseudorelativistic Nelson Hamiltonians
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Abstract
interior-boundary conditions (IBC's) allow for the direct description of the domain and the action of Hamiltonians for a certain class of ultraviolet-divergent models in Quantum Field Theory. The method was recently applied to models where nonrelativistic scalar particles are linearly coupled to a quantised field, the best known of which is the Nelson model. Since this approach avoids the use of ultraviolet-cutoffs, there is no need for a renormalisation procedure. Here, we extend the IBC method to pseudorelativistic scalar particles that interact with a real bosonic field. We construct the Hamiltonians for such models via abstract boundary conditions, describing their action explicitly. In addition, we obtain a detailed characterisation of their domain and make the connection to renormalisation techniques. As an example, we apply the method to two relativistic variants of Nelson's model, which have been renormalised for the first time by J. P. Eckmann and A. D. Sloan in 1970 and 1974, respectively.
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We consider a way of defining quantum Hamiltonians involving particle creation and annihilation based on an interior-boundary condition (IBC) on the wave function, where the wave function is the particle-position representation of a vector in Fock space, and the IBC relates (essentially) the values of the wave function at any two configurations that differ only by the creation of a particle. Here we prove, for a model of particle creation at one or more point sources using the Laplace operator as the free Hamiltonian, that a Hamiltonian can indeed be rigorously defined in this way without the need for any ultraviolet regularization, and that it is self-adjoint. We prove further that introducing an ultraviolet cut-off (thus smearing out particles over a positive radius) and applying a certain known renormalization procedure (taking the limit of removing the cut-off while subtracting a constant that tends to infinity) yields, up to addition of a finite constant, the Hamiltonian defined by the IBC.
We prove that a system of N fermions interacting with an additional particle via point interactions is stable if the ratio of the mass of the additional particle to the one of the fermions is larger than some critical m*. The value of m* is independent of N and turns out to be less than 1. This fact has important implications for the stability of the unitary Fermi gas. We also characterize the domain of the Hamiltonian of this model, and establish the validity of the Tan relations for all wave functions in the domain.
We consider a quantum mechanical three-particle system made of two identical fermions of mass one and a different particle of mass m, where each fermion interacts via a zero-range force with the different particle. In particular we study the unitary regime, i.e., the case of infinite two-body scattering length. The Hamiltonians describing the system are, by definition, self-adjoint extensions of the free Hamiltonian restricted on smooth functions vanishing at the two-body coincidence planes, i.e., where the positions of two interacting particles coincide. It is known that for m larger than a critical value m
∗ ≃ (13.607)−1 a self-adjoint and lower bounded Hamiltonian H
0 can be constructed, whose domain is characterized in terms of the standard point-interaction boundary condition at each coincidence plane. Here we prove that for m ∈ (m
∗,m
∗∗), where m
∗∗ ≃ (8.62)−1, there is a further family of self-adjoint and lower bounded Hamiltonians H
0,β
, β ∈ ℝ, describing the system. Using a quadratic form method, we give a rigorous construction of such Hamiltonians and we show that the elements of their domains satisfy a further boundary condition, characterizing the singular behavior when the positions of all the three particles coincide.
We consider the one-particle sector of the spinless Yukawa model, which
describes the interaction of a nucleon with a real field of scalar massive
bosons (neutral mesons). The nucleon as well as the mesons have relativistic
dispersion relations. In this model we study the dependence of the nucleon mass
shell on the ultraviolet cut-off . For any finite ultraviolet cut-off
the nucleon one-particle states are constructed in a bounded region of the
energy-momentum space. We identify the dependence of the ground state energy on
and the coupling constant. More importantly, we show that the model
considered here becomes essentially trivial in the limit
regardless of any (nucleon) mass and self-energy renormalization. Our results
hold in the small coupling regime.
It is shown that there exists a selfadjoint Hamilton operator corresponding to the interactionH
0(a)+H
0(b)+∫Φ
b
+(x)Φ
a
(x)Φ
b
−(x)d
3x, wherea andb denote two types of scalar particles. We discuss the scattering theory of this operator.
The Nelson Hamiltonian is unitarily equivalent to a Hamiltonian defined through a closed, semibounded quadratic form, the unitary transformation being explicitly known and due to Gross. In this paper we study mapping properties of the Gross-transform in order to characterize regularity properties of vectors in the form domain of the Nelson Hamiltonian. Since the operator domain is a subset of the form domain, our results apply to vectors in the domain of the Hamiltonian was well. - This work is a continuation of our previous work on the Fr\"ohlich Hamiltonian.
In the large polaron model of H. Froehlich the electron-phonon interaction is
a small perturbation in form sense, but a large perturbation in operator sense.
This means that the form-domain of the Hamiltonian is not affected by the
interaction but the domain of self-adjointness is. In the particular case of
the Froehlich model we are nevertheless able, thanks to a recently published
new operator bound, to give an explicit characterization of the domain in terms
of a suitable dressing transform. Using the mapping properties of this dressing
transform we analyze the smoothness of vectors in the domain of the Hamiltonian
with respect to the position of the electron.
We demonstrate the mathematical existence of a meson theory with nonrelativistic nucleons. A system of Schro¨dinger particles is coupled to a quantized relativistic scalar field. If a cutoff is put on the interaction, we obtain a well-defined self-adjoint operator. The solution of the Schro¨dinger equation diverges as the cutoff tends to infinity, but the divergence amounts merely to a constant infinite phase shift due to the self-energy of the particles. In the Heisenberg picture, we obtain a solution in the limit of no cutoff. We use a canonical transformation due to Gross to separate the divergent self-energy term. It is shown that the canonical transformation is implemented by a unitary operator, and that the transformed Hamiltonian, with an infinite constant subtracted, can be interpreted as a self-adjoint operator.
Hamiltonians for the polaron of fixed total momentum are defined using momentum cutoffs. A renormalized Hamiltonian of fixed total momentum is defined in two space dimensions by proving the strong convergence of the resolvents of the cutoff Hamiltonians. The Hamiltonian for the physical polaron is defined as the direct integral of the fixed momentum Hamiltonians.
The present paper deals with a phenomenological description of nuclear reactions by means of boundary conditions. The several stages of the reaction (initial particles, compound nucleus, etc.) are described in their respective configuration spaces; and the state representing the whole system is given by a wave function in Fock space. The wave equations and the boundary conditions are deduced with the help of an analogy to vibration problems. The interaction leading to the nuclear reaction appears as a boundary condition between the different components of the Fock type wave function. A discussion is given of elastic scattering and of two-particle nuclear reactions, showing that the present procedure gives the same dependence of the cross sections on energy as the Breit-Wigner resonance formula and its generalizations. A sketch is given of the extension of the present ideas to multiple particle reactions. The whole treatment is nonrelativistic.
We construct the resolvent for a nonrelativistic multiparticle Schrödinger Hamiltonian having point interactions, and accommodating annihilation and creation of, at most, three particles. The construction is similar to that for the simple point-interaction (pseudopotential) case and leads to Skorniakov-Ter Martirosian-type integral equations. Unlike the Skorniakov-Ter Martirosian equations, however, the integral equations considered here have a unique solution and, moreover, the resulting Hamiltonian is bounded below. Scattering theory for the operator is discussed briefly.
Self-adjoint extensions of the operator- Delta with the domain C0infinity (R3) in the space Ck(+)L2(R3) are described. Such operators are interpreted as Hamiltonians of a point interaction of a quantum particle with the vacuum. Bound states and scattering objects of these Hamiltonians are investigated.
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 interacting with neutral, scalar bosons. Special attention is paid to the so called one particle problem: the existence of dressed one electron states – corresponding to an isolated one particle shell in the spectrum of the energy-momentum operator (H, P) – is established for the models with massive bosons and the ones with bosons of rest mass zero and an infrared cutoff in the interaction.In dieser Arbeit studieren wir die Dynamik einer Klasse einfacher (sog. persistenter) Modelle, unter anderen diejenige des Modelles von Nelson [15].Solche Modelle beschreiben Systeme von erhaltenen, nicht relativistischen, spinlosen Elektronen, die mit neutralen, skalaren Bosonen wechselwirken.Besonders sorgfltig wird das Einteilchenproblem studiert: Für die Modelle mit Bosonen positiver Ruhemasse und diejenigen mit Bosonen der Ruhemasse 0 und Infrarot-Cutoff im Wechselwirkungsoperator beweisen wir die Existenz physikalischer Einelektronen Zustnde, oder, in anderen Worten, wir zeigen, daß das gemeinsame Spektrum von Energie- und Impulsoperator eine isolierte Einteilchenschale hat.
The massless Nelson model describes non-relativistic, spinless quantum
particles interacting with a relativistic, massless, scalar quantum field. The
interaction is linear in the field. We analyze the one particle sector. First,
we construct the renormalized mass shell of the non-relativistic particle for
an arbitrarily small infrared cut-off that turns off the interaction with the
low energy modes of the field. No ultraviolet cut-off is imposed. Second, we
implement a suitable Bogolyubov transformation of the Hamiltonian in the
infrared regime. This transformation depends on the total momentum of the
system and is non-unitary as the infrared cut-off is removed. For the
transformed Hamiltonian we construct the mass shell in the limit where both the
ultraviolet and the infrared cut-off are removed. Our approach is constructive
and leads to explicit expansion formulae which are amenable to rigorously
control the S-matrix elements.
The (non-Lorentz covariant) system consisting of a relativistic scalar Boson field ϕ interacting with a single spinless particle (relativistic polaron) with kinetic energy function (m
2+|p|2)1/2 is studied ind space demensions, whered≧3. The interaction Hamiltonian is taken to be ∝ Ψ (x)* Ψ(x) ϕ(x)dx where ϕ has a momentum cutoff. The physical one polaron Hilbert space ℋph
for this model, corresponding to no cutoff on ϕ, is constructed. The total renormalized HamiltonianH without cutoff is constructed as a semibounded self-adjoint operator on ℋpH
. The time zero physical Boson field is also constructed. First order estimates are established for the local (in momentum space) number operators in terms ofH.
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