We introduce an abstract class of bosonic QFT Hamiltonians and study their spectral and scattering theories. These Hamiltonians are of the form $H=\d\G(ω)+ V$ acting on the bosonic Fock space $\G(\ch)$, where $ω$ is a massive one-particle Hamiltonian acting on $\ch$ and $V$ is a Wick polynomial $\Wick(w)$ for a kernel $w$ satisfying some decay properties at infinity. We describe the essential spectrum of $H$, prove a Mourre estimate outside a set of thresholds and prove the existence of asymptotic fields. Our main result is the {\em asymptotic completeness} of the scattering theory, which means that the CCR representations given by the asymptotic fields are of Fock type, with the asymptotic vacua equal to the bound states of $H$. As a consequence $H$ is unitarily equivalent to a collection of second quantized Hamiltonians.
We present a general construction of KMS states in the framework of perturbative algebraic quantum field theory (pAQFT). Our approach may be understood as an extension of the Schwinger-Keldysh formalism. We obtain in particular the Wightman functions at positive temperature, thus solving a problem posed some time ago by Steinmann. The notorious infrared divergences observed in a diagrammatic expansion are shown to be absent due to a consequent exploitation of the locality properties of pAQFT. To this avail, we introduce a novel, Hamiltonian description of the interacting dynamics and find, in …
We construct Hamiltonians for systems of nonrelativistic particles linearly coupled to massive scalar bosons using abstract boundary conditions. The construction yields an explicit characterisation of the domain of self-adjointness in terms of boundary conditions that relate sectors with different numbers of bosons. We treat both models in which the Hamiltonian may be defined as a form perturbation of the free operator, such as Fröhlich's polaron, and renormalisable models, such as the massive Nelson model.
In (1+1)-dimensional quantum field theory, integrability is typically defined as the existence of an infinite number of local charges of different Lorentz spin, which commute with the Hamiltonian. A well known consequence of integrability is that scattering of particles is elastic and factorizable. These properties are the basis for the bootstrap program, which leads to the exact computation of S-matrices and form factors. We consider periodically-driven field theories, whose stroboscopic time-evolution is described by a Floquet Hamiltonian. It was recently proposed by Gritsev and Polkovnikov …
This paper is devoted to the analysis of an abstract formula describing quantum adiabatic charge pumping in a general context. We consider closed systems characterized by a slowly varying time-dependent Hamiltonian depending on an external parameter $α$. The current operator, defined as the derivative of the Hamiltonian with respect to $α$, once integrated over some time interval, gives rise to a charge pumped through the system over that time span. We determine the first two leading terms in the adiabatic parameter of this pumped charge under the usual gap hypothesis. In particular, in case t…
It is well-known that there exist infinitely-many inequivalent representations of the canonical (anti)-commutation relations of Quantum Field Theory (QFT). A way out, suggested by Algebraic QFT, is to instead define the quantum theory as encompassing all possible (abstract) states. In the present paper, we describe a quantization scheme for general linear (aka. free) field theories that can be seen as intermediate between traditional Fock quantization and full Algebraic QFT, in the sense that: * it provides a constructive, explicit description of the resulting space of quantum states; * it doe…
The aim of this paper is to create a large geometrical background on the dual 1-jet space J^{1*}(T,M) for a multi-time Hamiltonian approach of the electromagnetic and gravitational physical fields. Our geometric-physical construction is achieved starting only from a given quadratic Hamiltonian function of polymomenta H, which naturally produces a canonical nonlinear connection N, a canonical Cartan N-linear connection CΓ(N) and their corresponding local distinguished (d-) torsions and curvatures. In such a context, we construct some geometrical electromagnetic-like and gravitational-like field…
We study random Hamiltonians on finite-size cubes and waveguide segments of increasing diameter. The number of random parameters determining the operator is proportional to the volume of the cube. In the asymptotic regime where the cube size, and consequently the number of parameters as well, tends to infinity, we derive deterministic and probabilistic variational bounds on the lowest eigenvalue, i.e. the spectral minimum, as well as exponential off-diagonal decay of the Green function at energies above, but close to the overall spectral bottom.
For two self-adjoint operators $H,A$ we show that a general commutation relation of type $[H,\mathrm{i}A]=Q(H)+K$, in addition to regularity of $H$ and Kato-smoothness of $K$, guarantee pointwise in time decay rates of diverse order. The methodology is based on the construction of a modified conjugate operator $\tilde{A}$ that reduces the problem to previously developed estimates when $K=0$. Our results apply to energy thresholds and do not rely on resolvent estimates. We discuss applications for the Schrödinger equation (SE) with potential of critical decay, and for the free SE on an asymptot…
We prove some abstract Wegner bounds for random self-adjoint operators. Applications include elementary proofs of Wegner estimates for discrete and continuous Anderson Hamiltonians with possibly sparse potentials, as well as Wegner bounds for quantum graphs with random edge length or random vertex coupling. We allow the coupling constants describing the randomness to be correlated and to have quite general distributions.
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 constru…
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.
In this paper we prove a result of almost global existence for some abstract nonlinear PDEs on flat tori and apply it to some concrete equations, namely a nonlinear Schrödinger equation with a convolution potential, a beam equation and a quantum hydrodinamical equation. We also apply it to the stability of plane waves in NLS. The main point is that the abstract result is based on a nonresonance condition much weaker than the usual ones, which rely on the celebrated Bourgain's Lemma which provides a partition of the "resonant sites" of the Laplace operator on irrational tori.
This is an extended abstract for the conference "Microlocal2011 : Microlocal Methods in Mathematical Physics and Global Analysis Universitat Tubingen, June 14 - 18, 2011"
We consider a rather general version of ladder operator $Z$ used by some authors in few recent papers, $[H_0,Z]=λZ$ for some $λ\in\mathbb{R}$, $H_0=H_0^\dagger$, and we show that several interesting results can be deduced from this formula. Then we extend it in two ways: first we replace the original equality with formula $[H_0,Z]=λZ[Z^\dagger, Z]$, and secondly we consider $[H,Z]=λZ$ for some $λ\in\mathbb{C}$, $H\neq H^\dagger$. In both cases many applications are discussed. In particular we consider factorizable Hamiltonians and Hamiltonians written in terms of operators satisfying the gener…
We consider an abstract pair-interaction model in quantum field theory with a coupling constant $λ\in {\mathbb R}$ and analyze the Hamiltonian $H(λ)$ of the model. In the massive case, there exist constants $λ_{\rm c}<0$ and $λ_{{\rm c},0}<λ_{\rm c}$ such that, for each $λ\in (λ_{{\rm c},0},λ_{\rm c})\cup (λ_{\rm c},\infty)$, $H(λ)$ is diagonalized by a proper Bogoliubov transformation, so that the spectrum of $H(λ)$ is explicitly identified, where the spectrum of $H(λ)$ for $λ>λ_{\rm c}$ is different from that for $λ\in (λ_{{\rm c},0}, λ_{\rm c})$. As for the case $λ<λ_{{\rm c},0}$, we show t…
We overview the recent results on the shift of the spectrum and norm bounds for variation of spectral subspaces of a Hermitian operator under an additive Hermitian perturbation. Along with the known results, we present a new subspace variation bound for the generic off-diagonal subspace perturbation problem. We also demonstrate how some of the abstract results may work for few-body Hamiltonians.
We study the spectrum of the spin-boson model with two photons in $\mathbb{R}^d$ for arbitrary coupling $α>0$. It is shown that the discrete spectrum is finite and the essential spectrum consists of a half-line the bottom of which is a unique zero of a simple Nevanlinna function. Besides the simplicity and more abstract nature of our approach, the main novelty is the achievement of these results under "minimal" regularity conditions on the photon dispersion and the coupling function.
