Lieb–Robinson Bounds,Arveson Spectrum and Haag–Ruelle Scattering Theory for Gapped Quantum Spin Systems

We consider translation invariant gapped quantum spin systems satisfying the Lieb–Robinson bound and containing single-particle states in a ground state representation. Following the Haag–Ruelle approach from relativistic quantum field theory, we construct states describing collisions of several particles, and define the corresponding S-matrix. We also obtain some general restrictions on the shape of the energy–momentum spectrum. For the purpose of our analysis, we adapt the concepts of almost local observables and energy–momentum transfer (or Arveson spectrum) from relativistic QFT to the lattice setting. The Lieb–Robinson bound, which is the crucial substitute of strict locality from relativistic QFT, underlies all our constructions. Our results hold, in particular, in the Ising model in strong transverse magnetic fields.     

Gamow vectors for an unstable relativistic quantum field

We study a model of cubic interaction between two scalar fields with a scattering resonance. The resonance manifests as two poles of the analytic continuations of the Green function with respect to energy. The Gamow vectors associated to these resonances acquire meaning in suitable rigged Fock spaces. Finally, we discuss some properties of the S-matrix for unstable fields.     

Waves, particles and fields: an explicitly covariant approach

The idea of associating particle trajectories with wave propagation rays exploited in a previous paper in the context of general relativity with a synchronous gauge, here is examined with no assumptions on co-ordinate choice (no synchronous gauge condition on the metric). Identification of particle Hamilton–Jacobi equation with wave-sheet equation in a space–time with more than 4 dimensions, is performed in an explicitly covariant formulation, leading to a Kaluza–Klein type theory involving Klein–Gordon equation arising from dilaton field equations. De Broglie and Einstein-Planck quantum relations are also deduced in a natural way. Adding suitable Yang–Mills fields provides unification of gravitational, electromagnetic, weak and strong interactions into a $16$ dimensional space–time geometry. The electron mass gap is also avoided compactifying extra dimensional co-ordinates on fractalized closed paths.     

Scattering of S0 Lamb mode in plate with multiple damage

A study of scattering properties of S₀ mode Lamb wave in an infinite plate with multiple damage is presented. Plate theory and wave function expansion method are used to derive the analytical solutions for the scattering wave field in plate with a single damage, and by using the addition theorems of Bessel functions, interference phenomena between scattering wave fields from different damage is investigated. Measurements agree well between theoretical results and FE simulation study of plate with two damage and validity of the model is confirmed. Numerical results of scattering displacement field in plate with two and three damage are graphically presented and discussed. An assessment of effects of damage geometric properties on the scattering properties is made.     

Scattering theory for the Federbush,massless Thirring and continuum Ising models

The existence of the LSZ limits of the positive energy quantum fields of the Federbush, Thirring and Ising models is proved. The corresponding S-matrices are obtained explicitly, confirming previous formal results, and are shown to result from a large class of dynamics. The scattering theory for the Federbush and Thirring models on the classical and negative energy quantum levels is studied as well. The wave and scattering operators are shown to exist and are obtained explicitly, and they are proved to be shared by a large class of dynamics, containing in particular the polynomial conserved charges.     

The Infrared Problem in QED: A Lesson from a Model with Coulomb Interaction and Realistic Photon Emission

The scattering of photons and heavy classical Coulomb interacting particles, with realistic particle–photon interaction (without particle recoil) is studied adopting the Koopman formulation for the particles. The model is translation invariant and allows for a complete control of the Dollard strategy devised by Kulish–Faddeev and Rohrlich (KFR) for QED: in the adiabatic formulation, the Møller operators exist as strong limits and interpolate between the dynamics and a non-free asymptotic dynamics, which is a unitary group; the S-matrix is non-trivial and exhibits the factorization of all the infrared divergences. The implications of the KFR strategy on the open questions of the LSZ asymptotic limits in QED are derived in the field theory version of the model, with the charged particles described by second quantized fields: i) asymptotic limits of the charged fields, \({\Psi_{{\rm out}/{\rm in}}(x)}\), are obtained as strong limits of modified LSZ formulas, with corrections given by a Coulomb phase operator and an exponential of the photon field; ii) free asymptotic electromagnetic fields, \({B_{{\rm out}/{\rm in}}(x)}\), are given by the massless LSZ formula, as in Buchholz approach;   iii) the asymptotic field algebras are a semidirect product of the canonical algebras generated by \({B_{{\rm out}/{\rm in}}}\), \({\Psi_{{\rm out}/{\rm in}}}\);   iv) on the asymptotic spaces, the Hamiltonian is the sum of the free (commuting) Hamiltonians of \({B_{{\rm out}/{\rm in}}}\), \({\Psi_{{\rm out}/{\rm in}}}\) and the same holds for the generators of the space translations.     

Poincaré's recurrence theorem and the unitarity of the S-matrix

A scattering process can be described by suitably closing the system and considering the first return map from the entrance onto itself. This scattering map may be singular and discontinuous, but it will be measured preserving as a consequence of the recurrence theorem applied to any region of a simpler map. In the case of a billiard this is the Birkhoff map. The semiclassical quantization of the Birkhoff map can be subdivided into an entrance and a repeller. The construction of a scattering operator then follows in exact analogy to the classical process. Generically, the approximate unitarity of the semiclassical Birkhoff map is inherited by the S-matrix, even for highly resonant scattering where direct quantization of the scattering map breaks down.     

Gauge invariance and implementability of the S-operator for spin-0 and spin-12 particles in time-dependent external fields

It is proved that the classical S-operator for relativistic spin-0 and spin-$\text{1}\text{2}$ particles in time-dependent external fields is gauge invariant, and that S_+- and S_?+ are entire functions of the coupling constant in the Hilbert-Schmidt norm. As a result the Fock space S-operator exists for any real value of the coupling constant, and is gauge invariant. The external fields and the gauge function are assumed to be real-valued resp. complex-valued functions in S(R⁴).     

The 3-permutation orbifold of a lattice vertex operator algebra

Irreducible modules of the 3-permutation orbifold of a rank one lattice vertex operator algebra are listed explicitly. Fusion rules are determined by using the quantum dimensions. The S-matrix is also given.     

A quantum model of a real scalar field

A quantum model of a real scalar field with local operator gauge symmetry is discussed. In the localized theory, in order to keep the local operator gauge symmetry, an operator gauge potential BB _μ, is needed. By combining the constraint of operator gauge potentialB _μ, and the microscopic causality theorem, the usual canonical quantization condition of a real scalar field is obtained. Therefore, a quantum model of a real scalar field without the usual procedure of quantizing a related classical model can be directly constructed. Project supported in part by T.D. Lee’s NNSF Grant, National Natural Science Foundation of China, Foundation of Ph. D. Directing Programme of Chinese Universities and the Chinese Academy of Sciences.     

Dynamical Locality of the Free Scalar Field

Dynamical locality is a condition on a locally covariant physical theory, asserting that kinematic and dynamical notions of local physics agree. This condition was introduced in arXiv:1106.4785, where it was shown to be closely related to the question of what it means for a theory to describe the same physics on different spacetimes. In this paper, we consider in detail the example of the free minimally coupled Klein–Gordon field, both as a classical and quantum theory (using both the Weyl algebra and a smeared field approach). It is shown that the massive theory obeys dynamical locality, both classically and in quantum field theory, in all spacetime dimensions n ≥ 2 and allowing for spacetimes with finitely many connected components. In contrast, the massless theory is shown to violate dynamical locality in any spacetime dimension, in both classical and quantum theory, owing to a rigid gauge symmetry. Taking this into account (equivalently, working with the massless current) dynamical locality is restored in all dimensions n ≥ 2 on connected spacetimes, and in all dimensions n ≥ 3 if disconnected spacetimes are permitted. The results on the quantized theories are obtained using general results giving conditions under which dynamically local classical symplectic theories have dynamically local quantizations.     

Discrete Quantum Scattering Theory

We formulate quantum scattering theory in terms of a discrete L ₂-basis of eigen differentials. Using projection operators in the Hilbert space, we develop a universal method for constructing finite-dimensional analogues of the basic operators of the scattering theory: S- and T-matrices, resolvent operators, and Möller wave operators as well as the analogues of resolvent identities and the Lippmann–Schwinger equations for the T-matrix. The developed general formalism of the discrete scattering theory results in a very simple calculation scheme for a broad class of interaction operators.     

Quantization of formal classical dynamical r-matrices: the reductive case

In this paper we prove the existence of a formal dynamical twist quantization for any triangular and non-modified formal classical dynamical r-matrix in the reductive case. The dynamical twist is constructed as the image of the dynamical r-matrix by a L_∞-quasi-isomorphism. This quasi-isomorphism also allows us to classify formal dynamical twist quantizations up to gauge equivalence.     

An Indefinite Metric Model for Interacting Quantum Fields on Globally Hyperbolic Space-Times

We consider a relativistic ansatz for the vacuum expectation values (VEVs) of a quantum field on a globally hyperbolic space-time which is motivated by certain Euclidean field theories. The Yang-Feldman asymptotic condition w.r.t. an in-field in a quasi-free representation of the canonic commutation relations (CCR) leads to a solution of this ansatz for the VEVs. A GNS-like construction on a non-degenerate inner product space then gives local, covariant quantum fields with indefinite metric on a globally hyperbolic space-time. The non-trivial scattering behavior of quantum fields is analyzed by construction of the out-fields and calculation of the scattering matrix. A new combined effect of non-trivial quantum scattering and non-stationary gravitational forces is described for this model, as quasi-free in- fields are scattered to out-fields which form a non-quasi-free representations of the CCR. The asymptotic condition, on which the construction is based, is verified for the concrete example of de Sitter space-time. Communicated by Klaus Fredenhagen submitted 03/05/02, accepted: 18/07/03     

Local scale invariance,Cantorian space–time and unified field theory

We develop field theoretic arguments for the unification of relativistic gravity with standard model interactions on El Naschie's Cantorian space–time. The work proceeds by showing the equivalence between the fundamental principle of local gauge invariance and the local scale invariance of space–time and matter fields undergoing critical behavior on high-energy scales. We focus on the transition boundary between the classical and non-classical regimes, the latter being characterized by generalized scaling laws with continuously varying exponents. Both relativistic gravity and standard model interactions emerge from the underlying geometry of Cantorian space–time near this transition boundary.     

Unified Field Theory and Principle of Representation Invariance

The main objectives of this article are to postulate a new principle of representation invariance (PRI), and to refine the unified field model of four interactions, derived using the principle of interaction dynamics (PID). Intuitively, PID takes the variation of the action functional under energy-momentum conservation constraint, and PRI requires that physical laws be independent of representations of the gauge groups. One important outcome of this unified field model is a natural duality between the interacting fields (g,A,W ^a ,S ^k ), corresponding to graviton, photon, intermediate vector bosons W ^± and Z and gluons, and the adjoint bosonic fields $(\varPhi_{\mu}, \phi^{0}, \phi^{a}_{w}, \phi^{k}_{s})$ . This duality predicts two Higgs particles of similar mass with one due to weak interaction and the other due to strong interaction. The unified field model can be naturally decoupled to study individual interactions, leading to (1) modified Einstein equations, giving rise to a unified theory for dark matter and dark energy (Ma and Wang in Discrete Contin. Dyn. Syst., Ser A. 34(2):335–366, 2014), (2) three levels of strong interaction potentials for quark, nucleon/hadron, and atom respectively (Ma and Wang in Duality theory of strong interaction, 2012), and (3) two weak interaction potentials (Ma and Wang in Duality theory of weak interaction, 2012). These potential/force formulas offer a clear mechanism for both quark confinement and asymptotic freedom—a longstanding problem in particle physics (Ma and Wang in Duality theory of strong interaction, 2012).     

Bootstrap Equations in Effective Theories

In this paper, we consider general properties of effective field theories. We note that the freedom of fixing renormalization conditions in the effective field theory is not as large as it seems. Consideration of the minimal set of correctness requirements of the perturbative scheme based on the Dyson's formula for the S-matrix leads to severe restrictions on essential parameters of the theory and hence on the possible set of renormalization conditions. In the first part of this paper, we give a short review of the structure of localizable effective field theories. We discuss necessary conditions which ensure the correctness of the first step of the iterative scheme of calculation of the S-matrix, i.e., the construction of tree-level amplitudes. In the second part, we discuss examples which demonstrate the main stages of acquisition and analysis of the system of bootstrap equations. Bibliography: 17 titles.     

On Eisenbud's and Wigner's R-matrix: A general approach

The main objective of this paper is to give a rigorous treatment of Wigner's and Eisenbud's R-matrix method for scattering matrices of scattering systems consisting of two selfadjoint extensions of the same symmetric operator with finite deficiency indices. In the framework of boundary triplets and associated Weyl functions an abstract generalization of the R-matrix method is developed and the results are applied to Schrödinger operators on the real axis.