Quantum Field Models Research Articles

Multiplicity of ground states in quantum field models: applications of asymptotic fields

Ground states of Hamiltonian H of quantum field models are investigated. The infimum of the spectrum of H is in the edge of its essential spectrum. By means of the asymptotic field theory, we give a necessary and sufficient condition for that the expectation value of the number operator of ground states is finite, from which we give an upper bound of the multiplicity of ground states of H. Typical examples are massless GSB models and the Pauli–Fierz model with spin 1 / 2 .

Collapsing Bose-Einstein condensates beyond the Gross-Pitaevskii approximation

We analyse quantum field models of the bosenova experiment, in which $^{85}$Rb Bose-Einstein condensates were made to collapse by switching their atomic interactions from repulsive to attractive. Specifically, we couple the lowest order quantum field correlation functions to the Gross-Pitaevskii function, and solve the resulting dynamical system numerically. Comparing the computed collapse times with the experimental measurements, we find that the calculated times are much larger than the measured values. The addition of quantum field corrections does not noticeably improve the agreement compared to a pure Gross-Pitaevskii theory.

SYSTEMS OF CLASSICAL PARTICLES IN THE GRAND CANONICAL ENSEMBLE, SCALING LIMITS AND QUANTUM FIELD THEORY

Euclidean quantum fields obtained as solutions of stochastic partial pseudo differential equations driven by a Poisson white noise have paths given by locally integrable functions. This makes it possible to define a class of ultra-violet finite local interactions for these models (in any space-time dimension). The corresponding interacting Euclidean quantum fields can be identified with systems of classical "charged" particles in the grand canonical ensemble with an interaction given by a nonlinear energy density of the "static field" generated by the particles' charges via a "generalized Poisson equation". A new definition of some well-known systems of statistical mechanics is given by formulating the related field theoretic local interactions. The infinite volume limit of such systems is discussed for models with trigonometric interactions using a representation of such models as Widom–Rowlinson models associated with (formal) Potts models at imaginary temperature. The infinite volume correlation functional of such Potts models can be constructed by a cluster expansion. This leads to the construction of extremal Gibbs measures with trigonometric interactions in the low-density high-temperature (LD-HT) regime. For Poissonian models with certain trigonometric interactions an extension of the well-known relation between the (massive) sine-Gordon model and the Yukawa particle gas connecting characteristic and correlation functionals is given and used to derive infinite volume measures for interacting Poisson quantum field models through an alternative route. The continuum limit of the particle systems under consideration is also investigated and the formal analogy with the scaling limit of renormalization group theory is pointed out. In some simple cases the question of (non-) triviality of the continuum limits is clarified.

Transfer Matrix Spectrum for O(N) High Temperature Lattice Classical Ferromagnetic Spin Systems and Staggering Transformations

We obtain the low-lying energy-momentum spectrum for the imaginary-time lattice quantum field model associated with d-dimensional lattice ferromagnetic classical N-component vector spin systems at high temperature (0 < β << 1). Each system is characterized by a single site a priori spin probability distribution. The energy-momentum spectrum exhibits isolated dispersion curves which are identified as single particles and multi-particle bands. Our two-particle bound state analysis is restricted to a ladder approximation of the Bethe–Salpeter equation, and the existence of bound states depend on whether or not Gaussian domination for the four-point function is verified. It is known that two-particle bound states appear below the two-particle band if Gaussian domination does not hold. Here, we show that two two-particle bound states appear above the two-particle band if Gaussian domination is verified. We also show how the complete two-particle spectral pattern for these models can be understood by making a correspondence between the Bethe–Salpeter equation and a two-particle lattice Schrödinger Hamiltonian operator with attractive or repulsive spin-dependent delta potentials at the origin. A staggering transformation is used to relate the attractive and repulsive potential cases, as well as their associated Hamiltonians spectrum and eigenfunctions.

A new lattice approximation forthe Høegh-Krohn quantum field model

The lattice approximation for the exponential interaction model (Høbegh-Krohn quantum field model) with different lattice cutoffs a, a′ in the free and interacting parts is discussed. It is shown that the continuum limit exists under certain conditions on the dependence a′(a).

Commutators, C0-semigroups and resolvent estimates

We study the existence and the continuity properties of the boundary values on the real axis of the resolvent of a self-adjoint operator H in the framework of the conjugate operator method initiated by Mourre. We allow the conjugate operator A to be the generator of a C 0-semigroup (finer estimates require A to be maximal symmetric) and we consider situations where the first commutator [ H,i A] is not comparable to H. The applications include the spectral theory of zero mass quantum field models.

Fractional dynamics, Cantorian space–time and the gauge hierarchy problem

The gauge hierarchy problem in particle physics refers to the large numerical disparity between the value of the Planck mass (MPl≃1.22×1019 GeV) and the mass scale of the electroweak interaction (MEW≈102 GeV). Explaining the hierarchy paradox has been attempted so far in quantum field models based on supersymmetry or higher dimensional space–time with a large number of extra dimensions (brane theories). Despite several years of experimental search, there is currently no validation for either one of these models. We approach the hierarchy paradox using the methodology of fractal operators in four-dimensional space–time. It is found that departure from the inverse-square gravity in the high-energy regime emerges naturally from the fractional Helmholtz equation and offers a simple resolution to the problem. Our work makes an explicit connection between the hierarchy problem and Cantorian geometry of space–time on energy scales comparable to the Planck mass.

A Remark on different lattice approximations and continuum limits for -fields

Consider the lattice approximation of a -quantum field model with different lattice cutoffs a′ and a in the free and interacting parts, respectively. In [1] it was shown that the corresponding continuum limit measure exists if lima→0a′| log a|5/4 < ∞ and it coincides with the original - field measure if lima→0a′| log a|2 < ∞. In this paper, a result is given indicating that the new continuum limit measure might be different from the original one if a′ is too big compared with a.

Enhanced Binding in a General Class of Quantum Field Models

We consider, in an abstract form, a system of "quantum particles" coupled to a Bose field. It is shown that, under suitable hypotheses, the composed system can have a ground state even if the uncoupled particle system has no ground state.

Meson spectrum and analytic confinement

The basic dynamic properties of two-particle bound states of ``quarks'' and ``gluons'' are investigated within a simple relativistic quantum field model with the Yukawa interaction. Provided with an analytic confinement of the constituent particles and small coupling constant, this model explains qualitatively the experimental evidence: the quarks and gluons are confined, the final bound states are stable, massless gluons constitute the glueballs and the Regge trajectories of ``mesonic'' orbital excitations are asymptotically linear.

Dynamo in helical MHD turbulence: quantum field theory approach

A quantum field model of helical MHD stochastically forced by gaussian hydrodynamic, magnetic and mixed noices is investigated. These helical noises lead to an exponential increase of magnetic fluctuations in the large scale range. Instabilities, which are produced in this process, are eliminated by spontaneous symmetry breaking mechanism accompanied by creation of the homogeneous stationary magnetic field.

Hyperfunction quantum field theory: Analytic structure, modular aspects, and local observable algebras

This paper addresses the following problem of relativistic quantum field theory: Given a relativistic quantum field, construct a net of local observable algebras over space–time with “natural” properties. A few years ago we started a project which suggests to look at this problem in the framework of relativistic quantum field theory in terms of Fourier hyperfunctions. Accordingly we present the relevant analyticity results, some modular aspects of our theory for the Bisognano–Wichman argument, and a concrete suggestion for the definition of the local observable algebras. Finally, we construct a class of models of hyperfunction quantum fields for which the algebras of bounded operators assigned to nonempty regions in space–time are not trivial. It is remarkable that our models are not tempered quantum fields and do not admit “test functions” of compact support.

Infinities in Quantum Gravities

AbstractWe first present a review, intended for classical relativists, of the ultraviolet difficulties faced by local quantum gravity theories in both the usual Einstein versions and in their supergravity extensions, at least perturbatively. These problems, present in arbitrary dimensions, are traceable to the dimensionality of the Einstein constant.We then summarize very recent results about supergravity at the highest allowed dimension, D = 11, showing that also this unique model suffers from infinities already at 2 loops, despite its high degree of supersymmetry. The conclusion is that there is no viable nonghost quantum field model that includes general relativity.

Infinities in Quantum Gravities

We first present a review, intended for classical relativists, of the ultraviolet difficulties faced by local quantum gravity theories in both the usual Einstein versions and in their supergravity extensions, at least perturbatively. These problems, present in arbitrary dimensions, are traceable to the dimensionality of the Einstein constant. We then summarize very recent results about supergravity at the highest allowed dimension, D=11, showing that also this unique model suffers from infinities already at 2 loops, despite its high degree of supersymmetry. The conclusion is that there is no viable nonghost quantum field model that includes general relativity.

On the Absence of Eigenvectors of Hamiltonians in a Class of Massless Quantum Field Models without Infrared Cutoff

A class of models of quantized, massless Bose fields, called the generalized spin-boson model (A. Arai and M. Hirokawa, J. Funct. Anal.151 (1997), 455–503) is considered. Theorems on the absence of ground states and the other eigenvectors of the model without infrared cutoff (but with ultraviolet cutoff) are established with conditions in terms of correlation functions for some operators.

Map from one-dimensional quantum field theory to quantum chaos on a two-dimensional torus.

Dynamics of a class of quantum field models on a one-dimensional (1D) lattice in a Heisenberg picture is mapped onto a class of quantum chaotic one-body systems on a configurational 2D torus (or 2D lattice) in the Schrödinger picture. The continuum field limit of the former corresponds to the quasiclassical limit of the latter.

Travel time of a quantum particle through a given domain

A quantum field model is proposed for determining the travel time of a quantum particle through a given domain. The model involves a source and a detector of scalar particles. Using this model, the particle travel time between the source and detector, as well as the tunneling time, is determined. The travel time depends on the detector characteristics, and tunneling decreases the travel time.

The generalized continuous analog of Newton’s method for the numerical study of some nonlinear quantum-field models

A numerical method for studying nonlinear problems arising in mathematical models of physics is systematically described in this review. The unified basis for the development of numerical schemes is a generalization of the continuous analog of Newton’s method, which represents a qualitatively new development of the Newtonian evolution process on the basis of the integration of concepts from perturbation theory and the theory of evolution in parameters. The results are presented of numerical studies of quantum-field models of the polaron, the solvated electron, the binucleon, and also QCD potential models for some commonly used potentials.

DIFFEOMORPHISM GROUPS AND CURRENT ALGEBRAS: CONFIGURATION SPACE ANALYSIS IN QUANTUM THEORY

The constuction of models of non-relativistic quantum fields via current algebra representations is presented using a natural differential geometry of the configuration space Γ of particles, the corresponding classical Dirichlet operator associated with a Poisson measure on Γ, being the free Hamiltonian. The case with interactions is also discussed together with its relation to the problem of unitary representations of the diffeomorphism group on ℝd.

Remarks on some new models of interacting quantum fields with indefinite metric

We study quantum field models in indefinite metric. We introduce the modified Wightman axioms of Morchio and Strocchi as a general framework of indefinite metric quantum field theory (QFT) and present concrete interacting relativistic models obtained by analytical continuation from some stochastic processes with Euclidean invariance. As a first step towards scattering theory in indefinite metric QFT, we give a proof of the spectral condition on the translation group for the relativistic models.