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Volume 37, Issue 5, May 1996

Why the Thirring model cannot fulfill canonical anticommutation relations
View Description Hide DescriptionPerturbative and constructive arguments show that the Gross–Neveu model with N≥2 flavor degrees is asymptotically free. Assuming canonical anticommutation relations (CAR), which are quite plausible, and some technicalities, we show for the Thirring model, i.e., the case of N=1 flavor, that the fields are necessarily free. This explains from an axiomatic point of view the different behavior depending on the number of flavors.

The doublet representation of non‐Hilbert eigenstates of the Hamiltonian
View Description Hide DescriptionWe find the minimal mathematical structure to represent quantum eigenstates with complex eigenvalues with no need of analytic continuation. These eigenvectors build doublets in non‐Hilbert spaces. We construct exact solutions for the Friedrichs model that continuously join the ones of the free Hamiltonian. We extend the Wigner operator to these non‐Hilbert spaces and enlarge the concept of normalized vectors via the definition of the doublets. Making use of these doublets, we describe systems whose states have initial conditions out of Hilbert space.

Maximal localization in the presence of minimal uncertainties in positions and in momenta
View Description Hide DescriptionSmall corrections to the uncertainty relations, with effects in the ultraviolet and/or infrared, have been discussed in the context of string theory and quantum gravity. Such corrections lead to small but finite minimal uncertainties in position and/or momentum measurements. It has been shown that these effects could indeed provide natural cutoffs in quantum field theory. The corresponding underlying quantum theoretical framework includes small ‘‘noncommutative geometric’’ corrections to the canonical commutation relations. In order to study the full implications on the concept of locality, it is crucial to find the physical states of then maximal localization. These states and their properties have been calculated for the case with minimal uncertainties in positions only. Here we extend this treatment, though still in one dimension, to the general situation with minimal uncertainties both in positions and in momenta.

Discrete decay and continuous measurement
View Description Hide DescriptionIn this paper measurement is defined via an observation operator with simple, purely discrete spectrum. When the quantum Zeno effect holds, a continuously measured quantum process with fixed initial state ‘‘freezes’’ in that state. Such states are called ‘‘regular’’ states. If the final state is also fixed, then movement from the initial state to the final state is forced to occur, whether or not the initial state is regular. This forced movement is studied here in the context of a discrete state model which contains a distinguished nonregular state which mediates all transitions. In this model the forced movement between any two states satisfies a ‘‘least action’’ principle, wherein action is identified with change of state transitions. For completeness, the model is also studied when only the initial state is fixed under continuous measurement. In that case the distinguished state exhibits exact exponential decay at all times. The model is of interest independently of issues related to continuous measurement inasmuch as it is a ‘‘discrete’’ approximation to a ‘‘continuous’’ decay model. More precisely, the distinguished state exhibits exact exponential decay on a finite time interval which expands without bound as the discrete decay products densely approach the continuum. This unexpected result provides a striking confirmation of Fermi’s ‘‘golden rule.’’

Quantum Riemann surfaces for arbitrary Planck’s constant
View Description Hide DescriptionWe continue our study of quantum Riemann surfaces initiated in Refs. [S. Klimek and A. Lesniewski, Commun. Math. Phys. 146, 103–122 (1992); Lett. Math. Phys. 24, 125–139 (1992); 32, 45–61 (1994)]. We construct a one parameter family of deformations of compact Riemann surfaces of genus g≥2. Our construction does not require any discreteness condition on the value of Planck’s constant. It coincides with the construction of Ref. [Lett. Math. Phys. 24, 125–139 (1992)] in the case when Planck’s constant assumes the discrete set of values dictated by geometric quantization.

Radial Coulomb and oscillator systems in arbitrary dimensions
View Description Hide DescriptionA mapping is obtained relating analytical radial Coulomb systems in any dimension greater than one to analytical radial oscillators in any dimension. This mapping, involving supersymmetry‐based quantum‐defect theory, is possible for dimensions unavailable to conventional mappings. Among the special cases is an injection from bound states of the three‐dimensional radial Coulomb system into a three‐dimensional radial isotropic oscillator where one of the two systems has an analytical quantum defect. The issue of mapping the continuum states is briefly considered.

Long‐time approximation to the evolution of resonant and nonresonant anharmonic oscillators in quantum mechanics
View Description Hide DescriptionA simple normal‐form approach is used to obtain uniform long‐time approximations to the evolution of a resonant or nonresonant anharmonic oscillator system governed by a Hamiltonian H _{ε} which is a self‐adjoint operator acting in the Hilbert spaceH=L ^{2}(R ^{ν}) (ν≥2) and is given formally by H _{0}+εV. Here H _{0} denotes the Hamiltonian of ν one‐dimensional harmonic oscillators whose coupling is represented by εV, where V is an operator of multiplication by a smooth function of at most polynomial growth at infinity and ε≥0 is a small parameter. We consider the general situation in which ρ≥1 of the frequencies of these oscillators are rationally independent, imposing a standard diophantine condition on the independent frequencies if ρ≥2. Under these assumptions, which are stated in a mathematically precise way in the paper, an Nth‐order approximant ψ_{ N }(t,ε) to the exact solution ψ(t,ε) of the Schrödinger equation id ψ(t,ε)/dt=H _{ε}ψ(t,ε) satisfying the initial condition ψ(0,ε)=ψ_{0} is constructed inductively, ψ_{0} being an arbitrary ε‐independent member of a suitable family of smooth functions dense in H. Our main result is that ψ_{ N }(t,ε) differs from ψ(t,ε) in H‐norm by ≤const ε^{ N+1}(‖t‖+1) for all t∈R and all ε in an arbitrary compact interval [0, ε_{0}].

An equivalence proof of the background gauge field quantization and the conventional one
View Description Hide DescriptionThe background gauge field quantization is a convenient tool for studying weakly interacting gauge and matter fields or analyzing anomalous current conservation in fermionic structures. This method is from the mid 1970’s, but it is only today that it received renewed interest for investigating nonperturbative evolution equations in Yang–Mills theory, as well as gauge field effective action formulations. We reviewed, to start with, the general formulation and assumptions about this method, and we pointed out some critical observations concerning it. In particular, we focus on some of the most common equivalence proofs presently known in the literature. We attempted to give a most convincing demonstration of this equivalence as it stands between the background gauge field scattering operator and the conventional one. The result shown here clearly indicates these methods are indeed physically equivalent. In proving that, we neglected all the infrared problems afflicting the pure Yang–Mills gauge theory; as a matter of fact, they appear to be a parallel, but nonintersecting problem with respect to the present one, i.e., to prove the equivalence.

Transient waves in nonstationary media
View Description Hide DescriptionThis paper treats propagation of transient waves in nonstationary media, which has many applications in, for example, electromagnetics and acoustics. The underlying hyperbolic equation is a general, homogeneous, linear, first‐order 2×2 system of equations. The coefficients in this system depend on one spatial coordinate and time. Furthermore, memory effects are modeled by integral kernels, which, in addition to the spatial dependence, are functions of two different time coordinates. These integrals generalize the convolution integrals, frequently used as a model for memory effects in the medium. Specifically, the scattering problem for this system of equations is addressed. This problem is solved by a generalization of the wave splitting concept, originally developed for wave propagation in media which are invariant under time translations, and by an imbedding or a Green’s functions technique. More explicitly, the imbedding equation for the reflection kernel and the Green’s functions (propagator kernels) equations are derived. Special attention is paid to the problem of nonstationary characteristics. A few numerical examples illustrate this problem.

An ongoing big bang model in the special relativistic Maxwell–Dirac equations
View Description Hide DescriptionAn exact, analytical solution of the combined Maxwell–Diracequation is presented. The solution is regular inside a wedgelike domain of the flat space–time. On the boundary of that domain, the solution exhibits singular behavior.

Quantum random walk for U _{ q }(su(2)) and a new example of quantum noise
View Description Hide DescriptionWe derive the quantum version of the central limit theorem for sample sums of q‐independent q‐identically distributed quantum variables for q∈R ^{+}−{1}. In particular, we consider U _{ q } ( su(2)), for which we obtain in the limit a state Ψ on the algebra generated by q‐commuting q‐oscillators indexed by pairs of real numbers (r,s), where 0≤r<s, giving a new example of non‐additive quantum noise.

Universality classes for asymptotic behavior of relaxation processes in systems with dynamical disorder: Dynamical generalizations of stretched exponential
View Description Hide DescriptionThe asymptotic behavior of multichannel parallel relaxation processes for systems with dynamical disorder is investigated in the limit of a very large number of channels. An individual channel is characterized by a state vector x which, due to dynamical disorder, is a random function of time. A limit of the thermodynamic type in the x‐space is introduced for which both the volume available and the average number of channels tend to infinity, but the average volume density of channels remains constant. Scaling arguments combined with a stochastic renormalization group approach lead to the identification of two different types of universal behavior of the relaxation function corresponding to nonintermittent and intermittent fluctuations, respectively. For nonintermittent fluctuations a dynamical generalization of the static Huber’s relaxation equation is derived which depends only on the average functional density of channels, ρ[W(t′)]D[W(t′)], the channels being classified according to their different relaxation rates W=W(t′), which are random functions of time. For intermittent fluctuations a more complicated relaxation equation is derived which, in addition to the average density of channels, ρ[W(t′)]D[W(t′)], depends also on a positive fractal exponent H which characterizes the fluctuations of the density of channels. The general theory is applied for constructing dynamical analogs of the stretched exponential relaxation function. For nonintermittent fluctuations the type of relaxation is determined by the regression dynamics of the fluctuations of the relaxation rate. If the regression process is fast and described by an exponential attenuation function, then after an initial stretched exponential behavior the relaxation process slows down and it is not fully completed even in the limit of very large times. For self‐similar regression obeying a negative power law, the relaxation process is less sensitive to the influence of dynamical disorder. Both for small and large times the relaxation process is described by stretched exponentials with the same fractal exponent as for systems with static disorder. For large times the efficiency of the relaxation process is also slowed down by fluctuations. Similar patterns are found for intermittent fluctuations with the difference that for very large times and a slow regression process a crossover from a stretched exponential to a self‐similar algebraic relaxation function occurs. Some implications of the results for the study of relaxation processes in

Bi‐Hamiltonian structures of the coupled AKNS hierarchy and the coupled Yajima–Oikawa hierarchy
View Description Hide DescriptionThe Hamiltonian theory for the two‐component AKNS hierarchy and Yajima–Oikawa hierarchy is considered from the viewpoint of reduction. We show that the second Hamiltonian structures of the former is a Dirac reduction of the sl(3) current algebra, while the latter is related to the classical W ^{(3)} _{4}algebra.

The W _{1+∞}(gl _{ s })‐symmetries of the s‐component KP hierarchy
View Description Hide DescriptionAdler, Shiota, and van Moerbeke obtained for the KP and Toda lattice hierarchies a formula which translates the action of the vertex operator on tau‐functions to an action of a vertex operator of pseudodifferential operators on wave functions. This relates the additional symmetries of the KP and Toda lattice hierarchy to the W _{1+∞− }, respectively, W _{1+∞}×W _{1+∞− }algebra symmetries. In this paper we generalize the results to the s‐component KP hierarchy. The vertex operators generate the algebraW _{1+∞}(gl _{ s }), the matrix version of W _{1+∞}. Since the Toda lattice hierarchy is formally equivalent to the 2‐component KP hierarchy, the results of this article uncover in that particular case a much richer structure than the one obtained by Adler, Shiota, and van Moerbeke.

A new integrable symplectic map associated with lattice soliton equations
View Description Hide DescriptionA method is developed that extends the nonlinearization technique to the hierarchy of lattice soliton equations associated with a discrete 3×3 matrix spectral problem. A new integrable symplectic map and its involutive system of conserved integrals are obtained by the nonlinearization of spatial parts and the time parts of Lax pairs and their adjoint Lax pairs of the hierarchy. Moreover, the solutions of the typical system of lattice equations in the hierarchy are reduced to the solutions of a system of ordinary differential equations and a simple iterative process of the symplectic map.

New no‐scalar‐hair theorem for black holes
View Description Hide DescriptionA new no‐hair theorem is formulated which rules out a very large class of nonminimally coupled finite scalar dressing of an asymptotically flat, static, and spherically symmetric black hole. The proof is very simple and based on a covariant method for generating solutions for nonminimally coupled scalar fields starting from the minimally coupled case. Such a method generalizes the Bekenstein method for conformal coupling and other recent ones. We also discuss the role of the finiteness assumption for the scalar field.

Projective group representations in quaternionic Hilbert space
View Description Hide DescriptionWe extend the discussion of projective group representations in quaternionic Hilbert space that was given in our recent book. The associativity condition for quaternionic projective representations is formulated in terms of unitary operators and then analyzed in terms of their generator structure. The multi‐centrality and centrality assumptions are also analyzed in generator terms, and implications of this analysis are discussed.

Derivation of conservation laws from nonlocal symmetries of differential equations
View Description Hide DescriptionAn identity is derived which yields a correspondence between symmetries and conservation laws for self‐adjoint differential equations. This identity does not rely on use of a Lagrangian as needed to obtain conservation laws by Noether’s theorem. Moreover, unlike Noether’s theorem, which can only generate conservation laws from local symmetries, the derived identity generates conservation laws from nonlocal as well as local symmetries. It is explicitly shown how Noether’s theorem is extended by the identity. Conservation laws arising from nonlocal symmetries are obtained for a class of scalar wave equations with variable wave speeds. The constants of motion resulting from these nonlocal conservation laws are shown to be linearly independent of all constants of motion resulting from local conservation laws.

Projective representations of the 1+1‐dimensional Poincaré group
View Description Hide DescriptionThe unitary irreducible representations of the central extension of the Poincaré group in 1+1 dimensions are constructed by an application of the Kirillov theory. These are then lifted to projective unitary irreducible representations of the Poincaré group. The 1+1 Galilean group is treated separately in an appendix.

Quantum homogeneous spaces as quantum quotient spaces
View Description Hide DescriptionIt is shown that certain embeddable homogeneous spaces of a quantum group that do not correspond to a quantum subgroup still have the structure of quantum quotient spaces. A construction of quantum fibre bundles on such spaces is proposed. The quantum plane and the general quantum two‐spheres are discussed in detail.