On semipositive definiteness of 2n-degree forms

The aim of this paper is to present necessary and sufficient conditions for the semipositive definiteness of 2n-degree forms. These conditions allow to verify whether a given map Λ: $\text{A}$ → $\text{A}$ (where $\text{A}$ is the algebra of linear operators on the n-dimensional Hilbert space $\text{H}$) is a semipositive map.     

Phenomenology of local scale invariance: from conformal invariance to dynamical scaling

Statistical systems displaying a strongly anisotropic or dynamical scaling behaviour are characterized by an anisotropy exponent θ or a dynamical exponent z. For a given value of θ (or z), we construct local scale transformations, which can be viewed as scale transformations with a space–time-dependent dilatation factor. Two distinct types of local scale transformations are found. The first type may describe strongly anisotropic scaling of static systems with a given value of θ, whereas the second type may describe dynamical scaling with a dynamical exponent z. Local scale transformations act as a dynamical symmetry group of certain non-local free-field theories. Known special cases of local scale invariance are conformal invariance for θ=1 and Schrödinger invariance for θ=2.The hypothesis of local scale invariance implies that two-point functions of quasiprimary operators satisfy certain linear fractional differential equations, which are constructed from commuting fractional derivatives. The explicit solution of these yields exact expressions for two-point correlators at equilibrium and for two-point response functions out of equilibrium. A particularly simple and general form is found for the two-time autoresponse function. These predictions are explicitly confirmed at the uniaxial Lifshitz points in the ANNNI and ANNNS models and in the aging behaviour of simple ferromagnets such as the kinetic Glauber–Ising model and the kinetic spherical model with a non-conserved order parameter undergoing either phase-ordering kinetics or non-equilibrium critical dynamics.     

Invariance operator groups for a charged particle in an external electromagnetic field

In this paper a general group theoretical approach is given for the problem of a charged particle moving in an external electromagnetic field F. From a knowledge of the symmetry transformations of the field (Galilean or Poincaré), it is possible to explicitly construct groups of operators which commute with the operators of the equations of motion (classical, quantum mechanical, Klein-Gordon or Dirac) using the concept of compensating gauge transformations together with a uniquely chosen map π: F → A fixing the gauge of the potential A. Other choices of gauges give rise to isomorphic operator groups. The general structure of the possible symmetry groups of the fields is discussed and the corresponding invariance operator groups are explicitly given for (almost) arbitrary fields. The structure of these groups is then investigated and it is shown in particular that a large class of fields give rise to non-Type I groups, i.e. to groups which have (unitary continuous) representations whose corresponding von Neumann algebras have non-discrete factors. A general criterion for these pathological cases is given. As an application, we study the problem of a Bloch electron in arbitrary constant uniform electric and magnetic fields.     

An effective method of investigation of positive maps on the set of positive definite operators

Let $\text{A}$₁ be the algebra of linear operators on the n-dimensional Hilbert space $\text{H}$₁, and let $\text{A}$₂ be the algebra of linear operators of the m-dimensional Hilbert space $\text{H}$₂. Let $\text{L}$($\text{A}$₁, $\text{A}$₂) denote the complex space of linear maps from $\text{A}$₁ to $\text{A}$₂. By a positive map we mean an element of the space $\text{L}$($\text{A}$₁, $\text{A}$₂) (superoperator with respect to $\text{H}$₁) which maps positive definite operators in $\text{A}$₁ into positive definite operators in $\text{A}$₂. The aim of this paper is to present an effective method which allows to verify whether a given superoperator Λ∈$\text{L}$($\text{A}$₁, $\text{A}$₂) is a positive map. Besides that necessary and sufficient conditions for the positive definiteness of even-degree forms in many variables are given.     

Transformation groups for soliton equations: IV. A new hierarchy of soliton equations of KP-type

A new approach to soliton equations, based on τ functions (or Hirota's dependent variables), vertex operators and the Clifford algebra of free fermions, is applied to study a new hierarchy of Kadomtsev-Petviashvili type equations (the BKP hierarchy). The infinite-dimensional orthogonal group acts on the space of BKP τ-functions. The Sawada-Kotera equation is obtained as a reduction of BKP. Its infinitesimal transformations constitute the Euclidean Lie Algebra A₂⁽²⁾.     

Normal forms for classical and boson systems

The group of Bogoliubov transformations of annihilation and creation operators is a subgroup of U(n,n) where n is the number of distinct pairs of annihilation and creation operators. Here, it is established that this subgroup of U(n,n) is isomorphic to Sp(2n,$\text{R}$), which appears in classical dynamics as the group of linear canonical transformations on a 2n-dimensional phase space. Well-known results in classical dynamics are then to used to deduce the full set of normal forms for Boson Hamiltonians. These are classified using a para-eigenvalue notation applicable to both classical and Bose field systems. A simple sufficient condition is given for the non-removability of pairs of creation operators. Explicit normal forms have not previously been given for Hamiltonians with this pathology, which may occur even when the corresponding classical Hamiltonian can be diagonalized.     

Irreversibility, entropy and incomplete information

The open system has been proved to be a system with perfect accessibility represented as a probability space in which is defined a PA-measure. But, the PA-measure is not yet known; consequently, it is difficult to develop the statistical thermodynamics for an irreversible system. Here its integral expression is obtained in order to its use in the statistical thermodynamic analysis of the complex and irreversible systems.     

On the mixing-enhancing dynamical maps and mixing-enhancing evolution of a quantum system

The mixing-enhancing (in the sense of Uhlmann) dynamical maps and dynamical evolution is studied. We give a necessary and sufficient condition for a dynamical map (and dynamical evolution) of a quantum system to be mixing-enhancing. In the case of a finite- dimensional Hilbert space this condition is equivalent to the condition that the dynamical map (dynamical evolution) preserve the most mixed state and the von Neumann entropy be non- decreasing. It is proved that, in contrast with the finite-dimensional case, increasing of the von Neumann entropy under a dynamical map (for any initial state) does not imply that the dynamical map is mixing-enhancing. We also give a necessary and sufficient condition for an infinitesimal generator of a norm-continuous dynamical semigroup to be mixing-enhancing.     

Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions

We investigate one-dimensional discrete Schrödinger operators whose potentials are invariant under a substitution rule. The spectral properties of these operators can be obtained from the analysis of a dynamical system, called the trace map. We give a careful derivation of these maps in the general case and exhibit some specific properties. Under an additional, easily verifiable ypothesis concerning the structure of the trace map we present an analysis of their dynamical properties that allows us to prove that the spectrum of the underlying Schrödinger operator is singular and supported on a set of zero Lebesgue measure. A condition allowing to exclude point spectrum is also given. The application of our theorems is explained on a series of examples.     

A model of the holographic principle: Randomness and additional dimension

In recent years an idea has emerged that a system in a 3-dimensional space can be described from an information point of view by a system on its 2-dimensional boundary. This mysterious correspondence is called the Holographic Principle and has had profound effects in string theory and our perception of space-time. In this note we describe a purely mathematical model of the Holographic Principle using ideas from nonlinear dynamical systems theory. We show that a random map on the surface S² of a 3-dimensional open ball B has a natural counterpart in B, and the two maps acting in different dimensional spaces have the same entropy. We can reverse this construction if we start with a special 3-dimensional map in B called a skew product. The key idea is to use the randomness, as imbedded in the parameter of the 2-dimensional random map, to define a third dimension. The main result shows that if we start with an arbitrary dynamical system in B with entropy E we can construct a random map on S² whose entropy is arbitrarily close to E.     

Dunkl operator,integrability, and pairwise scattering in rational Calogero model

The integrability of the Calogero model can be expressed as zero curvature condition using Dunkl operators. The corresponding flat connections are non-local gauge transformations, which map the Calogero wave functions to symmetrized wave functions of the set of N free particles, i.e. it relates the corresponding scattering matrices to each other. The integrability of the Calogero model implies that any k-particle scattering is reduced to successive pairwise scatterings. The consistency condition of this requirement is expressed by the analog of the Yang–Baxter relation.     

Global attractors and invariant measures for non-invertible planar piecewise isometric maps

The authors investigate dynamical behaviors of discrete systems defined by iterating non-invertible planar piecewise isometries, which are piecewisely defined maps that preserve Euclidean distance. After discussing subtleties for these kind of dynamical systems, they have characterized global attractors via invariant measures and via positive continuous functions on phase space. The main result of this Letter is that a compact set A is the global attractor for a piecewise isometry if and only if the Lebesgue measure restricted to A is invariant, while it is not invariant restricted to any measurable set B which contains A and whose Lebesgue measure is strictly larger than that of A.     

Spin substructure of space-time

In a spin space, with noncommutative spinorial coordinatesC ^A which satisfyx ^AB =1/2{C ^A ,C ^*B }, we investigatex-dependent spin coordinate transformations which correspond to a local fermi-bose symmetry. A concept of a spin vector corresponding to these local transformations is established, and spin space is provided with a spinorial vierbein field which determines non-nil-potent line and volume elements which are direct generalizations of the conventional relativistic ones.     

Finite Hamiltonian systems: Linear transformations and aberrations

Finite Hamiltonian systems contain operators of position, momentum, and energy, having a finite number N of equally-spaced eigenvalues. Such systems are under the æis of the algebra su(2), and their phase space is a sphere. Rigid motions of this phase space form the group SU(2); overall phases complete this to U(2). But since N-point states can be subject to U(N) ?U(2) transformations, the rest of the generators will provide all N ² unitary transformations of the states, which appear as nonlinear transformations—aberrations—of the system phase space. They are built through the “finite quantization” of a classical optical system.     

Dynamical interpretation for the quantum-measurement projection postulate

An apparatus model with discrete momentum space suitable for the exact solution of the problem is considered. The special Hamiltonian of its interaction with the object system under consideration is chosen. In this simple case it is easy to illustrate how difficulties in constructing the dynamical interpretation of selective collapse can be overcome without any limiting procedure. For this purpose one can apply either averaging with respect to a nonquantum parameter or reduce the algebra of joint-system operators (i.e., pass from an algebraA of operators to a subalgebraA ₀). The latter procedure implies averaging with respect to apparatus quantum variables not belonging toA ₀.On leave of absence from Physics Department, Moscow State University, 119899 Moscow, Russia.     

State-dependent dynamical variables in quantum theory

State-dependent local dynamical variables (LDVs) sharply differ from the ordinary operators of quantum mechanics. The N-level model system shows the physical importance of such operators in the complex projective Hilbert state space CP(N?1). The process of quantum measurement in terms of LDVs is described.     

On the existence of joint probability distributions for arbitrary incompatible observables in quantum mechanics

It is shown that for any two operators A and C on any Hilbert space H it is possible to construct infinitely many positive operator valued measures which can serve as joint probability distributions for A and C.