Generalized Householder Transformations

The Householder transformation, allowing a rewrite of probabilities into expectations of dichotomic observables, is generalized in terms of its spectral decomposition. The dichotomy is modulated by allowing more than one negative eigenvalue or by abandoning binaries altogether, yielding generalized operator-valued arguments for contextuality. We also discuss a form of contextuality by the variation of the functional relations of the operators, in particular by additivity.     

Spatial Transformations of Atoms

Russian Physics Journal - On the basis of fundamental probabilistic arguments, a theory of spatial transitions of atoms in configurations with number of spatial dimensions D = 1, 2, and 3 is...     

D-Modules and Darboux Transformations

A method of G. Wilson for generating commutative algebras of ordinary differential operators is extended to higher dimensions. Our construction, based on the theory of D-modules, leads to a new class of examples of commutative rings of partial differential operators with rational spectral varieties. As an application, we briefly discuss their link to the bispectral problem and to the theory of lacunas.     

Three-Mode Nonlinear Bogoliubov Transformations

We introduce the three-mode nonlinear Bogoliubov transformations based on the work of Siena et al. (Phys. Rev. A 64:063803, 2001) and Ying Wu (Phys. Rev. A 66:025801, 2002) about nonlinear Bogoliubov transformations. We show that three-mode nonlinear Bogoliubov transformations can be constructed by the combination of two unitary transformations, a coordinate-dependent displacement followed by the standard squeezed transformation. Such decomposition turns all the nonlinear canonic coordinate-dependent Bogoliubov transformations into essentially linear problems as we shall prove and hence greatly facilitate calculations of the properties and the quantities related to the nonlinear transformations.     

Time Dependent Coordinate Transformations

The general formalism of time-dependent canonical transformations is applied to the case of coordinate transformations in classical and quantum mechanics.     

Transformations in special relativity

By using the principle of relativity alone (no assumptions about signals or light) it is shown that a relativisitic group of linear transformations of a spacetime plane is, if infinite, either Galilean, Lorentzian or rotational. The largest such finite group is a Klein 4-group, generated by space-reversal and time-reversal. In the infinite case an invariant of the group, denotedc, appears. Whenc is real, nonzero, noninfinite, then the group is a Lorentz group andc is identified with the speed of light. Lorentz transformations are represented through an algebra ofiterants that provides a link among Clifford algebras, the Pauli algebra and Herman Bondi'sK-calculus.     

Conformal Transformations as Observables

C denotes either the conformal group in 3+1 dimensions, PSO(4, 2), or in one chiral dimension, PSL(2, ). Let U be a unitary, strongly continuous representation of C satisfying the spectrum condition and inducing, by its adjoint action, automorphisms of a von Neumann algebra . We construct the unique inner representation of the universal covering group of C implementing these automorphisms. satisfies the spectrum condition and acts trivially on any U-invariant vector. This means in particular: Conformal transformations of a field theory having positive energy are weak limit points of local observables. Some immediate implications for chiral subnets are given. We propose the name Borchers–Sugawara construction.     

Transformations between quantum states

General properties and characterization of transformations between quantum states (in the conventional framework of quantum mechanics) are given. Some classes of such transformations are described in detail. Examples of transformations between quantum states in statistical mechanics are given.     

Squeezing and Quantum Canonical Transformations

In this paper we present an approach to quantum mechanical canonical transformations. Our main result is that time-dependent quantum canonical transformations can always be cast in the form of squeezing operators. We revise the main properties of these operators in regard to its Lie group properties, how two of them can be combined to yield another operator of the same class and how can also be decomposed and fragmented. In the second part of the paper we show how this procedure works extremely well for the time-dependent quantum harmonic oscillator. The issue of the systematic construction of quantum canonical transformations is also discussed along the lines of Dirac, Wigner, and Schwinger ideas and to the more recent work by Lee. The main conclusion is that the classical phase space transformation can be maintained in the operator formalism but the construction of the quantum canonical transformation is not clearly related to the classical generating function of a classical canonical transformation. We hereby propose the much more efficient method given by the squeezing operators. This method has also been proved to be very useful, by one of the authors, in the framework of the dynamical symmetries (Cerveró, J. M. (1999). International Journal of Theoretical Physics 38, 2095–2109).     

On Irreducible Second-Order Darboux Transformations

The second-order Darboux transform for a growing potential of a Schrödinger equation is discussed in detail. Restrictions on the transforming functions whose eigenvalues are greater than the energy of the ground state if the potential of the transformed Schrödinger equation is regular has been established. It has been shown that, along with the well-known elimination of two levels of the discrete spectrum, other opportunities can also be realized: (a) the transformed spectrum remains identical to the original one; (b) one level is eliminated from the spectrum; (c) one additional level is generated, and (d) two additional levels are generated.     

Contact Transformations in Classical Mechanics

Abstract

Transformations of coordinates of points in an infinite-dimensional graded vector space, the so-called contact transformations, are examined. An infinite jet prolongation of the extended configuration space of N spinless particles is the subspace of this vector space. The dynamical equivalence among Lagrangian N-body systems connected by an invertible jet transformation is established. As an example of the invertible jet transformations, a class of gauge transformations of Lagrangian variables is investigated. The method of contact transformations is applied to the Wheeler-Feynman electrodynamics for two point charges.     

Non-Hermitian Hamiltonians and Similarity Transformations

We show that similarity (or equivalent) transformations enable one to construct non-Hermitian operators with real spectrum. In this way we can also prove and generalize the results obtained by other authors by means of a gauge-like transformation and its generalization. Such similarity transformations also reveal the connection with pseudo-Hermiticity in a simple and straightforward way.     

Thermodynamic Transformations of Nonequilibrium States

We consider a macroscopic system in contact with boundary reservoirs and/or under the action of an external field. We discuss the case in which the external forcing depends explicitly on time and drives the system from a nonequilibrium state to another one. In this case the amount of energy dissipated along the transformation becomes infinite when an unbounded time window is considered. Following the general proposal by Oono and Paniconi and using results of the macroscopic fluctuation theory, we give a natural definition of a renormalized work. We then discuss its thermodynamic relevance by showing that it satisfies a Clausius inequality and that quasi static transformations minimize the renormalized work. In addition, we connect the renormalized work to the quasi potential describing the fluctuations in the stationary nonequilibrium ensemble. The latter result provides a characterization of the quasi potential that does not involve rare fluctuations.     

On Spatial Transformations of Atoms

Russian Physics Journal - Based on the experimentally established existence of atoms with spatially one- and two-dimensional electronic structures that can be paired with conventional...     

Schlesinger Transformations and Quantum R-Matrices

 Schlesinger transformations are discrete monodromy preserving symmetry transformations of a meromorphic connection which shift by integers the eigenvalues of its residues. We study Schlesinger transformations for twisted -valued connections on the torus. A universal construction is presented which gives the elementary two-point transformations in terms of Belavin's elliptic quantum R-matrix. In particular, the role of the quantum deformation parameter is taken by the difference of the two poles whose residue eigenvalues are shifted. Elementary one-point transformations (acting on the residue eigenvalues at a single pole) are constructed in terms of the classical elliptic r-matrix. The action of these transformations on the τ-function of the system may completely be integrated and we obtain explicit expressions in terms of the parameters of the connection. In the limit of a rational R-matrix, our construction and the τ-quotients reduce to the classical results of Jimbo and Miwa in the complex plane. Received: 19 December 2001 / Accepted: 20 May 2002 Published online: 14 October 2002     

Schlesinger Transformations for Linearisable Equations

A simple axiomatic characterization of the noncommutative Itô algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. It is proved that every Itô algebra with a quotient identity has a faithful representation in a Minkowski space and is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Lévy (Poisson) algebra. In particular, every quantum thermal noise of a finite number of degrees of freedom is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Lévy–Khinchin Theorem in the classical case. Two basic examples of noncommutative Itô finite group algebras are considered.     

Mechanoelectrical Transformations in Piezodielectric Materials

The role of piezoeffect in generating an electromagnetic response under impact excitation of piezodielectric materials is studied. The presence of piezoelements in these materials is shown to improve the efficiency of mechanoelectric transformations. It is found out that the high-frequency component of the electromagnetic response in a piezomaterial with inclusions undergoes transformations. A change in porosity of piezomaterials is shown to give rise to shifts of major spectral maxima in the electromagnetic response pattern.__________Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 77–80, March, 2005.