Jordanian solutions of simplex equations

We construct for all N a solution of the Frenkel-Moore N-simplex equation which generalizes the R-matrix for the Jordanian quantum group.     

The classical field limit of scattering theory for non-relativistic many-boson systems. I

We study the classical field limit of non-relativistic many-boson theories in space dimensionn≧3. When ?→0, the correlation functions, which are the averages of products of bounded functions of field operators at different times taken in suitable states, converge to the corresponding functions of the appropriate solutions of the classical field equation, and the quantum fluctuations are described by the equation obtained by linearizing the field equation around the classical solution. These properties were proved by Hepp [6] for suitably regular potentials and in finite time intervals. Using a general theory of existence of global solutions and a general scattering theory for the classical equation, we extend these results in two directions: (1) we consider more singular potentials, (2) more important, we prove that for dispersive classical solutions, the ?→0 limit is uniform in time in an appropriate representation of the field operators. As a consequence we obtain the convergence of suitable matrix elements of the wave operators and, if asymptotic completeness holds, of theS-matrix.     

Massless Spin–Zero Particle and the Classical Action via Hamilton–Jacobi Equation in G?del Universe

In this letter we investigate the separability of the Klein–Gordon and Hamilton–Jacobi equation in G?del universe. We show that the Klein–Gordon eigen modes are quantized and the complete spectrum of the particle’s energy is a mixture of an azimuthal quantum number, m and a principal quantum number, n and a continuous wave number k. We also show that the Hamilton–Jacobi equation gives a closed function for classical action. These results may be used to calculate the Casimir vacuum energy in G?del universe.     

Quantum Integrable Models and Discrete Classical Hirota Equations

The standard objects of quantum integrable systems are identified with elements of classical nonlinear integrable difference equations. The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. Elliptic solutions of Hirota's equation give a complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's Q-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to the Bethe ansatz are studied. The nested Bethe ansatz equations for A _k-1 -type models appear as discrete time equations of motions for zeros of classical τ-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation are analysed and a new determinant formula for eigenvalues of the quantum transfer matrices is obtained. Difference equations for eigenvalues of the Q-operators which generalize Baxter's three-term T−Q-relation are derived. Received: 15 May 1996 / Accepted: 25 November 1996     

A Meinardus Theorem with Multiple Singularities

We solve the quantum version of the A ₁ T-system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infinite-rank) quantum cluster algebra, and Laurent positivity follows from our solution. As an application we re-derive the corresponding quantum network solution to the quantum A ₁ Q-system and generalize it to the fully non-commutative case. We give the relation between the quantum T-system and the quantum lattice Liouville equation, which is the quantized Y-system.     

The 50% Advanced Information Rule of?the?Quantum Algorithms

The oracle chooses a function out of a known set of functions and gives to the player a black box that, given an argument, evaluates the function. The player should find out a certain character of the function (e.g. its period) through function evaluation. This is the typical problem addressed by the quantum algorithms. In former theoretical work, we showed that a quantum algorithm requires the number of function evaluations of a classical algorithm that knows in advance 50% of the information that specifies the solution of the problem. This requires representing physically, besides the solution algorithm, the possible choices of the oracle. Here we check that this 50% rule holds for the main quantum algorithms. In structured problems, a classical algorithm with the advanced information, to identify the missing information should perform one function evaluation. The speed up is exponential since a classical algorithm without advanced information should perform an exponential number of function evaluations. In unstructured database search, a classical algorithm that knows in advance n/2 bits of the database location, to identify the n/2 missing bits should perform O(2 ^n/2) function evaluations. The speed up is quadratic since a classical algorithm without advanced information should perform O(2 ⁿ ) function evaluations. The 50% rule allows to identify in an entirely classical way the problems solvable with a quantum sped up. The advanced information classical algorithm also defines the quantum algorithm that solves the problem. Each classical history, corresponding to a possible way of getting the advanced information and a possible result of computing the missing information, is represented in quantum notation as a sequence of sharp states. The sum of the histories yields the function evaluation stage of the quantum algorithm. Function evaluation entangles the oracle’s choice register (containing the function chosen by the oracle) and the solution register (in which to read the solution at the end of the algorithm). Information about the oracle’s choice propagates from the former to the latter register. Then the basis of the solution register should be rotated to make this information readable. This defines the quantum algorithm, or its iterate and the number of iterations.     

Solutions of Polyakov's loop equations and their contribution to loop averages

The solutions of the classical equations for the Wilson loop presented by Polyakov are studied forU(1) andSU(2) gauge theories on a lattice. All the solutions are found ford=2 space-time dimensions while a particular set is obtained ford-3,4. A possible application of this result to the computation of quantum loop averages in the weak coupling region is discussed: ind=2 a saddle point approximation is shown to be very accurate and simple in terms of loop variables.     

Critical Behavior of Two Interacting Linear Polymer Chains in a Good Solvent

A model of two interacting (chemically different) linear polymer chains is solved exactly using the real-space renormalization group transformation on a family of Sierpinski gasket type fractals and on a truncated 4-simplex lattice. The members of the family of the Sierpinski gasket-type fractals are characterized by an integer scale factorb which runs from 2 to ∞. The Hausdorff dimensiond _F of these fractals tends to 2 from below asb → ∞. We calculate the contact exponenty for the transition from the State of segregation to a State in which the two chains are entangled forb = 2-5. Using arguments based on the finite-size scaling theory, we show that forb→∞, y = 2 - v(b) d _F, wherev is the end-toend distance exponent of a chain. For a truncated 4-simplex lattice it is shown that the system of two chains either remains in a State in which these chains are intermingled in such a way that they cannot be told apart, in the sense that the chemical difference between the polymer chains completely drop out of the thermodynamics of the system, or in a State in which they are either zipped or entangled. We show the region of existence of these different phases separated by tricritical lines. The value of the contact exponenty is calculated at the tricritical points.     

Critical behaviour at the displacive limit of structural phase transitions

For a special critical point at zero temperature,T _c =0, which is called the displacive limit of a classical or of a quantum-mechanical model showing displacive phase transitions, we derive a set of static critical exponents in the large-n limit. Due to zero-point motions and quantum fluctuations at low temperatures, the exponents of the quantum model are different from those of the classical model. Moreover, we report results on scaling functions, corrections to scaling, and logarithmic factors which appear ford=2 in the classical case and ford=3 in the quantum-mechanical case. Zero-point motions cause a decrease of the critical temperature of the quantum model with respect to the classicalT _c , which implies a difference between the classical and the quantum displacive limit. However, finite critical temperatures are found in both cases ford>2, while critical fluctuations still occur atT _c =0 for 0<d≦2 in the classical case and for 1 <d≦2 in the quantum model. Further we derive the slope of the critical curve at the classical displacive limit exactly. The absence of 1/n-corrections to the exponents of the classical model is also discussed.     

Quantization of Hénon's map with dissipation II — numerical results

The master equation for a quantized version of Hénon's map with dissipation derived in a preceding paper is here solved numerically for the Wigner quasi-probability density, under conditions of period doubling and classical chaos both in the transient regime and in the dissipative steady state. Approximations of the quantum map by a classical stochastic process are also considered and compared with solutions incorporating non-classical quantum fluctuations.     

Generalizations of Quantum Mechanics Induced by Classical Statistical Field Theory

No Heading We show that the Dirac-von Neumann formalism for quantum mechanics can be obtained as an approximation of classical statistical field theory. This approximation is based on the Taylor expansion (up to terms of the second order) of classical physical variables – maps f : Ω → R, where Ω is the infinite-dimensional Hilbert space. The space of classical statistical states consists of Gaussian measures ρ on Ω having zero mean value and dispersion σ²(ρ) ≈ h. This viewpoint to the conventional quantum formalism gives the possibility to create generalized quantum formalisms based on expansions of classical physical variables in the Taylor series up to terms of nth order and considering statistical states ρ having dispersion σ²(ρ) = hⁿ (for n = 2 we obtain the conventional quantum formalism).     

Exact classical and quantum solutions for a covariant oscillator near the black hole horizon in Stueckelberg-Horwitz-Piron theory

We found exact solutions for canonical classical and quantum dynamics for general relativity in Horwitz general covariance theory. These solutions can be obtained by solving the generalized geodesic equation and Schrödinger-Stueckelberg-Horwitz-Piron (SHP) wave equation for a simple harmonic oscillator in the background of a two dimensional dilaton black hole spacetime metric. We proved the existence of an orthonormal basis of eigenfunctions for generalized wave equation. This basis functions form an orthogonal and normalized (orthonormal) basis for an appropriate Hilbert space. The energy spectrum has a mixed spectrum with one conserved momentum p according to a quantum number n. To find the ground state energy we used a variational method with appropriate boundary conditions. A set of mode decomposed wave functions and calculated for the Stueckelberg-Schrodinger equation on a general five dimensional blackhole spacetime in Hamilton gauge.     

Classification of bicovariant differential calculi on quantum groups of type A,B, C and D

Under the assumptions thatq is not a root of unity and that the differentialsdu _j ⁱ of the matrix entries span the left module of first order forms, we classify bicovariant differential calculi on quantum groupsA _n–1 ,B _n ,C _n andD _n . We prove that apart one dimensional differential calculi and from finitely many values ofq, there are precisely2n such calculi on the quantum groupA _n–1 =SL _q (n) forn3. All these calculi have the dimensionn ². For the quantum groupsB _n ,C _n andD _n we show that except for finitely manyq there exist precisely twoN ²-dimensional bicovariant calculi forN3, whereN=2n+1 forB _n andN=2n forC _n ,D _n . The structure of these calculi is explicitly described and the corresponding ad-invariant right ideals of ker are determined. In the limitq1 two of the 2n calculi forA _n–1 and one of the two calculi forB _n ,C _n andD _n contain the ordinary classical differential calculus on the corresponding Lie group as a quotient.     

Multifractal behaviour of n-simplex lattice

We study the asymptotic behaviour of resistance scaling and fluctuation of resistance that give rise to flicker noise in an n-simplex lattice. We propose a simple method to calculate the resistance scaling and give a closed-form formula to calculate the exponent, β _L, associated with resistance scaling, for any n. Using current cumulant method we calculate the exact noise exponent for n-simplex lattices.     

Correlation inequalities for vector spin models

Correlation inequalities forn-vector spin models (n 2) are reviewed. A relatively simple and unified derivation of the inequalities is achieved, using duplicate variable methods, for spin dimensionalitiesn=2 (plane rotator model),n=3 (classical Heisenberg model), andn=4. Although correlation inequalities are lacking forn > 4, new proofs are presented for the comparison inequalities relating correlations for systems with arbitrary spin dimensionality to corresponding correlations for systems with low spin dimensionality (n = 1 or 2).Research supported by National Science Foundation under Grant DMR 76-23071.     

Nonlinear lagrangians and Einstein spaces

It is shown that, for a Riemannian spaceV _d of dimensiond, solutions of the equation ((–g)^1/2 R ⁿ)/g^ab = 0 forn = (1/4)(d+2) may be interpreted as (d + 1)-dimensional Einstein spaces.     

Separable quantizations of Stäckel systems

In this article we prove that many Hamiltonian systems that cannot be separably quantized in the classical approach of Robertson and Eisenhart can be separably quantized if we extend the class of admissible quantizations through a suitable choice of Riemann space adapted to the Poisson geometry of the system. Actually, in this article we prove that for every quadratic in momenta Stäckel system (defined on 2n$2n$ dimensional Poisson manifold) for which Stäckel matrix consists of monomials in position coordinates there exist infinitely many quantizations–parametrized by n$n$ arbitrary functions–that turn this system into a quantum separable Stäckel system.     

SO q (n+1,n−1) as a real form of SO q (2n,ℂ)

Quantum pseudo-orthogonal groups SO _q (n+1,n–1) are defined as real forms of quantum orthogonal groups SO _q (n+1,n–1) by means of a suitable antilinear involution. In particular, the casen=2 gives a quantized Lorentz group.     

New rational solutions of Yang-Baxter equation and deformed Yangians

In this paper a new class of quantum groups, deformed Yangians, is used to obtain new matrix rational solutions of the Yang-Baxter equation (YBE). The deformed Yangians arise from rational solutions of the classical Yang-Baxter equation of the form c ₂/u + const. The image of the universal quantum R-matrix for the deformed Yangian in finite-dimensional representations gives these new matrix rational solutions of YBE.     

Boundary Solutions of the Classical Yang--Baxter Equation

We define a new class of unitary solutions to the classical Yang--Baxter equation (CYBE). These boundary solutions are those which lie in the closure of the space of unitary solutions of the modified classical Yang--Baxter equation (MCYBE). Using the Belavin--Drinfel'd classification of the solutions to the MCYBE, we are able to exhibit new families of solutions to the CYBE. In particular, using the Cremmer--Gervais solution to the MCYBE, we explicitly construct for all n 3 a boundary solution based on the maximal parabolic subalgebra of obtained by deleting the first negative root. We give some evidence for a generalization of this result pertaining to other maximal parabolic subalgebras whose omitted root is relatively prime to n. We also give examples of nonboundary solutions for the classical simple Lie algebras.