Table of contents

We exploit a well-known isomorphism between complex Hermitian 2 × 2 matrices and ⁴, which yields a convenient real vector representation of qubit states. Because these do not need to be normalized we find that they map onto a Minkowskian future cone in ^1,3, whose vertical cross-sections are nothing but Bloch spheres. Pure states are represented by light-like vectors, unitary operations correspond to special orthogonal transforms about the axis of the cone, positive operations correspond to pure Lorentz boosts. We formalize the equivalence between the generalized measurement formalism on qubit states and the Lorentz transformations of special relativity, or more precisely elements of the restricted Lorentz group together with future-directed null boosts. The note ends with a discussion of the equivalence and some of its possible consequences.

The integral of the Wigner function of a quantum-mechanical system over a region or its boundary in the classical phase plane, is called a quasiprobability integral. Unlike a true probability integral, its value may lie outside the interval [0, 1]. It is characterized by a corresponding selfadjoint operator, to be called a region or contour operator as appropriate, which is determined by the characteristic function of that region or contour. The spectral problem is studied for commuting families of region and contour operators associated with concentric discs and circles of given radius a. Their respective eigenvalues are determined as functions of a, in terms of the Gauss–Laguerre polynomials. These polynomials provide a basis of vectors in a Hilbert space carrying the positive discrete series representation of the algebra su(1, 1) ≈ so(2, 1). The explicit relation between the spectra of operators associated with discs and circles with proportional radii, is given in terms of the discrete variable Meixner polynomials.

Using modern tools from the geometric theory of Hamiltonian systems it is shown that electronic excitations in diatoms which can be modelled by the two-centre problem exhibit a complicated case of classical and quantum monodromy. This means that there is an obstruction to the existence of global quantum numbers in these classically integrable systems. The symmetric case of H⁺₂ and the asymmetric case of H He⁺⁺ are explicitly worked out. The asymmetric case has a non-local singularity causing monodromy. It coexists with a second singularity which is also present in the symmetric case. An interpretation of monodromy is given in terms of the caustics of invariant tori.

This is part 1 of a two-part review on wave operator theory and methods. The basic theory of the time-independent wave operator is presented in terms of partitioned matrix theory for the benefit of general readers, with a discussion of the links between the matrix and projection operator approaches. The matrix approach is shown to lead to simple derivations of the wave operators and effective Hamiltonians of Löwdin, Bloch, Des Cloizeaux and Kato as well as to some associated variational forms. The principal approach used throughout stresses the solution of the nonlinear equation for the reduced wave operator, leading to the construction of the effective Hamiltonians of Bloch and of Des Cloizeaux. Several mathematical techniques which are useful in implementing this approach are explained, some of them being relatively little known in the area of wave operator calculations. The theoretical discussion is accompanied by several specimen numerical calculations which apply the described techniques to a selection of test matrices taken from the previous literature on wave operator methods. The main emphasis throughout is on the use of numerical methods which use iterative or perturbation algorithms, with simple Padé approximant methods being found sufficient to deal with most of the cases of divergence which are encountered. The use of damping factors and relaxation parameters is found to be effective in stabilizing calculations which use the energy-dependent effective Hamiltonian of Löwdin. In general the computations suggest that the numerical applications of the nonlinear equation for the reduced wave operator are best carried out with the equation split into a pair of equations in which the Bloch effective Hamiltonian appears as a separate entity. The presentation of the theoretical and computational details throughout is accompanied by references to and discussion of many works which have used wave operator methods in physics, chemistry and engineering. Some of the techniques described in this part 1 will be further extended and applied in part 2 of the review, which deals with the changes which are required to extend wave operator theory to the case of a time-dependent Hamiltonian such as that which describes the interaction of a laser pulse with an atom or molecule.

A novel determinantal representation for matrix elements of local spin operators between Bethe wavefunctions of the one-dimensional s = 1/2 XXZ model is used to calculate transition rates for dynamic spin structure factors in the planar regime. In a first application, high-precision numerical data are presented for lineshapes and band edge singularities of the in-plane (xx) two-spinon dynamic spin structure factor.

Solutions of Bethe equations are found for different numbers of nodes in the Heisenberg chain for S = 1/2 and for chosen winding numbers. The computing procedure starts from asymptotic solutions. It is shown that the evolution of solutions has quasi-continuous form even for a wide range of nodes number N. However, it is observed that in some cases critical and limiting points appear.

Based on the Boltzmann equation, the detonation problem is dealt with on a mesoscopic level. The model is based on the assumption that ahead of a shock an explosive gas mixture is in meta stable equilibrium. Starting from the Von Neumann point the chemical reaction, initiated by the pressure jump, proceeds until the chemical equilibrium is reached. Numerical solutions of the derived macroscopic equations as well as the corresponding Hugoniot diagrams which reveal the physical relevance of the mathematical model are provided.

The free energy of the spin-½ Heisenberg antiferromagnet (HAF) on a squagome lattice is calculated within the Suzuki–Takano quantum decimation technique. The resulting specific heat exhibits an additional peak in the low-temperature region.

We propose a nonlinear model on a regular infinite one-dimensional lattice. It describes the three-component dynamical system with modulated on-site masses and is shown to admit a zero-curvature representation. The associated auxiliary spectral problem is basically of third order and gives rise to fairly complicated subdivision into domains of regularity of Jost functions in the plane of complex spectral parameter. As a result, both the direct and the inverse scattering problems turn out to be substantially nontrivial. The Caudrey version of the direct and inverse scattering technique for the needs of model integration is adapted. The simplest soliton solution is found.

The cubic vector nonlinear Schrödinger equation with an external trigonometric potential models a quasi-one-dimensional multi-component Bose–Einstein condensate trapped in a standing light wave. We construct families of exact stationary solutions for the more general case of an elliptic function potential. Some of these solutions degenerate to zero as the effect of the external potential disappears, whereas others limit to solutions of the free vector nonlinear Schrödinger equation. The stability of these solutions is examined both analytically and numerically. The stability results depend on the nature of the atomic interactions not only within the components but also between components. As in the scalar case (one component) with repulsive interaction, all linearly stable solutions are deformations of the ground state of the linear Schrödinger equation. Unlike the scalar case with attractive interaction, no solutions are found to be stable if there is any attractive interaction present.

A formula expressing the Laguerre coefficients of a general-order derivative of an infinitely differentiable function in terms of its original coefficients is proved, and a formula expressing explicitly the derivatives of Laguerre polynomials of any degree and for any order as a linear combination of suitable Laguerre polynomials is deduced. A formula for the Laguerre coefficients of the moments of one single Laguerre polynomial of certain degree is given. Formulae for the Laguerre coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its Laguerre coefficients are also obtained. A simple approach in order to build and solve recursively for the connection coefficients between Jacobi–Laguerre and Hermite–Laguerre polynomials is described. An explicit formula for these coefficients between Jacobi and Laguerre polynomials is given, of which the ultra-spherical polynomials of the first and second kinds and Legendre polynomials are important special cases. An analytical formula for the connection coefficients between Hermite and Laguerre polynomials is also obtained.

We propose nonlinear integral equations to describe the groundstate energy of the fractional supersymmetric sine-Gordon models. The equations encompass the N = 1 supersymmetric sine-Gordon model as well as the ϕ_id,id,adj perturbation of the SU(2)_L × SU(2)_K/SU(2)_L+K models at rational level K. A second set of equations is proposed for the groundstate energy of the N = 2 supersymmetric sine-Gordon model.

In this paper we report on a package, written in the Mathematica computer algebra system, which has been developed to compute the spheroidal wavefunctions of Meixner and Schäfke (1954 Mathieusche Funktionen und Sphäroidfunktionen) and is available online (physics.uwa.edu.au/~falloon/spheroidal/spheroidal.html). This package represents a substantial contribution to the existing software, since it computes the spheroidal wavefunctions to arbitrary precision for general complex parameters μ, ν, γ and argument z; existing software can only handle integer μ, ν and does not give arbitrary precision. The package also incorporates various special cases and computes analytic power series and asymptotic expansions in the parameter γ. The spheroidal wavefunctions of Flammer (1957 Spheroidal Wave functions) are included as a special case of Meixner's more general functions. This paper presents a concise review of the general theory of spheroidal wavefunctions and a description of the formulae and algorithms used in their computation, and gives high precision numerical examples.

The zero modes of the chiral SU(n) WZNW model give rise to an intertwining quantum matrix algebra generated by an n × n matrix a = (aⁱ_α), i, α = 1, ..., n (with noncommuting entries) and by rational functions of n commuting elements q^p_i satisfying ∏ⁿ_{i = 1}q^p_i = 1, q^p_ia^j_α = a^j_αq^{p_i+δj_i − 1/n}. We study a generalization of the Fock space () representation of for generic q (q not a root of unity) and demonstrate that it gives rise to a model of the quantum universal enveloping algebra U_q = U_q(sl_n), with each irreducible representation entering with multiplicity 1. For an integer (n) height h ( = k + n ≥ n) the complex parameter q is an even root of unity, q^h = −1, and the algebra has an ideal _h such that the factor algebra _h = /_h is finite dimensional. All physical U_q modules—of shifted weights satisfying p_1n ≡ p₁ − p_n < h—appear in the Fock representation of _h.

We introduce a class of finite-dimensional nonlinear superalgebras L = L + L providing gradings of L = gl(n) ≃ sl(n) + gl(1). Odd generators close by anticommutation on polynomials (of degree >1) in the gl(n) generators. Specifically, we investigate 'type I' super-gl(n) algebras, having odd generators transforming in a single irreducible representation of gl(n) together with its contragredient. Admissible structure constants are discussed in terms of available gl(n) couplings, and various special cases and candidate superalgebras are identified and exemplified via concrete oscillator constructions. For the case of the n-dimensional defining representation, with odd generators Q_a, ^b and even generators E^a_b, a, b = 1, ..., n, a three-parameter family of quadratic super-gl(n) algebras (deformations of sl(n/1)) is defined. In general, additional covariant Serre-type conditions are imposed in order that the Jacobi identities are fulfilled. For these quadratic super-gl(n) algebras, the construction of Kac modules and conditions for atypicality are briefly considered. Applications in quantum field theory, including Hamiltonian lattice QCD and spacetime supersymmetry, are discussed.

We study the gauge transformation between spectral problems and their adjoints for the Ishimori-II (IS-II) and Davey–Stewartson-II (DS-II) equations. The commutativity between the gauge and adjoint transformations is proved. The commutativity is used for spectral decomposition and surface representation theorems for the Ishimori-II equations.

We develop two new approximations for the generalized Bessel function that frequently arises in the analytical treatment of strong-field processes, especially in non-perturbative multiphoton ionization theories. Both these new forms are applicable to the tunnelling environment in atomic ionization, and are analytically much simpler than the currently used low-frequency asymptotic approximation for the generalized Bessel function. The second of the new forms is an approximation to the first, and it is the second new form that exhibits the well-known tunnelling exponential.

When a quantum system undergoes unitary evolution in accordance with a prescribed Hamiltonian, there is a class of states |ψ⟩ such that, after the passage of a certain time, |ψ⟩ is transformed into a state orthogonal to itself. The shortest time for which this can occur, for a given system, is called the passage time. We provide an elementary derivation of the passage time, and demonstrate that the known lower bound, due to Fleming, is typically attained, except for special cases in which the energy spectra have particularly simple structures. It is also shown, using a geodesic argument, that the passage times for these exceptional cases are necessarily larger than the Fleming bound. The analysis is extended to passage times for initially mixed states.

The structure of the Lindblad equation of motion of quantum states is discussed. General specifications for this motion to lead asymptotically into equilibrium states are given. Incomplete 'thermalization', i.e. the Lindblad motion of only a selected subset of quantum states leads to a reduced quantum system whose observables are explicitly constructed and seen to incorporate memory terms. It is shown that under rather general conditions the absolute value of Lindblad operators is given by the (inverse square root of the) grand-canonical probability distribution.

We investigate the properties of three entanglement measures that quantify the statistical distinguishability of a given state with the closest disentangled state that has the same reductions as the primary state. In particular, we concentrate on the relative entropy of entanglement with reversed entries. We show that this quantity is an entanglement monotone which is strongly additive, thereby demonstrating that monotonicity under local quantum operations and strong additivity are compatible in principle. In accordance with the presented statistical interpretation which is provided, this entanglement monotone, however, has the property that it diverges on pure states, with the consequence that it cannot distinguish the degree of entanglement of different pure states. We also prove that the relative entropy of entanglement with respect to the set of disentangled states that have identical reductions to the primary state is an entanglement monotone. We finally investigate the trace-norm measure and demonstrate that it is also a proper entanglement monotone.

The compatibility of standard and Bohmian quantum mechanics has recently been challenged in the context of two-particle interference, both from a theoretical and an experimental point of view. We analyse different setups proposed and derive corresponding exact forms for Bohmian equations of motion. The equations are then solved numerically, and shown to reproduce standard quantum-mechanical results.

We examine the validity of the approximation in which an α particle interacting with an atom is treated classically. In order to analyse such interactions, we perform a model simulation in which the α particle is considered as a particle in one dimension, and the atom as a quantum two-level system. The particle impinges on and excites the two-level system. We treat the particle in two ways: as a quantum mechanical wave packet, and as a classical particle. The classical particle may be a point or may have an extended structure. In each case we calculate the excitation probability P₂₁(t) as a function of time t. We focus on the situation in which the kinetic energy of the incident particle well exceeds the excitation energy of the two-level system. Although the finite-time behaviour of P₂₁(t) varies, P₂₁(∞) is remarkably insensitive to the size and shape of the incident wave packet in the quantum mechanical treatment. In the classical treatment, in contrast, we find that P₂₁(∞) is sensitive to the size of the particle. The classical point particle, however, yields nearly the same values of P₂₁(∞) as the quantum wave packet. Implications of the results on the interaction between an α particle and an atom are discussed.

Unitary transformations in an angular momentum Hilbert space H(2j + 1), are considered. They are expressed as a finite sum of the displacement operators (which play the role of SU(2j + 1) generators) with the Weyl function as coefficients. The Chinese remainder theorem is used to factorize large qudits in the Hilbert space H(2j + 1) in terms of smaller qudits in Hilbert spaces H(2j_i + 1). All unitary transformations on large qudits can be performed through appropriate unitary transformations on the smaller qudits.

We apply the method of algebraic deformation to N-tuple of algebraic K3 surfaces. When N = 3, we show that the deformed triplet of algebraic K3 surfaces exhibits a deformed hyper-Kähler structure. The deformation moduli space of this family of noncommutative K3 surfaces turns out to be of dimension 57, which is three times that of complex deformations of algebraic K3 surfaces.

The Maxwell–Dirac equations are the equations for electronic matter, the 'classical' theory underlying QED. The system combines the Dirac equations with the Maxwell equations sourced by the Dirac current. A stationary Maxwell–Dirac system has ψ = e^−iEtϕ, with ϕ independent of t. The system is said to be isolated if the dependent variables obey quite weak regularity and decay conditions. In this paper, we prove the following strong localization result for isolated, stationary Maxwell–Dirac systems,

• there are no embedded eigenvalues in the essential spectrum, i.e. −m ⩽ E ⩽ m;

• if |E| < m then the Dirac field decays exponentially as |x| → ∞;

• if |E| = m then the system is 'asymptotically' static and decays exponentially if the total charge is non-zero.

We show that lattice Boltzmann simulations can be used to model the attenuation-driven acoustic streaming produced by a travelling wave. Comparisons are made to analytical results and to the streaming pattern produced by an imposed body force approximating the Reynolds stresses. We predict the streaming patterns around a porous material in an attenuating acoustic field.