Home
People

Workshops

Seminars

Publications
Grants Received

PhD/Honours projects

Links

Michael Baake
Rigourous results in diffraction theory

Mathematical diffraction theory was developed after the discovery of quasicrystals in order to get a better understanding of the fingerprints of ordered structures between crystals and amorphous systems. Of interest is a characterization of systems with pure point diffraction and of systems with structural disorder. Examples of both situations will be described, with special emphasis on connections to exactly solvable models.

Murray Batchelor
Attractive interacting bosons

Recent progress in realising 1D quantum gases of ultracold atoms -- described as virtuoso triumphs of nanoengineering -- has drawn attention to the integrable model of interacting bosons. The attractive regime of this fundamental model has essentially been a no-go area for 40 years. Our results will be presented here.

Vladimir Bazhanov
Eight-vertex model and dilute polymers

Peter Bouwknegt
Dimensionally reduced Gysin sequences and applications to T-duality

We reexamine the results on the global properties of T-duality for principal circle bundles in the context of a dimensionally reduced Gysin sequence. We will then construct a Gysin sequence for principal torus bundles and argue from there that the T-dual of a principal torus bundle with background H-flux is in general a continuous field of noncommutative, nonassociative tori.

Tony Bracken
Symmetries and the use of complex numbers in quantum mechanics

Are complex numbers essential for quantum mechanics? In the usual Hilbert space formulation of quantum mechanics it certainly seems so. However, quantum mechanics can also be formulated in phase space with the help of the Groenewold-Moyal bracket, and then complex numbers are nowhere to be seen. In Hilbert space, symmetries are described by unitary or antiunitary, complex, and possibly projective representations of groups and Lie algebras, whereas in phase space they are described by unitary, real, true representations. The reconciliation of these apparently irreconcilable differences will be discussed.

Phil Broadbridge
Quantisation of unstable fields: de Sitter space

Philip Burton
Introduction to topological quantum computing

Progress in Quantum Computing is hindered by the difficulty of establishing and maintaining entanglement. Topological Quantum Computing (TQC) aims to overcome this by using physical processes that are topological in nature and which are less susceptible to environmental disturbance. TQC is based on quasi-particles called anyons that have fractional spin and charge and statistics that fall somewhere between bosons and fermions. A conventional quantum computer stores information in the quantum states of electrons, photons, atoms, etc. By contrast, a topological quantum computer uses the motion of anyons past each other, an effect called braiding (a term taken from knot theory). In an anyonic system, the wave function can 'remember' which path was taken when two anyons move past each other. Quantum information can be stored in patterns of braids and this corresponds to a qubit in a conventional quantum computer. The talk will discuss the connection between the physics of topological quantum computers and the mathematics of Hopf algebras, their representations, and the connection with knot theory.

Grant Cairns
Classification of a special family of graded Lie algebras

We classify those finite dimensional Lie algebras which have a basis $x_1,...,x_n$ with the following properties: \begin{enumerate} \item $[x_i,x_j]=c_{i,j}x_{i+j}$ for some constants $c_{i,j}$, \item $c_{1,j}\neq 0,$ for all $1 \lt j \lt n$. \end{enumerate} It turns out that there are only 6 such algebras of dimension $n \lt 7$. In each of the dimensions 7,8,9,10,11, there are infinitely many algebras, while in dimension $n \gt 11$ there are precisely 4 or 5 algebras, according to whether $n$ is even or odd respectively. This is joint work with Barry Jessup (U. Ottawa)

Harvey Cohen
Anatomy of a Collaboration: From Generalisations of the Foldy-Wouthuysen Transformation to Their Interpretation as SO(4,1) Transformations of the Dirac Equation.

How does a powerful theoretic concept embolden and elaborate an analysis.? This paper presents a case study, where a retrospective overview is given of the first papers of Tony Bracken, reporting research performed in collaboration with this presenter. As a postdoc in the Mathematical Physics Department at Adelaide, I presented a honours/postgrad course in relativistic quantum mechanics and quantum field theory. Fresh from a PhD with heavy computational use of the Dirac Gamma matrices, my second look at the Foldy-Wouthuysen transformation (FWT) of the Dirac equation,, lead me to recognise that the FWT was one of four similar transformations, leading to what I later termed t-special, as well as (new) x,y, and z special forms of this famous equation.. Shown these results, student Tony Bracken recognised that all these transformations spoke of Lie Groups. But just what group? Somewhat surprisingly, these transforms were found to be manifestation of the group SO(4,1), and a very elegant framework was developed. This Lie Group Theoretical framework had some further interesting consequences. This collaboration was published in a joint paper [1] published in the Journal of Mathematical Physics, and another journal. With two flow-ons: a major input to the body of work encompassed in Bracken’s PhD thesis [2]. And to this writer, after experiencing the challenge of teaching undergraduate physics, the first study in qualitative physics. [1] A. J. Bracken and H. A. Cohen. “On canonical SO(4 ,1) transformations of the Dirac Equation”, Journal of Mathematical Physics 1969. [2] A.J. Bracken, “Group-theoretical applications in a tri-local model for baryons” Ph.D. thesis, Mathematical Physics, University of Adelaide, 1970 [3] Harvey A. Cohen, “The Art of Snaring Dragons”, MIT Artificial Intelligence Laboratory Memo 338, May 1975.

John Corbett
The Pauli problem revisted, reconstruction and quantum real numbers

The Pauli problem: is a wavefunction uniquely determined by its position and momentum probability distributions? ie is a $L^2$ function uniquely determined by the absolute values of it and its Fourier transform? is still not completely resolved. I will discuss some attempts to answer it and comment on its importance in quantum mechanics. The reconstruction of quantum states from measurements of physical quantities, i.e. from their expectation values, is a natural development of the Pauli problem. Finally I will indicate how quantum real numbers, as the ontological values of physical quantities, can be used in the reconstruction problem.

Jan de Gier
Brauer loops and the commuting variety

A surprising connection is uncovered between an integrable lattice model based on the Brauer algebra and the variety of commuting matrices. This discovery hints at unsuspected links between intersection theory of varieties based on Lie algebras and previous conjectures, relating the Temperley-Lieb algebra to the combinatorics of alternating sign matrices.

David De Wit
Skein relations for the Links--Gould invariants

The Alexander--Conway polynomial $\Delta$ is obtainable via a particular one-variable reduction of each two-variable Links--Gould invariant $LG^{m,1}$. This follows as the reduction of $LG^{m,1}$ satisfies the defining skein relation of $\Delta$. The key to the proof of this involves determining the kernel of a quantum trace.

Jennifer Dodd
Quantum computation with many-body Hamiltonians

A quantum computer is like an ordinary computer, except that instead of bits it uses two-dimensional quantum systems (qubits), and the usual logic operations are replaced by unitary quantum dynamics. One of the most important early discoveries was that any quantum computation can be achieved with certain finite sets of unitaries. What general characteristic separates quantum dynamics that can perform universal quantum computation from those that cannot? I will describe some recent progress on this question that reveals a surprisingly intuitive structure with an interesting twist: Under physically motivated assumptions, any Hamiltonian that interacts all the qubits in the computer is universal for quantum computation, except when it only contains terms each of which interact an odd number of qubits.

Tony Dooley
Intertwining Operators and the Cayley Transform

Demos Ellinas
Quantum Walks, Tilings and Aggregations

Positive maps on density matrices and appropriate operator observables are introduced, to effectively describe processes such as: i) quantum random walks (QRW) on integers, ii) tiling of phase plane by operator valued measures related to volumes (VOVM) formed by the Wigner function, and iii) 1D aggregations formed recursively by VOVM with fractal support. Aspects of classical vs. quantum dichotomy are investigated and contrasted in these three processes, in terms of their dynamical behavior, their quantum information content and their fractal geometric structure. Finally some questions referring to implementation of these algorithms are addressed.

Bertfried Fauser
New branchings induced by plethysm

Based on Littlewoods description of branching rules which uses Schur function series, a theory of branchings can be derived using the outer Hopf algebra of symmetric functions acting on series of the form \{\pi\}\circ M of plethysms of the $M$-series. (joint work with P.D. Jarvis, Ron C. King, Brian G. Wybourne)

Robert Fawcett
Seasonal forecasting at the Australian Bureau of Meteorology

The Australian Bureau of Meteorology has been issuing climate outlooks for (seasonal) three-monthly rainfall totals since the late 1980s, and for seasonal maximum and minimum temperature since autumn 2000. Empirical statistical techniques (albeit informed by relevant climatological knowledge) of several different kinds have been employed over the years, although the currently used techniques have essentially been in place since 1997. This presentation will discuss the theory and practice of seasonal forecasting at the Bureau of Meteorology as it is currently implemented within the National Climate Centre, and provide a brief sketch of expected future developments.

Jamin Flohr
Symmetries of an extended Tokuoka Lagrangian

Tokuoka's equation is extended and suitable Lagrangians are discussed. Although the Hamiltonian is not hermitian, meaningful conserved currents are obtained. Creation and annihilation operators are developed leading to a positive semi-definite energy operator. Furthermore, appropriate momenta and angular momenta operators are obtained. Also, a concise way of deriving Belinfante's tensor for the Dirac spinor is acquired.

Omar Foda
KdV finite gap solutions and off critical minimal models

I would like to argue that the particle spectra of certain off critical integrable models are encoded in KdV finite gap solutions.

Doug Gray
Multitarget Tracking: PLST, PMHT and other acronyms

Tracking multiple targets is an optimisation problem involving both state estimation (continuous) and associations (discrete) and thus can be computationally complex. By allowing the associations to be treated as continuous random variables to be estimated the complexity can be avoided. An engineering generalisation of least squares will be given to illustrate the general idea before dealing with a more formal Bayesian approach. The advantage of the latter approach in allowing crude measurments of target type to be inlcuded in the estimation process will be illustrated.

Hendrik Grundling
Constraining quantum particles to submanifolds

We consider constrained quantum systems where the dynamics do not preserve the constraints. The problem then is how to adjust the dynamics (i.e. constrain it) to produce physically correct results, and to preserve the constraints. Our main example is to restrict a quantum particle in $R^n$ to a curved submanifold. We propose a method of constraining and dynamics adjustment which produces the right Hamiltonian on the submanifold when tested on known examples. The method generalises to other situations, but involves analytic pathologies. (Joint work with Angas Hurst.)

Xi-Wen Guan
Thermal and magnetic properties of spin-1 chain compounds

The ground state and thermodynamic properties of the integrable spin-$1$ chain are investigated. The analysis involves the Thermodynamic Bethe Ansatz and the High Temperature Expansion methods. For the spin-$1$ chain with large single-ion anisotropy, a gapped phase occurs which is significantly different from the valence-bond-solid Haldane phase. The theoretical curves for the magnetization, susceptibility and specific heat are compared with experimental data for a number of low-dimensional spin compounds.

Tony Guttmann
New developments in the two-dimensional Ising Model

Anthony Henderson
Nilpotent orbits and Kazhdan-Lusztig polynomials of type A

Various quantities arising in the representation theory of quantum affine algebras or affine Hecke algebras are known to be related to intersection cohomologies of closures of nilpotent orbits of linear quivers (or cyclic quivers in the root of unity case). These in turn are known to be related to Kazhdan-Lusztig polynomials for the symmetric group (or affine symmetric group). I will explain how the latter relation can be simplified, and explore the consequences for representation theory.

Andrew Hines
Entanglement, bifurcations and quantum phase transitions

The burgeoning field of quantum information science has provided us with new tools with which to study complex quantum many-body systems. Specifically, the entanglement characteristics of condensed matter systems have been considered in the study of quantum phase transitions. Quantum phase transitions, a qualitative change in the ground state of a quantum many-body system induced by the variation of some parameter, are characterized by the development of long range correlations. It is argued that the property responsible for such long range correlation is entanglement, making quantum phase transitions truly quantum mechanical. In a classical system, a qualitative change in the dynamical phase space structure as a parameter is varied corresponds to a bifurcation of fixed points. I will demonstrate how the entanglement in the ground state of bipartite quantum systems can be associated with a fixed point bifurcation in the classical analog. Using the example of coupled giant spins, we show that when the fixed point undergoes a supercritical pitchfork bifurcation, the corresponding quantum ground state achieves a maximum amount of entanglement. As well, I will discuss this phenomenon in spin-boson models, where it can be shown that the bifurcation corresponds to a quantum phase transition.

Angas Hurst
Looking backwards on perturbation expansions in quantum field theory

This talk is a short summary on what happened to the divergent expansions in quantum field theory since my PhD thesis. The talk will be at a very elementary level so as to be understandable to non-experts.

Peter Jarvis
Yangians and GL(N) - O(N) state labelling

We consider the problem of labelling irreducible representations of $GL(N)$ in an $O(N)$basis, in the framework of the Yangian and twisted Yangian algebras in the $GL(N)$ enveloping algebra. For the $N=3$ case we find a resolution in terms of the $O(3)$ invariant Bethe subalgebra of the twisted Yangian (joint work with R B Zhang, hep-th/0411026).

William Joyce
Recoupling Lie Algebra: going beyond non-associativity

We begin with the essential components for a recoupling theory and argue a case for admitting non--trivial phases for both commutativity and associativity. For example this freedom is required in order to develop an algebraic structure for $Z_{3}$ graded fermion algebra of quarks. We define recoupling Lie algebra which generalises graded Lie algebra. In order to construct a universal envelope one requires a generalisation of the notion of non-associative algebra which we call $\omega $-algebra. We construct the envelope and discuss its Hopf $\omega $--algebra structure. We briefly outline the proof of the corresponding Poincar\'e--Birkhoff--Witt theorem. Finally we indicate other important omegafield structures.

Peggy Kao
Poisson-Lie T-duality

T-Duality is an important tool in string theory whose rule is to link seemingly different string backgrounds. Recently much attention was on the study of Non-Abelian T-duality, and it was further discovered that the sigma models on the Lie group and its (co)algebra are indeed dual to each other from the point of view of the so-called Poisson-Lie T-duality. Poisson-Lie T-duality is a generalisation of the standard Abelian T-duality (Buscher rule), and applies the concepts of a Drinfel'd double and Manin triple.

Graham Legg
Group Invariant Solutions of Maxwell Dirac Equations

It is demonstrated how the Dirac equation can be rewritten in gauge independent tensor form, in terms of observable bilinear Dirac 'currents', their derivatives, and a gauge independent vector field. The method of group invariant solutions is demonstrated on the system consisting of the rewritten Dirac equation coupled with the Maxwell equation. Symmetries leading to possible ODE solutions are listed. It is shown how care must be taken dealing with the phase, which has been eliminated from the equations.

Max Lohe
An algebraic approach to quantum mechanics in fractional dimensions

I formulate an algebraic approach to quantum mechanics in fractional dimensions in which the momentum and position operators $P,Q$ satisfy the $R$-deformed Heisenberg relations, and find representations of $P,Q$ in which the angular momentum $\ell$ and the dimension $d$, which can be any real positive number, appear as parameters. These representations lead to corresponding representations of paraboson operators which can used, for example, to solve the time-dependent harmonic oscillator for any $d \gt 0$ using the method of Lewis and Riesenfeld. I develop algebraic properties of Weyl-ordered polynomials in $P,Q$ by viewing them as tensor operators with respect to the Lie algebra $\mathfrak{sl}_2$, and obtain specific forms as hypergeometric functions. I also discuss the $q$-analogue deformation of these results.

Vladimir Mangazeev
Eight-vertex model and non-stationary Lame equation

We study the ground state eigenvalues of Baxter's Q-operator for the eight-vertex model in a special case when it describes the off-critical deformation of the $\Delta=-1/2$ six-vertex model. We show that these eigenvalues satisfy a non-stationary Schrodinger equation with the time-dependent potential given by the Weierstrass elliptic function where the modular parameter $\tau$ plays the role of imaginary time.

Jim McGuire
A Discussion of Solvable Quantum Systems with Many Degrees of Freedom

A solvable or integrable quantum system is one in which the state function of stationary states may be explicitly constructed using finite algebra alone. A class of such systems will be presented. The question which arises and the principal subject of this discussion is: `What then?'

Alex Molev
Sympletic branching rules and Yangian representations

It is well known that the restriction of a finite-dimensional irreducible representation of the symplectic Lie algebra $sp(2n)$ to the subalgebra $sp(2m)$ with $m \lt n$ need not be multiplicity-free. Each multiplicity space turns out to be an irreducible representation of a quantized enveloping algebra $Y(sp(2n-2m))$ called the twisted Yangian. We give a description of this representation by calculating its highest weight and Drinfeld polynomials.

Stuart Morgan
Born Reciprocity and the Quaplectic Group

The Born reciprocity principle is the principle that the laws of physics are invariant under either of the transformations: $(t, e, q, p) \to (t, e, p, -q)$ or $(t, e, q, p) \to (-e, t, q, p)$. It suggests that rather than a nature in which there are four dimensions (three dimensions of space and one time dimension) there are in fact eight: three space directions, three momentum directions, one time and one energy. The Quaplectic group is the semi-direct product of the pseudo-unitary group $U(1, n)$ with the Heisenberg group $H(n + 1)$. It embodies Born reciprocity and contains four distinct Poincare subgroups. The field equations that are representations of the Casimir invariants of the Quaplectic group are computationally complex but they may be represented in a coordinate space corresponding to a diagonalisation of $(t, q), (e, p), (t, p)$ or $(e, q)$. This talk will give a brief discussion of these topics. In particular, the idea of the Quaplectic group as a candidate for a bigger, more fundamental group than the Poincare group will be discussed.

Avijit Mukherjee
D-branes and $\Pi$-stability

We consider a description of D-branes using the framework of derived categories of coherent sheaves on projective manifolds. In the context of geometric engineering based on certain compactifications of the type II string theory on a singular CY 3-fold, the dyonic spectrum of the $N = 2$ SYM theory is identified with B-type D-branes wrapped on holomorphic cycles. We discuss how the notion of $\Pi$ stability leads us to understand the allowed decays, stable bound states and the transitions of these D-branes and the string junctions. In the particular case at hand, the well known results on the derived category of coherent sheaves on $P^1$ lends a complete determination of the stable dyonic spectra at all points of the moduli space.

Giuseppe Mussardo
Quantum spin chain and O(3) non-linear sigma model with topological term

By using recent developed techniques on non-integrable quantum field theories, we analyse the mass spectrum of the O(3) non-linear sigma model by varying the coupling of its topological term. This model corresponds to the continuum limit of the SU(2) quantum spin chain.

Amnon Neeman
Classical invariant theory meets statistics

The identifiability of mixtures naturally gives rise to a question about polynomials invariant under the action of certain groups. We explain the statistical question, the mathematical problem it reduces to, and the solution to the problem. The work was joint with Ryan Elmore and Peter Hall.

Michael Nielsen
Quantum computation by quantum measurement alone

A quantum computer is a computing device exploiting the principles of quantum mechanics to solve computational problems thought to be infeasible on a conventional classical computer. Until recently, the operation of quantum computers has always been described through the use of unitary operations on the state space of the computer. This led to the belief that it is the ability to manipulate and preserve superpositions which is the key to a quantum computer's power. In this conventional picture, measurements are viewed as the bane of quantum computing, since they lead to a loss of coherence. In this talk I will discuss a recently proposed model of quantum computation in which this conventional wisdom is tipped on its head. In particular, it turns out to be possible to do quantum computation using only measurements, with no unitary dynamics necessary. In this talk I will describe how the measurement-based models of quantum computation work, and what the practical implications for quantum computing are.

Paul Norbury
Hamiltonian systems on polygons

A particle moving along straight lines in the interior of a region bounded by perfectly reflecting walls gives a Hamiltonian system known as billiards. I will describe a billiard-like Hamiltonian system, with two-dimensional regions given by disks or polygons, where the particle is repelled from the walls, so in particular does not move along straight lines. This problem arises from toric geometry and periodic orbits in the system correspond to minimal hypersurfaces in a toric surface.

Judy-anne Osborn
Lattice paths in a strip

A relatively new technique for solving lattice path counting problems is the Constant Term (CT) method. This approach has been used to enumerate single (weighted) paths in the half-plane, where it complements a raft of existing approaches but has the special feature that the algebra involved provides a signpost to a direct combinatorial proof. This strategy is also applicable to configurations of multiple paths in the half-plane, where in particular it is the only method to date which has provided an exact answer for any osculating path question involving more than three paths, outside of those defined by the special geometry associated with the alternating sign matrix. I will discuss the CT method with reference to a single path confined to a horizontal strip inside the half plane.

Aleks Owczarek
Sentitized floculation: an exactly solved model

Alison Parker
Homological aspects of Schur Algebras and Lie Theory

In this talk I will describe some of the general aspects of the representation theory of Schur algebras and the general linear group over fields of prime characteristic. In particular I will discuss some of the known results about homomorphisms between certain important modules known as Weyl modules for two by two and three by three matrices.

Paul Pearce
Boundary $S$ Matrices of Unitary Minimal Models

In this joint work with Rafael Nepomechie, we propose explicit expressions for the boundary $S$ matrices of massive scattering theories obtained by perturbing the $A_m$ unitary minimal models with $(r,s)$ boundary conditions by both bulk and boundary $\phi_{1,3}$ operators. The boundary $S$ matrices are constructed as direct sums of $A_{m-1}$ Behrend-Pearce solutions of the boundary Yang-Baxter equation and are consistent with the boundary bootstrap.

Arlei Prestes Tonel
Quantum dynamics of a model for two Josephson-coupled BECs

The experimental realisation of Bose-Einstein condensates (BECs) in trapped clouds opened up the exploration of quantum mechanics of mesoscopic systems. Here there is the possibility for studying dynamical regimes at the frontier between the quantum and classical scenarios where new macroscopic quantum phenomenon can emerge. In this way BECs are seen as one fo the main tools to investigate, verify and improve our understanding of many concepts and principles in quantum physics. In this talk, we will show a numerical study, using exact diagonalisation, for a simple model of two Josephson-coupled BECs. Basically, we will do a detailed and systematic analysis of the relative number of particles (imbalance population) in all regimes and using different initial states. As a result we found that this simple model shows up a diversified and rich dynamics which depends on the ratio of parameters and the initial state when the total number of bosons is maintained fixed.

David Ridout
D-Brane charges for Wess-Zumino-Witten models - Geometric Results

In the last decade, the notion of D-branes has revolutionised string theory and led to the serious consideration of its successor, M-theory. More recently, conserved charges for D-branes have been studied, and the groups of such charges have been linked to specific K-theory groups. In the relatively tractable case of Wess-Zumino-Witten models, a combination of algebraic conformal field theory considerations and heuristic dynamical principles from condensed matter theory allow the computation of the brane charge groups, and hence make non-trivial predictions regarding the appropriate K-groups. In this talk, this computation is revisited using aspects of the theory of compact simple Lie groups, without any reference to any dynamical principles. Interestingly, the results almost exactly match those obtained algebraically (with dynamics).

Nuno Romao
Statistical mechanics of gauged vortices

I will present some recent results on the statistical mechanics of a gas of gauged vortices in the canonical formalism. At critical coupling, and for low temperatures, it is has been argued that the configuration space of the vortex system approximately truncates to a finite-dimensional moduli space with a Kaehler structure. For the case where the vortices live on a large 2-sphere, I will explain how to make use of the geometry of the moduli space to compute exactly the partition function of the vortex gas interacting with a background potential.

Gerd Rudolph
On the Structure of the Observable Algebra of Lattice QCD

We discuss quantum chromodynamics (QCD) on a finite lattice in the Hamiltonian approach. First, we present the field algebra as comprising a gluonic part, with basic building block being a certain crossed product C*-algebra, and a fermionic (CAR-algebra) part generated by the quark fields. This algebra has a unique irreducible representation. Next, the algebra of internal observables is defined as the algebra of gauge invariant fields, satisfying the Gauss law. In order to take into account correlations of field degrees of freedom inside the lattice with the ``rest of the world'', we extend this algebra by tensorizing it with the algebra of gauge invariant operators at infinity. This way, we construct the full observable algebra. It is shown that its irreducible representations are labelled by boundary flux distributions with values in the center of $SU(3)$. Then, it is proved that there exist unitary operators (charge carrying fields), which intertwine between irreducible sectors leading to a classification of irreducible representations in terms of the global boundary flux (with values in the center of $SU(3)$). By the global Gauss law, these 3 inequivalent charge superselection sectors can be labelled in terms of the global colour charge (triality) carried by quark fields.

Ryu Sasaki
Classical and Quantum Integrability in Multi-particle Dynamics

The relationship (resemblance and/or contrast) between quantum and classical integrability in multi-particle dynamics, for example, the Calogero-Moser (C-M) and Ruijsenaars-Schneider (R-S) systems is addressed. The classical Calogero and Sutherland systems (based on {\em any\/} root system) at equilibrium have many remarkable properties; for example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all `integer valued'. These are related to the energy eigenvalues of the quantum Calogero and Sutherland systems. Similar features and results hold for the R-S type of integrable systems based on the {\em classical\/} root systems. The equilibrium positions of the C-S systems are known to be described by the zeros of classical orthogonal polynomials, the Hermite, Laguerre, Chebyshev and Jacobi polynomials. Similarly the equilibria of the R-S systems give rise to {\em deformation\/} of these classical polynomials and they are identified as members of the Askey-scheme of hypergeometric orthogonal polynomials. A universal theorem, valid for non-integrable systems too, relating the $o(\hbar)$ part of the quantum spectrum to the eigenvalues of the small oscillations at the classical equilibrium.

Hisham Sati
M-theory and topology

Witten has shown that the topological part of the M-theory partition function is encoded in an index of an E8 bundle in eleven dimensions. Diaconescu-Moore-Witten related this to the K-theoretic partition function of type IIA string theory obtained via dimensional reduction, and later Varghese Mathai and I generalized part of the construction to twisted K-theory. In this talk, after reviewing the above, I report on my recent work with Igor Kriz on the appearance of elliptic cohomology in this context.

Sergey Sergeev
New solutions of the tetrahedron equation

In the framework of the method of the local Yang-Baxter equation we have found a specific solution of the functional tetrahedron equation. This solution has a certain Poisson structure and therefore it was successfully quantized. Quantum algebra of observables is the local $q$-oscillator algebra. We construct $R$-matrix of the tetrahedron equation for generic $q$ in the Fock space representation. In $2$ layers this $3D$ $R$-matrix produces all finite spin $2D$ $R$-matrices of $U_q(\widehat{sl_2})$ including the six-vertex one. (Authors: V. Bazhanov, S. Sergeev)

Marni Sheppeard
Where do quantum numbers live? Categories and Logic

A topos is a category with properties that make it a natural home for intuitionistic logic. There is an interesting history in the application of toposes to causal structures, which we partly review before considering an example underlying the conventional notion of a quantum observable.

Junji Suzuki
Nonlinear integral equations revisited

The nonlinear integral equation approach to the finite size excited spectra of integrable spin chains will be addressed. The spin 1 case will be discussed in detail with a particular emphasis on its similarity to analysis based on the string hypothesis.

Peter Szekeres
Mathematical Physics at the University of Adelaide

A personal reminiscence on the people and activities of the Department of Mathematical Physics at the University of Adelaide from the days of Bert Green to the present day.

Toshiyuki Tanisaki
Hypergeometric systems and Radon transforms for generalized flag manifolds

I will talk about a generalization of Gelfand's hypergeometric system on generalized flag manifolds and related Radon transforms. A part of my talk is based on a joint work with C. Marastoni.

Billy Todd
The Arnold cat map and elongational flow

For many years the simulation of elongational (or extensional) flows by molecular dynamics simulation was deemed impossible due to the limitation that the simulation must cease once the length of the simulation box in the contracting dimension equals twice the interaction potential radius. The problem was always that a nonequilibrium steady-state could never be achieved for molecular systems, since their relaxation times are vastly greater than the time taken for this minimum extension to be reached. This limitation was removed in 1998 when it was shown [1-3] how to remove this time constraint by the use of novel periodic boundary conditions, first devised by Kraynik and Reinelt [4] in their studies of foams. However, the derivation of Kraynik and Reinelt is algebraically involved and is not all that easy to follow. In this talk we show how to vastly simplify the derivation by use of dynamical systems theory. In particular, we show that the famous 'Cat Map', devised by Arnold [5], can be used as periodic boundary conditions for molecular dynamics simulations of planar elongational flow. Using the formalism of dynamical systems theory one can compute the required parameters of these periodic boundary conditions in several lines of algebra, compared to the pages of algebra involved in the original derivation. We also present some simulation data for linear polymer melts undergoing planar elongation and demonstrate how useful these periodic boundary conditions are in the study of polymer rheology. 1. Todd, B. D. and Daivis, P. J., 1998, Phys. Rev. Lett., 81, 1118. 2. Todd, B. D. and Daivis, P. J., 1999, Comput. Phys. Commun., 117, 191. 3. Baranyai, A. and Cummings, P. T., 1999, J. Chem. Phys., 110, 42. 4. Kraynik, A. M. and Reinelt, D. A., 1992, Int. J. Multiphase Flow. 18, 1045. 5. Arnold, V. I. and Avez, A., 1968, Ergodic problems of classical mechanics (Benjamin, New York).

Ioannis Tsohantjis
Quantum Operations Induced by Classical Randomness:the case of qubit

The question of quantum computation gates and quantum dissipation maps generated by means of classical random walks is here addressed in operations involving qubit states. Classical variables determining the manifold of qubit state vectors and density matrices are left to perform a classical random walk formulated algebraically by means of Hopf algebras (such as $Z_N$ and $R\times R$).It is shown that this induces quantum operations on the qubits and density matrices which are further identified with e.g known unitary transformations and completely positive trace preserving maps, depending upon the kind of random walk chosen.

Zengo Tsuboi
Nonlinear integral equations and high temperature expansions

Thermodynamic Bethe ansatz equations contain, in general, an infinite number of unknown functions. As alternative equations, we explain some nonlinear integral equations (NLIE) with only a finite number of unknown functions, which give the free energy of integrable one dimensional spin chains. By using the NLIE, we calculate the high temperature expansions of the free energy. In particular for the XXZ model, we can calculate the coefficients up to order 99.

Guenter von Gehlen
3-dimensional approach to the Bazhanov-Stroganov model

The talk is about using techniques and concepts of the 3-dim. integrable Zamolodchikov-Baxter-Bazhanov model to obtain exact results for the 2-dim. Bazhanov-Stroganov model. After reviewing the Sergeev construction of the vertex version of the 3-dim. ZBB model, we show how the 2-dim. BS model L-operator and intertwining S-matrix arise within the 3-dim. ZBB framework. New parameterizations for the BS model and the closely related integrable Chiral Potts model are suggested. The intertwining relation for the L-operators of the classical version of the BS model is found using a technique adopted from the 3-dim. ZBB approach.

George Willis
Ergodic ${\mathbb Z}^d$-actions by automorphisms

The existence of an ergodic ${\mathbb Z}^d$-action by automorphisms of a locally compact group $G$ implies that $G$ has a compact normal subgroup $K$ such that $G/K$ is a direct sum of vector groups. In the special case when $d=1$, $G$ must be compact, as was conjectured by P. Halmos. The proofs of these assertions rely on the structure theory of locally compact groups. Approximation by Lie groups is used to deal with the connected factor of $G$ and new techniques that parallel Lie theory are used to deal with the totally disconnected factor. The talk will focus on the totally disconnected case by outlining these new techniques and showing how they are used.

James Wood
Bounds on integrals of the Wigner function: the hyperbolic case

Wigner functions play a central role in the phase space formulation of quantum mechanics. Although closely related to classical Liouville densities, Wigner functions are not positive definite and may take negative values on subregions of phase space. We investigate the accumulation of these negative values by studying bounds on the integral of an arbitrary Wigner function over noncompact subregions of the phase plane with hyperbolic boundaries. We show using symmetry techniques that this problem reduces to computing the bounds on the spectrum associated with an exactly-solvable eigenvalue problem and that the bounds differ from those on classical Liouville distributions. In particular, we show that the total `quasiprobability' on such a region can be greater than 1 or less than zero.

Junfang Zhang
Structural and dynamic properties of inhomogeneous non-equilibrium fluids

Back to workshop page.