Scientific Program of the XII Max Born Symposium

Book of abstracts:

V. Abramov: "On realizations of q-exterior calculus."
We study realizations of the $q$-exterior calculus with exterior differential $d$ satisfying $d^N = 0, \, N>2$ on the free associative algebra with one generator and on the generalized Clifford algebras. Analogues of the notions of connection and curvature are discussed in the case of the $q$-exterior calculus on the generalized Clifford algebra. We show that the $q$-exterior calculus on the free associative algebra with one generator is related to $q$-calculus on the braided line.
J.-P. Antoine: "Partial *-algebras: a retrospective"
A partial *-algebra is a vector space $\A$, equipped with a multiplication $(x,y) \mapsto x \cdot y \in \A$ which is defined only for some pairs $x,y\in \A$ (in which case $y$ is called a right multiplier of $x$). This partial multiplication is assumed to be distributive with respect to addition, but not necessarily associative. From these simple ingredients, a whole theory has been developed over the last 15 years, that we shall quickly overview in this lecture. We distinguish three different levels.
(1) At the {\em algebraic level}, spaces of multipliers build a complete involutive lattice, which encodes conveniently the partial multiplication structure. At this stage, an important development is a representation theory, based on a generalized Gel'fand-Naimark-Segal construction; the main tool here is the notion of positive sesquilinear form, further extended to generalized weights.
(2) The next step is to introduce a topology $\tau$ on $\A$, that makes it into a locally convex topological vector space $\A[\tau]$. Then $\A$ is called a {\em topological partial *-algebra } if it satisfies a number of conditions, which all amount to require that the topology $\tau$ fits with the multiplier structure of $\A$. We shall describe several classes of such objects, such as CQ*-algebras (developed in detail in the lecture of C. Trapani), unction spaces or partial *-algebras of operators (operators on a partial inner product space, partial O*-algebras).
(3) The last, and most interesting, class is that of partial {*-algebras} of closable operators in a Hilbert space (partial O{*-algebras})We will first give a quick review of the general properties and describe several types of partial O*-algebras, notably the so-called partial GW{*-algebras}, which are a natural unbounded generalization of von Neumann algebras. Then we will examine two recent lines of research. First, automorphisms and derivations of partial O{*-algebras}, and their mutual relationship; the central theme here is to find conditions that guarantee spatiality. Then, the extension of the Tomita-Takesaki theory of modular automorphisms on $\A$, with the aim of constructing KMS states on $\A$ (which would represent equilibrium states if $\A$ is the observable algebra of some physical system).
A. Borowiec: "Algebraic vector fields"
Vector fields play a very important role in the classical differential geometry on manifolds and can be defined in purely algebraic way. They provide contravariant tensors, while differential forms (the dual objects) are source for covariant one. In contrast, noncommutative geometry is exclusively based on covariant differential calculi. We propose a new notion of Cartan pairs as noncommutative substitute of vector fields.
J. Cislo: "On some expansion of $\pi$"
J. Czerwonko: "Thermodynamics of the BCS model at broken particle-hole symmetry"
It is shown that breaking particle-hole symmetry (PHS) in DOS of the BCS models leads to the first order phase transition. The latent heat of the transition from the superconducting to the normal state is always negative, by virtue of conditions of the thermodynamic stability, formulated in the paper. Moreover, these conditions lead to to the negative value of the specific heat in the subcritical region at reasonable value of the PH asymmetry parameter. Salvation from the paradox lies in the retardation of the electron-phonon interaction, disregarded in the BCS truncated Hamiltonian. Moreover, at unimportant retardation it is shown that there are such values of the parameters of the Hamiltonian that the superconductivity is stable at low temperatures but it is not at subcritical ones. In such a case the order parameter and the chemical potential should jump, the order parameter to zero, at the temperature smaller than the critical one. The discontinuity of the chemical potential excludes the coexistence of the superconducting and normal phases. Similar phenomena appear also for BWV pairing in He-3. The obtained results are exact for the system with BCS or BWV truncated Hamiltonian.
M. Gusiew-Czudzak and Z. Oziewicz: "Empirical laws of the classical mechanics as a Cartan exterior differential systems"
This subject is a history of mechanics. Lagrange formulated Newton mechanics with a help of one scalar function - Lagrangian. Hamilton found an alternative - Hamilton formalism. But it is only Hilbert who in 1900 introduced differential one-form, today known as Poincare-Cartan form, which is - on the Lagrange submanifold - an exterior derivative of Hamilton-Jacobi function. Klein was probably the first to realize that Lagrange or Hamilton mechanics is based on closed differential two-form. Monography by Souriau (1970) is devoted to this form. Poincare-Cartan one-form is closed on Lagrange submanifolds but mechanics does not deal with Lagrange manifolds only. Analogously, the differential two forms in the two well known Lagrange and Hamilton formalisms are closed, but perhaps there exists a universal two-form, non closed, such that it is closed when restricted to Lagrange or Hamilton, or some other alternative formalism.
R. Haag: "Trying to divide the universe"
"Phenomena. Concepts and mathematical structure in quantum physics"
Z. Haba: "Proper time dynamics in quantum field theory"
We discuss a quantization of relativistic wave equations which allows to treat particles and fields simultaneously. It is suggested that the method is capable of a description of interacting particles at a finite time. We consider a stochastic wave function whose dynamics is determined by a non-linear Schrödinger-type evolution equation in an additional time parameter. The correct classical limit requires the proper time interpretation of the time parameter. An average over the proper time leads to the conventional quantum field theory of particles, which are free at an infinite space separation. We consider models with scalar and vector fields on a pseudoriemannian manifold. It is pointed out that the method should be especially useful for a quantization on a Lorentzian manifold as well as for the quantization of Einstein gravity.
L. Halpern: "On the role of intrinsic Rotation in Gravitational Theory"
Historical reviews, in particular of Einstein, on the future of gravitational theory are compared with the presently prevailing doomsday outlook of black hole collapse.
A modernized version of the principle of inertia based on the theory of simple Lie groups and which includes also Einstein's equations and the motion of structurless and spinning test particles is generalized to the structure of a unified relativistic theory. The general relativistic and geometric structure of the equation of motion are presented.
O. Haschke and W. Rühl: "Is it possible to derive exactly solvable dynamical models?"
We develop a constructive method to derive exactly solvable quantum mechanical models of rational (Calogero) and trigonometric (Sutherland) type. This method starts from a linear algebra problem: finding eigen-vectors of triangular finite matrices. These eigenvectors are transcribed into eigenfunctions of a selfadjoint Schroedinger operator. We prove the feasibility of our method by constructing a new "AG3 model" of trigonometric type (the rational case was known before from Wolfes 1975, but not resumed in the list of Olshanetsky and Perelomov). In order to better understand features of our construction we exhibit the F4 rational model with our method.
R.S. Ingarden: "Modal interpretation of quantum mechanics and classical physical theories"
In 1990 Bas C. van Frassen defined the modal interpretation of quantum mechanics as consideration of it as `` a pure theory of the possible, with testable, empirical implications for what actually happens''. This is a narrow, traditional understanding of modality, only in the sense of the concept of possibility (denoted usually in logic by functor M or the C. I. Lewis' symbol $\Diamond$) and the defined by it concept of necessity (functor L or $\Box$). In modern logic, however, modality is understood in a much wider sense, as any intentional functor (i.e. non-extensional or determined not only by the truth value of a sentence). In the recent (independent of van Frassen) publications of the author (1997), it is tried to apply this wider understanding of modality to interpretations of classical and quantum physics. In the present lecture these problems are discussed on the background of a short review of logical approach to quantum mechanics in the recent 7 decades. In this discussion the new concepts of sub-modality and super-modality of many orders is used.
M. Klimek : "Conserved currents for linear equations on discrete, quantum Minkowski and braided linear spaces."
The general linear equations with constant coefficients on discrete and noncommutative spaces are considered and explicit formulae for their conserved currents are given. This class of equtaions includes Klein-Gordon, Dirac and wave equations. As an example we construct symmetry operators for Klein-Gorodon equations and obtain conserved currents connected with symmetries of these equations. In the discrete case we use the derived currents in construction of integrals of motion.
J. Lukierski: "From quantum groups to deformed field theory"
We shall consider the quantum deformations of relativistic symmetries, in particular so-called $\kappa$-deformation, introducing fundamental mass parameter as well as "quantum" time degree of freedom. The classical fields, in particular $\kappa$-deformed Klein-Gordon and Dirac equations, will be defined on noncommutative $\kappa$-Minkowski space. The local products of $\kappa$-deformed fields, describing interaction vertices will be introduced. The description of local field theory on noncommutative $\kappa$-Minkowski space will be given in terms of nonlocal conventional fields. The interpretation of $\kappa$-deformed fields will be presented as describing the inclusion of short distance quantum gravity effects.
D. Maison: "Solitons and gravitation"
I will give a survey on self-gravitating non-abelian monopoles and sphalerons, solving coupled Einstein-YM-Higgs-(Dilaton) equations. Besides everywhere regular solutions there exist also `coloured' black hole versions of these solutions giving rise to violations of the `No-Hair' Conjecture.
W. A. Majewski: "Detailed balance and quantum dynamical maps"
Two definitions of detailed balance conditions are presented. The relation of detailed balance condition to microscopic reversibility is discussed. An entropy estimate is given.
P. Maslanka:
W. Marcinek: "On generalized quantum statistics"
Quantum states of particles are describe in terms of the theory of monoidal categories with duality. The quantum statistics is determined by the so called cross symmetry instead of a braid symmetry. The noncommutative Fock space is an algebra in the category. In this approach it is possible to determine the statistic of one particle.
W. Piechocki : "Relativistic particle in 2-dimensional gravitational field"
A model of a relativistic particle moving in the Liouville field is investigated. Symmetry group of the system is SL(2,R)/Z2 . The corresponding dynamical integrals describe full set of classical trajectories. Dynamical integrals are used for the gauge-invariant Hamiltonian reduction. The new scheme is proposed for quantization of the reduced system. Obtained quantum system reproduces classical symetry. Physical aspects of the model are discussed.
H. Rechenberg: "Born and the theory of molecules"
Max Born was occupied with molecular problems in several periods of his scientific work. First in 1916-1918 in Berlin, later 1922-1924 in Göttingen, then again 1927-32. Also in exile and then in Edinburgh he returned to this , finally in connection with fluid theory after 1946. Thus it was a typical Born theme for which there is no history.
H. Reeh: "Summational invariants"
W. Rühl: "Is it possible to derive exactly solvable dynamical models?"
Th. W. Ruijgrok: "On localization in relativistic quantum mechanics"
In nonrelativistic quantum mechanics causality is violated in an obvious way. The hope that this acausality would disappear in relativistic theories, in which the speed of propagation is finite, has turned out to be an idle hope. A localized state spreads over all space under a time translation or a boost.
In this note it is suggested that this strange behaviour is actually a semantic problem. The eigenstates of the Newton-Wigner position operator will be considered as single particle states, which are localized with an accuracy equal to their Compton wavelength. Correspondingly the nonlocality of a two-particle potential will not extend beyond the Compton wavelengths of the particles and can therefore still be called local.
These ideas will be elaborated in the framework of a previously formulated relativistic quantum theory. With this theory it will be shown that the sharp edge of a hard sphere interaction between two particles can still be determined with any accuracy by measuring the cross section in a high energy scattering experiment.
E. Seiler: "Critical behavior of classical spin models and local cohomology"
Using reflection positivity as the main tool, we establish a connection between the existence of a critical point in classical spin models and the triviality of a certain local cohomology class related to the Noether current of the model in the coninuum limit. Furthemore, we find a relation between the location of the critical point and the momentum space aotocorrelation function of the Noether current..
P. Stichel: "Dynamical equivalence, commutation relations and noncommutative geometry"
We revisit Wigner's question about the admissible commutation relations for coordinate and velocity operators given their equations of motion (EOM). In more general terms we want to consider the question of how to quantize dynamically equivalent Hamiltonian structures. A unique answer can presumably be given in those cases, where we have a dynamical symmetry. In this case arbitrary deformations of the symmetry algebra should be dynamically equivalent. We illustrate this for the linear as well as the singular $1d$-oscillator. In the case of nonlinear EOM quantum corrections have to be taken into account. We present some examples thereof. New phenomena arise in case of more then one degree of freedom, where sometimes the interaction can be described either by the Hamiltonian or by nonstandard commutation relations. This may induce a noncommutative geometry (for example the $2d$-oscillator in a constant magnetic field). Also some related results from nonrelativistic quantum field theory applied to solid state physics are briefly discussed within this framework.
C. Trapani: "Banach partial *-algebras and quantum models"
As a partial *-algebra, a CQ*-algebra $\A$ has quite a simple structure: the lattice of multipliers consists of four elements only $ \{ \A, R\A, L\A, R\A \cap L\A \}$ where $R\A$ ($L\A$) is the set of the right (left) multipliers of $\A$. On the other hand, the topological structure of a CQ*-algebra is quite rich: $\A$ is a Banach space and $R\A$ ($L\A$) is a C*-algebra.
The relevant point for applications is that CQ*-algebras arise in a natural way by completing a given C*-algebra with respect to a weaker norm. For this reason they appear as the natural framework for the mathematical description of several quantum models, when the usual algebraic approach in terms of C*-algebras fails: this is the case, for instance, of quantum spin systems. As is known, in fact, also for reasonably simple interactions, the thermodynamical limit of the local Heisenberg dynamics does not converge in the usual spin C*-algebra but, often, it converges in the CQ*-algebras context.
A. Uhlmann: "Quantum channels of the Einstein-Podolski-Rosen kind"
J. Wess: "From supersymmetry to quantum groups"
Supersymmetry generalizes the concept of symmetries. This allows us to generalize a Lorentz covariant quantum field theory to a supersymmetric quantum field theory. Quantum groups again generalize the concept of supersymmetries. Field theories can be generalized as well based on this concept.
K. Zalewski: "Bose-Einstein correlations in multiple particle production processes"
Correlations among identical bosons, which are familiar from statistical physics, play an increasingly important role in high energy multiple particle production processes. They provide information about the region, where the particles are produced and, if Einstein's condensation can be reached, they can lead to spectacular new phenomena.
B. Zegarlinski: "Relative entropy estimates in statistical mechanics and field theory"
In this lecture we plan to review numerous applications of relative entropy bounds to the study of large interacting systems.
W. Zimmermann: