Showing papers in "Journal of Mathematical Physics in 1994"

The noncommutative geometry of the quantum Hall effect

Abstract: An overview of the integer quantum Hall effect is given. A mathematical framework using nonommutative geometry as defined by Connes is prepared. Within this framework, it is proved that the Hall conductivity is quantized and that plateaux occur when the Fermi energy varies in a region of localized states.

Uncertainty relation in quantum mechanics with quantum group symmetry

Abstract: The commutation relations, uncertainty relations, and spectra of position and momentum operators were studied within the framework of quantum group symmetric Heisenberg algebras and their (Bargmann) Fock representations. As an effect of the underlying noncommutative geometry, a length and a momentum scale appear, leading to the existence of nonzero minimal uncertainties in the positions and momenta. The usual quantum mechanical behavior is recovered as a limiting case for not too small and not too large distances and momenta.

On the self‐linking of knots

Abstract: This note describes a subcomplex F of the de Rham complex of parametrized knot space, which is combinatorial over a number of universal ‘‘Anomaly Integrals.’’ The self‐linking integrals of Guadaguini, Martellini, and Mintchev [‘‘Perturbative aspects of Chern–Simons field theory,’’ Phys. Lett. B 227, 111 (1989)] and Bar‐Natan [‘‘Perturbative aspects of the Chern–Simons topological quantum field theory,’’ Ph.D. thesis, Princeton University, 1991; also ‘‘On the Vassiliev Knot Invariants’’ (to appear in Topology)] are seen to represent the first nontrivial element in H0(F)—occurring at level 4, and are anomaly free. However, already at the next level an anomalous term is possible.

Four‐dimensional topological quantum field theory, Hopf categories, and the canonical bases

Abstract: A new combinatorial method of constructing four‐dimensional topological quantum field theories is proposed. The method uses a new type of algebraic structure called a Hopf category. The construction of a family of Hopf categories related to the quantum groups and their canonical bases is also outlined.

Hamiltonian formulation of the teleparallel description of general relativity

Abstract: The Hamiltonian formulation of the teleparallel description of Einstein’s general relativity is established. Under a particular gauge fixing the Hamiltonian of the theory is written in terms of first class constraints. The algebra of the Hamiltonian and vector constraints resembles that of the standard Arnowitt–Deser–Misner formulation. This geometrical framework might be relevant as it is known that in manifolds with vanishing curvature tensor but with nonzero torsion tensor it is possible to carry out a simple construction of Becchi–Rouet–Stora–Tyutin‐like operators.

Supersymmetric Yang–Mills theory on a four‐manifold

Abstract: By exploiting standard facts about N=1 and N=2 supersymmetric Yang–Mills theory, the Donaldson invariants of four‐manifolds that admit a Kahler metric can be computed. The results are in agreement with available mathematical computations, and provide a powerful check on the standard claims about supersymmetric Yang–Mills theory.

Anomaly cancellation in six dimensions

Abstract: It is shown that anomaly cancellation conditions are sufficient to determine the two most important topological numbers relevant for Calabi–Yau (CY) compactification to six dimensions. This reflects the fact that K3 is the only nontrivial CY manifold in two complex dimensions. The Green–Schwarz counterterms are explicitly constructed and sum rules for charges of additional enhanced U(1) factors are derived and the results with all possible Abelian orbifold constructions of K3 are compared. This includes asymmetric orbifolds as well, showing that it is possible to regain a geometrical interpretation for this class of models. Finally, some models with a broken E7 gauge group which will be useful for more phenomenological applications are discussed herein.

Spectral functions and zeta functions in hyperbolic spaces

Abstract: The spectral function (also known as the Plancherel measure), which gives the spectral distribution of the eigenvalues of the Laplace–Beltrami operator, is calculated for a field of arbitrary integer spin (i.e., for a symmetric traceless and divergence‐free tensor field) on the N‐dimensional real hyperbolic space (HN). In odd dimensions the spectral function μ(λ) is analytic in the complex λ plane, while in even dimensions it is a meromorphic function with simple poles on the imaginary axis, as in the scalar case. For N even a simple relation between the residues of μ(λ) at these poles and the (discrete) degeneracies of the Laplacian on the N sphere (SN) is established. A similar relation between μ(λ) at discrete imaginary values of λ and the degeneracies on SN is found to hold for N odd. These relations are generalizations of known results for the scalar field. The zeta functions for fields of integer spin on HN are written down. Then a relation between the integer‐spin zeta functions on HN and SN is obtained. Applications of the zeta functions presented here to quantum field theory of integer spin in anti‐de Sitter space–time are pointed out.

R‐matrix approach to lattice integrable systems

Abstract: An r‐matrix formalism is applied to the construction of the integrable lattice systems and their bi‐Hamiltonian structure. Miura‐like gauge transformations between the hierarchies are also investigated. In the end the ladder of linear maps between generated hierarchies is established and described.

Quantum logic and the histories approach to quantum theory

Abstract: An extensive analysis is made of the Gell‐Mann and Hartle axioms for a generalized ‘histories’ approach to quantum theory. Emphasis is placed on finding analogs of the lattice structure employed in standard quantum logic. Particular attention is given to ‘quasitemporal’ theories in which the notion of time‐evolution in conventional Hamiltonian physics is replaced by something that is much broader; theories of this type are expected to arise naturally in the context of quantum gravity and quantum field theory in a curved space–time. The quasitemporal structure is coded in a partial semigroup of ‘temporal supports’ that underpins the lattice of history propositions. Nontrivial examples include quantum field theory on a non‐globally‐hyperbolic space–time, and a possible cobordism approach to a theory of quantum topology. A key result is the demonstration that the set of history propositions in standard quantum theory can be realized in such a way that each such proposition is represented by a genuine projection operator. This gives valuable insight into the possible lattice structure in general history theories.

Quantum temporal logic and decoherence functionals in the histories approach to generalized quantum theory

Abstract: The recent suggestion that a temporal form of quantum logic provides the natural mathematical framework within which to discuss the proposal by Gell‐Mann and Hartle for a generalized form of quantum theory based on the ideas of histories and decoherence functionals is analyzed and developed herein. Particular stress is placed on properties of the space of decoherence functionals, including one way in which certain global and topological properties of a classical system are reflected in a quantum history theory.

η‐invariants and determinant lines

Abstract: The η‐invariant of an odd dimensional manifold with boundary is investigated. The natural boundary condition for this problem requires a trivialization of the kernel of the Dirac operator on the boundary. The dependence of the η‐invariant on this trivialization is best encoded by the statement that the exponential of the η‐invariant lives in the determinant line of the boundary. Our main results are a variational formula and a gluing law for this invariant. These results are applied to reprove the formula for the holonomy of the natural connection on the determinant line bundle of a family of Dirac operators, also known as the ‘‘global anomaly formula.’’ The ideas developed here fit naturally with recent work in topological quantum field theory, in which gluing (which is a characteristic formal property of the path integral and the classical action) is used to compute global invariants on closed manifolds from local invariants on manifolds with boundary.

Integrability of difference Calogero-Moser systems

Abstract: A general class of n‐particle difference Calogero–Moser systems with elliptic potentials is introduced. Besides the step size and two periods, the Hamiltonian depends on nine coupling constants. We prove the quantum integrability of the model for n=2 and present partial results for n≥ 3. In degenerate cases (rational, hyperbolic, or trigonometric limit), the integrability follows for arbitrary particle number from previous work connected with the multivariable q‐polynomials of Koornwinder and Macdonald. Liouville integrability of the corresponding classical systems follows as a corollary. Limit transitions lead to various well‐known models such as the nonrelativistic Calogero–Moser systems associated with classical root systems and the relativistic Calogero–Moser system.

On Schrödinger superalgebras

Abstract: Using the supersymplectic framework of Berezin, Kostant, and others, two types of supersymmetric extensions of the Schrodinger algebra (itself a conformal extension of the Galilei algebra) were constructed. An ‘I‐type’ extension exists in any space dimension, and for any pair of integers N+ and N−. It yields an N=N++N− superalgebra, which generalizes the N=1 supersymmetry Gauntlett et al. found for a free spin‐1/2 particle, as well as the N=2 supersymmetry of the fermionic oscillator found by Beckers et al. In two space dimensions, new, ‘exotic’ or ‘IJ‐type’ extensions arise for each pair of integers ν+ and ν−, yielding an N=2(ν++ν−) superalgebra of the type discovered recently by Leblanc et al. in nonrelativistic Chern–Simons theory. For the magnetic monopole the symmetry reduces to o(3)×osp(1/1), and for the magnetic vortex it reduces to o(2)×osp(1/2).

Geometry of canonical correlation on the state space of a quantum system

Abstract: A Riemannian metric is defined on the state space of a finite quantum system by the canonical correlation (or Kubo–Mori/Bogoliubov scalar product). This metric is infinitesimally induced by the (nonsymmetric) relative entropy functional or the von Neumann entropy of density matrices. Hence its geometry expresses maximal uncertainty. It is proven that the metric is monotone under stochastic mappings, however, an example shows that it is not the only such Riemannian metric. This fact is remarkable because in the probabilistic case, the Markovian monotonicity property characterizes the Fisher information metric. The essential difference appears in the curvatures of a classical state space and a quantum one. A conjecture is made that the scalar curvature is monotone with respect to the ‘‘more mixed’’ (statistical) partial order of density matrices. Furthermore, an information inequality resembling the Cramer–Rao inequality of classical statistics is established. The inequality provides a lower bound for the canonical correlation matrix (of an unbiased observable) and it is saturated when a (partial) observation level and the corresponding family of coarse‐grained states are considered.

Singularity analysis and localized coherent structures in (2+1)‐dimensional generalized Korteweg–de Vries equations

Abstract: In this article, a singularity structure analysis of a (2+1)‐dimensional generalized Korteweg–de Vries equation studied originally by Boiti et al., admitting a weak Lax pair, is carried out and it is proven that the system satisfies the Painleve property. Its bilinear form is constructed in a natural way from the P analysis and then it is used to generate ‘‘multidromion’’ solutions (exponentially decaying solutions in all directions). The same analysis can be extended to construct the multidromion solutions of the generalized Nizhnik–Novikov–Veselov (NNV) equation from which the NNV equation follows as a special case.

A Wigner-function approach to (semi)classical limits: Electrons in a periodic potential

Abstract: A rigorous derivation of the semiclassical Liouville equation for electrons which move in a crystal lattice (without the influence of an external field) is presented herein. The approach is based on carrying out the semiclassical limit in the band‐structure Wigner equation. The semiclassical macroscopic densities are also obtained as limits of the corresponding quantum quantities.

Algebraic q‐integration and Fourier theory on quantum and braided spaces

Abstract: An algebraic theory of integration on quantum planes and other braided spaces is introduced. In the one‐dimensional case a novel picture of the Jackson q‐ integral as indefinite integration on the braided group of functions in one variable x is obtained. Here x is treated with braid statistics q rather than the usual bosonic or Grassmann ones. It is shown that the definite integral ∫x∞−x∞ can also be evaluated algebraically as multiples of the integral of a q‐Gaussian, with x remaining as a bosonic scaling variable associated with the q‐deformation. Further composing the algebraic integration with a representation then leads to ordinary numbers for the integral. Integration is also used to develop a full theory of q‐Fourier transformation F The braided addition Δx=x⊗1+1⊗x and braided‐antipode S is used to define a convolution product, and prove a convolution theorem. It is also proven that F2=S. The analogous results are proven on any braided group, including integration and Fourier transformation on quantum planes associated to general R matrices, including q‐Euclidean and q‐Minkowski spaces.

Generalized intelligent states and squeezing

Abstract: The Robertson–Schrodinger uncertainty relation for two observables A and B is shown to be minimized in the eigenstates of the operator λA+iB, λ being a complex number. Such states, called generalized intelligent states (GIS), can exhibit arbitrarily strong squeezing of A or B. The time evolution of GIS is stable for Hamiltonians which admit linear in A and B invariants. Systems of GIS for the SU(1,1) and SU(2) groups are constructed and discussed. It is shown that SU(1,1) GIS contain all the Perelomov coherent states (CS) and the Barut and Girardello CS while the spin CS are a subset of SU(2) GIS. CS for an arbitrary semisimple Lie group can be considered as a GIS for the quadratures of the Weyl generators.

An algebraic extension of Dirac quantization: Examples

Abstract: An extension of the Dirac procedure for the quantization of constrained systems is necessary to address certain issues that are left open in Dirac’s original proposal These issues play an important role especially in the context of nonlinear, diffeomorphism invariant theories such as general relativity Recently, an extension of the required type was proposed using algebraic quantization methods In this paper, the key conceptual and technical aspects of the algebraic program are illustrated through a number of finite dimensional examples The choice of examples and some of the analysis is motivated by certain peculiar problems endemic to quantum gravity However, prior knowledge of general relativity is not assumed in the main discussion Indeed, the methods introduced and conclusions arrived at are applicable to any system with first class constraints In particular, they resolve certain technical issues which are present also in the reduced phase space approach to quantization of these systems

Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli

Abstract: Linear processes in media with dissipation arising in conductivity, optics, viscoelasticity, etc. are considered. Time‐periodic fields in such media are described by linear differential equations for complex‐valued potentials. The properties of the media are characterized by complex valued tensors, for example, by complex conductivity or complex elasticity tensors. Variational formulations are suggested for such problems: The functionals whose Euler equations coincide with the original ones are constructed. Four equivalent variational principles are obtained: two minimax and two minimal ones. The functionals of the obtained minimal variational principles are proportional to the energy dissipation averaged over the period of oscillation. The last principles can be used in the homogenization theory to obtain the bounds on the effective properties of composite materials with complex valued properties tensors.

A renormalization method for the evaluation of lattice sums

Abstract: A number of problems in solid state physics and materials science can be resolved by the evaluation of certain lattice sums (sums over all sites of an infinite perfect lattice of some potential energy function). One classical example, the calculation of lattice sums of circular and spherical harmonics, dates back to the last century, to Lord Rayleigh’s work on computing the effective conductivity of a simple composite. While Lord Rayleigh presented an efficient asymptotic method for two‐dimensional problems, he resorted to direct evaluation of the lattice sums in the three‐dimensional case. More recent methods, based on Ewald summation, have been developed by Nijboer and De Wette, Schmidt and Lee, and others. In this article, a fast method for evaluating lattice sums which is based on a new renormalization identity is described.

Discrete differential calculus graphs, topologies and gauge theory

Abstract: Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a ‘‘reduction’’ of the ‘‘universal differential algebra’’ and this allows a systematic exploration of differential algebras on a given set. Associated with a differential algebra is a (di)graph where two vertices are connected by at most two (antiparallel) arrows. The interpretation of such a graph as a ‘‘Hasse diagram’’ determining a (locally finite) topology then establishes contact with recent work by other authors in which discretizations of topological spaces and corresponding field theories were considered which retain their global topological structure. It is shown that field theories, and in particular gauge theories, can be formulated on a discrete set in close analogy with the continuum case. The framework presented generalizes ordinary lattice theory which is recovered from an oriented (hypercubic) lattice graph. It also includes, e.g., the two‐point space used by Connes and Lott (and others) in models of elementary particle physics. The formalism suggests that the latter be regarded as an approximation of a manifold and thus opens a way to relate models with an ‘‘internal’’ discrete space (a la Connes et al.) to models of dimensionally reduced gauge fields. Furthermore, a ‘‘symmetric lattice’’ is also studied which (in a certain continuum limit) turns out to be related to a ‘‘noncommutative differential calculus’’ on manifolds.

Vertex operators in solvable lattice models

Abstract: The basic properties of q‐vertex operators are formulated in the context of the Andrews–Baxter–Forrester (ABF) series, as an example of face interaction models, the q‐difference equations satisfied by their correlation functions are derived, and their connection with representation theory established. The q‐difference equations of the Kashiwara–Miwa (KM) series are discussed as an example of edge interaction models. Next, the Ising model, the simplest special case of both ABF and KM series, is studied in more detail using the Jordan–Wigner fermions. In particular, all matrix elements of vertex operators are calculated.

Exact results and universal asymptotics in the Laguerre random matrix ensemble

Abstract: For various values of the parameters in the Laguerre random matrix ensemble, the distribution of the smallest eigenvalue and the scaled n‐level distribution function are calculated exactly in terms of generalized hypergeometric functions. In certain cases these functions are expressed as multidimensional integrals, from which asymptotic formulas are calculated and predictions of universal behavior verified.

The Classification of decoherence functionals: An Analog of Gleason's theorem

Abstract: Gell‐Mann and Hartle have proposed a significant generalization of quantum theory with a scheme whose basic ingredients are ‘‘histories’’ and decoherence functionals. Within this scheme it is natural to identify the space UP of propositions about histories with an orthoalgebra or lattice. This raises the important problem of classifying the decoherence functionals in the case where UP is the lattice of projectors P(V) in some Hilbert space V; in effect, one seeks the history analog of Gleason’s famous theorem in standard quantum theory. In the present article the solution to this problem for the case where V is finite dimensional is presented. In particular, it is shown that every decoherence functional d(α,β), α,β∈P(V) can be written in the form d(α,β)=trV⊗V(α⊗βX) for some operator X on the tensor‐product space V⊗V.

Solution of the central field problem for a Duffin–Kemmer–Petiau vector boson

Abstract: In view of current interest in the use of the Duffin–Kemmer–Petiau (DKP) relativistic equation, the problem of the vector DKP boson in a central field is resolved and the system of first‐order coupled differential radial equations needed for an exact calculation of the eigenvalues as well as the full ten‐component spinor is derived. This is of practical importance for problems involving massive vector bosons in central fields. This formalism is applied to the free‐particle, spherically symmetric square well and Coulomb problems.

On the complete symmetry group of the classical Kepler system

Abstract: A rather strong concept of symmetry is introduced in classical mechanics, in the sense that some mechanical systems can be completely characterized by the symmetry laws they obey Accordingly, a ‘‘complete symmetry group’’ realization in mechanics must be endowed with the following two features: (1) the group acts freely and transitively on the manifold of all allowed motions of the system; (2) the given equations of motion are the only ordinary differential equations that remain invariant under the specified action of the group This program is applied successfully to the classical Kepler problem, since the complete symmetry group for this particular system is here obtained The importance of this result for the quantum kinematic theory of the Kepler system is emphasized

Quantum chaos: An entropy approach

Abstract: A new definition of the entropy of a given dynamical system and of an instrument describing the measurement process is proposed within the operational approach to quantum mechanics. It generalizes other definitions of entropy, in both the classical and quantum cases. The Kolmogorov–Sinai (KS) entropy is obtained for a classical system and the sharp measurement instrument. For a quantum system and a coherent states instrument, a new quantity, coherent states entropy, is defined. It may be used to measure chaos in quantum mechanics. The following correspondence principle is proved: the upper limit of the coherent states entropy of a quantum map as ℏ→0 is less than or equal to the KS‐entropy of the corresponding classical map. ‘‘Chaos umpire sits, And by decision more imbroils the fray By which he reigns: next him high arbiter Chance governs all.’’ John Milton, Paradise Lost, Book II

Completeness of the eigenfunctions for the Caudrey–Beals–Coifman system

Abstract: The direct and the inverse scattering problem for the first order linear systems of the type Lψ(x,λ)≡(i(d/dx)+q(x)−λJ)ψ(x,λ)=0, J∈h, q(x)∈gJ , which generalizes the Zakharov–Shabat system and the systems studied by Caudrey, Beals, and Coifman (CBC) is analyzed herein. Here J is a regular complex constant element of the Cartan subalgebra h⊆g of the simple Lie algebra g and the potential q(x) vanishes fast enough for ‖x‖ → ∞ taking values in the image gJ of adJ. The CBC results are generalized and the fundamental analytic solution m(x,λ) for any choice of the irreducible finite‐dimensional representation V of g is constructed. Four pairwise equivalent minimal sets of scattering data for L, invariant with respect to the choice of the representation of g, are extracted from the asymptotics of m(x,λ) for x → ±∞. From m(x,λ) the resolvent of L is constructed in the adjoint representation Vad and the completeness relation is proven for the eigenfunctions of L in Vad. It is also proven that the discrete spectrum ...