Showing papers in "Reviews in Mathematical Physics in 1997"

Symmetry Groups in Quantum Mechanics and the Theorem of Wigner on the Symmetry Transformations

Abstract: Various mathematical formulations of the symmetry group in quantum mechanics are investigated and shown to be mutually equivalent. A new proof of the theorem of Wigner on the symmetry transformations is worked out.

Continuity of Symplectically Adjoint Maps and the Algebraic Structure of Hadamard Vacuum Representations for Quantum Fields on Curved Spacetime

Abstract: We derive for a pair of operators on a symplectic space which are adjoints of each other with respect to the symplectic form (that is, they are sympletically adjoint) that, if they are bounded for some scalar product on the symplectic space dominating the symplectic form, then they are bounded with respect to a one-parametric family of scalar products canonically associated with the initially given one, among them being its "purification". As a typical example we consider a scalar field on a globally hyperbolic spacetime governed by the Klein–Gordon equation; the classical system is described by a symplectic space and the temporal evolution by symplectomorphisms (which are symplectically adjoint to their inverses). A natural scalar product is that inducing the classical energy norm, and an application of the above result yields that its "purification" induces on the one-particle space of the quantized system a topology which coincides with that given by the two-point functions of quasifree Hadamard states. These findings will be shown to lead to new results concerning the structure of the local (von Neumann) observable-algebras in representations of quasifree Hadamard states of the Klein–Gordon field in an arbitrary globally hyperbolic spacetime, such as local definiteness, local primarity and Haag-duality (and also split- and type III1-properties). A brief review of this circle of notions, as well as of properties of Hadamard states, forms part of the article.

Geometry of Quantum Principal Bundles II

Abstract: A general non-commutative-geometric theory of principal bundles is developed. Quantum groups play the role of structure groups and general quantum spaces play the role of base manifolds. A general conceptual framework for the study of differential structures on quantum principal bundles is presented. Algebras of horizontal, verticalized and "horizontally vertically" decomposed differential forms on the bundle are introduced and investigated. Constructive approaches to differential calculi on quantum principal bundles are discussed. The formalism of connections is developed further. The corresponding operators of horizontal projection, covariant derivative and curvature are constructed and analyzed. In particular the analogs of the basic classical algebraic identities are derived. A quantum generalization of classical Weil's theory of characteristic classes is sketched. Quantum analogs of infinitesimal gauge transformations are studied. Interesting examples are presented.

Functional Integral Representation of a Model in Quantum Electrodynamics

Abstract: This paper presents functional integral representations for heat semigroups with infinitesimal generators given by self-adjoint Hamiltonians (Pauli–Fierz Hamiltonians) describing an interaction of a non-relativistic charged particle and a quantized radiation field in the Coulomb gauge without the dipole approximation. By the functional integral representations, some inequalities are derived, which are infinite degree versions of those known for finite dimensional Schrodinger operators with classical vector potentials.

Low Energy Scattering for Nonlinear Schrödinger Equations in Fractional Order Sobolev Spaces

Abstract: We consider the scattering problem for the nonlinear Schrodinger equations with interactions behaving as a power p at zero. In the critical and subcritical cases (s≥n/2-2/(p-1)≥0), we prove the existence and asymptotic completeness of wave operators in the sense of Sobolev norm of order s on a set of asymptotic states with small homogeneous norm of order n/2-2/(p-1) in space dimension n≥1.

An Explicit Large Versus Small Field Multiscale Cluster Expansion

Abstract: We introduce a new type of cluster expansion which generalizes a previous formula of Brydges and Kennedy. The method is especially suited for performing a phase-space multiscale expansion in a just renormalizable theory, and allows the writing of explicit non-perturbative formulas for the Schwinger functions. The procedure is quite model independent, but for simplicity we chose the infrared $\ph^4_4$ model as a testing ground. We used also a large field versus small field expansion. The polymer amplitudes, corresponding to graphs without almost local two and for point functions, are shown to satisfy the polymer bound.

Continuous Quantum Measurement:. Local and Global Approaches

Abstract: In 1979 B. Menski suggested a formula for the linear propagator of a quantum system with continuously observed position in terms of a heuristic Feynman path integral. In 1989 the aposterior linear stochastic Schrodinger equation was derived by V. P. Belavkin describing the evolution of a quantum system under continuous (nondemolition) measurement. In the present paper, these two approaches to the description of continuous quantum measurement are brought together from the point of view of physics as well as mathematics. A self-contained deductions of both Menski's formula and the Belavkin equation is given, and the new insights in the problem provided by the local (stochastic equation) approach to the problem are described. Furthermore, a mathematically well-defined representations of the solution of the aposterior Schrodinger equation in terms of the path integral is constructed and shown to be heuristically equivalent to the Menski propagator.

Hilbert C*-Bimodules Over Commutative C*-Algebras and an Isomorphism Condition for Quantum Heisenberg Manifolds

Abstract: A study of Hilbert C*-bimodules over commutative C*-algebras is carried out and used to establish a sufficient condition for two quantum Heisenberg manifolds to be isomorphic.

Poisson spaces with a transition probability

Abstract: The common structure of the space of pure states ℘ of a classical or a quantum mechanical system is that of a Poisson space with a transition probability. This is a topological space equipped with a Poisson structure, as well as with a function p:℘×℘→[0,1], with certain properties. The Poisson structure is connected with the transition probabilities through unitarity (in a specific formulation intrinsic to the given context). In classical mechanics, where p(ρ,σ)=δρσ, unitarity poses no restriction on the Poisson structure. Quantum mechanics is characterized by a specific (complex Hilbert space) form of p, and by the property that the irreducible components of ℘ as a transition probability space coincide with the symplectic leaves of ℘ as a Poisson space. In conjunction, these stipulations determine the Poisson structure of quantum mechanics up to a multiplicative constant (identified with Planck's constant). Motivated by E. M. Alfsen, H. Hanche-Olsen and F. W. Shultz (Acta Math.144 (1980) 267–305) and F.W. Shultz (Commun. Math. Phys.82 (1982) 497–509), we give axioms guaranteeing that ℘ is the space of pure states of a unital C*-algebra. We give an explicit construction of this algebra from ℘.

Superselection Structures for C*-Algebras with Nontrivial Center

Abstract: We present and prove some results within the framework of Hilbert C*-systems with a compact group . We assume that the fixed point algebra of has a nontrivial center and its relative commutant w.r.t. ℱ coincides with , i.e. we have . In this context we propose a generalization of the notion of an irreducible endomorphism and study the behaviour of such irreducibles w.r.t. . Finally, we give several characterizations of the stabilizer of .

Quantizations of Flag Manifolds and Conformal Space Time

Abstract: In this paper we work out the deformations of some flag manifolds and of complex Minkowski space viewed as an affine big cell inside G(2,4). All the deformations come in tandem with a coaction of the appropriate quantum group. In the case of the Minkowski space this allows us to define the quantum conformal group. We also give two involutions on the quantum complex Minkowski space, that respectively define the real Minkowski space and the real euclidean space. We also compute the quantum De Rham complex for both real (complex) Minkowski and euclidean space.

The connes lott program on the sphere

Abstract: We describe a classical Schwinger-type model as a study of the projective modules over the algebra of complex-valued functions on the sphere. On these modules, classified by π2(S2), we construct hermitian connections with values in the universal differential envelope. Instead of describing matter by the usual Dirac spinors yielding the standard Schwinger model on the sphere, we apply the Connes–Lott program to the Hilbert space of complexified inhomogeneous forms with its Atiyah–Kahler structure. This Hilbert space splits in two minimal left ideals of the Clifford algebra preserved by the Dirac–Kahler operator D=i(d-δ). The induced representation of the universal differential envelope, in order to recover its differential structure, is divided by the unwanted differential ideal and the obtained quotient is the usual complexified de Rham exterior algebra with Clifford action on the "spinors" of the Hilbert space. The subsequent steps of the Connes–Lott program allow to define a matter action, and the field action is obtained using the Dixmier trace which reduces to the integral of the curvature squared.

The Moduli Space of Yang–Mills Connections Over a Compact Surface

Abstract: Yang–Mills connections over closed oriented surfaces of genus ≥1, for compact connected gauge groups, are constructed explicitly. The resulting formulas for Yang–Mills connections are used to carry out a Marsden–Weinstein type procedure. An explicit formula is obtained for the resulting 2-form on the moduli space. It is shown that this 2-form provides a symplectic structure on appropriate subsets of the moduli space.

Exponential Localization for Multi-Dimensional Schrödinger Operator with Random Point Potential

Abstract: We consider a Schrodinger operator -Δα(ω) on L2(ℝd)(d=2,3) whose potential is a sum of point potentials, centered at sites of ℤd, with independent and identically distributed random amplitudes. We prove the existence of the pure point spectrum and the exponential decay of the corresponding eigenfunctions at the negative semi-axis for certain regimes of the disorder. In order to prove localization results, we elucidate the structure of the generalized eigenfunctions of -Δα(ω) and the relation between its negative spectrum and the spectra of a family of infinite-order operators on l2(ℤd). We apply the multiscale analysis scheme to investigate the point spectrum of these operators. We also prove the absolute continuity of the integrated density of states of the operator on the negative part of its spectrum.

Some mathematical problems related to classical-quantum interactions

Abstract: A mathematical model describing the interaction between classical and quantum systems is proposed. The discrete case of a counter as well as the continuous case of the SQUID-tank model are discussed.

Z-Graded Extensions of Poisson Brackets

Abstract: A Z-graded Lie bracket { , }P on the exterior algebra Ω(M) of differential forms, which is an extension of the Poisson bracket of functions on a Poisson manifold (M,P), is found. This bracket is simultaneously graded skew-symmetric and satisfies the graded Jacobi identity. It is a kind of an 'integral' of the Koszul–Schouten bracket [ , ]P of differential forms in the sense that the exterior derivative is a bracket homomorphism: [dμ, dν]P=d{μ, ν}P. A naturally defined generalized Hamiltonian map is proved to be a homomorphism between { , }P and the Frolicher–Nijenhuis bracket of vector valued forms. Also relations of this graded Poisson bracket to the Schouten–Nijenhuis bracket and an extension of { , }P to a graded bracket on certain multivector fields, being an 'integral' of the Schouten–Nijenhuis bracket, are studied. All these constructions are generalized to tensor fields associated with an arbitrary Lie algebroid.

Supergeometry and Quantum Field Theory, or: What is a Classical Configuration?

Abstract: We discuss the conceptual difficulties connected with the anticommutativity of classical fermion fields, and we argue that the "space" of all classical configurations of a model with such fields should be described as an infinite-dimensional supermanifold M. We discuss the two main approaches to supermanifolds, and we examine the reasons why many physicists tend to prefer the Rogers approach although the Berezin–Kostant–Leites approach is the more fundamental one. We develop the infinite-dimensional variant of the latter, and we show that the superfunctionals considered in [44] are nothing but superfunctions on M. We propose a programme for future mathematical work, which applies to any classical field model with fermion fields. A part of this programme will be implemented in the successor paper [45].

The born oppenheimer approximation: straight-up and with a twist

Abstract: The problem of calculating asymptotic series for low-lying eigennvalues of Schrodinger operators is solved for two classes of such operators. For both models, a version of the Born–Oppenheimer Approximation is proven. The first model considered is the family in L2(ℝ,ℋ) where H(x):ℋ→ℋ has a simple eigenvalue less than zero. The second model considered is a more specific family ℍe=-e4Δ+H(r,ω) in where the eigenprojection P(ω) of is associated with a non-trivial, or "twisted," fibre bundle. The main tools are a pair of theorems that allow asymptotic series for eigenvalues to be corrected term by term when a family of operators is perturbed.

A Scaling Limit of a Hamiltonian of Many Nonrelativistic Particles Interacting with a Quantized Radiation Field

Abstract: This paper presents a scaling limit of Hamiltonians which describe interactions of N-nonrelativistic charged particles in a scalar potential and a quantized radiation field in the Coulomb gauge with the dipole approximation. The scaling limit defines effective potentials. In one-nonrelativistic particle case, the effective potentials have been known to be Gaussian transformations of the scalar potential [J. Math. Phys.34 (1993) 4478–4518]. However it is shown that the effective potentials in the case of N-nonrelativistic particles are not necessary to be Gaussian transformations of the scalar potential.

Energy dependent boundary conditions and few-body scattering problem

Abstract: An exactly solvable problem with energy dependent interaction is investigated in the present paper. The selfadjoint model operator describes the scattering problem for three one-dimensional particles. It is shown that this problem is equivalent to the diffraction problem in the sector with energy dependent boundary conditions. The problem is solved with the help of the Sommerfeld-Maluzhinetz representation, which transforms the partial differential equation for the eigenfunctions to a functional equation on the integral densities. The solution of the functional equation can be constructed explicitly in the case of identical particles. The three-body scattering matrix describing rearrangement and excitation processes is represented in terms of analytic functions. (Less)

Attractor for Dissipative Zakharov Equations in an Unbounded Domain

Abstract: In this paper the authors consider the Cauchy problem of dissipative Zakharov equations in R and prove the existence of the maximal attractor.

A Modified approach to the Doplicher-Roberts theorem on the construction of the field algebra and the symmetry group in superselection theory

Abstract: Let be a unital C*-algebra with trivial center . Let denote a tensorial category of unital endomorphisms of equipped with several properties to be explained in the text. Doplicher and Roberts have shown, among other things, that there is a C*-algebra and a compact group of automorphisms of ℱ such that ℱ is a Hilbert C*-system over w.r.t. , where is the fixed point algebra w.r.t. , and the objects are characterized as the canonical endomorphisms of certain algebraic -invariant Hilbert spaces ℋρ⊂ℱ, see Doplicher/Roberts [1, 2, 3]. The starting point of the approach presented in this paper to point out the mentioned result is an -leftmodule ℱ0:={∑ρ,jAρ,jΦρ,j}. ρ runs through a full system of irreducible and mutually disjoint objects of , j=1,2,…,d(ρ), where d(ρ) denotes the statistical dimension of is an orthonormal basis of a d(ρ)-dimensional Hilbert space. The system {Φρj}ρj forms a leftmodule basis of ℱ0, the coefficients Aρj are members of . The strategy is to equip successively ℱ0 with a bimodule structure, a product and a *-structure and finally with a C*-norm ||.||*. The symmetry group appears as the group of all automorphisms of the *-algebra ℱ0 leaving the -scalar product invariant, where F=∑ρ,jAρjΦρj, G=∑ρ,jBρjΦρj. The field algebra is then given by ℱ:= clo||.||*ℱ0.

On the Right Hamiltonian for Singular Perturbations: General Theory

Abstract: Let the pair of self-adjoint operators {A≥0,W≤0} be such that: (a) there is a dense domain such that is semibounded from below (stability domain), (b) the symmetric operator is not essentially self-adjoint (singularity of the perturbation), (c) the Friedrichs extension of is maximal with respect to W, i.e., . . Let be a regularizing sequence of bounded operators which tends in the strong resolvent sense to W. The abstract problem of the right Hamiltonian is: (i) to give conditions such that the limit H of self-adjoint regularized Hamiltonians exists and is unique for any self-adjoint extension of , (ii) to describe the limit H. We show that under the conditions (a)–(c) there is a regularizing sequence such that tends in the strong resolvent sense to unique (right Hamiltonian) , otherwise the limit is not unique.

On the Structure of Monodromy Algebras and Drinfeld Doubles

Abstract: We give a review and some new relations on the structure of the monodromy algebra (also called loop algebra) associated with a quasitriangular Hopf algebra H. It is shown that as an algebra it coincides with the so-called braided group constructed by S. Majid on the dual of H. Gauge transformations act on monodromy algebras via the coadjoint action. Applying a result of Majid, the resulting crossed product is isomorphic to the Drinfeld double . Hence, under the so-called factorizability condition given by N. Reshetikhin and M. Semenov–Tian–Shansky, both algebras are isomorphic to the algebraic tensor product H ⊗ H. It is indicated that in this way the results of Alekseev et al. on lattice current algebras are consistent with the theory of more general Hopf spin chains given by K. Szlachanyi and the author. In the Appendix the multi-loop algebras ℒm of Alekseev and Schomerus [3] are identified with braided tensor products of monodromy algebras in the sense of Majid, which leads to an explanation of the "bosonization formula" of [3] representing ℒm as H ⊗…⊗ H.

On Time Development of a Quasi-Quantum Particle in Quartic Potential (x2-a2)2/2g

Abstract: In this article, I have precisely considered the time development of a quantum particle (of excited states) in the quartic potential (x2-a2)2/2g by means of the semiclassical path integral method. Using the elliptic functions, I have evaluated the tunneling phenomena and the quasi-quantum fluctuation around the quasi-classical paths. I found that the quasi-quantum fluctuation is expressed by the Lame equation and was exactly solved. Then I have shown that the obtained kernel function is in agreement with exact solutions of the linear potential and the quadratic potential under certain limits as no time-development kernel function of the quartic potential has ever been found which contains the exact solution of the linear and the quadratic potential. It is natural because the classical motion in the quartic potential becomes those of the linear and the quadratic potential under the limits. Thus the obtained time-development kernel function also consists of the energy representation of the Green function of the quartic potential in the semiclassical path integral method given by Carlitz and Nicole (Ann. Phys.164 (1985) 411), which agrees with that of the WKB method in the operator formalism.

Entropy Density of One-Dimensional Quantum Lattice Systems

Abstract: We prove that the entropy density of a KMS state of one-dimensional quantum lattice systems is equal to the thermodynamical limit of the entropy of local Gibbs states.

Geometric Realizations of Representations of Finite Length

Abstract: Let G=H×ℝn be a semidirect product Lie group. We reduce the problem of deciding which indecomposable representations of G may be realized in subquotients of spaces of sections of vector bundles over infinitesimal neighborhoods of orbits of H in the dual of ℝn to a problem involving only representations of the H-stabilizers of the orbits.

Residues and Topological Yang–Mills Theory in Two Dimensions

Abstract: A residue formula which evaluates any correlation function of topological SUn Yang–Mills theory with arbitrary magnetic flux insertion in two-dimensions are obtained. Deformations of the system by two-form operators are investigated in some detail. The method of the diagonalization of a matrix-valued field turns out to be useful to compute various physical quantities. As an application we find the operator that contracts a handle of a Riemann surface and a genus recursion relation.

Quasivectors and Tomita–Takesaki Theory for Operator Algebras on Π1-Spaces

Abstract: We consider operator algebras, which are symmetric with respect to an indefinite scalar product. It is shown, that in the case when the rank of indefiniteness is equal to 1 there exists a working modular theory, and in particular a precise analogue of the Fundamental Tomita's Theorem holds: For any weakly closed J-symmetric operator algebra with identity on a Π1-space H which has a cyclic and separating vector, there is an antilinear J-involution j : H→H such that . The paper also contains a full proof of the Double Commutant Theorem for J-symmetric operator algebras on Π1-spaces.