The Magnetisation of Large Atoms¶in Strong Magnetic Fields

In this paper we study the asymptotic form of the magnetisation and current of large atoms in strong constant magnetic fields. We prove that the Magnetic Thomas–Fermi theory gives the right magnetisation/current for magnetic field strengths which satisfy B≤Z ^4/3. Received: 24 April 2000 / Accepted: 21 August 2000     

The Birth of the Infinite Cluster:¶Finite-Size Scaling in Percolation

We address the question of finite-size scaling in percolation by studying bond percolation in a finite box of side length n, both in two and in higher dimensions. In dimension d= 2, we obtain a complete characterization of finite-size scaling. In dimensions d>2, we establish the same results under a set of hypotheses related to so-called scaling and hyperscaling postulates which are widely believed to hold up to d= 6. As a function of the size of the box, we determine the scaling window in which the system behaves critically. We characterize criticality in terms of the scaling of the sizes of the largest clusters in the box: incipient infinite clusters which give rise to the infinite cluster. Within the scaling window, we show that the size of the largest cluster behaves like n ^d π _n , where π _n is the probability at criticality that the origin is connected to the boundary of a box of radius n. We also show that, inside the window, there are typically many clusters of scale n ^d π _n , and hence that “the” incipient infinite cluster is not unique. Below the window, we show that the size of the largest cluster scales like ξ ^d π_ξ log(n/ξ), where ξ is the correlation length, and again, there are many clusters of this scale. Above the window, we show that the size of the largest cluster scales like n ^d P _∞, where P _∞ is the infinite cluster density, and that there is only one cluster of this scale. Our results are finite-dimensional analogues of results on the dominant component of the Erdős–Rényi mean-field random graph model. Received: 6 December 2000 / Accepted: 25 May 2001     

The Discrete Spectrum¶of the Perturbed Periodic Schrödinger Operator¶in the Large Coupling Constant Limit

Let A be a periodic Schr?dinger operator and let V ₀≥ 0 be a decaying potential. We study the number of the eigenvalues of the operator A(α) =A−αV ₀ inside a fixed interval (λ₁,λ₂). We obtain an asymptotic formula for as α→∞. Received: 12 September 2000 / Accepted: 22 November 2000     

The Oscillator Representation and¶Groups of Heisenberg Type

We obtain the explicit reduction of the Oscillator representation of the symplectic group, on the subgroups of automorphisms of certain vector-valued skew forms | of "Clifford type"-equivalently, of automorphisms of Lie algebras of Heisenberg type. These subgroups are of the form G · \Spin(k), with G a real reductive matrix group, in general not compact, commuting with Spin(k) with finite intersection. The reduction turns out to be free of multiplicity in all the cases studied here, which include some where the factors do not form a Howe pair. If G is maximal compact in G, the restriction to K · \Spin(k) is essentially the action on the symmetric algebra on a space of spinors. The cases when this is multiplicity-free are listed in [R]; our examples show that replacing K by G does make a difference. Our question is motivated to a large extent by the geometric object that comes with such a |: a Fock-space bundle over a sphere, with G acting fiberwise via the oscillator representation. It carries a Dirac operator invariant under G and determines special derivations of the corresponding gauge algebra.     

The Extended Moduli Space of¶Special Lagrangian Submanifolds

It is well known that the moduli space of all deformations of a compact special Lagrangian submanifold X in a Calabi–Yau manifold Y within the class of special Lagrangian submanifolds is isomorphic to the first de Rham cohomology group of X. Reinterpreting the embedding data X⊂Y within the mathematical framework of the Batalin–Vilkovisky quantization, we find a natural deformation problem which extends the above moduli space to the full de Rham cohomology group of X. Received: 29 June 1998 / Accepted: 7 June 1999     

The Construction of Frobenius Manifolds¶from KP tau-Functions

Frobenius manifolds (solutions of WDVV equations) in canonical coordinates are determined by the system of Darboux–Egoroff equations. This system of partial differential equations appears as a specific subset of the n-component KP hierarchy. KP representation theory and the related Sato infinite Grassmannian are used to construct solutions of this Darboux–Egoroff system and the related Frobenius manifolds. Finally we show that for these solutions Dubrovin's isomonodromy tau-function can be expressed in the KP tau-function. Received: 1 September 1998 / Accepted: 7 March 1999     

The Action of Outer Automorphisms on Bundles¶of Chiral Blocks

On the bundles of WZW chiral blocks over the moduli space of a punctured rational curve we construct isomorphisms that implement the action of outer automorphisms of the underlying affine Lie algebra. These bundle-isomorphisms respect the Knizhnik–Zamolodchikov connection and have finite order. When all primary fields are fixed points, the isomorphisms are endomorphisms; in this case, the bundle of chiral blocks is typically a reducible vector bundle. A conjecture for the trace of such endomorphisms is presented; the proposed relation generalizes the Verlinde formula. Our results have applications to conformal field theories based on non-simply connected groups and to the classification of boundary conditions in such theories. Received: 11 May 1998 / Accepted: 17 April 1999     

Billiards in Tubular Neighborhoods of Manifolds¶of Codimension 1

I prove that in (sufficiently small) tubular ρ$-neighborhoods of a given C ³ manifold of codimension 1, any two points can be connected by a billiard trajectory, and that in addition there exists such a trajectory having at most collision points, for a suitable H>0, provided the manifold is of class C ³. Received: 19 June 1998 / Accepted: 7 April 1999     

Scaling Behaviour of Conserved Sites in Protein Families

Base on the database of families of structurally similar proteins, a statistical study is made on the scaling behaviour of occupying probabilities of conserved sites (Pc) in various protein families. A power-law decrease of Pc with the increasing protein-chain length Lf is found. This is related to the power-law scaling behaviour of the occurring probabilities of local contact interactions (Plocal) between residues. In addition, applying residue grouping, we find the same scaling behaviour when the number of residue types is more than 12, indicating that 12 residue types are enough to present the complexity of proteins.     

The Local Structure of Zero Mode¶Producing Magnetic Potentials

We consider the class of continuous magnetic potentials on ?³ which decay as o(|x|^{? 1}). Within this class it is shown that the set of potentials whose associated Weyl-Dirac operator produces zero modes with multiplicity m forms a smooth submanifold of co-dimension m ² when m= 0, 1, 2, and is contained in a smooth submanifold of co-dimension 2m? 1 when m≥ 3.     

Triviality of Hierarchical Ising Model¶in Four Dimensions

Existence of critical renormalization group trajectory for a hierarchical Ising model in 4 dimensions is shown. After 70 iterations of renormalization group transformations, the critical Ising model is mapped into a vicinity of the Gaussian fixed point. Convergence of the subsequent trajectory to the Gaussian fixed point is shown by power decay of the effective coupling constant. The analysis in the strong coupling regime is computer-aided and Newman's inequalities on truncated correlations are used to give mathematical rigor to the numerical bounds. In order to obtain a criterion for convergence to the Gaussian fixed point, characteristic functions and Newman's inequalities are systematically used. Received: 27 April 2000 / Accepted: 5 January 2001     

In Appreciation¶The Depth and Breadth of John Bell's Physics

This essay surveys the work of John Stewart Bell, one of the great physicists of the twentieth century. Section 1 is a brief biography, tracing his career from working-class origins and undergraduate training in Belfast, Northern Ireland, to research in accelerator and nuclear physics in the British national laboratories at Harwell and Malvern, to his profound research on elementary particle physics as a member of the Theory Group at CERN and his equally profound "hobby" of investigating the foundations of quantum mechanics. Section 2 concerns this hobby, which began in his discontent with Bohr's and Heisenberg's analyses of the measurement process. He was attracted to the program of hidden variables interpretations, but he revolutionized the foundations of quantum mechanics by a powerful negative result: that no hidden variables theory that is "local" (in a clear and well-motivated sense) can agree with all the correlations predicted by quantum mechanics regarding well-separated systems. He further deepened the foundations of quantum mechanics by penetrating conceptual analyses of results concerning measurement theory of von Neumann, de Broglie and Bohm, Gleason, Jauch and Piron, Everett, and Ghirardi-Rimini-Weber. Bell's work in particle theory (Section 3) began with a proof of the CPT theorem in his doctoral dissertation, followed by investigations of the phenomenology of CP-violating experiments. At CERN Bell investigated the commutation relations in current algebras from various standpoints. The failure of current algebra combined with partially conserved current algebra to permit the experimentally observed decay of the neutral pi-meson into two photons stimulated the discovery by Bell and Jackiw of anomalous or quantal symmetry breaking, which has numerous implications for elementary particle phenomena. Other late investigations of Bell on elementary particle physics were bound states in quantum chromodynamics (in collaboration with Bertlmann) and estimates for the anomalous magnetic moment of the muon (in collaboration with de Rafael). Section 4 concerns accelerations, starting at Harwell with the algebra of strong focusing and the stability of orbits in linear accelerators and synchrotrons. At CERN he continued to contribute to accelerator physics, and with his wife Mary Bell he wrote on electron cooling and Beamstrahlung. A spectacular late achievement in accelerator physics was the demonstration (in collaboration with Leinaas) that the effective black-body radiation seen by an accelerated observer in an electromagnetic vacuum - the "Unruh effect" - had already been observed experimentally in the partial depolarization of electrons traversing circular orbits.     

Theory of Ordered Spaces¶II. The Local Differential Structure

In this paper we investigate the conditions under which the ordered spaces defined in [1] are locally diffeomorphic to ℝ ^N . In Sect.~1 we give an introduction and an overview of the results. In Sect. 2 we show that the axioms of [1] do not suffice to make light rays locally homeomorphic to ℝ. We introduce this structure via the new connectedness axiom 2.13, and work out some of its immediate consequences. In Sect. 3 we give the (somewhat involved) construction of timelike curves in a D-set, which are basic to everything that follows. They are used in Sect. 4 to prove (i) a nested interval theorem for ordered spaces; (ii) the contractibility of order intervals in D-sets; and (iii) that order intervals in D-sets are star-shaped. The notion of D-countability (meaning that a D-set has a countable base in the subspace topology) is introduced in Sect. 5. The Urysohn lemma shows that a D-countable ordered space is locally metrizable. If this space is also locally compact, then it has finite topological dimension N; these results are established in Sect. 6. The local differential structure now follows from known results: the embedding of such spaces in ℝ^{2 n +1}, and the result that an open star-shaped region in ℝ ⁿ is diffeomorphic to ℝ ⁿ . In conclusion, we exhibit these inclusions in Fig. 3, and suggest the possibility that Wigner's position on the “Unreasonable effectiveness of mathematics in the natural sciences” may be open to reasonable doubt. The axioms of [1] are given in the Appendix. Received: 26 November 1997 / Accepted: 10 February 1999     

Travelling Waves in a Chain¶of Coupled Nonlinear Oscillators

In a chain of nonlinear oscillators, linearly coupled to their nearest neighbors, all travelling waves of small amplitude are found as solutions of finite dimensional reversible dynamical systems. The coupling constant and the inverse wave speed form the parameter space. The groundstate consists of a one-parameter family of periodic waves. It is realized in a certain parameter region containing all cases of light coupling. Beyond the border of this region the complexity of wave-forms increases via a succession of bifurcations. In this paper we give an appropriate formulation of this problem, prove the basic facts about the reduction to finite dimensions, show the existence of the ground states and discuss the first bifurcation by determining a normal form for the reduced system. Finally we show the existence of nanopterons, which are localized waves with a noncancelling periodic tail at infinity whose amplitude is exponentially small in the bifurcation parameter. Received: 10 September 1999 / Accepted: 15 December 1999     

The Absolute Continuity of the Integrated Density¶of States for Magnetic Schrödinger Operators¶with Certain Unbounded Random Potentials

The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schr?dinger operator with magnetic field and a random potential which may be unbounded from above and below. In case that the magnetic field is constant and the random potential is ergodic and admits a so-called one-parameter decomposition, we prove the absolute continuity of the integrated density of states and provide explicit upper bounds on its derivative, the density of states. This local Lipschitz continuity of the integrated density of states is derived by establishing a Wegner estimate for finite-volume Schr?dinger operators which holds for rather general magnetic fields and different boundary conditions. Examples of random potentials to which the results apply are certain alloy-type and Gaussian random potentials. Besides we show a diamagnetic inequality for Schr?dinger operators with Neumann boundary conditions. Received: 20 October 2000 / Accepted: 8 March 2001     

Singular Dimensions¶of the N= 2 Superconformal Algebras II:¶The Twisted N= 2 Algebra

We introduce a suitable adapted ordering for the twisted N= 2 superconformal algebra (i.e. with mixed boundary conditions for the fermionic fields). We show that the ordering kernels for complete Verma modules have two elements and the ordering kernels for G-closed Verma modules just one. Therefore, spaces of singular vectors may be two-dimensional for complete Verma modules whilst for G-closed Verma modules they can only be one-dimensional. We give all singular vectors for the levels , 1, and for both complete Verma modules and G-closed Verma modules. We also give explicit examples of degenerate cases with two-dimensional singular vector spaces in complete Verma modules. General expressions are conjectured for the relevant terms of all (primitive) singular vectors, i.e. for the coefficients with respect to the ordering kernel. These expressions allow to identify all degenerate cases as well as all G-closed singular vectors. They also lead to the discovery of subsingular vectors for the twisted N= 2 superconformal algebra. Explicit examples of these subsingular vectors are given for the levels , 1, and . Finally, the multiplication rules for singular vector operators are derived using the ordering kernel coefficients. This sets the basis for the analysis of the twisted N= 2 embedding diagrams. Received: Received: 15 March 1999 / Accepted: 12 November 2000     

Modular-Invariance of Trace Functions¶in Orbifold Theory and Generalized Moonshine

The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of the theory of rational orbifold models in conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms. Under a certain finiteness condition on a rational vertex operator algebra V which holds in all known examples, we determine the precise number of g-twisted sectors for any automorphism g of V of finite order. We prove that the trace functions and correlation functions associated with such twisted sectors are holomorphic functions in the upper half-plane and, under suitable conditions, afford a representation of the modular group of the type prescribed in string theory. We establish the rationality of conformal weights and central charge. In addition to conformal field theory itself, where our conclusions are required on physical grounds, there are applications to the generalized Moonshine conjectures of Conway–Norton–Queen and to equivariant elliptic cohomology. Received: 7 January 1999 / Accepted: 14 March 2000     

Multi-Interval Subfactors and Modularity¶of Representations in Conformal Field Theory

We describe the structure of the inclusions of factors ?(E)⊂?(E′)′ associated with multi-intervals E⊂ℝ for a local irreducible net ? of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion associated with two separated intervals is isomorphic to the Longo–Rehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of ?. As a consequence, the index of ?(E)⊂?(E′)′ coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is non-degenerate, namely the representations of ? form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry. Received: 7 July 1999 / Accepted: 13 January 2001     

The Split Property and the Symmetry Breaking¶of the Quantum Spin Chain

We consider the relationship between the symmetry breaking and the split property of pure states of quantum spin chains. We obtain a representation theoretic condition implying that the half-sided uniform mixing condition leads to symmetry breaking of translationally invariant pure states. This is a mathematical generalization of Dichotomy previously found by I. Affleck and E. Lieb and M. Aizenman and B. Nachtergaele for ground states of a special class of Hamiltonians. Received: 1 February 1999 / Accepted: 5 December 2000     

The Modulus of Continuity¶for Γ0(m)\? Semi-Classical Limits

We study the behavior of a large-eigenvalue limit of eigenfunctions for the hyperbolic Laplacian for the modular quotient SL(2;ℤ)\?. Féjer summation and results of S. Zelditch are used to show that the microlocal lifts of eigenfunctions have large-eigenvalue limit a geodesic flow invariant measure for the modular unit cotangent bundle. The limit is studied for Hecke–Maass forms, joint eigenfunctions of the Hecke operators and the hyperbolic Laplacian. The first modulus of continuity result is presented for the limit. The singular concentration set of the limit cannot be a compact union of closed geodesics and measured geodesic laminations. Received: 10 March 2000 / Accepted: 26 July 2000