Volume 51, Issue 1, January 2010
 SPECIAL ISSUE: FIFTIETH ANNIVERSARY SPECIAL ISSUE


Introduction to Special Issue: Journal of Mathematical Physics turns 50
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Space and time from translation symmetry
View Description Hide DescriptionWe show that the notions of space and time in algebraic quantum field theory arise from translation symmetry if we assume asymptotic commutativity. We argue that this construction can be applied to string theory.

Nonequilibrium, thermostats, and thermodynamic limit
View Description Hide DescriptionThe relation between thermostats of “isoenergetic” and “frictionless” kind is studied and their equivalence in the thermodynamic limit is proven in space dimension and for special geometries, .

Time asymptotics and entanglement generation of Clifford quantum cellular automata
View Description Hide DescriptionWe consider Clifford quantum cellular automata (CQCAs) and their timeevolution. CQCAs are an especially simple type of quantum cellular automata, yet they show complex asymptotics and can even be a basic ingredient for universal quantum computation. In this work we study the time evolution of different classes of CQCAs. We distinguish between periodic CQCAs, fractal CQCAs, and CQCAs with gliders. We then identify invariant states and study convergence properties of classes of states, such as quasifree and stabilizer states. Finally, we consider the generation of entanglement analytically and numerically for stabilizer and quasifree states.

Landau damping
View Description Hide DescriptionIn this note we present the main results from the recent work of Mouhot and Villani (“On the Landau damping,” arXiv:0904.2760), which for the first time establish Landau damping in a nonlinear context.

Singularities of complexvalued solutions of the twodimensional Burgers system
View Description Hide DescriptionWe consider the twodimensional viscous Burgers system without external forcing. For complexvalued solutions, due to the loss of maximum principle and energy estimates, smooth solutions can develop finitetime singularities. We construct an open set of sixparameter families of initial conditions such that the corresponding solutions exhibit blowups in finite time.

Hardsphere fluids with chemical selfpotentials
View Description Hide DescriptionThe existence, uniqueness, and stability of solutions are studied for a set of nonlinear fixed point equations which define selfconsistent hydrostatic equilibria of a classical continuum fluid that is confined inside a container and in contact with either a heat and a matter reservoir, or just a heat reservoir. The local thermodynamics is furnished by the statistical mechanics of a system of hard balls, in the approximation of Carnahan–Starling. The fluid’s local chemical potential per particle at is the sum of the matter reservoir’s contribution and a selfcontribution , where is the fluid density function and a nonnegative linear combination of the Newton kernel , the Yukawa kernel , and a van der Waals kernel . The fixed point equations involving the Yukawa and Newton kernels are equivalent to semilinear elliptic partial differential equations (PDEs) of second order with a nonlinear, nonlocal boundary condition. We prove the existence of a grand canonical phase transition and of a petit canonical phase transition which is embedded in the former. The proofs suggest that, except for boundary layers, the grand canonical transition is of the type “all liquid” while the petit canonical one is of the type “all drop with vapor atmosphere.” The latter proof, in particular, suggests the existence of solutions with interface structure which compromise between the allliquid and allgas density solutions.

Quasiperiodic motions in dynamical systems: Review of a renormalization group approach
View Description Hide DescriptionPower series expansions naturally arise whenever solutions of ordinary differential equations are studied in the regime of perturbation theory. In the case of quasiperiodic solutions the issue of convergence of the series is bedeviled by the socalled small divisor problem. In this paper we review a method recently introduced to deal with such a problem, based on renormalization group ideas and multiscale techniques. Applications to both quasiintegrable Hamiltonian systems [KolmogorovArnoldMoser (KAM) theory] and nonHamiltonian dissipative systems are discussed. The method is also suited to situations in which the perturbation series diverges and a resummation procedure can be envisaged, leading to a solution which is not analytic in the perturbation parameter: we consider explicitly examples of solutions which are only in the perturbation parameter, or even defined on a Cantor set.

A quantum central limit theorem for sums of independent identically distributed random variables
View Description Hide DescriptionWe formulate and prove a general central limit theorem for sums of independent identically distributed noncommutative random variables.

From operator algebras to superconformal field theory
View Description Hide DescriptionWe survey operator algebraic approach to (super)conformal field theory. We discuss representation theory, classification results, full and boundary conformal field theories, relations to supervertex operator algebras and Moonshine, connections to subfactor theory of Jones, and certain aspects of noncommutative geometry of Connes.

Twenty five years of twodimensional rational conformal field theory
View Description Hide DescriptionA review for the 50th anniversary of the Journal of Mathematical Physics.

Exact results for ionization of model atomic systems
View Description Hide DescriptionWe review recent rigorous results concerning the ionization of model quantum systems by timeperiodic external fields. The systems we consider consist of a single particle (electron) with a reference Hamiltonian having both bound and continuum states. Starting from an initially localized state , the system is subjected for to an arbitrary strength timeperiodic potential . We prove that for a large class of and , the wave function will delocalize as , i.e., the system will ionize. The only exceptions are cases where there are timeperiodic bound states of the Floquet operator associated with . These do occur (albeit rarely) when is not small. For spatially rapidly decaying and , is generally given, for very long times, by a power series in which we prove in some cases to be Borel summable. For the Coulomb potential in , we prove ionization for , for and for . For this model, if is compactly supported both in and in angular momentum, , we obtain that as .

Developments in the theory of universality
View Description Hide DescriptionRecently, a rigorous foundation of several aspects of the theory of universality for statistical mechanics models with continuously varying exponents (among which are interacting planar Ising models, quantum spin chains, and onedimensional Fermi systems) has been reached; it has its root in the mapping of such systems into fermionic interacting theories and uses the modern renormalization group methods developed in the context of constructive quantum field theory. No use of exact solutions is done and the analysis applies either to solvable or not solvable models. A review of such developments will be given here.

Batalin–Vilkovisky integrals in finite dimensions
View Description Hide DescriptionThe Batalin–Vilkovisky (BV) method is the most powerful method to analyze functional integrals with (infinitedimensional) gauge symmetries presently known. It has been invented to fix gauges associated with symmetries that do not close offshell. Homological perturbation theory is introduced and used to develop the integration theory behind BV and to describe the BV quantization of a Lagrangian system with symmetries. Localization (illustrated in terms of Duistermaat–Heckman localization) as well as anomalous symmetries are discussed in the framework of BV.

Almost commuting matrices, localized Wannier functions, and the quantum Hall effect
View Description Hide DescriptionFor models of noninteracting fermions moving within sites arranged on a surface in threedimensional space, there can be obstructions to finding localized Wannier functions. We show that such obstructions are theoretic obstructions to approximating almost commuting, complexvalued matrices by commuting matrices, and we demonstrate numerically the presence of this obstruction for a lattice model of the quantum Hall effect in a spherical geometry. The numerical calculation of the obstruction is straightforward and does not require translational invariance or introduce a flux torus. We further show that there is a index obstruction to approximating almost commuting selfdual matrices by exactly commuting selfdual matrices and present additional conjectures regarding the approximation of almost commuting real and selfdual matrices by exactly commuting real and selfdual matrices. The motivation for considering this problem is the case of physical systems with additional antiunitary symmetries such as timereversal or particlehole conjugation. Finally, in the case of the sphere—mathematically speaking, three almost commuting Hermitians whose sum of square is near the identity—we give the first quantitative result, showing that this index is the only obstruction to finding commuting approximations. We review the known nonquantitative results for the torus.

Algebraic complementarity in quantum theory
View Description Hide DescriptionThis paper is an overview of the concept of complementarity, the relation to state estimation, to Connes–Størmer conditional (or relative) entropy, and to uncertainty relation. Complementary Abelian and noncommutative subalgebras are analyzed. All the known results about complementary decompositions are described and several open questions are included. The paper contains only few proofs, typically references are given.

Spectral gap, and split property in quantum spin chains
View Description Hide DescriptionIn this article, we consider a class of ground states with spectral gap for quantum spin chains on an integer lattice and we prove that the factorization lemma of Hastings [“Topology and phases in fermionic systems,” J. Stat. Mech.: Theory Exp.2008, L01001] implies split property (weak statistical independence) of left and right semiinfinite subsystems.

Statics and dynamics of magnetic vortices and of Nielsen–Olesen (Nambu) strings
View Description Hide DescriptionWe review recent works on statics and dynamics of magnetic vortices in the Ginzburg–Landau model of superconductivity and of Nielsen–Olesen (Nambu) strings in the Abelian–Higgs model of particle physics.

The principle of locality: Effectiveness, fate, and challenges
View Description Hide DescriptionThe special theory of relativity and quantum mechanics merge in the key principle of quantum field theory, the principle of locality. We review some examples of its “unreasonable effectiveness” in giving rise to most of the conceptual and structural frame of quantum field theory, especially in the absence of massless particles. This effectiveness shows up best in the formulation of quantum field theory in terms of operator algebras of local observables; this formulation is successful in digging out the roots of global gauge invariance, through the analysis of superselection structure and statistics, in the structure of the local observable quantities alone, at least for purely massive theories; but so far it seems unfit to cope with the principle of local gauge invariance. This problem emerges also if one attempts to figure out the fate of the principle of locality in theories describing the gravitational forces between elementary particles as well. An approach based on the need to keep an operational meaning, in terms of localization of events, of the notion of spacetime, shows that, in the small, the latter must loose any meaning as a classical pseudoRiemannian manifold, locally based on Minkowski space, but should acquire a quantum structure at the Planck scale. We review the geometry of a basic model of quantum spacetime and some attempts to formulate interaction of quantum fields on quantum spacetime. The principle of locality is necessarily lost at the Planck scale, and it is a crucial open problem to unravel a replacement in such theories which is equally mathematically sharp, namely, a principle where the general theory of relativity and quantum mechanics merge, which reduces to the principle of locality at larger scales. Besides exploring its fate, many challenges for the principle of locality remain; among them, the analysis of superselection structure and statistics also in the presence of massless particles, and to give a precise mathematical formulation to the measurement process in local and relativistic terms; for which we outline a qualitative scenario which avoids the Einstein, Podolski, and Rosen paradox.

Rigorous meaning of McLennan ensembles
View Description Hide DescriptionWe analyze the exact meaning of expressions for nonequilibrium stationary distributions in terms of entropy changes. They were originally introduced by McLennan [“Statistical mechanics of the steady state,” Phys. Rev.115, 1405 (1959)] for mechanical systems close to equilibrium and more recent work by Komatsu and Nakagawa [“An expression for stationary distribution in nonequilibrium steady states,” Phys. Rev. Lett.100, 030601 (2008)] has shown their intimate relation to the transient fluctuation symmetry. Here we derive these distributions for jump and diffusionMarkov processes and we clarify the order of the limits that take the system both to its stationary regime and to the closetoequilibrium regime. In particular, we prove that it is exactly the (finite) transient component of the irreversible part of the entropy flux that corrects the Boltzmann distribution to first order in the driving. We add further connections with the notion of local equilibrium, with the Green–Kubo relation, and with a generalized expression for the stationary distribution in terms of a reference equilibrium process.
