asymptotic behaviour of bohmian trajectories in scattering situations we study the asymptotic behaviour of bohmian trajectories in a scattering situation with short range potential v and for wavefunctions with a scattering and a bound part it is shown that the set of possible trajectories splits into trajectories whose long time behaviour is governed by the scattering part of the wavefunction scattering trajectories and trajectories whose long time behaviour is governed by the bound part of the wavefunction bound trajectories furthermore the scattering trajectories behave like trajectories in classical mechanics in the limit t as an intermediate step we show that the asymptotic velocity exists almost surely and is randomly distributed with the density where is the outgoing asymptote of the scattering part of the wavefunction

comment on what bell did according to maudlin bell showed that it is the world which is non lcoal and not just some particular theories of it i argue that this conclusion is arrived at by taking for granted all assumptions of realism or classicality if these are taken into account the resulting conclusion that a classical world which allows for bell inequality violations must be non local is in good agreement with the mainstream perception of bell s theorem s this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

quantization of hamiltonian loop group spaces we prove a fredholm property for spin c dirac operators mathsf d on non compact manifolds satisfying a certain condition with respect to the action of a semi direct product group k ltimes gamma with k compact and gamma discrete we apply this result to an example coming from the theory of hamiltonian loop group spaces in this context we prove that a certain index pairing mathcal x cap mathsf d yields an element of the formal completion r infty t of the representation ring of a maximal torus t subset h the resulting element has an additional antisymmetry property under the action of the affine weyl group indicating mathcal x cap mathsf d corresponds to an element of the ring of projective positive energy representations of the loop group

string structures and the path fibration of a group we use the realisation of the universal bundle for the loop group as the path fibration of the group to investigate the string class that is the obstruction to a loop group bundle lifting to a kac moody group bundle in the case that the loop group bundle is constructed by taking loops into a principal bundle we show that the classifying map is the holonomy around loops and give an explicit formula for the string class relating it to the pontrjangin class of the principal bunble

existence of large solutions for a quasilinear elliptic problem via explosive sub supersolutions abstract we consider the boundary blow up quasilinear elliptic problems div u m 2 u u q m 1 k x g u in a c 2 bounded domain with boundary condition u where m 1 q 0 m m 1 and 0 under suitable growth assumptions on k near the boundary and on g both at zero and at infinity we show the existence of at least one solution in c 1 our proof is based on the method of explosive sub supersolutions which permits positive weights k x which are unbounded and or oscillatory near the boundary

nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints hamiltoniens sousquadratiques conditions aux limites infinies et donnees d explosion hamiltoniens superquadratiques formulation viscosite des conditions aux limites le probleme ergodique commande stochastique optimale avec contraintes d etat

norm square localization and the quantization of hamiltonian loop group spaces abstract in an earlier article we introduced a new definition for the quantization of a hamiltonian loop group space m involving the equivariant l 2 index of a dirac type operator d on a non compact finite dimensional submanifold y of m in this article we study a deformation of this operator similar to the work of tian zhang and ma zhang we obtain a formula for the index with infinitely many non trivial contributions indexed by the components of the critical set of the norm square of the moment map this is the main part of a new proof of the q r 0 theorem for hamiltonian loop group spaces

boundary blow up solutions for a class of elliptic equations on a bounded domain this paper deals with existence of solutions of the quasilinear elliptic equation du f x u u in w satisfying the boundary blow up condition u x x email protected where email protected r n is a bounded domain with smooth boundary email protected our main result applies to the existence of blow up solutions of a class of equations which appear in stochastic control theory namely du a x g u email protected u email protected x in w where a x g u j x are suitable functions and l s are nonnegative constants our approach employs an approximation procedure combined with both a non monotone iteration variant of the method of lower and upper solutions and fixed point arguments

scattering theory from microscopic first principles abstract we sketch a derivation of abstract scattering theory from the microscopic first principles defined by bohmian mechanics we emphasize the importance of the flux across surfaces theorem for the derivation and of randomness in the impact parameter of the initial wave function even for an inevitably inadequate orthodox derivation

surjectivity for hamiltonian loop group spaces let g be a compact lie group and let lg denote the corresponding loop group let x be a weakly symplectic banach manifold consider a hamiltonian action of lg on x and assume that the moment map mu rm x to l g is proper we consider the function left mu right 2 x to mathbb r and use a version of morse theory to showthat the inclusion map j mu 1 left 0 right to x induces a surjection j h g left x right to h g left mu 1 left 0 right right in analogywithkirwan s surjectivity theorem in the finite dimensional case we also prove a version of this surjectivity theorem for quasi hamiltonian g spaces

the yang mills heat flow on the moduli space of framed bundles on a surface we study the analog of the yang mills heat flow on the moduli space of framed bundles on a cut surface existence and convergence of the heat flow give a stratification of morse type invariant under the action of the loop group we use the stratification to prove versions of kahler quantization commutes with reduction and kirwan surjectivity

the asymptotic behaviour of solutions with boundary blow up for semilinear elliptic equations with nonlinear gradient terms abstract in this paper we show existence uniqueness and exact asymptotic behaviour of solutions near the boundary to nonlinear elliptic problems u u q k x g u x u where is a bounded domain with smooth boundary in r n 0 q 0 2 lim d x 0 k x d x c 0 0 2 c 0 0 g c 1 0 g 0 0 g is increasing on 0 and is karamata regular variation function at infinity with index 1 and 0

chaotic bohmian trajectories for stationary states in bohmian mechanics the nodes of the wave function play an important role in the generation of chaos however so far most of the attention has been on moving nodes little is known about the possibility of chaos in the case of stationary nodes we address this question by considering stationary states which provide the simplest examples of wave functions with stationary nodes we provide examples of stationary wave functions for which there is chaos as demonstrated by numerical computations for one particle moving in three spatial dimensions and for two and three entangled particles in two dimensions our conclusion is that the motion of the nodes is not necessary for the generation of chaos what is important is the overall complexity of the wave function that is if the wave function or rather its phase has a complex spatial variation it will lead to complex bohmian trajectories and hence to chaos another aspect of our work concerns the average lyapunov exponent which quantifies the overall amount of chaos since it is very hard to evaluate the average lyapunov exponent analytically which is often computed numerically it is useful to have simple quantities that agree well with the average lyapunov exponent we investigate possible correlations with quantities such as the participation ratio and different measures of entanglement for different systems and different families of stationary wave functions we find that these quantities often tend to correlate to the amount of chaos however the correlation is not perfect because in particular these measures do not depend on the form of the basis states used to expand the wave function while the amount of chaos does

on singular boundary value problems for the monge ampere operator abstract we consider different types of singular boundary value problems for the monge ampere operator the approach is based on existing regularity theory and a subsolution supersolution method nonexistence and uniqueness results are also given

on the global existence of bohmian mechanics we show that the particle motion in bohmian mechanics given by the solution of an ordinary differential equation exists globally for a large class of potentials the singularities of the velocity field and infinity will not be reached in finite time for typical initial values a substantial part of the analysis is based on the probabilistic significance of the quantum flux we elucidate the connection between the conditions necessary for global existence and the self adjointness of the schrodinger hamiltonian

a bohmian approach to quantum fractals a quantum fractal is a wavefunction with a real and an imaginary part continuous everywhere but differentiable nowhere this lack of differentiability has been used as an argument to deny the general validity of bohmian mechanics and other trajectory based approaches in providing a complete interpretation of quantum mechanics here this assertion is overcome by means of a formal extension of bohmian mechanics based on a limiting approach within this novel formulation the particle dynamics is always satisfactorily described by a well defined equation of motion in particular in the case of guidance under quantum fractals the corresponding trajectories will also be fractal

quantization from an exponential distribution of infinitesimal action abstract a statistical model of quantization based on an exponential distribution of infinitesimal action is proposed a trajectory which does not extremize the action along an infinitesimal short segment of path is allowed to occur with a very small probability following an exponential law planck s constant is argued to give the average deviation from the infinitesimal stationary action

a necessary and sufficient condition for existence of large solutions to semilinear elliptic equations we consider the semilinear equation u p x f u on a domain rn n 3 where f is a nonnegative nondecreasing continuous function which vanishes at the origin and p is a nonnegative continuous function with the property that any zero of p is contained in a bounded domain in such that p is positive on its boundary for bounded we show that a nonnegative solution u satisfying u x as x exists if and only if the function s s0 f t dt satisfies 1 s 1 2 ds for unbounded including rn we show that a similar result holds where u x as x within and u x as x if p x decays to zero rapidly as x

quantum equilibrium and the origin of absolute uncertainty what is randomness probability certainty knowledge these are old and difficult questions and we shall not focus on them here nonetheless we shall obtain sharp striking conclusions concerning the relationship between these concepts

lie group valued moment maps we develop a theory of quasi hamiltonian g spaces for which the moment map takes values in the group g itself rather than in the dual of the lie algebra the theory includes counterparts of hamiltonian reductions the guillemin sternberg symplectic cro

simple proof for global existence of bohmian trajectories we address the question whether bohmian trajectories exist for all times bohmian trajectories are solutions of an ordinary differential equation involving a wavefunction obeying either the schrodinger or the dirac equation some trajectories may end in finite time for example by running into a node of the wavefunction where the law of motion is ill defined the aim is to show under suitable assumptions on the initial wavefunction and the potential global existence of almost all solutions we provide an alternative proof of the known global existence result for spinless schrodinger particles and extend the result to particles with spin to the presence of magnetic fields and to dirac wavefunctions our main new result is conditions on the current vector field on configuration space time which are sufficient for almost sure global existence

on the distribution of the wave function for systems in thermal equilibrium for a quantum system a density matrix that is not pure can arise via averaging from a distribution of its wave function a normalized vector belonging to its hilbert space while itself does not determine a unique additional facts such as that the system has come to thermal equilibrium might it is thus not unreasonable to ask which if any corresponds to a given thermodynamic ensemble to answer this question we construct for any given density matrix a natural measure on the unit sphere in denoted gap we do this using a suitable projection of the gaussian measure on with covariance we establish some nice properties of gap and show that this measure arises naturally when considering macroscopic systems in particular we argue that it is the most appropriate choice for systems in thermal equilibrium described by the canonical ensemble density matrix 1 z exp h gap may also be relevant to quantum chaos and to the stochastic evolution of open quantum systems where distributions on are often used

mixed hodge polynomials of character varieties we calculate the e polynomials of certain twisted gl n character varieties mathcal m n of riemann surfaces by counting points over finite fields using the character table of the finite group of lie type text gl n mathbb f q and a theorem proved in the appendix by n katz we deduce from this calculation several geometric results for example the value of the topological euler characteristic of the associated pgl n character variety the calculation also leads to several conjectures about the cohomology of mathcal m n an explicit conjecture for its mixed hodge polynomial a conjectured curious hard lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver we prove these conjectures for n 2

the biinvariant diagonal class for hamiltonian torus actions abstract suppose that an algebraic torus g acts algebraically on a projective manifold x with generically trivial stabilizers then the zariski closure of the set of pairs x y x x y g x for some g g defines a nonzero equivariant cohomology class g h g g x x we give an analogue of this construction in the case where x is a compact symplectic manifold endowed with a hamiltonian action of a torus whose complexification plays the role of g we also prove that the kirwan map sends the class g to the class of the diagonal in each symplectic quotient this allows to define a canonical right inverse of the kirwan map

microscopic origins of irreversible macroscopic behavior time asymmetric behavior as embodied in the second law of thermodynamics is observed in individual macroscopic systems it can be understood as arising naturally from time symmetric microscopic laws in accord with the ideas of thompson maxwell and boltzmann alternative explanations based on equating irreversible macroscopic behavior with the mixing type of behavior already present in the time evolution of ensembles probability distributions of certain systems having only a few degrees of freedom are in my opinion unnecessary misguided and misleading

logical reformulation of quantum mechanics i foundations the basic rules of quantum mechanics are reformulated they deal primarily with individual systems and do not assume that every ket may represent a physical state the customary kinematic and dynamic rules then allow to construct consistent boolean logics describing the history of a system following essentially griffiths proposal logical implication is defined within these logics the multiplicity of which reflects the complementary principle only one interpretative rule of quantum mechanics is necessary in such a framework it states that these logics providebona fide foundations for the description of a quantum system and for reasoning about it one attempts to build up classical physics including classical logic on these quantum foundations the resulting theory of measurement needs not to statea priori that the eigenvalues of an observable have to be the results of individual measurements nor to assume wave packet reduction both these properties can be obtained as consequences of the basic rules one also needs not to postulate that every observable is measurable even in principle a proposition calculus is obtained allowing in principle the replacement of the discussion of problems concerned with the practical interpretation of experiments by due calculations

schrodinger s paradox and proofs of nonlocality using only perfect correlations we discuss proofs of nonlocality based on a generalization by erwin schrodinger of the argument of einstein podolsky and rosen these proofs do not appeal in any way to bell s inequalities indeed one striking feature of the proofs is that they can be used to establish nonlocality solely on the basis of suitably robust perfect correlations first we explain that schrodinger s argument shows that locality and the perfect correlations between measurements of observables on spatially separated systems imply the existence of a non contextual value map for quantum observables non contextual means that the observable has a particular value before its measurement for any given quantum system and that any experiment measuring this observable will reveal that value then we establish the impossibility of a non contextual value map for quantum observables without invoking any further quantum predictions combining this with schrodinger s argument implies nonlocality finally we illustrate how bohmian mechanics is compatible with the impossibility of a non contextual value map

consistent histories and the interpretation of quantum mechanics the usual formula for transition probabilities in nonrelativistic quantum mechanics is generalized to yield conditional probabilities for selected sequences of events at several different times called consistent histories through a criterion which ensures that within limits which are explicitly defined within the formalism classical rules for probabilities are satisfied the interpretive scheme which results is applicable to closed isolated quantum systems is explicitly independent of the sense of time i e past and future can be interchanged has no need for wave function collapse makes no reference to processes of measurement though it can be used to analyze such processes and can be applied to sequences of microscopic or macroscopic events or both as long as the mathematical condition of consistency is satisfied when applied to appropriate macroscopic events it appears to yield the same answers as other interpretative schemes for standard quantum mechanics though from a different point of view which avoids the conceptual difficulties which are sometimes thought to require reference to conscious observers or classical apparatus

witten s formulas for intersection pairings on moduli spaces of flat g bundles abstract in a 1992 paper j geom phys 9 1992 303 witten gave a formula for the intersection pairings of the moduli space of flat g bundles over an oriented surface possibly with markings in this paper we give a general proof of witten s formula for arbitrary compact simple groups and any markings for which the moduli space has at most orbifold singularities

the two bell s theorems of john bell many of the heated arguments about the meaning of bell s theorem arise because this phrase can refer to two different theorems that john bell proved the first in 1964 and the second in 1976 his 1964 theorem is the incompatibility of quantum phenomena with the dual assumptions of locality and determinism his 1976 theorem is the incompatibility of quantum phenomena with the unitary property of local causality this is contrary to bell s own later assertions that his 1964 theorem began with that single and indivisible assumption of local causality even if not by that name while there are other forms of bell s theorems which i present to explain the relation between jarrett completeness fragile locality and epr completeness i maintain that bell s two versions are the essential ones although the two bell s theorems are logically equivalent their assumptions are not and the different versions of the theorem suggest quite different conclusions which are embraced by different communities for realists the notion of local causality ruled out by bell s 1976 theorem is motivated implicitly by reichenbach s principle of common cause and explicitly by the principle of relativistic causality and it is the latter which must be forgone operationalists pay no heed to reichenbach s principle but wish to keep the principle of relativistic causality which bolstered by an implicit principle of agent causation implies their notion of locality thus for operationalists bell s theorem is the 1964 one and implies that it is determinism that must be forgone i discuss why the two camps are drawn to these different conclusions and what can be done to increase mutual understanding

spin and the thermal equilibrium distribution of wave functions consider a quantum system s weakly interacting with a very large but finite system b called the heat bath and suppose that the composite s b is in a pure state with participating energies between e and e with small then it is known that for most the reduced density matrix of s is approximately equal to the canonical density matrix that is the reduced density matrix is universal in the sense that it depends only on s s hamiltonian and the temperature but not on b s hamiltonian on the interaction hamiltonian or on the details of it has also been pointed out that s can also be attributed a random wave function whose probability distribution is universal in the same sense this distribution is known as the scrooge measure or gaussian adjusted projected gap measure we regard it as the thermal equilibrium distribution of wave functions the relevant concept of the wave function of a subsystem is known as the conditional wave function in this paper we develop analogous considerations for particles with spin one can either use some kind of conditional wave function or more naturally the conditional density matrix which is in general different from the reduced density matrix we ask what the thermal equilibrium distribution of the conditional density matrix is and find the answer that for most the conditional density matrix is approximately deterministic in fact approximately equal to the canonical density matrix