symplectic covariance properties for shubin and born jordan pseudo differential operators among all classes of pseudo differential operators only the weyl operators enjoy the property of symplectic covariance with respect to conjugation by elements of the metaplectic group in this paper we show that there is however a weaker form of symplectic covariance for shubin s dependent operators in which the intertwiners no longer are metaplectic but still are invertible non unitary operators we also study the case of born jordan operators which are obtained by averaging the operators over the interval 0 1 such operators have recently been studied by boggiatto and his collaborators we show that metaplectic covariance still hold for these operators with respect top a subgroup of the metaplectic group

the weighted hardy s inequality for nonincreasing functions the purpose of this paper is to give an alternative proof of recent results of m arino and b muckenhoupt 1 and e sawyer 8 concerning hardy s inequality for nonincreasing functions and related applications to the boundedness of some classical operators on general lorentz spaces our approach will extend the results of 1 8 to the values of the parameters which are inaccessible by the methods of these papers

time frequency representations of wigner type and pseudo differential operators we introduce a dependent wigner representation wig e 0 1 which permits us to define a general theory connecting time frequency representations on one side and pseudo differential operators on the other the scheme includes various types of time frequency representations among the others the classical wigner and rihaczek representations and the most common classes of pseudo differential operators we show further that the integral over t of wig yields a new representation q possessing features in signal analysis which considerably improve those of the wigner representation especially for what concerns the so called ghost frequencies the relations of all these representations with respect to the generalized spectrogram and the cohen class are then studied furthermore a characterization of the l p boundedness of both pseudo differential operators and t wigner representations are obtained

hermitian einstein metrics from noncommutative u 1 instantons we show that hermitian einstein metrics can be locally constructed by a map from anti self dual two forms on euclidean r4 to symmetric two tensors introduced in yang and salizzoni phys rev lett 96 201602 2006 e print arxiv hep th 0512215 this correspondence is valid not only for a commutative space but also for a noncommutative space we choose u 1 instantons on a noncommutative c2 as the self dual two form from which we derive a family of hermitian einstein metrics we also discuss the condition when the metric becomes kahler we show that hermitian einstein metrics can be locally constructed by a map from anti self dual two forms on euclidean r4 to symmetric two tensors introduced in yang and salizzoni phys rev lett 96 201602 2006 e print arxiv hep th 0512215 this correspondence is valid not only for a commutative space but also for a noncommutative space we choose u 1 instantons on a noncommutative c2 as the self dual two form from which we derive a family of hermitian einstein metrics we also discuss the condition when the metric becomes kahler

noncommutative instantons and twistor transform recently n nekrasov and a schwarz proposed a modification of the adhm construction of instantons which produces instantons on a noncommutative deformation of 4 in this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces we exhibit one to one correspondences between three classes of objects framed bundles on a noncommutative 2 certain complexes of sheaves on a noncommutative 3 and the modified adhm data the modified adhm construction itself is interpreted in terms of a noncommutative version of the twistor transform we also prove that the moduli space of framed bundles on the noncommutative 2 has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative 2 the natural complex structures on the two moduli spaces do not coincide but are related by an so 3 rotation finally we propose a construction of instantons on a more general noncommutative 4 than the one considered by nekrasov and schwarz a q deformed 4

space time foam from non commutative instantons we show that a u 1 instanton on non commutative open image in new window corresponds to a non singular u 1 gauge field on a commutative kahler manifold x which is a blowup of open image in new window at a finite number of points this gauge field on x obeys maxwell s equations in addition to the susy constraint f0 2 0 for instanton charge k the manifold x can be viewed as a space time foam with b2 k a direct connection with integrable systems of calogero moser type is established we also make some comments on the non abelian case

metric adjusted skew information we extend the concept of wigner yanase dyson skew information to something we call metric adjusted skew information of a state with respect to a conserved observable this skew information is intended to be a non negative quantity bounded by the variance of an observable in a state that vanishes for observables commuting with the state we show that the skew information is a convex function on the manifold of states it also satisfies other requirements proposed by wigner and yanase for an effective measure of information content of a state relative to a conserved observable we establish a connection between the geometrical formulation of quantum statistics as proposed by chentsov and morozova and measures of quantum information as introduced by wigner and yanase and extended in this article we show that the set of normalized morozova chentsov functions describing the possible quantum statistics is a bauer simplex and determine its extreme points we determine a particularly simple skew information the skew information parametrized by a 0 1 and show that the convex cone this family generates coincides with the set of all metric adjusted skew informations

hilbert schemes separated variables and d branes we explain sklyanin s separation of variables in geometrical terms and construct it for hitchin and mukai integrable systems we construct hilbert schemes of points on t for c c or elliptic curve and on c 2 and show that their complex deformations are integrable systems of calogero sutherland moser type we present the hyperkahler quotient constructions for hilbert schemes of points on cotangent bundles to the higher genus curves utilizing the results of hurtubise kronheimer and nakajima finally we discuss the connections to physics of d branes and string duality

local uncertainty principles for the cohen class abstract in this paper we analyze time frequency representations in the cohen class i e quadratic forms expressed as a convolution between the classical wigner transform and a kernel with respect to uncertainty principles of local type more precisely the results we obtain concerning the energy distribution of these representations show that a too large amount of energy cannot be concentrated in a too small set of the time frequency plane in particular for a signal f l 2 r d the energy of a time frequency representation contained in a measurable set m must be controlled by the standard deviations of f 2 and f 2 and by suitable quantities measuring the size of m

extension of the conley zehnder index a product formula and an application to the weyl representation of metaplectic operators the aim of this paper is to express the conley zehnder index of a symplectic path in terms of an index due to leray and which has been studied by one of us in a previous work this will allow us to prove a formula for the conley zehnder index of the product of two symplectic paths in terms of a symplectic cayley transform we apply our results to a rigorous study of the weyl representation of metaplectic operators which plays a crucial role in the understanding of semiclassical quantization of hamiltonian systems exhibiting chaotic behavior

quantum covariance quantum fisher information and the uncertainty relations in this paper the relation between quantum covariances and quantum fisher informations is studied this study is applied to generalize a recently proved uncertainty relation based on quantum fisher information the proof given here considerably simplifies the previously proposed proofs and leads to more general inequalities

bypassing the groenewold van hove obstruction on mathbb r 2n a new argument in favor of born jordan quantization there are known obstructions to a full quantization of in the spirit of dirac s approach the most known being the groenewold and van hove no go result we show following a suggestion of s k kauffmann that it is possible to construct a well defined quantization procedure by weakening the usual requirement that commutators should correspond to poisson brackets the weaker requirement consists in demanding that this correspondence should only hold for hamiltonian functions of the type this reformulation leads to a non injective quantization of all observables which when restricted to polynomials is the rule proposed by born and jordan in the early days of quantum mechanics

on a correspondence between regular and non regular operator monotone functions abstract we prove the existence of a bijection between the regular and the non regular operator monotone functions satisfying a certain functional equation as an application we give a new proof of the operator monotonicity of certain functions related to the wigner yanase dyson skew information

on bounds for weighted norms for matrices and integral operators in this note we consider inequalities of the form ax q bx v p where a and b are matrices or integral operators x decreasing sequence or function and and v are weights obtained results are generalizations of results of g bennett linear algebra appl 82 1986 81 and p e renaud bull aust math soc 34 1986 225

covariance and fisher information in quantum mechanics variance and fisher information are ingredients of the cramer rao inequality we regard fisher information as a riemannian metric on a quantum statistical manifold and choose monotonicity under coarse graining as the fundamental property of variance and fisher information in this approach we show that there is a kind of dual one to one correspondence between the candidates of the two concepts we emphasize that fisher information is obtained from relative entropies as contrast functions on the state space and argue that the scalar curvature might be interpreted as an uncertainty density on a statistical manifold

born jordan quantization and the uncertainty principle the weyl correspondence and the related wigner formalism lie at the core of traditional quantum mechanics we discuss here an alternative quantization scheme the idea of which goes back to born and jordan and which has recently been revived in another context namely time frequency analysis we show in particular that the uncertainty principle does not enjoy full symplectic covariance properties in the born and jordan scheme as opposed to what happens in the weyl quantization

lower bounds for matrices abstract in a recent paper 3 lyons has discovered an interesting lower bound for the cesaro matrix c his result says that cx 6 x for every x l 2 satisfying x 1 x 2 0 the purpose of this note is to establish analogous lower bounds for arbitrary matrices with nonnegative entries acting on arbitrary l p spaces 1 p

noncommutative instantons revisited we find a new gauge in which u 1 noncommutative instantons are explicitly non singular on noncommutative r 4 we also present a pedagogical introduction into noncommutative gauge theories

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

continuity properties for modulation spaces with applications to pseudo differential calculus i we discuss continuity for weighted modulation spaces andprove that many such spaces can be obtained in a canonicalway from the corresponding standard modulation spaces we also discussthe trace operator a a 0 acting on modulationspaces the results are used to get inclusions betweenmodulation spaces and besov spaces and proving continuityfor pseudo differential operators and toeplitz operators

wigner functions on non standard symplectic vector spaces we consider the weyl quantization on a flat non standard symplectic vector space we focus mainly on the properties of the wigner functions defined therein in particular we show that the sets of wigner functions on distinct symplectic spaces are different but have non empty intersections this extends previous results to arbitrary dimension and arbitrary constant symplectic structure as a by product we introduce and prove several concepts and results on non standard symplectic spaces which generalize those on the standard symplectic space namely the symplectic spectrum williamson s theorem and narcowich wigner spectra we also show how wigner functions on non standard symplectic spaces behave under the action of an arbitrary linear coordinate transformation

an informational characterization of schrodinger s uncertainty relations heisenberg s uncertainty relations employ commutators of observables to set fundamental limits on quantum measurement the information concerning incompatibility non commutativity of observables is well included but that concerning correlation is missing schrodinger s uncertainty relations remedy this defect by supplementing the correlation in terms of anti commutators however both heisenberg s uncertainty relations and schrodinger s uncertainty relations are expressed in terms of variances which are not good measures of uncertainty in general situations e g when mixed states are involved by virtue of the wigner yanase skew information we will establish an uncertainty relation along the spirit of schrodinger from a statistical inference perspective and propose a conjecture the result may be interpreted as a quantification of certain aspect of the celebrated wigner araki yanase theorem for quantum measurement which states that observables not commuting with a conserved quantity cannot be measured exactly

cohen class of time frequency representations and operators boundedness and uncertainty principles abstract this paper presents a proof of an uncertainty principle of donoho stark type involving e concentration of localization operators more general operators associated with time frequency representations in the cohen class are then considered for these operators which include all usual quantizations we prove a boundedness result in the l p functional setting and a form of uncertainty principle analogous to that for localization operators

uncertainty principle and quantum fisher information ii heisenberg and schrodinger uncertainty principles give lower bounds for the product of variances var a var b if the observables a b are not compatible namely if the commutator a b is not zero in this paper we prove an uncertainty principle in schrodinger form where the bound for the product of variances var a var b depends on the area spanned by the commutators i a and i b with respect to an arbitrary quantum version of the fisher information

two aspects of the donoho stark uncertainty principle abstract we present some forms of uncertainty principle which involve in a new way localization operators the concept of e concentration and the standard deviation of l 2 functions we show how our results improve the classical donoho stark estimate in two different aspects a better general lower bound and a lower bound in dependence on the signal itself

some uncertainty principles for time frequency transforms of the cohen class the question of how the time frequency concentration of transforms of the cohen class is affected by fundamental properties is addressed the investigated transform properties are the marginal property the property of energy conservation and moyal s relation we show that each of these transform properties imply restrictions on the time frequency concentration of the cohen class and is thus related to an uncertainty principle different notions of time frequency concentration adapted to the property under consideration are applied

uncertainty principle positivity and lp boundedness for generalized spectrograms abstract in this paper we are concerned with the properties of positivity uncertainty principle and continuity in l p spaces of a generalized spectrogram in particular we study the connections of a generalized spectrogram as a subclass of the cohen class with the rihaczek and the wigner representations we also consider the behavior of the generalized spectrogram with respect to the positivity and the l p boundedness of the corresponding localization operators

how to distinguish quantum covariances using uncertainty relations abstract to each operator monotone function it is possible to associate a quantum version of the classical covariance we show that i only for regular quantum covariances one can prove non trivial uncertainty relations ii the usual quantum covariance gives the best inequalities in this setting

equivalence of projections as gauge equivalence on noncommutative space projections play crucial roles in the adhm construction on noncommutative 4 in this article a framework for the description of equivalence relations between projections is proposed we treat the equivalence of projections as gauge equivalence on noncommutative space we find an interesting application of this framework to the study of the u 2 instanton on noncommutative 4 a zero winding number configuration with a hole at the origin is gauge equivalent to the noncommutative analog of the bpst instanton thus the gauge transformation in this case can be understood as a noncommutative resolution of the singular gauge transformation in ordinary 4

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

time frequency analysis of born jordan pseudodifferential operators abstract born jordan operators are a class of pseudodifferential operators arising as a generalization of the quantization rule for polynomials on the phase space introduced by born and jordan in 1925 the weak definition of such operators involves the born jordan distribution first introduced by cohen in 1966 as a member of the cohen class we perform a time frequency analysis of the cohen kernel of the born jordan distribution using modulation and wiener amalgam spaces we then provide sufficient and necessary conditions for born jordan operators to be bounded on modulation spaces we use modulation spaces as appropriate symbols classes

the parametrisation of quantisation rules equivalent to operator orderings and the effect of different rules on the physical spectrum the author recalls the problem of choosing an ordering for the terms in a hamiltonian of the form h p q of finite dimension in p q he demonstrates that most orderings which provide one with a hermitian operator are in fact redundant and he gives a general form for ordering rules in terms of arbitrary parameters he concludes that differences in the choice of an ordering contribute differences to the physics which are only of order 2 and higher he provides an example of a hamiltonian for which the spectrum is explicitly dependent on the choice of quantisation rule operator ordering he compares the method of characterising operator ordering rules with that of cohen 1966 1970 finally he generalises cohen s method to the case of all linear quantisation rules and shows how his rule is a special case

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation

norm estimates for τ pseudodifferential operators in wiener amalgam and modulation spaces abstract we study continuity properties on modulation spaces for pseudodifferential operators op a with symbols a in wiener amalgam spaces we obtain boundedness results for 0 1 whereas in the end points 0 and 1 the corresponding operators are in general unbounded furthermore for 0 1 we exhibit a function of which is an upper bound for the operator norm the continuity properties of op a for any 0 1 with symbols a in modulation spaces are well known here we find an upper bound for the operator norm which does not depend on the parameter 0 1 as expected key ingredients are uniform continuity estimates for wigner distributions