the classical obstacle problem for nonlinear variational energies abstract we develop the complete free boundary analysis for solutions to classical obstacle problems related to nondegenerate nonlinear variational energies the key tools are optimal c 1 1 regularity which we review more generally for solutions to variational inequalities driven by nonlinear coercive smooth vector fields and the results in focardi et al 2015 concerning the obstacle problem for quadratic energies with lipschitz coefficients furthermore we highlight similar conclusions for locally coercive vector fields having in mind applications to the area functional or more generally to area type functionals as well

from geometric quantization to moyal quantization it is shown how the moyal product of phase space functions and the weyl correspondence between symbols and operator kernels may be obtained directly using the procedures of geometric quantization applied to the symplectic groupoid constructed by doubling the phase space

time dependent transformations in deformation quantization we study the action of time dependent canonical and coordinate transformations in phase space quantum mechanics we extend the covariant formulation of the theory by providing a formalism that is fully invariant under both standard and time dependent coordinate transformations this result considerably enlarges the set of possible phase space representations of quantum mechanics and makes it possible to construct a causal representation for the distributional sector of wigner quantum mechanics

finite tame and wild actions of parabolic subgroups in gl v on certain unipotent subgroups abstract let p be a parabolic subgroup of some general linear group gl v where v is a finite dimensional vector space over an infinite field the group p acts by conjugation on its unipotent radical pu and via the adjoint action on p u the lie algebra of pu more generally we consider the action of p on the lth member of the descending central series of p u denoted by p l u let l p u denote the nilpotency class of pu in our main result we show that p acts on p l u with a finite number of orbits precisely when l p u 4 for l 0 or l p u 5 2l for l 1 moreover in case the field is algebraically closed we consider the modality mod p p l u of the action of p on p l u we show that mod p p l u grows linearly in the minimal cases which admit infinitely many orbits i e l p u 5 for l 0 or l p u 6 2l for l 1 whereas the corresponding modality grows quadratically in all other infinite cases these results are obtained by interpreting the orbits of p on p l u as isomorphism classes of good modules over certain quasi hereditary algebras and by a detailed inspection of the representation types of these algebras

husimi transform of an operator product it is shown that the series derived by mizrahi giving the husimi transform or covariant symbol of an operator product is absolutely convergent for a large class of operators in particular the generalized liouville equation describing the time evolution of the husimi function is absolutely convergent for a large class of hamiltonians in contrast the series derived by groenewold giving the weyl transform of an operator product is often only asymptotic or even undefined the result is used to derive an alternative way of expressing expectation values in terms of the husimi function the advantage of this formula is that it applies in many of the cases where the anti husimi transform or contravariant symbol is so highly singular that it fails to exist as a tempered distribution

linear canonical transformations and quantum phase a unified canonical and algebraic approach the algebra of generalized linear quantum canonical transformations is examined in the perspective of schwinger s unitary canonical operator basis formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the canonical transformations in the quantum phase space the representations of the wigner function in the generalized action angle unitary operator pair for certain hamiltonian systems with dynamical symmetry is examined this generalized canonical formalism is applied to the quantum harmonic oscillator to examine the properties of the unitary quantum phase operator as well as the action angle wigner function

conditions for the convolution of two wigner distributions to be itself a wigner distribution the convolution of two wigner distribution functions wdf s is always non negative but not always a wdf as jagannathan et al phys lett a 120 161 1987 have shown in this paper conditions are given that are sufficient and probably necessary for such a convolution to be a wdf and a new characterization of gaussian wdf s is obtained as a by product

on the porosity of free boundaries in degenerate variational inequalities abstract in this note we consider a certain degenerate variational problem with constraint identically zero the exact growth of the solution near the free boundary is established a consequence of this is that the free boundary is porous and therefore its hausdorff dimension is less than n and hence it is of lebesgue measure zero

when is the wigner function of multidimensional systems nonnegative it is shown that for systems with an arbitrary number of degrees of freedom a necessary and sufficient condition for the wigner function to be nonnegative is that the corresponding state wavefunction is the exponential of a quadratic form this result generalizes the one obtained by hudson rep math phys 6 249 1974 for one dimensional systems

regularity results for solutions to obstacle problems with sobolev coefficients abstract we establish the higher differentiability of solutions to a class of obstacle problems of the type min f x d v x d x v k where is a fixed function called obstacle k v w loc 1 p r v a e in and the convex integrand f satisfies p growth conditions with respect to the gradient variable we derive that the higher differentiability property of the weak solution v is related to the regularity of the assigned under a suitable sobolev assumption on the partial map x d f x the main novelty is that such assumption is independent of the dimension n and this in the case p n 2 allows us to manage coefficients in a sobolev class below the critical one w 1 n

entanglement bell inequality and all that we start from the geometrical observation that a finite set of pure states correspond to some points on a sphere and their convex span cannot be the whole set of states if we call the left over entangled we can pursue this picture from the simplest case of a two dimensional hilbert space to the usual alice and bob game of entangled states and then move to bigger systems and finely to quantum field theory where almost everything is entangled on the way we encounter more or less known old friends up from the shell structure of states to the monogamy of squashed entanglement we study how entanglement can be concentrated on a small slice and how it depends on the particular factorization of the hilbert space

quantum symmetries and the weyl wigner product of group representations in the usual formulation of quantum mechanics groups of automorphisms of quantum states have ray representations by unitary and antiunitary operators on complex hilbert space in accordance with wigner s theorem in the phase space formulation they have real true unitary representations in the space of square integrable functions on phase space each such phase space representation is a weyl wigner product of the corresponding hilbert space representation with its contragredient and these can be recovered by factorizing the weyl wigner product however not every real unitary representation on phase space corresponds to a group of automorphisms so not every such representation is in the form of a weyl wigner product and can be factorized the conditions under which this is possible are examined examples are presented

quantum harmonic analysis on phase space relative to an irreducible representation of the canonical commutation relations convolutions between quantum mechanical operators and between functions and operators are defined for which the usual weyl transform acts as a fourier transform basic properties of these operations are developed in close analogy to harmonic analysis on r2n using the quantum version of wiener s approximation theorem a natural one to one correspondence between the closed phase space translation invariant subspaces of classical and quantum observables is established

a symplectic extension map and a new shubin class of pseudo differential operators for an arbitrary pseudo differential operator a s rn s rn with weyl symbol a s r2n we consider the pseudo differential operators a s rn k s rn k associated with the weyl symbols a a 12k s where 12k x 1 for all x r2k and s is a linear symplectomorphism of r2 n k we call the operators a symplectic dimensional extensions of a in this paper we study the relation between a and a in detail in particular their regularity invertibility and spectral properties we obtain an explicit formula allowing to express the eigenfunctions of a in terms of those of a we use this formalism to construct new classes of pseudo differential operators which are extensions of the shubin classes hg m1 m0 of globally hypoelliptic operators we show that the operators in the new classes share the invertibility and spectral properties of the operators in hg m1 m0 but not the global hypoellipticity property finally we study a few examples of operators that belong to the new classes and which are important in mathematical physics

regularity results for a class of non autonomous obstacle problems with p q growth abstract we study some regularity issues for solutions of non autonomous obstacle problems with p q growth under suitable assumptions our analysis covers the main models available in the literature

the moyal lie theory of phase space quantum mechanics a lie algebraic approach to the unitary transformations in weyl quantization is discussed this approach being formally equivalent to the quantization is an extension of the classical poisson lie theory and can be used as an efficient tool in the quantum phase space transformation theory

weyl manifolds and deformation quantization this paper deals with non commutative objects based on the weyl algebra from the differential geometric point of view we propose to extend familiar notions from manifolds to non commutative objects in this algebraic approach the basic object is not a point of a space but certain sections of bundles

prehomogeneous spaces for parabolic group actions in classical groups abstract let g be a reductive linear algebraic group p a parabolic subgroup of g and p u its unipotent radical we consider the adjoint action of p on the lie algebra p u of p u richardson s dense orbit theorem says that there is a dense p orbit in p u we consider some instances when p acts with a dense orbit on terms p u l of the descending central series of p u in particular we show in good characteristic that a borel subgroup b of a classical group acts on b u l with a dense orbit for all l further we give some families of parabolic subgroups p such that p u l contains a dense p orbit for all l

geometry and uncertainty in the context of the wigner weyl phase space formulation of quantum mechanics a version of the uncertainty relations invariant under affine canonical transformations is derived for a fixed wigner distribution function possessing a finite covariance directions of minimal uncertainty are found the geometry of the wigner ellipsoid and its legendre transform the dual wigner ellipsoid both of which are associated with the same covariance is discussed the results obtained are generalizations of the well known fact that for one degree of freedom the area of the wigner ellipse must be of order or larger instead of area which is an invariant only when n 1 these results involve poincare invariants of certain curves and surfaces

the construction of a quantum markov partition we present a method for constructing a quantum markov partition its elements are obtained by quantizing the characteristic function of the classical rectangles the result is a set of quantum operators which behave asymptotically as projectors over the classical rectangles apart from edge and corner effects we investigate their spectral properties and different methods of construction the quantum partition is shown to induce a symbolic decomposition of the quantum evolution operator in particular an exact expression for the traces of the propagator is obtained having the same structure as the gutzwiller periodic orbit sum

bounds on integrals of the wigner function the integral of the wigner function over a subregion of the phase space of a quantum system may be less than zero or greater than one it is shown that for systems with 1 degree of freedom the problem of determining the best possible upper and lower bounds on such an integral over an possible states reduces to the problem of finding the greatest and least eigenvalues of a hermitian operator corresponding to the subregion the problem is solved exactly in the case of an arbitrary elliptical region these bounds provide checks on experimentally measured quasiprobability distributions

wigner functions for noncommutative quantum mechanics a group representation based construction this paper is devoted to the construction and analysis of the wigner functions for noncommutative quantum mechanics their marginal distributions and star products following a technique developed earlier viz using the unitary irreducible representations of the group gnc which is the three fold central extension of the abelian group of 4 these representations have been exhaustively studied in earlier papers the group gnc is identified with the kinematical symmetry group of noncommutative quantum mechanics of a system with two degrees of freedom the wigner functions studied here reflect different levels of non commutativity both the operators of position and those of momentum not commuting the position operators not commuting and finally the case of standard quantum mechanics obeying the canonical commutation relations only

wigner functions with boundaries we consider the general wigner function for a particle confined to a finite interval and subject to dirichlet boundary conditions we derive the boundary corrections to the stargenvalue equation and to the time evolution equation these corrections can be cast in the form of a boundary potential contributing to the total hamiltonian which together with a subsidiary boundary condition is responsible for the discretization of the energy levels we show that a completely analogous formulation in terms of boundary potentials is also possible in standard operator quantum mechanics and that the wigner and the operator formulations are also in one to one correspondence in the confined case in particular we extend baker s converse construction to bounded systems finally we elaborate on the applications of the formalism to the subject of wigner trajectories namely in the context of collision processes and quantum systems displaying chaotic behavior in the classical limit

weyl wigner formulation of noncommutative quantum mechanics we address the phase space formulation of a noncommutative extension of quantum mechanics in arbitrary dimension displaying both spatial and momentum noncommutativities by resorting to a covariant generalization of the weyl wigner transform and to the darboux map we construct an isomorphism between the operator and the phase space representations of the extended heisenberg algebra this map provides a systematic approach to derive the entire structure of noncommutative quantum mechanics in phase space we construct the extended star product and moyal bracket and propose a general definition of noncommutative states we study the dynamical and eigenvalue equations of the theory and prove that the entire formalism is independent of the particular choice of the darboux map our approach unifies and generalizes all the previous proposals for the phase space formulation of noncommutative quantum mechanics for concreteness we rederive these proposals by restricting our formalism to some two dimensional spaces

entropy power uncertainty relations towards a tight inequality for all gaussian pure states we show that a proper expression of the uncertainty relation for a pair of canonically conjugate continuous variables relies on entropy power a standard notion in shannon information theory for real valued signals the resulting entropy power uncertainty relation is equivalent to the entropic formulation of the uncertainty relation due to bialynicki birula and mycielski but can be further extended to rotated variables hence based on a reasonable assumption we give a partial proof of a tighter form of the entropy power uncertainty relation taking correlations into account and provide extensive numerical evidence of its validity interestingly it implies the generalized rotation invariant schrodinger robertson uncertainty relation exactly as the original entropy power uncertainty relation implies heisenberg relation it is saturated for all gaussian pure states in contrast with hitherto known entropic formulations of the uncertainty principle

quantum bochner s theorem for phase spaces built on projective representations bochner s theorem gives the necessary and sufficient conditions on a function such that its fourier transform corresponds to a true probability density function in the wigner phase space picture quantum bochner s theorem gives the necessary and sufficient conditions on a function such that it is a quantum characteristic function of a valid and possibly mixed quantum state and such that its fourier transform is a true probability density we extend this theorem to discrete phase space representations which possess enough symmetry more precisely we show that discrete phase space representations that are built on projective unitary representations of abelian groups with a slight restriction on admissible two cocycles enable a quantum bochner s theorem

homogeneous fedosov star products on cotangent bundles i weyl and standard ordering with differential operator representation in this paper we construct homogeneous star products of weyl type on every cotangent bundle t q by means of the fedosov procedure using a symplectic torsion free connection on t q homogeneous of degree zero with respect to the liouville vector field by a fibrewise equivalence transformation we construct a homogeneous fedosov star product of standard ordered type equivalent to the homogeneous fedosov star product of weyl type representations for both star product algebras by differential operators on functions on q are constructed leading in the case of the standard ordered product to the usual standard ordering prescription for smooth complex valued functions on t q polynomial in the momenta where an arbitrary fixed torsion free connection 0 on q is used motivated by the flat case t n another homogeneous star product of weyl type corresponding to the weyl ordering prescription is constructed the example of the cotangent bundle of an arbitrary lie group is explicitly computed and the star product given by gutt is rederived in our approach

convergence of the wick star product we construct a frechet space as a subspace of c omega mathbb c n where the wick star product converges and is continuous the resulting frechet algebra mathcal a is studied in detail including a representation of mathcal a in the bargmann fock space and a discussion of star exponentials and coherent states

the orthogonal lie algebra of operators ideals and derivations abstract we study in this paper the infinite dimensional orthogonal lie algebra o c which consists of all bounded linear operators t on a separable infinite dimensional complex hilbert space h satisfying c t c t where c is a conjugation on h by employing results from the theory of complex symmetric operators and skew symmetric operators we determine the lie ideals of o c and their dual spaces we study derivations of o c and determine their spectra these results complete some results of p de la harpe and provide interesting contrasts between o c and the algebra b h of all bounded linear operators on h

generating all wigner functions in the context of phase space quantization matrix elements and observables result from integration of c number functions over phase space with wigner functions serving as the quasiprobability measure the complete sets of wigner functions necessary to expand all phase space functions include off diagonal wigner functions which may appear technically involved nevertheless it is shown here that suitable generating functions of these complete sets can often be constructed which are relatively simple and lead to compact evaluations of matrix elements new features of such generating functions are detailed and explored for integer indexed sets such as for the harmonic oscillator as well as continuously indexed ones such as for the linear potential and the liouville potential the utility of such generating functions is illustrated in the computation of star functions spectra and perturbation theory in phase space

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

the quantum state vector in phase space and gabor s windowed fourier transform representations of quantum state vectors by complex phase space amplitudes complementing the description of the density operator by the wigner function have been defined by applying the weyl wigner transform to dyadic operators linear in the state vector and anti linear in a fixed window state vector here aspects of this construction are explored and a connection is established with gabor s windowed fourier transform the amplitudes that arise for simple quantum states from various choices of windows are presented as illustrations generalized bargmann representations of the state vector appear as special cases associated with gaussian windows for every choice of window amplitudes lie in a corresponding linear subspace of square integrable functions on phase space a generalized born interpretation of amplitudes is described with both the wigner function and a generalized husimi function appearing as quantities linear in an amplitude and anti linear in its complex conjugate schrodinger s time dependent and time independent equations are represented on phase space amplitudes and their solutions described in simple cases

formal gns construction and states in deformation quantization in this paper we develop a method of constructing hilbert spaces and the representation of the formal algebra of quantum observables in deformation quantization which is an analog of the well known gns construction for complex c algebras in this approach the corresponding positive linear functionals states take their values not in the field of complex numbers but in a suitable extension field of the field of formal complex laurent series in the formal parameter by using the algebraic and topological properties of these fields we prove that this construction makes sense and show in physical examples that standard representations such as the bargmann and schrodinger representation come out correctly both formally and in a suitable convergence scheme for certain hamiltonian functions contained in the gel fand ideal of the positive functional a formal solution to the time dependent schrodinger equation is shown to exist moreover we show that for every kahler manifold equipped with the fedosov star product of wick type all the classical delta functionals are positive and give rise to some formal bargmann representation of the deformed algebra

quantum magnetic algebra and magnetic curvature the symplectic geometry of the phase space associated with a charged particle is determined by the addition of the faraday 2 form to the standard dp dq structure on in this paper we describe the corresponding algebra of weyl symmetrized functions in operators satisfying nonlinear commutation relations the multiplication in this algebra generates an associative product of functions on the phase space this product is given by an integral kernel whose phase is the symplectic area of a groupoid consistent membrane a symplectic phase space connection with non trivial curvature is extracted from the magnetic reflections associated with the stratonovich quantizer zero and constant curvature cases are considered as examples the quantization with both static and time dependent electromagnetic fields is obtained the expansion of the product by the deformation parameter written in the covariant form is compared with the known deformation quantization formulae

hudson s theorem for finite dimensional quantum systems we show that on a hilbert space of odd dimension the only pure states to possess a non negative wigner function are stabilizer states the clifford group is identified as the set of unitary operations which preserve positivity the result can be seen as a discrete version of hudson s theorem hudson established that for continuous variable systems the wigner function of a pure state has no negative values if and only if the state is gaussian turning to mixed states it might be surmised that only convex combinations of stabilizer states give rise to non negative wigner distributions we refute this conjecture by means of a counterexample further we give an axiomatic characterization which completely fixes the definition of the wigner function and compare two approaches to stabilizer states for hilbert spaces of prime power dimensions in the course of the discussion we derive explicit formulas for the number of stabilizer codes defined on such systems

symbolic calculus on the time frequency half plane this study concerns a special symbolic calculus of interest for signal analysis this calculus associates functions on the time frequency half plane f 0 with linear operators defined on the positive frequency signals full attention is given to its construction which is entirely based on the study of the affine group in a simple and direct way the correspondence rule is detailed and the associated wigner function is given formulas expressing the basic operation star bracket of the lie algebra of symbols which is isomorphic to that of the operators are obtained in addition it is shown that the resulting calculus is covariant under a three parameter group which contains the affine group as subgroup this observation is the starting point of an investigation leading to a whole class of symbolic calculi which can be considered as modifications of the original one

moyal quantization with compact symmetry groups and noncommutative harmonic analysis the phase space approach to quantization of systems whose symmetry group is compact and semisimple is developed from two basic principles covariance and traciality this generalizes results and methods already implemented for spin systems j c varilly and j m gracia bondia ann phys 190 107 1989 the twisted product of phase space functions is shown to be the image of group convolution in the context of a novel fourier theory on the coadjoint orbits

bounds on integrals of the wigner function the hyperbolic case wigner functions play a central role in the phase space formulation of quantum mechanics although closely related to classical liouville densities wigner functions are not positive definite and may take negative values on subregions of phase space we investigate the accumulation of these negative values by studying bounds on the integral of an arbitrary wigner function over noncompact subregions of the phase plane with hyperbolic boundaries we show using symmetry techniques that this problem reduces to computing the bounds on the spectrum associated with an exactly solvable eigenvalue problem and that the bounds differ from those on classical liouville distributions in particular we show that the total quasiprobability on such a region can be greater than 1 or less than zero c 2005 american institute of physics

normal form decomposition for gaussian to gaussian superoperators in this paper we explore the set of linear maps sending the set of quantum gaussian states into itself these maps are in general not positive a feature which can be exploited as a test to check whether a given quantum state belongs to the convex hull of gaussian states if one of the considered maps sends it into a non positive operator the above state is certified not to belong to the set generalizing a result known to be valid under the assumption of complete positivity we provide a characterization of these gaussian to gaussian not necessarily positive superoperators in terms of their action on the characteristic function of the inputs for the special case of one mode mappings we also show that any gaussian to gaussian superoperator can be expressed as a concatenation of a phase space dilatation followed by the action of a completely positive gaussian channel possibly composed with a transposition while a similar decomposition is shown to fail in the multi mode scenario we prove that it still holds at least under the further hypothesis of homogeneous action on the covariance matrix

algebraic index theorem we prove the atiyah singer index theorem where the algebra of pseudodifferential operators is replaced by an arbitrary deformation quantization of the algebra of functions on a symplectic manifold

generalized weyl wigner map and vey quantum mechanics the weyl wigner map yields the entire structure of moyal quantum mechanics directly from the standard operator formulation the covariant generalization of moyal theory also known as vey quantum mechanics was presented in the literature many years ago however a derivation of the formalism directly from standard operator quantum mechanics clarifying the relation between the two formulations is still missing in this article we present a covariant generalization of the weyl order prescription and of the weyl wigner map and use them to derive vey quantum mechanics directly from the standard operator formulation the procedure displays some interesting features it yields all the key ingredients and provides a more straightforward interpretation of the vey theory including a direct implementation of unitary operator transformations as phase space coordinate transformations in the vey idiom these features are illustrated through a simple example

mixed states with positive wigner functions the wigner distribution function of a pure quantum state is everywhere positive if and only if the state is coherent according to a result of hudson the characterization of mixed states with a positive wigner function is a special case of the problem of determining functions satisfying a twisted positive definiteness condition for a prescribed set of twisting parameters i e functions with given wigner spectrum in the sense of narcowich if a state is a convex combination of coherent states it has the property that the wigner spectrum contains the unit interval which in turn implies that the wigner function is positive it is shown by explicit examples that the converses of both implications are false the examples are taken from a low dimensional section of the state space in which all wigner spectra can be computed in this set counterexamples to a conjecture by narcowich concerning the wigner spectrum of products are also found as well as a state whose wigner spectrum is a convergent sequence of discrete points

variation on a theme of richardson abstract we consider the structure of parabolic subgroups p in general linear groups the group p acts on its unipotent radical p u and on all members of the descending central series p u l via conjugation by a fundamental theorem due to richardson p acts on p u with an open dense orbit in fact this density theorem holds for any reductive algebraic group in this note we investigate the question of the existence of a dense p orbit on p u l for l 1 using only most elementary methods despite the fact that for special p it is the case that p operates on p u l with such a dense orbit for all l 0 in general however this fails we present a counterexample in gl 15 k besides the general linear groups we also study this question for other reductive algebraic groups

moyal nahm equations various aspects of the nahm equations in three and seven dimensions are investigated the residues of the variables at simple poles in the seven dimensional case form an algebra a large class of matrix representations of this algebra is constructed the large n limit of these equations is taken by replacing the commutators by moyal brackets and a set of nontrivial solutions in a generalized form of wigner distribution functions is obtained

the wigner distribution of n arbitrary observables we study a generalization of the wigner function to arbitrary tuples of hermitian operators we show that for any collection of hermitian operators a1 an and any quantum state there is a unique joint distribution on rn with the property that the marginals of all linear combinations of the ak coincide with their quantum counterparts in other words we consider the inverse radon transform of the exact quantum probability distributions of all linear combinations we call it the wigner distribution because for position and momentum this property defines the standard wigner function we discuss the application to finite dimensional systems establish many basic properties and illustrate these by examples the properties include the support the location of singularities positivity the behavior under symmetry groups and informational completeness

wigner measures in noncommutative quantum mechanics we study the properties of quasi distributions or wigner measures in the context of noncommutative quantum mechanics in particular we obtain necessary and sufficient conditions for a phase space function to be a noncommutative wigner measure for a gaussian to be a noncommutative wigner measure and derive certain properties of the marginal distributions which are not shared by ordinary wigner measures moreover we derive the robertson schrodinger uncertainty principle finally we show explicitly how the set of noncommutative wigner measures relates to the sets of liouville and commutative wigner measures

phase space observables and isotypic spaces we give necessary and sufficient conditions for the set of neumark projections of a countable set of phase space observables to constitute a resolution of the identity and we give a criteria for a phase space observable to be informationally complete the results will be applied to the phase space observables arising from an irreducible representation of the heisenberg group

distributions of ℏ positive type and applications states in classical mechanics are probability measures and their fourier transforms are continuous functions of positive type states in the phase space formulation of quantum mechanics are wigner distribution functions and their symplectic fourier transforms have been characterized by kastler commun math phys 1 14 1965 and loupias and miracle sole commun math phys 2 31 1966 ann insto h poincare 6 39 1967 as being continuous functions of positive type in this paper schwartz distributions of positive type are defined and studied it is shown that if such distributions are bounded on a certain sequence of test functions then they are symplectic fourier transforms of wigner distribution functions these results are applied to a variety of problems ranging from ones involving the quantum liouville equation to a problem in signal analysis

richardson elements for classical lie algebras parabolic subalgebras of semi simple lie algebras decompose as p m circle plus n where m is a levi factor and n the corresponding nilradical by richardson s theorem r w richardson bull london math soc 6 1974 21 24 there exists an open orbit under the action of the adjoint group p on the nilradical the elements of this dense orbits are known as richardson elements in this paper we describe a normal form for richardson elements in the classical case this generalizes a construction for gl n of briistle et al algebr represent theory 2 1999 295 312 to the other classical lie algebra and it extends the authors normal forms of richardson elements for nice parabolic subalgebras of simple lie algebras to arbitrary parabolic subalgebras of the classical lie algebras k baur represent theory 9 2005 30 45 as applications we obtain a description of the support of richardson elements and we recover the bala carter label of the orbit of richardson elements c 2005 elsevier inc all rights reserved

wigner distribution functions and the representation of a non bijective canonical transformation in quantum mechanics in a previous paper with a similar title the authors showed that for bijective i e one to one onto canonical transformations they can obtain their quantum mechanical representations in the form of a kernel in the phase space of wigner distribution functions and recover the classical expectation for this kernel in the limit to 0 for non bijective canonical transformations the situation is more complex as the phase space acquires either a riemann surface structure or requires the introduction of the concept of an ambiguity group by discussing the non bijective canonical transformation taking them from an oscillator of frequency kappa 1 where kappa is an integer to one of unit frequency they see how the kernel is generalised to include the indices associated with the irreducible representations of the ambiguity group i e the ambiguity spins they obtain the kernel explicitly and show that in the limit to 0 they recover the form that they expect in the riemann surface picture of their phase space while they illustrate their analysis through the representation in wigner distribution phase space of a specific non bijective canonical transformation the procedure is clearly extendable to the representations of general canonical transformations of this type

cotangent bundle quantization entangling of metric and magnetic field for manifolds of noncompact type endowed with an affine connection for example the levi civita connection and a closed 2 form magnetic field we define a hilbert algebra structure in the space and construct an irreducible representation of this algebra in this algebra is automatically extended to polynomial in momenta functions and distributions under some natural conditions this algebra is unique the non commutative product over is given by an explicit integral formula this product is exact not formal and is expressed in invariant geometrical terms our analysis reveals that this product has a front which is described in terms of geodesic triangles in the quantization of functions induces a family of symplectic reflections in and generates a magneto geodesic connection on this symplectic connection entangles on the phase space level the original affine structure on and the magnetic field in the classical approximation the 2 part of the quantum product contains the ricci curvature of and a magneto geodesic coupling tensor