coherent states on the circle a careful study of the physical properties of a family of coherent states on the circle introduced some years ago by de bievre and gonzalez in 1992 semiclassical behaviour of the weyl correspondence on the circle group theoretical methods in physics vol i madrid ciemat is carried out they were obtained from the weyl heisenberg coherent states in by means of the weil brezin zak transformation they are labelled by the points of the cylinder and they provide a realization of by entire functions similar to the well known fock bargmann construction in particular we compute the expectation values of the position and momentum operators on the circle and we discuss the heisenberg uncertainty relation

quantum cohomology of flag manifolds and toda lattices we discuss relations of vafa s quantum cohomology with floer s homology theory introduce equivariant quantum cohomology formulate some conjectures about its general properties and on the basis of these conjectures compute quantum cohomology algebras of the flag manifolds the answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a toda lattice

analytic representations with theta functions for systems on ℤ d and on an analytic representation with theta functions on a torus for systems with variables in d is considered another analytic representation with theta functions on a strip for systems with positions in a circle x1d54a and momenta in is also considered the reproducing kernel formalism for these two systems is studied wigner and weyl functions in this language are also studied

localized wavefunctions in quantum systems with multiwell potentials one dimensional systems with the multiwell potential v x 2 1cos x 2 2cos x are considered localized wavefunctions in the case when the energy is greater than the maximum value of the potential are presented wigner and weyl functions corresponding to the wavefunctions of these systems are also studied it is shown that they are very sensitive to the value of the parameter

eigenvectors and functions of the discrete fourier transform a method is presented for computing an orthonormal set of eigenvectors for the discrete fourier transform dft the technique is based on a detailed analysis of the eigenstructure of a special matrix which commutes with the dft it is also shown how fractional powers of the dft can be efficiently computed and possible applications to multiplexing and transform coding are suggested

coherent states for a quantum particle on a circle the coherent states for a quantum particle on a circle are introduced the bargmann representation within the actual treatment provides the representation of the algebra where u is unitary which is a direct consequence of the heisenberg algebra but it is more adequate for the study of circular motion

chaos revealing multiplicative representation of quantum eigenstates the quantisation of the two dimensional toric and spherical phase spaces is considered in analytic coherent state representations every pure quantum state admits therein a finite multiplicative parametrisation by the zeros of its husimi function for eigenstates of quantised systems this description explicitly reflects the nature of the underlying classical dynamics in the semiclassical regime the distribution of the zeros in the phase space becomes one dimensional for integrable systems and highly spread out conceivably uniform for chaotic systems this multiplicative representation thereby acquires a special relevance for semiclassical analysis in chaotic systems

infrared singular fields and nonregular representations of canonical commutation relation algebras infrared singular variables which often enter in the formulation of models in quantum field theory many body theory and quantum statistical mechanics are described in terms of nonregular representations of canonical commutation relations algebras general properties of such representations are discussed in this way one also obtains insight into the construction and representation of charged field algebras a as canonical extensions of a neutral observable algebra a0 the charged state representations of a0 are obtained through nonregular vacuum representations of the extended field algebra a the above structures are proven to work in several explicit models

t equivariant k theory of generalized flag varieties let g be a kac moody group with borel subgroup b and compact maximal torus t analogous to kostant and kumar kostant b kumar s 1986 proc nati acad sci usa 83 1543 1545 we define a certain ring y purely in terms of the weyl group w associated to g and its action on t by dualizing y we get another ring 1 which we prove is canonically isomorphic with the t equivariant ktheory kt g b of gib now kt g b apart from being an algebra over kt pt a t also has a weyl group action and moreover kt g b admits certain operators dw w similar to the demazure operators defined on a t we prove that these structures on kt g b come naturally from the ring y by evaluating the a t module w at 1 we recover k g b together with the above mentioned structures we believe that many of the results of this paper are new in the finite case i e g is a finite dimensional semisimple group over c

coherent states for the quantum mechanics on a compact manifold we review our recent observations concerning quantum mechanics on such compact manifolds as a circle torus and sphere

coherent evolution in a multiply connected space in the presence of a fractional magnetic flux the back reaction effect electric charges on a circle in the presence of a fractional magnetic flux are considered a winding operator and a flux operator are defined and the corresponding phase space is shown to describe the global properties of the system the heisenberg weyl group of discrete displacements and the sl 2 z q group of discrete bogoliubov transformations in the s w phase space are studied they describe coherent evolution of the system going beyond the external field approximation and taking into account the back reaction of the electric charge on the magnetic flux when q is the power of a prime z q is a galois field and stronger results can be proved

discrete fourier transform and jacobi θ function identities classical theta function identities are derived using properties of eigenvectors corresponding to the discrete fourier transform 2 in particular an extended watson addition formula is obtained corresponding to 2 and the corresponding result for the discrete fourier transform 3 is obtained

quantum cohomology of flag manifolds g b and quantum toda lattices let g be a connected semi simple complex lie group b its borel subgroup t a maximal complex torus contained in b and lie t its lie algebra this setup gives rise to two constructions the generalized nonperiodic toda lattice 28 29 and the flag manifold g b the toda lattice for g b t is the dynamical system on the cotangent bundle t lie t endowed with the canonical holomorphic symplectic form and the holomorphic hamiltonian function we consider in this paper

a conjectural peterson isomorphism in k theory abstract we state a precise conjectural isomorphism between localizations of the equivariant quantum k theory ring of a flag variety and the equivariant k homology ring of the affine grassmannian in particular relating their schubert bases and structure constants this generalizes peterson s isomorphism in co homology we prove a formula for the pontryagin structure constants in the k homology ring and we use it to check our conjecture in few situations

a criterion for the simplicity of the lyapunov spectrum of square tiled surfaces we present a galois theoretical criterion for the simplicity of the lyapunov spectrum of the kontsevich zorich cocycle over the teichmuller flow on the mathrm sl 2 mathbb r orbit of a square tiled surface the simplicity of the lyapunov spectrum has been proved by a avila and m viana with respect to the so called masur veech measures associated to connected components of moduli spaces of translation surfaces but is not always true for square tiled surfaces of genus geqslant 3 we apply our criterion to square tiled surfaces of genus 3 with one single zero conditionally to a conjecture of delecroix and lelievre we prove with the aid of siegel s theorem on integral points on algebraic curves of genus 0 that all but finitely many such square tiled surfaces have simple lyapunov spectrum

quasiprobability distribution functions for periodic phase spaces i theoretical aspects an approach featuring s parametrized quasiprobability distribution functions is developed for situations where a circular topology is observed for such an approach a suitable set of angle angular momentum coherent states must be constructed in an appropriate fashion

fractional discrete q fourier transforms the discrete fourier transform dft matrix has a manifold of fractionalizations that depend on the choice of its eigenbases one prominent basis is that of mehta functions here we examine a family of fractionalizations of the dft stemming from q extensions of this basis although closed expressions are given many results of our analysis derive from numerical computation and display thus we suggest that the account of fractional fourier transformations applied on signals as presented by other authors typically of a centred rectangle function may be biased because the support of the function lies in the central part of the domain only the phase and amplitude of the whole fractional dft matrices reveal the location of departures from the continuous kernel of the fractional fourier integral transform whose phase and constant amplitude are well known

finite dimensional hilbert space and frame quantization the quantum observables used in the case of quantum systems with finite dimensional hilbert space are defined either algebraically in terms of an orthonormal basis and discrete fourier transformation or by using a continuous system of coherent states we present an alternative approach to these important quantum systems based on the finite frame quantization finite systems of coherent states usually called finite tight frames can be defined in a natural way in the case of finite quantum systems novel examples of such tight frames are presented the quantum observables used in our approach are obtained by starting from certain classical observables described by functions defined on the discrete phase space corresponding to the system they are obtained by using a finite frame and a klauder berezin toeplitz type quantization semi classical aspects of tight frames are studied through lower symbols of basic classical observables

non orthogonal wigner like mapping we have developed a general method for construction of wigner like mapping to the phase space associated with a non orthogonal set of mutually unbiased equally separated bases explicit expressions of the wigner functions in terms of measurements in non orthogonal bases are given and the limit cases are analyzed

quantum mechanics on a circle husimi phase space distributions and semiclassical coherent state propagators we discuss some basic tools for an analysis of one dimensional quantum systems defined on a cyclic coordinate space the basic features of the generalized coherent states the complexifier coherent states are reviewed these states are then used to define the corresponding quasi densities in phase space the properties of these generalized husimi distributions are discussed in particular their zeros furthermore the use of the complexifier coherent states for a semiclassical analysis is demonstrated by deriving a semiclassical coherent state propagator in phase space

jacobi ϑ functions and discrete fourier transforms properties of the jacobi 3 function and its derivatives under discrete fourier transforms are investigated and several interesting results are obtained the role of modulo n equivalence classes in the theory of functions is stressed an important conjecture is studied

on the position uncertainty measure on the circle new position uncertainty delocalization measures for a particle on the circle are proposed and illustrated in several examples where the previous measures based on 2 periodic position operators appear to be unsatisfactory the new measures are suitably constructed using the standard multiplication angle operator variances they are shown to depend solely on the state of the particle and to obey uncertainty relations of the schrodinger robertson type

simplicity of lyapunov spectra proof of the zorich kontsevich conjecture we prove the zorich kontsevich conjecture that the non trivial lyapunov exponents of the teichmuller ow on any connected component of a stratum of the moduli space of abelian differentials on compact riemann surfaces are all distinct by previous work of zorich and kontsevich this implies the existence of the complete asymptotic lagrangian flag describing the behavior in homology of the vertical foliation in a typical translation surface

extended cahill glauber formalism for finite dimensional spaces i fundamentals the cahill glauber approach for quantum mechanics on phase space is extended to the finite dimensional case through the use of discrete coherent states all properties and features of the continuous formalism are appropriately generalized the continuum results are promptly recovered as a limiting case the jacobi theta functions are shown to have a prominent role in the context

systems with outer constraints gupta bleuler electromagnetism as an algebraic field theory since there are some important systems which have constraints not contained in their field algebras we develop here in ac context the algebraic structures of these the constraints are defined as a groupg acting as outer automorphisms on the field algebra g aut g inn and we find that the selection ofg invariant states on is the same as the selection of states onm g m g mathop times limits alpha f by ug 1 g g whereug m g m g mathop times limits alpha f are the canonical elements implementing g these states are taken as the physical states and this specifies the resulting algebraic structure of the physics inm g m g mathop times limits alpha f and in particular the maximal constraint free physical algebra a nontriviality condition is given for to exist and we extend the notion of a crossed product to deal with a situation whereg is not locally compact this is necessary to deal with the field theoretical aspect of the constraints next thec algebra of the ccr is employed to define the abstract algebraic structure of gupta bleuler electromagnetism in the present framework the indefinite inner product representation structure is obtained and this puts gupta bleuler electromagnetism on a rigorous footing finally as a bonus we find that the algebraic structures just set up provide a blueprint for constructive quadratic algebraic field theory

quantizations on the circle and coherent states we present a possible construction of coherent states on the unit circle as configuration space our approach is based on borel quantizations on s1 including the aharonov bohm type quantum description coherent states are constructed by perelomov s method as group related coherent states generated by weyl operators on the quantum phase space because of the duality of canonical coordinates and momenta i e the angular variable and the integers this formulation can also be interpreted as coherent states over an infinite periodic chain for the construction we use the analogy with our quantization and coherent states over a finite periodic chain where the quantum phase space was the coherent states constructed in this work are shown to satisfy the resolution of unity to compare them with canonical coherent states some of their further properties are also studied demonstrating similarities as well as substantial differences this article is part of a special issue of journal of physics a mathematical and theoretical devoted to coherent states mathematical and physical aspects

unbounded elements affiliated with c algebras and non compact quantum groups the affiliation relation that allows to include unbounded elements operators into thec algebra framework is introduced investigated and applied to the quantum group theory the quantum deformation of the two fold covering of the group of motions of euclidean plane is constructed a remarkable radius quantization is discovered it is also shown that the quantumsu 1 1 group does not exist on thec algebra level for real value of the deformation parameter

quantum k theory of grassmannians we show that equivariant k theoretic 3 point gromov witten invariants of genus zero on a grassmann variety are equal to triple intersec tions computed in the equivariant k theory of a two step flag manifold thus generalizing an earlier result of buch kresch and tamvakis in the process we show that the gromov witten variety of curves passing through 3 general points is irreducible and rational our applications include pieri and giambelli formulas for the quantum k theory ring of a grassmannian which determine the multiplication in this ring we also compute the dual schubert basis for this ring and show that its structure constants satisfy s3 symmetry our for mula for gromov witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of chaput manivel and perrin

equivariant k theory of affine flag manifolds and affine grothendieck polynomials we study the equivariant k group of the affine flag manifold with respect to the borel group action we prove that the structure sheaf of the infinite dimensional schubert variety in the k group is represented by a unique polynomial which we call the affine grothendieck polynomial

an angular momentum approach to quadratic fourier transform hadamard matrices gauss sums mutually unbiased bases the unitary group and the pauli group the construction of unitary operator bases in a finite dimensional hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of su 2 a single formula for the bases is obtained from a polar decomposition of su 2 and is analyzed in terms of cyclic groups quadratic fourier transforms hadamard matrices and generalized gauss sums weyl pairs generalized pauli operators and their application to the unitary group and the pauli group naturally arise in this approach

on the quantum differential calculus and the quantum holomorphicity under some natural assumptions less restrictive than in the paper by wess and zumino preprint cern th 5697 90 lapp th 284 90 differential calculi on the quantum plane are found and investigated complex structure complex derivatives and holomorphic functions on the quantum plane are defined generalized cauchy riemann equations are given and solved

classification of rauzy veech groups proof of the zorich conjecture we classify the rauzy veech groups of all connected components of all strata of the moduli space of translation surfaces in absolute homology showing in particular that they are commensurable to arithmetic lattices of symplectic groups as a corollary we prove a conjecture of zorich about the zariski density of such groups

a reconstruction theorem in quantum cohomology and quantum k theory a reconstruction theorem for genus 0 gravitational quantum cohomology and quantum k theory is proved a new linear equivalence in the picard group of the moduli space of genus 0 stable maps relating the pull backs of line bundles from the target via different markings is used for the reconstruction result examples of calculations in quantum cohomology and quantum k theory are given

quantum systems with finite hilbert space galois fields in quantum mechanics a galois quantum system in which the position and momentum take values in the galois field gf pl is considered it is comprised of l component systems which are coupled in a particular way and is described by a certain class of hamiltonians displacements in the gf pl gf pl phase space and the corresponding heisenberg weyl group are studied symplectic transformations are shown to form the sp 2 gf pl group wigner and weyl functions are defined and their properties are studied frobenius symmetries which are based on frobenius automorphisms in the theory of galois fields are a unique feature of these systems for l 2 if they commute with the hamiltonian there are constants of motion which are discussed an analytic representation in the l sheeted complex plane provides an elegant formalism that embodies the properties of frobenius transformations the difference between a galois quantum system and other finite quantum systems where the position and momentum take values in the ring is discussed

quantum cohomology of g p and homology of affine grassmannian let g be a simple and simply connected complex algebraic group p g a parabolic subgroup we prove an unpublished result of d peterson which states that the quantum cohomology qh g p of a flag variety is up to localization a quotient of the homology h gr g of the affine grassmannian gr g of g as a consequence all three point genus zero gromov witten invariants of g p are identified with homology schubert structure constants of h gr g establishing the equivalence of the quantum and homology affine schubert calculi

quantum k theory on flag manifolds finite difference toda lattices and quantum groups let x y be homogeneous coordinates on cp a degree d holomorphic map cp 1 cp is uniquely determined up to a constant scalar factor by n 1 relatively prime degree d binary forms f0 x y fn x y omitting the condition that the forms are relatively prime we compactify the space of degree d holomorphic maps cp 1 cp to a complex projective space of dimension n 1 d 1 1 we denote this compactification of the space of maps by cp d this construction defines the compactification d r i 1cp ni 1 di of the space of degree d d1 dr maps from cp 1 to composing degree d holomorphic maps from cp 1 to the flag manifold x with the plucker embedding we embed the space of such maps into d the closure qmd of this space in d is often referred to as the drinfeld s compactification of the space of degree d maps from cp 1 to x and will be called the space of quasimaps following 14 it is a generally speaking singular irreducible projective variety of complex dimension dimx 2d1 2dr the flag manifold is a homogeneous space of the group slr 1 c and of its maximal compact subgroup sur 1 the action of these groups on x lifts naturally to the spaces d and qmd in addition to this action the spaces d and the subspaces qmd carry the circle action induced by the rotation of cp 1 defined by x y 7 x ey thus the product group g s sur 1 and its complex version gc c slr 1 c act on the quasimap spaces we will see later that qmd have g equivariant desingularizations qmd

gromov witten invariants for and pontryagin product for ω we give an explicit formula for t equivariant 3 pointed genus zero gromov witten invariants for g b we derive it by finding an explicit formula for the pontryagin product on the equivariant homology of the based loop group k

wigner distribution function for finite systems we construct a wigner distribution function for finite data sets it is based on a finite optical system a linear wave guide where the finite number of discrete sensors is equal to the number of modes which the guide can carry the dynamical group for this model is su 2 and the wave functions are sets of n 2l 1 data points the wigner distribution function assigns classical c numbers to the operators of position momentum and wave guide mode

zorich conjecture for hyperelliptic rauzy veech groups we describe the structure of hyperelliptic rauzy diagrams and hyperelliptic rauzy veech groups in particular this provides a solution of the hyperelliptic cases of a conjecture of zorich on the zariski closure of rauzy veech groups

analytic representation of finite quantum systems a transform between functions in r and functions in zd is used to define the analogue of number and coherent states in the context of finite d dimensional quantum systems the coherent states are used to define an analytic representation in terms of theta functions all states are represented by entire functions with growth of order 2 which have exactly d zeros in each cell the analytic function of a state is constructed from its zeros results about the completeness of finite sets of coherent states within a cell are derived

eigenvalues and eigenvectors of the finite fourier transform the eigenvalues and eigenvectors of the n n unitary matrix of finite fourier transform whose j k element is 1 n 1 2 exp 2 i n jk i 1 1 2 is determined in doing so a multitude of identities some of which may be new are encountered a conjecture is advanced

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

the classical limit of a state on the weyl algebra this paper considers states on the weyl algebra of the canonical commutation relations over the phase space r 2n we show that a state is regular iff its classical limit is a countably additive borel probability measure on r 2n it follows that one can reduce the state space of the weyl algebra by altering the collection of quantum mechanical observables so that all states are ones whose classical limit is physical

the wigner function associated with the rogers szego polynomials a wigner function associated with the rogers szego polynomials is proposed and its properties are discussed it is shown that from such a wigner function it is possible to obtain well behaved probability distribution functions for both angle and action variables defined on the compact support and for m 0 respectively the width of the angle probability density is governed by the free parameter q characterizing the polynomials

coherent states on spheres we describe a family of coherent states and an associated resolution of the identity for a quantum particle whose classical configuration space is the d dimensional sphere sd the coherent states are labeled by points in the associated phase space t sd these coherent states are not of perelomov type but rather are constructed as the eigenvectors of suitably defined annihilation operators we describe as well the segal bargmann representation for the system the associated unitary segal bargmann transform and a natural inversion formula although many of these results are in principle special cases of the results of hall and stenzel we give here a substantially different description based on ideas of thiemann and of kowalski and rembielinski all of these results can be generalized to a system whose configuration space is an arbitrary compact symmetric space we focus on the sphere case in order to carry out the calculations in a self contained and explicit way

zero lyapunov exponents and monodromy of the kontsevich zorich cocycle we describe all the situations in which the kontsevich zorich kz cocycle has zero lyapunov exponents confirming a conjecture of forni matheus and zorich we find this only occurs when the cocycle satisfies additional geometric constraints we also show that the connected components of the zariski closure of the monodromy must be from a specific list and the representations in which they can occur are described the number of zero exponents of the kz cocycle is then as small as possible given its monodromy

q extension of mehta s eigenvectors of the finite fourier transform for q a root of unity it is shown that the continuous q hermite polynomials for q a root of unity have simple transformation properties with respect to the classical fourier transform this result is then used to construct q extended eigenvectors of the finite fourier transform in terms of these polynomials