conformal invariance and finite one dimensional quantum chains based on previous work of cardy 1984 the authors show in a systematic way how using conformal invariance one can determine the anomalous dimensions of various operators from finite quantum chains with different boundary conditions the method is illustrated in the case of the three and four state potts models where the anomalous dimensions of the para fermionic operators are found

automorphisms of the fine grading of sl n c associated with the generalized pauli matrices we consider the grading of sl n c by the group n of generalized pauli matrices the grading decomposes the lie algebra into n2 1 one dimensional subspaces in the article we demonstrate that the normalizer of grading decomposition of sl n c in n is the group sl 2 zn where zn is the cyclic group of order n as an example we consider sl 3 c graded by 3 and all contractions preserving that grading we show that the set of 48 quadratic equations for grading parameters splits into just two orbits of the normalizer of the grading in 3

weyl groups of fine gradings on matrix algebras octonions and the albert algebra given a grading a g gag on a nonassociative algebra a by an abelian group g we have two subgroups of aut a the automorphisms that stabilize each component ag as a subspace and the automorphisms that permute the components by the weyl group of we mean the quotient of the latter subgroup by the former in the case of a cartan decomposition of a semisimple complex lie algebra this is the automorphism group of the root system i e the so called extended weyl group a grading is called fine if it cannot be refined we compute the weyl groups of all fine gradings on matrix algebras octonions and the albert algebra over an algebraically closed field of characteristic different from 2 in the case of the albert algebra

grading refinements in the contractions of lie algebras and their invariants all mutually inequivalent toroidal gradings of the lie algebra are constructed each of them provides several different graded contractions to other eight dimensional lie algebras these can be either continuous or discrete the continuous ones are related to generalized wigner inonu contractions the behaviour of casimir operators under the graded contractions is studied and the casimir and generalized casimir operators of the contracted lie algebras are presented

on lie gradings ii abstract to obtain fine gradings of gl n c it is necessary to study maximal abelian subgroups mad subgroups of diagonizable automorphisms in a ut gl n c we described two finite sets h n and k n such that each mad subgroup consisting only from inner automorphisms is conjugated to some element of h n and each mad subgroup containing at least one outer automorphism is conjugated to some element of k n using these results concerning k n we then easily find all mad subgroups in a ut o n c for n 8 and in a ut sp 2n c

the weyl group of the fine grading of sl n c associated with tensor product of generalized pauli matrices we consider the fine grading of sl n c induced by tensor product of generalized pauli matrices in the paper regard pgl n c as the inner automorphism group of sl n c based on the classification of maximal abelian subgroups of pgl n c consisting of diagonalizable automorphisms by havlicek et al we prove that any such subgroup k of pgl n c is a symplectic abelian group and its weyl group which describes the symmetry of the fine grading induced by the action of k is just the isometry group of the symplectic abelian group k for a finite symplectic abelian group it is also proved that its isometry group is always generated by the transvections contained in it

multicomplementary operators via finite fourier transform a complete set of d 1 mutually unbiased bases exists in a hilbert space of dimension d whenever d is a power of a prime we discuss a simple construction of d 1 disjoint classes each one having d 1 commuting operators such that the corresponding eigenstates form sets of unbiased bases such a construction works properly for prime dimension we investigate an alternative construction in which the real numbers that label the classes are replaced by a finite field having d elements one of these classes is diagonal and can be mapped to cyclic operators by means of the finite fourier transform which allows one to understand complementarity in a similar way as for the position momentum pair in standard quantum mechanics the relevant examples of two and three qubits and two qutrits are discussed in detail

galois quantum systems irreducible polynomials and riemann surfaces finite quantum systems in which the position and momentum take values in the galois field gf pl are studied ideas from the subject of field extension are transferred in the context of quantum mechanics the frobenius automorphisms in galois fields lead naturally to the frobenius formalism in a quantum context the hilbert space splits into frobenius subspaces which are labeled with the irreducible polynomials associated with the ypl y the frobenius maps transform unitarily the states of a galois quantum system and leave fixed all states in some of its galois subsystems where the position and momentum take values in subfields of gf pl an analytic representation of these systems in the l sheeted complex plane shows deeper links between galois theory and riemann surfaces

a perturbative algorithm for quasi periodic linear systems close to constant coefficients a perturbative procedure is proposed to formally construct analytic solutions for a linear differential equation with quasi periodic but close to constant coefficients the scheme constructs the necessary linear transformations involved in the reduction process up to an arbitrary order in the perturbation parameter it is recursive can be implemented in any symbolic algebra package and leads to accurate analytic approximations sharing with the exact solution important qualitative properties this algorithm can be used in particular to carry out systematic stability analyses in the parameter space of a given system by considering variational equations

galois quantum systems a finite quantum system in which the position and momentum take values in the galois field gf p is constructed from a smaller quantum system in which the position and momentum take values in using field extension the galois trace is used in the definition of the fourier transform the heisenberg weyl group of displacements and the sp 2 gf p group of symplectic transformations are studied a class of transformations inspired by the frobenius maps in galois fields is introduced the relationship of this galois quantum system with its subsystems in which the position and momentum take values in subfields of gf p is discussed

global anomalies on orbifolds we consider global anomalies for heterotic string theory formulated on orbifolds the vanishing of certain characteristic classes in group cohomology provides sufficient conditions for the absence of global anomalies for abelian orbifolds level matching implies these cohomology conditions so suffices for the absence of anomalies for nonabelian orbifolds level matching does not suffice and there are additional constraints we give some examples to illustrate these new constraints

unitary transformations depending on a small parameter we formulate a unitary perturbation theory for quantum mechanics inspired by the liedeprit formulation of canonical transformations the original hamiltonian is converted into a solvable one by a transformation obtained through a magnus expansion this ensures unitarity at every order in a small parameter a comparison with the standard perturbation theory is provided we work out the scheme up to order ten with some simple examples

bmo rational approximation and one dimensional hausdorff content let x c c be compact and let f e vmo c we give necessary and sufficient conditions on f and x for flx to be the limit of a sequence of rational functions without poles on x in the norm of bmo x the space of functions of bounded mean oscillation on x we also characterize those compact x c c with the property that the restriction to x of each function in vmo c which is holomorphic on x is the limit of a sequence of rational functions with poles off x our conditions involve the notion of onedimensional hausdorff content as an application a result related to the inner boundary conjecture is proven 0 introduction and statement of results let x be a compact subset of the complex plane c and let 9ix denote its inner boundary that is the set of points in ax which are not in the closure of a single connected component of c x the starting point of this paper is the following theorem theorem 0 let x c c be compact and assume that 9ix has finite onedimensional hausdorff measure let f be a continuous function on the riemann sphere which is holomorphic on x then we can find a sequence of functions fn in bmo c each holomorphic in a neighborhood of x such that f f in bmo c to explain the motivation for the above result let us begin by recalling that the problem of qualitative uniform rational approximation is controlled by a set function a called continuous analytic capacity see 3 for its definition and basic properties it turns out that no metric description of ca is known so that geometric sufficient conditions for rational approximation have been lacking and a number of natural conjectures are still open to get precise statements for two of them let us recall the definition of the two classical algebras r x and a x associated to the compact x the latter is the set of continuous functions on x which are holomorphic on x and the former is the uniform closure in c x of the rational functions with poles off x vitushkin discovered that in the problem of deciding whether r x a x a decisive role is played by the structure of 9ix accordingly he formulated 19 the so called inner boundary conjecture if a aix 0 then r x a x it is by now clear that the inner boundary conjecture is closely linked to the difficult problem of the semiadditivity of ca see 2 to get rid of the complications received by the editors may 16 1984 1980 mathematics subject classification 1985 revision primary 30e10 secondary 42b99

quantum phase corrections from adiabatic iteration the phase change gamma acquired by a quantum state psi t rangle driven by a hamiltonian h 0 t which is taken slowly and smoothly round a cycle is given by a sequence of approximants gamma k obtained by a sequence of unitary transformations the phase sequence is not a perturbation series in the adiabatic parameter epsilon because each gamma k except gamma 0 contains epsilon to infinite order for spin frac 1 2 systems the iteration can be described in terms of the geometry of parallel transport round loops c k on the hamiltonian sphere non adiabatic effects transitions must cause the sequence of gamma k to diverge for spin systems with analytic h 0 t this happens in a universal way the loops c k are sinusoidal spirals which shrink as epsilon k until k sim epsilon 1 and then grow as k the smallest loop has a size exp 1 epsilon comparable with the non adiabaticity

the pauli matrices in n dimensions and finest gradings of simple lie algebras of type a n 1 properties of the lie algebra gl n c are described for a basis which is a generalization of the 2 2 pauli matrices the 3 3 case is described in detail the remarkable properties of that basis are the grading of the lie algebra it offers each grading subspace is one dimensional and the matrix group it generates it is a finite group with the center of sl n c as its commutator group

graded contractions of lie algebras and central extensions we use the concept of grading of lie algebras to investigate the appearance of central charges during the contraction process as for the usual graded contractions one finds simultaneously the central extensions of classes of algebras rather than specific lie algebras to illustrate the method we consider in detail two physical applications the kinematical algebras of spacetime and the u n bosons limits of some lie algebras

fine grading of sl p2 c generated by tensor product of generalized pauli matrices and its symmetries study of the normalizer of the mad group corresponding to a fine grading offers the most important tool for describing symmetries in the system of nonlinear equations connected with contraction of a lie algebra one fine grading that is always present in any lie algebra sl n c is the pauli grading the mad group corresponding to it is generated by generalized pauli matrices for such mad group we already know its normalizer its quotient group is isomorphic to the lie group sl 2 zn z2 in this paper we deal with a more complicated situation namely that the fine grading of sl p2 c is given by a tensor product of the pauli matrices of the same order p p being a prime we describe the normalizer of the corresponding mad group and we show that its quotient group is isomorphic to sp 4 fp z2 where fp is the finite field with p elements

the quantum normal form and its equivalents a quantum analog called the quantum normal form of the classical birkhoff gustavson normal form is presented the algebraic relationship between the quantum and birkhoff gustavson normal forms has been established by developing the latter using lie transforms it is shown that the birkhoff gustavson normal form can be obtained from the quantum normal form using an anharmonic oscillator and a henon heiles system as test cases the equivalence between the quantum normal form and the rayleigh schrodinger perturbation method is shown this equivalence provides an algebraic connection between the birkhoff gustavson normal form and the rayleigh schrodinger perturbation approach the question of weyl and torus quantizations of the birkhoff gustavson normal form is discussed in the light of the quantum normal form

the construction of solvable lie algebras from equidimensional nilpotent algebras abstract the study of gradings of solvable lie algebras l of finite dimensionover a field f of zero characteristic led the authors to the discovery of equidimensional nilpotent algebras l uniquely determined by l up to f isomorphy conversely for any finite dimensional lie algebra l over f an algorithm is developed which yields in parametric form all solvable lie algebras l determining l as the corresponding nilpotent algebra the exposition is independent of lie grading theory it organizes in a novel way the classification of solvable lie algebras of given dimension around the same task for nilpotent algebras every isomorphy class of solvable f algebras is obtained in this way

variations on a theme of heisenberg pauli and weyl the parentage between weyl pairs the generalized pauli group and the unitary group is investigated in detail we start from an abstract definition of the heisenberg weyl group on the field and then switch to the discrete heisenberg weyl group or generalized pauli group on a finite ring the main characteristics of the latter group an abstract group of order d3 noted pd are given conjugacy classes and irreducible representation classes or equivalently lie algebra of dimension d3 associated with pd leaving the abstract sector a set of weyl pairs in dimension d is derived from a polar decomposition of su 2 closely connected to angular momentum theory then a realization of the generalized pauli group pd and the construction of generalized pauli matrices in dimension d are revisited in terms of weyl pairs finally the lie algebra of the unitary group u d is obtained as a subalgebra of the lie algebra associated with pd this leads to a development of the lie algebra of u d in a basis consisting of d2 generalized pauli matrices in the case where d is a power of a prime integer the lie algebra of su d can be decomposed into d 1 cartan subalgebras

solvable lie algebras of dimension 4 over perfect fields abstract the solvable lie algebras of dimension not greater than four over a perfect field of reference are described in terms of their nilpotent frame counted over finite fields of q elements and classified over the real and complex number fields the number dq n of nonisomorphic solvable lie algebras of dimension n over f q roughly speaking grows as a polynomial in n with qn 2 as highest term but there are smaller additional terms depending on the residue class of q modulo n 1 if n 4

there is no generalization of known formulas for mutually unbiased bases in a quantum system having a finite number n of orthogonal states two orthonormal bases ai and bj are called mutually unbiased if all inner products ai bj have the same modulus 1 n this concept appears in several quantum information problems the number of pairwise mutually unbiased bases is at most n 1 and various constructions of such n 1 bases have been found when n is a power of a prime number we study families of formulas that generalize these constructions to arbitrary dimensions using finite rings we then prove that there exists a set of n 1 mutually unbiased bases described by such formulas if and only if n is a power of a prime number

wigner functions and separability for finite systems a discussion of discrete wigner functions in phase space related to mutually unbiased bases is presented this approach requires mathematical assumptions which limits it to systems with density matrices defined on complex hilbert spaces of dimension pn where p is a prime number with this limitation it is possible to define a phase space and wigner functions in close analogy to the continuous case that is we use a phase space that is a direct sum of n two dimensional vector spaces each containing p2 points this is in contrast to the more usual choice of a two dimensional phase space containing p2n points a useful aspect of this approach is that we can relate complete separability of density matrices and their wigner functions in a natural way we discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd special attention is required for two qubits p 2 and our technique fails to establish the separability property for more than two qubits finally we give a brief discussion of hamiltonian dynamics in the language developed in the paper

discrete wigner function for finite dimensional systems a phase space approach to finite dimensional systems is developed from basic principles for a system describable by a hilbert space of dimension d we define a one to one correspondence between operators and functions on a discrete and finite phase space with points valid for any dimension d the properties fulfilled by this correspondence and its uniqueness are examined this formalism is applied to the number difference and phase difference of a two mode field this case is compared with the marginal distribution for these variables arising from a two mode wigner function for number and phase

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

removable singularities for analytic functions in the little zygmund space using estimates of the law of the iterated logarithm type a sharp sufficient condition for a compact set to be removable for analytic functions in the little zygmund space is given

conjugate variables in finite phase plane we construct two pairs of quasicoordinates and quasimomenta in a finite phase plane which form sets of conjugate variables in such a plane the coordinate x is quantized with a step c and the momentum p with a step where mc is the size of the phase plane in the x direction the construction depends crucially on the possibility of writing m m1m2 with m1 and m2 relatively prime the conjugate variables are applied to harper like hamiltonians it is shown how to design physical systems with energy spectra containing any desired number of discrete energy levels say m1 each of them having a prescribed degeneracy m2

feynman s path integral and mutually unbiased bases our previous work on quantum mechanics in hilbert spaces of finite dimension n is applied to elucidate the deep meaning of feynman s path integral pointed out by g svetlichny he speculated that the secret of the feynman path integral may lie in the property of mutual unbiasedness of temporally proximal bases we confirm the corresponding property of the short time propagator by using a specially devised n n approximation of quantum mechanics in applied to our finite dimensional analogue of a free quantum particle

partial order and a t0 topology in a set of finite quantum systems a whole part theory is developed for a set of finite quantum systems n with variables in z n the partial order subsystem is defined by embedding various attributes of the system m quantum states density matrices etc into their counterparts in the supersystem n for m n the compatibility of these embeddings is studied the concept of ubiquity is introduced for quantities which fit with this structure it is shown that various entropic quantities are ubiquitous the sets of various quantities become t0 topological spaces with the divisor topology which encapsulates fundamental physical properties these sets can be converted into directed complete partial orders by adding top elements the continuity of various maps among these sets is studied

quantum averaging for driven systems with resonances abstract we discuss the effects of resonances in driven quantum systems within the context of quantum averaging techniques in the floquet representation we consider in particular iterative methods of kam type and the extensions needed to take into account resonances the approach consists in separating the coupling terms into resonant and nonresonant components at a given scale of time and intensity the nonresonant part can be treated with perturbative techniques which we formulate in terms of kam type unitary transformations that are close to the identity these can be interpreted as averaging procedures with respect to the dynamics defined by effective uncoupled hamiltonians the resonant parts are treated with a different kind of unitary transfomations that are not close to the identity and are adapted to the structure of the resonances they can be interpreted as a renormalization of the uncoupled hamiltonian that yields an effective dressed hamiltonian around which a perturbation expansion can be developed the combination of these two ingredients provides a strongly improved approximation technique

birkhoff gustavson normal form in classical and quantum mechanics a quantum analogue of the classical transformations to the birkhoff gustavson normal form is derived with unitary transformations taking over the role of canonical transformations the author finds a striking similarity to lie transformations in the classical case the author suggests that the quantum normal form is identical to the rayleigh schrodinger perturbation series and thus usually only asymptotic to the eigenvalues the convergence question of both classical and quantum normal forms is discussed in some detail

factorizations and physical representations a hilbert space in m dimensions is shown explicitly to accommodate representations that reflect the decomposition of m into prime numbers representations that exhibit the factorization of m into two relatively prime numbers the kq representation zak j 1970 phys today 23 51 and related representations termed q1q2 representations together with their conjugates are analysed as well as a representation that exhibits the complete factorization of m in this latter representation each quantum number varies in a subspace that is associated with one of the prime numbers that make up m

histories of adiabatic quantum transitions the way in which the transition amplitude to an initially unoccupied state increases to its exponentially small final value is studied in detail in the adiabatic approximation for a 2 state quantum system by transforming to a series of superadiabatic bases clinging ever closer to the exact evolving state it is shown that transition histories renormalize onto a universal one in which the amplitude grows to its final value as an error function rather than via large oscillations as in the ordinary adiabatic basis the time for the universal transition is of order surd hslash delta where delta is the small adiabatic slowness parameter in perturbation theory the pre exponential factor of the final amplitude renormalizes superadiabatically from the incorrect value textstyle frac 1 3 pi for the ordinary adiabatic basis to the correct value unity the various histories could be observed in spin experiments

quantum averaging iii time dependent systems we present new perturbation algorithms for time dependent quantum systems these new methods make use of a time dependent averaging method and are based on analogies with the poincare von zeipel pvz and kolmogorov arnold moser kam expansions application to a simple example shows that the pvz as well as the kam expansion are superior to the standard quantum mechanical time dependent perturbation method dyson expansion

joint approximation in bmo a characterization of mergelyan sets for the space of harmonic functions whose boundary values are in bmo vmo is obtained the main step of the proof is the use of certain vmo functions which are related to the sharpness of the johnnirenberg inequality

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

graded contractions of the lie algebra e 2 1 the graded contractions of pseudo euclidean lie algebra are studied the non equivalent gradings of of type and a are extended to the entire lie algebra using the action of on the abelian ideal the translations the graded contractions embed into a large family of six dimensional lie algebras the family includes solvable nilpotent and nonsolvable lie algebras both decomposable and indecomposable ones the distinction between graded and inonu wigner contractions is analysed the physically most interesting lie algebras obtained by the contractions are the inhomogeneous galilei and pseudo galilei

weak mutually unbiased bases quantum systems with variables in are considered the properties of lines in the phase space of these systems are studied weak mutually unbiased bases in these systems are defined as bases for which the overlap of any two vectors in two different bases is equal to d 1 2 or alternatively to one of the d 1 2i 0 where di is a divisor of d apart from d 1 they are designed for the geometry of the phase space in the sense that there is a duality between the weak mutually unbiased bases and the maximal lines through the origin in the special case of prime d there are no divisors of d apart from 1 d and the weak mutually unbiased bases are mutually unbiased bases

symmetries of finite heisenberg groups for multipartite systems a composite quantum system comprising a finite number k of subsystems which are described with position and momentum variables in i 1 k is considered its hilbert space is given by a k fold tensor product of hilbert spaces of dimensions n1 nk the symmetry group of the respective finite heisenberg group is given by the quotient group of certain normalizer this paper extends our previous investigation of bipartite quantum systems to arbitrary multipartite systems of the above type it provides detailed description of the normalizers and the corresponding symmetry groups the new class of symmetry groups represents a very specific generalization of symplectic groups over modular rings as an application a new proof of existence of the maximal set of mutually unbiased bases in hilbert spaces of prime power dimensions is provided

weyl groups of some fine gradings on e6 abstract the weyl groups of the fine gradings on the exceptional lie algebra e 6 whose universal grading group is infinite are computed

the angle angular momentum quantum phase space the angle angular momentum quantum phase space is considered and the corresponding heisenberg weyl group is studied when 2j 1 is a power of a prime z 2j 1 is a galois field and stronger results can be proved sl 2 z 2j 1 transformations are explicitly constructed and various properties of the displacement operators are studied central extensions of the abelian group by z 2j 1 are studied and they provide all the ways of constructing the heisenberg weyl group from g and z 2j 1

factorization in finite quantum systems unitary transformations in an angular momentum hilbert space h 2j 1 are considered they are expressed as a finite sum of the displacement operators which play the role of su 2j 1 generators with the weyl function as coefficients the chinese remainder theorem is used to factorize large qudits in the hilbert space h 2j 1 in terms of smaller qudits in hilbert spaces h 2ji 1 all unitary transformations on large qudits can be performed through appropriate unitary transformations on the smaller qudits

graded contractions of the pauli graded sl 3 c the lie algebra sl 3 c is considered in the basis of generalized pauli matrices corresponding grading is the pauli grading here it is one of the four gradings of the algebra which cannot be further refined the set s of 48 contraction equations for 24 contraction parameters is solved our main tools are the symmetry group of the pauli grading of sl 3 c which is essentially the finite group sl 2 z3 and the induced symmetry of the system s a list of all equivalence classes of solutions of the contraction equations is provided among the solutions 175 equivalence classes are non parametric and 13 solutions depend on one or two continuous parameters providing a continuum of equivalence classes and subsequently continuum of non isomorphic lie algebras solutions of the contraction equations of pauli graded sl 3 c are identified here as specific solvable lie algebras of dimensions up to 8 earlier algorithms for identification of lie algebras given by their structure constants had to be made more efficient in order to distinguish non isomorphic lie algebras encountered here resulting lie algebras are summarized in tabular form there are 88 indecomposable solvable lie algebras of dimension 8 77 of them being nilpotent there are 11 infinite sets of parametric lie algebra which still deserve further study

symplectic transformations and quantum tomography in finite quantum systems quantum systems where the position and momentum are in the ring zd d is an odd integer are considered symplectic transformations are studied and the order of sp 2 zd is calculated quantum tomography is also discussed it is shown that measurements used in the inverse radon transform need to be made on j2 d lines where j2 d is the jordan totient function

the symmetries of the fine gradings of sl nk c associated with direct product of pauli groups a grading of a lie algebra is called fine if it could not be further refined for a fine grading of a simple lie algebra we define its weyl group to describe the symmetry of this grading it is already known that the weyl group of the fine grading of sl n c induced by the action of the group n of the generalized pauli matrices of rank n is sl 2 zn where zn is the cyclic group of order n in this paper we consider the fine grading of sl nk c induced by the action of the group of k fold tensor product of the generalized pauli matrices of rank n we prove that its weyl group is sp 2k zn and is generated by transvections therefore this generalizes the previous result

hjelmslev geometry of mutually unbiased bases the basic combinatorial properties of a complete set of mutually unbiased bases mubs of a q dimensional hilbert space with p being a prime and r a positive integer are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective hjelmslev plane defined over a galois ring of characteristic p2 and rank r the q vectors of a basis of correspond to the q points of a so called neighbour class and the q 1 mubs answer to the total number of pairwise disjoint neighbour classes on the conic

mutually unbiased bases generalized spin matrices and separability abstract a collection of orthonormal bases for a complex d dimensional hilbert space is called mutually unbiased mub if for any two vectors v and w from different bases the square of the inner product equals 1 d v w 2 1 d the mub problem is to prove or disprove the existence of a maximal set of d 1 bases it has been shown in ann phys 191 1989 363 that such a collection exists if d is a power of a prime number p we revisit this problem and use d d generalizations of the pauli spin matrices to give a constructive proof of this result specifically we give explicit representations of commuting families of unitary matrices whose eigenvectors solve the mub problem additionally we give formulas from which the orthogonal bases can be readily computed we show how the techniques developed here provide a natural way to analyze the separability of the bases the techniques used require properties of algebraic field extensions and the relevant part of that theory is included in the appendix

group theoretical construction of mutually unbiased bases in hilbert spaces of prime dimensions mutually unbiased bases in hilbert spaces of finite dimensions are closely related to the quantal notion of complementarity an alternative proof of existence of a maximal collection of n 1 mutually unbiased bases in hilbert spaces of prime dimension n is given by exploiting the finite heisenberg group also called the pauli group and the action of on finite phase space implemented by unitary operators in the hilbert space crucial for the proof is that for prime is also a finite field

discrete and continuous graded contractions of representations of lie algebras simultaneous grading of lie algebras and their representation spaces is used to develop a new theory of grading preserving contractions of representations of all lie algebras admitting the chosen grading the theory is completely different from the traditional ways of contracting representations the graded contractions fall naturally into two classes discrete and continuous ones related respectively to 2 cocycles and coboundaries of the grading group

geometrical approach to the discrete wigner function in prime power dimensions we analyse the wigner function in prime power dimensions constructed on the basis of the discrete rotation and displacement operators labelled with elements of the underlying finite field we separately discuss the case of odd and even characteristics and analyse the algebraic origin of the non uniqueness of the representation of the wigner function explicit expressions for the wigner kernel are given in both cases

discrete and continuous graded contractions of lie algebras and superalgebras grading preserving contractions of lie algebras and superalgebras of any type over the complex number field are defined and studied such contractions fall naturally into two classes the wigner inonu like continuous contractions and new discrete contractions a general method is described for any abelian grading semigroup and any lie algebra or superalgebra admitting such a grading all contractions preserving z2 z3 and z2 z2 gradings are found examples of these gradings and contractions for the simple lie algebra a2 affine kac moody algebra a 1 and the simple superalgebra osp 2 1 are shown

the four sets of additive quantum numbers of su 3 the four maximal sets of additive quantum numbers and related fine gradings of the lie algebra sl 3 c are described in detail the quantum numbers are determined by a grading of the lie algebra fine gradings providing maximal sets of them two sets of additive quantum numbers are equivalent precisely if the corresponding gradings are equivalent under a transformation from the automorphism group of the lie algebra