prime decomposition and correlation measure of finite quantum systems under the name prime decomposition pd a unique decomposition of an arbitrary n dimensional density matrix into a sum of separable density matrices with dimensions determined by the coprime factors of n is introduced for a class of density matrices a complete tensor product factorization is achieved the construction is based on the chinese remainder theorem and the projective unitary representation of by the discrete heisenberg group the pd isomorphism is unitarily implemented and it is shown to be co associative and to act on as comultiplication density matrices with complete pd are interpreted as group like elements of to quantify the distance of from its pd a trace norm correlation index is introduced and its invariance groups are determined

renormalization of gauge invariant operators and anomalies in yang mills theory a long standing conjecture on the structure of renormalized gauge invariant integrated operators of arbitrary dimension in yang mills theory is established the general solution of the consistency condition for anomalies with sources included is also derived this is achieved by computing explicitly the cohomology of the full unrestricted becchi rouet stora tyutin operator in the space of local polynomial functionals with ghost number equal to zero or one the argument does not use power counting and is purely cohomological it relies crucially on standard properties of the antifeld formalism

the batalin fradkin and vilkovisky formalism for higher order theories the generalized batalin fradkin and vilkovisky bfv formalism was developed as a method for determining the ghost structure for theories such as gravity and supergravity whose hamiltonian formalism has constraints not related to a lie algebra action previously the classical dynamical content of the bfv description of yang mills theory was investigated there it was found that this approach had a homological interpretation derived from the lie algebra cohomology of the gauge group which allowed one to understand the construction in terms of the dirac approach to constrained systems in this paper the dynamical consequences of the generalized bfv formalism are investigated it is found that even though one no longer has a lie algebra structure associated with the constraints one can still develop a homology theory that reproduces the dirac analysis and from which the generalized bfv formalism can be derived some of the consequences of this approach are discussed

geometrical description of quantal state determination under the assumption that every quantal measurement may give data about the post measurement state of the inspected ensemble the problem of the state determination is reconsidered it is shown that orthogonal decomposition of the set of complex n n hermitian matrices into the commutative subsets allows operators to be found such that post measurement information on these observables allows a partial in some cases total determination of the pre measurement state to be effected

generalized symmetries in mechanics and field theories generalized symmetries are introduced in a geometrical and global formalism such a framework applies naturally to field theories and specializes to mechanics generalized symmetries are characterized in a lagrangian context by means of the transformation rules of the poincare cartan form and the generalized nother theorem is applied to obtain conserved quantities first integrals in mechanics in the particular case of mechanics it is shown how to use generalized symmetries to study the separation of variables of hamilton jacobi equations recovering standard results by means of this new method supersymmetries wess zumino model are considered as an intriguing example in field theory

the simple harmonic urn we study a generalized polya urn model with two types of ball if the drawn ball is red it is replaced together with a black ball but if the drawn ball is black it is replaced and a red ball is thrown out of the urn when only black balls remain the roles of the colors are swapped and the process restarts we prove that the resulting markov chain is transient but that if we throw out a ball every time the colors swap the process is recurrent we show that the embedded process obtained by observing the number of balls in the urn at the swapping times has a scaling limit that is essentially the square of a bessel diffusion we consider an oriented percolation model naturally associated with the urn process and obtain detailed information about its structure showing that the open subgraph is an infinite tree with a single end we also study a natural continuous time embedding of the urn process that demonstrates the relation to the simple harmonic oscillator in this setting our transience result addresses an open problem in the recurrence theory of two dimensional linear birth and death processes due to kesten and hutton we obtain results on the area swept out by the process we make use of connections between the urn process and birth death processes a uniform renewal process the eulerian numbers and lamperti s problem on processes with asymptotically small drifts we prove some new results on some of these classical objects that may be of independent interest for instance we give sharp new asymptotics for the first two moments of the counting function of the uniform renewal process finally we discuss some related models of independent interest including a poisson earthquakes markov chain on the homeomorphisms of the plane

deformations of global symmetries in the extended antifield formalism it is outlined how deformations of field theoretical rigid symmetries can be constructed and classified by cohomological means in the extended antifield formalism special attention is devoted to deformations referring only to a subset of the rigid symmetries of a given model and leading to a nontrivial extension of the graded lie algebra associated with that subset the method is illustrated for a d 4 n 2 supersymmetric model where the central extension of the supersymmetry algebra emerges via a deformation deformations of gauge fixed actions with a brst symmetry are discussed too and illustrated by the curci ferrari model

noether s second theorem in a general setting reducible gauge theories we prove noether s direct and inverse second theorems for lagrangian systems on fibre bundles in the case of gauge symmetries depending on derivatives of dynamic variables and parameters of an arbitrary order the appropriate notions of a reducible gauge symmetry and noether identity are formulated and their equivalence by means of a certain intertwining operator is proved

the antifield koszul tate complex of reducible noether identities a generic degenerate lagrangian system of even and odd fields is examined in algebraic terms of the grassmann graded variational bicomplex its euler lagrange operator obeys noether identities which need not be independent but satisfy first stage noether identities and so on we show that if a certain necessary and sufficient condition holds one can associate to a degenerate lagrangian system the exact koszul tate complex with the boundary operator whose nilpotency condition restarts all its noether and higher stage noether identities this complex provides a sufficient analysis of the degeneracy of a lagrangian system for the purpose of its bv quantization

homological perturbation theory and the algebraic structure of the antifield antibracket formalism for gauge theories the algebraic structure of the antifield antibracket formalism for both reducible and irreducible gauge theories is clarified this is done by using the methods of homological perturbation theory hpt a crucial ingredient of the construction is the koszul tate complex associated with the stationary surface of the classical extremals the koszul tate differential acts on the antifields and is graded by the antighost number it provides a resolution of the algebraa of functions defined on the stationary surface namely it is acyclic except at degree zero where its homology group reduces toa acyclicity only holds because of the introduction of the ghosts of ghosts and provides an alternative criterion for what is meant by a proper solution of the master equation the existence of the brst symmetry follows from the techniques of hpt the classical lagrangian brst cohomology is completely worked out and shown to be isomorphic with the cohomology of the exterior derivative along the gauge orbits on the stationary surface the algebraic structure of the formalism is identical with the structure of the hamiltonian brst construction the role played there by the constraint surface is played here by the stationary surface only elementary quantum questions general properties of the measure are addressed

existence theorem for gauge algebra a general gauge action is defined by postulating a minimum of its properties necessary for the existence of loop expansion in the quantum theory the structure of the general gauge algebra is derived from these postulates the proof of existence of the structure functions lacking in previous works is given the question of uniqueness is also completely cleared up some unsolved problems are discussed in connection with the covariant quantization conjecture

tomographically complete sets of orthonormal bases in finite systems quantum systems where the position and momentum are in the ring d is an odd integer are considered a tomographically complete set of bases i e a set of bases such that probabilities from tomography experiments corresponding to these bases can be used for the calculation of an arbitrary density matrix is considered such a set of bases is constructed by considering a factorization of a system with dimension d p1p2 where p1 p2 are prime numbers in terms of two subsystems with dimensions p1 and p2 appropriate combination of the mutually unbiased bases in the subsystems leads to a tomographically complete set of bases in the full system this set leads to probabilities along all the lines through the origin 0 0 in the phase space with exactly d points it is shown that the number of these lines is d the dedekind function the general theory is exemplified with examples

covariant differential identities and conservation laws in metric torsion theories of gravitation i general consideration arbitrary diffeomorphically invariant metric torsion theories of gravity are considered it is assumed that lagrangians of such theories contain derivatives of field variables tensor densities of arbitrary ranks and weights up to a second order only the generalized klein noether methods for constructing manifestly covariant identities and conserved quantities are developed manifestly covariant expressions are constructed without including auxiliary structures like a background metric in the riemann cartan space the following manifestly generally covariant results are presented a the complete generalized system of differential identities the klein noether identities is obtained b the generalized currents of three types depending on an arbitrary vector field displacements are constructed they are the canonical noether current symmetrized belinfante current and identically conserved hilbert bergmann current in particular it is stated that the symmetrized belinfante current does not depend

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

existence uniqueness and cohomology of the classical brst charge with ghosts of ghosts a complete canonical formulation of the brst theory of systems with redundant gauge symmetries is presented these systems includep form gauge fields the superparticle and the superstring we first define the koszul tate differential and explicitly show how the introduction of the momenta conjugate to the ghosts of ghosts makes it acyclic the global existence of the brst generator is then demonstrated and the brst charge is proved to be unique up to canonical transformations in the extended phase space which includes the ghosts finally the brst cohomology in classical mechanics is investigated and shown to be equal to the cohomology of the exterior derivative along the gauge orbits as in the irreducible case this is done by re expressing the exterior algebra along the gauge orbits as a free differential algebra containing generators of higher degree which are identified with the ghosts of ghosts the quantum cohomology is not dealt with

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

billiards in a general domain with random reflections we study stochastic billiards on general tables a particle moves according to its constant velocity inside some domain fancyscript d subset mathbb r d until it hits the boundary and bounces randomly inside according to some reflection law we assume that the boundary of the domain is locally lipschitz and almost everywhere continuously differentiable the angle of the outgoing velocity with the inner normal vector has a specified absolutely continuous density we construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location velocity we mainly focus on the case of bounded domains then we prove exponential ergodicity of these two markov processes we study their invariant distribution and their normal gaussian fluctuations of particular interest is the case of the cosine reflection law the stationary distributions for the two processes are uniform in this case the discrete time chain is reversible though the continuous time process is quasi reversible also in this case we give a natural construction of a chord picked at random in fancyscript d and we study the angle of intersection of the process with a d 1 dimensional manifold contained in fancyscript d

hamiltonian brst anti brst theory the hamiltonian brst anti brst theory is developed in the general case of arbitrary reducible first class systems this is done by extending the methods of homological perturbation theory originally based on the use of a single resolution to the case of a biresolution the brst and the anti brst generators are shown to exist the respective links with the ordinary brst formulation and with thesp 2 covariant formalism are also established

brst cohomology in classical mechanics the classical non quantum cohomology of the becchi rouet stora tyutin brst symmetry in phase space is defined and worked out no group action for the gauge transformations is assumed the results cover therefore the general case of an open algebra and are valid off shell each cohomology class contains all brst invariant functions with fixed ghost number an integer which differ from each other by a brst variation if the ghost number is negative there is only the trivial class whose elements are equivalent to zero if the ghost number is positive or zero there is a bijective correspondence between the brst classes and those of the exterior derivative along the gauge orbits these gauge orbits lie in the phase space surface on which the gauge generators are constrained to vanish the brst invariant functions of ghost numberp are then related to closedp forms along the orbits the addition of a brst variation corresponds to the addition of an exact form some comments about the quantum case are included the physical meaning of the classes with ghost number greater than zero is not discussed

on knudsen flows within thin tubes the dynamics of a test particle in an infinite tube is investigated it is proven that the evolution tends to that of a diffusion process as the radius of the tube decreases to zero this justifies the hypotheses underlying an experiment of clausing 1930

local brst cohomology in the antifield formalism i general theorems we establish general theorems on the cohomologyh s d of the brst differential modulo the spacetime exterior derivative acting in the algebra of localp forms depending on the fields and the antifields sources for the brst variations it is shown thath k s d is isomorphic toh k d in negative ghost degree k k 0 where is the koszul tate differential associated with the stationary surface the cohomology grouph1 d in form degreen is proved to be isomorphic to the space of constants of the motion thereby providing a cohomological reformulation of noether s theorem more generally the grouph k d in form degreen is isomorphic to the space ofn k forms that are closed when the equations of motion hold the groupsh k d k 2 are shown to vanish for standard irreducible gauge theories the grouph2 d is then calculated explicitly for electromagnetism yang mills models and einstein gravity the invariance of the groupsh k s d under the introduction of non minimal variables and of auxiliary fields is also demonstrated in a companion paper the general formalism is applied to the calculation ofh k s d in yang mills theory which is carried out in detail for an arbitrary compact gauge group

asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains we study stochastic billiards in infinite planar domains with curvilinear boundaries that is piecewise deterministic motion with randomness introduced via random reflections at the domain boundary physical motivation for the process originates with ideal gas models in the knudsen regime with particles reflecting off microscopically rough surfaces we classify the process into recurrent and transient cases we also give almost sure results on the long term behaviour of the location of the particle including a super diffusive rate of escape in the transient case a key step in obtaining our results is to relate our process to an instance of a one dimensional stochastic process with asymptotically zero drift for which we prove some new almost sure bounds of independent interest we obtain some of these bounds via an application of general semimartingale criteria also of some independent interest

a note on local brst cohomology of yang mills type theories with free abelian factors we extend previous work on antifield dependent local becchi rouet stora tyutin brst cohomology for matter coupled gauge theories of yang mills type to the case of gauge groups that involve free abelian factors more precisely we first investigate in a model independent way how the dynamics enters the computation of the cohomology for a general class of lagrangians in general spacetime dimensions we then discuss explicit solutions in the case of specific models our analysis has implications for the structure of characteristic cohomology and for consistent deformations of the classical models as well as for divergences counterterms and for gauge anomalies that may appear during perturbative quantization

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

noether s second theorem for brst symmetries we present noether s second theorem for graded lagrangian systems of even and odd variables on an arbitrary body manifold x in a general case of brst symmetries depending on derivatives of dynamic variables and ghosts of any finite order as a preliminary step noether s second theorem for lagrangian systems on fiber bundles y x possessing gauge symmetries depending on derivatives of dynamic variables and parameters of arbitrary order is proved

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

cohomology of the infinite order jet space and the inverse problem cohomology of the bicomplex of exterior forms on the infinite order jet space of a smooth fiber bundle of a field model is computed this provides a solution of the global inverse problem of the calculus of variations in lagrangian field theory and time dependent mechanics in the case of an affine fiber bundle the outcomes to brst theory are discussed we show that there is no topological obstruction to constructing global descent equations in the even field sector of brst theory

random walk with barycentric self interaction we study the asymptotic behaviour of a d dimensional self interacting random walk x n n 1 2 3 which is repelled or attracted by the centre of mass g n n 1 sum i 1 n x i of its previous trajectory the walk s trajectory x 1 x n models a random polymer chain in either poor or good solvent in addition to some natural regularity conditions we assume that the walk has one step mean drift mathbb e x n 1 x n mid x n g n mathbf x approx rho mathbf x beta hat mathbf x for and 0 when 0 we show that x n is transient with a limiting random direction and satisfies a super diffusive law of large numbers n 1 1 x n converges almost surely to some random vector when 0 1 there is sub ballistic rate of escape when 0 and we give almost sure bounds on the norms x n which in the context of the polymer model reveal extended and collapsed phases

perturbation theory and locality in the field antifield formalism the batalin vilkovisky formalism is studied in the framework of perturbation theory by analyzing the antibracket becchi rouet stora tyutin brst cohomology of the proper solution s0 it is concluded that the recursive equations for the complete proper solution s can be solved at any order of perturbation theory if certain conditions on the classical action and on the gauge generators are imposed the solution can be taken local

on the notion of gauge symmetries of generic lagrangian field theory general lagrangian theory of even and odd fields on an arbitrary smooth manifold is considered its nontrivial reducible gauge symmetries and their algebra are defined in this very general setting by means of the inverse second noether theorem in contrast with gauge symmetries nontrivial noether and higher stage noether identities of lagrangian theory can be intrinsically defined by constructing the exact koszul tate complex the inverse second noether theorem that we prove associates with this complex the cochain sequence with the ascent operator whose components define nontrivial gauge and higher stage gauge symmetries these gauge symmetries are said to be algebraically closed if the ascent operator can be extended to a nilpotent operator the necessary conditions for this extension are stated the characteristic examples of yang mills supergauge theory topological chern simons theory gauge gravitation theory and topological background field bf theory are presented

quantum theory the church turing principle and the universal quantum computer it is argued that underlying the church turing hypothesis there is an implicit physical assertion here this assertion is presented explicitly as a physical principle every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means classical physics and the universal turing machine because the former is continuous and the latter discrete do not obey the principle at least in the strong form above a class of model computing machines that is the quantum generalization of the class of turing machines is described and it is shown that quantum theory and the universal quantum computer are compatible with the principle computing machines resembling the universal quantum computer could in principle be built and would have many remarkable properties not reproducible by any turing machine these do not include the computation of non recursive functions but they do include quantum parallelism a method by which certain probabilistic tasks can be performed faster by a universal quantum computer than by any classical restriction of it the intuitive explanation of these properties places an intolerable strain on all interpretations of quantum theory other than everett s some of the numerous connections between the quantum theory of computation and the rest of physics are explored quantum complexity theory allows a physically more reasonable definition of the complexity or knowledge in a physical system than does classical complexity theory

lyapunov functions for random walks and strings in random environment we study two typical examples of countable markov chains in random environment using the lyapunov functions method random walk and random string in random environment in each case we construct an explicit lyapunov function investigating the behavior of this function we get the classification for recurrence transience ergodicity we obtain new results for random strings in random environment though we simply review well known results for random walks using our approach