universality results for the largest eigenvalues of some sample covariance matrix ensembles for sample covariance matrices with i i d entries with sub gaussian tails when both the number of samples and the number of variables become large and the ratio approaches one it is a well known result of soshnikov that the limiting distribution of the largest eigenvalue is same that of gaussian samples in this paper we extend this result to two cases the first case is when the ratio approaches an arbitrary finite value the second case is when the ratio becomes infinite or arbitrarily small

on the existence of local observables in theories with a factorizing s matrix the scaling limit of the two dimensional ising model above the critical temperature is considered as an example for relativistic quantum theories on two dimensional minkowski space exhibiting a factorizing s matrix in this model a recently proposed criterion for the existence of local quantum field theories with a prescribed factorizing scattering matrix is verified thereby establishing a new constructive approach to two dimensional quantum field theory in a particular example the existence proof is accomplished by analysing the nuclearity properties of certain specific subsets of fermionic fock spaces and yields as a byproduct also a verification of the energy nuclearity condition of buchholz and wichmann in models of free fermions in four spacetime dimensions

standard forms of local nets in quantum field theory nets of von neumann algebras with local algebras associated with all compact convex causally complete subsets of spacetime are constructed assuming a particular kind of net considered by bisognano and wichmann the properties of such nets are discussed within the general framework of algebraic quantum field theory with an emphasis on the relation between the geometry of spacetime and the structure of the nets an intersection property for the local algebras is stated and proved the structure of subnets of such nets is discussed a number of particular possible features of the nets are shown to be hereditary in the sense that the subnets have the same features possible further extensions of the domain of definition of these nets are discussed briefly

on the continuity of the map ϑ ϑ from the predual of a w algebra abstract let be elements in the predual of a w algebra for their absolute value parts the estimate 2 1 2 is obtained

an analogue of the kac wakimoto formula and black hole conditional entropy a local formula for the dimension of a superselection sector in quantum field theory is obtained as vacuum expectation value of the exponential of the proper hamiltonian in the particular case of a chiral conformal theory this provides a local analogue of a global formula obtained by kac and wakimoto within the context of representations of certain affine lie algebras our formula is model independent and its version in general quantum field theory applies to black hole thermodynamics the relative free energy between two thermal equilibrium states associated with a black hole turns out to be proportional to the variation of the conditional entropy in different superselection sectors where the conditional entropy is defined as the connes stormer entropy associated with the dhr localized endomorphism representing the sector the constant of proportionality is half of the hawking temperature as a consequence the relative free energy is quantized proportionally to the logarithm of a rational number in particular it is equal to a linear function of the logarithm of an integer once the initial state or the final state is taken fixed

kramers equation and supersymmetry hamilton s equations with noise and friction possess a hidden supersymmetry valid for time independent as well as periodically time dependent systems it is used to derive topological properties of critical points and periodic trajectories in an elementary way from a more practical point of view the formalism provides new tools to study the reaction paths in systems with separated time scales a reduced current which contains the relevant part of the phase space probability current is introduced together with strategies for its computation

a note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices recently johansson and johnstone proved that the distribution of the properly rescaled largest principal component of the complex real wishart matrix x x xtx converges to the tracy widom law as n p the dimensions of x tend to in some ratio n p 0 we extend these results in two directions first of all we prove that the joint distribution of the first second third etc eigenvalues of a wishart matrix converges after a proper rescaling to the tracy widom distribution second of all we explain how the combinatorial machinery developed for wigner random matrices in refs 27 38 and 39 allows to extend the results by johansson and johnstone to the case of x with non gaussian entries provided n p o p1 3 we also prove that max n1 2 p1 2 2 o p1 2 log p a e for general 0

extremal eigenvalues and eigenvectors of deformed wigner matrices we consider random matrices of the form h w lambda v lambda in mathbb r where w is a real symmetric or complex hermitian wigner matrix of size n and v is a real bounded diagonal random matrix of size n with i i d entries that are independent of w we assume subexponential decay of the distribution of the matrix entries of w and we choose lambda sim 1 so that the eigenvalues of w and lambda v are typically of the same order further we assume that the density of the entries of v is supported on a single interval and is convex near the edges of its support in this paper we prove that there is lambda in mathbb r such that the largest eigenvalues of h are in the limit of large n determined by the order statistics of v for lambda lambda in particular the largest eigenvalue of h has a weibull distribution in the limit n rightarrow infty if lambda lambda moreover for n sufficiently large we show that the eigenvectors associated to the largest eigenvalues are partially localized for lambda lambda while they are completely delocalized for lambda lambda similar results hold for the lowest eigenvalues

hilbert space structure in classical mechanics i in this paper we study the hilbert space structure underlying the koopman von neumann kvn operatorial formulation of classical mechanics kvn limited themselves to study the hilbert space of zero forms that are the square integrable functions on phase space they proved that in this hilbert space the evolution is unitary for every system in this paper we extend the kvn hilbert space to higher forms which are basically functions of the phase space points and the differentials on phase space we prove that if we equip this space with a positive definite scalar product the evolution can turn out to be nonunitary for some systems vice versa if we insist in having a unitary evolution for every system then the scalar product cannot be positive definite identifying the one forms with the jacobi fields we provide a physical explanation of these phenomena we also prove that the unitary nonunitary character of the evolution is invariant under canonical transformations

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

fluctuations of matrix entries of regular functions of wigner matrices we study the fluctuations of the matrix entries of regular functions of wigner random matrices in the limit when the matrix size goes to infinity in the case of the gaussian ensembles goe and gue this problem was considered by a lytova and l pastur j stat phys 134 147 159 2009 our results are valid provided the off diagonal matrix entries have finite fourth moment the diagonal matrix entries have finite second moment and the test functions have four continuous derivatives in a neighborhood of the support of the wigner semicircle law moreover if the marginal distributions satisfy the poincare inequality our results are valid for lipschitz continuous test functions

conformal quantum field theory and half sided modular inclusions of von neumann algebras letn be von neumann algebras on a hilbert space a comon cyclic and separarting vector assume to be cyclic and separating also forn denote byj jn the modular conjugations to and n the associated modular operators if open image in new window and open image in new window these data define in a canonical way a conformal quantum field theory in a cricle conversely the chiral part of a conformal quantum field theory in two dimensions always yields such data in a natural way

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

solution of the similarity problem for cyclic representations of c algebras in a paper from 1955 13 kadison considered the following problem is any bounded non selfadjoint representation 77 of a c algebra a on a hilbert space h similar to a representation i e does there exist an invertible operator t e b h such that tht t is a representation of a this problem is a natural counterpart to a problem considered by dixmier in 1950 9 is any bounded representation of a group g on a hilbert space similar to a unitary representation the group question was solved in the negative by kunze and stein in 1960 14 which in turn was based on previous work of ehrenpreis and mautner 21 the similarity problem for representations of c algebras is still open although a number of partial results have been obtained using the deep results of connes on infective von neumann algebras 7 and the characterization of nuclear c algebras given in 3 one gets the result fairly easily that every bounded representation of a nuclear c algebra is similar to a representation cf 2 th 3 5 and 5 th 4 1 the similarity problem for bounded cyclic representations of arbitrary c algebras was first considered by barnes in 1 he proved that when 77 is a bounded representation of a c algebra a on a hilbert space h such that 7v a and v a have a common cyclic vector i then there exists a closed infective operator t on h such that t and t are densely defined and such that x tht x t is a representation of a bunce sharpened this in 2 by proving that t can be chosen bounded but not necessarily with bounded inverse he also removed the condition that is cyclic for 7v a finally christensen proved in 5 that every irreducible bounded representation of a c algebra is similar to a representation these two results of bunce and christensen rely on an inequality due to pisier 16 cor 2 3 stating that when s is a bounded linear map from a c algebra a into a c algebra b then for all n e n and a an e a

edge universality of beta ensembles we prove the edge universality of the beta ensembles for any beta ge 1 provided that the limiting spectrum is supported on a single interval and the external potential is fancyscript c 4 and regular we also prove that the edge universality holds for generalized wigner matrices for all symmetry classes moreover our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class fancyscript c 4

natural energy bounds in quantum thermodynamics given a stationary state for a noncommutative flow we study a boundedness condition depending on a parameter 0 which is weaker than the kms equilibrium condition at inverse temperature this condition is equivalent to a holomorphic property closely related to the one recently considered by ruelle and d antoni zsido and shared by a natural class of non equilibrium steady states our holomorphic property is stronger than ruelle s one and thus selects a restricted class of non equilibrium steady states we also introduce the complete boundedness condition and show this notion to be equivalent to the pusz woronowicz complete passivity property hence to the kms condition

the dimension free structure of nonhomogeneous random matrices let x be a symmetric random matrix with independent but non identically distributed centered gaussian entries we show that begin aligned mathbf e vert x vert s p asymp mathbf e bigg bigg sum i bigg sum j x ij 2 bigg p 2 bigg 1 p bigg end aligned for any 2 le p le infty where s p denotes the p schatten class and the constants are universal the right hand side admits an explicit expression in terms of the variances of the matrix entries this settles in the case p infty a conjecture of the first author and provides a complete characterization of the class of infinite matrices with independent gaussian entries that define bounded operators on ell 2 along the way we obtain optimal dimension free bounds on the moments mathbf e vert x vert s p p 1 p that are of independent interest we develop further extensions to non symmetric matrices and to nonasymptotic moment and norm estimates for matrices with non gaussian entries that arise for example in the study of random graphs and in applied mathematics

on dilation symmetries arising from scaling limits quantum field theories at short scales can be approximated by a scaling limit theory in this approximation an additional symmetry is gained namely dilation covariance to understand the structure of this dilation symmetry we investigate it in a nonperturbative model independent context to that end it turns out to be necessary to consider non pure vacuum states in the limit these can be decomposed into an integral of pure states we investigate how the symmetries and observables of the theory behave under this decomposition in particular we consider several natural conditions of increasing strength that yield restrictions on the decomposed dilation symmetry

construction of quantum field theories with factorizing s matrices the subject of this thesis is a novel construction method for interacting relativistic quantum field theories on two dimensional minkowski space employing the algebraic framework of quantum field theory it is shown under which conditions an algebra of observables localized in a wedge shaped region of spacetime can be used to construct model theories a crucial input in this context is the modular nuclearity condition for wedge algebras which implies the existence of local observables as an application of the new method a rigorous construction of a large family of models with factorizing s matrices is obtained in an inverse scattering approach a given factorizing scattering operator is used to define certain semi localized wightman fields associated to it with the help of these fields a wedge algebra can be defined which determines the local observable content of a well defined quantum field theory in this approach the modular nuclearity condition translates to certain analyticity and boundedness conditions on the formfactors of wedge local observables these conditions are shown to hold for a large class of underlying s matrices including the scattering operators of the sinh gordon model and the scaling ising model as special examples the so constructed models are investigated with respect to their scattering properties they are shown to solve the inverse scattering problem for the underlying s matrices and a proof of asymptotic completeness for these models is given

wigner random matrices with non symmetrically distributed entries we show that the spectral radius of an n n random symmetric matrix with i i d bounded centered but non symmetrically distributed entries is bounded from above by 2 o n 6 11 e where 2 is the variance of the matrix entries and e is an arbitrary small positive number our bound improves the earlier results by z furedi and j komlos 1981 and van vu 2005

causal independence and the energy level density of states in local quantum field theory within the general framework of local quantum field theory a physically motivated condition on the energy level density of well localized states is proposed and discussed it is shown that any model satisfying this condition obeys a strong form of the principle of causal statistical independence which manifests itself in a specific algebraic structure of the local algebras split property it is also shown that the proposed condition holds in a free field theory

local law and tracy widom limit for sparse random matrices we consider spectral properties and the edge universality of sparse random matrices the class of random matrices that includes the adjacency matrices of the erd s renyi graph model g n p we prove a local law for the eigenvalue density up to the spectral edges under a suitable condition on the sparsity we also prove that the rescaled extremal eigenvalues exhibit goe tracy widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included for the adjacency matrix of the erd s renyi graph this establishes the tracy widom fluctuations of the second largest eigenvalue when p is much larger than n 2 3 with a deterministic shift of order np 1

quasisymplectic integrators for stochastic differential equations two specialized algorithms for the numerical integration of the equations of motion of a brownian walker obeying detailed balance are introduced the algorithms become symplectic in the appropriate limits and reproduce the equilibrium distributions to some higher order in the integration time step comparisons with other existing integration schemes are carried out both for static and dynamical quantities

particle weights and their disintegration ii the first article in this series presented a thorough discussion of particle weights and their characteristic properties in this part a disintegration theory for particle weights is developed which yields pure components linked to irreducible representations and exhibiting features of improper energy momentum eigenstates this spatial disintegration relies on the separability of the hilbert space as well as of the c algebra neither is present in the gns representation of a generic particle weight so that we use a restricted version of this concept on the basis of separable constructs this procedure does not entail any loss of essential information insofar as under physically reasonable assumptions on the structure of phase space the resulting representations of the separable algebra are locally normal and can thus be continuously extended to the original quasi local c algebra

interpolation by type i factors and the flip automorphism abstract given two von neumann algebras a b we study the relation between the existence of an interpolating type i factor f namely a f b the implementability of the flip automorphism of a a by a unitary in b b and the statistical independence of a and b a and b generate a w tensor product as an application in q f t we derive in a natural way a structure theorem of buchholz for the von neumann algebras of local observables associated to free fields

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

quantum inequalities from operator product expansions quantum inequalities are lower bounds for local averages of quantum observables that have positive classical counterparts such as the energy density or the wick square we establish such inequalities in general possibly interacting quantum field theories on minkowski space using nonperturbative techniques our main tool is a rigorous version of the operator product expansion

nuclear maps and modular structures ii applications to quantum field theory a correspondence between spectral properties of modular operators appearing in quantum field theory and the hamiltonian is established it allows to prove the distal split property for a wide class of models conversely any model having this property is shown to satisfy the haag swieca compactness criterion the results lead to a new type of nuclearity condition which can be applied to quantum field theories on arbitrary space time manifolds

measurement theory in local quantum physics in this paper we aim to establish foundations of measurement theory in local quantum physics for this purpose we discuss a representation theory of completely positive cp instruments on arbitrary von neumann algebras we introduce a condition called the normal extension property nep and establish a one to one correspondence between cp instruments with the nep and statistical equivalence classes of measuring processes we show that every cp instrument on an atomic von neumann algebra has the nep extending the well known result for type i factors moreover we show that every cp instrument on an injective von neumann algebra is approximated by cp instruments with the nep the concept of posterior states is also discussed to show that the nep is equivalent to the existence of a strongly measurable family of posterior states for every normal state two examples of cp instruments without the nep are obtained from this result it is thus concluded that in local quantum physics not every cp instrument represents a measuring process but in most of physically relevant cases every cp instrument can be realized by a measuring process within arbitrary error limits as every approximately finite dimensional von neumann algebra on a separable hilbert space is injective to conclude the paper the concept of local measurement in algebraic quantum field theory is examined in our framework in the setting of the doplicher haag roberts and doplicher roberts theory describing local excitations we show that an instrument on a local algebra can be extended to a local instrument on the global algebra if and only if it is a cp instrument with the nep provided that the split property holds for the net of local algebras

half sided modular inclusions of von neumann algebras letn m be von neumann algebras on a hilbert spaceh a common cyclic and separating vector denote m n resp jm jn the associated modular operators and conjugations assume m it n it n fort 0 we call such an inclusion half sided modular then we prove the existence of a oneparameter unitary groupu a onh a r with generator frac 1 2 pi ln delta cal n ln delta m mathbin lower 3ex hbox buildrel over smash scriptstyle vphantom x 0 and relations 1 itu a it it nit u a nit u e 2 ta for alla t r 2 jmjn u 2 3 nit u 1 it u 1 for allt r 4 n u 1 u 1

beta boundedness semipassivity and the kms condition the proof of a recent result by guido and longo establishing the equivalence of the kms condition with complete beta boundedness 2 is shortcut and generalized in such a way that a covariant version of the theorem is obtained

local aspects of superselection rules we study a theory of short range forces in terms of local observable quantities among the superselection structure determined by the algebra of all local observables to each additive independent charge we associate local observables having a meaning analogous to the regularized integrals of charge density fields over a finite volume among other assumptions we require that parastatistics are absent from the theories considered

normal product states for fermions and twisted duality for ccr and car type algebras with application to the yukawa2 quantum field model we present sufficient conditions that imply duality for the algebras of local observables in all abelian sectors of all locally normal irreducible representations of a field algebra if twisted duality obtains in one of these representations it is verified that the yukawa2 model satisfies these conditions yielding the first proof of duality for the observable algebra in all coherent charge sectors in this model this paper also constitutes the first verification of the assumptions of the axiomatic study of the structure of superselection sectors by doplicher haag and roberts in an interacting model with nontrivial sectors the existence of normal product states for the free fermi field algebra and thus the verification of the funnel property for the associated net of local algebras are demonstrated

nuclear maps and modular structures i general properties abstract we establish a correspondence between the split property of inclusions a b of von neumann algebras and nuclearity properties of the natural embeddings p l a lp b p 1 2 given by the modular theory

a proof of haag swieca s compactness property for elastic scattering states it is proved that in any massive relativistic quantum field theory satisfying two particle asymptotic completeness all the bounded energy components in the elastic two particle range of all subsets of states which are excitations of the vacuum state by uniformly bounded observables localized in a given finite region of spacetime are compact in the hilbert space of states this result which is in agreement with haag swieca s conjecture is also given a more precise form in terms of the rate of decrease of the n dimensional thickness or approximation number of such sets of states when n tends to infinity a similar computation valid at arbitrarily high energies is also given for the massive free field case

complex classical fields an example abstract we study complex classical scalar fields within a new framework introduced in a previous work we replace the usual functional integral by a complex functional arising from a boosted hamiltonian we generalize the feynman kac relation to this setting and use it to establish the spectral condition on a cylinder we consider also positive temperature states

supersymmetric probability distributions we use anticommuting variables to study probability distributions of random variables that are solutions of langevin s equation we show that the probability density always enjoys worldpoint supersymmetry the partition function however may not we find that the domain of integration can acquire a boundary which implies that the auxiliary field has a non zero expectation value signalling spontaneous supersymmetry breaking this is due to the presence of fermionic zeromodes whose contribution cannot be canceled by a surface term this we prove by an explicit calculation of the regularized partition function as well as by computing the moments of the auxiliary field and checking whether they satisfy the identities implied by wick s theorem nevertheless supersymmetry manifests itself in the identities that are satisfied by the moments of the scalar whose expressions we can calculate for all values of the coupling constant we also provide some quantitative estimates concerning the visibility of supersymmetry breaking effects in the identities for the moments and remark that the shape of the distribution of the auxiliary field can influence quite strongly how easy it would be to mask them since the expectation value of the auxiliary field does not coincide with its typical value

the largest eigenvalues of sample covariance matrices for a spiked population diagonal case we consider large complex random sample covariance matrices obtained from spiked populations that is when the true covariance matrix is diagonal with all but finitely many eigenvalues equal to 1 we investigate the limiting behavior of the largest eigenvalues when the population and the sample sizes both become large under some conditions on moments of the sample distribution we prove that the asymptotic fluctuations of the largest eigenvalues are the same as for a complex gaussian sample with the same true covariance the real setting is also considered

transition path sampling algorithm for discrete many body systems we propose a monte carlo method for efficiently sampling trajectories with fixed initial and final conditions in a system with discrete degrees of freedom the method can be applied to any stochastic process with local interactions including systems that are out of equilibrium we combine the proposed path sampling algorithm with thermodynamic integration to calculate transition rates we demonstrate our method on the well studied two dimensional ising model with periodic boundary conditions and show agreement with other results for both large and small system sizes the method scales well with the system size allowing one to simulate systems with many degrees of freedom and providing complementary information with respect to other algorithms

operator product expansions as a consequence of phase space properties the paper presents a model independent nonperturbative proof of operator product expansions in quantum field theory as an input a recently proposed phase space condition is used that allows a precise description of point field structures based on the product expansions we also define and analyze normal products in the sense of zimmermann

sparse regular random graphs spectral density and eigenvectors we examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs we find that when the degree sequence of the graph slowly increases to infinity with the number of vertices the empirical spectral distribution converges to the semicircle law moreover we prove concentration estimates on the number of eigenvalues over progressively smaller intervals we also show that with high probability all the eigenvectors are delocalized