pseudounitary symmetry and the gaussian pseudounitary ensemble of random matrices employing the currently discussed notion of pseudo hermiticity we define a pseudounitary group further we develop a random matrix theory that is invariant under such a group and call this ensemble of pseudo hermitian random matrices the pseudounitary ensemble we obtain exact results for the nearest neighbor level spacing distribution for 2 x 2 pt invariant hamiltonian matrices that have forms approximately sln 1 s near zero spacing for three independent elements and approximately s for four independent elements this shows a level repulsion in a marked distinction with an algebraic form s beta in the wigner surmise we believe that this paves the way for a description of varied phenomena in two dimensional statistical mechanics quantum chromodynamics and so on

pseudo hermiticity versus pt symmetry iii equivalence of pseudo hermiticity and the presence of antilinear symmetries we show that a diagonalizable non hermitian hamiltonian h is pseudo hermitian if and only if it has an antilinear symmetry i e a symmetry generated by an invertible antilinear operator this implies that the eigenvalues of h are real or come in complex conjugate pairs if and only if h possesses such a symmetry in particular the reality of the spectrum of h implies the presence of an antilinear symmetry we further show that the spectrum of h is real if and only if there is a positive definite inner product on the hilbert space with respect to which h is hermitian or alternatively there is a pseudo canonical transformation of the hilbert space that maps h into a hermitian operator

on the pseudo hermitian nondiagonalizable hamiltonians we consider a class of possibly nondiagonalizable pseudo hermitian operators with discrete spectrum and we establish for such a class the equivalence between the pseudo hermiticity property and the existence of an antilinear involutory symmetry moreover we prove that this class actually coincides with the one of possibly nondiagonalizable weak pseudo hermitian operators and that in no case unless they are diagonalizable and have a real spectrum they are hermitian with respect to a definite inner product finally we show that a typical degeneracy of the real eigenvalues which reduces to the well known kramers degeneracy in the hermitian case occurs whenever a fermionic possibly nondiagonalizable pseudo hermitian hamiltonian admits an antilinear symmetry like the time reversal operator t some consequences and applications are briefly discussed

alternative descriptions and bipartite compound quantum systems we analyse some features of alternative hermitian and quasi hermitian quantum descriptions of simple and bipartite compound systems we show that alternative descriptions of two interacting subsystems are possible if and only if the metric operator of the compound system can be obtained as a tensor product of positive operators on component spaces some examples also show that such property could be strictly connected with symmetry properties of the non hermitian hamiltonian

pseudo hermiticity and generalized pt and cpt symmetries we study certain linear and antilinear symmetry generators and involution operators associated with pseudo hermitian hamiltonians and show that the theory of pseudo hermitian operators provides a simple explanation for the recent results of bender brody and jones quant ph 0208076 on the cpt symmetry of a class of pt symmetric non hermitian hamiltonians we present a natural extension of these results to the class of diagonalizable pseudo hermitian hamiltonians h with a discrete spectrum in particular we introduce generalized parity p time reversal t and charge conjugation c operators and establish the pt and cpt invariance of h

hyperbolic cone manifolds short geodesics and schwarzian derivatives given a geometrically finite hyperbolic cone manifold with the cone singularity sufficiently short we construct a one parameter family of cone manifolds decreasing the cone angle to zero we also control the geometry of this one parameter family via the schwarzian derivative of the projective boundary and the length of closed geodesics

coexistence on reflecting hyperplane in generalized probability theories the coexistence of effects in a certain class of generalized probability theories is investigated the effect space corresponding to an even sided regular polygon state space has a central hyperplane that contains all the nontrivial extremal effects the existence of such a hyperplane called a reflecting hyperplane is closely related to the point symmetry of the corresponding state space the effects on such a hyperplane can be regarded as the generalized unbiased effects a necessary and sufficient condition for a pair of unbiased effects in the even sided regular polygon theories is presented this result reproduces a low dimensional analogue of known results of qubit effects in a certain limit the coexistence of effects in a certain class of generalized probability theories is investigated the effect space corresponding to an even sided regular polygon state space has a central hyperplane that contains all the nontrivial extremal effects the existence of such a hyperplane called a reflecting hyperplane is closely related to the point symmetry of the corresponding state space the effects on such a hyperplane can be regarded as the generalized unbiased effects a necessary and sufficient condition for a pair of unbiased effects in the even sided regular polygon theories is presented this result reproduces a low dimensional analogue of known results of qubit effects in a certain limit

alternative descriptions in quaternionic quantum mechanics we characterize the quasianti hermitian quaternionic operators in quaternionic quantum mechanics by means of their spectra moreover we state a necessary and sufficient condition for a set of quasianti hermitian quaternionic operators to be anti hermitian with respect to a uniquely defined positive scalar product in a infinite dimensional right quaternionic hilbert space according to such results we obtain two alternative descriptions of a quantum optical physical system in the realm of quaternionic quantum mechanics while no alternative can exist in complex quantum mechanics and we discuss some differences between them

nonlinear pseudo supersymmetry in the framework of fold supersymmetry we recall the importance of recognizing the different mathematical nature of various concepts relating to symmetric quantum theories after clarifying the relation between supersymmetry and pseudo supersymmetry we prove generically that nonlinear pseudo supersymmetry recently proposed by sinha and roy is just a special case of fold supersymmetry in particular we show that all the models constructed by these authors have type a two fold supersymmetry furthermore we prove that an arbitrary one body quantum hamiltonian which admits two local solutions in closed form belongs to type a two fold supersymmetry irrespective of whether or not it is hermitian symmetric pseudo hermitian and so on

geometrization of 3 dimensional orbifolds this paper is devoted to the proof of the orbifold theorem if o is a compact connected orientable irreducible and topologically atoroidal 3 orbifold with nonempty ramification locus then o is geometric i e has a metric of constant curvature or is seifert fibred as a corollary any smooth orientationpreserving nonfree finite group action on s3 is conjugate to an orthogonal action

exact pt symmetry is equivalent to hermiticity we show that a quantum system possessing an exact antilinear symmetry in particular pt symmetry is equivalent to a quantum system having a hermitian hamiltonian we construct the unitary operator relating an arbitrary non hermitian hamiltonian with exact pt symmetry to a hermitian hamiltonian we apply our general results to pt symmetry in finite dimensions and give the explicit form of the above mentioned unitary operator and hermitian hamiltonian in two dimensions our findings lead to the conjecture that non hermitian cpt symmetric field theories are equivalent to certain nonlocal hermitian field theories

pseudounitary operators and pseudounitary quantum dynamics we consider pseudounitary quantum systems and discuss various properties of pseudounitary operators in particular we prove a characterization theorem for block diagonalizable pseudounitary operators with finite dimensional diagonal blocks furthermore we show that every pseudounitary matrix is the exponential of i 1 times a pseudo hermitian matrix and determine the structure of the lie groups consisting of pseudounitary matrices in particular we present a thorough treatment of 2 2 pseudounitary matrices and discuss an example of a quantum system with a 2 2 pseudounitary dynamical group as other applications of our general results we give a proof of the spectral theorem for symplectic transformations of classical mechanics demonstrate the coincidence of the symplectic group sp 2n with the real subgroup of a matrix group that is isomorphic to the pseudounitary group u n n and elaborate on an approach to second quantization that makes use of the underlying pseudounitary dynamical groups

finite dimensional pt symmetric hamiltonians this paper investigates finite dimensional pt symmetric hamiltonians it is shown here that there are two ways to extend real symmetric hamiltonians into the complex domain i the usual approach is to generalize such hamiltonians to include complex hermitian hamiltonians ii alternatively one can generalize real symmetric hamiltonians to include complex pt symmetric hamiltonians in the first approach the spectrum remains real while in the second approach the spectrum remains real if the pt symmetry is not broken both generalizations give a consistent theory of quantum mechanics but if d 2 a d dimensional hermitian matrix hamiltonian has more arbitrary parameters than a d dimensional pt symmetric matrix hamiltonian

generalized probability theories what determines the structure of quantum theory the framework of generalized probabilistic theories is a powerful tool for studying the foundations of quantum physics it provides the basis for a variety of recent findings that significantly improve our understanding of the rich physical structure of quantum theory this review paper tries to present the framework and recent results to a broader readership in an accessible manner to achieve this we follow a constructive approach starting from a few basic physically motivated assumptions we show how a given set of observations can be manifested in an operational theory furthermore we characterize consistency conditions limiting the range of possible extensions in this framework classical and quantum theory appear as special cases and the aim is to understand what distinguishes quantum mechanics as the fundamental theory realized in nature it turns out that non classical features of single systems can equivalently result from higher dimensional classical theories that have been restricted entanglement and non locality however are shown to be genuine non classical features

pseudo hermiticity versus pt symmetry ii a complete characterization of non hermitian hamiltonians with a real spectrum we give a necessary and sufficient condition for the reality of the spectrum of a non hermitian hamiltonian admitting a complete set of biorthonormal eigenvectors

pseudo hermitian hamiltonians indefinite inner product spaces and their symmetries we extend the definition of generalized parity p charge conjugation c and time reversal t operators to nondiagonalizable pseudo hermitian hamiltonians and use these generalized operators to describe the full set of symmetries of a pseudo hermitian hamiltonian according to a fourfold classification in particular we show that t p and ct p are the generators of the p antiunitary symmetries moreover a necessary and sufficient condition is provided for a pseudo hermitian hamiltonian h to admit a p reflecting symmetry which generates the p pseudounitary and the p pseudoantiunitary symmetries finally a physical example is considered and some hints on the p unitary evolution of a physical system are also given

variation of cone metrics on riemann surfaces we discuss the variational properties of the unique conical metric of constant curvature 1 associated to a compact riemann surface together with a weighted divisor 2005 elsevier inc all rights reserved

a canonical form for matrices under consimilarity abstract square complex matrices a b are said to be consimilar if a sb s 1 for some nonsingular matrix s consimilarity is an equivalence relation that is a natural matrix generalization of rotation of scalars in the complex plane we survey the known forms to which a given complex matrix may be reduced by unitary consimilarity and describe a canonical form to which it may be reduced by a general consimilarity we derive a useful criterion for two matrices to be consimilar and show that every matrix is consimilar to its own conjugate transpose and adjoint to a real matrix and to a hermitian matrix

quasi fuchsian manifolds with particles we consider 3 dimensional hyperbolic cone manifolds which are convex cocompact in a natural sense with cone singularities along infinite lines such singularities are sometimes used by physicists as models for massive spinless point particles we prove an infinitesimal rigidity statement when the angles around the singular lines are less than any infinitesimal deformation changes either these angles or the conformal structure at infinity with marked points corresponding to the endpoints of the singular lines moreover any small variation of the conformal structure at infinity and of the singular angles can be achieved by a unique small deformation of the cone manifold structure these results hold also when the singularities are along a graph i e for interacting particles

on solutions of the matrix equations x axb c and x axb c this paper studies the solutions of complex matrix equations x axb c and x axb c and obtains explicit solutions of the equations by the method of characteristic polynomial and a method of real representation of a complex matrix respectively

pseudo hermiticity for a class of nondiagonalizable hamiltonians we give two characterization theorems for pseudo hermitian possibly nondiagonalizable hamiltonians with a discrete spectrum that admit a block diagonalization with finite dimensional diagonal blocks in particular we prove that for such an operator h the following statements are equivalent 1 h is pseudo hermitian 2 the spectrum of h consists of real and or complex conjugate pairs of eigenvalues and the geometric multiplicity and the dimension of the diagonal blocks for the complex conjugate eigenvalues are identical 3 h is hermitian with respect to a positive semidefinite inner product we further discuss the relevance of our findings for the merging of a complex conjugate pair of eigenvalues of diagonalizable pseudo hermitian hamiltonians in general and the pt symmetric hamiltonians and the effective hamiltonian for a certain closed frw minisuperspace quantum cosmological model in particular

quasistationary quaternionic hamiltonians and complex stochastic maps we show that the complex projections of time dependent quasianti hermitian quaternionic hamiltonian dynamics are complex stochastic dynamics in the space of complex quasi hermitian density matrices if and only if a quasistationarity condition is fulfilled i e if and only if is an hermitian positive time independent complex operator an example is also discussed

weak pseudo hermiticity and antilinear commutant we inquire into some properties of diagonalizable pseudo hermitian operators showing that their definition can be relaxed and that the pseudo hermiticity property is strictly connected with the existence of an antilinear symmetry this result is then illustrated by considering the particular case of the complex morse potential

prescribing curvature on compact surfaces with conical singularities we study the berger nirenberg problem on surfaces with conical singularities i e we discuss conditions under which a function on a riemann surface is the gaussian curvature of some conformal metric with a prescribed set of singularities of conical types