conformal invariance and vector operators in the o n model it is widely expected that for a large class of models scale invariance implies conformal invariance a sufficient condition for this to happen is that there exists no integrated vector operator invariant under all internal symmetries of the model with scaling dimension 1 in this article we compute the scaling dimensions of vector operators with lowest dimensions in the o n model we use three different approximation schemes epsilon expansion large n limit and third order of the derivative expansion of non perturbative renormalization group equations we find that the scaling dimensions of all considered integrated vector operators are always much larger than 1 this strongly supports the existence of conformal invariance in this model for the ising model an argument based on correlation functions inequalities was derived which yields a lower bound for the scaling dimension of the vector perturbations we generalize this proof to the case of the o n model with n in left 2 3 4 right

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

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

brunn minkowski inequality for multiplicities this is a convex subset of the real vector space p zr in fact it is a convex polytope see br where this polytope is discussed from an algebraic point of view it is known that the total mass of m is a polynomial in m for su ciently large m denote by k its degree and m k m m weak d where d is the lebesgue measure supported on and the density is a piecewise polynomial function we will not use this piecewise polynomiality in this paper recall that a real function f de ned on a convex subset u of a vector space v is called concave if

nonuniqueness of the mathcal c operator in mathcal p mathcal t symmetric quantum mechanics the operator in symmetric quantum mechanics satisfies a system of three simultaneous algebraic operator equations and these equations are difficult to solve exactly so perturbative methods have been used in the past to calculate the usual approach has been to express the hamiltonian as h h0 h1 and to seek a solution for in the form where q q q p is odd in the momentum p even in the coordinate q and has a perturbation expansion of the form q q1 3q3 5q5 in previous work it has always been assumed that the coefficients of even powers of in this expansion would be absent because their presence would violate the condition that q p q is odd in p in an earlier paper it was argued that the operator is not unique because the perturbation coefficient q1 is nonunique here the nonuniqueness of is demonstrated at a more fundamental level it is shown that the perturbation expansion for q actually has the more general form q q0 q1 2q2 in which all powers and not just odd powers of appear for the case in which h0 is the harmonic oscillator hamiltonian q0 is calculated exactly and in closed form and it is shown explicitly to be nonunique the results are verified by using powerful summation procedures based on analytic continuation it is also shown how to calculate the higher coefficients in the perturbation series for q

a physical interpretation for the non hermitian hamiltonian we explore a way of finding the link between a non hermitian hamiltonian and a hermitian one based on the analysis of bethe ansatz solutions for a class of non hermitian hamiltonians and the scattering problems for the corresponding hermitian hamiltonians it is shown that a scattering state of an arbitrary hermitian lattice embedded in a chain as the scattering center shares the same wavefunction with the corresponding non hermitian tight binding lattice which consists of the hermitian lattice with two additional on site complex potentials no matter whether the non hermitian is broken symmetry or even non symmetry exactly solvable models are presented to demonstrate the main points of this paper

the minimally anisotropic metric operator in quasi hermitian quantum mechanics we propose a unique way to choose a new inner product in a hilbert space with respect to which an originally non self adjoint operator similar to a self adjoint operator becomes self adjoint our c

some general results concerning the critical exponents of percolation processes in this paper we will give some results concerning the critical exponents of percolation processes which are valid for any model these results show that in several respects the behavior which occurs for percolation on the binary tree provides bounds on one side for what happens in general these results and their proofs are closely related to their analogues for the ising model

correlation length and its critical exponent for percolation processes some critical exponent inequalities are given involving the correlation length of site percolation processes on d in particular it is shown thatv 2 d which implies that the critical exponentv cannot take its mean field value for the three dimensional percolation processes

metric operator in pseudo hermitian quantum mechanics and the imaginary cubic potential we present a systematic perturbative construction of the most general metric operator and positive definite inner product for quasi hermitian hamiltonians of the standard form h 1 2 p 2 v x in one dimension we show that this problem is equivalent to solving an infinite system of iteratively decoupled hyperbolic partial differential equations in 1 1 dimensions for the case that v x is purely imaginary the latter have the form of a nonhomogeneous wave equation which admits an exact solution we apply our general method to obtain the most general metric operator for the imaginary cubic potential v x i x 3 this reveals an infinite class of previously unknown cpt as well as non cpt inner products we compute the physical observables of the corresponding unitary quantum system and determine the underlying classical system our results for the imaginary cubic potential show that unlike the quantum system the corresponding classical system is not sensitive to the choice of the metric operator as another application of our method we give a complete characterization of the pseudo hermitian canonical quantization of a free particle moving in r that is consistent with the usual choice for the quantum hamiltonian finally we discuss subtleties involved with higher dimensions and systems having a fixed length scale

j self adjoint operators with mathcal c symmetries an extension theory approach a well known tool in conventional von neumann quantum mechanics is the self adjoint extension technique for symmetric operators it is used e g for the construction of dirac hermitian hamiltonians with point interaction potentials here we reshape this technique to allow for the construction of pseudo hermitian j self adjoint hamiltonians with complex point interactions we demonstrate that the resulting hamiltonians are bijectively related to the so called hypermaximal neutral subspaces of the defect krein space of the symmetric operator this symmetric operator is allowed to have arbitrary but equal deficiency indices n n general properties of the operators for these hamiltonians are derived a detailed study of operator parametrizations and krein type resolvent formulae is provided for j self adjoint extensions of symmetric operators with deficiency indices 2 2 the technique is exemplified on 1d pseudo hermitian schr dinger and dirac hamiltonians with complex point interaction potentials

pt symmetry and its spontaneous breakdown in three dimensions the mechanism of introducing non hermiticity to non central symmetric potentials through both the azimuth and polar angles is discussed generalizing the results of a previous work it is shown that this can be done also through the polar angle part if appropriate potentials such as the scarf i or rosen morse i potentials are used in the eigenvalue equation of the polar component it is shown that the spontaneous breakdown of symmetry can also occur in these non central potentials several potentials are proposed in the azimuthal eigenvalue equation too where the use of periodic boundary conditions is essential possible generalizations of the results are outlined

scaling laws at the phase transition of systems with divergent order parameter and or internal length the example of dna denaturation we used the transfer integral method to compute with an uncertainty smaller than 5 the six fundamental characteristic exponents of two dynamical models for dna thermal denaturation and investigate the validity of the scaling laws doubts concerning this point arise because the investigated systems i have a divergent internal length ii are described by a divergent order parameter and iii are of dimension 1 we found that the assumption that the free energy can be described by a single homogeneous function is robust despite the divergence of the order parameter so that rushbrooke s and widom s identities are valid relations josephson s identity is instead not satisfied this is probably due to the divergence of the internal length which invalidates the assumption that the correlation length is solely responsible for singular contributions to thermodynamic quantities fisher s identity is even more wrong we showed that this is due to the d 1 dimensionality and obtained an alternative law which is well satisfied at dna thermal denaturation

quantization of diffeomorphism invariant theories of connections with local degrees of freedom quantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphism constraint is solved the space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions this provides in particular a quantization of the husain kucha model the main results also pave the way to quantization of other diffeomorphism invariant theories such as general relativity in the riemannian case i e signature the approach appears to contain all the necessary ingredients already in the lorentzian case it will have to be combined in an appropriate fashion with a coherent state transform to incorporate complex connections

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

partition function zeros of the one dimensional blume capel model in transfer matrix formalism zeros of the partition function of the one dimensional ferromagnetic and antiferromagnetic blume capel models have been studied by using the transfer matrix method in the thermodynamic limit and for finite size chains the equation for the distribution of zeros of the partition function in the thermodynamic limit is derived the distribution of the yang lee and fisher zeros are studied for a variety of values of the parameters of the model densities of the yang lee and fisher zeros are investigated and a singular behavior of the corresponding densities of zeros at the edge points is shown the edge singularity exponents are calculated analytically for the densities of the yang lee and fisher zeros it is found that for both cases edge singularity exponents are universal and equal to sigma 1 2

metric operators for quasi hermitian hamiltonians and symmetries of equivalent hermitian hamiltonians we give a simple proof of the fact that every diagonalizable operator that has a real spectrum is quasi hermitian and show how the metric operators associated with a quasi hermitian hamiltonian are related to the symmetry generators of an equivalent hermitian hamiltonian

new inequalities for ising ferromagnets it is shown that for ising ferromagnets which obey the lee yang theorem the ursell functions or cumulants of the magnetization variable at nonzero external field satisfy series of inequalities several relations connecting ursell functions with nonzero and zero field are derived

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

okounkov bodies and toric degenerations let vardelta be the okounkov body of a divisor d on a projective variety x we describe a geometric criterion for vardelta to be a lattice polytope and show that in this situation x admits a flat degeneration to the corresponding toric variety this degeneration is functorial in an appropriate sense

pt symmetric cubic anharmonic oscillator as a physical model there is a factor of 2 error in equation 61 of this paper correcting this error leads to minor changes in equations 62 and 63 please see pdf for details

moyal products a new perspective on quasi hermitian quantum mechanics the rationale for introducing non hermitian hamiltonians and other observables is reviewed and open issues identified we present a new approach based on moyal products to compute the metric for quasi hermitian systems this approach is not only an efficient method of computation but also suggests a new perspective on quasi hermitian quantum mechanics which invites further exploration in particular we present some first results which link the berry connection and curvature to non perturbative properties and the metric

mean field bound and ghs inequality a new proof of the mean field bounds for magnetizations is presented it applies to any single component spin system which allows ghs inequality and to ann vector model forn 3 and to ann solid sphere model for all values ofn provided that the interactions are ferromagnetic and translation invariant

fluctuation susceptibility relations for classical spin systems we prove identities between integrated ursell functions and derivatives of the pressure in the thermodynamic limit for multicomponent classical spin systems which obey the lee yang theorem and some form of gaussian domination when the susceptibility is finite t tc following refs 3 and 4 we view the moment generating function of the magnetization as the inverse of an infinitely divisible characteristic function fluctuation susceptibility relations of all orders then follow by bounding the corresponding cumulants taken in zero external field high order cumulants are bounded in terms of the susceptibility using gaussian and simon s inequalities for short range interactions

pt symmetry cartan decompositions lie triple systems and krein space related clifford algebras gauged quantum mechanics ptqm and corresponding krein space setups are studied for models with constant non abelian gauge potentials and extended parity inversions compact and noncompact lie group components are analyzed via cartan decompositions a lie triple structure is found and an interpretation as symmetrically generalized jaynes cummings model is possible with close relation to recently studied cavity qed setups with transmon states in multilevel artificial atoms for models with abelian gauge potentials a hidden clifford algebra structure is found and used to obtain the fundamental symmetry of krein space related j self adjoint extensions for ptqm setups with ultra localized potentials

delta function potential with a complex coupling we explore the hamiltonian operator where is the dirac delta function and z is an arbitrary complex coupling constant for a purely imaginary z h has a spectral singularity at for re z 0 h has a real positive continuous spectrum that is free from spectral singularities for this latter case we construct an associated biorthonormal system and use it to perform a perturbative calculation of a positive definite inner product that renders h self adjoint this allows us to address the intriguing question of the nonlocal aspects of the equivalent hermitian hamiltonian for the system in particular we compute the energy expectation values for various gaussian wave packets to show that the non hermiticity effect diminishes rapidly outside an effective interaction region

solvable non hermitian discrete square well with closed form physical inner product a new hermitizable quantum model is proposed in which the bound state energies are real and given as roots of an elementary trigonometric expression while the wave function components are expressed as superpositions of two chebyshev polynomials as an site lattice version of square well with complex robin type two parametric boundary conditions the model is unitary with respect to the hilbert space metric which becomes equal to the most common dirac s metric in the conventional textbook hermitian hamiltonian limit this metric is constructed in closed form at all

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

lattice systems with a continuous symmetry we prove that the expansion in powers of the temperature t of the correlation functions and the free energy of the plane rotator model on a d dimensional lattice is asymptotic to all orders in t the leading term in the expansion is the spin wave approximation and the higher powers are obtained by the usual perturbation series we also prove the inverse power decay of the pair correlation at low temperatures for d 3

on the reality of the eigenvalues for a class of symmetric oscillators we study the eigenvalue problem with the boundary conditions that decays to zero as z tends to infinity along the rays where is a real polynomial and we prove that if for some we have for all then the eigenvalues are all positive real we then sharpen this to a larger class of polynomial potentials

hyperscaling inequalities for percolation a set of critical exponent inequalities for independent percolation which saturate under the hyperscaling hypothesis is proved one of the consequences of the inequalities is the lower bounddc 6 for the upper critical dimension the proof is based on a rigorous version of the finite size scaling argument which extends easily to other systems such as ising ferromagnets

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

okounkov bodies on projectivizations of rank two toric vector bundles abstract the global okounkov body of a projective variety is a closed convex cone that encodes asymptotic information about every big line bundle on the variety in the case of a rank two toric vector bundle e on a smooth projective toric variety we use its klyachko filtrations to give an explicit description of the global okounkov body of p e in particular we show that this is a rational polyhedral cone and that p e is a mori dream space

one dimensional point interaction with three complex parameters for a pair of non hermitian hamiltonian h and its hermitian adjoint h there are situations in which their eigenfunctions form a biorthogonal system we illustrate such a situation by means of a one particle system with a one dimensional point interaction in the form of the fermi pseudo potential the interaction consists of three terms with three strength parameters gi i 1 2 and 3 which are all complex this complex point interaction is neither hermitian nor invariant in general the s matrix for the transmission reflection problem constructed with h or with h in the usual manner is not unitary but it conforms to the pseudo unitarity that we define the pseudo unitarity is closely related to the biorthogonality of the eigenfunctions the eigenvalue spectrum of h with the complex interaction is generally complex but there are cases where the spectrum is real in such a case h and h form a pseudo hermitian pair

generalized swanson models and their solutions we analyse a class of non hermitian quadratic hamiltonians which are of the form where are real constants with and and are generalized creation and annihilation operators respectively thus these hamiltonians may be classified as generalized swanson models it is shown that the eigenenergies are real for a certain range of values of the parameters a similarity transformation mapping the non hermitian hamiltonian h to a hermitian one h is also obtained it is shown that h and h share identical energies as explicit examples the solutions of a couple of models based on the trigonometric rosen morse i and the hyperbolic rosen morse ii type potentials are obtained we also study the case where the non hermitian hamiltonian is symmetric

rigorous proof of the high temperature josephson inequality for critical exponents i give a rigorous proof of the high temperature josephson inequalitydv 2 following the original ideas of josephson the proof is applicable to a class of models including the ferromagnetic ising model and the 4 lattice field theory

calculation of the metric in the hilbert space of a cal p cal t symmetric model via the spectral theorem in a previous paper 1 we introduced a very simple symmetric non hermitian hamiltonian with a real spectrum and derived a closed formula for the metric operator relating the problem to a hermitian one in this paper we propose an alternative formula for the metric operator which we believe is more elegant and whose construction based on a backward use of the spectral theorem for self adjoint operators provides new insights into the nature of the model

complete set of inner products for a discrete pt symmetric square well hamiltonian a discrete n point runge kutta version h n of one of the simplest non hermitian square well hamiltonians with real spectrum is studied its possible hermitizations mediated by nontrivial often called non dirac metrics i are considered as a source of nonequivalent standard probabilistic interpretations of this quantum model a complete set of these alternative multiparametric metrics a b n defining all the eligible hamiltonian dependent representations of the physical hilbert space of states is constructed in closed form for any coupling 1 1 and for any matrix dimension n

space of state vectors in symmetric quantum mechanics the space of states of symmetric quantum mechanics is examined the requirement that eigenstates with different eigenvalues must be orthogonal leads to the conclusion that eigenfunctions belong to a space with an indefinite metric self consistent expressions for the probability amplitude and the average value of operator are suggested further specification of the space of state vectors yields a superselection rule redefining the notion of the superposition principle an expression for the probability current density satisfying the equation of continuity and vanishing for the bound state is proposed

unbounded mathcal c symmetries and their nonuniqueness it is shown that if the operator for a symmetric hamiltonian with simple eigenvalues is not unique then it is unbounded the fact that the operator is unbounded is significant because while there is a formal equivalence between a symmetric hamiltonian and a conventionally hermitian hamiltonian in the sense that the two hamiltonians are isospectral the hilbert spaces are inequivalent this is so because the mapping from one hilbert space to the other is unbounded this shows that symmetric quantum theories are mathematically distinct from conventional hermitian quantum theories this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

transition elements for a non hermitian quadratic hamiltonian the non hermitian quadratic hamiltonian h a a a2 a 2 is analyzed where a and a are harmonic oscillator creation and annihilation operators and and are real constants for the case that 2 4 0 it is shown using operator techniques that the hamiltonian possesses real and positive eigenvalues a generalized bogoliubov transformation allows the energy eigenstates to be constructed from the algebra and states of the harmonic oscillator the eigenstates are shown to possess an imaginary norm for a large range of the parameter space finding the orthonormal dual space allows the inner product to be redefined using the complexification procedure of bender et al for non hermitian hamiltonians transition probabilities governed by h are shown to be manifestly unitary when the complexification procedure is followed a specific transition element between harmonic oscillator states is evaluated for both the hermitian and non hermitian cases to identify the differences in time evolution

on the critical behavior of the magnetization in high dimensional ising models we derive rigorously general results on the critical behavior of the magnetization in ising models as a function of the temperature and the external field for the nearest neighbor models it is shown that ind 4 dimensions the magnetization is continuous att c and its critical exponents take the classical values 3 and 1 2 with possible logarithmic corrections atd 4 the continuity and other explicit bounds formally extend tod 3 1 2 other systems to which the results apply include long range models ind 1 dimension with 1 x y couplings for which 2 1 replacesd in the above summary the results are obtained by means of differential inequalities derived here using the random current representation which is discussed in detail for the case of a nonvanishing magnetic field

critical exponent of susceptibility for a class of general ferromagnets in d 4 dimensions a rigorous proof is presented that the critical exponent of the susceptibility takes its mean field value in d 4 dimensions for ferromagnets with single spin measure in the brydges frohlich spencer class modulo a numerical calculation of a certain function i d of dimension d this class of ferromagnets contains for example the ising spin model and lattice scalar 4 model

infinitely divisible distribution functions of class l and the lee yang theorem it is shown that the free energy density of a large class of ferromagnets satisfying the lee yang property is to be connected with the limit characteristic function of a suitably renormalized sum of independent and non identically distributed random variables using the canonical representation formulae of such characteristic functions various chains of inequalities are derived for the ursell functions

more inequalities for critical exponents a variety of rigorous inequalities for critical exponents is proved most notable is the low temperature josephson inequalitydv 2 2 others are 1 1 v 1 1 1 d 1 1 for d dv 3 for d 4 and 2m 2m 2 form 2 the hypotheses vary all inequalities are true for the spin 1 2 ising model with nearest neighbor ferromagnetic pair interactions