Showing papers on "Ladder operator published in 2013"
TL;DR: In this paper, the quantum dimensions of vertex operator algebras are defined and their properties are discussed systematically, and a criterion for simple current modules of a rational vertex operator algebra is given.
Abstract: The quantum dimensions of modules for vertex operator algebras are defined and their properties are discussed systematically. The quantum dimensions of the Heisenberg vertex operator algebra modules, the Virasoro vertex operator algebra modules and the lattice vertex operator algebra modules are computed. A criterion for simple current modules of a rational vertex operator algebra is given. The possible values of the quantum dimensions are obtained for rational vertex operator algebras. A full Galois theory for rational vertex operator algebras is established using the quantum dimensions.
107 citations
TL;DR: In this article, a model operator approach to calculations of the QED corrections to energy levels in relativistic many-electron atomic systems is developed, where the Lamb shift operator is represented by a sum of local and non-local potentials which are defined using the results of ab initio calculations of diagonal and nondiagonal matrix elements of the one-loop QED operator with H-like wave functions.
Abstract: A model operator approach to calculations of the QED corrections to energy levels in relativistic many-electron atomic systems is developed. The model Lamb shift operator is represented by a sum of local and nonlocal potentials which are defined using the results of ab initio calculations of the diagonal and nondiagonal matrix elements of the one-loop QED operator with H-like wave functions. The model operator can be easily included in any calculations based on the Dirac-Coulomb-Breit Hamiltonian. The efficiency of the method is demonstrated by comparison of the model QED operator results for the Lamb shifts in many-electron atoms and ions with exact QED calculations.
102 citations
TL;DR: In this article, it was shown that the Kirchberg conjecture is equivalent to the statement that every operator system that is (min,er)-nuclear is also (el,c)-nuclear.
98 citations
TL;DR: In this paper, the type III Hermite Xm exceptional orthogonal polynomial family is generalized to a double-indexed one and corresponding rational extensions of the harmonic oscillator are constructed by using second-order supersymmetric quantum mechanics.
Abstract: The type III Hermite Xm exceptional orthogonal polynomial family is generalized to a double-indexed one (with m1 even and m2 odd such that m2 > m1) and the corresponding rational extensions of the harmonic oscillator are constructed by using second-order supersymmetric quantum mechanics. The new polynomials are proved to be expressible in terms of mixed products of Hermite and pseudo-Hermite ones, while some of the associated potentials are linked with rational solutions of the Painlev? IV equation. A novel set of ladder operators for the extended oscillators is also built and shown to satisfy a polynomial Heisenberg algebra of order m2 ? m1 + 1, which may alternatively be interpreted in terms of a special type of (m2 ? m1 + 2)th-order shape invariance property.
67 citations
TL;DR: In this article, the authors present a framework where first-principles calculations of jet modification may be carried out in a nonperturbative thermal environment, where the leading-order contribution to the transverse momentum broadening of a high-energy (near on-shell) quark in a thermal medium is computed.
Abstract: We present a framework where first-principles calculations of jet modification may be carried out in a nonperturbative thermal environment. As an example of this approach, we compute the leading-order contribution to the transverse momentum broadening of a high-energy (near on-shell) quark in a thermal medium. This involves a factorization of a nonperturbative operator product from the perturbative process of scattering of the quark. An operator product expansion of the nonperturbative operator product is carried out and related via dispersion relations to the expectation of local operators. These local operators are then evaluated in quenched $SU(2)$ lattice gauge theory.
63 citations
TL;DR: In this paper, a ladder operator for a rational extension of the harmonic oscillator associated with type III Hermite exceptional orthogonal polynomials and characterized by an even integer m was constructed.
Abstract: New ladder operators are constructed for a rational extension of the harmonic oscillator associated with type III Hermite exceptional orthogonal polynomials and characterized by an even integer m. The eigenstates of the Hamiltonian separate into m + 1 infinite-dimensional unitary irreducible representations of the corresponding polynomial Heisenberg algebra. These ladder operators are used to construct a higher-order integral of motion for two superintegrable two-dimensional systems separable in cartesian coordinates. The polynomial algebras of such systems provide for the first time an algebraic derivation of the whole spectrum through their finite-dimensional unitary irreducible representations.
47 citations
TL;DR: In this article, the authors construct the holonomy-flux operator algebra in the recently developed spinor formulation of loop gravity and show that the familiar grasping and Wilson loop operators are written as composite operators built from the gauge-invariant ''generalized ladder operators'' recently introduced in the $\mathrm{U}(N)$ approach.
Abstract: We construct the holonomy-flux operator algebra in the recently developed spinor formulation of loop gravity. We show that, when restricting to SU(2)-gauge invariant operators, the familiar grasping and Wilson loop operators are written as composite operators built from the gauge-invariant ``generalized ladder operators'' recently introduced in the $\mathrm{U}(N)$ approach to intertwiners and spin networks. We comment on quantization ambiguities that appear in the definition of the holonomy operator and use these ambiguities as a toy model to test a class of quantization ambiguities which is present in the standard regularization and definition of the Hamiltonian constraint operator in loop quantum gravity.
44 citations
TL;DR: In this article, the relation between the Kaneko-Zagier equation and the Mathur-Mukhi-Sen classification was studied and extended to the case of solutions with logarithmic terms, which correspond to pseudo-characters of non-rational vertex operators.
Abstract: We study the relation between the Kaneko–Zagier equation and the Mathur–Mukhi–Sen classification, and extend it to the case of solutions with logarithmic terms, which correspond to pseudo-characters of non-rational vertex operator algebras. As an application, we prove a non-existence theorem of rational vertex operator algebras.
34 citations
TL;DR: In this article, a ladder operator for a rational extension of the harmonic oscillator associated with type III Hermite exceptional orthogonal polynomials and characterized by an even integer $m was constructed.
Abstract: New ladder operators are constructed for a rational extension of the harmonic oscillator associated with type III Hermite exceptional orthogonal polynomials and characterized by an even integer $m$. The eigenstates of the Hamiltonian separate into $m+1$ infinite-dimensional unitary irreducible representations of the corresponding polynomial Heisenberg algebra. These ladder operators are used to construct a higher-order integral of motion for two superintegrable two-dimensional systems separable in cartesian coordinates. The polynomial algebras of such systems provide for the first time an algebraic derivation of the whole spectrum through their finite-dimensional unitary irreducible representations.
33 citations
TL;DR: In this article, all spin operators for a Dirac particle satisfying the following very general conditions: (i) spin does not convert positive (negative) energy states into negative (positive) states, (ii) spin is a pseudovector, and (iii) eigenvalues of the projection of a spin operator in an arbitrary direction are independent of this direction.
Abstract: We find all spin operators for a Dirac particle satisfying the following very general conditions: (i) spin does not convert positive (negative) energy states into negative (positive) energy states, (ii) spin is a pseudovector, and (iii) eigenvalues of the projection of a spin operator in an arbitrary direction are independent of this direction (isotropy condition). We show that there are four such operators and all of them fulfill the standard su(2) Lie algebra commutation relations. Nevertheless, only one of them has a proper nonrelativistic limit and acts in the same way on negative and positive energy states. We show also that this operator is equivalent to the Newton-Wigner spin operator and Foldy-Wouthuysen mean-spin operator. We also discuss another operator proposed in the literature.
31 citations
TL;DR: In this paper, the covariant relativistic spin operator is shown to be equivalent to the spin operator commuting with the free Dirac Hamiltonian, which implies that it is a good quantum observable.
Abstract: We have shown that the covariant relativistic spin operator is equivalent to the spin operator commuting with the free Dirac Hamiltonian. This implies that the covariant relativistic spin operator is a good quantum observable. The covariant relativistic spin operator has a pure quantum contribution that does not exist in the classical covariant spin operator. Based on this equivalence, reduced spin states can be clearly defined. We have shown that depending on the relative motion of an observer, the change in the entropy of a reduced spin density matrix sweeps through the whole range.
TL;DR: In this article, the first and certain second order supersymmetric partners of the initial Hamiltonian possess third-order differential ladder operators and are linked with the Painleve IV equation.
Abstract: Supersymmetry transformations of first and second order are used to generate Hamiltonians with known spectra departing from the harmonic oscillator with an infinite potential barrier. It is studied also the way in which the eigenfunctions of the initial Hamiltonian are transformed. The first and certain second order supersymmetric partners of the initial Hamiltonian possess third-order differential ladder operators. Since systems with this kind of operators are linked with the Painleve IV equation, several solutions of this non-linear second-order differential equation will be simply found.
TL;DR: In this article, the meaning of time in an open quantum system is considered under the assumption that both, system and environment, are quantum mechanical objects, and the Hamilton operator of the system is non-Hermitian.
Abstract: The meaning of time in an open quantum system is considered under the assumption that both, system and environment, are quantum mechanical objects. The Hamilton operator of the system is non-Hermitian. Its imaginary part is the time operator. As a rule, time and energy vary continuously when controlled by a parameter. At high level density, where many states avoid crossing, a dynamical phase transition takes place in the system under the influence of the environment. It causes a dynamical stabilization of the system what can be seen in many different experimental data. Due to this effect, time is bounded from below: the decay widths (inverse proportional to the lifetimes of the states) do not increase limitless. The dynamical stabilization is an irreversible process.
TL;DR: In this paper, the form of the spin operator in relativistic quantum mechanics was derived for the case when the states with negative energies are admitted. But the form was not shown to be invariant to the number of states in the system.
TL;DR: In this article, Tremblay, Turbiner and Winternitz construct a symmetry algebra for the 3D Kepler-Coulomb system and show that the symmetry algebra closes algebraically.
Abstract: The quantum Kepler–Coulomb system in three dimensions is well known to be second order superintegrable, with a symmetry algebra that closes polynomially under commutators. This polynomial closure is also typical for second order superintegrable systems in 2D and for second order systems in 3D with nondegenerate (four-parameter) potentials. However, the degenerate three-parameter potential for the 3D Kepler–Coulomb system (also second order superintegrable) is an exception, as its symmetry algebra does not close polynomially. The 3D four-parameter potential for the extended Kepler–Coulomb system is not even second order superintegrable, but Verrier and Evans (2008 J. Math. Phys. 49 022902) showed it was fourth order superintegrable, and Tanoudis and Daskaloyannis (2011 arXiv:11020397v1) showed that, if a second fourth order symmetry is added to the generators, the symmetry algebra closes polynomially. Here, based on the Tremblay, Turbiner and Winternitz construction, we consider an infinite class of quantum extended Kepler–Coulomb three- and four-parameter systems indexed by a pair of rational numbers (k1, k2) and reducing to the usual systems when k1 = k2 = 1. We show these systems to be superintegrable of arbitrarily high order and determine the structure of their symmetry algebras. We demonstrate that the symmetry algebras close algebraically; only for systems admitting extra discrete symmetries is polynomial closure achieved. Underlying the structure theory is the existence of raising and lowering operators, not themselves symmetry operators or even defined independent of basis, that can be employed to construct the symmetry operators and their structure relations.
TL;DR: In this paper, the authors characterize topologically mixing cosine operator functions, generated by unilateral and bilateral weighted shifts on l p ( N 0 ) and l p( Z ) respectively.
TL;DR: Using the recently discovered N=1 supersymmetric extension of the conformal fourth-order scalar operator (introduced originally by Fradkin and Tseytlin and also known as the "Paneitz operator" or "Riegert operator"), the authors derived a new representation for the nonlocal action generating the super-Weyl anomalies.
Abstract: Using the recently discovered N=1 supersymmetric extension of the conformal fourth-order scalar operator (introduced originally by Fradkin and Tseytlin and also known as the "Paneitz operator" or "Riegert operator"), we derive a new representation for the nonlocal action generating the super-Weyl anomalies.
TL;DR: In this paper, the ladder operators associated with the pseudoharmonic oscillator were introduced by solving the corresponding Schrodinger equation by using the factorization method, and the obtained generalized raising and lowering operators naturally lead us to the Dirac representation space of the system which is much easier to work with, in comparison to the functional Hilbert space.
Abstract: In this paper we try to introduce the ladder operators associated with the pseudoharmonic oscillator, after solving the corresponding Schrodinger equation by using the factorization method. The obtained generalized raising and lowering operators naturally lead us to the Dirac representation space of the system which is much easier to work with, in comparison to the functional Hilbert space. The SU (1, 1) dynamical symmetry group associated with the considered system is exactly established through investigating the fact that the deduced operators satisfy appropriate commutation relations. This result enables us to construct two important and distinct classes of Barut—Girardello and Gilmore—Perelomov coherent states associated with the system. Finally, their identities as the most important task are exactly resolved and some of their nonclassical properties are illustrated, numerically.
TL;DR: Upper estimates for the low-lying eigenvalues of the Dirichlet realization of the magnetic Schrödinger operator H in the semiclassical limit are estimated and the existence of an arbitrary large number of spectral gaps inThe semiclassicals limit in the corresponding periodic setting is proved.
TL;DR: In this article, the ladder operators and associated compatibility conditions for types I and II multiple orthogonal polynomials were obtained, which can be used to derive the differential equations satisfied by multiple OOPs with exponential weights and cubic potentials.
Abstract: In this paper, we obtain the ladder operators and associated compatibility conditions for types I and II multiple orthogonal polynomials. These ladder equations extend known results for orthogonal polynomials and can be used to derive the differential equations satisfied by multiple orthogonal polynomials. Our approach is based on Riemann–Hilbert problems and the Christoffel–Darboux formula for multiple orthogonal polynomials, and the nearest-neighbor recurrence relations. As an illustration, we give several explicit examples involving multiple Hermite and Laguerre polynomials, and multiple orthogonal polynomials with exponential weights and cubic potentials.
TL;DR: In this article, the authors study lifting maps that are on one hand equivariant with respect to divergenceless vector fields, and on the other hand with values in self-adjoint or anti-self-adjoint operators, and apply it to the study of diff(M)-equivariant liftings.
Abstract: Let Δ be a linear differential operator acting on the space of densities of a given weight λ0 on a manifold M. One can consider a pencil of operators Π(Δ)={Δλ} passing through the operator Δ such that any Δλ is a linear differential operator acting on densities of weight λ. This pencil can be identified with a linear differential operator Δ acting on the algebra of densities of all weights. The existence of an invariant scalar product in the algebra of densities implies a natural decomposition of operators, i.e., pencils of self-adjoint and anti-self-adjoint operators. We study lifting maps that are on one hand equivariant with respect to divergenceless vector fields, and, on the other hand, with values in self-adjoint or anti-self-adjoint operators. In particular, we analyze the relation between these two concepts, and apply it to the study of diff (M)-equivariant liftings. Finally, we briefly consider the case of liftings equivariant with respect to the algebra of projective transformations and descri...
TL;DR: In this paper, the quotient of a Boolean algebra by a cyclic group is proven to have a symmetric chain decomposition, which generalizes earlier work of Griggs, Killian and Savage on the case of prime order, giving an explicit construction for any order, prime or composite.
Abstract: The quotient of a Boolean algebra by a cyclic group is proven to have a symmetric chain decomposition. This generalizes earlier work of Griggs, Killian and Savage on the case of prime order, giving an explicit construction for any order, prime or composite. The combinatorial map specifying how to proceed downward in a symmetric chain is shown to be a natural cyclic analogue of the $\mathfrak{sl}_2$ lowering operator in the theory of crystal bases.
TL;DR: In this paper, it was shown that the spectrum of a Schrodinger's operator is equal to the generalized spectrum of two bounded operators, which is the same spectrum of the harmonic oscillator.
Abstract: We show that the spectrum of a Schrodinger’s operator is equal to the generalized spectrum of two bounded operators. Using an approximation method of integral operator, based on regularization by convolution and Fourier series, we approach perfectly the spectrum of the harmonic oscillator.
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TL;DR: In this paper, the ladder operators associated with the pseudoharmonic oscillator were introduced, after solving the corresponding Schr\"{o}dinger equation by using the factorization method.
Abstract: In this paper we try to introduce the ladder operators associated with the pseudoharmonic oscillator, after solving the corresponding Schr\"{o}dinger equation by using the factorization method. The obtained generalized raising and lowering operators naturally lead us to the Dirac representation space of the system which is very easier to work with, in comparison to the functional Hilbert space. The SU(1,1) dynamical symmetry group associated with the considered system is exactly established through investigating the fact that the deduced operators satisfy appropriate commutation relations. This result enables us to construct two important and distinct classes of Barut-Girardello and Gilmore-Perelomov coherent states associated with the system. Finally, their identities as the most important task are exactly resolved and some of their nonclassical properties are illustrated, numerically.
TL;DR: In this article, the action of the dilatation operator on the basis of local operators constructed from elements of the walled Brauer algebra, with non-planar corrections fully taken into account, is studied.
TL;DR: In this article, a three-dimensional generalized isotonic oscillator whose radial part is the newly introduced generalized ISO whose bound state solutions have been shown to admit the recently discovered X1-Laguerre polynomials was investigated.
Abstract: We explore squeezed coherent states of a three-dimensional generalized isotonic oscillator whose radial part is the newly introduced generalized isotonic oscillator whose bound state solutions have been shown to admit the recently discovered X1-Laguerre polynomials. We construct a complete set of squeezed coherent states of this oscillator by exploring the squeezed coherent states of the radial part and combining the latter with the squeezed coherent states of the angular part. We also prove that the three-mode squeezed coherent states resolve the identity operator. We evaluate Mandel?s Q-parameter of the obtained states and demonstrate that these states exhibit sub-Possionian and super-Possionian photon statistics. Furthermore, we illustrate the squeezing properties of these states, both in the radial and angular parts, by choosing appropriate observables in the respective parts. We also evaluate the Wigner function of these three-mode squeezed coherent states and demonstrate the squeezing property explicitly.
TL;DR: In this paper, a non-Hermitian generalized oscillator model was studied in the framework of R-deformed Heisenberg algebra and the spectrum was obtained using algebraic technique.
Abstract: A non-Hermitian generalized oscillator model, generally known as the Swanson model, has been studied in the framework of R-deformed Heisenberg algebra. The non-Hermitian Hamiltonian is diagonalized by generalized Bogoliubov transformation. A set of deformed creation annihilation operators is introduced whose algebra shows that the transformed Hamiltonian has conformal symmetry. The spectrum is obtained using algebraic technique. The superconformal structure of the system is also worked out in detail. An anomaly related to the spectrum of the Hermitian counterpart of the non-Hermitian Hamiltonian with generalized ladder operators is shown to occur and is discussed in position dependent mass scenario.
TL;DR: In this article, a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric is developed, where the spectrum is symmetric about zero and zero itself is a double eigen value.
Abstract: Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formula and the eta invariant.
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TL;DR: In this article, the perturbed periodic Maxwell operator in d = 3 can be seen as a pseudodifferential operator and the behavior of M_0 and the physical initial states at small crystal momenta $k$ and small frequencies |omega| is characterized.
Abstract: As a first step to deriving effective dynamics and ray optics, we prove that the perturbed periodic Maxwell operator in d = 3 can be seen as a pseudodifferential operator. This necessitates a better understanding of the periodic Maxwell operator M_0. In particular, we characterize the behavior of M_0 and the physical initial states at small crystal momenta $k$ and small frequencies |\omega|. Among other things, we prove that generically the band spectrum is symmetric with respect to inversions at k = 0 and that there are exactly 4 ground state bands with approximately linear dispersion near k = 0.
TL;DR: In this paper, a description of all pairs of Hermitian operators X and Y, which satisfy the condition −Y ≤ X ≤ Y ≤ Y, is given, and it is shown that this inequality does not necessarily imply the inequality |X| ≤ ZY Z* for any operator Z, ‖Z ≤ 1.
Abstract: We obtain a description of all pairs of Hermitian operators X and Y, which satisfy the condition −Y ≤ X ≤ Y. We give the examples of such operator pairs. Each of the presented examples leads us to the new weak majorization for the Hermitian operator pair. It is shown that this inequality does not necessarily imply the inequality |X| ≤ ZY Z* for any operator Z, ‖Z‖ ≤ 1. We prove that invertibility of Y follows from invertibility of operators X and A* A for Hermitian operators X and Y, Y ≥ 0 and an arbitrary operator A such that −AY A* ≤ X ≤ AY A*. We discuss one analog of triangle inequality found by the author in one earlier paper for pairs of Hermitian operators.