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Showing papers on "Ladder operator published in 1993"


Journal ArticleDOI
TL;DR: In this paper, a four-gluon partial wave amplitude was used to study the deep inelastic limit of the Gribov-Levin-Ryskin evolution equation.
Abstract: Starting from a four-gluon partial-wave amplitude which provides the first unitarity corrections to the Lipatov Pomeron we study the deep inelastic limit. We find mixing between the four-gluon operator and a twistfour piece of the two-gluon operator. We calculate the diagonal elements of the matrix of the anomalous dimensions and the coefficient function of the singular part (nearn=1) of the four-gluon operator. Several disagreements between our results and the Gribov-Levin-Ryskin evolution equation are found.

139 citations


Journal ArticleDOI
TL;DR: A unitary operator representing the exponential of the phase difference between two modes of the electromagnetic field is introduced, and the eigenvalue spectrum has a discrete character that is fully analyzed.
Abstract: We introduce a unitary operator representing the exponential of the phase difference between two modes of the electromagnetic field. The eigenvalue spectrum has a discrete character that is fully analyzed. We relate this operator with a suitable polar decomposition of the Stokes parameters of the field, obtaining a natural classical limit. The cases of weakly and highly excited states are considered, discussing to what extent it is possible to talk about the phase for a single-mode field. This operator is applied to some interesting two-mode fields.

116 citations


Journal ArticleDOI
TL;DR: The spectrum of a one-dimensional chain of SU(n) spins positioned at the static equilibrium positions of the particles in the corresponding classical Calogero system with an exchange interaction inversely proportional to the square of their distance is studied in this article.
Abstract: The spectrum of a one-dimensional chain of SU(n) spins positioned at the static equilibrium positions of the particles in the corresponding classical Calogero system with an exchange interaction inversely proportional to the square of their distance is studied. As in the translationally invariant Haldane-Shastry model the spectrum is found to exhibit a very simple structure containing highly degenerate 'super-multiplets'. The algebra underlying this structure is identified and several sets of raising and lowering operators are given explicitly. On the basis of this algebra and numerical studies the authors give the complete spectrum and thermodynamics of the SU(2) system.

88 citations


Journal ArticleDOI
TL;DR: The spectrum of a one-dimensional chain of $SU(n)$ spins positioned at the static equilibrium positions of the particles in a corresponding classical Calogero system with an exchange interaction inversely proportional to the square of their distance is studied in this paper.
Abstract: The spectrum of a one-dimensional chain of $SU(n)$ spins positioned at the static equilibrium positions of the particles in a corresponding classical Calogero system with an exchange interaction inversely proportional to the square of their distance is studied. As in the translationally invariant Haldane--Shastry model the spectrum is found to exhibit a very simple structure containing highly degenerate ``super-multiplets''. The algebra underlying this structure is identified and several sets of raising and lowering operators are given explicitely. On the basis of this algebra and numerical studies we give the complete spectrum and thermodynamics of the $SU(2)$ system.

69 citations


Journal ArticleDOI
TL;DR: In this article, a suitable ordering of phase exponential operators has been compared with the antinormal ordering of the annihilation and creation operators of a single mode optical field in the enlarged Hilbert space.
Abstract: A suitable ordering of phase exponential operators has been compared with the antinormal ordering of the annihilation and creation operators of a single mode optical field. The extended Wigner function for number and phase in the enlarged Hilbert space has been used for the derivation of the Wigner function for number and phase in the original Hilbert space.

45 citations


Journal ArticleDOI
TL;DR: Finite quantum systems are considered and dual quantities are defined with a finite Fourier transform and Ladder operators that translate the eigenstates of these quantities are shown to form a finite Weyl group.
Abstract: Finite quantum systems are considered and dual quantities are defined with a finite Fourier transform. Ladder operators that translate the eigenstates of these quantities are shown to form a finite Weyl group. Dual measurements are introduced and shown to obey certain entropic inequalities. A factorization of these systems into subsystems with the use of number-theoretic results is also presented

41 citations


Posted Content
TL;DR: In this paper, the authors describe a general method for approximating the spectrum of an operator A using the eigenvalues of large finite dimensional truncations of A. The results of several papers are summarized which imply that the method is effective in most cases of interest.
Abstract: This paper deals with mathematical issues relating to the computation of spectra of self adjoint operators on Hilbert spaces. We describe a general method for approximating the spectrum of an operator A using the eigenvalues of large finite dimensional truncations of A. The results of several papers are summarized which imply that the method is effective in most cases of interest. Special attention is paid to the Schrodinger operators of one-dimensional quantum systems. We believe that these results serve to make a broader point, namely that numerical problems involving infinite dimensional operators require a reformulation in terms of C ∗ -algebras. Indeed, it is only when the given operator A is viewed as an element of an appropriate C ∗ -algebra A that one can see the precise nature of the limit of the finite dimensional eigenvalue distributions: the limit is associated with a tracial state on A. For example, in the case where A is the discretized Schrodinger operator associated with a one-dimensional quantum system, A is a simple C ∗ -algebra having a unique tracial state. In these cases there is a precise asymptotic result.

35 citations


Journal ArticleDOI
TL;DR: In this paper, the authors investigated the relationship between the Ψ-space and probability-operator measure (POM) approaches and showed that the relation between non-orthogonal POMs and probability has to be accepted in the latter method as a postulate.
Abstract: In the Ψ-space limiting procedure limits of expectation values, rather than operator or states, are found as the state space dimensionality tends to infinity. This approach has been applied successfully to the calculation of the phase properties of various states of light, but its status as a valid quantum mechanical thory equivalent to the usual infinite Hilbert space (H-space) approach has not yet been fully accepted. Here we address this issue by investigating the formal relationship between the two approaches. We establish the consistency between the Ψ-space and H-space approaches for observables which are amenable to an H-space treatment. Such observables are represented in H by operators which are strong limits of Ψ-space operators and which obey the same algebra as the corresponding Ψ-space operators. The phase operator, however, exists in H only as a weak limit of a Ψ-space operator. For such limits the Ψ-space operator algebra is not preserved, which is the fundamental reason for the difficulties in constructing a consistent quantum description of phase in H. We show that for the phase observable the Ψ-space approach is consistent with the probability-operator measure (POM) method with the important distinction that, whereas the relation between non-orthogonal POMs and probability has to be accepted in the latter method as a postulate, the corresponding relation is derived in the Ψ-space approach. We conclude that the Ψ-space approach is not only equivalent to, but is also more fundamental than both the H-space and POM approaches.

32 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the Selberg zeta function ζs(s) yields an exact relationship between the periodic orbits of a fully chaotic Hamiltonian system (the geodesic flow on surfaces of constant negative curvature) and the spectrum of the Laplace-Beltrami operator on the same manifold.

31 citations


Journal ArticleDOI
TL;DR: In this article, the deformed commutation relations of raising and lowering operators that transform under the SUq(n)−quantum group get deformed decomposition relations and are represented as adjoint operators on a Hilbert space of noncommutative holomorphic functions.
Abstract: Raising and lowering operators that transform under the SUq(n)‐quantum group get deformed commutation relations. They are represented as adjoint operators on a Hilbert space of noncommutative holomorphic functions. Through the algebraically defined integral on this function space, every operator on the Fock space can also be represented as an integral kernel. The Green function for free harmonic oscillators and spin‐1/2’s in a constant magnetic field is given. Further on it is studied how such systems react on the switching on of a driving force. Calculating the vacuum–vacuum transition amplitude, it is found that the deviations from the undeformed case grow with the strength of the driving force. Throughout all calculations the bosonic and the fermionic cases are considered simultaneously.

29 citations


Journal ArticleDOI
Masashi Ban1
TL;DR: In this paper, a quantum mechanical phase operator is presented in terms of the relative number states and defined on the direct product space of the two Hilbert spaces, and it is shown that the phase operator gives the same results as are obtained using the Pegg-Barnett phase operator.

Journal ArticleDOI
TL;DR: In this paper, the coefficient functions in the products of φ2 and φ4 to first order in the parameter λ were derived and two-loop beta functions were obtained.

Journal ArticleDOI
TL;DR: In this article, a lower bound for the largest root of the continuous Hahn polynomials Sn(x) is given. But this lower bound is not applicable to the case where the state vectors form a natural basis for state vectors.
Abstract: Continuous Hahn polynomials Sn(x) appear in a formulation of quantum mechanics on a discrete time lattice, where they form a natural basis for the state vectors. In this paper we derive some of their generating functions, the expression of the raising and lowering operators and give a lower bound for the largest root of the equation Sn(x)=0.

Journal ArticleDOI
TL;DR: In this article, it was shown that an anti-self-adjoint operator with quaternion defined spectrum in [0,∞] has a symmetrical effective spectrum, from the point of view of functional analysis, under quite general conditions.
Abstract: Following the spectral methods established by Adler in his treatment of scattering and decay in the quaternionic quantum theory, it is shown that an anti‐self‐adjoint operator with quaternion defined spectrum in [0,∞) has a symmetrical effective spectrum, from the point of view of functional analysis, under quite general conditions. In the case of an operator with absolutely continuous spectrum in [0,∞), the effective spectrum is absolutely continuous in (−∞,∞), and a canonically conjugate operator exists. If the anti‐self‐adjoint operator is the generator of motion in time (Hamiltonian), the conjugate operator is a ‘‘time operator.’’ Moreover, according to a theorem of Misra, Prigogine, and Courbage, such a system may admit a Lyapunov operator, and therefore describe irreversible behavior. It is shown directly that no evident contradiction (as is found in the semibounded case in complex quantum mechanics) arises from the definition of a Lyapunov operator in quaternionic quantum mechanics.

Journal ArticleDOI
TL;DR: In this article, the quantum mechanics of charged, massive, spin-1 bosons in the presence of a homogeneous magnetic field (HMF) were studied using a six-component wave function formalism.
Abstract: The quantum mechanics of charged, massive, spin-1 bosons in the presence of a homogeneous magnetic field (HMF) is studied using a six-component wavefunction formalism. The energy eigenvalues are compared with those previously obtained via other formalisms, the equations of motion of certain operators are given, and the positive and negative energy eigensolutions are obtained by the use of a ladder operator method. The six-component current for the case of general external electromagnetic fields is also displayed and finally, the employment of the eigensolutions and current in a study of a spin-1 boson-antiboson plasma is discussed.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the spectrum of the Frobenius-Perron operator for a class of piecewise-linear maps that satisfy an intertwining relation between the derivative operator and the FPN operator.

Journal ArticleDOI
Xian-Geng Zhao1
TL;DR: In this paper, a general closed-form solution of the time-evolution operator is obtained, from which the results for any special case can be derived, which may be useful for future studies on a variety of fields in quantum physics.

Journal ArticleDOI
TL;DR: In this article, the Schrodinger operator with constant full-rank magnetic field, perturbed by an electric potential which decays at infinity, and has a constant sign was considered, and the asymptotic behaviour for large values of the electric-field coupling constant of the eigenvalues situated in the gaps of the essential spectrum of the unperturbed operator was studied.
Abstract: We consider the Schrodinger operator with constant full-rank magnetic field, perturbed by an electric potential which decays at infinity, and has a constant sign. We study the asymptotic behaviour for large values of the electric-field coupling constant of the eigenvalues situated in the gaps of the essential spectrum of the unperturbed operator.

Journal ArticleDOI
TL;DR: In this enlarged space of the problem, a phase operator is found that reproduces previous candidates to represent a well-behaved phase operator in the quantum domain and makes possible the unitarity of its representations in quantum optics.
Abstract: The well-known difficulties of defining a phase operator of an oscillator are considered from the point of view of the canonical transformation to action and phase-angle variables. This transformation turns out to be nonbijective, i.e., it is not a one-to-one onto mapping. In order to make possible the unitarity of its representations in quantum optics we should enlarge the Hilbert space of the problem. In this enlarged space we find a phase operator that, after projection, reproduces previous candidates to represent a well-behaved phase operator in the quantum domain.

Journal ArticleDOI
TL;DR: In this paper, a series of decomposition theorems are presented, leading to a description of the structure of the exponents and symmetries of an arbitrary operator stable measure.
Abstract: A series of decomposition theorems are presented, leading to a description of the structure of the exponents and symmetries of an arbitrary operator stable measure. An example of a full operator stable measure with a one parameter group as its symmetry group is presented. Connections between the structure results and the generalized domain of attraction problem and tail behavior of the operator stable measure are described.

Journal ArticleDOI
TL;DR: In this article, the authors considered the problem of polar decomposition of a nonhermitean operator in a linear space and proposed a phase representation for the Boson operator in the extended space.
Abstract: In this paper I consider the general mathematical problem of the polar decomposition of an operator in a linear space. Extending the space makes it possible to define a unitary operator related to the original nonhermitean one. By Stone's theorem this guarantees the existence of a phase operator in the extended space. The connection with supersymmetry is pointed out. Applying the general results to harmonic oscillator creation and annihilation operators we regain a phase description originally introduced by Newton. Projecting the phase operator from the extended space to the original one, we find a phase representation for the Boson operators. Introducing the conjugate rotation operator, one can describe the oscillator dynamics in the phase representation. The connection with the Barnett–Pegg phase operator is pointed out.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the wave operator with damping strictly less than the basic operator does not generate a strongly continuous semigroup on the product space H x H, contrary to a common belief.

Journal ArticleDOI
TL;DR: In this article, three algebraic structures of quantum projective [sl(2, ℂ)-invariant] field theory are studied: the operator algebra Vert(sl( 2, 2), the infinite-dimensional R matrix Rproj(u), and the deformation of weighted shift operators.
Abstract: Systematic studies are made of three algebraic structures of quantum projective [sl(2, ℂ)-invariant] field theory: the operator algebra Vert(sl(2, ℂ)), the infinite-dimensionalR matrixRproj(u), and the deformationTℏ(ℝ) of the algebraT(ℝ) of weighted-shift operators, which is associated with expansion of the renormalized pointwise product of vertex operator fields.

Book ChapterDOI
01 Jan 1993
TL;DR: In this paper, the Navier-Stokes operator splitting problem is solved explicitly and in which operator splitting converges, and a one-dimensional model with Euler/Stokes and Chorin-Marsden product formula is presented.
Abstract: Operator splitting applied to the Navier-Stokes equations, in which alternately the nonlinearity and the diffusion are ignored, requires the accommodation of incompatible initial conditions at each time step. A one-dimensional model problem which can be solved explicitly and in which operator splitting converges is presented. Euler/Stokes and the Chorin-Marsden product formula are discussed. Convergence results are reviewed.

Journal ArticleDOI
TL;DR: In this paper, an adiabatic core polarization procedure of Christiansen is expanded to derive an REP-based relativistic core/valence correlation potential (RCVPP) operator.

Journal ArticleDOI
Zurong Yu1
TL;DR: In this paper, the relations between the q-photon phase operator and q-spin coherent states are discussed, and new results are given for the relation between the two states.

Journal ArticleDOI
TL;DR: Exact wave functions for non-Abelian Chern-Simons (NACS) particles are obtained by the ladder operator approach because the two distinct base states of the NACS particles are multivalued and are defined in terms of path-ordered line integrals.
Abstract: Exact wave functions for [ital N] non-Abelian Chern-Simons (NACS) particles are obtained by the ladder operator approach. The same method has previously been applied to construct exact wave functions for multianyon systems. The two distinct base states of the NACS particles that we use are multivalued and are defined in terms of path-ordered line integrals. Only strings of operators that preserve the monodromy properties of these base states are allowed to act on them to generate new states.

Journal ArticleDOI
TL;DR: In this article, it was shown that there is only one operator having some minimal properties enabling it to be a one-photon position operator, and the solution is shown to be the photon position operator proposed by Pryce.

Journal ArticleDOI
01 Feb 1993
TL;DR: In this paper, the integration operator in two variables on L2[0, 1]2 is considered, and the multiplicity and reducing subspaces of the integrator are determined.
Abstract: In this paper we consider the integration operator in two variables on L2[0, 1]2, determine its multiplicity and reducing subspaces, and make some observations about its invariant and hyperinvariant subspaces.

Journal ArticleDOI
TL;DR: In this paper, the authors revisited quantum kinematics as a group-theoretic quantization procedure within the regular representation of non-Abelian non-compact r-dimensional Lie groups.
Abstract: Quantum kinematics is revisited, as a group-theoretic quantization procedure within the regular representation of non-Abelian non-compact r-dimensional Lie groups. The set of r basic quantum-kinematic invariant operators is exhibited; generalized Heisenberg commutation relations and the structure of the closed generalized Weyl-Heisenberg algebra of the quantized group are also discussed. Then it is shown how these structures yield a complete set of r 'annihilation' and 'creation' boson operators, which give rise to several intrinsic (i.e. embedded) Lie algebras, obtained in the standard way, within the quantized group model. As a miscellaneous example, these features are discussed within the quantum-kinematic theory of the Poincare group P+up arrow (1,1), and some interesting possibilities for elementary particle theory are conjectured in the light of the attained results.