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Showing papers on "Operator (computer programming) published in 1968"



Journal ArticleDOI
TL;DR: In this paper, the development of the time-translation operators in a matrix element of an arbitrary operator is examined, and a connection is established between the real-time Green's functions and the Green's function defined along the path, thereby yielding a perturbation expansion for the realtime functions and a justification of the interaction terms in the equations of motion for realtime quantities.
Abstract: The development of the time‐translation operators in a matrix element of an arbitrary operator is examined. It is noted that we may interpret time as evolving from some remotely early time (t0) to a time in the far future (t∝) and then back to (t0). Using this interpretation, a perturbation expansion is developed for Green's functions defined along this path and a separation of the two‐particle interaction terms into self‐energy parts and single‐particle Green's function terms is justified for quantities on this path. A connection is established between the real‐time Green's functions and the Green's function defined along the path, thereby yielding a perturbation expansion for the real‐time functions and a justification of the separation of the interaction terms in the equations of motion for the real‐time quantities. The transport equations of Kadanoff and Baym are derived without resorting to an analytic continuation from imaginary times and without the correction terms of Fujita.

140 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered the initial value problem and showed that the decay constant X is an eigenvalue of the Boltzmann operator A. The exact domain of definition of A is dependent on the space of functions where A acts, and will be described in Section 2.

137 citations


Journal ArticleDOI
TL;DR: In this paper, the relationship between the mass-shell singularities and the asymptotic states of the theory is studied by considering the nature of the intermediate states that can contribute to the corresponding discontinuity functions.
Abstract: In paper II of this series, the mass-shell singularities of the Green's functions of quantum electrodynamics were investigated. In this paper, the relationship between these singularities and the asymptotic states of the theory is studied by considering the nature of the intermediate states that can contribute to the corresponding discontinuity functions. The basic principle underlying this work is that the asymptotic states of the theory should not be specified a priori but should be determined from the structure of the Green's functions themselves. The pure soft-photon asymptotic states, which can be created from the vacuum by operators constructed from the soft-photon part of the electromagnetic field, are studied first. These states are defined by appropriate weak limits and are shown to span a space with the same structure as in the noninteracting case. Next, states containing a single particle (massive particle or hard photon), together with soft photons, are investigated. These states can appear as intermediate states in the two-point function. They are again defined by weak limits, and are shown to be stable in the absence of external currents. It is demonstrated that the near-mass-shell components of the field operator, acting on the vacuum or on a soft-photon coherent state, yield a state containing one particle and a soft-photon coherent state. Finally, the analysis is extended to two-particle and multiparticle states. The only essentially new feature here is the appearance of factors related to the "Coulomb phases." General reduction formulas are obtained that permit matrix elements between arbitrary asymptotic states to be extracted from the Green's functions. In effect, these matrix elements may be identified with the coefficients not of poles but of branch-point singularities.

126 citations


Journal ArticleDOI
TL;DR: For a large class of tensor operators in U(n), a surprisingly simple diagrammatic calculus of patterns is shown to exist; to each operator of this class a pattern may be assigned in terms of which the complete algebraic formula for all matrix elements may be read off directly as discussed by the authors.
Abstract: For a large class of tensor operators inU(n), a surprisingly simple diagrammatic calculus of patterns is shown to exist; to each operator of this class a pattern may be assigned in terms of which thecomplete algebraic formula for all matrix elements may be read off directly. The class of operators includes all fundamental, elementary and extremal Wigner operators in allU(n). Application of the pattern calculus toward the explicit determination of all tensor operators is discussed.

124 citations


Journal ArticleDOI
TL;DR: One parameter operator imbedding to modify Newton method for solution of nonlinear equations was proposed in this article, where one parameter operator was imbedded with a modified Newton method.
Abstract: One parameter operator imbedding to modify Newton method for solution of nonlinear equations

110 citations




Journal ArticleDOI
TL;DR: In this paper, a self-adjoint operator satisfying an exponentiated form of the equation Na* =a*(N+I), wherea* is an arbitrary creation operator.
Abstract: Anumber operator for a representation of the canonical commutation relations is defined as a self-adjoint operator satisfying an exponentiated form of the equationNa*=a*(N+I), wherea* is an arbitrary creation operator. WhenN exists it may be chosen to have spectrum {0, 1, 2, ...} (in a direct sum of Fock representations) or {0, ±1, ±2, ...} (otherwise). Examples are given of representations having number operators, and a necessary and sufficient condition is given for a direct-product representation to have a number operator.

56 citations


Journal ArticleDOI
TL;DR: The relation between the response of a macroscopic system to an external perturbation and time-dependent correlations in the equilibrium ensemble is examined in detail in this article, where ten correlation functions are constructed from commutators of the five operators appearing in the microscopic conservation laws.

47 citations


Journal ArticleDOI
TL;DR: In this paper, the relation between this algebra and the algebra of partial differential operators over Rn is compared with a relation between a quotient field and its generating ring, and various resulting advantages are essentially derived from the fact that a pseudo differential operator can be inverted within the algebra, if at all.
Abstract: Introduction. The conventional pseudo differential operators were introduced by Kohn and Nirenberg [8] and extensively studied in a generalized version by Hdrmander [7]. In either case they appear as operators acting on C, -functions of a differentiable manifold; their definition is designed to obtain a class of linear operators containing all linear partial differential operators with coefficients behaving like constants at infinity, and having properties like such differential operators. Many other important classes of linear operators are also special classes of pseudo differential operators over Rn: a) singular integral operators with C' -kernels; b) singular integro-differential operators of the form 7 KYDa IaI _N studied earlier by Dynin [4] and Seeley [10], with coefficients Ka being singular integral operators; c) regular and singular convolution operators, especially convolutions with functions having a singularity like I x 1-t at zero, as investigated earlier by Zarantonello [14]; d) integral operators with kernels like Green's functions. The Kohn-Nirenberg pseudo differential operators over Rn form an algebra, and the relation between this algebra and the algebra of partial differential operators over Rn might be compared with the relation between a quotient field and its generating ring. The various resulting advantages are essentially derived from the fact that a pseudo differential operator can be inverted within the algebra, if at all. All pesudo differential operators have an order, defined as a real number t such that the operator possesses extensions which map every L2-Soboleff space 1 continuously into >'~t. An operator of all orders is said to have order -oo. Also to every pseudo differential operator over Rn there is asso-

Journal ArticleDOI
TL;DR: In this article, the Weyl transform is applied in quantum dynamics to derive and extend Moyal's statistical theory of phase-space distributions for noncommuting coordinate and momentum operators, and the distinction is made between Weyl transforms in Schrodinger and Heisenberg pictures; the general case of time-dependent Hamiltonians is considered.
Abstract: The Weyl transform is applied in quantum dynamics to derive and extend Moyal's statistical theory of phase‐space distributions for noncommuting coordinate and momentum operators. The distinction is made between Weyl transforms in Schrodinger and Heisenberg pictures; the general case of time‐dependent Hamiltonians is considered. The Wigner function for the probability distribution in a phase space of Cartesian coordinates Q and momenta K propagates according to a conditional probability P(t, Q, K | t0, Q0, K0), which is exhibited as a Feynman path integral in phase space. Properties of P(t, Q, K | t0, Q0, K0) are developed; it is expressed in terms of the quantum generalization of the classical Liouville operator. The Weyl transform of a Heisenberg operator propagates according to P(t, Q0, K0 | t0, Q, K) which is also given as a Feynman path integral. An equation for the time evolution of Weyl transforms of Heisenberg operators is obtained, according to which the transform of Heisenberg coordinate and mome...

Patent
Farid J Neema1, John F. Graham1
20 Nov 1968
TL;DR: In this article, a method and apparatus for indicating the occurrence of, and procedure for correcting an error in an information processing device, by classifying various types of errors in accordance with various error recovery procedures to be used by an operator to correct such errors, and providing indications of such errors which specify which of a plurality of errors which are to be corrected.
Abstract: A method and apparatus for indicating the occurrence of, and procedure for correcting an error in an information processing device, by classifying various types of errors in accordance with various error recovery procedures to be used by an operator to correct such errors, and providing indications of such errors which specify which of a plurality of error recovery procedures is to be used to correct said errors.


Journal ArticleDOI
TL;DR: In this article, the rate of two-photon absorption from an arbitrary state of the radiation field, and its dependence on the statistical properties of the field, were studied, and it was found that when the density operator of a field factors into a product of single-mode density operators (in which case the modes are statistically independent), the rate depends on the spectral composition of the spectrum.
Abstract: The rate of two-photon absorption from an arbitrary state of the radiation field, and its dependence on the statistical properties of the field, are studied. It is found that when the density operator of the field factors into a product of single-mode density operators (in which case the modes are statistically independent), the rate depends on the spectral composition of the field. When the density operator is nonfactorable, the process depends on a photon doublet density. Therefore, the information obtained from two-photon experiments in the first case can, in principle, also be obtained from single-photon counting, while in the second case it would require coincidence counting.


Journal ArticleDOI
TL;DR: In this article, some new properties of self-adjoint operators are given, which are of some general interest by themselves, since they do not refer explicitly to eigenvalues.
Abstract: In the present note some new properties of selfadjoint operators are given. These results came to light in our investigation of Weinstein's new maximum-minimum theory of eigenvalues [ l ] , [2], but may be of some general interest by themselves, since they do not refer explicitly to eigenvalues. For the theory of unbounded operators, as used here, see the recent book of Goldberg [3]. Let T be any selfadjoint linear operator on a dense subspace 3) of a Hubert space 3C. Let

Journal ArticleDOI
TL;DR: In this paper, a brief summary of recently derived general results relating to the mapping of functions of noncommuting operators on functions of c-numbers is given which describe the time evolution of the c-number equivalents (phase-space representations) of the density operator and of a Heisenberg operator.
Abstract: After a brief summary of recently derived general results relating to the mapping of functions of noncommuting operators on functions of c-numbers, equations are given which describe the time evolution of the c-number equivalents (phase-space representations) of the density operator and of a Heisenberg operator. The evaluation of time-ordered functions of operators by c-number techniques is also briefly discussed.

Patent
24 Oct 1968
TL;DR: In this paper, a fluid flow measuring system that senses well input and output flows and converts them into compared electrical analogues for signaling the operator upon the occurrence of imbalance is described.
Abstract: For well drilling operations, a fluid flow measuring system that senses well input and output flows and converts them into compared electrical analogues for signaling the operator upon the occurrence of imbalance.

Journal ArticleDOI
TL;DR: In this article, the Riemann-Liouville fractional integro-differentiation operator was generalized to a generalized RiemANN fractional integral differential operator for the representation of analytic functions in the interior of a circle.
Abstract: In this paper we formulate operators which constitute an essential generalization of the Riemann-Liouville fractional integro-differentiation operator. With the aid of these operators fundamentally new analogues of the classical formulas of Cauchy, Schwarz, and Poisson for the representation of analytic and harmonic functions in the interior of a circle are established. These formulas enable us to give a complete structural representation for the broad classes of harmonic and analytic functions associated with the generalized operators. Finally, in this paper we also establish sufficient conditions for the solvability of the Hausdorff and Stieltjes moment problems for some general families of sequences of positive numbers.

Journal ArticleDOI
TL;DR: In this article, the authors considered a polynomial-time interaction Hamiltonian with smeared Fermion field operators (the smearing function providing an ultraviolet cut-off) and showed that time-displacements define a one-parameter group of automorphisms of the field algebra att=0, which acts continuously in the time-parameters.
Abstract: We consider a Euclidean invariant interaction Hamiltonian which is a polynomial in smeared Fermion field operators (the smearing function providing an ultraviolet cut-off). By considering Guenin's perturbation series for the time-development of the theory, we show that time-displacements define a one-parameter group of automorphisms of the field algebra att=0, which acts continuously in the time-parameter. Results are obtained for any dimension of space and for both relativistic and nonrelativistic forms for the free Hamiltonian. In special cases the total Hamiltonian is a positive self-adjoint operator in Fock space, thus defining a concrete non-relativistic quantized field with non-trivial particle production.

Journal ArticleDOI
TL;DR: In this article, the complete quantum-dynamical solutions for a free particle moving in one dimension and for free particles moving on a ring are obtained in terms of classical path sums.
Abstract: The complete quantum-dynamical solutions for a free particle moving in one dimension and for a free particle moving on a ring are obtained in terms of classical path sums. The propagators and Green operator kernels, and, in particular, the kernel of the spectral operator IE = ?(E - H) are expressed explicitly in this way. Relations between quantum and classical mechanics which hold for low quantum numbers and relatively small values of the classical action are discussed. They are the correspondence identities.




Journal ArticleDOI
TL;DR: In this paper, the authors considered the case of a perturbation Y of arbitrary finite rank and gave sufficient conditions under which an embedded eigenvalue h, of a self-adjoint operator T, of simple multiplicity vanishes under perturbations by I' (Theorem I), and the main tool in the proof was the formula (1.0.6) for the inverse of the Weinstein-Aronszajn matrix W(z).

Patent
29 Jul 1968

Patent
15 Oct 1968
TL;DR: A self-propelled combine has a forward transverse engine adjacent the operator's station, which drives various combine functions through several drive systems having inputs mounted on the engine output shaft and covered by an enclosure with a raisable top on which the operator seat is mounted as mentioned in this paper.
Abstract: A self-propelled combine has a forward transverse engine adjacent the operator''s station. The engine drives the various combine functions through several drive systems having inputs mounted on the engine output shaft and covered by an enclosure with a raisable top on which the operator''s seat is mounted. The seat is also shiftable rearwardly to permit standup operation of the combine, which is facilitated by an adjustable steering column. The handrail at the front of the operator''s station is attached to and moves with the steering column.


Patent
23 Dec 1968
TL;DR: An elevated cabin is provided at the rear of a passenger automobile to separate the operator from the passenger space and to provide him with an all-around view as discussed by the authors, which may be part of the original construction of the automobile or may be made to be attached to the rear.
Abstract: An elevated cabin is provided at the rear of a passenger automobile to separate the operator from the passenger space and to provide him with an all-around view. The cabin may be part of the original construction of the automobile or may be made to be attached to the rear of a completed automobile.