New many body perturbation method and the Hubbard model

By adapting the functional derivative method developed by Kadanoff and Baym to the Hubbard model, a new perturbation method is formulated. The unperturbed state is defined by the two equations which yield Hubbard's results, while the remainder is given by functional derivatives of the Green's functions which are shown to generate a complete perturbation series. Advantages of this method are discussed.     

An eikonal perturbation theory having applications in hadron dynamics,I: Genesis

We develop a Lagrangian field-theoretic laboratory where one can rigorously investigate ideas and problems in high-energy hadronic interactions. In this paper (the first of a series) the general field-theoretic framework is outlined in the oversimplified model of a scalar-scalar Yukawa interaction. Functional methods are used to cast all Green's functions in an “operator eikonal” form. The eikonal approximations (EA's) in Lagrangian relativistic quantum mechanics are reviewed and discussed. We then derive an exact eikonal equation in quantum field theory. The perturbation theoretic solution of this equation leads to a new kind of eikonal perturbation theory (EPT) which generalizes simultaneously the EA's as well as the ordinary perturbation theory (OPT). Some salient features of Green's functions in the EPT are as follows: (i) the lowest-order EPT amplitudes correspond to a kind of semiclassical approximation; (ii) the lowest-order four-point amplitudes contain the high-energy part of the full radiatively corrected crossed ladder series, without vacuum polarization effects; (iii) for spin-one gluons, the latter amplitude develops diffractive behavior in the direct channel and, for spin-one and spin-zero gluons, Regge behavior in the crossed channel; (iv) for vanishing gluon mass, this amplitude develops poles, in the direct channel, corresponding to a positronium-like bound-state spectrum. Properties (i)–(iv) are generalized to EPT from EA's and are absent in OPT. Unlike in the case of EA's we also have that (v) the EPT is a quantum field theory, which properly includes selfinteraction effects; (vi) the EPT is an iterative perturbation theoretic scheme, which shares with OPT the properties of renormalizability.     

Asymptotic nature of perturbation theory in the many-Fermion problem

It is shown, using Watson's lemma, that, given a suitable definition of normal systems, the loop expansion and hence the perturbation series in many-fermion series are asymptotic and Borel summable.     

Calculation of the vertical ionization potentials of HCNO,HNCO, HOCN,and HN3 by perturbation corrections to Koopmans'Theorem

Vertical ionization potentials (VIP's) have been calculated for HCNO, HNCO, HOCN, and HN₃, using Rayleigh—Schrödinger perturbation theory (RSPT) to determine corrections to Koopmans' Theorem. The calculated VIP's are used to resolve discordances between previous assignments of the PE spectra of HCNO, HNCO, and HN₃. Very little is known about HOCN. The present work contains predicted VIP's for this molecule, which it is hoped will aid in its identification in the laboratory. A diagram is presented correlating the VIP's of the isoelectronic series N₂O, HN₃, HOCN, HNCO, HCNO, and CO₂. It is shown that the variation in the magnitudes of the VIP's associated with the ionization of π electrons, and in the splittings induced in these VIP's by deviations from a linear geometry, can be accounted for qualitatively using a simple Hückel model.     

Summation of perturbation series in the generalized Hubbard model

The cumulant expansion is proposed for the summation of perturbation series in the generalized Hubbard model, which considers strongly correlated d-electrons hybridized with nearly free conduction electrons. It is shown, that summation of the principle series for the Green's functions leads to the Kondo renormalization of d-level hybridized with s-band.     

Formulation of the theory of turbulence in an incompressible fluid

The theory of turbulence in an incompressible fluid is formulated using methods similar to those of quantum field theory. A systematic perturbation theory is set up, and the terms in the perturbation series are shown to be in one to one correspondence with certain diagrams analogous to Feynman diagrams. From a study of the diagrams it is shown that the perturbation series can be rearranged and partially summed in such a way as to reduce the problem to the solution of three simultaneous integral equations for three functions, one of which is the second order velocity correlation function. The equations have the form of infinite power series integral equations, and the first few terms in the power series are derived from an analysis of the diagrams to sixth order. Truncation of the integral equations at the lowest nontrivial order yields Chandrasekhar's equation, and truncation at a higher order yields the equations discussed by Kraichnan.     

The exact nonlinear driven response of a one-dimensional gas

The exact nonlinear driven response of a one-dimensional gas of hard rods to a time dependent uniform external field is obtained in terms of the single particle Green's function. Limiting cases of very large and very small field are examined and, for the latter, comparison is made between the linear approximation to the exact response and the response obtained from the linear response theory perturbation approximation. Strong disagreement between these results is observed. This supports van Kampen's contention that a macroscopic linear response to an external perturbation is not a manifestation of microscopic linearity.     

A note on Keldysh's perturbation formalism

The expansion theorem of quantum field theoy relating Heisenberg operators to asymptotic free-field operators is rewritten by means of the time-path technique, originally due to Schwinger, which to date has only found application in statistical mechanics. The theorem is combined with Bogoliubov's initial condition of vanishing correlations in the infinite past to rederive Keldysh's perturbation scheme for non-equilibrium statistical Green's functions.     

Pulse compression techniques in ultrasonic non-destructive testing

The development of a pulse compression system applicable to ultrasonic flaw detection is described. The use of pulse compression technique permits a pulse-echo detection system to operate with long transmitted pulses for increased sensitivity, but without sacrificing resolution. The described system is economically implemented using an aluminium strip dispersive delay line. The operating system achieves a time-bandwidth product of 84 and sidelobe levels of —25 dB. Experimental evidence of the system's capabilities is presented.     

Wilson-Zimmermann expansions and the normal-product algorithm for the Ward identity of the axial-vector current in meson-nucleon dynamics

Zimmermann's normal-product algorithm is developed for the renormalizable γ₅ interaction of meson-nucleon dynamics. With the help of Zimmermann's algebraic identities Wilson's short-distance expansion and the Zimmermann identity are verified in perturbation theory for a fermion-fermion operator product. We demonstrate the power of this method in clarifying problems which seemed to appear in earlier, less rigorous, treatments of a generalized Ward-identity for the axial vector in such a field theory.     

串联式电光开关实现光脉冲宽度大范围可调

为了获得大范围脉宽可调的光脉冲,以连续激光为光源,采用两个外腔调制型电光开关,将该连续激光调制成单脉冲光源.其中两个外腔调制型电光开关采用串联方式工作,两者之间的检偏方向相互垂直,由第一级电光开关实现激光脉冲的快速开启,第二级电光开关实现激光脉冲的快速关闭.通过延时同步机对两个电光开关的动作时间进行控制,从而实现光脉冲宽度大范围可调.实验研究获得了最窄脉冲宽度40 ns,最宽脉冲宽度400 μs的532 nm激光脉冲.     

Class of Hamiltonians with one degree-of-freedom allowing application of Kruskal's asymptotic theory in closed form. I

In this paper, apart from a small restriction, all time-dependent Hamiltonians with one degree-of-freedom are determined, for which Kruskal's nice variables can be found by a sort of partial separation of the variables in the equations in question. These Hamiltonians allow an application of Kruskal's perturbation method in closed form, in a way similar to Lewis' treatment of the time-dependent harmonic oscillator. For those “appropriate” Hamiltonians, a connection is further established, between the invariant J following from Kruskal's theory, and an invariant that can be calculated equivalently from Hamilton-Jacobi theory.     

Definite paths and upper bounds on regular regions of velocity phase space

Hamilton's principle states that the path integral of the Langrangian is stationary with respect to variations of a classical path. It does not distinguish between a local minimum, a local maximum or a saddle point in path space. A simple algorithm is devised which provides strict and useful upper bounds on the region of velocity phase space occupied by paths that are either local maxima or local minima. The technique is illustrated graphically for the standard map. It is found that the bounds provide accurate numerical upper estimates for the region of velocity phase space filled by the rotational KAM surfaces at arbitrarily chosen values of the perturbation parameter.     

Dynamic stability of thin walled beams under traveling follower load systems

It is found that the perturbation equation of motion of a thin walled beam under a traveling follower load system becomes Hill's equation and that parametrically excited unstable coupled vibration occurs. The boundary frequency equations of the simple parametric resonance, from which the unstable regions are estimated, are obtained by Bolotin's method. Stability maps of a simply supported beam are shown, with account taken of the effects of load mass and damping.     

Analysis of car-following model considering driver’s physical delay in sensing headway

An extended car-following model is proposed by taking into account the delay of the driver’s response in sensing headway. The stability condition of this model is obtained by using the linear stability theory. The results show that the stability region decreases when the driver’s physical delay in sensing headway increases. The KdV equation and mKdV equation near the neutral stability line and the critical point are respectively derived by applying the reductive perturbation method. The traffic jams could be thus described by soliton solution and kink-antikink soliton solution for the KdV equation and mKdV equation respectively. The numerical results in the form of the space-time evolution of headway show that the stabilization effect is weakened when the driver’s physical delay increases. It confirms the fact that the delay of driver’s response in sensing headway plays an important role in jamming transition, and the numerical results are in good agreement with the theoretical analysis.     

Ground state of an exciton in a three-dimensional parabolic quantum dot: Convergent perturbative calculation

Working in the effective-mass approximation, we apply a powerful convergent perturbative technique of Turbiner's to the calculation of the ground state energy and the wave function of an exciton confined to a three-dimensional parabolic quantum dot. Unlike the usual Rayleigh–Schrödinger perturbation theory, Turbiner's approach works well even in the regime of strong coupling and does not require the knowledge of the full solution to the undisturbed problem. The second-order convergent calculation presented below is in excellent agreement with the results of exact numerical simulations for a wide range of system's confinement parameters.     

Transient motions of bi-lattices

We develop formal solutions for the propagation of transient pulses on a variety of bi-lattice models. The lattices are composed of a finite homogeneous chain connected in series with a different semi-infinite homogeneous chain at a common location occupied by a single mass which is different from the masses of both chains. Exact analytic solutions of this general case are not possible. Some analytic solutions are, however, possible for a variety of special cases. The general solutions are illustrated by numerically inverting the Laplace transform functions. The exact solutions are found to correlate very well with the numerical inversion scheme. Such correlations give confidence in the numerical scheme's predictions of the solutions of the more complicated chains.     

The dynamic analysis of circular plates and shallow spherical shells

In the investigation reported here an attempt has been made to study the influence of Berger's approximation on the non-linear transient response of circular plates and shallow spherical shells. The governing equations of motion obtained from Berger's approximation are solved by using the rapidly converging Chebyshev series spacewise and the Houbolt scheme for integration in the time domain. Results calculated when using Berger's approximation are compared with exact results. It is shown that Berger's method yields very accurate values for plates and shells under transient loading, in the case of immovable edge conditions.     

Impact-induced vibrations of a simple viscoelastic column model

A two-degree-of-freedom model for an almost-axially impacted viscoelastic cantilever column is analyzed. The impact load is produced by a mass striking the free end of the column. Under the assumption of small displacements two second-order non-linear ordinary differential equations for the coupled longitudinal and transverse vibrations of the column are derived. In the absence of damping these equations of motion are reduced to Mathieu's equation through the use of a perturbation method. The excitation parameters are (i) the natural frequency of small amplitude transverse vibrations of the undamped column and (ii) the initial velocity of the end of the column. The boundaries of two unstable regions are obtained. In the stable regions the solution of Mathieu's equation for the transverse displacement is close to that of the original non-linear equations of motion. In the first unstable region there is agreement only for early time. With increasing damping the peak of the maximum transverse displacement in the first unstable region decreases or even vanishes.     

Green's function equations of motion for a many-fermion system: II. Inhomogeneous system at finite temperature

The equations of motion for many-time causal Green's functions are extended to an inhomogeneous many-fermion system at finite temperature. The boundary condition that the perturbation vanishes in the remote past and distant future (adiabatic hypothesis) is used to determine the unperturbed propagator. The temperature enters the theory only as a parameter. Thus there is no need for analytic continuations in the complex temperature-time plane. The theory is used to derive thermal Hartree-Fock theory and Wick's theorem at finite temperature. A linked cluster perturbation expansion at finite temperature is obtained by iterating the equations of motion, without unlinked disconnected diagrams even appearing. After integration over frequency, the present theory gives the perturbation theory rules in terms of global propagators that Baym and Sessler obtained from the imaginary-time theory.