Time-dependent Green Function Research Articles

Statistical mechanics of a system of interacting bosons

The equilibrium properties of a system of interacting bosons are studied by the methods of quantum field theory. The problem is formulated in terms of a density matrix in an irreducible representation so as to be able to treat the degenerate as well as the nondegenerate phase. The quantities of basic interest are considered to be the one-particle, time-dependent Green's functions of the system from which the free energy can be obtained through suitable prescriptions. Explicit expressions are derived for the various thermodynamic functions in the weak-interaction approximation for the self-energy parts. To the first order in the interaction v 0, the behaviour of the thermodynamic functions is such as would correspond to a second-order phase transition across the line of condensation. To the order (v 3 2 0) , the specific heat diverges proportional to ( n - n c ) -1 2 as the density n approaches the critical density n c from the degenerate phase. The inverse compressibility on the other hand becomes negative for densities sufficiently close to the critical density indicating that the weak interaction results, or the formally equivalent low density results, are valid in the neighbourhood of the transition line in an asymptotic sense only.

The Time-Dependent Green's Function for Electromagnetic Waves in Moving Simple Media

This paper treats the problem of radiation from sources of arbitrary time dependence in a moving medium. The medium is assumed to be lossless, with permittivity ε and permeability μ, and to move with constant velocity v̄ with respect to a given inertial reference frame xyz. It is shown how the Maxwell-Minkowski equations for the electromagnetic fields in the moving medium can be integrated by means of a pair of vector and scalar potential functions analogous to those commonly used with stationary media. The wave equation associated with these potential functions is derived, and a scalar Green's function is defined to satisfy the same type of equation, with a delta-function source term δ(r − r′) δ(t − t′), and the casuality condition. The solution for the Green's function is derived in closed form, by means of a Fourier integral method. The resulting Green's function is useful not only for calculating the fields from arbitrary sources in moving media, but also for its pedagogical value. It is simpler to understand the phenomenon of Cerenkov radiation using this method than it is from the conventional approach to the Cerenkov problem.

Aharonov-Bohm scattering of a wave packet of finite extension

The scattering of an electron from a thin, shielded solenoid (Aharonov-Bohm effect) is studied by means of the time-dependentGreen's function of the problem. The structure of theGreen's function reveals in a lucid way the nonidentity of canonical and kinetic angular momentum as the origin of the Aharonov-Bohm scattering. The time development of a Gaussian wave packet is studied in some detail to investigate the validity of certain objections against the original work ofAharonov andBohm. In an appendix some results are given for the case of finite radius of the solenoid and finite shielding potential.

Theory of vibrations of defects using green's functions

Genetic variability among subspecies, ecotypes and populations is an important component of biodiversity, particularly when habitats are characterized by steep environmental gradients across short distances as in the landscape of the European Alps. In Poa alpina population differentiation is high due to polyploidy, facultative apomixis and the occurrence of seed- and bulbil-producing plants. We analyzed microsatellite diversity of seed- and bulbil-producing populations from two nearby alpine grassland sites in the Swiss Alps and transplanted them reciprocally to test the hypothesis that established plants are locally adapted. The seed- and bulbil-producing populations differed molecularly strongly from each other despite their proximity. The seed-producing plants from the rich, calcareous grassland site performed well at both the home and the away site. Conversely, the bulbil-producing plants from the species-poor grassland had a comparatively low productivity at both sites, but performed better at the high-quality away site compared to its home site. Thus, local adaptation of established plants could not be confirmed by reciprocal transplantation. Results suggest that the distribution of seed- and bulbil-producing P. alpina is a result of habitat quality, differences in plant performance, and the requirements for successful offspring recruitment. The prevailing occurrence of bulbil-producing plants at low productivity sites is probably due to different regeneration niches for seeds vs. bulbils, i.e. an advantage of bulbils at cold sites with poor soil and short growing seasons at this elevation.Genetische Variabilität zwischen Unterarten, Ökotypen und Populationen ist ein wichtiger Aspekt von Biodiversität, besonders, wenn die Habitate einer Art durch steile Umweltgradienten über kurze Distanzen getrennt sind, wie dies in der Landschaft der Europäischen Alpen der Fall ist. Bei Poa alpina ist die Differenzierung zwischen Populationen als Folge von Polyploidie, fakultativer Apomixis und durch die Bildung von Nachkommen aus Samen oder vegetativen Bulbillen besonders gross. Wir untersuchten mit Mikrosatelliten die molekulare Diversität von zwei Populationen nah benachbarter alpiner Rasen, von denen die eine mit Samen und die andere mit Bulbillen reproduziert. Die beiden Populationen wurden reziprok transplantiert, um auf lokale Anpassung etablierter Pflanzen zu testen. Trotz ihrer engen Nachbarschaft waren die Populationen molekular stark differenziert. Die Samen produzierenden Pflanzen des nährstoff- und artenreicheren Kalkrasens wuchsen sowohl am Heim- als auch am fremden Standort sehr gut. Umgekehrt zeigten die Bulbillen produzierenden Pflanzen des ärmeren Standorts ein vergleichsweise geringeres Wachstum an beiden Orten, wuchsen jedoch am fremden Standort höherer Qualität besser als an ihrem Heim-Standort. Lokale Anpassung etablierter Pflanzen konnte folglich durch reziproke Verpflanzung nicht nachgewiesen werden. Die Ergebnisse legen nahe, dass das unterschiedliche Vorkommen von Samen bzw. Bulbillen produzierenden Pflanzen eine Folge ist der verschiedenen Habitatsqualität, der unterschiedlichen Wüchsigkeit der Pflanzen aus den beiden Populationen sowie der spezifischen Anforderungen an die erfolgreiche Etablierung von Nachkommen. Das ausschliessliche Vorkommen von Bulbillen produzierenden Pflanzen in Habitaten geringerer Produktivität ist wohl in erster Linie eine Folge unterschiedlicher Etablierungsnischen von Samen und Bulbillen, insbesondere, dem Vorteil von Bulbillen an kühleren Standorten nährstoffarmer Böden mit einer sehr kurzen Wachstumsperiode.

Perturbation-Theory Rules for Computing the Self-Energy Operator in Quantum Statistical Mechanics

Convenient rules are given for the general term in the time-independent perturbation-theory expansion for the self-energy operator of quantum statistical mechanics. The rules are derived by starting from the usual formalism involving time-dependent Green's functions.

Infrared Divergencies in Statistical Mechanics of Electron Plasma

The problem of infrared divergencies which appear in the perturbation series in quantum statistical mechanics of electron gas is discussed generally. It is shown that no infrared divergency appears in the exact partition function. The suppression of infrared divergency is connected with the existence of plasma oscillation. The dispersion relation of the plasma oscillation is obtained as a pole of the time dependent Green function.

Analytic properties of the time green functions in the statistical mechanics of many-body systems

Expressions for the Gibbs potential in many-body systems have been derived by Luttinger and Ward using the methods of second quantization and by Klein using the timedependent statistical Green functions. In this investigation the results of Luttinger and Ward are deduced from those of Klein by making use of the analytic properties of the statistical Green function in the complex time plane. (H.D.R.)

Dispersion relations and the causality concept

It is well known that the scattering functionS(x) associated with a cut potential has certain analytic properties that make it satisfy dispersion relations. It is of interest to see how these analytic properties are modified when the potentials are not cut-off at a certain point, but continue to infinity, going asymptotically to zero there. The discussion is first carried using a causality condition enunciated as follows : The wave function associated with any initial wave packet remains bounded for all time. As a consequence of the causality condition, we obtained that it is no longer theS(x), but a new function, which we call the dispersion function, that satisfies the analytic properties that imply dispersion relations. We also check these analytic properties directly from the Schrodinger equation. Finally, to discuss the significance of the poles of the dispersion and scattering functions, we analize in detail the scattering by the Eckart potential, obtaining the time dependent Green function in terms of basic interaction Green (BIG) functions associated with the poles of the dispersion function. From the behaviour of the BIG functions, as functions of time, we can also obtain restrictions on the analytic behaviour of the dispersion function.

The green's function method in quantum statistics

A Green's function method in quantum statistics is developed. It is shown that the equations obtained contain in a simple approximation various methods encountered in statistical physics and in the theory of many particles as well as their extensions to cases of non-vanishing temperature (e.g., the Debye-Hückel, Hartree-Fock, Thomas-Fermi, Gell-Mann-Brueckner methods). A transition to time-dependent Green's functions is considered and a method for the determination of the energy spectrum of the system is proposed.

Perturbation Theory for an Infinite Medium of Fermions

The ground-state energy and low-energy excitations of single-particle character of an infinite medium of fermions are discussed with the aid of time-dependent Green's functions, which are convenient generalizations of the exact particle correlation functions in the ground state of $N$ particles. The power series development for the one- and two-particle functions, under the restriction to two-body forces, is derived and described by means of Feynman diagrams. The derivation of the linked-cluster expansion for the energy then follows immediately. The equivalence to previous versions is established. The one-particle function is examined in particular detail, and it is shown that the poles of its space-time Fourier transform studied as a function of the energy variable, for fixed momentum, determined the ($N+1$)-particle and ($N\ensuremath{-}1$)-particle excited states which have single-particle character. For a reasonable assumption about the full spectrum of excited states, it is found that for the interacting system, single-particle excitations with a real energy occur only at the Fermi momentum. It is pointed out that the corresponding energy, termed the perturbed Fermi energy, equals the binding energy per particle in the ground state of $N$ particles for a saturating system at equilibrium density.It is shown, finally, that the entire structure of the theory may be carried over to the case of finite temperature, requiring only a redefinition of the Green's functions. The analogy is constructed from a discussion of the internal energy.