Application of conventional equation of motion methods to the Tomonaga model

It is shown that the equation of motion technique provides a very concise way of calculating Green's functions for the Tomonaga-Luttinger model of a 1-d electron gas. The spectral function of the single-particle Green's function is worked out for the most general version of this model and for finite temperature. Extensions of the model are briefly discussed.     

Dynamical pictures and causality in finite temperature Green's functions

Real-time finite temperature Green's functions are discussed on the basis of the definition of new dynamical pictures. Causality appears explicitly. Feynman diagrams are formally identical to the ones of zero temperature. The vanishing of disconnected diagrams follows naturally.     

Perturbation and renormalization in thermo field dynamics

A systematic renormalization procedure used in the perturbative calculation of the real-time causal Green's functions at finite temperature is presented. The formalism of thermo field dynamics is employed throughout, permitting the use of Feynman diagram techniques. The renormalizability by means of the temperature-independent counterterms is proved.     

Green's function formalism and quantum correction for finite-temperature baryon matter

A relativistic temperature Green's function formalism is used to find the one-loop quantum correction for infinite baryon matter with scalar and vector interactions. Effects on the equation of state and the baryon effective mass are analyzed.     

Generating functional approach to the Green's functions diagrammatic technique for spin and pseudospin lattice systems: I. General theory

The theory of the irreducible many-point Green's functions, describing spin and pseudospin lattice systems, is formulated with the help of the generating functional approach. The diagrammatic technique for the generating functional is also developed. Special attention is paid to the construction and summation of the diagrammatic series for the one- and two-point Green's functions. Closed formulae for the one-point Green's function and the generalized Vaks-Larkin- Pikin equation are obtained. The $\text{1}\text{z}$ expansion scheme near the critical temperature of the order-disorder phase transition, is discussed, where z denotes the effective number of nearest- neighbours for a given site in a crystal lattice.     

Field-theoretic approach to the properties of the solid state

The techniques of quantum field theory are used to investigate the thermodynamic ion displacement correlation function—or Green's function of the phonon field—in a crystal and especially in a metal. The structure of thermodynamic Green's functions is outlined and the method for solving for them at finite temperature is fully discussed.The analytic structure of the phonon Green's function is then considered. This function is shown to be bounded and invertible everywhere off the real axis; a spectral form is derived for its inverse. The symmetries imposed by the point group of the crystal are then discussed.Assuming small ionic oscillations, we find the inverse of the phonon Green's function as a linear function of the electronic contribution to the dielectric response function of the metal. This dielectric function is shown to be simply related to the longitudinal part of the conductivity tensor that gives the response of the electrons to the effective electric field in the metal. The assumption of translational invariance then leads to an explicit expression for the phonon Green's function in terms of this conductivity.The deformations in the lattice induced by an arbitrarily time varying external force are calculated in terms of the retarded phonon Green's function. In the static long wavelength limit the phonon Green's function yields the macroscopic elastic constants of the crystal. Their relation to the conductivity is exhibited, and several elastic constants are estimated. We also see that the complete phonon spectrum and the lifetimes of the phonon states may be calculated from this Green's function. A relation between the long wavelength acoustic attenuation in metals and the de conductivity is derived, which is in good agreement with recent experiments. Furthermore, the ions in a metal are shown to have a high-frequency oscillation along with the electrons, at essentially the electron plasma frequency.     

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.     

Wave propagation according to higher order field equations I

A detailed study is made of wave propagation according to a sixth-order partial differential equation with complex masses proposed by Swieca and Marques, which presents a kind of generalized Klein-Gordon equation. The choice of definite Green's functions in the corresponding Yang-Feldman integral equation corresponds to a certain choice of boundary conditions for the allowed solutions of the corresponding partial differential equation. The advanced and retarded Green's functions used possess the anomalous feature of having non-zero values in the neighbourhoods of those, past or future parts of the light cone, for which traditional advanced and retarded Green's functions are zero. However, it is shown that a suitable averaging procedure provides the possibility of defining sets of functions, such that solutions of the Yang-Feldman equations belonging to this set possess the property that the future behaviour of the solution is determined by its asymptotic initial conditions. Certain features of the wave propagation, according to the equations considered, can be usefully compared with the properties of the solutions of the ordinary differential equation - and corresponding integral equation - which represents the equation of motion of a charged particle including the force for radiation reaction. The particle then has a certain “size”. Analogously the “non-local field equations” have solutions characterized by a certain “fundamental length” indicating the space-time distances for which averaging occurs. The admitted solutions of the field equations seem to represent a relativistic field with a “finite a number of degrees of freedom” within a finite volume.     

Coupling parameter trajectories

Results are presented about singular surfaces of Green's functions in the complex manifold (g², t) of the coupling parameter g² and a dynamical variable t.     

Ferromagnetic 3d-metals: On the definition of “local bands”

The basic concept of a picture for itinerant ferromagnetism is discussed. The central point is that a local exchange splitting and local moments, exist even above the transition temperature T_c. Transverse fluctuations (and not magnitude fluctuations, as in Stoner theory) are the dominant source for the phase transition to the paramagnetic state. The author's Green's function method is extended to the use of a full bandstructure including hybridization and general electron-electron interactions. Spin waves are also discussed.     

On the energy-momentum tensor in gauge field theories

The energy-momentum tensor in spontaneously broken non-Abelian gauge field theories is studied. The motivation is to show that recent results on the finiteness and gauge independence of S-matrix elements in gauge theories extends to observable amplitudes for transitions in a gravitational field. Path integral methods and dimensional regularization are used throughout. Green's functions Γ_μν^(j)(q; p₁,…,p_j) involving the energy-momentum tensor and j particle fields are proved finite to all orders in perturbation theory to zero and first order in q, and finite to one loop order for general q. Amputated Green's functions of the energy momentum tensor are proved to be gauge independent on mass shell.     

The classical limit of temperature Green's function expansions

Rules are obtained for calculating the classical limit of Green's function diagrammatic expansions. The classical cluster expansion is derived by calculating the classical limit of the exact Green's function. Other operators of interest in linear response theory may be calculated in the classical limit. The retarded real-time spin density correlation function, proportional to the magnetic susceptibility, is shown to be exactly proportional to the density in this limit. The relation of this work to other approaches is discussed.     

Evaluation of the spin wave Green's functions for rutile structure Heisenberg antiferromagnets with exchange between ions both on the same and opposite sublattices

A straightforward method is presented for the evaluation of the spin wave Green's function appropriate to the Raman scattering in the xy and xz geometry in rutile structure Heisenberg antiferromagnets with exchange between ions both on the same and on opposite sublattices and arbitrary local anisotropy. The analytical asymptotic behaviour of the Green's functions near the singularities is explicitly given and the problem of the numerical evaluation of their real and imaginary parts is discussed. Tables of the imaginary parts, calculated at the points corresponding to a Gaussian quadrature procedure in the appropriate interval, are supplied on request.     

Generalized density expansion in the kinetic theory and the resistivity of liquid metals

For the system of electrons and immovable interacting centers an exact equation for averaged electron Green's function is formulated. The expansion of self-energy part over the one-particle t-matrices and explicit Green's functions is derived. It represents a kind of a generalized density series containing the correlation functions of the centres. In the low approximation over t-matrix, the transition probability (t)²S in the kinetic equation is obtained (S = the structure factor of centers).     

Biorthogonality and radiative transfer in finite slab atmospheres

It is shown that the exact solution of transfer problems of polarized light in finite slab atmospheres can be obtained from an eigenmode expansion, if there is a known set of adjoints defined appropriately to treat two-point, half-range boundary-value problems. The adjoints must obey a half-range biorthogonality relation.The adjoints are obtained in terms of Case's eigenvectors and the reflection or the transmission matrices. Half-range characteristic equations for the eigenvectors and their adjoints are derived, where the kernel functions of the integral operators are given by the boundary values of the source function matrix of the slab albedo problem. Spectral formulae are obtained for the surface Green's functions. A relationship is noted between the biorthogonality concept and some half-range forms of the transfer equation for the surface Green's functions and their adjoints. Linear and non-linear functional equations that are well known from an invariance approach, are derived from a new point of view. The biorthogonality concept offers the opportunity for a better understanding of mathematical structures and the nonuniqueness problem for solutions of such functional equations.     

The one-loop Green’s functions of dimensionally reduced gauge theories

We apply the dimensional regularization technique as well as that by dimensional reduction to the calculation of the regularized one-loop Green’s functions ind ₀-dimensional Yang-Mills theory with real massless scalars and spinors in arbitrary (real) representations of a gauge groupG. As a particular example, the super-symmetrically regularized one-loop Green’s functions of theN=4 supersymmetric Yang-Mills model are derived.     

The t2 ← e transition at the Cu impurity in ZnS crystal

The excitation energy for the localized t₂ ← e transition is calculated by the Green's function method in CNDO approximation using a parent system model.     

Green's functions for gravitational multi-instantons

The Green's functions for scalar fields propagating on the self-dual gravitational multi-instantons and multi-Taub-NUT metrics are given explicitly in closed form. The special cases for flat space, Taub-NUT and the Eguchi-Hanson instanton are listed. A construction is described for obtaining the Green's functions for fields of arbitrary spin.     

Biological Bose condensation and the time threshold for biological effects

Fröhlich's model of Bose condensation in biological systems is analyzed using finite temperature Green's function techniques. The appropriate biological system excitation lifetime is calculated and represents the time needed to produce the condensed state. The possible relevance of this calculated lifetime to experiments concerning the microwave irradiation of biological organisms is discussed.     

Thermodynamic calculations in relativistic finite-temperature quantum field theories

A Minkowski space formalism of finite-temperature quantum field theory is used to compute static thermodynamic quantities in the one- and two-loop approximation in an elegant and straightforward way using a generalization of Weinberg's tadpole method of calculating effective potentials. Systematic diagrammatic techniques for low- and high-temperature expansions are developed. Renormalizability by zero-temperature symmetric counterterms is proven for all orders in the loop expansion and demonstrated explicitly to two loops. Many useful computational techniques applicable to general finite temperature calculations are explained.