The nucleon Green function in pseudoscalar meson theory. II

1955 ◽  
Vol 232 (1190) ◽  
pp. 377-388 ◽  
Keyword(s):  
Green Function ◽  
Pseudoscalar Meson ◽  
Preceding Paper ◽  
Functional Integral ◽  
Coupling Constants ◽  
Intermediate Coupling ◽  
Coupled Equations ◽  
Meson Theory ◽  
Coupling Approach ◽  
The Green Function

Following the formal work of the preceding paper, a method is proposed to evaluate the functional integral which has been derived for the Green function of one nucleon interacting with a pseudoscalar meson field. The method is basically that of stationary phase taken to its second approximation, and since this approximation where applicable is accurate in the limits of strong and weak coupling constants, it is assumed good in general. There are several difficulties involved in the evaluation, and as far as possible these are isolated and discussed with the aid of models each showing one difficulty alone. Combining these separate points, the evaluation of the functional integral is thereby expressed in terms of the solution of a set of coupled equations which provide a basis for a covariant intermediate coupling approach to the problem. The solution of these equations is not attempted in this paper.

The nucleon Green function in pseudoscalar meson theory. I

1955 ◽  
Vol 232 (1190) ◽  
pp. 371-376 ◽  
Keyword(s):  
Green Function ◽  
Pseudoscalar Meson ◽  
Functional Integration ◽  
Functional Integral ◽  
Charged Scalar ◽  
Polarization Effects ◽  
Scalar Mesons ◽  
Meson Theory ◽  
Nucleon Recoil ◽  
The One

The method of functional integration has been previously applied to the evaluation of Green functions, in particular to the one-nucleon Green function in the cases of neutral scalar and charged scalar mesons interacting with a static nucleon. This work is extended in this and subsequent papers to pseudoscalar meson theory allowing nucleon recoil. In this paper the formal extension, in particular the inclusion of vacuum polarization effects, is made and the resulting forms discussed. In particular, the comparison between forms arising from the usual interaction of fermions and bosons with the analogous boson-boson forms striking dissimilarity. The reduction of the evaluation of the functional integral is discussed in part II.

Field equations in functional form

1954 ◽  
Vol 224 (1156) ◽  
pp. 24-33 ◽  
Keyword(s):  
Green Function ◽  
External Field ◽  
Closed Form ◽  
Functional Equations ◽  
Scalar Meson ◽  
Functional Form ◽  
Radiative Corrections ◽  
Field Equations ◽  
Meson Theory ◽  
The Green Function

Starting from the functional equations governing the Green function of a single nucleon moving in an external field with radiative corrections, as given by Schwinger, a formulation is developed which relates this Green function to that of a nucleon moving in an arbitrary external field, without radiative corrections. In the case of neutral scalar meson theory in which the recoil of the nucleon is neglected, the Green function is obtained in closed form. Mass and Green-function renormalizations are easily done completely, and the singularities of the solution investigated, proving to be an interesting illustration of the expected behaviour in more realistic cases.

The nucleon Green function in charged meson theory

1955 ◽  
Vol 228 (1174) ◽  
pp. 411-424 ◽  
Keyword(s):  
Green Function ◽  
External Field ◽  
Scalar Meson ◽  
Functional Integration ◽  
Functional Integral ◽  
Approximate Evaluation ◽  
Charged Scalar ◽  
Meson Theory ◽  
Charged Meson ◽  
General Method

The method of functional integration is used on the problem of obtaining the nucleon Green function including radiative corrections in charged scalar meson theory without nuclear recoil. An explicit functional integral for the solution in the presence of an external field is given, and since this cannot be evaluated explicitly a general method is proposed for an approximate evaluation. The method is illustrated by solutions for weak and strong coupling, and the structure of these solutions is discussed.

Exponentially Convergent Approximation to the Elliptic Solution Operator

2006 ◽  
Vol 6 (4) ◽  
pp. 386-404 ◽  
Author(s):  
Ivan. P. Gavrilyuk ◽  
V.L. Makarov ◽  
V.B. Vasylyk
Keyword(s):  
Green Function ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Solution Operator ◽  
Hyperbolic Operator ◽  
Elliptic Solution ◽  
Elliptic Boundary Value ◽  
Sine Family ◽  
The Green Function

AbstractWe develop an accurate approximation of the normalized hyperbolic operator sine family generated by a strongly positive operator A in a Banach space X which represents the solution operator for the elliptic boundary value problem. The solution of the corresponding inhomogeneous boundary value problem is found through the solution operator and the Green function. Starting with the Dunford — Cauchy representation for the normalized hyperbolic operator sine family and for the Green function, we then discretize the integrals involved by the exponentially convergent Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [0, 1].

The Green function method for the two-surface problem

1970 ◽  
Vol 8 (13) ◽  
pp. 1069-1071 ◽  
Author(s):  
F. Flores ◽  
F. Garcia-Moliner ◽  
J. Rubio
Keyword(s):  
Green Function ◽  
Function Method ◽  
Green Function Method ◽  
The Green Function

Electrostatic oscillations in cold inhomogeneous plasma I. Differential equation approach

1971 ◽  
Vol 5 (2) ◽  
pp. 239-263 ◽  
Author(s):  
Z. Sedláček
Keyword(s):  
Differential Equation ◽  
Green Function ◽  
Normal Mode Analysis ◽  
Inhomogeneous Plasma ◽  
Mode Analysis ◽  
Complex Frequency ◽  
Plasma Oscillations ◽  
The Laplace Transform ◽  
Collective Oscillations ◽  
The Green Function

Small amplitude electrostatic oscillations in a cold plasma with continuously varying density have been investigated. The problem is the same as that treated by Barston (1964) but instead of his normal-mode analysis we employ the Laplace transform approach to solve the corresponding initial-value problem. We construct the Green function of the differential equation of the problem to show that there are branch-point singularities on the real axis of the complex frequency-plane, which correspond to the singularities of the Barston eigenmodes and which, asymptotically, give rise to non-collective oscillations with position-dependent frequency and damping proportional to negative powers of time. In addition we find an infinity of new singularities (simple poles) of the analytic continuation of the Green function into the lower half of the complex frequency-plane whose position is independent of the spatial co-ordinate so that they represent collective, exponentially damped modes of plasma oscillations. Thus, although there may be no discrete spectrum, in a more general sense a dispersion relation does exist but must be interpreted in the same way as in the case of Landau damping of hot plasma oscillations.

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