Read an Excerpt
INTRODUCTION
In the past few years the many body problem in the non-relativistic quantum mechanics has been discussed from many points of view and approximations suited to different physical situations have been developed. At the present moment there does not exist any single framework in which all these ideas may be fitted. Perhaps the most important stimulus to the recent developments has come from perturbation theory which also provides a comprehensive framework for the discussion of a large number of the existing theories. The main approaches can be well exemplified by considering the development of the nuclear many body problem and a fairly complete story can thus be made concerning the topics mentioned in the title.
However, in my view the situation is not quite so satisfactory as has often seemed from the first reading of many papers on the subject. What we have available is only a set of ideas and techniques which lead to results apparently having the right form. Questions of convergence are as intractable here as anywhere else. Actually the situation is far worse than in the case of quantum electrodynamics. In quantum electrodynamics there exists a unique way of separating the divergences, and after this has been done the resulting series is an expansion in powers of a small quantity. Strictly speaking, of course, even this is not any guarantee for convergence. In the non-relativistic many body problem, however, the infinities have to be, as it were, patched up and the resulting expressions are not expansions in powers of any small parameter. Often in the literature one comes across certain discussions which concern the numerical magnitudes of the first few terms and are supposed to be 'qualitative' arguments concerning the convergence. Also one comes across certain allusions to the connection between different approaches. Many such things are vague and in a larger context unhelpful. No doubt this state of affairs has resulted partly from the fact that the subject is difficult and the literature has grown very rapidly. Therefore I have felt that in writing a book about the subject at this time a different style should be adopted. Thus, vague so-called 'qualitative', discussions concerning the connection between different methods and their validity have been deliberately left out. On the other hand, some pains have been taken to explain the meaning of the highly picturesque and often mystifying language of the modern literature. Such an approach strips the subject of its glamour and gives a certain brittleness to the narrative. It might also cause a feeling of discomfort but it is hoped that one will in that way be closer to an appreciation of the actual situation. Many of the papers on these subjects have been superseded and even the best papers contain some statements and results which are vague or incorrect. This makes it very difficult and equally unprofitable to provide a complete critical evaluation of the literature. However, it is hoped that the following will provide a fairly complete and critical statement of the present situation in the field.
The book is primarily meant as an introduction to those who wish to specialise. We have not tried to enforce a complete uniformity of notation because often different notations are convenient for different purposes and also because this will facilitate the transition to journal literature. For instance, when one comes to the non-zero temperature case one finds a large number of different types of diagrams and conventions. It is desirable that one should get used to such practices. We have, therefore, simultaneously used two types of diagrams in the text. Although it does give the figures an appearance of complexity it will be seen that it does not make things any more difficult.
Some portions of text dealing with more difficult aspects of theory or with questions of detail have been put in small print. A list of main symbols has been appended to each chapter for easy reference.
After this explanation of the general point of view we turn to a brief description of the plan of the book.
It was considered appropriate to start in Chapter I with a discussion of the basic ideas of the perturbation theory. What are the conditions under which the perturbation theory may be set up and what are the various forms of the perturbation expansions?
The problem of existence and convergence of the series associated with the first part of the question has been answered only for a few cases which are not particularly helpful in the many body problem. We can do no more than call attention to this important question. The connection between different forms of expansions is well brought out by following the method of Riesenfeld and Watson (R1) where different choices of a certain operator lead to different expansions. One assumes that these series exist and are meaningful and then tries to compare the relative rates of decrease of the first few terms in order to decide which series to prefer. Also in the same spirit one compares the results of a perturbation calculation to a given order with that of a variational principle suggested by it or by certain freedoms associated with the splitting of the total Hamiltonian in an unperturbed part and a perturbation. If it were possible to carry any of these methods to the limit of the order n -> ∞, one would know about the main question of convergence also. Apart from this the methods have an interest of their own as techniques for evaluation of effects up to finite orders; since it is to be hoped that after the questions of convergence are settled, the forms of the final expressions will be similar to those from finite order calculations. To a large extent the specific nature of the interaction does not play any part in these discussions but in the last section we introduce two-body interactions to show how new properties are seen in the expansions on introducing these details. The methods used in the first chapter are purely algebraic and it will be seen that even though the diagrammatic methods of later chapters provide improvements in classification of terms the basic questions of convergence are still very similar.
Further progress in many body perturbation theory has depended upon diagrammatic methods, which provide the most convenient method of classifying the terms. The formal expressions for energies and wave functions can be greatly simplified and it becomes possible to discuss the dependence of various quantities on the volume or number of particles in the system. Various rearrangement methods which depend on the selection and summation of a class of terms having certain desirable characteristics are also greatly facilitated and often suggested by diagrammatic representation of the terms. Therefore, we have considered these topics in detail in Chapters II and III. In Chapter II we start by showing how a simple reduction of the terms of the Rayleigh-Schrodinger series by means of determinantal (Fermion) wave functions into two-particle matrix elements gives rise to the idea of the diagrams. This straightforward approach serves to connect the results of later second quantisation methods to those of Chapter I. At the same time it helps us to make some points concerning some exceptional situations. The rest of this chapter is concerned with a detailed description of the techniques used in second quantisation methods. Expressions for energies and wave function are derived and the important general theorems regarding the systems are proved. Practically none of the results are universally true and about the mathematical rigour one can say very little indeed. We have tried to indicate in the footnotes as much of the limitations as are known to us. It is hoped that this expedient will suffice till we have the final version of the theory. The discussion of rearrangement methods in Chapter III follows the same pattern. We look at these both from the purely algebraic and from the diagrammatic points of view. The case we consider is, of course, the one most useful in the nuclear studies, viz., the t-matrix and associated propagator modifications, but it is obvious that other rearrangements are also possible, and in other problems they may be more strikingly important.
Chapter IV is concerned mainly with techniques of solving the t-matrix equation and other equations that arise in the nuclear many body problem. All methods described in this chapter are of necessity approximate. The intuitive reasoning behind each of them is given, and in the systematic development it is easy to see how they are connected. But the question of their relative merits or their validity can not be easily discussed. We have given references to papers in which such discussions are given but, as mentioned earlier, in the very nature of things, any discussion of the magnitudes of a few terms can not be very helpful. We have further elaborated upon this remark at appropriate places in the text and especially in the section on the accuracy of calculations (Ch. IV, Sec. 6d). For the same reason and because of the ever changing parameters of the two-body nuclear potential we have not tried to give any numerical results but the well-established qualitative features of our understanding have been stated. It is often said that this theory has provided a justification for the shell model. A more accurate statement would be that it has provided a scheme according to which calculations of nuclear properties may be performed with finite, well behaved two-body potentials. It gives, in principle, a prescription for finding these effective potentials from the actual two-body potential which need not be well behaved and, in particular, shows that the treatment of 'configuration-mixing' should be different from that according to earlier ideas but fortunately the same as it was in earlier practice. This topic is dealt with in the last section of Chapter IV.
In Chapter V we have collected various other methods of approaching the many body problem and show how the results may be compared with those of previous chapters. Many of these connections, in particular that with the theory of superconductivity, are not fully worked out. However, it seems that a way has been found to circumvent the most glaring inconsistencies. Whether these procedures are fully justified is a different question whose answer will depend on the successful resolution of some of the questions raised in the following, particularly in Chapters I and V.
Note on numbering of equations
In each section equations are numbered anew in the usual way, giving the section number and the equation number. When reference is made to the same section the section number is omitted. When reference is made to a different chapter the number of the chapter in which the equation occurs is prefixed to the usual number.
(Continues…)
Excerpted from "Perturbation Theory and the Nuclear Many Body Problems"
by .
Copyright © 2017 Kailash Kumar.
Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
