AlllDD TfiTtlO

NAI L INS I Uh SI ANUAHUS S lEtlM R.I.C. NBS

PUBLICATIONS A1 11 00989610

Danos, MIchael/RelatlvlstIc many-body bo QC100 .U556 V147;1975 C.I NBS-PUB-C 1977

Q NBS MONOGRAPH 147

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade,

and (4) technical services to promote public safety. The Bureau consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology,

the Institute for Computer Sciences and Technology, and the Office for Information Programs.

THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United

States of a complete and consistent system of physical measurement; coordinates that system

with measurement systems of other nations; and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation's scientific community, industry,

and commerce. The Institute consists of a Center for Radiation Research, an Office of Measurement Services and the following divisions: Applied Mathematics — Electricity — Mechanics — Heat — Optical Physics — Nuclear

Sciences - — Applied Radiation " — Quantum Electronics " — Electromagnetics ' — Time

and Frequency " — Laboratory Astrophysics — Cryogenics "'.

THE INSTITUTE FOR MATERIALS RESEARCH conducts materials research leading to improved methods of measurement, standards, and data on the properties of well-characterized

materials needed by industry, commerce, educational institutions, and Government; provides

advisory and research services to other Government agencies; and develops, produces, and

distributes standard reference materials. The Institute consists of the Office of Standard

Reference Materials and the following divisions:

Analytical Chemistry — Polymers — Metallurgy — Inorganic Materials — Reactor

Radiation — Physical Chemistry.

THE INSTITUTE FOR APPLIED TECHNOLOGY provides technical services to promote

the use of available technology and to facilitate technological innovation in industry and

Government; cooperates with public and private organizations leading to the development of

technological standards (including mandatory safety standards), codes and methods of test; and provides technical advice and services to Government agencies upon request. The Institute consists of a Center for Building Technology and the following divisions and offices: Engineering and Product Standards — Weights and Measures — Invention and Innovation — Product Evaluation Technology — Electronic Technology — Technical Analysis — Measurement Engineering — Structures, Materials, and Life Safety * — Building

Environment * — Technical Evaluation and Application * — Fire Technology.

THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts research

and provides technical services designed to aid Government agencies in improving cost effec- tiveness in the conduct of their programs through the selection, acquisition, and effective utilization of automatic data processing equipment; and serves as the principal focus within

the executive branch for the development of Federal standards for automatic data processing

equipment, techniques, and computer languages. The Institute consists of the following

divisions: Computer Services — Systems and Software — Computer Systems Engineering — Information Technology.

THE OFFICE FOR INFORMATION PROGRAMS promotes optimum dissemination and

accessibility of scientific information generated within NBS and other agencies of the Federal

Government; promotes the development of the National Standard Reference Data System and

a system of information analysis centers dealing with the broader aspects of the National

Measurement System; provides appropriate services to ensure that the NBS staff has optimum

accessibility to the scientific information of the world. The Office consists of the following

organizational units:

Office of Standard Reference Data — Office of Information Activities — Office of Technical

Publications — Library — Office of International Relations.

>- Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D.C. 20234.

" Part of the Center for Radiation Research.

•> Located at Boulder, Colorado 80302.

« Part of the Center for Building Technology.
^'lo&aoil Earsou oi Slandoicis

Relativistic Many-Body Bound SystemT**

MAY 191975 - Iie9.

. . (J ^

Michael Danos *

, . . ^ no . ( H '7

Institute for Basic Standards ^

National Bureau of Standards —

Washington, D.C. 20234

and

Vincent Gillet

Service de Physique Theorique

and

Departement de Physique Nucleaire

Centre d'Etudes Nucle'aires de Saclay

BP n°2.91190 Gif-sur-Yvette-France

U.S. DEPARTMENT OF COMMERCE

, S. NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, D/recfor

Issued ApriM 975
Library of Congress Cataloging in Publication Data

Danos, Michael.

Relativistic many-body bound systems.

(National Bureau of Standards Monograph; 147)

Bibliography : p. Supt. of Docs. No.: C 13.44:147

1. Field theory (Physics). 2. Quantum field theory. 3. Problem

of many bodies. 4. Particles (Nuclear physics). I. Gillet, Vincent,

joint author. II. United States. National Bureau of Standards.

III. Title. IV. Series: United States. National Bureau of Standards. Monograph; 147. QC100.U556 No. 147 [QC793.3.F5] 389'.08s [530.1'44] 74-26975

National Bureau of Standards Monograph 147

Nat. Bur. Stand. (U.S.), Monogr. 147, 149 pages (Apr. 1975)

CODEN: NBSMA6

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1975

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

(Order by SD Catalog No. C13.44:147).

Price $2.15
Foreword

When Yukawa introduced the mesons in order to explain the short

range character of the nuclear forces, the field of nuclear physics

split into two parts: nuclear structure and nuclear forces. Nuclear

structure developed into non-relativistic nuclear physics and led

to the creation of various models to describe the emerging wealthof nuclear data. The field of nuclear forces developed into high

energy particle physics with its own immense body of phenomena and

data.

For most nuclear phenomena the non-relativistic framework is

fully adequate. However, this framework is too narrow in phenomena

associated with exchange currents and with high momentum transfers.

In such cases the presence of particles other than protrons and

neutrons in the nuclei must be explicitly accounted for, and

one must take recourse to the field of high energy physics. This

monograph is devoted to the merger of nuclear and high energy

physics, and to the formulation of the quantum field theory

of nuclei. The main emphasis in this work is on providing the

mathematical tools needed to obtain solutions to specific problems

in a fully relativistic , consistent manner and up to a known,

predetermined accuracy. I expect that this pioneering work should

be of value to all who are involved in calculations of nuclear

structure.

Richard W. Roberts

Director

iii
i
ABSTRACT

The principles and the mathematical details of a fully relativistic nuclear

theory are given. Since the concept of nuclear forces is a strictly non-relativistic construct, it must be abandoned and the forces must be replaced explicitly

by their physical origin, i.e., by the interaction between nucleons and mesons.

Thus, in this monograph the description of a nucleus has been formulated as a problem of relativistic quantum field theory which is solved by nuclear physics

methods. To wit: The physics is described by specifying a Lagrangian which is

a functional of the constituent fields ( = of the parton fields). The solutions

for the physical systems then are obtained in a time-ind^ endent treatment as expansions in the parton fields: both particles and nuclei are composite systems,

made up of- parton configurations, which define a representation of the Hamiltonian

(associated with the specified Lagrangian). The Hamiltonian is truncated by

omitting all configurations having a diagonal element exceeding that of the lowest

configuration by a pre-determined value, Emax) is diagonalized. All formulae

needed to carry out this program are derived and given in full detail for spin

0, 1/2, and 1 parton fields for PS, PV, and cp^ interactions. Particular attention

is devoted to the center-of -mass position coordinate which in relativistic kinematics is a non-separable many-body operator. Finally, the configurations up to

Emax - ^ GeV are listed for the nuclecci, the deuteron, and the pion.

Key words: Composite particles; interacting quantum fields; nuclear structure;

particle structure; relativistic bound systems; relativistic nuclear physics.

CONTENTS

Pape

Chapter I - OVERVIEW AND GENERAL FORMALISM 1

1.1 - Introduction 1

1.2 - Survey of the work 4

1.3 - Field theory for stationary systems 5

1.3.1 - The Schrodinger picture 5

1.3.2 - State vectors 7

1.3.3 - Wave functions 9

1.3.4 - The secular problem 10

1.3.5 - Nature of the solutions 13

1.3.6 - Comparison with time dependent perturbation treatments 14

1.4 - Extraction of the center of mass motion 15

Chapter II - DEFINITIONS AND FORMULAE 19

II. 1 - Phases 19

II. 1.1 - Tensorial sets 19

II. 1.2 - Creation and annihilation operators 20

II. 2 - Invariant states and matrices 22

11. 3 - A graphical representation of angular coupling 24

11. 4 - Vector algebra 28

11. 5 - States and matrix elements in the angular momentum coupled

Fock representation 33

11. 5.1 - Commutators in a coupled scheme 33

11.5.2 - State vectors and wave functions 34

11. 5.3 - Matrix elements of Fermion- Boson systems 39

Chapter III - FREE FIELD DISCRETIZED EXPANSIONS AND ENERGIES 41

III.l - Spin 0 field 41

111. 1.1 - Field equations 41

111. 1.2 - Multipole expansion of the field 42

III. 1.3 - Discretization of the field 43

III. 1.4 - The discretizing unitary transformation 44

vii
III. 2 - The free field energy for spin 0 particles 45

111. 3 - Spin 1/2 field 46

111. 3.1 - Field equations 46

111. 3. 2 - The plane wave solutions 48

111. 3. 3 - Multipole expansion and discretization 52

111. 4 'f. The free field energy for spin 1/2 particles 54

111. 5 - Spin 1 field 56

111. 5.1 - Field equations 56

III. 5. la - The w field Hamiltonian 57

III. 5. lb - The p field Hamiltonian 60

III. 5.1c - Equations of motion and the Maxwell

equations 61

III. 5. Id - Quantization 62

111. 5.2 - Representation of the Lorentz transformation and

boost 62

111. 5. 3 - The plane wave solutions 65

111. 5.4 - Multipole expansion 69

111. 5. 5 - Multipolarities and discretization 72

III. 6 - The free field energy for spin 1 particles 76

Chapter IV - CENTER OF MASS 77

IV. 1 - The center of mass pseudo-Hamiltonian 77

IV, 2 - The center of mass momentum 77

IV. 3 - The center of mass coordinate 84

Chapter V - THE INTERACTION HAMILTONIAN 92

V.l - The pion-nucleon interaction 92

V.1.1 - Pseudo- scalar coupling 92

V.1.2 - Pseudo-vector coupling 93

V.1.3 - Pion-nucleon invariant matrix elements 95

V.1.4 - Momentum conservation 98

V.2 - The pion-pion interaction 99

4

V.2.1 - The K interaction 99

V.2. 2 - Momentum conservation 103

V.2.3 - Pion-pion invariant matrix elements 103

V.3 - The nucleon-spin 1 meson interaction 105

V.3.1 - The oiSIN interaction 105

V.3. 2 - The pNN interaction 109

viii
V.4 - The spin 0 - spin 1 meson interaction 110

V.4.1 - The pTui interaction 110

3

V.4. 2 - The con interaction 113

Chapter VI - MODELS OF HADRONS 117

VI. 1 - The Hamiltonian matrix 117

VI. 2 - The partons 118

VI. 3 - The nucleon configurations 120

VI. 4 - The deuteron configurations 121

VI. 5 - The pion configurations 123

Appendix - RELATIVISTIC KINEMATICS OF CENTER OF MASS 124

- BOOSTED STATES 131 - SCATTERING STATES 134

- THE PHYSICAL VACUUM 138

Acknowledgements « 140

References 141

ix

I.l

CHAPTER I

OVERVIEW AND GENERAL FORMALISM

I.l - INTRODUCTION

With the advent of accelerators for hadrons and electrons in the

0.2-2. GeV range, the development of intermediate energy nuclear physics opens

a whole new field of studies. Namely the postulate of the microscopic non- relativistic theory of nuclei, that is nuclei are made up only of protons and

neutrons interacting via two-body potentials, has to be replaced by an explicit

description of the mesic degrees of freedom and of the admixture of baryonic

resonances in the nuclear medium.

In the present work the mathematical tools and the equations needed

to compute fully relativistically many- body bound hadronic systems will be

written down. As is well known the relativistic description of nuclei entails

inescapably the dynamical description of the low energy structure of the nucleons

and the mesons themselves. Thus the hadrons with baryonic number 0 or 1 will

have to be described in a consistent manner in terras of constituent particles

the nature of which (quantum numbers, masses, coupling constants ...) constitutes the various phenomeno logical models one may want to develop.

Since the nuclear physicist is interested in the characteristics

(form factors, magnetic and electric multipole moments...) of bound systems, our

[2-7 1 basic tool will be the Lagrangian field theory solved in a time independent formalism in order to obtain relativistic stationary wave functions. The

Hamiltonian matrix will be constructed in a truncated Hilbert space on discretized many-body configurations of correct relativistic behaviour, proper statistics and given total angular and isospin quantum numbers. The truncation

- 1 -
1.2

criteriuin will be provided by the energy: all those configurations are included which yield a diagonal element of the Hamiltonian less or equal to the truncation energy.

The usual divergences of strong interaction theories will show up in

the present treatment via the dependence of the solutions upon the cut-off

energy chosen to truncate the functional space. Hence the background assumptions of this work are the following. As is well known, owing to vacuum fluctur

ations, a physical particle has necessarily a composite structure. In usual

field theory this structure cannot be easily described because the relevant

expressions diverge. One has to resort to renormalization techniques which by

definition preclude even the posing of the question of the structure of a free

particle. In this point of view the only observables are the S-matrix elements

and energies. On the other hand it could be that the non- relativistic concept

of a wave function in a truncated space together with the use of effective mas-

ses and coupling constants may be generalizable also to a fully relativistic

system. Thus our work re sis on the following basic hypothesis: there is a dimension of the conf igurational space on the one hand large enough to render

the main physical characteristics of the solutions little dependent upon the

truncation energy (although the input masses and coupling constants will be

of course a function of the cut-joff energy) and on the other hand, small enough

so that the problem is still of manageable magnitude. If this assumption should

be proven out, then the discussion in the present framework of the properties of

bound hadronic systems (mesons, nucleons, deuteron, nuclei,...) such as form

factors, electromagnetic characteristics, coherent photoproduction, photodisintegration, etc .

, . . . will be analogous (using the language of nuclear physicists)

to the treatment of nuclei with dressed particle energies and effective forces

within the non-relativistic microscopic theory of nuclear structure.

Thus in short the physical hadrons, i.e., the mesons, the nucleons and

also the light nuclei will be described in terms of a set of Boson and Fermion

fields, the parton fields, which interact via the usual interaction Lagrangians

of field theory. In other words the physical particles will be composite systems, i.e., configuration mixtures. The configurations are made up of different

combinations of partons coupled to have the strong quantum numbers of the system. In the present model the partons carry the quantum numbers of the physical

hadrons. The high energy microscopic structure of the partons (e. g., whether or

not they are made up of quarks) cannot be accounted for within the limitations

of the truncated space and are contained in the model masses and coupling

- 2 -
1.3

constants. For example the wave function of the physical proton contains the

following parton configurations :

i) a single "proton,"

ii) a "proton" plus one or several "rr-mesons," "p-mesons, " "tw-mesons". . .

iii) a "proton" plus one or several "nucleon"-"anti-nucleon" pairs etc.,

... . Here the quotation marks denote the parton which has the quantum numbers

of the corresponding physical particle. From now on, for typographical convenience, we shall adopt the convention of dropping the quotation marks and, when

necessary, of adding the word "physical" if the physical particle rather than

the parton is meant.

The overall procedure which will be followed thus is : write down the

Lagrangian in terms of the parton masses and coupling constants ; expand the parton fields in sets of discrete orthonormal modes ; construct the correctly sjmimetrized and anti- sjmimetrized many- body relativistic state vectors with the quantum numbers of the physical system under consideration; compute and diagonalize

the secular Hamiltonian matrix in the time independent Schrddinger picture

using the representation defined by this set of states. The partons of the theory

are partially undressed. The higher the truncation energy, the more they are undressed. For a truncation energy of say 1 GeV above the nucleon ground state,

it will be shown later that this choice leads already to configurations containing up to 6 pions in intermediate states. All irreducible diagrams which can be

constructed with up to 6 pions are thus treated to all orders by the secular problem with only the 1 GeV restriction on virtual intermediate state energies.

The dimensions of the discretized energy matrix here are of ~120 for reasonable basis size parameter. It will then be interesting to test how much of say

the anomalous magnetic moment, the form factors etc... of the physical nucleon

will be described by this non perturbative approach. The question then is , to

which accuracy it is possible to describe in a consistent way the physical data

in terms of parton dynamics on the basis of a field theoretic relativistic Lagrangian. Of course, the answer to this question can only be given by the amount

of low energy data which will be reproduced with a given set of effective masses

and coupling constants.

The present work intends to lay the mathematical groundwork for such

a phenomenological program. We treat in detail the spin 0, 1/2 and 1 fields and

their interactions. For low values of the truncation energies it is improbable

that the A^^ will emerge as a physical pion-nucleon scattering resonance or that

- 3

-
1.4

the effect of that state in hadronic systems can be accounted for. In order to

achieve this, one has very likely to use a (3 /2 ,3 /2) parton field in the Lagrangian.

Such a field is not included here. With the present tools a large number of

applications and calculations must be carried out before either dismissing the

postulates of the present description or proving that it gives a reasonable

line of attack on the formidable problem of intermediate energy nuclear dynamics.

1.2 - SURVEY OF THE WORK

In the next sections of this introductory chapter, we shall go into

more detail concerning the present theoretical approach to hadron dynamics.

First, in Section 1,3, we review a time- independent formulation of Lagrangian

field theory, which is, of course, non-covariant but which yields a secular problem particularly well suited to the description of stationary states. In this

connection one has to use discretized fields, i.e., wave packets over the energy.

This in turn brings in one of the fundamental problems of the treatment of com- posite systems: the center of mass motion. The difficulty is augmented in the case of systems made of relativistic particles, because, as shown in the Appendix,

r gi

the CM. coordinate becomes a many-body operator. The removal of the CM.

spurious energy is discussed in Section 1.4.

In order to allow the use of the well-known angular momentum tech-

[9- 121 niques for many-body systems, the free field expansion will have to be

constructed in tensorial form both in angular and isospin space. This requires

a number of definitions and basic formulae which are given in Chapter II. In particular, a consistent set of phases is defined for the creation and annihilation

operators, invariant vectors and matrix elements. This way powerful graphical

techniques can be used to construct invariant quantities. The angular momentum

coupling diagrams, whose rules are briefly recalled, vastly simplify the calculations. Their use throughout this work shall help the reader to follow and

check the results in a convenient and concise way.

Chapter III is devoted to the construction with wave packets over the

energy variable of the discretized expansions of the free fields in tensorial

forms. The invariant matrix elements of the free Hamiltonian in the new basis

are also computed in this chapter. The obtained tensorial forms of the field

expansions are checked to verify the proper commutation (ant i- commutation)

- 4 -
1.5

relations and equations of motion. In this work we treat the cases of spin 0,

1/2 and 1 fields, namely the partons with the quantum numbers of the nucleons

(1/2,1/2), the n-mesons (0,1), the p-mes,ons (1,1) and the oj-meson (1,0). For

rl3 1 the spin 1 field we adopt the framework of the 8-component theory of Hayward"-

in which the reduction of the relativistic spin 1 field into its tensorial components of spin 1 and spin 0 is simple.

In Chapter IV the treatment of the center of mass motion is carried

through in detail. This entails the calculation of the invariant matrix elements

of the C.M. momentum and position operators. The latter being a many-body operator a method is given to transform it into a sum of products of one-body operators.

In Chapter V we calculate on the orthonormal discrete basis sets of

Chapter III the invariant matrix elements of the simplest Hamiltonians of the

usual strong interaction Lagrangians, namely rtNN (PS and PV couplings), (JSIN, pNN,

3 4

pKK and con (V coupling) and n .

As illustrations of the many applications of this general formalism.

Chapter VI presents some possible parton models of the nucleon, the deuteron

and the pion. The partons constituting these systems are assumed to have the

quantum numbers of N,n, p and co. The lists of all the configurations which can

be built up to the truncation energy of ~ 1 GeV above the ground states of the

physical systems are given. It turns out that the dimensions of the various secular problems are quite reasonable ( ~ 60 X 60 for the deuteron, 120 X 120 for

the nucleon) although they may involve as many as six pions. Furthermore in these

spaces the symmetrization of the bosons presents no difficulty since the most

3 complicated redundant configurations are of the typq (Ip) . All other configurations with a number of bosons larger than 3 involve at most the trivial (Os)'^

structure which is treated in detail in Chapter II. All the matrix elements

needed for these models have been given in the preceding chapters.

1.3 - FIELD THEORY FOR STATIONARY SYSTEMS

1.3.1 - The Schrb'dinger picture

The general relations of field theory needed throughout this work can

be found in any textbook on the subject, see for example refs. [2-7],

- 5 -
1.6

In field theory, as is well known, one works with field operators and

state vectors. The wave functions of quantum mechanics are matrix elements of

field operators between state vectors. The dynamics of the system is contained

in the Lagrangian which is a functional of the field operators and their derivatives. Conserved quantities are constructed from the field operators.

One utilizes three pictures, viz. the Heisenberg, the Schrodinger and

the interaction pictures. In the Schrodinger picture the dynamical behaviour of

(S) a system is described by a time dependent state vector (t) ) which is a solution of the Schrodinger equation :

n I'J'f ^^^)> = - i I'P^^^t)) . (1.1)

The Hamiltonian is defined as the component T^^ of the energy momentum tensor,

and is given for the cases of interest here in Chapter III. In the Schrodinger

picture, it does not depend on time for closed systems. In the Heisenberg picture

(H) the state vectors do not depend on time while the dynamical variables exhibit the time dependence. For the Hamiltonian

H(«)(t) = e^«^'^'H(S> e-^«^'^^ ^ H^^) . (1.2)

From now on we set H = H^^^ = H^^^ (t) . Any solution of Eq.(l.l) is of the

form

|^^^(t)> = e-^«^|>.f>> = e-^^nt,^(H)> ^

'

In treating interacting fields we have to decompose the Hamiltonian

into two parts

H = H + V (1.4)

o

where H^ is the Hamiltonian of the free fields and V the interacting Hamiltonian

In terms of Hamiltonian densities

H^ =-]* d^x(T^(x))^ = Jd\jf^(x) , (1.5)

V = - J

d^x(Tj(x))^ = J d\ v(k) . (1.6)

- 6 -
1.7

In these expressions the energy densities 3f^(x) and j;(x) are combination of the

basic field functions taken at a certain fixed instant of time, let us say

t = 0.

In the Schrodinger picture the free field operators can be expanded as

$^^^x) = (cp. (x)a. + 9*(x)a+) (1.7)

1

for Bosons and

v|/^^(x) = y (u (x)b + V (x)c+) + c.c. (1.8)

j

for Fermions. Here the sums are understood to be over the possible discrete

quantum numbers of the fields (magnetic and isospin projection quantum numbers)

and over the continuous field energy variable.

The Boson operators at, a^ create or annihilate a particle in state i

respectively, while the Fermion operators b^, c^ annihilate a particle or

create an anti-particle in state j respectively. They fulfill the relations

[a. ,a1;,l_ = 5... , (1.9)

[bj,b;.]^ =6... . (1.10)

In the Heisenberg picture the corresponding free field operator expansions are

f^"^x,t) = [(e ^^^%.(x)a. + eV''>*'(p*(x)a^^ , (1.11)

i

( -ie.t ie.t

v|/^"^(x,t) = [

(e ^ Uj(x)bj+e ^ v^ (x)c^ ) + c. c. (1.12)

j

These fields are solutions of the equations of motion derived from the free

field Lagrangian and they will be constructed in tensorial form in Chapter III.

1.3.2 - State vectors

Because of the separation of the total Hamiltonian, Eq.(1.4), the free

fields play a special role. They are used to construct a basis set of state

vectors, i.e., to define a Fock space. We shall define now the basis state

- 7

-
1.8

vectors |a(t))^^ and the basis conf igurational state vectors )v(t))^^ in this

(S) (S) Fock space, as well as the corresponding wave functions r (t) and R (t) a V in the ordinary position-momentum space,

i) Basis state vectors

A particular state of a many- body system for V = 0 can be specified

in the occupation number representation by the basis state vector

•i H t

,(H) ^3(H) a

|„(t)>('> = sf)(t)|0> = e'^"°^ sf)|0> , (1.14)

(h) where s^ is a product of Fermion and Boson creation operators. The index a. stands for all the quantum nximbers which are needed to specify the occupied

single particle states. For Fermions, as usual, this product of creation opera^

tors is given in a standard or lexicographic order. The ket |0^ denotes the

vacuum and with the definition

<OjO> = 1 , (1.15)

the basis state vectors ( 1. 13) and (1,14) fom a complete orthonormal basis of the

Fock space. Simple examples are for a single Fermion in state a

s^^^lO) = b+)0> , (1.16)

- i e t s^^(t)|0> = e " b+IO) , (1.17)

and for a system made of a Boson in state i and a Fermion in state j

sf^|0> = <b+|0> , (1.18)

sf\t^,t2)|0> = e"''^'^ ''j'^ a^b+|0> , (1.19)

etc... . Later on we shall discretize the Hilbert space by introducing, as already mentionned, wave packets over the energies e^^, Ej ... . At this point

however the indices i, j, as shown, still specify discrete and energy quantum

numbers.

ii) Basis configurational state vectors

They are the proper linear combination of the above basis state vectors needed to construct eigenstates |v)^^^ of simultaneously H^, (the

- 8 -
1.9

total angular momentum),!^ ( the total isospin) and all other relevant constants

of the motion

|,>(H) = ^(H),o> . y c« .<H)|o> . (1.2G)

a

The coefficients C are the elements of a unitary transformation constructed to

V yield an orthonormal basis

^^\n|v>^^^ = 6 . (1.21)

These coefficients are related of course to angular momentum coupling algebra

and to coefficients of fractional parentage (CFP) as shown in Section II. For

example the basis configurational state vectors for a system of total angular

momentum I and total isospin T made of a Boson with quantum numbers

j^T^ and a Fermion 32^2

K--\K-2 A- +

X a; |0> . (1.22)

(H) Since in the linear combinations entering the sums are only over magnetic

quantum numbers, when going to the Schrddinger picture the time dependence is

simply

-i y E.t.

L J J |v(t^,t2,...t^)>^^^ = <sf\tj^,t2,. . . t^) |0> = e j ^^^^)|0> . (1.23)

1.3.3 - Wave functions

The Schrodinger basis wave function corresponding for example to a basis state vector with M Bosons and N - M Fermions is given by

(S)

'^a ^^l'^l'^2*^2---^^M' ^lS«-l--*^V

= <0|$^^^x^) ...$(^^x^) ^^'^x^+P ...^^'^V =r<h"---V|0>

= <0 |$(«) (x^t^) (x^t^) (^+it^+i) . . . ^^^^ (x^t^) ) |0> (1.24)

and likewise for a basis conf igurational wave function

- 9

-
1. 10

,(S),„. _ /nl.(S),„N ,,,(S), . ,,(8) Rv^^'CVi--- Vn^ = <o|s' •••^ Vl°> • (^-25)

These wave functions are automatically symmetric (or antisymmetric) under the

exchange of the Boson (or Fermion) coordinates.

The basis wave functions thus constructed are more general than we

need, in that each particle has its own time coordinate t^. For our purposes

it is sufficient to have them defined at a single common time, i.e., on a single

space-like hypersurface, see for example ref.[7 ], chapter II. In other words,

we shall require the description of the system in a single frame of reference,

viz. the laboratory system. At this point the treatment ceases to be fully co- variant, biit, of course, it remains fully relativistic. In principle one can com- pute quantities which are relativistically correct with the resulting non-covariant wave functions, provided of course that the time-independent formulation is

used throughout. This is in contrast to the usual time -dependent formulation wh

where one would require the full information contained in Eq. (1.25).

Hence, from now on in the Schrb'dinger picture all times are made

equal

t. = t for all i

so that with the definition of the conf igurational energy

M + N

i = 1

the time dependence of the conf igurational state vectors and wave functions in

the Schrb'dinger picture become respectively

|v(t)>^^^ = sl^\t) = e ^^"^|0> , (1.27)

and

~ i. (o* t \^'^^^l---^ + N'^^ = ^ Mol^^'^xp ...v^^^^x^^^) ^f>|0> .(1.28)

1.3.4 - The secular problem

An approximation scheme can now be developed, based on the Schrbdinger

representation, along similar lines as in ordinary non- relativistic quantum me-

(S) chanics. A formal solution (t) of the Schrodinger equation with the complete

- 10 -
1. 11

Hamiltonian (1.4) is assumed to be obtainable from an expansion on the complete

(S)

set of the configurational wave functions (t)

= I

x;;(t) Rf)(t) = I

x;;(t) e"""^' R^«) . (1.29)

V V

or in state vector form

l^l^ht)) = I

x;;(t) e"^^"' ^f^lo> . (1.30)

V

(h) This expansion generally involves basis state vectors |0) with a

given number of baryons but with different numbers M of mesons. The indices v

thus stand for N,M as well as for all the other quantum numbers defining the

basis state. The strong quantum numbers in v (total J,T, baryonic number, parity,

parity, etc.,,) are of course those of the composite system under consideration.

The Ansatz (1,30) means that the solution of the problem of interacting

fields is represented by an expansion in terms of free fields. As is well known

this procedure, strictly speaking, may not be legitimate in relativistic field

theory in contrast to non- relativistic quantum mechanics because of the problem

of divergences. We shall return to this point below.

For a stationary solution the Schrodinger equation

- iH^^^ |vj/^^\t)> '= - iEj^^^ht)} (1.31)

entails that the amplitudes x'^(t) must be of the form

V -i(E -^^)t

x^(t) = e , (1.32)

Inserting (1.32) into Eq. (1.31) we obtain the secular equation

,(H)V„ .(H) in\ . <0|e8^"^y H^^?^ |0> x'" = E x", - (1.33)

I V namely

I ^"\v |H-E^|v'>^^^ % = 0 . (1.34)

V

This is a continuous matrix since the sum over v' contains an integration over

the free field energies

- 11 -
• 1. 12

Since we are aiming at a formulation of the secular problem which

will lead to the diagonalization of finite matrices, we must now re-express Eq.

(1.30) in terms of discretized basis states.

Before doing it let us note that discretization of relativistic fields

is somewhat different from the non- relativistic case. Namely in non- relativistic

theories one can use a fictitious potential to generate a complete set of eigenstates with discrete energy eigenvalues. The solution of the true Hamiltonian

then can be expanded in terms of these discrete states. In a relativistic theory

one cannot use the non- relativistic concept of potential. Consequently one has

to start from the free field solutions themselves for basis states, as it has

been done above. The way to achieve a discrete secular problem is then to construct wave packets over the energy. This will be done in detail in chapter III.

by applying a unitary transformation on the field operators,

a"*" = r de f (e) a"*"

/< J /< E (1.35)

b"*" = r de f (e) b"*" , etc. .. ,

where the energy functions f (e) with discrete index k form an orthonormal set

r de f (e) f ,(e) = 6 , J K. K. KK. (1.36)

y f (e) f (e') = 5(e-e')

K. A convenient choice is for example harmonic oscillator functions. Inserting the

discretized fields (1.35) into the definitions of basis state vectors, section

1.3.2,, by means of the inverse transformations

a = ) f (e) A^

K

b"*" = y f (e) b"*" , etc.,

E L, K K. K we shall obtain discretized basis conf igurational states |r) . The corresponding

amplitudes X^^"^ for the solution 'P^^^ (t) are now given by the completely

^ n n discretized system

^{<r|H^-Ejr'>+ <r|v|r'>}x^^') = 0 . (1.37)

r'

In words, the time independent description of quantum field theory leads to a

secular equation which is formally indistinguishable from that of the classical

theory.

- 12 -
1.13

1.3.5 - Nature of the solutions

(S)

The solution 1^ (t) ^ contains an undefined number of hadrons, i.e.

' n ' it is expanded on configurational states with different numbers of Bosons. Unless

the employed truncation energy is above the baryon - antibaryon threshold, the

number of baryons of course is identical in all components, but their intrinsic

quantum numbers may be different. The meaning of a stationary solution is then

one in which the relative admixtures of parton components do not change in time.

A pictorial representation of the effect of the creation and annihilation operators in time is given in figure 1.1. There a system is described as a mixture

of a single Fermion f with different numbers of Bosons b. Thus at time t the state

(S) vector 1^ (t)) is a mixture of the components f, fb, fbb ... with amplitudes

(r)

(r= 1,2,3 ... respectively). After the infinitesimal time At, several processes have taken place due to the time evolution Hamiltonian, where a single

Boson has been created or absorbed. These processes are associated with the

field interaction matrix elements (r ' |V

| r) . At time t+At the conf igurational

mixture miist, however, remain unchanged in order for the state vector to be

stationary. We have

¥^^^(t+At) = ¥^^^(t) - iAtH^P^^^(t) . (1.38)

Substituting the expansion (1.30) we get

^^)^ ^^n^^ ^-^(r) . g = X^^^-iAt y <r|vjr'> X^^'^ , (1.39) n n n' n ^^I'^n'

r'

which is the relation between the amplitudes given by Eq.(1.37). Assuming for

simplicity to be diagonal on the representation r

Enn ^iX«+<X|v|2>xf' In^ii/n »,

E X^^^= <2|V|1> X^^+ ^„ X^\ <2|V|3> X„^^^ etc... (1.40)

Graphically (see fig, 1.1) when going from t to t+At the component r = 1 may

either go into the component r = 1 via the free field Hamiltonian H with the

(1) ° amplitude (f, X^ ^^or into r = 2 via an interaction V where a Boson is created. In

(amplitude <2|V|l> X^^. Likewise the component r = 2 may either go to r = 1

via the absorption of a meson (amplitude {l|v|2) into r = 2 via H or

(2\

° into r = 3 via the creation of a Boson (amplitude <(3|v|2) X^ 0 etc. In order

for the composite state to be stationary all the amplitude flows (indicated by

the arrows) must be such that they leave all the ratios X^^V X^^*^ unchanged.

n n

- 13 -
1.14

This is precisely the meaning of the linear relations (1.37) or (1..40),

,(1) t+At

I I U

etc.

I I

I

X(2f

ii

!i

y(3) 0 fb fbb

Figure 1.1

1.3.6 - Comparison with time dependent perturbation treatments

There are fundamental differences between the present approximation

scheme, based on the hierarchy of the basis conf igurational states in the ex- pansion of Eq. (1. 29) with a cut-off energy or truncation of the Hilbert space,

and the usual time- dependent perturbation treatment generally carried out in the

interaction picture

i) The energy matrix of the secular problem requires only the calculation of the interaction operator V at a single time point , between of course many-body basis state vectors . The proper statistics between the particles is

readily achieved with well known techniques. This is in contrast to the ill-defined

nature of the chronological products of interaction operators at contact points,

see for example ref.[7 1 page 220, and to the rapidly increasing difficulty in

calculating irreducible diagrams of increasing order.

ii) Upon diagonalization of the energy matrix all processes which can

exist in the chosen configuration space are automatically generated and treated

to all orders, with only the restriction of the cut-off energy. In a timedependent treatment only selected irreducible graphs are calculated.

- 14 -
1.15

iii) The solutions of the Schrodingei" equation are of course unitary in

the truncated space.

iv) The divergences of the strong interaction theories which in the timedependent treatments show up when integrating over the four-momenta of the intermediate virtual states, appear here as a divergent dependence of the solutions

upon the cut-off energy of the truncated space.

1.4 - EXTRACTION OF THE CENTER OF MASS MOTION

As usual when working in a truncated Hilbert space, the solutions of

the secular problem of Eq. (1.37) are each a mixture of states with different

center of mass motion. In order to obtain solutions with a well-defined C.M.

motion, viz. Os, a standard procedure of non-relativistic quantum mechanics consists in adding an artificial C.M. energy operator to the Hamiltonian

^C.M. = i C +

^^^^^ , (1.^1)

where P and R are the center of mass momentum and position operators. This

C.M. Hamiltonian is used without ascribing a physical meaning to it but as a

device to split apart the solutions of

^''"^CM ^^^^ groups with each a given

C.M. motion. This splitting increases with increasing values of the parameter

5 . Non-relativistically this procedure is exact besides the difficulties associated with truncation . Note that a large value of ^ does not imply a high

velocity of the C.M. motion. The parameter ^ only changes the scale of the

C.M. level spacing. It is the frequency or size parameter fi which determines the

velocity of the C.M. motion.

We shall use this method in the present work. Relativistically the

C.M. momentum is simply

P = Pj^ + P2 + P3 .. . (1.-^2)

and the evaluation of the matrix elements of ? on the relativistic basis state

vectors presents no particular difficulty. However in relativistic kinematics '•^ ^

the relation

^ = Q m^ X. ^y^M (1.43)

i

- 15 -
1.16

M = m. (1.44)

i

is replaced as shown in the appendix by

R = (I (I ^k) ' (1-^^)

i / k

where the e^'s are the energies of the partons making up the configuration

One sees that the operator

o

(.e. X.; e.e. X. .X.

1' = y +

y

-^^-^

(1.47)

1 k k J. ^ J k ^

in the C.M, Hamiltonian (1.41) is in fact not only a sum of many-body operators

2 because of the denominators ( E e, ) but is non- separable in the individual

k

^ energy variables . This typically relativistic difficulty can be circumvented by

the use of the transformation

dz e E e, k

^

-z(^e^)

00 CO

(E ej ^o ^o ^

- (z^12k +z„) (e e, ) dz2 e , (1.48)

k

which factorizes, after substitution, each term of (1.47) into products of oneor two-body operators. Thus (z = +

^'<^> = i(n ^'"'){^"^"^'\')

i k i

i ^ j k 7^ i, j

The invariant matrix elements of the pseudo-C.M. Hamiltonian (1.41) can thus be

evaluated simply as shown in detail in Chapter IV.

- 16 -
1.17

There is, however, another difficulty which arises with relativistic

kinematics. This is linked to retardation effects. To be specific, let us consider a problem in which we want to describe a composite particle (for example

the physical nucleon) in its ground state. For high value of ^ the center of

mass motion of the solutions of the pseudo Hamiltonian

= H + (1.50)

is well defined. However, this motion is not free but confined by the artificial

potential (1.41). In such a potential the particle is accelerated and, because

of retardation effects, the different parts of the particle experience different accelerations ( analogous difficulties beset the classical model of an extended electron). In non- relativistic mechanics the potential can act simultaneously on all parts of the particle: the particle can be accelerated "as a

whole." Because of the inevitable retardation effects, in relativistic mechanics

the difference in the acceleration of the different parts of the particle will

lead to deformation of the particle, i.e., to admixture of excited states (of

the baryon resonances if the particle is a baryon)to the ground state of the

particle. Thus relativistically, if no care is exercised, the purification of

the C.M. motion by the adjunction of the potential (1.41) in turn entails the

mixing in of intrinsic excited states. The magnitude of these admixtures will

be small in the limit of a slow C.M. motion, i.e., for small values of the parameter Q in ( 1.41) .

Thus the solutions of the pseudo Hamiltonian (1.50)

3f = , (1.53)

owing to completeness are in principle of the general form

a, a

where m are a set of wave functions of the C.M. motion while the v are a set

of wave functions for the internal motion. Now from the above discussion we make

the hypothesis that for a sufficiently high value of ^ and low value of Q, the

lowest eigenstates are of the form

~ ^o \i

17 -

(1.55)
1.18

where 9^ describes a Os C.M. motion. Of course, the errors introduced by the

hypothesis (1.55) as well as the independency of the final results as a function

of the parameters C and n must be discussed on the numerical solutions.

Finally we define (E^ is the energy of the physical vacuum;cf (D.ll))

= \ (1.56)

2 2 In the limit of a slow C.M. motion P « M , we obtain

n

'

E = M + \ ^ - T ^ (1-57) n n 2 M 3 ^ .3 n (2 M ) n

i.e., upon inversion

= - + ... (1.58)

2 En

which yields the mass spectrum of the composite system, which is the physical

quantity of interest.

- 18 -
II. 1

CHAPTER II

DEFINITIONS AND FORMULAE

II. 1 - PHASES

II. 1.1 - Tensorial sets

The tensorial harmonics are defined as contrastandard tensors [9]

Yy](0(p) = vjf^ = (-i)^Y. (ecp) (2.1)

with

yJ, Me9) = y Yj,V(e'9') ®:. («Py) . (2.2)

m

Here the rotation matrix is defined as in Fano- Racah and Edmonds

,

^, («3V) = e'"'v dJ (p) e^"" (2.3)

where the coordinate system Ox'y'z' is obtained from Oxyz by a rotation through

the Euler angles a,p and y in that order.

with

The time reversed tensors are the standard sets

Y(^>(ecp) = ^^'^ = ^y+^^^n

m ^' m ^ ' -m

(2.4)

= I ^m- <*m(«Pv) . (2.5)

m

Thus for the case of vector operators we have

0 z

= -fe (i J - J ) 1 /2 X y

- 19 -
II.

These forms fulfill the relations

[J , J 1 = i J

X y-" z

fJ , J ] = i J

z' x-" y

^^y' 'z^ = ^ ^x

Hi ' ^-1 J "

1-1 oo -11 ^ ' y

Next we then define the hermitian conjugate tensorial set cp^^^ (x) of

r 1 TM

the normalized set o (x) as m

For example for the spin 1/2 functions, from their explicit form

xf/f' - Co) : - (?) <^.io>

and from the defining relation,

r- ~(l/2) [1/2] . V - ^1^

I

Xs («) Xg. («) = 5^3- (2.11)

a

we get the hermitian conjugate spin functions -^f^ ~(l/2)

s

^1/f

^ = (10) ; 3c.^}/2) = (0 1) . (2.12)

II. 1.2 - Creation and annihilation operators

We now give a consistent set of definitions for creation and annihilation operators of the usual field formalism which incorporate these transformation properties of standard and contrastandard tensors as well as the hermitian

conjugation. Therefore we demand

,(j)++ = ,(J) . ^(j)H-+

= j(j)

m m m m

^2 \o\

m m ' m m

where on the one hand a^-'^ mmand a'-^^ are respectively tr J the standard and

- 20 -
II.

contrastandard annihilation operators which transform according to

m m m ' m

m'

(2.14)

m ^ m m m

m'

and where on the other hand the contrastandard creation operators a^^ are dem

fined as the hermitian conjugates of the standard annihilation operators. This

definition results from the fact that hermitian conjugation changes the rotational transformation properties as it includes a time reversal operation. Thus

SUJ , (-)2J^<3) +

m m

and

,(3) , (.)23S[3J +

m m

These definitions are chosen to be consistent with our phase convention given

above, i.e.

,

a(J> = (-)j-*TMafjl ; a ^ ^ ^ = (-)j-"^a^j^ m -m ' m -m

~(j) ^ (-)j-^afj] ; a"! = (-)j-"^a^j> m -m ' m -m

These relations may also be written

(2.17)

m -m m -m

3[(j)+ = (-)j'"a^j^ ; afjl"^ = (O^'^'aljl

(2.18)

m -m m -m

which shows that with our choice of phases the creation and annihilation operators

transform exactly like tensorial sets.

- 21 -
II.4

II. 2 - INVARIANT STATES AND MATRICES

The most general invariant state vector with amplitudes W^-'^ and norr^i . ^ . TM

malized components is, with the notation j = / 2j+l

^ = I"m'^ ^m

^ = ? r^fj] itJlifO] ^ (2.19)

m

The normalization of ^ imposes

wfJljfOl = i . (2.20)

j

Likewise the most general invariant operator n(J) of multipolarity J is

o(J) = l^'> = J nl-"it°l (2.21)

M

with normalized amplitudes

J

Its matrix elements between the noinnalized states and are

1 f

which defines the invariant matrix element [^'^'^ ^ |v|/f'^^ 1. Its connexion with

ordinary matrix elements is given by

<*^i' 1=^''

> = I

= (.)J+'» (-)W + '< (_W (+(J1|,[J1|^W,

, (2.24)

and with the reduced matrix elements of Edmonds ,Fano-Racah and Messiah flO-12] by

(iplUQf-^llKfl) = (-)j-'^-S^|^l|nr'l|H'fll . (2.25)

We shall need the following list of useful invariant matrix elements

,^tl/21|„rll|^rl/21, , , (2.27)

- 22 -
[Y

K.] Kol KoJ l/2(^ +/ +^ ) ^T^/^ /, ^

1

|Y ^ |Y ] = (-) 1 ^

1 Z 3jj / 1 2 3\

Likewise defining

cpf^J(r) = f . (r) Y[^hr) am Of / m

II.

(2.28)

(2.29)

(2.30)

together with the unit vectors in coordinate and momentum spaces

^flJ

= (W3)^/2 ^y[^J(^)

m m

m

i V

[1]

we get the invariants

m

(2.31)

(2.32)

r.t"|p'''|.^'-^'i

a P

or P

= 1 "a/ ' r ar I p/

i /m <f

^+2 ^ ^

"a/ ' r ar I p^f+l-

(2,33)

(2.34)

The radial functions f will frequently be spherical Bessel functions j^(ar)

Of c t in which case using the relations

we get

with the factor a which will often be used thereafter

^ K

0

if A. = <f+l

if A. = ^

if A, = if-l

(2.35)

(2.36)

(2.37)

- 23 -
II. 6

II. 3 - A GRAPHICAL REPRESENTATION OF ANGULAR MOMENTUM COUPLING

Considering four tensors, the basic recoupling transformation is

hi

- a b c

-

d e f

- h i g .

[g]

(2.38)

where the square 9-j symbol is related to the ordinary 9-j coefficient by

ra b c d e f

h i g J

^ ^ ^

= c f h i {a b c N

d e f

h i gj

(2.39)

[9] This transformation is represented by the basic diagram of figure 2.1'- , where

summation over all new quantum number appearing in the diagram is implied (in

the present case the summation must go over h and i) .

t

Fig. 2.1

Recoupling transformations for three tensors, viz. of the 6-j type, are represented by the same diagram with a dashed line representing a mock tensor of

rank zero, as shown in figures 2.2a and 2.2b .

g

-a 0 a bed

-e c g

Fig. 2.2a

- 24 -
II.

a g

0_ V0 a a bed

Lb f g

II.

Fig. 2.2b

We frequently shall need the relations and special values of Table

-a b c

" "a d e

-

d d 0 b d f

_e f C - _c 0 c .

"a a 0

- "a b e

"

b c d a c f _e f d . ^0 d d .

e f , .a+f+c+d

(-)

d

e f , K e+c+a+d

a

r a b c I f e d j

ra b e Id f c j

c f (-)

f+d+b+c ra b c le d f j

d

-

cf .c e (-)

c+e+b+d ra b c If d e j

-a a 0- 'a 0 a" "a b c

"

0 b b b b 0 0 b b

_a c b . -C b a _ _a 0 a _ a

(-)

a+b- c

a a 0

b a c

.c 0 c ^2

a

(-)

a+c-b

0 a a- b b 0

b c a

-0 b ba b c -a 0 a -

c

a b

-a a 0"

a b c _0 c c -

"a b c

"

a b c -0 0 0 .

^1

c a b

a

b

c .

0

0

0 0 0 J

= 1

a a 0

0 a a La 0 a i (.)2a

a

Table II.

- 25 -
II.

The last line of Table II. 1 corresponds to the common situation of

final coupling into two invariant matrix elements. Finally for the interchange

in the order of coupling two tensors we shall "use the simplified diagram of

figure 2,3.

Fig. 2.3

which of course corresponds to the special square 9-j value

[0 a a -| b 0 b =

b a c J

(-1)

a+b-c

(2.40)

When working both in angular momentum and isospin spaces we shall

rJTl

employ the double coupling and recoupling notation. Thus we shall write v|/ ^ m T in that order and define the diagram of figure 2.4.

\ ^3^3 t

t -^2^2

53. X , ^4^4

.ij 1 ^—1

h ^2

h ^4 ^2

'3 ^4 ' .

^3 \ \

LT3 T^ T_

Fig. 2.4

- 26 -
II.

We now turn to the diagrammatic representation of the invariant matrix

elements which are given in figures 2.5a, or when working both in angular momentum and isospin spaces in figures 2.5b.

Fig. 2.5a

Fig. 2.5b

We have explicitly shown on these diagrams that the upper line corresponds to

the bra function. The ~ sign will in general be omitted from the diagram with

the understanding that the bra line must always enter a hatched box (invariant

matrix element) at the top. An example of this rule, which in fact yields a phase only in the case of half-integer spins, is given below.

Finally we also need the insertion of a complete normalized set of

states which is represented by the diagram of figure 2.6 which fulfills the

L a 'a

fa

Fig. 2.6

completeness relation of figure 2.7. Here a denotes all other quantum numbers

(besides f) which define the basis set of states.

- 27 -
II, 10

Big. 2.7

The crossing sign box in Figure 2.7 is not needed for integer angular momentum.

We also will frequently need the reduction formula :

[Y ' (r) Y (r)l = m 12

/ 1 l/2(^ +<• +^)

- ^ V2 (o 0 o) <-' ' ' «")

1 z y^Tt

and which is represented by the diagram of figure 2.8 .

\\\

Y

Y

Fig. 2.8

II. 4 - VECTOR ALGEBRA

Here we collect the invariant forms of vector algebra and vector analysis. They are based on the invariant representations

V = Ve +Ve +Ve = \ V^'^^'^e'-''"^ XX yy zz TM

m

= 1 [V^^l e[^l][°l = t [ef^l vC^ljf°] (2.42)

^ . t ^V^l] e[l]]f01 (2.43)

- 28 -
11.11

where e'^''"^ is the unit vector related to the Cartesian unit vector by

.{^1 = (ie^-e)//^

e^^] = - i (2.44)

-i^ = (-i e^-e ) //2

Thus we get for div, rot. and grad the following expressions :

1) div V = V.V = VV+VV+VV XX y y z z

= t [Vf^j v[^]][°J , (2.45)

i.e. the divergence of a vector is obtained by replacing in the vector

expression (2.42) the components of the unit vector by the respective components

of the gradient. Hence for

F = [ ef^^ ] (2.46)

we get generally

div F = [ yf^^

] . (2.47)

ii) rot V = '^V = (^V) e + (^V) e + (^ XV ) e xx yy 'zz

(2.48)

where we have used the Fano-Racah notation for the invariant triple product.

The factor y2 arises because of the different normalization implied by the

Cartesian cross product and the coupling of two vectors to / = 1 . Generally

—

for the above function F we get

rot F = /2 r... re^^Wf^l] f^^..] _ (2,49)

iii) grad f = ^ f = 1 [V^^^^ f e^^^jt^J . (2.50)

- 29 -
II. 12

iv) Special cases

We now give the formulae for a few special cases which will be relevant

latter on. Let us define the function

9^^^ = j. (Pr) Y^^hr) ^pm X m

and the vector spherical multipole J with amplitudes A

m

We get from Eq. (2.47)

div 7 rAf^^ vf^i ^M^\0]

which diagrammatically is represented by the diagram of figure 2.9

(2.51)

(2.52)

(2.53)

V

9

1 \J

0 J

JJ] |„rl] I. J^lJOlrAfJ] .J-^IJO]

Fig. 2.9

which yields

. , (2.54)

From now on we shall read off directly from the diagram the completeness summations and integrations, carry out directly the triangular condition (J 'JO) =

5jj

and the integration over the continuous variable q which leads to p = q .

div p = ) J J' 0 J

- 30 -
11.13

Likewise we get

rot; . [At^l[e[^lv[ll,(ll<pW,C01

_

which is evaluated with the diagram of figure 2.10 .

(2.55)

9

1 \^

V

'A

a

vK

Fig. 2.10

Thus we get

rot p = J

UlvJJ LvOvJ

for X = J + 1 ,

= - p /T[Af'^le[^Vf''"'^]^°^+/3+IrAt'le[^]^f^-^]]f°]

for X = J - 1

for A, = J

(2.56)

We have made use of the explicit values of the 6-j symbol entering the expression

(2.56)

V = J, J + 1 (2.57)

- 31 -
11.14

In a similar fashion

rot rot; = 2 J [Af-JJt[[eCl]v[^l]C^lv[^3]rilcpf^3]f-^l]f°l

which from the diagram of figure 2.11 yields

(2.58)

rot rot

-1 1 1 ]=

" V V 0

- -1 1 1-

0

-1

X X

V J ^ V

V0VV _ P a

vX

0

.1

V VJ -

-M. |JL 0 -1 0 H M- P a

Lm- 0 n J

fA[Jle[l]^[^]][0] (2.59)

1 . 1

Fig. 2.11

i

After simplification this expression becomes

rot rot p = 2_

/j(/r[Ar'^ie[^icpf^-^^]]f°U/jTr

X fA"- 'e if A, = J + 1

(2J + 1) [A[^letM^[Jlif°l

^[01

J

if X, = J

if X, = J - 1 (2.60)

- 32 -
11.15

Finally the gradient is given by the formula

X = J + 1

II. 5 - STATES AND MATRIX ELEMENTS IN THE ANGULAR MOMENTUM COUPLED FOCK REPRE -

SENTATION

II. 5.1 - Commutators in a coupled scheme

Symmetrization or antisymmetrization of many- particle states of good

angular momentum is carried out by fractional parentage coefficients. In field

theory on the other hand the observable quantities are given in terms of field

operators expressed with annihilation and creation operators well adapted to a

single Slater determinant representation . Thus we must now give the tools to

cast the fields into a form suitable to the angular momentum representation.

The creation and annihilation operators of section II. 1.2 obey the commutation or anticommutation relations

fa(j>, a^f\ = 5 , , (2.62)

L m ' m J+ mm '

or identically

m ' m J+ mm

In order to go to a coupled scheme we note that this expression may be written

as

~JV . - , (-)^^(-)^+TM J^J' afJ' + a^, . (2.64)

with T] = - 1 for Fermions and +1 for Bosons. After summation over the magnetic

quantirai numbers, with the restriction m = m'

rJj] ir[jl][01 = 0U Jj]^[0] ^ (_)2j ^ ^ (2.65)

since (-)^^ = t] . The commutator or anticommutator bracket notation

|^ ^ ^ in

eq.(2.63) is here of course to distinguish it from the coupling brackets f J

. Taking away the restriction m = m' and coupling to an angular momentum I 0

we get

[a"' SUJJPI = , [^Hl alJljP] +

(-)^J J 6,0 (2.66)

- 33 -
11.16

2

* with of course r\ (-) = 1 . This commutator or anticommutator relation in an- gular momentum representation shall be represented graphically by figure 2.12,

j \ I

j

Fig. 2.12

where the square boxes are the usual crossing phases and the slash in the first

one represents the factor t] . (Recall that the overlap box. Fig. 2. 5a, implies a Kronecker factor S-j-q.)

II. 5.2 - State vectors and wave functions

The basis conf igurational state vectors for n particles, defined in Eq.

(1.20), in the occupation number representation, shall be constructed and used

in the invariant form

l^f >0.N. ' f [W'''?fjl'°'|0> , (2.67)

where |0) is the reference vacuum state, s'^''"-' a linear combination of products

-Ml of creation operators a ^ chosen such as to yield a state of quantum number

la . These state vectors are normalized according to

Hence

since

l^r'>O.N. = «ap ^11

^ ' na n(i I / rvR TI ' ap II'

(2.68)

(2.69)

or (2.70)

M

In the R representation the corresponding invariant wave function is

$f (R) = #<0|{Y}^ I [Wf^l ?f^l]f°l|0>

, (2.71)

where ^ is the field operator and

M = ¥(x^) 'P(x^_^)... Y(x^) (2.72)

We consider now a few special cases.

- 34 -
11.17

i) One particle state vector

In the occupation nimber representation it is

0.„.<^Jl = f<0|rKP'aPlj[°J . (2.73)

The factor I ensures normalization as shown by the norm diagram of figure 2,13

1

W

a X X

. I

I

Fig. 2.13

In the R representation

$J(x) = #I<0|vl/(x) [Wf^^ af^l]f°l|0>

(2.74)

ni The norm JV is equal to unity if the single particle wave function cp (x) entering

in the field ^'(x) « a cp + is assumed to be normalized. This expression

can be read off the diagram of figure 2.14. We have introduced the cross- hatched box for the termination of the amplitudes having a value 1/1 according

to 2.20.

W

w

Fig. 2.14
11.18

ii) Two particle state vector

The state vector in the O.N. representation for two particles in shells

j and k respectively is

0.».<'2l = (-)J*'^-%t<0|[wf^l[aW aHIlPljtO] . (2.75)

The norm is given by the norm diagram (2.15).

- 36 -
11.19

Hence

(2.76)

or in our notation

^2

/l + 5.

1 taf^] aC^]][I] (2.77)

iii) The s Boson state vector

The case of n identical bosons in a single s shell is very simple and

will play later on an important role. Here only the isospin part needs symmetrization, and for isospin 1 particles all possible states are uniquely specified

by the particle number and the total isospin quantum number T. This happens because each state (n,T) is generated at most by two states, namely the states

(n-1, T-1) and (n-1, T+1) which yield , together with the normalization condition

imposed on the two coefficients, a single possible state.

Thus in order to generate the successive (n, T) states let us consider

the recursion in isospin quantum numbers

(^n^[TJ,o> = -^J,^(T)[(A-br^-l] At^l][^Up^(T) f (A^ ^

[^-^^ ^ A 1

]

^ j

|0>

/n+l! ^

^

+ p^(T-i)r(A-^f^iA[^iir^-^iAr^ii[^i}

+ P,+l(T){a,(T+l) [(A-^

+ p^(T+i)r(A-Vr^-^2iA[^ii[^+iU[^i]E^i]]j |o> (2.78)

These four terms can be recoupled so as to bring out the S37mmetry character of the two last particles. It must be symmetric, namely the recoupled

terms where the two last particles have t = 1 must be zero which yields the

recursion relation

- 1 1 T-1' Oil

_T t T J

T 1 T+1' Oil

.T t T J

= 0 (2.79)

- 37 -
11.20

We now get the coefficients of the state together with the normalization condition

" «n+l<^> + Pn+1^^) = ^

•

For example for n = 3

a3(T = l) = ^ , p^(T = l) = I ,

a3(T=3) = 1 , p^(T = 3) = 0 . (2.80)

iv) The general n particle state vector

Let us first consider the case of Fermions and denote (i ^) a single

n-1

normalized Slater determinant for n-1 particles. The n particle antisymmetrized

state in the R representation is

J = ^ ) (2.81) n i^m L n-1

i

where is the appropriate permutation operator and phase. The normalization is

< )| )> - n . (2.82) n ' n

A correctly antisymmetrized normalized coupled state is a linear combination of

the form

$fj^(R) = y (Ljpl )Ia) [9'^^^(x) ($P ^ f^'jf^l

, (2.83) n a L n- J. PL

where the single particle CFP (| }) has been introduced. In the occupation num- ber representation the equivalent expression of Eq.(2.81) is

I (\)>0.N„ = ^m ^ Vl|0> ' (2.84)

with S , a normalized product of n-1 creation operators. Since all indivin-1

dual state indices (j'm') in S^ ^ are such that (j'm') (jm) the norm is

<0|S , afj^ af^l S J0> = 1 . (2.85)

^ ' n-1 m m n-1 ' ^

Comparing with the expression (2.82) in R space, the coupled state vector in the

occupation number representation is thus seen to be

- 38 -
11.21

S^^j|0> = I

(LjPl [afj] S^^Jp]r^]|0> (2.86)

PL

where the s/n factor corrects for the noinnalization of the usual CFP coefficients

calculated in tables with the definitions of Eq.(2.82) and Eq.(2.83).

For Bosons the same result applies. This can be most easily seen from

the fact that the norm is determined by the number of ways a particular coupling

scheme appears. In that respect the only difference between Bosons and Fermions is

the appearance of a minus sign in the contraction, i.e.) the f] factor in Eq.(2,66),

The number of permutations (and the corresponding coupling schemes) are the same

for both cases. Hence the relations (2.81) and (2.85) hold also for Boson and

consequently the result (2.86) remains true. Of course the CFP's are different for

Bosons and Fermions. In particular an arbitrary number of Bosons can occupy a given

N/ j shell in contrast to Fermions for which the CFP's vanish for n > 2j+l.

II. 5.3 - Matrix elements of Fermion- Boson systems

The field operators which will be constructed in the following chapters

are of the general invariant form (see Eq.(2.21)).

where F(x,y ; X\xX' se^) is the invariant matrix element for a given point process

of multipolarities <e and ^ in angular momentxam and isospin spaces, b'-^^^ and

^ are respectively linear combination of products of Boson and Fermion

operators of multipolarities X, x' and isospin |i, For simplicity we shall

limit here our expression to a scalar operator, for example the energy, for

which se^Oy sr = 0 and \ = X' ,

|j, = |-l'.

0

0°

. ^F(.,y;X,)[Bf^" Af*^!]'"' . (2.88)

F(x,y ; X\i) is an invariant matrix element whose calculation is given in Chapter

III (free field energy). Chapter IV (center of mass energy),^ Chapter VI (field

interactions)

We consider now an initial state |i,IT) made of many Fermions coupled

to J.T. and many Bosons coupled to J.'T.' :

|i> = fi[r.f"'ryr"V'^'^'3'"']'"'lo> .

- 39 -
11.22

where Y and X represent linear combinations of Fermion and Boson operators which

are property orthonormalized as discussed above, defining completely a configuration i in a given coupling scheme. We have a similar expression for the final

state |f) and the matrix eleAent is here,

<f|o°°|i> = [F(x,y;x^i) i^t^

<0| [wt"l [Y^'^'^' X^'^^^^ ] B

f'^' Af^^^

]

I

[w

[IT]

^ ^

f'^i^i 1 ~ f^i^i ^ [IT] ^ [00]

Xi J (2.90)

The recoupling diagram is given in figure 2.16.

VZZA

Fig. 2.16

This diagram yields,

„00|

<f|o""|i> = [ F(x,y;Xn)

xyXfj,

\ X 0

J. J.' I

T^ T^ t'

H

0

T T ^ T Ll X JUL 1 -1 11 11 J. J.' t.t;

<o|^X^'^'^|Bt^^^|x'^'^^f°°l|o>

<0l[ Y^'^""'! At^^l|Y''^'^f°°^|0> (2.91)

Thus after evaluation of the basic invariant matrix elements F, the many body problem appears as the usual evaluation of mean values of complicated products of

creation and annihilation operators according to well known technique. A few simple

examples will be worked out in the following chapters.

- 40 -
III.l

CHAPTER III

FREE FIELD DISCKETIZED EXPANSIONS AND ENERGIES

III.l - SPIN 0 FIELD

III, 1.1 - Field equations

We consider a complex field $'(r,t) of spin 0, isospin 1 and charge /c.

K, The Lagrangian is

i? = - y a, d. $' + j'"^ . (3.1)

For K = 0, f'"^ = $ . The conjugate field is

» o o °

<(r,t) = -ffr = i'"^ (3.2a)

/C

K''^(r,t) = = . (3.2b)

O $

K

We shall work with real fields $(r, t) defined as

$ = + , (3^3)

which yields, deleting the k index

1 r 2 2-1 ,^

se = - 2 I 9^ $ $ + m $ J

(3.4)

and

n = -n" = i § . (3.5)

The equations of motion (Klein-Gordon) are

2 2

- a i + m $ = 0 (3.6)

A* and the Hamiltoniari

H=:n$-iP: = :-a^$a^f-Jf: (3.7)

X 2 2 = 2

• {" ^4 ^ ^4 ^ ^x ^ \ * " ^ } • *

- 41 -
III. 2

where x = 1, 2, 3.

III. 1.2 - Multlpole expansion of the field

The plane wave solutions of the free field equations are

1 ri(p?-Et) (1) [1]^

fe) U''M^' %j <

*

kP " ^

In this expression 'n''^^ is the isospin wave function normalized according to

(n^'^ nflh = 6 . . (3.9) \'k 'k / KK.

V

• /

The creation (annihilation) operators a^^2 {a^^2 ) transform as scalars in orbiK P K p

tal space and as vectors in isospin space. They verify the commutation relations,

ra<» , a»C} - a^P-?') . (3.10)

/c p /c p —

The normalization in (3.8) is chosen to achieve the equal time commutator,

[$(r,t), n(?',t)j = i35^(?-?') . (3.11)

The factor 3 corresponds to the summation over the isospin projection. This way

each particle of a given charge k. is normalized to unity.

We introduce the multipole expansion of the field at t = 0,

'(r) = /— ) \ d p —=. ) 1 j.(pr) ^p ' r"-

J a ' •n'-

^

, <-)'

^'V ^:") . (3.12)

/,m

kP

where we use the notation.

^K] = ](^ = (-i)^ Y. (^ . (3.13) m m cm

We define multipole creation and annihilation operators by

. (^1) ,. p ,2^ (1) ^(/) ,„ ... =

J

' P Pm <3.14)

Atf^(p) = C^^P) , (3.15)

- 42 -
III.

which obey the commutation relations

{t^'^^iv), = 6 . 6,,. 6^. 1^ . (3.16) L m K m K J KK mm ^ — P

Herewith the field becomes

<?m/c

X fA<'» (p) t'" ,[^1 + (-)' Xf'll(p)

, (3.17) Lm/c m'K — m/c m'Kj'' v./

or in a coupled form both in orbital and isospin space

Mr) = /I Jp dp ^

(i)^ j^(pr) . 1

III. 1.3 - Discretization of the field

We now want to go from the plane wave basis of continuous index p to

a discrete basis with a discrete index v. To that aim we introduce a set of

orthogonal functions f (p) which obey the relations :

orthogonality jpdpf^^(p) f^^^Cp) = 6^^ , (3.19)

completeness ^ ^vZ-^^^ ^v^^^'^

^^^'V )

^

We thus can introduce discretized creation and annihilation operators

A.^'''''-' defined as

Af''^^(p) = ^

A^t^^l f^^(p) , (3.21)

V or conversely

py^^ = J

p^dp Af''^](p) f^^(p) . (3.22)

They fulfill the commutation relations

fA^^^^\ A^f^V .1 = 5 , 5^, 5... 5 , . (3.23) L V m k' V m K J /c/c mm vv \ • /

- 43 -
III. 4

Substituting the definition (3.21 ) into the multipole expansion (3.18 ) we get

for the field.

+ (->' tAf'l f['l ^itllll""!) , (3.24)

- (-)' (At'll ff) ,rll,t-i3

, (3.25)

where we have introduced the functions.

By a straigthforward calculation one may demonstrate on these discretized forms

of the fields that the field commutation relations (3.11) are indeed fulfilled,

using the result

I

= f H P'dpp'%' j/pr) j/p'r-) f^^(p) f^^(p')

= . (3.28)

r

The wave function for the spin 0 particle with quantum numbers v,

m, K is obtained from the field expansion,

^^'"hr) = <oi.Ar^^^]|o> = ^ a/ g (r) ^;[^] ,fi]

, (3.29)

which is normalized according to

i r d^r (pt'L^]*(?) ^ (pff 'V ,(?) = 6 , 6.,, 5^, 6 , . (3.30)

J vm/c 3t V m K vv mm k/c *

3

* Here we have the usual notation f — g = f g - f g

ot

111,1.4 - The discretizing unitary transformation

We choose for the discrete basis f^^(p) harmonic oscillator functions,

^ , s 3/2 ^ / 1 2 2n . _(/ + l/2), 2 2. ... f^^(p) = a ' C^^ expf- 2 a P J(ap) (a P ) , (3.31)

where a specifies the scale and where v = 0, 1, 2, 3 ... etc, with

- 44 -
III. 5

1/4

^ y!

(2v + 2/ + l)! !

-, 1/2

(3.32)

The harmonic oscillator principal quantum number is N = 2v + ^

III. 2 - THE FREE FIELD ENERGY FOR SPIN 0 PARTICLES

The Boson Hamiltonian is of the form, eq.(3.7)

J

dr^ 3fg = I

: J

d^r 9^ $ 3^ $ + 3^ $ 3^ $ +m^

j

12 1 r j3 i <^^2 p 2, p ,2,

X (pr)f^ ^ (p) 2(-) - [A^ - r ^ J ]

J ( . a^^^ - +m^)

l-l

X [A r T)^

J J, (p'r)f (p')

^2^2

(3.33)

where a factor 2 comes from the two identical contributions :AA: and :AA: . Using

(-V^+m^)j^(pr) Y^'^^?) = (p^+m^j^(pr) Yf''](?) , (3.34)

we get,

and for the matrix element, defined between single particle states as in eq.(2.23)

^ ^ xs 2

^2 ^1

K 1] K 1] [oo] 1] 1]

' ^ ^ l'<0|[W ^.LW,. ]|0> ' A ' ] fd^.^^

] Jp^dpEf^^^ (p) f^^^ (p) 6,^,^, . (3.35)

Thus, the free field energy invariant matrix element for spin 0 particle is

[-iM I

" ^ ^bIV21 = ^1^1 P ^P E fv^./P) fv^^/(P) (3.36)

45
III. 6

III. 3 - SPIN 1/2 FIELD

II I . 3 . 1 - Field equations

We consider a field \|/ for spin 1/2 and isospin 1/2 particles. Its com- ponents are denoted ^^^^^^j^) where a stands for the spinor index and k for the

isospin projection.

The Lagrangian is

with

" '•''^ Y4 • (3.38)

The conjugate field is

n = i Y, ) = i / . (3.39)

The equation of motion (Dirac equation) is.

^ = 0 = yd^ + m^ , (3.40)

and the Hamiltonian

H = : 'J' - 5P : = : ^ ^ ^ + m ^ ^ : . (3.41)

i = l,2,3

We choose the following representation for the 4x4 representation

of the Y matrices.

_/0 -iaM _/I 0

Yi=x,y,z -

0 j \o -I

1

with the usual definition ,

cO l\ /O -i\ /I 0 \ ^ /I 0

(3.42)

a

X

=

(? J) = G o) = (o -l) ^ = (o ?) •

(3 ^3) Thus we have the anticommutation relations,

YY+YY=26 ; y^ = + I . (3.44)

- 46 -
III.

We work with the metric : /l 0 0

0 1 0

0 0

•1 V 0 0

Hence the four vector scalar product y.x shall be denoted by

Y X =

i = 1,2,3

or

Y X

(3.45)

i = l,2,3

with = it, x^ = t and Yq ~ " ^ Y4 • Likewise the time derivative is

(3.46)

(3.47)

(3.48)

Let us recall here a few useful relations for the 4x4 matrices

(i» j»k = x,y,2 , cyclically) : ^i^j - 5j5i = 2iai^(e. .j^) , (3.49)

^iSj+^A = 2 6,j ' ^3.50)

o, = - i YjY,^ • (3.51)

The matrices

S.1

verify

Z Y,Yj) = |(o J ^'-'^^

(\'Sjl= ^ \ ^iik • (3.53)

Hence our representation of spin is

« US ;) •

We shall also need the matrix

Y5 = ^VyY2Y4 = (°i 0^) (3.55)

which verifies the anticommutators

Y5YV + VyY5 = 0 . (3.56)

We define now the spherical tensor representation of these various matrices, which will be needed to carry out the calculations in angular momentum

Hilbert space,

- 47 -
III.

CT^^^ = - i CT = - Y Y , (3.57)

.^-i^=7=(VW = Tt^-Vz-'^W • (3.59)

We can check from the definitions,

[1] _ l-m [1] - ^ ^-1 » (3.60)

£?m^^' -n^0+ " °

^"'^ ^^'^^P*' ' (3-61)

[af^^af^n =-2 . L^o ' ^o J_j_

ni Likewise the spherical ^ are given by

(3.62)

m

and

^Yj^^''> Yj![^''^_|_ ~ ^ except (3.64)

III. 3. 2 - The plane wave solutions

The free field plane wave expansion of the solutions of the Dirac

equation (3.40) is,

I /- r j3 /ET r (1/2 1/2) [1/2],. [1/2] ^^.v

^a^'^'^^ = fe) PP/E lbs.? J(P) ^

J expaCpr-Et))

K, S

s K p s a '/c J

ri/2 1/21 where a^ ^

denotes the annihilation operator for a particle of spin 1/2,

spin projection s, isospin 1/2, isospin projection k, described by a plane wave

of linear momentiam p corresponding to the energy

= p^ + . (3.67)

We have also introduced for the same energy the creation operator for an anti-

—

particle of same quantvim numbers s, k, p.

- 48 -
~[l/2 1/2] ^ ^(1/2 1/2)+

S K P S K P

III. 9

(3.68)

2S+2T

Note that the phase (-) from Eq, ( 2.15) is here equal to +1 . The isospin

1/2 functions are normalized according to

K 'K. KK.

The spinors are solutions of the equations of motion ( 3.40),

(3.69)

(iy^p^ + m) 'Ul'^^^h?) = 0

(-i Yj^Pj^+ m) U^'^^^hp) = 0 .

In a spherical representation they are of the form.

(3.70)

(3.71)

^['/'^(P)

/E +m

/ 2m

[1/2]

V.[1/2]

E + m

—* —

g . P

. s / E +Trm I E+m ^ ^s

[1/2]

Xs

[1/2]

(3.72)

(3.73)

The explicit form of the spin 1/2 vectors x^^^^^ representation are given

^~(1 /2) in equation (2.10). The conjugate functions defined by the normalization

(2,11) are given in equation (2,12).

With the chosen normalization the spinors obey the following relations

(the a indices here denote the spinor indices)

i) Orthogonality

I

a

+

( ) ^[1/2] = E

s a s a m ss (3.74)

- 49 -
In order to demonstrate this result we note that

III. 10

^ Ll 1/2 1/2 J Ll/2 0 1/2-1

X [X/2|c|l/2][p-Ill x'^'^'li''"' (3.75)

from the coupling diagram of figure 3.1, which makes use of the graphical representation of completeness in spin space according to figure 2.7.

Figure 3.1

Furthermore

C2 - [l/2]. + ,-» -» [1/21. , 1 rr-»~-.rl/2] ,[1/2] To]

(<^.P Xy. ) (o.v Xg ) = figs' Y ^

p2

5^s. , (3.76)

hence

y ^^^/^^'(P) ^f^/^J(p) = E+mp [1/2]+ [1/2] .ij: [l/2]N+/iJ: [l/2]v-|

I %ci %' a 2m L'^s' ^s ^l.E+m'^s' j^E+m^s J J

2 ^

fl +^-t) 5 . = ^ 6 , . (3.77)

V (E+m)^>' TM

- 50 -
III. 11

Likewise

a

a = l,2a = 3,4

Qr = l,2 a = 3,4

a = l,2 a = 3,4

i_j o Of o Ct a

ii) Completeness

s

= 6 X f J "

o' f (3.83)

ap L-1 for a = 3, 4

s

These two relations (3,84) and (3.85) make use of,

(_

- -» E -CT.PV

(3.86)

CT.p E /

It follows that the normalized field \|/ Eq.(3.66) and its conjugate

(3.39) fulfill the equal time anti commutation relations

fv|/ (r), K rr')l = 2i 6 „ 6^ (?-?') , (3.87)

where the factor 2 comes from the summation over the isospin components.

- 51 -
III. 3.3 - Multipole expansion and discretization

III. 12

The multipole expansion of the field at time t=0 is,

, /2 p ,3 V r ,.s<C r (1/2 1/2) [1/2], , . , .

+ (./ bf^/a 1/2] ~(i/2^ ) -K] (p,) ~(i/2).

' s /CP s a ^ '^m m -^^ ' k J

'

or in a coupled form, separating large and small components.

" J/1 (i) 2 J

/ j = / + l/2

[JO], /eT^ U'^/^I ?I'll'^'j/pr)'

[1/2 1/2] ^[^] [1/2]

p Ti

+ (-)' n-[l/2 1/2] ^[.] ~[l/2]nfj°]/-^ f^^'^'^ ?f'^]"^j,(pr)

P ^ J (^^^[l/2].[.]^fj].^(p^^

where X = / + lfor j=/' + l/2 and X = <? - 1 f or j = / - 1/2

[1/2]

(3.88)

[0]

[0]

(3.89)

The angular algebra which is used to obtain the small components is

given in figure 3.2 neglecting isospin.

X

r

X X

ri/2|a|l/2]

Figure 3 ,

- 52 -
111,13

We define again multipole creation and annihilation operators as in

eq.(3.14)

which obey the anticommutation relations

and which upon discretization are substituted with (cf . eq. (3.21))

- A 6(P-P')

ij mm K.K I

[j 1/2]

(3.91)

(3.92)

omitting the m and k indices. We have similar expressions for the antipartlcle

operators.

Furthermore we define the tensorial multipoles

1/2 /m (r) = [X

[1/2] 0K]Jj]

m

?lj] (-) = r-[i/2] -[^],[j]

n/2^m'^^'' J„ , m

(3.93)

(3.94)

and the functions,

= /! Ip'^p/¥^.(p> j.^P'^) '

(3.95)

(3.96)

which are normalized such that.

(3.97)

Subs'-.ituting these definitions in Eq.(3.89), we finally obtain the discretized

field expansion.

(r) . — [ (x) JO

[j 1/2] [1/2]^^°]/ ^1/2 V^'^^

[0]

+ (-)" pty/2] ^tl/2]^C ]

j

,(^) v^,^(r)

;[j]

1/2 ^' (^) u^/r)

[0]

(3.98)

53
III. 14

where the C's denote the antiparticle field. In each term the first coupling to

[0] corresponds to total isospin and the second one to total angular momentum.

It may be verified on this discretized form, together with the similar expression for

' the conjugate field, and using the relation (3. 97) that the anticommutation relations are fulfilled.

HI. 4 - THE FREE FIELD ENERGY FOR SPIN 1/2 PARTICLES

The Fermion Hamiltonian is of the form (omitting the antiparticle part)

J

d^r ji^ = J

d"^r \|/ y.V^ + mv|/\|/

i J 9 ~[j 1/2]/

y^e^

:[i/2]

{/I

[00]

^ ~ ^ _m ' CT.P

B

'ri^-i

Y.fjJ ,

(r) u . (r)

[j l/2t ^'^ ^2 V2

V2^2

[00]

.[1/2]

-^1/2 .2^^^ V2'^2^'^'

1 y

1 2 L 2

j r r dr m(u . (r)u (r) - v (r)v (r))6. . 6

J ^11 ^2 1 '"i l l ^2 11 12 '^i '^J

- (i)

1/2 /'2 j

1/2 j

L 1 1 0 J

r'^2 [1/2 I g I 1/21

1/2 j

1/2 ^2 j

L 1 1 OJ

r v^^^i-fi]

L v/^x.^ I \4 ^1

fB" y^] 3[j 1/2]. [00]

L vj_/^ V2^2 -I (3.99)

In order to get the ^/^^^2 first contribution we have used the fact

that 5x,-|^X2 ^ given j = ^2 determines uniquely the / value. The matrix

elements of the first term are given by.

2 2m

r r^dr m(u . (r)u, , (r) - v . ^ (r)v (r)) =

f

p dp — f (p)f (p)

J ^1^1 ^2^1 ^I'^lh ^2^lH J ^ "Vl "11

(3 . 100) while in the second term.

- 54 -
]

=

when j = X,2 + l/2, ~ ^2 ~ ^2

V

V2^2^2

8 , "2 + ^

8r r •V2^2^2 ^2 H^2

when j = \„ - 1/2, = = X,, - 1. Likewise

^2^-

L v^^^X^ 'P I V2^2

when j = \^ + l/2, = = ^2 ^'^^

= - i/T^TT

when j = X - 1/2, = ^2 ^2

"

Utilizing the relations (2.35) we get

ar

" r

3 , "2 + ^

ar r V2<?2^2'

ar r ^

^2

ar r

4

^2^2^ J E v^^^ V2^2 ^1^2

Furthermore since

- 1/2 j r

1/2 j = «

_ 1 1 0 .

1 ^ ^= jl ifi? = X + l

j 1 if £ = X - \

III. 15

(3.101)

(3.102)

(3.103)

(3.104)

(3.105)

(3.106)

(3.107)

(3.108)

the second term of eq.(3.99) has the same form for both cases, i.e.

and we finally get.

(3.109)

e = X.±l,

- 55 -
III. 16 ^ 2

^ 4

1 p , P

Its matrix element between single particle states of the form (2.81) is

and its invariant matrix element (cf. the defining equation (2.23))

III. 5 - SPIN 1 FIELD

III. 5.1 - Field equations

We consider now the p and cj meson fields, the quantum numbers of which

are

J- - i ; T = 1 ; even ^ parity for the p meson ^

l"^ = 1 ; T = 0 ; odd <3 parity for the w meson .

In order for the equations of motion to yield back the Maxwell equations at zero mass, we shall work within the 8 component theory framework of

Hayward '''"^ In this framework the cj and p free field solutions are 4 component

vectors, 3 components transforming like those of a spin 1 and the fourth like

a spin 0,

- 56 -
III. 17

w =

P _

/

'2

n \

P3

(3.113)

(3.114)

III. 5. la - The co field Hamiltonian

For the neutral field we have*

0) = CO (3.115)

In general

+ - 1 -»

to = (o), co^, 0, 0) , to = T] (to, -to^, 0, 0)-[\ , — + -

to = to Y4 = T] (to, to^, 0, 0)ti

In the Lorentz gauge t] = I . However in the Feynman gauge defined as

^ -* ^ + d A , 2 2

where (Q+m)A=0, (P+m)aA=0 , — -1 + - + - + x|/ = Ti ^ Y4 ^ such that = Y4 > = ^Ya , — +

^Time ~ ~ ^T^4 necessary for X invariant in the transformation .

Here the indices se, 3", Time denote the longitudinal, transversal and time- like

solutions respectively.

- 57 -
III. 18

The free Lagrangian for the w field is

1 r <—:r T> 1 2 2-)

2 |j. 'v V 2 J

The Y are 8x8 matrices defined as

y = - i 3 a

(3.116)

(3.117)

• • •

-1 7^

1 !

'

1 . /• • • •

^3

1 .

\

^4

1 . .-1

.-1

• a

-1

[11

Y5

-1

-1

,-1

, .-1

,-1

(3.118)

These definitions yield.

Y 9

94 .

.

. 8^ .9^

-^1

-82

-93

83 . -8^

•

"^l"^2"^3 -94 • • •

-83 . 8^ «

-94 •

• -94

. . -94

^1

82

^3

• • • ^1 ^2 ^3 ^4

(3.119)

and (V =V, ; W = W, W^) (V4 = i ; W4 = i W^)

/ 84V - grad + rot W ^

79 r

div V - 84V^

- rot V + grad - 84W

\

div W + 84W^

/

(3.120)

- 58 -
III. 19

The resulting canonical momentum is then

/ i d^Z - i ^. W4 \

dse i

) =

\

i V. to + i S^w^

0

0

/

(3.121)

We choose now the Lorentz gauge which eliminates the component,

= 0 Lorentz gauge ,

i.e.^ the fourth component of the field is given by

(3.122)

4 4 to

and 7i=7x, n = K ,

Let us define the "electric" and "magnetic" fields.

E =

—> _^ i (9^^ - grad w^) = - n

B = rot o)

(3.123)

(3.124)

(3.125)

which yields for the Lagrangian the well-known form (taking into account the

Lorentz condition).

= -

-J I

(9^00) - 29^0) grad + (grad to^) + (rot w) + m (oj - uj^)

|

= -

-J j^-E + B + m (w -u)^)} . (3.126)

Likewise the Hamiltonian is

2 . , . .2

= :ttw - a': = 2*{"(34") + ^8'^^*' "4) + + m (u - w^) J

: . (3.127)

Hence for transverse solutions for which

0 (3.128)

we get the usual form

1 , „2 , 2 ^2.

SifL. = -7 (E +B +m w ) (3.129)

while for the longitudinal part, for which rot 0, there is no such

simple counterpart.

- 59
III. 20

111.5.1b - The P field Hamiltonian

The charged field is complex and we denote it with a prime index p'

tcmake the distinction with the real field p that we shall construct later on.

The free Lagrangian is

I P y a y d p - m pp (3.130)

where the summation is carried out over the three charge states with isospin projection K . This summation and the index k are deleted hereafter. We have

^ 0

ap-

= 1

+ grad

pI'^ \

div p - a^p^

\P

. I _ = 1

0

0

/V' - gi^a^ pI\

div p' + a^p^

0

0

/

(3.131)

(3.132)

(3.133)

We again make use of the Lorentz gauge = 0

div p'"*" - a^p^"*" = div ^' + a^p^ = 0 ,

and we introduce the fields

E' = - i(a^p' - grad p^)

B ' = rot p' ,

which finally yields for the Lagrangian

B = rot p

if = {^a^p'"*" a^7' - 94?'"*" grad p^ + grad p'^ a^p' - grad p'^ grad p^

+ rot p'"*" rot p' + m^(p''^' - p]^ pI)^

- I^-E E + B B + m (p p - p^ p^) j

- 60 -

(3.134)

E'"^ = - i(a^;''*" + grad pI^) , (3.135)

(3.136)

(3.137)
III. 21

The Hamiltonian is thus.

^ = :k' y + Tx' p' - ^: . (3.138)

+ + + +

Note that for the complex boson field :p p: = p p ; :pp:=pp.We have the

results

,

Ti'p' = - a^;*"^ dj' - grad p^^ dj' , (3.139)

n'y = - a^;' a^;-^ + grad p' a^^'"^ ; (3.140)

3^ = V'"^ ^4?' - g'^ad p^"^ grad p^

+ rot ^'"^ rot + J-Cp'^ - p'^ pp] : . (3.141)

We get again for the transverse solutions (p^ = 0)

X = E'"^' + B-'^B' + p-'^p . (3.142)

For the longitudinal ones we must use the expression (3.141) with rot ^' = 0.

For convenience we shall however work now with real fields,

p = p-'^+p' = (p'^^+p' ; p'^+pI ; 0 ; 0) , (3.143) - = + = Cp'^ + 7 ;-p;"^-p' ; 0 ; 0) . (3.144)

4

Hence _ ^

P = P '

The Lorentz condition is thus

(3.145)

div p = a^Cp^"*" - p^) = - d^p^ , (3.146)

where

P4 = i Po ' (3.147)

P4 = - i Po " ^4 ' (3.148)

which are the relations used for cj^. We can now define ^P, tx and ^ for the charged

real field p as in 111.5.1a for the neutral field, equations (3.126), (3.121)

and (3.127).

111.5.1c - Equations of motion and the Maxwell equations

For the real co and p fields the Klein Gordon equations are,

-» -» 2 dydyi>i-mLi = 0 , (3.149)

-» -» 2

9^Y^ ^v^v P - m P = 0 . (3.150)

- 61 -
III. 22

From the result of eq. (3.120) we get successively

3 Y to V V

- B

\ o/

I a^w - grad 0)^^

0

\

rot to 0

i E

/

0

\ 0

- rot S

- i div E

i rot E - i 3^5

= m

/to

0

\ o/

(3.151)

(3.152)

- div B

In the limit of zero mass these expressions yield back the Maxwell equations.

The equations (3.151) and (3.152) hold also for the p- field.

III. 5. Id - Quantization

After quantization the conjugate fields to and n or p and n obey the

the equal time commutators

fto (x), K ,(x')^ = 6 , 5(x-x') ,

fp (x), n , ,(x')^ =5,5, 6(x-x') ,

where a, a' are the spinor indices.

^4 = ^^4

^4 = ^4

(3.153)

(3.154)

III. 5. 2 - Representation of the Lorentz transformation and boost

Consider the Lorentz transform from a reference system X to a system X'

which is moving along Ox relatively to X with the velocity v = p . The transformation induces the change of coordinates

X'

0

1

4 1 1

v = p

X

Xj^ = x^ cose - x^ sinG = Y - i PYX4

^2 '*~ ^2 ^3 ^^3

x^ = x^ sine + x^ cos

or conversely

e = i Pyx{ + Y

x^ = Xj^ cose + x^ sine

x^ = - -x.^ sine + x^ cose (3.155)

- 62 -
III. 23

where

Y = / ^ , . (3.156)

sine = ipY > cose = y . (3.157)

Let us define the infinitesimal Lorentz transformation on a four

vector (x) as

A'(x') = (l+^i y EflS ) A (x) , (3.158)

with the set of 4x4 matrices.

. . 1 . . .-1

23 -

I . i . . I "31 - 1-i .' .' ; ^12 = I • •

1 s., = I :

• • •

1 s - ' ^

••••

. (3.159)

Comparing the definition (3.155) with the infinitesimal transformation (3.158)

where y ~ 1, we get

= sine^^ " ^Pl4 • (3.160)

—

-» Q For a finite Lorentz transformation, setting B. . , = — , we obtain

' ^infinitesimal n

'

A'(x') = (l-^t) A (x) = e'"^ A (x) . (3.161)

\i V n V V

Instead of the transformation from the system X to the system X', we

low tl

a velocity p ,

consider now the infinitesimal boost of a 4-vector A (x) in the system X to

V

a;(x) - (I - il

j;

.„pSj A^W . (3.162)

ap

and the finite boost is given by

A;(x) = (l+f?) A^(x)

= (1 - Si, + ? sinhQ + S,, coshH) A (x) = e " A (x) . (3.163)

- 63 -
III. 24

For example a particle of spin 1 at rest is represented by

A,

(«)(x) = JV e (3.164)

where the are the spin orientation of the cartesian components. Upon boosting

along Ox with velocity p, we transform - i mt into i px in the exponent since.

. . .-1

;!i.h(::::)-"^(:?o:)

1 . . . N

coshfi

0

n 0

im imy

p

0

0

iE

and noting that (3.165)

ipx = Kp^x^ + P2X2 + p^x^ + p^x^) = i(0 + 0 + 0+ (im) (it)) ,

iJP(px) = i(mpYX]^+0 + 0+ (imy) (it)) = i(?.5?-Et) ,

- - ^ , T,2 2 2 2,, 2, 2

since E = my, p = m p y '^^^ verifies that E-p =m y {1 - ^ ) = va.

the cartesian boost along Ox on a cartesian four vector yields,

coshQ . . -i sinhn \ /A,

1 .

. 1

JV e

- i mt I ^2

= JV e

1 px

^2

i sinhn coshQ \i Aj^Py'

(3.166)

(3.167)

Thus

(3.168)

We can also verify that the Lorentz gauge 8 A , see Eq. (3.122), is

[1 (-1 conserved either in a Lorentz transformation, where both a and A are transformed or in a Lorentz boost, where only A changes. In the latter case, for example

the expression

d e

- i mt

= 0 (3.169)

entails for the boosted vector along Ox

a e

1 px

/

"1

A,

AiY

2

XiA^PY

= i p Aj^Y i E ^iPy = 0 (3.170)

64
III. 25

III. 5.3 - The plane wave solutions

The vector part of the real fields after quantization is of the form,

p J

[a^ e"- M J^ , (3.172)

^-i(pr-Et) ~[1]^[1]

4-e - --'[ai^^^i^t^J^'J] ] . (3.173)

for isospin 1 and 0 respectively. N is a normalization constant. The fourth

time- like scalar component is defined by the Lorentz condition (3.134)

3/2 . -^2 ^ ^

/- V ^ V / 1 \ r j3 /¥ 1 r i (p r- Et) , m] , -»rl] [1] , [001

P4(r,t) = - j a^V.p(r,t) = (2^) j

d p E b

'[^^''^iP^ \^

,-i(pr-Et) ~[ll],.;.[l]~rl]

P

and a similar expression for cj^(r,t). We have here introduced the notation

, pvt^?) ./^P?:^' . (3.174)

(P'^'?'^'l'°'= ip^ . (3.175)

We have also used the result (2. 54) i. e. div [. . e

^ . . ] '^ ^= [. .V '•''].]

. The conjugate field is obtained from the equations (3.132) and (3.135), where E and E

denote of course respectively the electric field and the energy, namely for isospin 1 -f = K(?,t) = i(a,r-fv,J) = ,(lsf"^,\/^^^^

r -i(pr-Et)/ ~[ii] ri]~[i] roo]1 V p E P ^

- e^^P^-^'^>(C.C.)}

. f^\'^ ? /nE :2 r -i(p7- Et) / ~[11] [l]~[l]/f°°^

P

L

Here we have made use of grad [...]f°^= t [e^^Vf^^J^^^ [...]'^°^ cf. Eq.(2.50).

The summation over L contains only L = 0, 2 since p X p = 0 .

- 65 -
III. 26

X fi

3-i(?r-Et) . i ^+i(?r-Et)jJll]^[l]Jl]^[l]^f°0].

^ ^ (3.177)

In this expression we have utilized the relation (cf. Eq.(2.48))

JO]

7XB /T l[vtl]B[l]er^]J ' , (3.178)

and the reordering of a^'''^ and p^''"-' introduces an overall sign.

P

The expressions of the conjugate fields of isospin 0 are similar, with

rl 1 a 1 factor missing and suppression of the isospin function .

We compute now the commutation relations and the normalization.

The equal time commutator is

{^(^.t). n(r..t)} = i(J./jJaVp-/^

X r3i[(?r-p'r')-(E-E')]t . Jll ] Jl ] ^

[1 ]

^ [00]

^ f fai^^e ]

j [00]

P ^ P'

-ill [[;i"iet^i,ti',^°i[?tii?f'i,!^i]"")3 -c.c.}

^ L

^

3 .3„ _2

= i(2t + l) (2s + l) 5(?-?') . (3.179)

We have s =» 1 and t*l for the p-field, and S 1, t=0 for the arfield.

Furthermore here we have set the normalization constant to be

2 2

N » —P-T = ? 9

. (3.180)

3E'^-P^ 2E^+m

The calculation of the commutator made use of the result,

f[ar'e[^],f^J]t°°], rai^l]e[^];irl]][0°U = 6(?-?') , (3.181)

(- p p' J.

which follows from

rJU]^ ~ril] ^.5.5. 5(?-p') . (3.182)

'^rap T'a'p'-'-

and from the result.

- 66 -
III. 27

fr [11] [1] [l]i[00] rr~[ll] [LO] [Lln^^S j^[a_;^ Jti^ J, |^[a_|^^ ^e^ ^ti^ ']' ^ [p ^ J]^

J i

" Y

(3 183^ from the recoupling diagram of figure 3.3

a

e

a

e

•n

-»P 2

Figure 3.3.

As an exercice we can check that the various equations of motion are fulfilled by the free field solutions we have constructed. For simplification

of the notation we set

' = (i/^' P'p/^ ' (3.186)

and we consider the fields of isospin 0. We get successively

- 67 -
III. 28

-* 2 ^

i) 3^. E - rot B = m CO . (3.185)

E-rot stj;e-^<?-">(E^Se][°l-i[S?,[°)-2[fp-]fll[ep-i[^l]'°')4. ...}

J-j j-i u

= st e-i(P^-^'^)(E2-p2)[Se][°] = , (3.186)

where we have used the following results

L = l [ptl]-[l]][l]

= 0 , (3.187)

L = 2

L.O [p-[llp-[lllf°l = 4 . (3.189)

1

ii) i div E = - m cj^ . (3.190)

i div E-^= S t E (e-i(P^-^'^>([I(-i^)]i°J - ^

[Xp]f°][ -ip-'pl[°J)+ ...}

= -iS 1 i(e-^^P^-^^\E^-p2)[apl[°U ...} =-m\ . (3.191)

iii) rot E = - a^B . (3.192)

rotE= S /I IE [e-^^P^-^*^^[aer][°b + ...}=- a,B . (3.193)

iv) div B = 0 . (3.194)

This result comes trivially from the fact that B is a rotational

field.

and

div B = S /T 1 (e-^^P^-^^^af^lp-^r^Vf'^]f°] + ...} . 0.195)

- 68 -
III. 29

111.5,4 - Multipole expansion

— The multipole expansion of the field vector potential p is

7(?,t) = /I

J

1^

I a/ 7 j/pr) ([^Ki^r^i^co]

— p

nJ^P/2f^I^'^ J [[[a^ ^G;J(pr)J +(-)...],

(3.197)

with the notation,

Gl'hpr) = fe[^]?[^]][^J j^(pr) . (3.198)

The scalar potential is given by the Lorentz condition (3,134),

P:(?.t) = - .4 =,J ^,^t<r.t) - /|jd% ^

(-1)' t J/pr)

x([a[^^M^]7i[^3]f°°][pf^]?[^]][°] - (-)^C.C.} . (3.199)

Thus,

n = i(a,/.V p\, - iyfJ d^yf E 1 2 ^ (-i)^ 7 j^(pr)

X

([ir^^lef^l^tl]] [00] -[/]^[.']

J [0] ^ ^ ref^Vr^l ] [0]

fa Ivt^ ]

j [^0]

E J

x{[[S'"lf"l^''llTM Gf^'(p?)]t°l - (-/CC.) . (3.200)

The V p^ term in k has been calculated with the diagram of figure 3.4 .

- 69 -
III. 30

eV

aV

P

r

P aJ/

Figure 3.4

The multipole expansion of the magnetic part of the field is likewise

B(r,t) = rot -p+(r,t) = J /f t'p [ (i)'(-l)'

from the diagram of figure 3.5.

Figure 3.5

- 70 -
III. 31

Now we can check the commutator {p, n) on these multipole expansion

forms. We immediately get a factor 6 . and we have to evaluate the commutator,

A.*

= 6jj, 6,,, 6^?-?') k

f'l'ljlOl J/pr)J/pr') , (3.202)

from the diagram of figure 3.6.

^777?

Figure 3.6

Note that after the double recoupling box the space and Isospin parts are

completely disentangled. Furthermore we are making use in figure 3.6 of a simplified graphical representation to simultaneously represent the couplings of

a'-'''^-' both in angular momentum space (to ^^^^) and in isospin space (to q'-''"-').

This notation will often be used in other figures.

- 71 -
III. 32

We thus get, for the equal time commutator

2

5 {7(r,t),n(r ,t)] = X - dpdpy2;^E 1 ^ {^(x) J^'^l+IJ

^ J

^//•^JJ-r^^''' r'[^J][°]j^(pr)j^(pr') 6^?-?') + C.C.}

= i 9 6^(r-r') = i(2s+l) (2t+l)5^(?-?') , (3.203)

since we have

N ^ J = 1 , (3.204)

J=/ + l ^

if we set the normalization constant equal to the value given in Eq. (3.180). For

the 0) field the result is similar with t = 0.

III. 5. 5 - Multlpolarities and discretization

In analogy to the usual electromagnetic field expansion, we want now

to define electric, magnetic and longitudinal multipoles such that each is a solution of the equation of motion with given total angular momentum and given

parity. We consider here the w field. Introduction of the isospin for the p field

is readily made.

The magnetic, electric and longitudinal multipoles are given by the

unitary transformation,

A^Cp) - Jdf (.tH ^m^m _ (3.205)

a|/1(p) = Jdf [a^^l (i^3±I-[J-l] . ^

jU+llj-jr-Jl

, (3.206)

Afjl(p) = Jdp[ai^] (^pf^-lU^^ff^-^^])]C^l . (3.207)

p J J

—> —

With these definitions the field u)(r, t)

Za.t) - /I I

(o''J([lafllf"ll''l g[J1(p?)]'°' + (-)'c.C.) ,

^ (3.208) can be expressed in term of its multipoles

- 72 -
III. 33

J

where the normalization of the multipoles is chosen such that each multipole

field by itself obeys a commutation relation with weight 1 for isospin 0 and

weight 3 for isospin 1,

Utilizing the formulas (2.61) and (2.63), it can be checked that

these different parts obey

div u)„ = 0 , hence oj =0 (Lorentz) , (3.210) M

div w„ = 0 , hence w =0 (Lorentz), (3.211)

rot tJg, = 0 > (3.212)

and

div

(3.213)

^4 %

Hence

(3.214)

The condition stated above for the normalization thus yields.

= 1 , = 1 , = 4 . (3.215)

m

We now obtain the canonical conjugate fields from the relations (3.124)

and (3.125). Thus for isospin 0 the electric and the magnetic fields of the magnetic

multipoles are respectively

7t„ = - E M M

(3.216)

(3.217)

- 73 -
III. 34

while for the electric and magnetic field of the electric multipoles we get

- (-)"^C.C.] (3.218)

J

Finally the longitudinal part is,

= 0 . (3.220)

It can be checked directly on these forms through a lengthy but simple

calculation that these fields with the chosen normalization do fulfill the equations of motion and the commutation relations.

Finally all these fields are discretized by using the substitution,

k^^hv) = I

A^-^] f^^(p) . (3.221)

V

The various multipoles of the discretized form of w(r, t) are thus,

Z^Cr.ty -l 11^7 [lAlJhl'] (?) ] V. W ^ (-)^C.C.} , (3.222)

V J (3.223)

V J

ir,t) . I I

i'"-' J ([A].^j tt^l, t°l (r) + (.)^«C.C. )

(3.225)

V J

and the conjugate fields.

V ^ roi

^ ^ (3.227)

- 74 -
III. 35

V J (3.228)

JO*i'v^rjl""<..^(-)'-'c-C.) , (3.229)

V J (3.230) - 0 . (3.231)

we have introduced the vector spherical harmonics and the functions :

^^v'j.W = /f I

P'''!' /F =fvJ<P) Jx(P'> . 0.233)

V J \

(r) - /I Jp'dp/^pf^^(p) j^(pr) , (3.234)

= Ajp'^P/lf l^.(P> Ja(P^)

= /FJP'^P/J ^^J^P) ^J^P'^) • (3.236)

It can be checked again that these discretized fields fulfill indeed the commu- tation relation

^

ju^(r), (?•)]_ = i 3 63(?-r'), k ^ M, SB, (3.237)

K where we made use of the result

2

I VfJ'jjWxfj/r') +^(wtjj.i(r)<„.i(r') +<j_,.,(r)afj,.l(r'))

J r(3.238)

—

For the charged field p , we have similar expressions where the isospin

functions v^'^ must be coupled with the charged operators a'-'^'''^, and the usual

1 factor introduced. This yields a factor 9 in the commutator instead of 3.

- 75 -
III. 36

III. 6 - THE FREE FIELD ENERGY FOR SPIN 1 PARTICLES

The free field energy is given by equation (3.127) or more simply for

the transverse case by equation (3.129). Thus the contribution of the magnetic

multipoles is in the case of the t^r field

[01

vv

= n?

( J

p'«iP E £,,(P) £^,,(P)) [A^'l A^;i/°' , (3.239)

vv ' J

with a similar expression for the electric multipoles, since there the expressions

for E and 15 are just interchanged. For the longitudinal part we start from expression (3.127), and we get again a form similar to Eq. (3.239) with the longitudinal

A [J]

operators •

For the p- fields the energy operators are identical with the isospin

coupling added, namely

vv' J

The invariant matrix elements are easily obtained from these expressions as in

the case of the pion field, see Eq.(3.35) and Eq.(3.36).

- 76 -
IV. 1

CHAPTER IV

CENTER OF MASS

IV. 1 - THE CENTER OF MASS PSEUDO-HAMILTONIAN

According to the discussion of Section 1.4 our treatment of the cen- ter of mass motion requires the calculation of the invariant matrix elements of

the pseudo-Hamiltonian

= c(P^+ n^^h , (4.1)-

which is added to the Hamiltonian of the system. The square of the center of

^ 2

mass momentum operator P will require the evaluation of one and two-body ma- trix elements (viz. p? and p.p.). On the other hand the square of the center of

mass coordinate R is a sum of many-body operators which we will transform into products of one-and two-body operators as discussed in Section 1.4 , Eqs.

(1.48) and (1.49).

IV. 2 - THE CENTER OF MASS MOMENTUM

For a system made of Bosons and Fermions the C.M. momentum ? is

given in terms of the field operators (tp and \|/ respectively) by

2 r r ,3 +/ N . a -2 , . p^3 ,+ , ,-»2

1 J

,3 ,3

+

[ J

d\.dV[vl;+(x^)p^vj/(x.)].[/(Xj) Pj v|/(xj)]

+

[ J

d\.dV[9'^(xpi p^cp(xp].[/(x.) p. Mx.)] . (4.2)

In that expression note that cp"*" and cp represent respectively the creation and

the annihilation part of the real Boson fields, Eqs. (3.24, 3.25).

- 77 -
IV. 2

^ 2

i - The quadratic terms

p^^

In order to evaluate the one -body invariant matrix elements of the

quadratic terms (diagonal in the total angular momentum of the particle) we

introduce the number operator N(p) in the p representation

J2

" +

d p a_^ a

P P

For the spin 0 Boson field it is

»(p) - [ f^/p) f^,,(p) i? [A/'llAlflljI^l . (4.3)

vvV

_,2 Thus in terms of fields the single particle momentum operator p is

?^ - I

[v.|?^vV,rtt^l|,[M,(lt'iJ,Kil,[oo] _

VV f

where the invariant matrix element is

rv/|p^|vV] = ? Jp^dp f^^(p) p2 f^,/p) . (4.5)

For the spin 1/2 Fermion field the number operator is (on similar li- nes as Section III. 4)

N(P) = i

I I? [itJ 1/2] 3[J^l/2]jtOO]^^,2^^, ^^2^^

vv

'

- /ilSifU j^(p., j^(p.., f,,(p).,.,(p')}

K — ^ J

vv Vj

which yields

vvVj

- 78 -
IV. 3

with

[v/j|p^|vVj] = j

[v^\p^ \v'n , (4.7)

in terms of the matrix element of Eq.(4.5).

For spin 1 Bosons with isospin t and multipolarity k we obtain along

the same line as in Eq. (3.239) for the number operator

N

K

vJ

and

-^2 V r T |-»2 I ,^ ir~[tl| [tK rT[-Jt] .[-Jtl fOO]

vv'

where for all multipolarities

[vJk|p^ |v'Jk] = [vJjp^ |v'JJ , (4.9)

of Eq. (4.5).

The radial integrals entering expression (4.5) are readily evaluated

when using the harmonic oscillator basis (3.31) with parameter a = (.m tj) ^

J P% f^^(p) f^.^(p)

C C

K C ^ 2 %',v-l C^. , 2 %',v+lj

= - ^ /v(v + / + l/2) 5^,^_^ + / (H-1) (v + /'+3/2) 6^, . (4.10)

The coefficients C^^ are defined in Eq.(3.32)

ii - The bi-linear terms P^.Pj

We evaluate now the invariant two- body matrix elements by expressing

the scalar product Pj^-Pj terms of a product of invariant operators

- 79 -
IV. 4

We consider first the single particle linear momentum operator p ex- pressed in terms of the field operators.

For the spin 0 Boson case

where

as shown in figure 4. 1 .

(4.12)

(4.13)

A —

P777]

1 . 0

10

i

.00

[XeA] [00]

Figure 4.1

- 80 -
IV. 5

P =

For the spin 1/2 Fermion case

y

~[l/2], [1/2] ,^[1], irB^-^^^^^^et^lB''^^^^^^ f°°]

where

t^l^l^llP^ ^ lv2^2j2J = ^ ^1 ^2 ^ I

P ^^./P) %.2^P)

<f„ 1/2 \„ 1/2 X,, j,

(4.15)

The geometry is shown on figure 4.2. The coefficients are defined in Eq.

(2.37).

Figure 4.2

We are making use on this diagram of the same simplified graphical representation as in figure 3.6, to represent the coupling of the ti's to the isospin

part of the B's.

- 81 -
IV. 6

For the spin 1 Boson field with isospin t we get for the different

multipolarities k

[00]

(4.16)

where for the magnetic multipoles

1 J,-

1 J2 J2

L 0 1 1

l«j J I fp^^P ^

J ^P>P^ J

(P> ' ("^-17)

"^12 '-' 11 2 2

according to the diagram of figure 4.3

0

0

I X

—f:

:2I

7

[AeA] [0]

[J9I

Figure 4.3

- 82 -
IV. 7

For the electric multipoles we obtain likewise

2

J +1-J

[v^J^<f |p f^^|v2J2<^] = (-i) ^ ^

^A r p dp p

J-|^J2 <J

X J y J^(J2 + 1)

1 + 1 J^'

1 J2 - 1 J2

0 1 1 J

"j^+l,J2-l ^Vj^J^+1 ^V2J2-1

1 + 1 J^'

1 J2 + I J2

LO 1 1 J

°J^+1,J2+1 ^v^J^+1 ^V2J2+1

+ y (J^ + 1)(J2 + 1)

1 J^-1 J^'

I J2-I ^2

0 1 1 J

"j^-l,J2-l

^'^I'^i'^

^^2'^2~^

+ y (j^+i)j.

1 j^-i j^-

1 J2+1 J

2

Lo 1 1

"j^-l,J2+l ^v^J^-1 ^V2>J2-1 (4.18)

and for the longitudinal part

J-(_+l-J2 i^rili ...i z Ii p^,2, mPp

[v^J^^Ip L J |V2J2^] = (-0 ^

J

P T

"1 J^+1 J^'

X < / (J^+1) J2

1 Jj^+1

1 J2-I J2

.0 1 1

/ (J^+1)(J2+1)

1 J^+1

1 J2+I J2

Lo 1 1

"j^+l,J2-l ^Vj^J^+1 ^V2J2-1

°'jj^+l,J2+l ^Vj^J^+1 ^V2J2+1

1 J^-1 Jj^

1 J2-I ^2

0 1 1 .

a f f

J^-1,J2-1 v-|^J-|^-l ^2'^2"''"

+ y J^(J2+1)

1 J^-1

1 J2+I ^2

0 1 1

°'jj^-l,J2+l ^v-^J-^-l ^V2J2+^ (4.19)

Finally the two-body invariant matrix elements of the bilinear terms

obtaine

of spin 0 Bosons

p^.Pj are obtained from standard recoupling procedures. For example in the case

- 83 -
IV, 8

/ +/'2+I+T

IT|p^P2|v3^3V4/'^IT] = I T (-)

[IT] [00] _^ ^

] ] |Pi-Pil[W

[IT] ~[''3lL[^4ll [IT] [00]

[A

^4

1 1

I 1 ^3 ^4 ^

L 1 1 oJ

rVilP^^^lV3Hv2^2lP^^^lV4]

l>

(4.20)

IV. 3 - THE CENTER OF MASS COORDINATE

For an N- body system of relativistic Bosons and Fermions, the center

^ 2 of mass coordinate term R in the expression (4.1) is a sum of many- body operators. In terms of the fields it is of the form (see Appendix, Eq.(A.l))

••• ^"^V ^yTTji [ I ^i ^i' +

I ^i^j ^i^j } • (^-^i)

i, 2 2 1/2 where the E.'s are the free field energies, E. = (p. + m.)

The fields have been commuted (anticommuted) so as to be in corresponding pairs

with same coordinate x^ . No sign is introduced by this operation since anticommutation always arises between pairs of Fermion operators. The i(6^/9t) factors

operate on the immediate neighbouring Boson fields.

_^ 2

The non -separability character of R originates in the denominator

N

I

i = 1 step by employing the transformation

an a 1

^ ^ E^^ . In order to carry out the calculation, we go through an intermediate

This way R is obtained from the double integration of a sum of separable operators which are functions of the variable z = z, + z„ ,

- 84 -
IV. 9

r2 3 3 3

+ ^ "^^1+ ^ -ZE2 _^ ~^^N

(9 (Xj^)i — 9(x^)e ) (9 (X2)i — 9(x2)e ) ... (\|/ (x^^) \j/(Xj^)e )

i = l

Thus we get, for each term of this sum, products of single particle operators

which are solely functions of the energy (namely operators exp (- zE^))and of

single and two-body operators which are functions of both the energy and posi2 —> 2 -* tion coordinates (namely the operators exp(-zE^)E^ and exp (- z E^) E^x^

. exp(- z Ej)E^Xj ) .

i - Energy dependent terms

For the first kind of factors which depend only upon the energy we can

use the number operator N(p) of section IV. 2,

J d\ (9'^(x) i 9(x))e" = Jp%N(p)e"^^ . (4.23)

We shall need in fact the more general expression, where n = 0, 1, 2

J

d^x (9'^(x) i 9(x))e" e"" = J p% N(p)e" e"" , (4.24)

J

d^x (v^'^(x)v(.(x))e" e"" = J

p^dp N(p)e"^^E'' . (4.25)

The corresponding invariant matrix elements for spin s = 0, 1 Bosons and spin

s = 1/2 Fermions are given by the expressions (4.5), (4.7) and (4.9) where the

2 zE n factor p in the integrand of Eq.(4.5) is replaced by e E , i.e.,

[v^|e"^^ e'^IvV] = / J

p^dp f^^(p) e"^^ E^" f^,^(p) . (4.26)

ii - Mixed energy- coordinate terms

In order to evaluate the mixed energy- coordinate terms we shall first

consider the simpler case of the spin 0 Boson fields. The extension of the calculatlonal method to the other fields will be readily made thereafter.

- 85 -
IV. 10

Let ^ X be a mixed energy coordinate operators, where <f is a function

of the energy. This mixed operator must be sjmimetrized since <f and x do not

necessarily commute. Thus we have to evaluate in terms of the field operators

for Bosons

^ X = i

J d\ J

{^9'^(x) ? + X <f) n(x) - n'^(x) (,?x + x^ )cp(x) J

. (4.27)

Each of the terms of the right hand expression is then separated into two parts

by inserting the unit operator. For example

J

d X (p (x)^ X n:(x) = Jdxjdytp (x)^ 6 (x-y)y n(y) . (4.28)

For spin 0 Boson the unit operator is

,3

K^-y) = 6 (x-y) = ^ JPi dp^ JP2 / f^/P^ f,/P2)

X j^Cp^x) j^(p2y) ^ [xf''^ . (4.29)

It differs from the non-relativistic expression by the square root of the energies associated with the orthogonality relation (3.30) or equivalently the com- mutators (3.28). Of course instead of the factor y/E^jE^ one can employ Je^I^-^'

and we use this fact in evaluating (4.30). Thus for example

|- ^

d^x((p^(x)^x Tt(x) - Ti''"(x)<f x (p(x) =

1 ^ -rii ~rii rii -f^^l rii

\ I

[v^^^|,f |v2^2M^f^|Tl^^l[A^^' \^ ef^]] . (4.30)

In the field momentum representation the mixed energy- coordinate invariant matrix

element is

tViK-^'^|v2^2] = I^'- (I) JP>iIP2'<^P2i (^V^-^ >/V^>

X i

^ ^ f . (pj j. (p,x)[x ^xt^Jx ^ ] f (p )j (p x) . (4.31)

Vj^* i c ^

L ^2 2 2

After insertion of the unit operator (4.29) we obtain the separated form

- 86 -
IV. 11

V

\

((/E^E^/E^Ep + C/EiE^/E^E-)) i

^ 1 f (p^) j,^

(p^x) ^ f (p^ j^^(pjx)

[0] >^[^]-.rii'^t''2^ tO]

In this expression the matrix elements of the energy functions <f have been given

in Eq.(4.26). We are left with the calculation of the matrix elements of the

coordinate operator •

In terms of field operators the coordinate x is,

X = ^ d^x cp^(x) i <p(x)x = i

J

d^x((p^(x)x jt (x) - 7t"(x)x (p(x)) , (4.33)

for Bosons and

X = J

d^x ^"^(x) v^(x) X , (4.34)

for Fermions. Thus in the case of spin 0 Boson fields

X [j. (Pix)x ^ |x[^]|j (px)x ]hf'^|Tif']]X [a/ a/ eC^]]

I CVll^ IV2H^^ %^^][\^ \2

^^^] (4.35)

^1^2^/2

where

i

^ ^ [j (p x)x ^ |x (p x)x ] =

1 ^ 2

^12 pj^"* '^l'^2 ^1^2

-> The invariant matrix element of x is then

- 87 -
2 2-

"^12

f . (P) f . (P) ^1^ V2

IV. 12

a 12 (f . (P)f . (P) - f . (p)f' . (p))l 2 v^/j^ V2^2 ^1^1 V2 J

(4.37)

The geometrical recouplings are given in figure 4.4.

1 00

9

Figure 4.4

Here again as in f iga re 4.2 the isospin part separates simply and we are usinp

a schematic graphical representation for the isospin coupling.

- 88 -
IV. 13

For the spin 1/2 Fermion field we get likewise

X =

ri/21 ri/21 ->rn ~[jii/2] rii [^2^/2] [oo]

^2^2J2

with

[v^/^j^|x[^]|v,^j,l = i

2 ^ (-1)

'2 2-^2

^2"2 ^ . Jl J2

1 -'l (E +m)

2E \ e^e

2E

(4.38)

For the spin 1 Boson fields with isospin t the position operator for

the different multipolarities k is

X, r ~rti rti -.rn -^'^i''] rii ^-^2^1 [°°]

^1-^1

V2J2

Thus the invariant matrix elements are, for the magnetic multipoles

= i 1

1 J,

1 J2 J2

0 1 1 J

P 2, ^1^2

(P) ,

with

Of, ^1^2

, ,

' -^1-^2 J,J2

^1^1 ^2" 2 Vl ^2^2

(4.39)

v^J^ V2

- 89 -
IV. 14

for the electric multipoles

[V J2«^] = (-

"1 J^+1

X /j^(J2+l) 1 J2-I

-0 1 1-

'I'l 1 J1J2 Jp^dp

Vn1''2 V 0 1

1 J2+I 1

.0 1 1-1

^1^2

^J^+1,J2+1^P^

+ /(J^+1)(J2+1)

1 J^-1

1 J2-I ^2

L 0 1 l-i

v^V2

''Jt-1,J.-1 (P)

+ /(J^+1) (J2)

1 J^-1

1 ^2+^ ^2

LO 1 1-1

and for the longitudinal multipoles

^1^2

^J^-1,J2+1^P^ (4.41)

[v^J^5P| X |v,J,^] = (-i)

^ '2 2

1 J,+l J

i p 2^ mP

/(J^+1)J, 1 J2-I ^2

.0

y(J^+l)(J2+l)

1 1-1

1 J,+l J

^J^+1,J2-1^P^

1 J2+I J2

LO 1

1 J^-1

1 J2-I J2

LO 1 1

1

2U

^1^2

^J^+1,J2+1^P^

v^V2

^J^-l,J2-l^P^ + ^/V^2+l)

r 1 j^-i

1 J2+1 J2

,01 1

v^V2

^J^-1,J2+1^P^

(4.42)

We can now write straightforwardly the mixed energy- coordinate factors of

^ 2

the invariant matrix elements of the C.M. squared coordinate operator R , 2 2E 2

Eq.(4.22). For example, we get for the quadratic operator E e" x , spin 0

case

[^1^i|eV^^x^|v2^21 = ^[vi^i|E^^"^^l^^l][i'''ll^^Vll^^ • ^^-^^^

V

— 2

Here the quadratic operator x is again expressed in terms of the linear ones

by introducing the relativistic unit operator of Eq.(4.29)

- 90 -
IV. 15

= I

Likewise the invariant matrix elements of the bilinear terms are

(4.44)

- 2 E. z E . [Vj^/^V2<?2 I T |E .e ^S. . E .e ^x . \v^e^y^ei^ I T]

1

I 1

. 1 1 0 J

[v^Z-^lE.e ^ xL MV3][V2 ^j^ ^ ^ IV4]

(4.45)

Finally we sketch on figure 4. 5 the angular coupling diagram of one

^ 2

of the term of the invariant many-body matrix element of R .We have chosen

the case of identical particles in a same shell j,t. The calculation involves

now the well-known technique of angular momentum coupling for many- particle

systems of given statistics. On figure 4.5 the <f's denote the energy factors and

we have adopted an obvious graphical representation for the single particle

CFP's .

Figure 4.5

- 91 -
V.I

CHAPTER V

THE INTERACTION HAMTLTONIAN

V. 1 - THE PION-NUCLEON INTERACTION

We shall consider successively the pseudoscalar and the pseudo vector

couplings.

^PS =

i Ss(nNN) = I ^.a „p

(r^) (5.1)

and

KK \i

^PV PV(rNN) • y (y,Y^) o (t ) , v|/

, a $ : , (5.2)

KK \i ap

where a, p are spinor indices, /c, /c' isospin indices and |j. denotes the components

in 3- dimensional isospin space of the pion field. As before we use the notation

{ ] ^^^^ for double coupling in orbital and isospin spaces.

V.1.1 - Pseudoscalar coupling

The pseudoscalar interaction Hamiltonian is ^

^ ^^PS =-^^Ss(.NN)7f I' ' ^J- Jdr

n[j"i/2] /?[P^,.(^) u^„^„(r)'

[00].

vv V

j'j"

,11/2]

(5.3)

We have performed here the scalar product t.$ , hence the replacement of the

isospin vectors 'n'^''"-' by t in the boson field. The t matrices being hermitian

we have

92
~[1] , ,[1]

V.2

(5.4)

In order to evaluate the spin and isospin summation and the 3- dimensional space

integral we have the following recoupling diagram of figure 5.1

i" l/2t 00

1/2

A

r —

j'1/2

1

1/2 ^:[l/2|rf^l|l/2l

Figure 5.

We have introduced again double recoupling coefficients which are the product

of recoupling coefficients in j and t spaces, as defined in Chapter 2, figure

2.4.

We finally obtain for the pseudoscalar interaction Hamiltonian

'^PS(tiNN) " ^ 2 Ss(txNN) \

vv'v"

j'j"

V"' j" 1/2

j' X' /

^, , ,j'+X'+l/2

1 J J (-)

(5.5)

V.1.2 - Pseudo-vector coupling

The pseudo vector coupling interaction yields two terms one space-like,

one time- like :

- 93 -
V.3

PV {rem) -1 d r

PV(7tNN)

i G

PV (nNN)

d r

(5.6)

We tackle the time-part first, where (Eq.(3.5))

= - i n (5.7)

The recoupling diagram is essentially similar to the pseudoscalar one and we get

^PV(nNN)^^^"'^-^^^^^ = '^^PV(nNN) I

vv'v"

j'j"

-/"j"l/2

•/"+/'+l

j' j" (-)

j'+\'+l/2

L V V V V V f J

(5.8)

The (-) sign in front of A comes from the time derivative.

The space- like part has a geometry given by the recoupling diagram

of figure 5,2. (Here we are using again a simplified notation for the coupling

of B's both in angular and isospin space, see the remark at the bottom of Fig.

3.6.)

X

rl/2|a''' |l/2)

rx"|>.|x'i

[1/2|t'^'|1/2]

Figure 5.2

- 94 -
The space -like part is thus

VV V A,A, A,

j'j"

^11 U/2 X' j'j

X[X"|X|\'] {Jr^dr \€\ g^^ (r)u^„^„ (r)u^ , (r) 5^„^„ 5^,^,

. X [B[?;y2](Af^ ^] + (-)^ ^%V''f'' . (5.9)

In this expression the gradient matrix element is given by, see Eq.(2.34)

(5.10)

[m|vf^l|/] g^^(r)

2

(5.11)

V.1.3 - Pion-nucleon invariant matrix elements

We evalxiate now the matrix elements of the two processes of figure 5.3

(v3/3j3l/2) (v,.,l)

(v3/'3j3iy2)

(a) (b)

Figure 5.3

- 95 -
V.5

The states are here defined with amplitude vectors W,

IV3J3^/2>f = j3 |[W,^ B^^^^ ]

|0> , (5.12)

^

[j3l/2] ^[j 1/2] [j 1/2] [00]

, |(v,V2V2J2>j3^/2>i = I'-l^ ^ ]

(5.13)

and processes(a) and (b) correspond respectively to the contractions (the + (-)

sign is associated to the space (time) part),

/oi l/2|rBfj"^/2](Af^ ^] + (^1^^ ^hB[j'^/2]lf°°J lU v/^ i l/2\ f\V3J3'-'^ I ^\ - ^ ^ V

^ vV ^ I'-^lVl 2J2-'J3-'-/2>i

^[j3l/2] [j3l/2] = [Wf W ]5 6 . 6 „ 6,. 5-,. 6.,,- 5... 6.,,. , (5.14)

i 1 vvj^ V V2 V V3 ^/j^ 2 3 -^2 -"3

«Vi^V2J2>J3V2|IbK"'''< ± <-)'\'' ^')B^i;y^J,[~l|v3.3J3l/a>,

j2+j3 ~[j3l/2] rj3l/2] = ± (-) [Wf W ]5 5 , 5 „ 5.. 5.,. 5.,,. 5,,. 6.,,. (5.15)

t 1 ' vvj^ V V3 V V2 «^3 <^ J ^3 ^2

according to the following recoupling diagrams, of figure 5.4 and figure 5.5sWhere

we are using for the contraction sjrmbols of figure 2.12 a double notation as in

figure 2.4 or 2.5b to include isospln.

and

f

Figure 5.4

- 96 -
V .6

Figure 5.5

We finally get for the pion-nucleon matrix element of process (a) for

example the expression (5.16), It is of course hermitic and one may check simply

from the average value (5.15) that the matrix element (b) is the complex conjugate of (a).

- 97 -
V.7

j2+\2+l/2 ^ IT , r^3 ^3

C f SS(.NN) ( (j2 .2 ^1 3 \H Kll J

-

<^'^

n j2 ^2 ^ 1^2 l^ll I ^vi/,"v2''2''''3^3'^3) ~ ""^^^^^N)

X, jo 1/2

3^'2^'l

3^3''V2^2

^3 j3 1/2

(C ^^}r.3l^2K.i J—ViV3^2V2- C,;.; ,^)f^3K2K.]

e ^Li/2 jH

xj r^dr h . u . V

^1^1 V2 ^3^3^3) + (-1)

j 2+^2+1/2 3

/2 PV(nNN)

r 1/2 Xo j3-j 2 rn

V.1,4 - Momentum conservation

In these expansions the integration over r brings in the momentum conservation requirements. Thus we have, see ref . [14]

g . (r)

•^'''^

V2'^V3S^'^ = JP^Pl JP^P2 JP^P2 (!)

3/2

'VI

V2^3

^ ^PlP2P3 I

^

(E2+m) (E^-m)

v^/^ 1 V2^2 2 ^3'^3 ^

(5.17)

where

/l^2^3

P1P2P3

= Jr^dr j^^(p^r) j^^(p2r) ^(P3r)

,A(p^£VV^

4 p^p2P3 //-^ ^2 ^3) ^ ^0 0 0^* "

(/^-m) ! (/2+m) ! 1

L(/^+m)! (/2-m)!

{ £ £ ^ (m' -m 0) ^^^^^°^^13> ^;2^^°'^23> (5.18)

- 98 -
V.8

with

A(pj^ V>2 P3) =

"

1 when P-j^P2P3 foTM ^ non degenerate triangle , 1/2 when Pj^P2P3 foTM ^ degenerate triangle,

< 0 when P-j^P2P3 ^° "^^^ form a triangle ,

cos 0

2 2 2

1 Pi - P2 - P3

23 2 P2P3

cos 6 13

2 2 2

1 P2 ' Pi " P3

2 P1P3

(5.19)

(5.20)

Likewise the integrals with a gradient are explicitly given by,

^ ''H[.^1|v[^]k|/Vi^'^V2^'^^V3'^^ = I

Pl^Pl I

P2^P2 I

P3^P3 (I)

f

- Pi \

V 1 1 Pi

if^+l ^2 -^3

P2 P3 +m) (E^ +m)

pi P2 P3 ^

f (p )f (p )f (p ) , 11 ^2 2 ^3 3

(5.21)

and

3/2

X <

Pi %\

I ^

Pl Ap^

Pl P2 P3

P2 P3

(E2 - m) (E3 - m)

E^E^E^ ^^/^^PP^2^2^P2>^3^3^P3> •

(5.22)

V.2 - THE PION-PION INTERACTION

4

V.2.1 - The n interaction

4

The $ operator without overall angular coupling is

^ (4n) p2, ... arfB+vH-6 ^ ^ ^ ^ ,<^. 4

^(4n)= ~4"

J

^ g„(^)gp(^)gY(^)g6^^)

i°^^ ^ a P Y 6 (D

X r { } I ) { } { } , (5.23)

J a p Y 0

where

a a

- 99 -
We rewrite this operator in the various couplings approriate to the possible

processes

i) First the four pion annihilation term is given in figure 5.6 .

V.9

(5.25)

Figure 5.6

L R=0,2

where I (aPv^) = J

^ g^(r)gp(r)g (r)g^(r) . (5.26)

In this expression we have R = 0,2 because with a point interaction,

the relative two pion states are all even (symmetric under interchange of r)

We have introduced the notation

L /7- ^ /'« P L^ r n1/2(cH-P+L)

where the coefficient Q''^„ is defined in Eq. (2.41) and the corresponding square

ap

sjnnbol appearing in figure 5.6 is defined in figure 2.8 .

- 100 -
V.IO

We note in Eq.(5.25) the selection rules resulting from the symmetry in isospin

space which corresponds to the symmetry in space imposed by a point interaction.

We now give the various other terms of the pion-pion interaction operator, as shown in figure 5.7.

,J1

P/

A

yI \ 6

V

/ \

^ _ • / \

/ \ ^ /Y \5

irl ^ —N/ '

J T

(c)

Figure 5.

- 101 -
V.ll

ii) The creation of four pions, figure 5,8a

L R=0,2

X C ^"U "rI

[[tX'"'lS'P'll''''''2:'^'']'''l2:I''l][~' . (5.27)

iii) Scattering with one creation, figure 5.8b

L R=0.2

iv) Scattering with one absorption, figure 5.8c

(4.) -Vie X0.ve)i<-^V.«Q^,Q;,

L R=0,2

v) Pion-pion scattering, figure 5.8d

(4.) - 4^1 I

X(„,,a) l<-^« Q^^

L R=0,2

X 74;; q^^

[CAf'lXtPU, [J^l [^tyllAtell, [JT,-|[00,

_ (3_3„)

- 102 -
V.12

V.2.2 - Momentvim conservation

In order to calculate the integral I of Eqs. (5.23)- (5.30),

r r^dr g (r)g (r)g (r)g (r) =

J ^1^1 ^2'^2 ^3^3 ^^4

-4_p 2, p 2, p 2, p 2, ^\^2^2fk 1

2j p^dp, J

p^dp, J

P3dp3 J

p,dp,

^ V/Pl>V2''2>fv3.3<P3>V4''^^ >

(5.31)

we have to evaluate the discontinuous function which represents the momentiam

conservation law

°V2P3pt ^ J

"'"'^ j/i^Pl'^>^//P2'^>j/3(P3'^>j/^(P4'^) • <5.32)

This is done by applying twice the definition (5.18)

J

r^dr j^(pr)j^ (Pj^r)j^ (P2r) = Jr^dr

J

q^dq N(p^P2p) j^(pr) j^(qr)

Hence ^ ^ (5.33)

f

N(p p p) = A„„\^ , (5.34)

^ J- ^ P Pi ^2

and

U = \ - p dp A A . (5.35) P1P2P3P4 J 7x ^ p Pj^p 2 T> P3 P4 v^.^J-*

^l^l^^fh oil, ^^1^2 ^ ''a ^4

Here x, must obey the triangular rules and {\,£^y€^ since otherwise (5.18) is indeterminate and useless. Of course, only such cases will in

fact arise as all triangularities are guaranteed by the angular integrations.

V.2.3 - Pion-pion invariant matrix elements

4

We give now a few relevant matrix elements of the f interaction, as

examples.

i) The 4w pion annihilation operator yields a particular simple result

4

for the case of the s configuration. Here the unique state vector is of the

form, see section II. 5. 2.3

\') = I

S[[Af°']Sf°']]f°^J [At°l]Ar°^]]m]^°°V> . (3.36)

R = 0,2

- 103 -
V.13

The coefficients C^^ are given by the condition that the state |s ) be of even

T in all recoupled pairs, namely

|-fI[0l]s[01]j[0K] ~[01]j[0l]^[0K]^[°°]|^^

= 0 , (5.37)

if K = 1. This condition yields together with normalization

" 1 1 0" -

1 2

-

1 1 0 1 1 2

- K K 0 - -K K 0 - )

/5 2_

3

(5.38)

We have used the value

r 1 1 2

112

- 1 1 0

Then

Q

l°Ky 1 = -f^KOOOO)

I

8 [ + ^},(5.40)

R=0,2 K=0,2 ^

from the recoupling diagram, in isospin space, of the average value

(5.39)

[

C^<0|[[AA]f^][AA]f^]]f°] [[S]f^][AA]r^]][°]|0> , (5.41)

KR

which is given in figure 5.8 .

Figure 5.8

- 104 -
V.14

ii) In the case of the pton-pion scattering, Eq.6.30) we get for a 2 pion

system (JT) „

1

, j'i^ <0 I [W ] [A U ] ]

]

((l +

5j^P (1 + 6..))

X

[[Af«^]A[P^]]rJT] [A[Yl]AfSl]][JT]j[°°] w{^T][A[^]srj5]f^^l|0>

^(M.T/- -1 .('i-j+k+/) n-J n-J Z" ^ ^0

X

^ Jjx 4 ; T Itjf-^TI W.f^^l] . (5.42)

((l + 6^p(l + 6,.))^/2

In this case the invariant matrix element is :

T T 0 T T O'^"

X ^11 ^11 ^TT <i QjJ ^ ^ • (5-^3)

V.3 - THE NUCLEON-^IN 1 MESON TNTERACTTON

V.3.1 - The oiNN interaction

Let us first consider the absorption or emission of an u meson by the

nucleon. The adopted interaction is

tjNN cjNN

where the second (time- like) part comes only from the longitudinal field u). This form is hermitic since : co^ = - co^ ; co = oj ; YxY4 ~ ~ Y4YX ' Y4 ~ ^ ^'^^

(^''"y4Y, ^' <*>,)'*' = w*" "^^ Y4Y,,Y4 = ' ^ Y,, ^ w, • (5.45)

The space- like part of the energy requires the calculation of

105
V.15

ye u)NN (.,..[1] . )^'^ ]

(5.46)

•ICT^ ^ 0

or more explicitly for the magnetic multipole contribution of (3.222)

(l-<f^+<?.+J)

f 1 ^ 2

JiJf I

^ ^ ^vJJ^'^^

X

V J

V. j. ^.

Jf

,_^[Jfl/2]^^^/2]

( ^/^^f Vf

f^f

(r)

1 11

]

[00]

(5.47)

We see that the geometry for this contribution as well as those of the electric

and space-like longitudinal parts is of the form of figure 5.9

Figure 5.9

- 106 -
V.16

S (K) =

K

Thus let us define the quantity (K) for multipole K, k = M,^,^

3 f X I

V J

V. j. 1 .X .1 Vf

ri/2 jfl

1 K J

Ll/2 X. j.J

1 1

>^i J

r dr W^jj, "vj^j ^^i^iH

1/2 jfl

1 K J

L 1/2 j.

[Xj|K|A]

f 2 , K

I r dr W V

vJK v^^^X^ ^i^i

(5.48)

Then the interaction with the magnetic oj field (3.222) is

toNN r - r fJl J~rJl ~fjf^/2] [jil/2]

[00]

With the electric one, Eq. (3.223), it is

(5.49)

G y f /j+r s (j-i) + /j s (j+i)i r (A^^^^ + (O'^'^^Af'^b

^[jjl/2] [j. 1/2] [00]

X B B

^f^f , ]

(5.50)

and with the longitudinal space-like part Eq. (3.224) it is

'u)NN

(space)

= - 1

^a>NN I

Cv^S(J-l) - /K4 S (J+1)) [(A^J^' + (O'-^^Af^b

X B

.[j^l/2] [j. 1/2] [00]

^f^'f ]

(5.51)

We finally have the time-like longitudinal contribution, (which corresponds to

a spin 0 transfer), Eq. (3.225)

^""^"^^

. (-/.+/'. +J+1) <-2 _

V J

V. j. 1 .1 .1

- 107 -
V.17

L. V'f f

-Y " V .

/ Y, . u

1

[jf] I J

(5.52)

the geometry of which is given in figure 3.10 .

B

»1

B

r

Figure 5.10

Hence,

toNN (time )

(-/^+if.+j)

f 1

'1/2

^f jf"

< 1/2 h

0 J J . f^f

1/2 jj"

1/2 j,

- 0 J J . Vf^f

^[jfl/2] [j 1/2] [00]

B ^ 1

\'i

(5.53)

- 108 -
V.18

V.3.2 - The PNN interaction

For the p field the expressions are similar except for an isospin con- tribution which is given in figure 5. 11 ,

where as usual the isospin function t|

Figure 5.11

[1] in the P field is replaced by the isoIn order

to use with it the previous expressions for the coNN interaction, one must furtherspin transition operator t'-'''-'. This diagram yields a factor -i/(|-)

more^ multiply the latter by 1 for isospin coupling in the P field expansion and

by 2 to cancel the previous isospin contribution which is given by figure 5.12.

B

B

1/2 X

1/2/

We finally get

Figure 5.12

uNN G

pNN

(5.54)

109
V.19

V.4 - THE SPIN 0 - SPIN 1 MESON INTERACTION

V.4.1 - The pTXK interaction

The p meson must interact with two pions in order to conseirve parity.

The simplest form for this interaction is,

se = - G p (T)a $

pTui pim '^[1 \i

= -G^^(,^l tef^Mlljtoj ,2 + ,^(;)a^,2} . (5.55)

Here p (t) is the p field with substitution of its isospin wave function ti [1]

by the operator t^''"''. The gradient must be a S3r[nmetrized operator in the two pion

coordinates. The various possible processes are shown in figure 5.13.

\ nf \k I I \ I \

Figure 5.13

They give the same geometry since they yield invariant triple products in the operators. For example using the result,

1 [ef^lvf^ljf^] [Af-^^Yf^Jlf^l = [Af'^Y^]]f°] , (5.56)

where

= fvf^J^f^Jlf''^ , (5.57)

we get for the various multipoles of the p field expressions the structure of which

is of the form (x = ^, jH, ^p)

P 3 1 r "+''i-«'2> - ^3

pTUI J pKK

V J

^1 'l

"2 ^2

x^^' ^,.,<'> ^''T' * •••)

X ([c:;^'' ,i^>]'°°' . (-)%.c.) f"^' . (-,%.c.),

with recoupling diagrams of the type given xn figure O.i-H.

- 110 -
V.20

V

1

r

r

1

X

Figure 5.14

Here we have made a change of notation to distinguish the p and n crea- tion operators : we use C for the p mesons. Furthermore product of invariant ma- X

trix elements [V] . and [L] . is given in detail by

X Jp^dp p^dp^ p2dp2 /^^^//Pl>%2^2^P2)fvj(P> ^P Pi P2

Finally we note the identity

(5.59)

rilrf^'ll] = [l/2|r^'^|l/2] [1]

since

[JIsI^^^Ij] = i J/J(J+1) and S = |J | ct .

(5.60)

(5.61)

Thus we define the symmetrized function (with Sy^ = \lj2 if ^\^\^^2^2 °^

12 1 if ^iV^-/'2V2),

- Ill -
V.21

T^(L) =

1/2 _ (J+^j^+/2-l)

^ © I J X.

1 L J

L X L

1 L J

'

X

if v^e^ V2^2 ' ^^-^^^

and the various contributions are.

J V

^2 "2

pTxn M \_ i;:^! , + (-)' A^^'bcc + (-) ^

)

X (C ^ + (.) ^ C ^ )1

"2 ^2 -I (5.63)

3if y -i G ( /j+r T (J-D + /T T^(J+l))

J V ^1 ^1 r rJii J -rJii f^^^ ^1

X (C ^ + (-) C ^ )] , (5.64)

"2 "2 -I

pTTTT ^ pTTTt (space)

J V

^2 ^2

r rjii J -rJii ^^1^^

X (C ^ + (-) ^ C Ml . (5.65)

"2 "2

The time- like part of the longitudinal p field yields

^ p 3 2 1 0+^-^+^2'^^^ ^ ^ ^ ^3

3^ =\drG p, a, i=G ) i J/t/„1Si„ pKK J pnK '^4 4 pTXTx ^ L 1 2 12

(time) V J

-1 'i

"2 '2

>{(t4'v"^'"-'"i'°°' + (-)'-''c.c.) (rc^'i''?''i\[i],f°°i + (;)(-)%.c.)

'<([cf''y''\f'')'°°i + (:>(-)\c.)} P('i) . (5. 66)

- 112 -
V.22

with a sum over the two lines if v-^^-^ i' ^2^2 antisymmetrization purposes.

The factor F (-n.^ ) is E2

(E2) = I

P^^P P^Pl P^P2 (e^) /if /^/f ^J^P^^^^^^Pp

J ^ ^2

X f

V2^2, (Po) A. „ „

2 Pi P2

(5, 67)

After integration over angles we get

3 1/2 0+/' ) . = (2) I ^ ' ^12 [-^1^11^2] ^(e2)

(time) V J

^2 ^2

r rjii j~rJii [^^^ - ^1 -Ki^l

xf^^ - (->^l >(c ^ +( + )(-) c '

)

' ^ +('^) (-) ^ C ^ )1

"2 -I X (c (5.68)

V. 4,2 - The ojtt: interaction

The cj can interact with at least three pions for ^ parity conservation

and we adopt the following form for the interaction, where we have performed the

—» —

scalar product $.t in isospin space (hence the replacement of the isospin vectors

T\^^^ by t''''"^ in one of the boson field as explained in Eq.(5.3) )

can

G „ w d $(t) $

3 [x \x

2 2

G 3

j^o)^ 9^ $(t) $ + 3^ $(t) $ J

ton (5.69)

where the derivative must be symmetrized between the three pion coordinates. The

evaluation of the energy goes along the same lines as above. We first note that

the isospin part yields an invariant triple product of the operators in isospin

space according to figure 5.15.

$(t)

$(ti) 2

11

i(Ti) 3

1^1 ^2 S J

Figure 5.15

- 113 -
V.23

while the geometry of the angular momentum part is of the form shown on figure 5. 16.

A

V

r

r

Hence we define the function.

Figure 5.16

123

V (LR) X

1 L J'

Lx L R.

Q^2^3 [V],.^ i^f^ ' (5.70)

where

[V]

2 2 2 2 f^]\R " [^I'^ki] [^|L|R] J

P dp p^dp^ P2dp2 P3dp3

J ^2 ^3

and the symmetrized sum

V (LR) X

i jk

(L)

59(1 jk)

2trizat:

teraction energy operators in a way similar to the ojNN interaction :

(5.71)

(5.72)

where s^^^^ is the proper sjrmmetrization weight , Finally we can define the in-

- 114 -
V.24

= G

UK

' 3 I I ^V^'^>

V J R

^ ^2 ^3

'^l ^2 ^3

L V2 V2 V. v„ /J J i

(5.73)

i G

ton ton ^ [/j+T V^(J-1,R) + yj V^(J+l,R)]

x{[K:'-<-)'"Ar)< }

(5.74)

= - 1

ton (space)

G 3 ^

(v/rV^(J-l,R) - ym"v^(J+l,R)]

(5.75)

Finally the longitudinal time-like contribution is

^ ~ G

ton 3 r d r to^ $(t)$ . (5.76)

ton (time)

Here the isospin summation is the same as above while the geometry in orbital

space is giyen straightforwardly by the diagram of figure 5.17

r

r

r

A

Figure 5. 17

- 115 -
As above we define a function,

and the integral (E^^ = E^, Y.^, E^)

= J

p^dp p^dp^ p^dp2 p5dp3 /I P

2E m

.^2-3

€ £ €

X f (p)f (p )f (p )f (p ) ^ ^ ^

We get, with proper sjmraietrization,

jj, ^ _ ^ O \ \ \ \ o T.T J ^ - ''.n^ 1 III V «> ^»

(time) V J R ijk

^1^2^3 (even permutation

/ / ^ of 123)

'^l 2^3

rjol j~rJOi Kii] [^^]

<[[(a];J°] - (-)-^A^°b(c ' - (-) " ]

X [(C ^ + (-) J )(c + (-) c )] ]

- 116 -
VI. 1

CHAPTER VI

MODELS OF HADRONSVI. 1 - THE HAMILTONIAN MATRIX

As an example, for a system of nucleons and pions the pseudo Hamiltonian

(1.50) in terms of the field operators could be

3f = H^ + j^^^ + a/ + ... + I

|(P^+fi^R^) (6.1)

where H^ are the free field Hamiltonians of Chapter III and where we have introduced the artificial kinematical center of mass Hamiltonian (Chapter IV) in order

to extract the intrinsic states. With a configuration space limited to four pions

and one nucleon the energy matrix looks for example as shown on figure 6.1

N N2r N 3n N 4k

N H + f o

^ A A

Nn Y

H

0 +

Al A

N 2r Vi1 1

t 1 JV

° l/\

; ii /Ml

N 3n

1

1 Yii

H + E o

^ Alii / Mil

N 4n H + P o

^

Figure 6.1

- 117 -
VI. 2 - THE PARTONS

By now it is a generally accepted idea that hadrons are composite systems, the constituents of which have been called partons by Feynman. In this language our fields represent the partons. We will want to test for example a model

based on the following partons :

B T S P G

N 1 1/2 1/2 +

A 1 3/2 3/2 +

n 0 1 0

P 0 1 1

0) 0 0 1

etc

We limit this list to the fields with quantum numbers corresponding to those of

the first few lowest physical particles. For completeness we have included the A,

although the spin 3/2 field is not treated in this work.

In the calculation the parton masses are parameters to be adjusted for

fitting the physical masses. The parton conf igurational energies are used as a criterium to truncate the functional space. These energies are evaluated by taking N in the Os state as the reference energy, and roughly the energy scale of

the harmonic oscillator basis to be 'ti w = 2m =» 280 MeV, which yields a level n

2

spacing for the unperturbed Hamiltonian (which contains p only) of the order

of magnitude of the pion mass. This way A in the Os state is at an energy of

about 2 effective quanta and the jt, p, to at energies of 1, 4, 4 effective quanta

respectively. These values are utilized below for the truncation.

n^

The total list of the distributions (j^ ^2 (i^i the sense of

occupied orbits) for the pion cloud up to 6 effective quanta with n^ particles

in state etc... and total angular momentum J is given in Table VI.

first column n denotes the number of effective quanta , i.e. the harmonic oscil-^

lator total principal quantum number N of the configuration plus the corresponding number of pion masses.

- 118 -
o

7J C3

1—1 o

K o in lOp

CO o

CMo^—

CMo

CM

CO —o

CM

r—

K o

<f

O

<r

tn O

^—

\

a

o

CO

CO o

o

CO

CO O

O

CO

T-H ro

CO O

CMo,—- CM CO

Cu O CM

CO O

CM 1—t CL ° a G. --^ K o i-H CO CM

CO o

o/—

\

CM i-j

CO O CM

'

*i CO t-H CO ^-N /— ID O* o

o

P.

1—

CO moo

O fX CO CM CM CM O r-l (A CO CO CO CO CO o— o o O O O

y

—

' -—' ^

CM

* CO CM CM O -—

•o /-N rH CO o CO .11-1 a. CM O 'O O o CO ^ O

O p. ^ ^ r-l O CM r-l P. /-V

y—

y N CM O CM

•o a ^ ^ ai K O n i-i

C>J CO O

CO OT-c CM ^

CO O CO n ^ o CM O CO r-H o « —

/—\ O CO cw Cfl 00 IH o f-J O rH O O CM M tfl CO CO tx CO a -o O o o O O O O o

O I-t

CO pi

C^I CM o I-l CM CO

. s "O K rH 1—

— t-i rH

o i-t CO cn

y—

.—

\

CO D- 13 00 O O O o o o

C CVI CO -* in

k 119 -
VI.

VI. 3 - THE NUCLEON CONFIGURATIONS

1 1 Thus for the ( j ^) system with B = 1 we get up to a truncation

energy of 1 GeV above the nucleon mass the configurations of Table VI. 2 :

N, N+n, N + 27X N + p, N + p + tx A+ k, A+2netc ... .

N + n + 2:n. . + 3k +4k + 5n +6n

(Os) (Od)

dp)

(2p)

(Os^)

2 (Osls),(Op )

(0s2s),(0plp)

(Is^), (Od^)

(Os^Op)

(Os^lp)

(OsOpOd)

(Op-^)/OslsOp)

(Os"")

(Os^^ls) 2 2

• (Os Op )

(Os^Op) (Os^)

(Op) (Os)

ds),(Od)

dd),(2s)

(OsOp), (Oslp)

(OpOd), (Opls)

(Os^), (Os^ls)

(OsOp^)

(Os^Op) (Os^)

(Is) (Od) (Od)

(Of),(lp)

(OsOp) , (Osls)

(OsOd), (Op^)

(Os^Op) (Os^)

(lp),(Of) (Os)

(Od),(ls)

(OsOp) (Os^)

N

Os

Is

2s

3s

(2s)

(Id)

(Op)

N + P + p,n

(Os)

(Op)

(Od)

(Os)

(Os,Os)

A + n + 2n + 3 n + P

(Os) (Op)

dp)

(Op^) , (OsOd) (Os^Op) (Os)

(Op) (Os)

(Od)

(OsOp) (Os^)

(Is)

(Od)

(Op) (Os)^

dp) (Os)

Table VI. 2

We have included the A, which couples strongly to the

one pion system (s^j^^^ ~ ^% n

^^'^ ^ meson which couples extremely strongly

to the pion system (g ~ 5g^, ) . prni Ntui

- 120 -
VI.

VI. 4 - THE DEUTERON CONFIGURATIONS

For the deuteron case we consider separately the configurations where

the two nucleon orbitals yield an even or odd space symmetry respectively. For

the even case all the listed 2N configurations can contribute without any pion

except (OsOg) . For the odd case at least one pion must be present. We also give

between parenthesis the number of quanta for the NN and the nn configurations.

The configurations are limited to six "tiu and by the parity and angular momentum

requirements. The even space sjmimetry configurations are given in Table VI. 3 and

the odd space ones in Table VI. 4. If the oj and p are added we get in addition the

Table VI. 5 .

2N + n

(2) (4) (6) (2)

+ 2k

(6)

+ 3n

(4) (6)

+ 4n

(4) (6)

(0) (Os^)

(2) (OsOd)

(2) (Op')

(4) (Osld)

(4) (OsOg)

(4) (Oplp)

(4) (OpOf)

(4) (Od^)

(4) (Odls)

(4) (Is^)

(6) (Os2d)

(6) (Op2p)

(6) (Oplf)

(6) (Od2s)

(6) (OdOg)

(6) (ls2s)

(6) (Of^)

(6) (Oflp)

(6) (Ip^)

(Op) dp) (2p)

(Of)

(Os^)

(Op) dp)

(Of)

(Op) dp)

(Of)

(Op)

(Op)

(Op)

(Op)

(Op)

(Op)

(Op)

(Osls)

(OsOd)

(Op^)

(Os2s)

(Osld)

(IsOd)

(Od^)

ds^)

(Oplp)

(Os^Op) (Os^lp)

(Op^)

COs'^)

(Os^) (Osls)

(OsOd)

(Op^)

(Os^Op)

(Os^) (Osls)

(OsOd)

(Op^)

(Os^Op)

(Os^)

(Os^)

(Os^)

(Os^)

(Os^)

(Os^)

(Os^)

2N + 5a

(6)

+ 6n

(6)

(0) (Os^) (Os^Op) (OsS

(OsS

(Os^

(Os-^ls)

(Os'^Od)

2 2 (OS Op )

Table VI.

- 121 -
VI. 6

2N + K

(1) (3) (5)

+ 2n

(3) (5)

+ 3k

(3) (5)

(1) (OsOp)

(3) (Oslp)

(3) (OsOf)

(3) (OpOd)

(3) (Opls)

(5) (Os2p)

(5) (Opld)

(5) (Op2s)

(5) (Odlp)

(5) (OdOf)

(5) (Islp)

(Os) (Is) (2s)

(Od) (Id)

(Os) (Is)

(Od)

(Os) (Is)

(Od)

(Os) (Is)

(Od)

(Os) (Is)

(Od)

(Os)

(Os)

(Os)

(Os)

(Os)

(Os)

(OsOp) (Oslp)

(Opls)

(OpOd)

(Os-^) (Os^ls)

(OsOp^)

(OsOp)

(OsOp)

(OsOp)

(OsOp)

(Os^)

(Os^)

(Os^)

(Os^)

2N + 4k

(5)

+ 5k

(5)

(1) (OsOp) (Os"^Op) (Os^)

Table VI.

2N + P + pn + tOK

(Os^) (Op) (OsOs) (Op) (OsOs)

(OsOp) (Os) (Os)

Table VI.

- 122 -
VI.

VI. 5 - THE PION CONFIGURATIONS

For the pion system we get likewise the configurations of Table VI.

for a cut-off energy of about 1 GeV with our chosen oscillator parameter.

n (Os) (Is) (2s) (3s)

3k (Os^)

(Osls2s)

(Os^ls)

(Op^Od)

(Os^2s)

(OsOp^)

(Od^Os)

(OsOplp)

(Osls^)

5k (Os^) (Os^ls)

Kp (OsOp) (OpOs)

Table VI.

Thus we see on these various examples that the truncated spaces up to

six pion masses (~ 1 GeV) are of moderate size. The complications introduced by

the symmetrization problem are limited essentially to the s"' case, see section

II. 5. 2 . The only redundant conf igurational set which appears in the above ta3

bles is the very simple p case. Of course the effect of the cut-off energy on

the low energy properties of the systems (form factors, electromagnetic moments

etc...) has to be studied numerically.

- 123 -
A.l

APPENDIX

RELATIVISTIC KINEMATICS OF THE CENTER OF MASS

We demonstrate in this appendix the result of Eq,(l,45), namely the

—

CM. coordinate R for a system of relativistic particles of energies E. all con-

[8] ^ sidered at an equal time is given by

I

E.

X 1

R = . (A.l)

E. I k

The discussion shall be carried out by considering successively the case of a single particle of mass M which decays into two others of masses m^ and m2, then

the case of two distinct particles of coordinates (Xj^t-j^) and (x2t2).

In the first case the C.M. trajectory is given by the trajectory of the

initial particle M and we simply must compute the geometrical relations between

that trajectory and the trajectories of the daughter particles.

In a given reference system (S), see figure A.l, considering only the

projections on the (xt) plane and assuming that the trajectory of the initial

particle goes through the origin, the CM trajectory is

X = V t , (A. 2)

with

V = P^/E , (A.3)

E = /p^ +. P^ +. P^ + M^ . (A.4) X y 2

- 124 -
A. 2

Figure A.

At the point (x^^jt^) the particle M decays into the two daughter particles

and m2 the positions of which at time t are

and

^1 = X + V, (t-T ) , (A. 5)

^2 = ^o + ^2^^"V . ^^-^^ with

^1 = "i" ' ^1

1 X

/"i 2 2

2~ = VPi + + Pi + ; (A. 7)

P2

V2 = , = /P2. + pj + P2 + ; (^-8)

2 X y z

+ E2 = E , p +

p

2 = ? . (A, 9)

- 125 -

/I 2 2 2 = /Po. +
A. 3

Hence at the equal time t, the ratio of the distances to the C.M. coordinate x

given in Eq.(A.2), is

Pi Pi +P2

X _ X X

Ax^ x^ - X

' + E^

AX2 X2 - X Po Pl+P2x (A. 10)

E2 E^ + E2

which for the non relativistic limit goes into the usual relation

(A. 11)

Ax^ m2

AX

2

^

Now in some other system S' the CM and the two daughter particles have coordinates i, and ^2 corresponding to x, x^ and X2 with now however non equal

times T, and T2. Nevertheless these three world points being connected by the

above physical relations still retain the character that t) is the position

of the CM for the particles m^^ and m2 at positions (Cj^Tj^) ^^2^2^ moving now

with velocities v| and V2. On the other hand the point t) is also the CM of

the particles m^ and m2 at the equal time t in S', i.e.^ at the positions (^|t)

and (C2T) as shown on figure A.l. Conversely going back to the initial system

(S), the world point (xt) (corresponding to (^t) in (S')) is also the CM of

the particles m^ and m2 at (x|tp and (x^tp (corresponding to d^T-^) and it,2'''2^

in (S') respectively). Thus we see that the C.M. in relativity is associated

with a whole family of particle positions with space-like separations, i.e.^ all

the positions obtained at the crossing points of any straight line containing

(xt) with the particle trajectories. To each of these straight line corresponds

a Lorentz transform into a reference system in which the three points have equal

times.

We now turn the problem around : given two particles with arbitrary

4-momenta (p^,E^) and (p2,E2) in (S), find the center of mass if the particles

have the positions (^j^t^) and (r2t2) . The C.M. energy and momentum are again

given by Eq. (A. 9)

Restricting ourselves again to the x-t plane one can find the crossing

of the world lines by drawing lines through the points (x^^t^) and (x2t2) with

slopes (see figure A, 2)

Pi P2

V = , V = . (A. 12)

^1 X ^2

- 126 -
A.4

X

Figure A.

Let us call the crossing point contrast to the above example of the

decay of a particle, here the point (^^t^) has no physical meaning, since in

general the world lines 1 and 2 do not cross, i.e. the projections of the world

lines on the x-t, y-t and z-t planes cross at different times. We now can construct the trajectory of the CM as it goes through the point (x^t^) with the slope

P

V = Y <^-13)

and finally, the CM corresponding to the given points (x-j^^j^) given by the

crossing of the CM trajectory with the line connecting the two points. After some algebra we find, considering the x-t plane, for the CM coordinates

ix^- K^) (E^-K^+E^K^) - (tj^ - t2) (Pj^ ^1+P2 ^2^

" (x^-x^E - (t^-tpP^ ' ^^-^^^

(K^-K^)(.E^t^+E^t^) - (t^-t^)(p^ '=1+P2 ^2^

^ " (x^-x^E - (t^- tpP^ • ^^-^^^

For the y-t plane, we have

(y^ - (Ej^y^+E2y2) - (tj^ - t2) (Pi yj^ +P2

(y^ -y2)(Eit^ + E2t2) - (t^-t2)(Pi t^+p2 t2)

(y^ - y2)E - (t^ - t2)p,

- 127 -

t = r„ ^ .. X. _ . ^ ^

, (A. 17)
A. 5

and, for the z-t plane

; (Zj^ - 22)(Ej^tj^+E2t2) - (tj^-t2)(Pi tj^+p2 t2)

• .

• ' . • ' • » . Of course, one could express these formulae in terms of the trajectories of the particles. We shall, however, not go into further detail since we

will need only the case ~ ^2 ' which we immediately obtain the Eq.(1.45).

We only note that the CM coordinates are not given by the ansatz

1 = J

(.r^ + r^) , ^ = 2 ^'^l ^2^ ' ^^-^^^

which is used in the Bethe-Salpeter equation. Finally, there holds

which can be verified by direct calculation from (A. 14) . Thus, formally, the

commutation relations [P^,x]= i . . . are fulfilled.

This procedure for finding the CM coordinate can be continued to more

particles, for example in analogy to the Jacobi coordinates of non- relativistic

kinematics. Thus one first defines the CM for particles 1 and 2, x^2' ^12'

(A. 14), (A. 15). Now one adds the third particle by replacing in (A. 14), (A. 15)

^1 "* ^12» ^2 "* ^3' ^1 ~* ^"12' ^2 ~* ^3 ' -^^^ obtains x^2 3* ^12 3 ^°

on. As to be expected, the general expressions are cumbersome. However in the

present work we are interested only in the equal time case (t^^ = t2 = t^ = . . .

)

for which the expression for the CM coordinate is simply

E^Xj^+E2X2

x,<, „(t,-t_—1„) - =12^2+^3 ^^'^^^^

( E^+E2 V3

12,3^"1 2

3' ^12'''^3 ^l'^^2'^^3

^1^1+V2+V3

(A. 22)

as asserted in (1.4^

Finally we only point out that the definition of the non equal -time

C.M. coordinate used above in the case of space-like seprations can be analytically

- 128 -
A. 6

continued to yield the definition of the CM coordinate for time like separations,

i.e.^ for slopes of the line connecting (x^t^^) and iyi2^2^ > figure A. 3, larger than

45°. This case arises in covariant treatments of many body systems, e.g.^ in the

Bethe- Salpeter equation. Therefore the CM of two points (x^t^) and (x2t2) is the

intersection of the straight line connecting these points with the trajectory

of the CM defined in (S).

Figure A.

An example of such points is the pair (x^t^^), (x2t2) of figure A. 3. In fact, by

a Lorentz transform one can always achieve that a pair of points with a timelike separation is parallel to the time-axis. An observer in that coordinate

system then will note that first the particle 2, then the CM, then the particle

1 fly by his observation post. This way the straight line criterion acquires a physical meaning. With this definition we have a uniform description for the

center of mass of a system of two particles.

For completeness we list the following relations. To re-write the

familiar CM-separa tion

i(p.r. p^r^) i(P R + p r)

e ^ ^ ^ ^ = e (A.23)

in relatlvistic kinematics we have, In the equal time case

- 129 -
A.

^^1 ^2^ ^1 ^2

E r r ^ = E^E (A.26)

1 2

r = - (A. 27)

and the inverse formulae

_^ _^ E

P2 = ^E^^ - P (^-29)

r = R^r ^

(A. 30)

1 E^+E^

E, r = R-r ^

(A. 31)

2 E^+E^

They all arise from the non-relativistic formulae by the replacement

^i'

- 130 -
B.l

B, BOOSTED STATES

When computing, say, the electron scattering form factors one needs the wave

functions of the system at momenta and = + q, which may correspond to

relativistic velocities. To that end one will have to boost the solutions of the

composite physical particle. With these boosted functions one will obtain a

result, correct within the model, i.e., no relativistic or recoil corrections will

have to be applied. In the procedure outlined in this monograph, the wave functions of the system are obtained as configuration mixtures in a discretized

basis (cf. Eq. 1.29, or 1.30, together with 1.32 and 1.35) in the form

-iE t

_^ Y (r.,t) = e " S X t (r.) (B.l) n 1 V

^ 1

The amplitudes X^^'^^ are obtained upon diagonalization of the secular matrix

(1.37). We have here changed the notation in writing instead of |r> of

(1.37). According to (1.55) the CM motion of each configuration f^C^-j^) can be

separated off using

t (r.) = y (?.) CP (R ) (B.2)

'v 1 '^v 1 o V ,

-> 2 2 -(R /2R ) CP^CR,) = e ^ ° (B.3)

The CM coordinate R^ is a function of the coordinates r^ and is given by (1.45)

Since the number of particles is different in the different configurations

the explicit form of R^ depends on the index v. Owing to (1.55) the functions

)(^(r^) in fact depend only on relative coordinates, which, however, are not

specified explicitly.

Now to describe a system in free motion having momentum P, it is best to

introduce a moving coordinate system:

?i = R + (B.4)

-131-
B.2

Then we have, beginning with the non-relativistic case

X^(P,§.) = e 5 ° ^n^P'V

Here is given by (1.45) uoon the replacement r^ ^ (recall, the are not

explicitly relative coordinates J ) and,

2 E = or + P ) (B.7)

with M from Eq . 1,58,

n The physical meaning of (B,4) through (B,7) is; the wave function of the

system is translated to the point R from the origin by (B,4), (The point ^. = 0

in (B.5),(B,6) corresponds to the point r^ = 0 in (B,l), (B.2)). The CM wave

function is divided out by (B.6). A new CM wave function is supplied by the

plane wave in (B.5). The energy of the new system is now given by (B.7). Note

that for non-relativistic velocities X>^(P.b^) in fact is independent of P, For

relativistic velocities, however, the coordinates undergo a Lorentz contraction

-> this way the functions \^ acquire a dependence on P. Otherwise the form of (B.5)

-> and (B.6) remains unchanged. Because of the Lorentz contraction of the ?^ the

- L-t] spherical multipole functions i (kr)r in the basia ' functions v/ are replaced

^ n n

by spheroidal functions. (Recall that a sphere in the moving frame becomes an

ellipsoid in the lab frame.) The spheroidal functions can be expanded in terms

11/

of spherical multipole functions. As can be seen from the tables in Ref, 15,

this expansion converges rapidly for not too highly relativistic velocities.

Inserting this expansion into the basis functions given in Chapter III, remembering to boost the spin of the spin carrying fields one can write down the boosted

functions in a straightforward, if perhaps tedious manner. Somewhat more

2 2 involved is the boost of the factors exp(R-/2R^) appearing in (B.6). This way

we have

-132-
B,3

(B.8)

We here shall not derive the explicit form of the boost vectors b (p). We only

note that they depend on the structure of the boosted system.

-133-
C.l

C. SCATTERING STATES

The diagonalization procedure of this monograph in principle yields solutions

which approach the exact solutions of the model specified by the chosen Langrangian,

upon enlargement of the set of basis states, i.e., upon letting the truncation

2 energy E^^^ grow, and taking the limit Q -* 0 in the CM pseudo-Hamiltonian (4.1),

Therefore the solutions having an energy which lies above the threshold for

particle emission in fact directly describe scattering states. This is in contrast to R-matrix type theories in which the problem is solved in the "inside"

region, together with suitable artificial boundary conditions. A scattering state

then must be expanded in terms of the "inside" solutions. In the present treatment, on the other hand, one still has to analyse the solutions to extract the

parameters of the S-matrix. This is similar to the description of the continuum

in a shell-model framework^ ''"^'^ We shall discuss this procedure first for the

case of a single open channel, and we use the system B = 1, J = 1/2 , i.e., a

nucleon, as an example.

We expect the following spectrum: a single bound state, i^e., the state of

the free nucleon, and after a gap corresponding approximately to the mass of

Tr-»meson a discrete spectrum of states which correspond to the p-wave rr-mesonnucleon scattering continuum. Above the 2-Tr threshold a new set of states would

appear which correspond to the 3-body system tt-tt-N. As long as no photons are

included in the set of basis states no radiative pion capture can take place and

below the 2tt threshold the meaning of the states is unambiguous.

Let us now consider specifically a state, say Y, obtained by diagonalization

of the Hamiltonian, From the energy eigenvalue one derives by means of (1.56),

(1.58) the intrinsic energy of the system. Using the previously obtained rest

mases of the nucleon and the pion one can derive the discrete relative momentum

of the scattering state, and one can verify that ¥ belongs to the single channel

region. Thus this solution must have the following character: at small |x^| the

solution describes a complicated compound system; at jx^j > R^ + R^, i.e., in the

-134-
C.2

"as)nnptotic" region, the system must break apart into a pion and a nucleon. Here

and are the nucleon and pion radii.

We now discuss the form of the solution at this asymptotic separation. It

can be constructed as follows. Consider the states ^„ and f of the nucleon

N n

-> ^

and TT -meson, boosted to p and -p respectively. According to (B.8) the boosted

intrinsic parts of the configurations are given by

(c.i)

-> Here and below p and r are the relative momentum and coordinate, respectively.

_ -» They are obtained by putting in (A. 24) through (A. 31) - t^-^ = nucleon CM

_ -> -» position and ~

^tt ~ P^°n CM position, etc., and taking R = P = 0 since here

we are considering the scattering state in the intrinsic, i.e., the CM system.

-> -> Finally, the F. are measured from r„ and the from r . The relative motion is

' ^1 N 'J provided by the expansion of a plane wave into multipoles in which in the radial

part j^(pr) is replaced by

F^^(pr) = cos6j^^ h^^^^ "\(P^) (^-3)

S are the phase shifts for a given scattering state. A particular

^ J system then can be extracted by angular projection.

Thus, a general intrinsic scattering state (the CM wave function is still

absent) is of the form

-135-
c.3

With the abbreviation, in analogy to, say, (3.14)

We have

(C,6)

and thus' the intrinsic conf igurational scattering state of a given multipolarity

is of the form

(C.7)

where „1 is a normalization constant. Replacing now the coordinates ^£>Sj, by

means of (A. 3), (A.32),(B.3) by the parton laboratory coordinates, equation (C.7)

is already almost in the foirm (B.2). To achieve that form one must rewrite the

multipoles making up the nucleon system and the pion system: they are still

^17]

written about the points r,, for the nucleon and r for the pion. They can be

rewritten about the center of mass point by means of the translation operator.

The state which one obtains in the diagonalization of the Hamiltonian, say Y^,

thus is (implicitly) identical to the state, say Yg , which one obtains from the

rewritten equation (C.7) when multiplying each configuration (C.7) with the CM

functions 'V^i^^^) , multiplying with the amplitudes X^^^ and X^^^(cf. Eq. 1.37)

of the solutions for the nucleon and pion, respectively, and summing over the

configurations:

y = s xf,^^ x^^^ x^^.^\?.,?:) CP (t ) (C.8)

-136-
C,4

One then obtains cos6^^ and sin6T^j by computing Y^(r) at two values of r, say

r and r , which are suitable roots of

a p

j;^(pr^) = 0 (C.9)

n^(prp) = 0 (C.IO)

and computing the overlaps of (f Y (r )) and (Y Y (r„)). The ratio of these ^ " n' s a n' s

two overlaps equals -cot6 .; cf.(C.3). (Of course, in our example j = 1/2,

\ = 1,) We recall here that this is true only for non-relativistic CM motion.

The relative motion between the scattering particles, however, is allowed to be

relativistic . Note, that for non-relativistic relative motion no Lorentz contraction cakes place and the sum over k, in Eq. (C.7) collapses into a single

term (k = 1/2, = 0 for the considered nucleon-pion system).

We shall not discuss here the analysis of in the case of several open

channels since no new problems arise in that case. Finally, of course, the

scattering states are correct as far as the CM motion is concerned. No recoil

or relativistic corrections have to be made.

-137-
D.l

D. THE PHYSICAL VACUUM

" The lowest eigenstate, say |v), obtained in diagonali zing the pseudoHamiltonian for a sj'Stem having the quantum numbers of the vacuum is the (not

manifestly covariant) physical vacuum state of the model field theory. All

energies have to be measured from the expectation value of the Hamiltonian for

this state:

= (v|h|v) (D.l)

This has been incorporated in (1.56), Chapter I.

The vacuum state arises as a consequence of the matrix elements of the kind

of Fig. 5.7(a) and its time reversed form, or of the similar matrix elements of

Fig. 5.13, i.e., of those matrix elements which connect the "ground configuration"

jo) with a configuration containing partons. (These matrix elements thus are

proportional to the P = 0 component of the parton configuration.) This situation

is familiar in the non-relativistic shell model in which the grounu state of a

many-body system is not equal to the ground configuration.

Owing to the finite size of the model Hilbert space it is always possible

to construct a unitary operator which allows to represent the eigenstates on the

basis of the physical vacuum state, |v), instead of on the basis of the ground

configuration 1 0> , Wfe now list the principal characteristics of |V) :

|V^ = v^|0^ (D.2)

v|0> = 0 (D.3)

<0|vv"^|0) = 1 (D.4)

3^v"^|0) = e^v'^IO) (D.5)

yiv^v'\Q>) i \v'^v"^;0) (D.6)

Thus, for an eigen - state vector, say | s) = s"*"] O) , we have

Is) = s"^ |0) = ct"^|v> (D.7)

This defines implicitly the physical state vector a"*". Note that an explicit

-138-
D.

definition is not easy since only {o|[v,v"''] |0> = 1 while [v,v''']_ is not a

c-number. At any rate, the physical state vector a"^ shows which parton configu

rations in excess of the phys ical vacuum make up the physical particle. (Note,

that in non-relativistic physics the definition of an operator anologous to

(D.7) is possible, and, in fact, particularly in nuclear physics might be an

interesting quantity to investigate.)

In the model of Chapter VI, | v) will have the configurations: |0), | it )

,

I jt ) , |n ), I Jtuu) , |n p) . At higher truncation energy more configurations will

participate. If a neutral scalar meson field (a-model) is included, |v) would

2 3 contain likewise terms of the form | ct) , |ct ), | ct ) , etc. In writing out the

configuration mixture of | v) one must supply a particular creation operator,

v^ , to generate the amplitude of the ground configuration |o) in | V) , in order

to achieve (D.3) and (D.4). Thus one must write

v"*" = Av^ + Ba"'"a"'' + Ca"'"a''"a''"a"'' + ... (D.8)

where v"*" and v obey the usual Boson commutation relations

o o

[v^,v+] = 1 (D.9)

Equation (D.8) above has been written symbolically, omitting the integrations

over momenta and summations over spin and isospin, etc.

In time -dependent perturbation theory the structure of the physical vacuum

can be eliminated from explicit treatment by omitting the disconnected graphs

and by ignoring the Pauli principle within the diagrams, and by dividing the

S-matrix elements by the vacuum expectation value of the S-matrix. (See, e.g.,

Ref. 5, Chapter 7.2.)

-139-
ACKNOWLEDGEMENTS

The authors acknowledge useful discussions with many colleagues too numerous

to be listed individually. They are thankful to R. W. Hayward for communication

of his 8-component representation of spin 1 fields prior to publication, and for

permission to use this formalism in the present work. They gratefully acknowledge

the help of the Documentation Group of the Service de Physique Theorique at Saclay

in preparing the text. They particularly wish to thank Mile Daniele Dubost who

carried out the typing of the manuscript in a form suitable for direct reproduction.

- 140 -

I
R.1

REFERENCES

1 - DANOS M. - in High-energy Physics and Nuclear Structure, S. Devons ed.,

p. 811, Plenum Press (1970)

2 - WENTZEL G. - Quantum Theory of Wave Fields, Interscience, New York (1949)

3 - BOGOLIUBOV N.N. and SHIRKOV D.V. - Introduction to the Theory of Quantized

Fields, Interscience, New York (1959)

4 - GASIOROWICZ S. - Elementary Particle Physics, Wiley, New York (1966)

5 - LURIE D. - Particles and Fields, Interscience, New York (1968)

6 - BERNSTEIN J. - Elementary particles and their Currents, Freeman, San

Francisco (1968)

7 - ROMAN P. - Introduction to Quantum Field Theory, Wiley, New York (1969)

8 - PRYCE M.H.L. - Proc. Roy. Soc. A195, 62 (1948)

9 - DANOS M. - Ann. Phys. New York 63, 319 (1971)

10 - FANO U. and RACAH G. - Irreducible Tensorial Sets, Academic Press, New York

(1959)

11 - EDMONDS A.R. - Angular Momentum in Quantum Mechanics, Princeton University

Press, Princeton (1957)

12 - MESSIAH A. - Mecanique Quantique vol.2. Appendix A, Dunod, Paris (1959)

13 - HAYWARD R.W. - Private communication and to be published. We thank Dr. R.W.

Hajward for permitting us to use his results prior to publication.

14 - JACKSON A.D. and MAXIMON L.C. - SIAM J. Math. Anal. 3, 446 (1972)

15 - STRATTON, MORSE, CHU, LITTLE and CORBATO - Spheroidal Wavefunctions , MIT

Technology ^ress and Wiley, New York (1956).

16 - MAHAUX, C. and WEIDENMULLER, H. A. • Shell Mpdel Approach tp Nuclear

Reactions, North Holland, Amsterdam (1969).

17 - DANOS, M. and MAXIMON, L. C. - J. Math. phys. 6, 766 (1965).

USCOMM-NBS-DC

- 141 -
NBS-114A (REV. 7-73) -2_

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBS Monograph 147

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Reiativistic Many- Body Bound Systems

5. Publication Date April 1975

6. Performing Organization Code

7. AUTHOR(S)

Michael Danos and Vincent Gillet

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2400104

1 1. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP) 13. Type of Report & Period

Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number 75-26975

16. ABSTRACT (A 200-word or less {actual summary ol most si^ificant information. If document includes a significant

bibliography or literature survey, mention it here.)

The principles and the mathematical details of a fully reiativistic nuclear

theory are given. Since the concept of nuclear forces is a strictly non-relativistic construct, it must be abandoned and the forces must be replaced explicitly

by their physical origin, i.e., by the interaction between nucleons and mesons.

Thus, in this monograph the description of a nucleus has been formulated as a problem of reiativistic quantum field theory which is solved by nuclear physics

methods. To wit: The physics is described by specifying a Lagrangian which is

a functional of the constituent fields ( = of the parton fields). The solutions

for the physical systems then are obtained in a time-independent treatment as expansions in the parton fields: both particles and nuclei are composite systems,

made up of parton configurations, which define a representation of the Hamiltonian

(associated with the specified Lagrangian). The Hamiltonian is truncated by

omitting all configurations having a diagonal element exceeding that of the lowest

configuration by a pre -determined value, Emaxi ^^i^ is diagonalized. All fomulae

needed to carry out this program are derived and given in full detail for spin

0, 1/2, and 1 parton fields for PS, PV, and cp^ interactions. Particular attention

is devoted to the center-of -mass position coordinate which in reiativistic kinematics is a non-separable many-body operator. Finally, the configurations up to

^ax ~ ' listed for the nucleon, the deuteron, and the pion.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Conposite particles ; interacting quantum fields; nuclear structure;

paj-ticLe structure; reiativistic bound systems; reiativistic nuclear physics.

18. AVAILABILITY [X] Unlimited

I ! For Official Distribution. Do Not Release to NTIS

I X! Order From Sup. of Doc, U.S. Government Printing Office

Washington, D.C. 20402, SD Cat. No. C13>, rrinting u 44:H7

I I Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151

19. SECURITY CLASS

(THIS REPORT)

UNCL ASSIFIED

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

149

22. Price

$2.15

USCOMM-DC 29042-P74

.1
NBS TECHNICAL PUBLICATIONS

PERIODICALS JOURNAL OF RESEARCH reports National Bureau

of Standards research and development in physics,

mathematics, and chemistry. It is published in two sections, available separately:

• Physics and Chemistry (Section A)

Papers of interest primarily to scientists working in these fields. This section covers a broad range of physical and chemical research, with major emphasis on

standards of physical measurement, fundamental constants, and properties of matter. Issued six times a

year. Annual subscription: Domestic, $17.00; Foreign,

$21.25.

• Mathematical Sciences (Section B)

Studies and compilations designed mainly for the mathematician and theoretical physicist. Topics in mathematical statistics, theory of experiment design, numerical analysis, theoretical physics and chemistry, logical

design and programming of computers and computer

systems. Short numerical tables. Issued quarterly. Annual subscription: Domestic, $9.00; Foreign, $11.25.

DIMENSIONS/NBS (formerly Technical News Bulletin) —This monthly magazine is published to inform

scientists, engineers, businessmen, industry, teachers,

students, and consumers of the latest advances in science and technology, with primary emphasis on the work at NBS. The magazine highlights and reviews such

issues as energy research, fire protection, building tech- nology, metric conversion, pollution abatement, health and safety, and consumer product performance. In addition, it reports the results of Bureau programs in measurement standards and techniques, properties of matter and materials, engineering standards and services, instrumentation, and automatic data processing.

Annual subscription: Domestic, $9.45; Foreign, $11.85.

NONPERIODICALS

Monographs—Major contributions to the technical liter- ature on various subjects related to the Bureau's scientific and technical activities.

Handbooks—Recommended codes of engineering and

industrial practice (including safety codes) developed

in cooperation with interested industries, professional

organizations, and regulatory bodies.

Special Publications—Include proceedings of confer- ences sponsored by NBS, NBS annual reports, and other

special publications appropriate to this grouping such

as wall charts, pocket cards, and bibliographies.

Applied Mathematics Series—Mathematical tables,

manuals, and studies of special interest to physicists,

engineers, chemjsts, biologists, mathematicians, computer programmers, and others engaged in scientific and technical work.

National Standard Reference Data Series—Provides

quantitative data on the physical and chemical properties of materials, compiled from the world's literature and critically evaluated. Developed under a world-wide

program coordinated by NBS. Program under authority

of National Standard Data Act (Public Law 90-396).

NOTE: At present the principal publication outlet for

these data is the Journal of Physical and Chemical

Reference Data (JPCRD) published quarterly for NBS

by the American Chemical Society (ACS) and the American Institute of Physics (AIP). Subscriptions, reprints,

and supplements available from ACS, 1155 Sixteenth

St. N. W., Wash. D. C. 20056.

Building Science Series—Disseminates technical infor- mation developed at the Bureau on building materials,

components, systems, and whole structures. The series presents research results, test methods, and performance criteria related to the structural and environmental functions and the durability and safety characteristics of building elements and systems.

Technical Notes—Studies or reports which are complete

in themselves but restrictive in their treatment of a

subject. Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject area. Often serve as a vehicle for final reports of work performed at NBS under the sponsorship of other

government agencies.

Voluntary Product Standards—Developed under pro- cedures published by the Department of Commerce in Part 10, Title 15, of the Code of Federal Regulations.

The purpose of the standards is to establish nationally

recognized requirements for products, and to provide

all concerned interests with a basis for common understanding of the characteristics of the products. NBS

administers this program as a supplement to the activities of the private sector standardizing organizations.

Federal Information Processing Standards Publications

(FIPS PUBS)—Publications in this series collectively constitute the Federal Information Processing Standards Register. Register serves as the official source of information in the Federal Government regarding stand- ards issued by NBS pursuant to the Federal Property

and Administrative Services Act of 1949 as amended,

Public Law 89-306 (79 Stat. 1127), and as implemented

by Executive Order 11717 (38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR (Code of Federal

Regulations).

Consumer Information Series—Practical information,

based on NBS research and experience, covering areas

of interest to the consumer. Easily understandable

language and illustrations provide useful background

knowledge for shopping in today's technological

marketplace.

NBS Interagency Reports (NBSIR)-—A special series of interim or final reports on work performed by NBS for

outside sponsors (both government and non-government). In general, initial distribution is handled by the sponsor; public distribution is by the National Technical

Information Service (Springfield, Va. 22161) in paper

copy or microfiche form.

Order NBS publications (except NBSIR's and Biblio- graphic Subscription Services) from: Superintendent of Documents, Government Printing Office, Washington,

D.C. 20402. BIBLIOGRAPHIC SUBSCRIPTION SERVICES

The following current-awareness and literature-survey

bibliographies are issued periodically by the Bureau: Oyogenic Data Center Current Awareness Service

A literature survey issued biweekly. Annual subscription: Domestic, $20.00; foreign, $25.00.

Liquefied Natural Gas. A literature survey issued quarterly. Annual subscription: $20.00.

Superconducting Devices and Materials. A literature

survey issued quarterly. Annual subscription: $20.00.

Send subscription orders and remittances for the pre- ceding bibliographic services to National Technical

Information Service, Springfield, Va. 22161.

Electromagnetic Metrology Current Awareness Service

Issued monthly. Annual subscription: $100.00 (Special rates for multi-subscriptions). Send subscription

order and remittance to Electromagnetics Division,

National Bureau of Standards, Boulder, Colo. 80302.
U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Washington, D.C. 20234

OFFICIAL BUSINESS

Penalty for Private Use. S300

POSTAGE AND FEES PAID

U.S. DEPARTMENT OF COMMERCE

COM-215

FOURTH CLASS MAIL