Table Of ContentPARTICLES AND NUCLEI
Volume 1,. Part 2
PARTICLES AND NUCLEI
Volume 1, Part 1
Elastic Scattering of Protons by Nucleons in the Energy Range 1-70 Ge V
V. A. Nikitin
Probability Description of High-Energy Scattering and the Smooth Quasi-potential
A. A. Logunov and O. A. Khrustalev
Hadron Scattering at High Energies and the Quasi-potential Approach in Quantum Field Theory
V. R. Garsevanishvili, V. A. Matveev, and L. A. Slepchenko
Interaction of Photons with Matter
Samuel C. C. Ting
Short-Range Repulsion and Broken Chiral Symmetry in Low-Energy Scattering
V. V. Serebryakov and D. V. Shirkov
CP Violation in Decays of Neutral K-Mesons
S. M. Bilen'kii
Nonlocal Quantum Scalar-Field Theory
G. V. Efimov
Volume 1, Part 2
The Model Hamiltonian in Superconductivity Theory
N. N. Bogolyubov
The Self-Consistent-Field Method in Nuclear Theory
D. V. Dzholos and V. G. Solov'ev
Collective Acceleration of Ions
E.
I. N. Ivanov, A. B. Kuznetsov, A. Perel'shtein, V. A. Preizendorf, K. A. Reshetnikov, N. B. Rubin,
S. B. Rubin, and V. P. Sarantsev
Leptonic Hadron Decays
E. I. Mal'tsev and I. V. Chuvilo
Three:Quasiparticle States in Deformed Nuclei with Numbers between 150 and 190 (EfT)
K. Va. Gromov, Z. A. Usmanova, S.1. Fedotov, and Kh. Shtrusnyi
Fundamental Electromagnetic Properties of the Neutron
Yu. A. Aleksandrov
Volume 2, Part 1
Self-Similarity, Current Commutators, and Vector Dominance in Deep Inelastic Lepton-Hadron Inter
actions
V. A. Matveev, R. M. Muradyan, and A. N. Tavkhelidze
Theory of Fields with Nonpolynomial Lagrangians
M. K. Volkov
Dispersion Relationships and Form Factors of Elementary Particles
P. S. Isaev
Two-Dimensional Expansions of Relativistic Amplitudes
M. A. Liberman, G.1. Kuznetsov, and Va. A. Smorodinskii
Meson Spectroscopy
K. Lanius
Elastic and Inelastic Collisions of Nucleons at High Energy
K. D. Tolstov
PARTICLES AND NUCLEI
N. N. Bogolyubov
Editor-in-Chief
Director, Laboratory for Theoretical Physics
Joint Institute for Nuclear Research
Dubna, USSR
/
A Translation of Problemy Fiziki Elementarnykh Chastits i Atomnogo Yadra
(Problems in the Physics of Elementary Particles and the Atomic Nucleus)
Volume 1, Part 2
®
CONSULTANTS BUREAU • NEW YORK-LONDON • 1972
Editorial Board
Editor-in-Chief
N. N. Bogolyubov
Associate Editors
A. M. Baldin Nguen Van Heu V. G. Solov'ev
Secretary
I. S. Isaev
K. Aleksander N. Kroo
D. I. Blokhintsev R. M. Lebedev
V. P. Dzhelepov M. M. Lebedenko
G. N. Flerov M. G. Meshcheryakov
I. M. Frank I. N. Mikhailov
V. G. Kadyshevskii S. M. Polikanov
Kh. Khristov Shch. Tsitseika
A. Khrynkevich A. A. Tyapkin
The original Russian text, published by Atomizdat in Moscow in 1971
for the Joint Institute for Nuclear Research in Dubna, has been revised
and corrected for the present edition. This translation is published under
an agreement with Mezhdunarodnaya Kniga, the Soviet book export
agency.
PROBLEMS IN THE PHYSICS OF ELEMENTARY PARTICLES AND THE ATOMIC NUCLEUS
PROBLEMY FIZIKI ELEMENTARNYKH CHASTITS I ATOMNOGO YADRA
n p06J1eMbi IjJH3HKH 3J1eMeHTapHblX 'faCTH~ H aTOMHoro HApa
Library of Congress Catalog Card Number 72-83510
ISBN-13: 978-1-4684-7661-3 e-ISBN-13: 978-1-4684-7659-0
DOl: 10.1007/978-1-4684-7659-0
© 1972 Consultants Bureau, New York
Softcover reprint of the hardcover I st edition 1972
A Division of Plenum Publishing Corporation
227 West 17th Street, New York, N. Y. 10011
United Kingdom edition published by Consultants Bureau, London
A Division of Plenum Publishing Company, Ltd.
Davis House (4th Floor), 8 Scrubs Lane, Harlesden,
London NWlO 6SE, England
All rights reserved
No part of this publication may be reproduced in any form without
written permission from the publisher
CONTENTS
Volume 1, Part 2
Eng'/Russ.
The Model Hamiltonian in Superconductivity Theory-N. N. Bogolyubov ........... . 1 301
The Self-consistent-Field Method in Nuclear Theory -R. V. Dzholas
and V. G. Solov'ev ............ , ............................ . 53 390
Collective Acceleration of Ions-I. N. Ivanov, A. B. Kuznetsov, E. A. Perel'shtein,
V. A. Preizendorf, K. A. Reshetnikov, N. B. Rubin, S. B. Rubin,
and V. P. Sarantsev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 391
Leptonic Hadron Decays - E. I. Mal' tsev and I. V. Chuvilo .................... . 105 443
Three-Quasiparticle states in Deformed Nuclei with Mass Numbers between 150
and 190 - K. Ya. Gromov, Z. A. Usmanova, S. I. Fedotov,
and Kh. Shtrusnyi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 525
Fundamental Electromagnetic Properties of the Neutron-Yu. A. Aleksandrov ...... . 170 547
THE MODEL HAMILTONIAN IN SUPERCONDUCTIVITY THEORY
N. N. Bogolyubov
A system of fermions with attraction described by the model Hamiltonian in superconduc
tivity theory with separable interaction is considered. Asymptotically exact estimates (as
V - 00) for the minimal eigenvalue of the Hamiltonian, correlation functions, and Green's
functions are obtained.
§ 1. Statement of the Problem
The simplest model system considered in superconductivity theory is characterized by a Hamiltonian
in which only the interaction between particles having opposite momenta and spins is retained:
H -- .~..;. T(t)a,+ af- 2VI .,~. A(f)A(j , )a,+ a+_, a_f, a,., (1.1)
f f. f'
where j = (p, s), s = ±1; p is the momentum vector. For fixed v= L3,
2lt
PZ=L nz,
nx' ny, nz are integers; -j = (-p, -s).
Finally, TV) = (p2/2m)-!l, where!l > 0 is the chemical potential,
-Ill
J'B(S) for I :: ~~,
A(f) = (
o I ::- Ill>~;
for
E (s) = ± I, J = const.
The application of the Bardeen-Cooper-Schrieffer method [1] and the method of compensation of dan
gerous diagrams leads to the identical result in the case given. Moreover, in [2] it was shown that a Ham
iltonian of the type (1.1) is of great methodological interest, since here we have one of the very few com
pletely solvable problems in statistical physics.
In the paper mentioned it is established that for this problem we may obtain an asymptotically exact
(for V - 00) expression for the free energy.
This result was found there in the following manner. The Hamiltonian (1.1) was partitioned into two
parts Ho and H1 in a special manner. The problem with the Hamiltonian Ho was solved exactly. Perturba
tion theory was used to consider the effect of H1• It was shown that any n-th term of the corresponding ex
panSion becomes asymptotically small for V-oo, in connection with which it was concluded that the effect
of H1 may in general be neglected after the transition in the limit V-oo. Of course, reasoning of this kind
,.
Joint Institute for Nuclear Research, Dubna. Translated from Problemy Fiziki Elementarnykh
Chastits i Atomnogo Yadra, Vol. 1, No.2, pp.301-364, 1971.
Cl1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York,
N, Y. 10011, All rights reserved, This article cannot be reproduced for any purpose whatsoever without
permission of the publisher. A copy of this article is available from the publisher for $15.00.
1
cannot pretend to mathematical rigor, but it should nevertheless be underlined that in statistical physics
problems still cruder devices are often used. For example, approximate methods based on selective sum
mation of "principal terms" (in some sense) of the perturbation-theory series are very widely used; here
the remaining terms are discarded even though they do not vanish even for V- 00.
Doubts of the validity of the results of [2] also arise in connection with the fact that various attempts
I-I
at using conventional Feynman diagram techniques (without allowance for "anomalous pairings" at a-f'
I-I
a2:f at. to which canonical 11-, v-transformation leads) did not yield the expected result. Furthermore,
based on the summation of a certain class of Feynman diagrams, Prange [3] obtained a solution which dif
fered in principle from the solution obtained in [1. 2] and assumed that the latter papers were wrong.
In [4] a study was made of a chain of linked equations for the Green's function without the use of per
turbation theory. It was shown there that the Green's function for the Hamiltonian Ho satisfied this entire
chain of equations for the exact Hamiltonian H = Ho + Hl with an error of order 1/V. This substantiates the
results of [2] and reveals the "inefficiency" of the correction H1•
However, one can also dwell on the purely mathematical point of view. As soon as we have fixed the
Hamiltonian, say in the form (1.1), we have an already fully defined mathematical problem which should be
rsolved rigorously without any "physical assumptions." In this case, the approximate expressions satisfy
the exact equations with an error of order 1/V, and we should estimate the difference between the most
exact and approximate expressions.
Having in mind complete parity in the problem of the behavior of a dynamic system having the Hamil
tonian (1.1), we shall take precisely such a purely mathematical viewpoint in this paper.
We shall study the Hamiltonian (1.1) at a temperature () = 0 and demonstrate rigorously that the rela
tive difference (E-Eo) lEo between the lowest energy levels Hand Ho' and likewise between the correspond
ing Green's functions, tends to vanish for V - 00; we shall obtain estimates for the order of decrease.
Based on .methodological concepts it is convenient to cons ider a somewhat more general Hamiltonian
containing terms which represent sources of creation and annihilation of pairs:
3f=.v., .T(f) a,+ a,-'Y~~- 2A -(f)( a_,a,+a,+ a+_I) -2V1 ~~,,.. <f) ,1. . (t')a ,+ a+_I a_I' a,., (1.2)
" '0
l'
where v is a parameter which we shall assume to be greater than or equal to zero.
Let us note that the case v < 0 need not be considered, since it can be reduced to the case v> 0 by a
trivial change in the gauge of the Fermi operators:
+ . +
a, --+ -1 a, .
Let us emphasize the fact that the case v >0 will be considered exclusive of those notions that it is of in
terest in understanding the situation in the actual case v = o.
For the investigation undertaken we shall not need those specific properties of the functions ?.(f),
T(f) of which we spoke above. It will be quite sufficient if the following general conditions are satisfied:
1) the functions ?.(f) and T(f) are real, piecewise continuous, and have the ~ymmetry conditions
A(-f)= -AU); T (-f) = T (f);
2) ?.(f) is uniformly bounded throughout the entire space, and T(f)-oo for Ifl-oo;
3) ~ ~ IA(f)I~const for V-oo;
V I
2
4) lim J..- ~ A' (f) > I for sufficiently small positive x.
v .... oo 2V f YA2 (f) x+ T' (f)
Let us represent ;;e (1.2) in the form
(1.3)
where
(1.4)
(1.5)
Here u is a certain complex number.
Let us note that if u is determined from the condition for the minimum of the least eigenvalue {ffo,
while {f{l is discarded, we arrive at the well-known approximate solution which was considered in [1,2,4].
Here our problem will consist in finding the estimates for the deviation of the minimal eigenvalues 3(0, J£
and for the deviation of the corresponding Green's functions. Let us show that these deviations will vanish
in the process of the transition in the limit V- 00.*
§ 2. The General Properties of the Hamiltonian
1. In this section we shall establish certain general properties of the model Hamiltonian ;;e (1.2).
Let us consider the occupancy numbers nj = ajaj and let us show that the differences nj-o._j are integrals
of motion. Actually,
and likewise
therefore,
Consequently,
ddt (n, (I) - n_, (I)) = O. (2.1)
2. Let us show that for the wave function <D .Yt', corresponding to the least eigenvalue of the Hamil
tonian ;;e, we may place
(2.2)
for any j.
In order to prove this let us assume the opposite. Since (nj-n-j) commutates with;;e (and with one
another) one can always choose <D.1t' in such a way that it is an eigenfunction for all these operators:
* Recently papers have appeared [7-12] in which new methods have been developed for finding asymptotically
exact expressions for multitemporal correlation functions (Green's functions) in the case of arbitrary tem
e.
peratures Estimates were likewise constructed for finding expressions for the free energies in model
systems of the BCS type which are exact for v-co. Based on an analysis and generalization of the papers,
it was possible to formulate a new prinCiple-the minimax principle [12]-for an entire class of model prob
lems in statistical physics.
3
Let us use Ko, K_, K+, respectively to denote the ensemble of subscripts f for which
(n,-n_,) cD.:ft' = 0, f E Ko;
(n,-n_,- I)CD.rt=O, f EK+;
(n,-n_,+ 1) cD.rt =0, f EK-.
This assumption can be reduced to the proposition that the sets K+, K_ are not empty and that*
for any function cp.
Further we shall require that cp satisfy the additional conditions
(2.3)
Let us note now that if f E K+, then nf =1 , I1-f =0 , while if f E K_, then nf =0 ; n_f =1 . Therefore, cD.rt
may be represented in the form of the direct product
where
while cI>Ko is a function of only those nf for which f E Ko:
cDx.=F( ... n, ... ); (cDt,cDx) = 1, fEKo'
Let us note further that
a_, a, 6 (n,- 1) 6 (n_,) = 0; a_, at 6 (n,) 6 (n_,- 1) = 0;
at a~t 6(n,-I) 6(n_f)=0; at a~f b(nf) 6(n_f-l)=O,
and therefore that
if f E K+ for K_. Consequently,
And thus,
* The symbol <cI>*w> will be used to denote the scalar product of the functions cI> and w.
4
--V1 "~" """ A(f)A(f,) a,+ a+_ ,a_"ar }cDK"/,
2 fEKo,'EK, 0
Let us now partition the set K+ + K_ into two sets
in such a way that Q+ will include those subscripts j from K+ + K _ for which T(f) 2: 0, while Q_ will include
those subscripts from K++ K_ for which T(f)< o. In view of the symmetry of T(f) = T(-j) the subscript j
will always be included in Q+ and Q_ simultaneously with -j.
We have
+/"< DK. 'II "~" T (t)nf- -\I "~" /1i oU ) (a_f a,+ a+f a+_ f)
fEK, 2 fEK,
Let us now construct the function q; likewise in the form of a simple product, having placed
where
(Here it is preclsely essential that j belong to Q+ or Q_ simultaneously with -j.) For such a function
As is evident,
On the other hand, the method of construction of q; satisfies all of the additional conditions (3), and we have
arrived at a contradiction with Eq. (2). Thus, our statement has been proved. From the statement (2.2) it
follows, in particular, that the total momentum for cD 3t' is equal to zero:
(2.4)
As is evident from what has been said earlier, the eigenfunction cJ> for the least eigenvalue :Je may always
be sought in the class of functions q; which are governed by the additional conditions (2.3). Let us note that
for this special class of q; satisfying the conditions (2.3) the Hamiltonian :Je may be expressed in terms
of Pauli amplitudes.
Let us consider the operators
bf = a_f af; b+f = a+f a+_ f'
5
