Collective Effects in Quantum Statistics of Radiation and Matter
Material particles, electrons, atoms, molecules, interact with one another by means of electromagnetic forces. That is, these forces are the cause of their being combined into condensed (liquid or solid) states. In these condensed states, the motion of th
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MATHEMATICAL PHYSICS STUDIES A SUPPLEMENTARY SERIES TO LETTERS IN MATHEMATICAL PHYSICS
Editors:
M. FLATO, Universite de Dijon, France E. H. LIEB, Princeton University, U.sA. W. THIRRING, Institut fUr Theoretische Physik der Universitiit Wien, Austria A. TRAUTMAN, Institute o/Theoretical Physics, Warsaw, Poland Editorial Board:
H. ARAKI, Kyoto University, Japan M. CAHEN, Universite Libre de Bruxelles, Belgium A. CONNES, IH.E.S., France L. FADDEEV, Steklovlnstitute o/Mathematics, Leningrad, U.S.s.R. B. NAGEL, K.TH., Stockholm, Sweden R. RACZKA, Institut Badan Jadrowych, Warsaw, Poland A. SALAM, International Centre for Theoretical Physics, Trieste, Italy W. SCHMID, Harvard University, U.sA. J. SIMON, Universite de Dijon, France D. STERNHEIMER, College de France, France I. T.TOOOROV, Institute o/Nuclear Research. Sofia, Bulgaria J. WOLF, University o/California, Berkely, u.sA.
VOLUME 9
Collective Effects in Quantum Statistics of Radiation and Matter V. N. Popov Leningrad Branch of V. A. Steklov Mathematical Institute of the Academy of Sciences of the U.S.S.R.
and
v. S. Yarunin S. I. Vavilov State Optical Institute
Translated by G. G. Gould
KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON
library of Congress Cataloging in Publication Data Popov. V. N. 'V'~lCr N' ~ o l n v' C~1 I Kol le kl ,,,nye effekl. v ~vanlovQ i sta"."k e 'l lu~~.,nliio vesn~~n1Va . Eno ll Sh] Collect'.,.., .,Ih,on ,n qu .n l .... SUI , sl,es o~ roa,a t 'on ana u lter by u.N . Popov Ind V.S. Var"n,n. p. co. - - 'Matn .. atlca l phys , es st ud,,: , y . g . Tra ns l allOn 0 1 Ko l le~llvnyt efft~1"/ " kVilnl0vo\ SIi1I ' SIIk e Incluoes ,n oex. 8 , o l, o\lr.phy p. IS6/I: 902772735)( 1. I n "gral>o n . Fun~"o na l. 2. Coll e etlv . ,xc"."ons . 3. ~ a., ttr --Eff~ct 01 racut\c n on. 4. Sup,rl lu, a " •. 3. p ~u, Ir,n s loru t'ons n=1,2 ... , 0, n=O,
'" >
nln> = nln>, = 11>, '1'11> = 10>,
~11>=
'1'10>=0.
To construct the coherent states of the Fermi oscillator we use the elements ~ and ~* of the two-dimensional Grassmann algebra, which anticommute with the Fermi operators and ~ and satisfy the relations [20]
'I'
{~,~} = {~., ~.} = {~, ~.}
= 0
(1.6)
Taking into account the equalities: ~
I0> =
~·~, ~ 11>
= ~,
=
I0> -~ 11 >,
. d
o We now turn to the Fermi statistics. Using the above method and formula (1.9) for the Fermi coherent states, we have
trexp(-Ci>~r'l') = fd~d~·eXp(~·~)~r'l')I~> = =
fd~ d~· exp(~·~)n d~id~i x i=1
The commutation relations for the Grassmann elements with different indices have the form {~r,~s} = O{~~,~s}- Taking into account the approximation
8
CHAPTER 1
when n is large. we obtain the following formula for the finite-multiplicity approximation of the partition function: tr exp(-
Jn-,
ro~n) ~ rrd~id~i d~ d~· expCl>. i-1
L [- ~i~i+( 1- OO!) ~i+,~a. no'
=
(1.15)
The boundary values of the Grassmann elements in the latter sum are defined by the equalities ~, = -~. ~~ = ~•.
(1.16)
On passing from the discrete enumeration of the Grassmann elements to their continuous parametriz
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