05153 - Linear Representations of the Lorentz Group [Naimark].pdf

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OFTHE
LORENTZ GROUP
M. A. NAIMARK
Translated
by
ANN SWINFEN
and
O.
J.
MARSTRAND
Tl'tlII31ation edited
by
H. K FARAHAT
DEPARTMENT OF MATHEMATICS
THE UNIVERSITY OF SHEFFIELD
PERGAMON PRESS
OXFORD. LONDON - EDINBURGH· NEW YORK
PARI S - FRANK FUR T
1964
PERGAMON PRESS LTD.
Headington Hill Hall, Oxford
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dl:
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Fitzroy Square, London W.I
PERGAMON PRESS ('SCOTLAND) LTD.
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&
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Distributo::d
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THE MACMILLAN COMPANY· NEW YORK
pun.uant
to
d.
special a:mmgemept with
Pergamon Pn-"s Limited
Corvright
@,
1964.
Pergamon Press Ltd.
This jra'lsiallOn
has
been made from M. A. Naimark's
book entitled lIHHF,itHbIE
npE~CTAB.)1EHHH
rpy
nnu J10PEHQA (Lineinyye pr'!{l..tavieniya g·uppy
Loremsa) published in
1958
by Fizf1Ul/giz, Moscow
Library
of
Congre~s
Catalog Card Number 63--10025
PRlNnD IN POLAND
PWN'-DRP
*
E
CONTENTS
PREFACE
CHAPTER
1.
THB THREE-DIMENSIONAL ROTAl [ON GROUP
AND THE LORENTZ GROUP
1.
THE
'THREE-DIMENSIONAL ROTATION GROUP
l.
General detinit.on of a group.
2. Defirntion of the three-tilI!leD.slOnal rotation group.
3. Description of rotations
by
meaDS of orUl'Jgonal ITlatnces,
4.
Eulerian angles.
5. The description of rotation by means of unitary
matfice~.
6. The invariant Integral over
th~
rotation group.
7.
The invariant integral on the unitary group.
2. THE
LoRENTZ GROUl'
I.
The general Lorentz group.
XI
I
I
1
2
2
5
7
13
17
18
18
2..
The complete u..rentz group and the proper Lorentz group.
CHAPTER
II. THE
REPRE'~ENTATIONS
Of-
DIMENSIONAL ROTATION GROUP
23
THE
THREE-
25
25
25
27
3.
THE BASK CoNc-EPTS
OF
THE.
THEoRY Of FINITE-DIMENSIONAL
REPRESENTATIOr s
t.
Linear spaces.
2.
Linear
operators.
3.
Definition
wf
a finit,,-Jimensional representation of a group.
4. Continuous finite-dimensionaI repre<;entations of the three-
dimensional rotation group.
5.
Ur>itary representations.
4.
IRREDucmLE RrPRESF'ITATIOM OF
THE.
THREE-DIMENSIONAL ROfATION
:8
:>9.
30
GROUP IN
I.
2.
3.
4.
5.
6.
INfTI
ITESIMAL FORM
Ddl'erentiability of representations of lhe group
Go
Basic
infinitolSimal
matrice~
of the group
Go'
Basic infinitesimal operators of a representation of
tm
grOlIP
Gt>.
Relations between the bask inlinitesimal operators of a repre-
sematioo of the group
Go.
The conditi.m for a represer'ltation to be unitary-
General form of the basic
inf•
.a.itesimal operators
or
thE irre-
duclb:e
represen~tlons
of the group G
o-
v
31
31
:"13
35
39
41
43
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