Table Of Contenthttp://dx.doi.org/10.1090/trans2/006
AMERICAN MATHEMATICAL SOCIET Y
TRANSLATIONS
Series 2
Volume 6
Five Papers on Algebra
and Group Theory
E. B. Dynkin P . K. Rasevskii
M. A. Naimark N . Ya. Vilenkin
Published by the
AMERICAN MATHEMATICA L SOCIET Y
Providence, Rhode Island
1957
Published under a grant from the National Science Foundation
First printing 195 7
Second printing 197 0
International Standard Book Number 0-8218-7106-X
Library of Congress Catalog Number 51-559
Printed in the United States of America
TABLE OF CONTENTS
Page
Rasevskii, P. K. Th e theory of spinors. [PailieBCKMM , II. K. TeopM H
ClMHopoB.] Uspeh i Mat. Nauk (N.S.) 10, no. 2(64), 3-110 (1955) 1
Dynkin, E. B. Semisimpl e subalgebra s of semisimple Lie algebras.
[/IbiHKMH, E. B . IlojiynpocTbi e noflajire6pbi nojiynpocTbi x ajire6 p
JIM.] Mat . Sbornik N.S. 30(72), 349-462 (1952) Il l
Dynkin, E. B. Maxima l subgroup s of die classical groups . [/IblHKMH ,
E. B . MaKCMMajibHbi e noArpynnbi KJiaccMqecKMx rpynn.] Trud y
Moskov. Mat. Obsc. 1, 39-166(1952) 24 5
Naimark, M. A. Linea r representations of the Lorentz group. [HaMMapK,
M. A. JIwHeiiHbi e npeflCTaBJieHwa rpynnbi JIopeHija. ] Uspeh i Mat.
Nauk (N.S.) 9, no. 4(62), 19-93 (1954). 37 9
Vilenkin, N. Ya. O n die classification o f zero-dimensional locall y
compact abelian group s with an everywhere dense set of elements
of finite order. [BiiJieHKMH , H. fl. K KJiacCM(J3MKaiyiM HyjibMepHblx
jioicajibHo KOMnaKTHbix aSejieBbix rpynn co BCiofly IUIOTHMM MHO)KecT-
BOM 3jieMeHTQB KOHeqHoro nopaflKa.] Mat . Sbornik N.S. 34(76),
55-80(1954) 45 9
iii
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http://dx.doi.org/10.1090/trans2/006/01
THE THEORY OF SPINORS
P. K. RASEVSKII
Introduction
The theory of spinors, especially in several dimensions, is only weakly rep-
resented in our mathematical literature. Th e present article aims to remedy thi s
deficiency t o a certain extent.
From our point of view, th e theory of spinors is in the first instance th e theo-
ry of a linear representation of a Clifford algebra, and only incidentally th e theory
of a linear representation of a rotation group. Our exposition is base d upon thi s
principle. We first study in detail th e geometrized Clifford algebra in the comple x
Euclidean space of several dimensions R + (§ § 1-8) , and in particular we examin e
rotations in /?_ + from the point of view of the Clifford algebra (§ 6). We next prove
the fundamental theore m on linear representation of a Clifford algebra, in connec -
tion with which we introduce the concept of a spinor space (§§ 9-12). We consider
fundamental spi n tensors, which appear in a spinor space in connection with the
fundamental automorphism s of the corresponding Clifford algebra (§§ 13-16), and
also the spinor representation of rotations in R (§ § 17-19) . Furthermore, in R*
we single out real Euclidean or pseudo-Euclidean spac e /? „ o f one or another
signature, and specialize and complete the foregoing constructions for this spac e
(§§ 20-26). We establish th e connection with the apparatus of spinors used in
physics (§ 25). Finally, in § 27, we derive in the ^-dimensional cas e certain deli-
cate properties of fundamental spi n tensors, which were previously proved only
for the case n = 2 an d n - 4. I n § 28, we show how the theory of spinors in an
odd number of dimensions can be reduced, in a known sense, t o the even-dimen-
sional cas e (which has occupied us up to this point).
The exposition bears a geometrical character , an d relies upon invariant prop-
erties of the spinor space. Fro m this point of view, th e present article may possibl
be useful also for theoretical physicists who wish to make a thoroughgoing stud y
of the notion of spinor used in physics. In fact, even in the best treatises on quan-
tum mechanics, we constantly encounter heavy emphasis on non-invariant (which
is t o say possessing no physical significance ) properties of the "matrices" tha t
are under consideration. At the same time, their invariant properties are insuffi -
ciently explained. Th e reason for this is tha t in the writings of physicists, th e
colorless pseudonym "matrix" hides a multitude of spin tensors of widely varyin g
nature, the properties of which remain almost entirely in the dark. Even more unde-
sirable is th e fact that relations among spinor quantities are often written in non-inva-
riant form, a circumstance which naturally does not permit the disclosure of their
1
2 P . K. RASEVSKII
physical meaning .
In our exposition, we have tried to establish a direct connection with the ap-
paratus of spinors in physics and to reveal the invariant character of the quanti -
ties ordinarily used in this theory. In this connection, w e have given specia l at -
tention to the space of the special theor y of relativity, an d we interpret all of our
general results specifically fo r this case. However, in addition to this, we have
always tried to illustrate our general results by means of simple specia l cases .
We suppose that the reader is familia r with linear algebra, is able t o carry out
computations with matrices, and also understands the fundamentals of tensor alge-
bra. References t o the relevant literature are found in the text. No other specia l
information is required.
§ 1. Comple x Euclidean space R*
Complex Euclidean space of n dimensions , denoted by R* i s most simply
defined on the basis of the axioms of vector spaces. In fact, w e consider two
classes of objects, "points " and "vectors", t o which we ascribe th e ordinary prop-
erties of 3-dimensiona l vector algebra, with, however, th e following differences :
1) th e operation of multiplying a vector by a number can be carried out for an ar-
bitrary vector and an arbitrary complex number ; 2) th e operation of scalar multi-
plication (formation of the inner or scalar product) yields in general a complex
number (the operation of vector multiplication is not defined); 3 ) th e largest pos-
sible number of linearly independent vectors is not 3, a s in the case of ordinary
vector algebra, bu t is an arbitrary natural number n. (More details about this mat-
ter are given in [1], §§21-23. )
Nevertheless, in the sequel we shall require only vectors o f the space i ? +
(we could in fact completely ignore points of this space) .
As usual, w e shall cal l any set e, , • • •, e o f n linearl y independent vec -
tors of R + a basis.
n
Bases may be chosen, of course, with a great degree of arbitrariness, and in
fact the vectors e, , • • •, e ca n be subjected to an arbitrary non-singular linear
transformation (with complex coefficients) :
V=4'V < L1)
Here the index i denote s summation from 1 t o n (similarl y in the sequel); the e. '
are the vectors of a new basis; the matrix A1.' i s non-singular (here and in the
sequel):
Det\A\i\*0. (1.2 )
Every vector a ca n be written as a linear combination of basis elements :
a=a'e.; (1.3 )
THE THEORY OF SPINORS 3
the numbers a , • • •, a n ar e called the co-ordinates of the vector. In going to a
new basis , th e co-ordinates of a given vector are transformed according to the formula
al'=/t;:'a\ (1.4 )
where A 1, i s th e matrix inverse to A 1, i. Thu s
AlA)' = l), 4 ' 4 = 8;;- (1-5 )
Formula (1.4) is very simple to establish (see for example [1], p. 97). On the other
hand, if for every basis ther e is given a system of numbers a , • • •, an tha t are
transformed according to formula (1.4) upon going from one basis to another, the n
the vector a = ale. i s invariant, sinc e we have
i i
a e . / = a e. ,
as an elementary calculatio n shows . Thus , to define a vector a is the same thing
as to define in each basis a system of numbers a , • • •, an that transform to for-
mula (1.4).
On this basis, we shall also call such a system of numbers a vector. (In a
more general context , it is referred to as a 1 -contravariant tensor. )
§2. Polyvector s
We now consider a system of numbers a 1^ depending upon two indices
{i, j=l,2, • • •, n), given in every basis and trans forming according to the formula
ai'i'^Al'A'/a1' (2.1 )
upon going from one co-ordinate system to another. (Tha t is, th e transformatio n
law (1.4) is repeated for each index.) Such a system of numbers is called a twice
contravariant tensor, and the numbers a l1 themselve s are called the co-ordinates
(or components) of the tensor. We shall be interested only in the special cas e
where the matrix a 1^ is skew-symmetric :
ai>=-a>i. (2.2 )
It is easy t o show that this property is invariant under the transformations (2.1) .
A system of numbers a 1^ - - a*' , given in every basis and transforming according
to the law (2.1) , is called a bivector.
Of particular interest is the special cas e of a bivector that is formed from two
vectors p l an d ql (i n that order) by means of the formula
ij i i _ y. (2.3 )
a = p q p
In this case, th e bivector a 1^ is sai d to be simple. A non-zero simple bivecto r in
real Euclidea n space has an obvious geometric significance. I n fact, specifyin g
such a bivector is equivalent to specifying th e 2-dimensiona l directio n goin g
through p l an d ql, wit h area equal t o the area of the parallelogram spanned by p l
and q J an d with that orientation in the plane p l, q J coincidin g with the direction
of rotation from p l t o ql throug h an angle < n. W e shall sometimes denote a bi -
4 P . K. RASEVSKII
vector a ll b y the symbol a . In *his case, w e denote the formation of a simple bi -
vector a fro m the vectors p an d q b y means of (2.3) with the expressio n
a = P[p,q]. (2.4 )
It is evident that
P[p,q] = -P[q,p]. (2.5 )
We remark that the vectors p , q defining a simple bivector in accordance wit h
(2.4) are determined by P[p, q] t o within a unimodular linear transformatio n
p-apH-Pq, q = 7 P + ^q > |"£ | = 1 . (2.6 )
(For more details on this, se e [1], §§ 35, 37. )
Linear operations on bivectors can be carried out in an obvious way: the bi -
vector c lJ forme d from the bivectors a lJ an d b lJ b y the formula
c1' = a1' + 6l7, (2.7 )
is called the sum of these bivectors .
Furthermore, the bivector c lJ forme d from the bivector a lJ an d the arbitrary
(invariant) number b b y the formula
cij = bali, (2.8 )
is called the product o f die bivector a lJ b y the number b. One verifies in an ob-
vious way that the system of numbers c ll forme d according to formula (2.7) or (2.8)
in each basis, actually forms a bivector. Tha t is, it is skew-symmetri c and trans-
forms according to (2.1).
We now consider the basis bivectors e» » = P[e», e,], tha t is, the simple bi -
vectors that are formed from basis vectors taken in pairs. It follows easil y fro m
(2.3) that the only non-zero co-ordinates a lJ o f the bivector e , , ar e
akl = 1, a lk = -l. (2.9 )
We note also that, as (2.5) shows ,
*ki = -eik- < 2-10>
An arbitrary bivector a (a lJ) ca n be written as a linear combination of basis bivec -
tors, according to the formula
a = 2>i;e , (2.11 )
i;
where the summation is taken over all pairs of indices 1, 2, • • •, n. (The same con-
vention will be followed in the sequel.) The validity of (2.11) follows at once upon
comparing the corresponding co-ordinates of the bivectors on the rigjit and left sides.
It is not difficult t o pass from a bivector to a poly vector o f arbitrary valenc e
k. I n fact, a k-contravariant tensor is a system of numbers a 1 2 k (£ { .. .
?
i, -1, 2, • • •, n), given in every basis and transforming according to the law
THE THEORY OF SPINORS
a1™ •••* = ,ii\Ail. ..A^a™ -'".. _ _ _ (2.12 )
under a change from one basis to another. Th e numbers a 1 2 k calle d co-
ar e
ordinates (or components) of the tensor.
We shall be interested only in the case in which a 7 2 ^ is skew-symmetri c
in all indices, tha t is, is multiplied by - 1 under every odd permutation of indices
(and hence does not change in value under an even permutation). In this case, th e
A;-contra variant tensor a 7 2 k calle d a polyvector (or a A;-vector).
ls
For k > n, all polyvectors are equal to zero, since the n the indices i,, i , • • •,
2
i, mus t include som e that are equal, and in the presence of skew symmetry, thi s
implies that all a 7 2 k equa l to zero. Therefore, w e shall consider poly-
ar e
vectors only for k - 0, 1, 2, • • •, n.
We agree that a polyvector a 7 2 k f k = 0 is simpl y a scalar a . Fo r
or
k = 7, we obtain a vector a 7, for k = 2 a bivector a 7 2 and so on. Fo r k - n ,
?
the polyvector a 7 2 n ha s only one "essential " co-ordinat e a ' " * n. (Th e
other co-ordinates a' 1*2 "''n on e except for a possible chang e of
agre e wit h ±i s
sign iji "••*! , i s a permutation of 7 2 • • • n, or are equal to zero if there are rep-
2
etitions among the indices t ' i • • • i . )
2
Addition of two polyvectors of the same valence and multiplication of a poly-
vector by a number are defined just as in (2.7) and (2.8):
cilm"ik = il'"ik + bil"fik, (2.13 )
a
c'V'^A^fca'V''1"*. (2.14 )
A simple polyvecto r a 7 2 k define d in the following way. Le t
ls
PJ(PJ), P (P2^ ' - * » Pfc(pjfc)> (2 '15)
2
be & vectors, written in a fixed order. Then define
a^-^^pPp'P .. . /4* 1. (2.16 )
The square brackets mean that alternation i s carrie d out on the indices i, i , • • •,
y 2
ii. Tha t is, all possible permutations are carried out on them, and + sign s are af -
fixed when th e permutation is even and - sign s are affixed whe n the permutation
is odd. The resulting k ! expression s are then added together. W e may write thi s
in the following form:
p\x Pi' • • • P\ k
pii
lk _ P\* • • • P[> (2.17)
. ih
Pk p'h • P
We will also write formula (2.16) in the abbreviated form
a=P[ p ... ]. (2.18 )
Pi 2 Pfc
