of real antisymmetr[e tensors of a special form, from the spinor components sA .... A spinor representation of a proper orthogonal group of transformations in the.
REPRESENTATION TENSOR
OF
SPINORS
BY
REAL
AND
COMPLEX
AGGREGATES V.
A.
Zhelnorovich
All independent complex and r e a l c h a r a c t e r i s t i c s of a s c a l a r and tensor nature are determined for spinors in n-dimensional (generally, complex) Euclidian space. A o n e - t o - o n e c o r r e s p o n dence is established between the components of a spinor and of a proposed special complex tensor aggregate C, which is defined by the spinor. In real Euclidian spaces, a homomorphic relation between the components of a spinor and the components of a real tensor aggregate D, defined by the spinor, is given. All the formulas and relationships between a spinor and the a g g r e g a t e s C and D a r e given in p a r t i c u l a r for f o u r - d i m e n s i o n a l Minkowskispace. The r e sultant theory allows one to a r r i v e at some conclusions about the possibility of a m e t r i c d e s cription of the interaction between fermion and gravitational fields. INTRODUCTION This paper is devoted to a study of the geometric and algebraic properties of spinors in n-dimensional + (generally speaking, complex) Euclidian space R n. Section 1 gives the basic definitions and some information needed for the subsequent discussion about spiaor representations of orthogona[ transformation groups in the space R +n. All the independent complex and real characteristics of a scalar and tensor nature, which are invariant with respect to the complex affine transformation group of the space R +n' are established for spinors in R + in Sections 2 and 3. It is shown that in n-dimensional Euclidian space, a spinor with complex components which are defined except for sign is equivalent to a complex tensor aggregate C consisting of aatisymmetric tensors of a special form, the components C 0~I~2" ' " c~k of which are biiinear algebraic functions of the spinor components, and a one-to-one relationship is established between the spinor components and the tensor aggregate C. In accordance with the latter, it is found possible to write down spinor equations in equivalent form as tensor equations in the components of the aggregate C. In real Euclidian (pseudo-Euclidian) spaces, one can form a real tensor aggregate D, which consists of real antisymmetr[e tensors of a special form, from the spinor components sA and from the complex conjugate components cA. In Section 3, it is shown that the real aggregate D defines the components of a spinor except for a common factor exp(i~), and a homomorphism between the spinor components and the components of the tensors D is established. All the general formulas and relations between spinors and the tensor aggregates C and D, and between the tensor aggregates C and D areparticularizedforthecaseoffourdimensional Minkowski space, which is important from the point of view of application. The theory developed in this paper is of particular interest for the clarification and deepening of the physical meaning of basic premises and for the generalization to non-Euclidian space of various physical theories. It should also be pointed out there is a widespread, but erroneous, idea that a description of the interaction between fermion fields and the gravitational field is only possible through the introduction of a tetrad (i. e., nonholonomic, orthonormal coordinate systems). The theory we have developed indicates that the interaction of fermion fields with a gravitational field can also be described in the framework of a metric formalism. This deduction allows one, in particular, to solve in positive fashion the question of the possibility of constructing geometrodynamics for particles with half-integer spin. In regard to this point, it should be pointed out that geometrodynamics for an electromagnetic field was considered unsatisfactory
Institute of Mechanics, Moscow State University. T r a n s l a t e d from T e o r e t i c h e s k a y a i M a t e m a t i c h e skaya Fizika, Vol. 2, No. i, pp. 87-102, January, 1970. Original article submitted June 10, 1969.
9 Consultants Bureau~ a division of Plenum Publishing Corporation, 227 West 17th Street: New York: N. Y. lOOll. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. /1 copy of this article is available from the publisher for $15.00.
66
b e c a u s e of fundamental lack of c l a r i t y with r e s p e c t to the p o s s i b i l i t y of c o n s t r u c t i n g such a t h e o r y for p a r ticles which a r e d e s c r i b e d by s p i n o r equations (for example, see [3]). The e s s e n t i a l d i f f e r e n c e between our r e s u l t s and t h o s e of o t h e r a u t h o r s [4-7] lies in the indication of t e n s o r a g g r e g a t e s in n - d i m e n s i o n a l s p a c e which a r e c o m p l e t e l y equivalent to a s p i n o r and in the e s t a b l i s h m e n t of a o n e - t o - o n e c o r r e s p o n d e n c e between t h e m w h e r e a s only the c a s e of f o u r - d i m e n s i o n a l s p a c e was c o n s i d e r e d p r e v i o u s l y and, m o r e i m p o r t a n t l y , a o n e - t o - o n e c o r r e s p o n d e n c e was not e s t a b l i s h e d b e t w e e n the c o m p o n e n t s of the s p i n o r s and t e n s o r s . Many p r o m i n e n t s p e c i a l i s t s questioned the e x i s t e n c e of a o n e - t o - o n e c o r r e s p o n d e n c e between t e n s o r and s p i n o r c o m p o n e n t s and m a d e e r r o n e o u s s t a t e m e n t s ( W h e e l e r [3], Ivanenko [9], M u e l l e r [10], and others). In our work, methods have been developed for the replacement of spinor equations by equivalent tensor equations; these are methods which are applicable to arbitrary spinor equations while in the work of other authors, certain particular transformations of the Dirac equation in four-dimensional pseudo-Euclidian space proved possible only because of a special form of the Dirac equation, thus making the particular methods discussed in them inapplicable to more general equations (an equivalent tensor description even of the Dirac equation was not obtained previously) [i, 2]. 1.
Spinor
Representation
of Orthogonal
Groups
We f i r s t c o n s i d e r an e v e n - d i m e n s i o n a l c o m p l e x E u c l i d i a n s p a c e R +, n = 2v, r e l a t i v e to the o r t h o n o r real b a s i s ec~ (c~ = 1, 2 . . . . . 2v). Let Yc~ = IlYAc~BII be m a t r i c e s of d i m e n s i o n a l i t y 2u, which by definition s a t i s f y the equation (1.1)
~o.W + Y~u = 2b~].
H e r e 6c~p is the K r o n e c k e r delta, J is the unit m a t r i x of d i m e n s i o n a l i t y 2u.
We introduce the notation
As is usual, p e r m u t a t i o n s a r e c a r r i e d out o v e r the s u b s c r i p t s in the s q u a r e b r a c k e t s . F r o m definitions (t.1) and (1.2), it follows that the s q u a r e of the m a t r i x "/c~lc~... C~k is p r o p o r t i o n a l to the unit m a t r i x and that the t r a c e of m a t r i x yozic~2.., c~k equals z e r o k(k--l)
(~,~o. % ) ~ = ( _ ~ )
2 j, tr7~:...%=O.
(1.3)
Using (1.3), it is easy to show that the matrix system Y, Y~, ~&,~, . . . . 7~...... %
(1.4)
is l i n e a r l y independent. As is well known, any two solutions Yc~, Yc~' of Eq. (1.1) a r e r e l a t e d through the equality y~' .~- Ty~T -1,
det T =# 0.
(1.5)
If y~ is some solution of Eq. (i.i), it is obvious that the transposed matrices ~/c~T are also sotutions of (i.i); it therefore follows from (1.5) that there exists a matrix C = I]CABII such that "V~r ~ C7~C-1,
det C =# 0.
(1.6)
We also define a m a t r i x E E = ]] cAB II ~ i'~(~'-1)/2(7~,~-1... ?,)rC.
(1.7)
Since the m a t r i c e s ~/ce and C a r e not d e g e n e r a t e , the m a t r i x E is also not d e g e n e r a t e as follows f r o m the definition (1.7). We shall d e s i g n a t e the c o m p o n e n t s of the i n v e r s e m a t r i x E -1 as e AB, so that e A B e B c = 5cA. F r o m (1.6) and (1.7), we have [12] CT = (--i)~(v-1)~g,
ET _.= (--1)v(,+1)~E.
(1.8)
Using (1.7) and (1.8), we also obtain
~(~+l)+k(k+l)
(1.9) 67
S i n c e the m a t r i x E is n o n d e g e n e r a t e and the m a t r i c e s J and ~/c~1o~2* . . c~k a r e l i n e a r l y i n d e p e n d e n t , the s y s t e m of m a t r i c e s E, Eye, Eya,~., . . . . EYa,a.~...%
(1.10)
is a l s o l i n e a r l y i n d e p e n d e n t . We now c o n s i d e r the g r o u p L = ][l~ [[ of p r o p e r o r t h o g o n a l t r a n s f o r m a t i o n s
in the s p a c e R+n (1.11)
e~' ~ l%e=.
T h e s e t of u n h n o d u l a r m a t r i c e s
S d e f i n e d by the e q u a t i o n yf~ =
forms a group which realizes L.
representation
l ~ S y~S -~,
(1.12)
of the g r o u p L, c a l l e d the s p i n o r r e p r e s e n t a t i o n of the g r o u p
A g e o m e t r i c o b j e c t r with c o n t r a v a r i a n t c o m p o n e n t s ~bA d e f i n e d e x c e p t for sign, which t r a n s f o r m s + l i k e the r e p r e s e n t a t i o n S, is c a l l e d a s p i n o r of the f i r s t r a n k in the s p a c e R n. By d e f i n i t i o n , the c o v a r i a n t c o m p o n e n t s g'B of t h e s p i n o r ~b a r e g i v e n by the e q u a l i t y ~ ~ e B ~ A, w h e r e eBA a r e the c o m p o n e n t s of the m a t r i x (1.7).
(1.13)
O b v i o u s l y , cA = eABcB"
It f o l l o w s f r o m Eq. (1.12) that the c o m p o n e n t s y A a B of t h e m a t r i x ~ can be c o n s i d e r e d a s c o m p o n e n t s of a s p i n t e n s o r with one c o n t r a v a r i a n t and one c o v a r i a n t index which is i n v a r i a n t u n d e r t h e t r a n s f o r m a t i o n s (1.11). S i m i l a r l y , t h e c o m p o n e n t s eAB(eBA) a r e c o v a r i a n t ( c o n t r a v a r i a n t ) c o m p o n e n t s of a n i n v a r i a n t s p i n o r of the s e c o n d r a n k . In the s p a c e R +, we s e p a r a t e out a p s e u d o - E u c l i d i a n s p a c e R~S) of i n d e x S, s p e c i f y i n g a b a s i s ie 1, ie 2, -+ . . . . i e s , e s + 1. . . . . e n in t h e s p a c e R n. We i n t r o d u c e a m a t r i x II = [[II]~A[[, which is d e f i n e d b y t h e e q u a tions
v~ = ~: ~ f i [ - ~ , ]]fr = (.~)~-~)~ .... ~)/~j. H e r e , the m i n u s s i g n i s u s e d f o r ez = 1, 2, . . . . S, and the p l u s s i g n f o r cz = s + 1 , . . . , T h e r e f o l l o w s f r o m d e f i n i t i o n (1.14)
(1.14) n.
The dot a b o v e
a l e t t e r i n d i c a t e s the c o m p l e x c o n j u g a t e .
(E H) ~ = T
[(#HI) y~. . . . k] . . .
T
= (--
(Eli)',
.~)k(}+l)'2
[(EH)Ty ........ a]'.
(1.15)
S i n c e the m a t r i x (EH) T is n o n d e g e n e r a t e and the m a t r i x s y s t e m (1.10) i s l i n e a r l y i n d e p e n d e n t , the matrix system (Eli) T, (EII) T y ~ , . . . , (EII)Tya,... %
(1.16)
is a l s o l i n e a r l y i n d e p e n d e n t . The spinor ~+with contravariant components r ~+A is c a l l e d c o n j u g a t e w i t h r e s p e c t to ~. b y m e a n s of t h e m e t r i c s p i n o r eAB
A
(1.17)
The c o v a r i a n t components ~A+of the conjugate spinor a r e defined
~ . . . .eAB ~'~A'~'/~,) = f [ ~ l ~ . d-
We now consider the o d d - d i m e n s i o n a l s p a c e s Rn, n = 2u + 1. . . . . Y2v 9 In this case, m a t r i c e s with an even number of indices
(1.18) We use the notation Y2u+ i = iuYIY2,
7, 'Y~.:',. . . . . , Y........ ~v a~ ~ t, 2 . . . . . 2v -}- i
(1.19)
a r e l i n e a r l y i n d e p e n d e n t . A s p i n o r r e p r e s e n t a t i o n of a p r o p e r o r t h o g o n a l g r o u p of t r a n s f o r m a t i o n s in the s p a c e R2+ +1 is g i v e n by a g r o u p of m a t r i c e s S d e f i n e d b y Eq. (1.12) in which the i n d i c e s c~ and fl a r e t a k e n o n
68
values f r o m 1 to 2u + 1. E = [l eAB [I
The c o v a r i a n t c o m p o n e n t s CA of the s p i n o r a r e defined by the m e t r i c s p i n o r (-- t) , u
The conjugate spinor r
Spinor
Eu
is defined in the s p a c e R!S)+t by m e a n s of the i n v a r i a n t spinor 11 = I[IIgA][
Here, the m i n u s sign is used for ce = 1, 2 . . . . . 2.
=
Representation
by
S.
a System
of Complex
Tensors
We c o n s i d e r a c o m p l e x m a t r i x ~ = [IcABIt of d i m e n s i o n a l i t y r. cAB a r e r e p r e s e n t e d in the f o r m
It is obvious that if the c o m p o n e n t s (2.1)
the cAB s a t i s f y the equalities (2.2) Of the Eqs. (2.2), one can take as independent the r ( r - 1) equations (2.3)
q)AAqvBC= ~ a , ~ a c ,
~Aa =/= 0,
B, C :~ A.
Indeed, all the Eqs. (2.2) can be obtained f r o m Eqs. (2.3) and (2.4).
(2 ~ A s s u m i n g CEE ~ 0, we have
C o n v e r s e l y . if s o m e c o m p o n e n t s CAB s a t i s f y Eq. (2.3), t h e r e e x i s t s a s y s t e m of r c o m p o n e n t s CA, d e fined except for s'ign, s u c h that r AB = r162B. In fact, if all the diagonal e l e m e n t s of the m a t r i x IIr ABll a r e z e r o (~BB = 0 for all B), it then follows f r o m (2.3) that r AB = 0 f o r all values of the indices A and B. In this case, we a s s u m e r A = 0. If t h e r e should be one diagonal e l e m e n t of the m a t r i x IIcABI] which is not equal to z e r o , r
r 0, then
we s e t q~A :=
~,A • V ~V~"
(2,5)
B e c a u s e of (2.3), such a definition of the c o m p o n e n t s CA does not depend on the value of the index B. Indeed, if CBB ~ 0 and c E E ~ 0 (E # B), we then obtain, keeping Eq. (2.3) in mind,
~Bx
~EE~;~a
~BZ~Ea
~A
We nowsuppose that the r AB are components of the object q, which transforms in accordance with the representation S • S, where S is any representation of a certain group, and we let Eq. (2.3) and (2.4) be invariant with respect to the group S • S. Then the components C A, defined by (2.5), transform like the representation S. It then follows from (2.5) that the transformation of C A is determined uniquely by thetransformation of CAB. It is obvious that the iavariance of Eqs. (2.3) and (2.4) is preserved if r A transforms like the representation S and consequently can only transform like the representation S because of the uniqueness of C A. Thus the object q~ = IIcABII, which transforms like the representation S • S and which satisfies the i n v a r i a n t equations (2.3) and (2.4), is functionally equivalent to the t w o - v a l u e d o b j e c t C = {cA}, which t r a n s f o r m s like the r e p r e s e n t a t i o n S.
69
In p a r t i c u l a r , the d i s c u s s i o n indicates that a s p i n o r o f s e c o n d r a n k with c o m p o n e n t s ~AB which s a t i s f y the t n v a r i a n t equations (2.3) and (2.4) is functionally equivalent to the s p i n o r r A. B e c a u s e of the c o m p l e t e n e s s and l i n e a r independence of the m a t r i x s y s t e m s (1.4) and (1.19), any m a t r i x of d i m e n s i o n a l i t y 2u, in p a r t i c u l a r the m a t r i x c A B can be r e p r e s e n t e d in the f o r m of a Hnear c o m b i n a tion of m a t r i c e s (1.4) or of m a t r i c e s (1.19): ~v
~A~
=
Ce~S + ~ (-- ~)*-K
2 2v
Here, "y(~1o~2.. .AB
~"
~--~]
'
~=i
(~2k = eBC~'catcq...A Ok"
l"
k~l
In e x p r e s s i o n (2.6a), the s u m m a t i o n is c a r r i e d out f r o m 1 to 2v
o v e r the indices O~k, and in e x p r e s s i o n (2.6b), the s u m m a t i o n is f r o m 1 to 2v + 1. In o r d e r to d e t e r m i n e the c o e f f i c i e n t s C a p p e a r i n g in Eq.(2.6a), we m u l t i p l y (2.6a) by y A . . -~ .y~fil o ~ 9 9 fir and s u m o v e r the indices A and B f r o m 1 to 2v.
We then obtain
J~
=
Ce % V
fir = eAC
(2.7)
j.
+ 2, --roT--',.
We now note that t h e r e follows f r o m the p r o p e r t i e s (1.3) and definition (1.2) of the s p i n t e n s o r s * 7 a l e 2 9
e~k 7A~
,~i,..~r=O,
if
a 1 ,.. c~kYAB
,~P,P
.~AB .... [Jk : atc~2... ale A B
k=p:r,
(2.8) ( - - 5 )k 2vk! tJ[~t'J~.~- . 8ale]. ""
Using (2.8), we d e t e r m i n e the c o e f f i c i e n t s C f r o m (2.7) (2.9)
C . . . . . . . ak = YA~....... k~ AB.
C = (--~)"(v*l)/2eAB ~AB,
In the s a m e m a n n e r , we find the coefficients C in the e x p a n s i o n (2.6b) a r e a l s o d e t e r m i n e d by f o r m u las (2.9). If the c o m p o n e n t s ~AB a r e c o n t r a v a r i a a ~ c o m p o n e n t s of a s p i n o r of the s e c o n d r a n k in the s p a c e R~; 9( R ~ + l), it is obvious that C is an invariant and the c o m p o n e n t s C~ ' 9 9 O~k f o r m i n t h e s p a c e R+v (R+~+0 the c o m p o n e n t s of a t e n s o r of r a n k k which is a n t i s y m m e t r i c in all the indices cq, ~2 . . . . . ok. It t h e r e f o r e follows f r o m (2.6a) that a s p i n o r of the s e c o n d r a n k in the s p a c e R+u is equivalent to the t e n s o r a g g r e g a t e C C = {C, C", C ~'~, . . . .
(2.10)
C ~'~ ..... ~"}
F r o m (2.6b), it a l s o follows that a spinor of s e c o n d r a n k in the s p a c e l ~ + i s equivalent to a t e n s o r a g g r e g a t e c o n s i s t i n g of a n t i s y m m e t r i c t e n s o r s of even r a n k C={C,C
....
,...,C
.......
~'~}.
If the s p i n o r c o m p o n e n t s ~;AB s a t i s f y the invariant equations (2.3) t h e r e then follows f r o m definition (2.9) and the s y m m e t r y p r o p e r t i e s (1.9) 2
is odd,
~
O,
if
2
is odd.
* R e l a t i o n (2.8) can also be obtained by p e r m u t i n g o v e r the indices A and B the equality ,,~A
,D
k __ ~
p (2k-p-l)
Z)~...%~Blt~...pz -- ~p=0 (-- i)
zp6 ap+ 1 r
X 5 [ a ~ .,. ~Ctp
... ~ak] c~k 5 [ ~
which is r e a l i z e d b e c a u s e of (1.1) and (1.2). = [(k + l - 2v + 1 ) / 2 ] . 70
2 5
"'" ~p
kl Z! /~! ( k - p)l (l --p)! 5~p~ ...6Gp ~p
PP8[Jp~-iPp~-I "'" ~ l ] pl X 'VB~p+I A "" ~kPP +1 "'" Pl '
The n u m b e r s h and 0 a r e defined by h = m i n (k, l) and 0
If the spinor components ~bA B satisfy the invariant equations (2.4) the components C, C c~2" "" a k satisfy 2 y (2~ - i) independent bilinear invariant equations which can be obtained by multiplying Eqs0 (2.6) D
~
written for the indices A B and C D and then contracting the result with the splntensor TAD" " ~PVBI~" and taking (2.4) into consideration (for R2p )+
Ch . . ~ p C ~ . - . % _ 2i~-~ ~' ~" ~~, (__~)~t k'l' C~ -k:o
/:0
"
$
(Yq
(2.11)
.... ~C%"'% (~... a k i ~,,CD ' l "'" r ?~"'~V?B~"%> AD
'
and for a s p a c e of 2v + 1 d i m e n s i o n s , v
C~l...~2pC(yl...(y2q
= T2
v
~k~,~
0
'~
= (2k)i (2/)!
C(ll .... 2kcPl...p2t(AB
CD
O~...~2p,%2Gl...(~,~q ~
Y~...%~Y%...02~YAO
Here. zero indices are omitted (C a0 = CP0 = C) and it is a s s u m e d y A B = c A B 9
,uc
(2.12)
,"
7C? = eCD
0! 0
Thus the object q, = {~bAB}, on the one hand, is equivatent to a tensor aggregate C, and on the other, to a spinor ~ = {cA}. Hence the following theorem.
T H E O R E M . A s p i n o r ~ = {cAB} in the s p a c e R +2u(-R 2v + +l) is equivalent to a c o m p l e x t e n s o r a g g r e g a t e C c o n s i s t i n g of the a n t i s y m m e t r i c t e n s o r s (2.9) and (2.10) which s a t i s f i e s 2v ( 2 v _ 1) b i l i n e a r i n v a r i a n t e q u a tions (2.11) -(2.12). In p a r t i c u l a r ,
this t h e o r e m leads to the fact that any s p i n o r equation can be w r i t t e n in e q u i v a l e n t
f a s h i o n a s an e q u a t i o n in c o m p o n e n t s of the t e n s o r s C, 3.
Spinor
Representation
by
Systems
of
C OLl~2.
" "
Real
C~k
Tensors
We c o n s i d e r the r - d i m e n s i o n a l c o m p l e x m a t r i x 45 = lie ~Bll. r e p r e s e n t e d in the f o r m
We a s s u m e the c o m p o n e n t s r
are
(3.1) T h e n the ~b~-B s a t i s f y the equations (3.2) a m o n g which t h e r e a r e the (r - 1) 2 independent equations ( c o n s i d e r i n g c o m p l e x equations twice) (~22~)" = ~i~A,
(3.3)
~AA~;~C = ~;,c g~A"
(3.4)
Indeed, all the Eqs. (3,2) a r e c o n s e q u e n c e s of Eqs, (3.3) and (3.4)
If the components cA define a matrix 45, it is obvious that the components cA exp iq) (~ is an arbitrary real number), and only they, define the s a m e matrix 45. Conversely, if s o m e components pAB satisfxEqs. (.3.3) and (3.4), there exists a system of components ~A defined except for the phase exp iq~ such that r A B = ~ A c B In point of fact, if all the diagonal elements of the matrix 45 are equal to zero (r = 0 for all B), it follows in general from (3.3) and (3.4) that all the elements of the matrix 45 equal zero, ~bA B = 0. Ii that ease, we a s s u m e cA = 0. If there should be a single diagonal element of the matrix 4) not equal to zero, r
~A ~ A - - ],/~B exp(iq)).
r 0, we then set
(3.5)
71
B e c a u s e of (3.3) and (3.4), s u c h a definition of the set {cA} is independent of the value of the index B. In fact, a s s u m i n g that eBB ~ 0, r a 0 (B ~ C), we obtain
Since m o d ~bl~c/Jr I~Cr tended:
= 1, one can s e t r
CB = expiq~ ', and the equality (3.6) can be ex-
= --exp
i (q) + q~').
(3.7)
We now a s s u m e that the c o m p o n e n t s ~AB a r e c o m p o n e n t s of an object which t r a n s f o r m like the r e p r e s e n t a t i o n S" x S, w h e r e S is any r e p r e s e n t a t i o n of a c e r t a i n group, and let Eqs. (3.3) and (3.4) be invariant with r e s p e c t to the g r o u p S" x 8. It is then obvious that one can point out a t r a n s f o r m a t i o n law for ~ such t h a t the c o m p o n e n t s ~ A will t r a n s f o r m like the r e p r e s e n t a t i o n S. Thus the a s s i g n m e n t of the c o m p o n e n t s sAB of an object which t r a n s f o r m like the r e p r e s e n t a t i o n S" x S and which s a t i s f y the i n v a r i a n t equations (3.3) and (3.4) and the a s s i g n m e n t of the p h a s e ~0 for one of the c o m p o n e n t s of cA c o m p l e t e l y defines the object cA which t r a n s f o r m s hike the r e p r e s e n t a t i o n S. B e c a u s e of the c o m p l e t e n e s s and l i n e a r independence of the m a t r i x s y s t e m (1.16), any m a t r i x $A.B of d i m e n s i o n a l i t y r can be r e p r e s e n t e d as ~AB =
j-
(D H BA + '~, k~---.,~'(k-'>/''a" ~.:~.... %J'~ 1) ..... leYa,a
2 ~
(3.8a)
=
~;.aB : _ _ i( DII Bj* q- k~=,l(~k)~..k(2k-3)~a t L, ....... ~leBi y ....... %le]~ , 2~
(3.8b)
w h e r e T~, ~ A e CD YDo~lo~2" B oelo~2.., oek = I IC .. ~ k.
In f o r m u l a (3.8a), the indices o~k take values f r o m 1 to 2v, and
in f o r m u l a (3.8b), values f r o m 1 to 2v + 1.
P e r f o r m i n g a c a l c u l a t i o n s i m i l a r to (2.6)-(2.9), one c a n d e t e r -
m i n e uniquely the c o e f f i c i e n t s D, D ~
" "~
in the e x p a n s i o n s (3.8a) and (3.8b) : D = I!AB~A~, (3.9)
P a~a~ "'" % ~
1,'k(le+l)/2"~ABa~a2... ate~ A B .
c~ic~2...ce k _ liD ,CalC9 9 ..C~k Here, 7 ~ B =~CD Ar B 9 F o r m u l a s (3.8) and (3.9) lead to the fact that the object %AB which t r a n s f o r m s like the r e p r e s e n t a t i o n S' • S in the s p a c e R6 S) is equivalent to the a g g r e g a t e D: D = {D, O ~, D .... , . . . . D~'a' "'%'},
(3.10)
(s).
and in the s p a c e R2v+I , to the a g g r e g a t e D: O = {D, D .... , . . . . D ~....... '"}. If the c o m p o n e n t s r
(3.11)
s a t i s f y Eqs. (3.3), the H e r m i t i a n p r o p e r t i e s of the m a t r i x HT~j~cq" " "c~ k[ l (1.15)
c a u s e the coefficients, D, D c~l~
" " ~k to be r e a l .
If the c o m p o n e n t s $~-B t r a n s f o r m like the d i r e c t p r o d u c t of spinor r e p r e s e n t a t i o n s S'• 8, it is obvious
(s) ,.(s)
that D is an invariant and the c o m p o n e n t s D c~lc~2" " " a k f o r m in the s p a c e R2v ~*'2v +l) the c o m p o n e n t s of a t e n s o r of r a n k k which is a n t i s y m m e t r i e in all indices c~l, a 2. . . . . a k.
72
If the components r s a t i s f y the Eqs, (3.4), the components D, Da 1(~2" " " O~k s a t i s f y (2 v - 1 ) 2 r e a l equations which can be obtained by multiplying Eqs. (3.8) written for the indices AB and ~D, and c o n t r a c t i n g the r e s u l t with the spintensor 7
. . aq taking (3.4) into account (in the s p a c e ~'2v
.. ~,/.
k(k--3)+~(l--3) p(p-}-;)-~-q(q-}-l) 2v
D,% ... ~pD Z~... ,;q =
2 ~"
2v
_ _ k! l!
k~O I~9
D~.-%D~...
~ (.y%..a~,~.~ yDb~...~zyADr~... t!.~yS~ '" ~q)
(3.12)
and in the s p a c e R~vS)+i
D[t~...l:~pD%...~q_
~. ip(2p+D+q(2q+l) Z Z
D~"'a:~D'~"'~(v':~
(2k)! (2l)i
~
v~~P~'~'"%~
(3.13)
F o r s i m p l i c i t y , it is a s s u m e d h e r e vB.~ : II B-/~, 0~ 0
It is c l e a r f r o m the d i s c u s s i o n that the a s s i g n m e n t of an a g g r e g a t e D which s a t i s f i e s the bilinear equations (3.12) and of the a r g u m e n t ~0 for one of the components r A completely defines a spinor. It then follows that spinor equations can be written equivalently in t e r m s of components of the a g g r e g a t e D and ~. Eliminating the p h a s e ~8 f r o m such equations, one can obtain a closed s y s t e m of equations in components of the a g g r e g a t e D. If the components ~bAB and cAB satisfy Eqs. (2.2) and (3.2) it is obvious that r AB and r the equations
also s a t i s f y
thus the components of the a g g r e g a t e s C and D, which s a t i s f y the bilinear equations (2.11) and (3.12), a r e also r e l a t e d by c r o s s i n g equations which can be obtained by multiplying Eqs. (2.6) and (3.8) and c o n t r a c t i n g the r e s u l t with the s p i n t e n s o r s 7, p(P4-t) 2~v
k(I~--3) 2v
2v
~'~ l=3 ~k=o
D~ .... ~C~... ~z~ u~
k! l!
CD
~ .. ~VV~...%).
(3,15)
An equation of another type is obtained by multiplication of equation (2.6) by the complex conjugate of Eq. (2.6) and subsequent contraction with the spintensor y ~ " D ~1"'"~ D ~1"'"~q = ~22~ i P(~+~)+r ~
" fiPYcIB"'
~" ( ~ ~)k.~ C,~...%C,_.~(yAB ~~ ~, k=0l=0
%:
al...a~
k ! l!
~CD py~A..~,f.t.-%~ PI""
I
[CB
(3.16)
/"
Similar equations can be written for the s p a c e R~Sv'+l. r To write spinor equations in tensor form, one can, in addition to f o r m u l a s (2.5) and (3.5), also use the f o r m u l a ~;A =
q/~A
J=Y
and f o r m u l a s (2.6) and (3.8), which define the components r s o r a g g r e g a t e s C and D.
(3.17)
and cAB through the components of the ten-
B e c a u s e of Eqs. (2.3), (2.4), (3.3), and (3.4), the right side of (3.10) is independent of a specified value of the index B. This f o r m u l a m a k e s it possible to obtain a closed s y s t e m of tensor equations in the cornportents of the a g g r e g a t e s C and D. As follows f r o m what has gone before, such a s y s t e m of t e n s o r equations will be equivalent to the original s y s t e m of spinor equations.
73
4.
Spinors
in Minkowski
Space
In Minkowski s p a c e (a f o u r - d i m e n s i o n a l p s e u d o - E u c l i d i a n s p a c e of index three), the m a t r i c e s ya, which s a t i s f y Eq. (1.1), have a d i m e n s i o n a l i t y of four and a r e c a l l e d D i r a c m a t r i c e s . A s s u m i n g that the H e r m i t i a n m a t r i c e s 72 and 74 a r e s y m m e t r i c , and the m a t r i c e s yt and 3/3 a r e a n t i s y m m e t r i c :
we find that the s p i n t e n s o r s E and II introduced by f o r m u l a s (1.7) and (1.14) a r e d e t e r m i n e d in this c a s e by the e q u a t i o n s * E ----y472,
II == iy~/:t~,
(4.2)
with m a t r i x E being a n t i s y m m e t r i c and the m a t r i x Eli being H e r m i t i a n .
We a l s o obtain f r o m (1.12) and
(i. 21)
E~ = --E,
(Ew) ~ = Ew,
( E ~ ) ~ = (En)',
(EWe) ~ = Ewe,
[(En)~wl~=--[(EH)~W]',
(4.3)
[ (En) ~ v ~ ] ~ = - [ (En) ~w~]', [ (EII) ~ W ~ ] ~"= [ (En) ~W~o] ", [ (El-f) ~"W~] ~ = [ (EII) TW~. 1", w h e r e , a e c o r d i n g to (1.2) (4.4) In p a r t i c u l a r ,
one can take as the m a t r i c e s 7~ the following
--i
]
i In that case,
i
IoO
t
I
(4.5)
we have
o_ i
E=
~0t
,
H=
(4.6)
i --i
F r o m the s y m m e t r y p r o p e r t i e s of (4.3), it follows that the t e n s o r a g g r e g a t e C c o n s i s t s of a v e c t o r and an a n t i s y m m e t r i c t e n s o r of the s e c o n d rank, the c o m p o n e n t s of which a r e given by the f o r m u l a s C a ~ y ~ A B ~ A ~ B, C ~ -= y A B ~ A ~ B (4.7) o r , in m a t r i x f o r m , C a -----~ r E ~ ,
C~ = ~rE~.
U s i n g the identities (2.11), one can show that the c o m p o n e n t s C a and Carl s a t i s f y the following i n v a r i ant e q u a t i o n s : C , C ~ = O,
C~C ~ ~
O,
Civic ~
~-
0, C ~ C ~ ~ 0, C[~C~ o] ---- 0, C~C~ -4- Cv~C~ = 0.
(4.8)
Thus a s p i n o r in Minkowski s p a c e is equivalent to the t e n s o r a g g r e g a t e {C a , carl} which s a t i s f i e s the six independent i n v a r i a n t equations (4.8). A r e l a t i o n b e t w e e n the s p i n o r c o m p o n e n t cA and the t e n s o r c o m p o n e n t s C a and C aft is given by the formulas
9 V~
:T
(- ~ ' ~
t~-~.
~).
(4.9)
* In the p s e u d o - E u c l i d i a n s p a c e R4(1) of index one, the s p i n t e n s o r s E and H, in a c c o r d a n c e with (1.14) a r e defined by the equations E = Y4T2, II = Y2 (if the Ya s a t i s f y Eqs. (4.1)). 74
The r e a l tensor aggregate D for Minkowski space consists of the s c a l a r f~, the vector j a , and the a n t i s y m m e t r i c t e n s o r s Me/?, saC?a, and NaP cry, the components of which a r e given by the f o r m u l a s
or, in m a t r i x form,
where r + = {CA~} = (EIIr One finds from (2.1) that the components of the t e n s o r s (4.8) satisfy the following [nvariant equations:
][~S~-~ = 0, S ~ ] a = ~ M ~ @ -~ N ~ a ' l ] f ~ , l
(4.12)
i
The relation between the spinor components cA and the tensor aggregate D in the p a r t i c u l a r ease under consideration is given by the f o r m u l a s @a q)~ ----- y@_~ exp (@),
(4.13)
=~-T h e r e also exists algebraic relations between the tensor a g g r e g a t e s C and D. occur :
The following equalities
4f~ = Cc,d c~-- ~ C ~ d e~, 4~Qj~ = i (C~r '~ -- r
~N ~
