A NOTE ON THE a-SYMBOLS

By HAIuSH-CHANDRA

(J. H. Bhabha Student, Cosmic Ray Research Unit, Indian Institute of Science, Bangalore)

Received June 15, 1945

(Communicated bv Prof. H. J. Bhabha, F.A.SC., F.R.S.)

(1)

(2 a)

-k- ka .=a.

p.). ).p.

).p. ). i m).p. na a. = 8 g - - £0 a a.kilL" "kl 2 klmn IL"

As is well known (see Van der Waerden, 1932) the symbols ak

• are used).p.

to set up a connection between tensors and spinors for transformationsof the Lorentz group. k is a tensor index running from 0 to 3 while,\ and 1J- are spinor indices which can take the values 1 and 2 only.*Hitherto it has been usual to prescribe the numbers a

k. explicitly and to

).p.

show that they remain unaltered when subjected to a Lorentz transforma­tion and the associated spinor transformation simultaneously. In fact thespinor transformation associated to a given Lorentz transformation is, ineffect, defined by this condition. In the present paper no use will bemade of an explicit representation of the a's. All their properties will bededuced from the defining equations (1) and (2). Besides compactness,this procedure has the advantage that the same equations and all theirconsequences remain valid even when the most general transformations notincluded in the Lorentz group are admitted. They can therefore bedirectly taken over to the general theory of relativity (cf. Infeld and Vander Waerden, 1933).

For the present the space-time is assumed flat and the metric tensoris taken to be g/d=O k=l=l, gOO=-gl1=-g2z=-gaa=1. Similarly the

antisymmetric spinors £o"p., £oi..iJ., £o>.w £O>.~ used for raising and lowering the

spinor indices are given by £012= £012 = I, £oi2 = £oli = 1. For any spinor aIL

a>' = £o"p. ap.> a>. = ap.£op.x

with similar relations for the dotted spinors. The a's are defined by the twofollowing conditions:

* In this paper Latin alphabets shall always denote tensor indices the Greek alphabetsbeing reserved for spinor indices.

152

A Note on the o-Symbols 153

Here the bar denotes conjugate-complex and €klmn is a tensor antisymmetricin all the four indices with €Ol2'l=- 1. From (1) and (2a) it follows that

~ ~ i m~ nU <1 • = S. g +" e IT a. (2b)

k I, ILl' I' kl 2 klmn ILl'

The fact that the usual representation of o's satisfies (2) becomes obviouswhen, in conformity with the usual method, one regards cfJ as the two-rowedunit matrix and 0 1, 0 2, cfJ as the three Pauli matrices and compares theircommutation rules with (2). Some results, which are already well known(Infeld and Van der Waerden, 1933; Fierz and Pauli, 1939) followimmediately from (2)

Also

. .k 1fl.I' I kp.I' 2~I' u

'J • 0 + 0 . 0 = (1 g>.p. >.p. .>'

. ..kIp.. I kp.II 2 ~II J:l

a. 0 +0. a = (1 g>.p. >.p. >.

>.p. I 2~1o a. = 0k p.. ·k

k Ip.II I It;" . .klm1l ,.."0.0 -0.0 =-IE 0'.0'

Ap. JI,.. m>.p. n

k 1p.I' kp.". klmlt p."0'. a-a. a = I e 0'. a

1Ip. 1Ip. m>.p. n

(3 a)

(3 b)

(4 a)

(4 b)

From the irreducibility of the Pauli matrices it follows that (Fierz andPauli, 1939)

. .)..p. k _ 2~>' ~/Io

00'.-0.0k"p p"

(5)

However a direct proof based on (2) can be given as follows. Consider

It Ip.·" m0' • 0' 0'. Notice that from (2)

1Ip. "p

k I~p m i kIp'! )..p mE 0'.00"=-2-£ £ 0 ·u u.klmn p.).. p" klmn P, p).. '! p.

- 2i (s sP - 0' . i P) urn. from (3)mn II- n, /10).. m PII

=- 6i 0'''' p.~ from (2). (6)

154 Harish-Chandra

Also from (2)

~IL 1 I ) A ~CI CI . CI . + CI . CI. = a. CI + 3. CI •

k (I, p.v alI I, ILfl av )1 k, af3 f3 s, av

Now notice that+ i 111, AIL (to I 't I),.,€ CI O.CI.+CI.CI. (1)

~ k/11Ilt p.v af3 p.fJ av

111 It "' " 111 n, l' 111 to, fJCI. CI.-Cl. CI.=E 0'. a. +E.. a .. a

Ap. af3 af3 1Ip. All yp. f3 p.f3 IIp 11(8)

Thereforem~p' "I ,,1

E CI (CI.CI.+O'.a.)klmn p.v 118 p.8 all

.i, In, i\ (n I, p. .. I, p. .)="2" € CI CI. a. +CI. 0'. v

kllnn a vp. f3 f3p... .

~ [(~A 111 It, pp. I mlt,.\Y I )=4E o.a.CI CI.-a.CI a.

klmn "ap p.f3 "l' af3. .

+ ( ~ .\ 111 n, pp. I mIt, Al' i)Jo.a.O' CI .-0'.0' a.

f3 ap p." f3y a"., .

(.\ ,\) ~ ,n.\p.(1t / It I)=!i 8. CI . + 8. a . + 4 E a' CI. CI . + CI .0' .II ..e, at> f3 Ie a" k/t,t1t p." af3 p.6 a"

from (6)Therefore

m, ~p. (It i n I) .. ~ .\E CI CI . 0' . +CI • 0' • = 2z (3. CI . -l-- 3. CI . )kim.. 11." af3 p.8 a" y k, afl f3 Ie, all

From (7) and ~9) it follows that

1IP.1, I t)a \CI . a • + a . fT • = O.

k t, p.IJ af3 t, p.f3 a"

Multiplying by /t. and using (2) one getsp)t

I i0' . 0' . + CI . a •= O.

I, p." Qf3 I, p.f3 QIJ

(9)

(10 a)

Now

(10b)

I I Ip'CI . CI . - a • 0' • = E•• 0' . fT

I, p.""'f3 I, p.f3 aIJ vf3 I, p.p a

= 4E •• E from (2)liP p.a

(5) follows immediately from (10).

A few other useful relations can be derived from (2). Put

Ie I, ~p ~ ARltltu. a a.= .

p.A f1P p.IJ

A Note Oil the u-Symbols

Then from (3)

Ak

:m

= Ak

:1Jl

/LIJ /LIJ

Alkm + 2 kl 1Jl= - . g u./LIJ /LIJ

_ A"nk+ 2 kim 2 km 1- . g (]'.- g (]'./LIJ /LIJ /LIJ

_ A mlk + 2 kl 11l 2 km I 2 111/ k-- . g o .- g o .,+ g a :/LIJ /LIJ /LIJ flo"

155

=-A~I/LIJ

2km 1

21mk

- g u.+ g a :/LIJ /LIJ

+2lnt s

g U.IJ,IJ

Adding up the various expressions on the right one gets

6 Aklm _ klmn Apqr + 6 (kl m km I + I" k)• - - E E • g u. - g u. g U.

/LIJ npqy /LIJ iLlJ iLII iLII

so that from (6)

k I, ~P ". _ kl m kin I + 1m k + . klmno . a o . - g a : - g a : g U. 'E a

iL), PIJ iLlJ iLII iLII n, iLII

The conjugate-complex equation is

k I, ),p' m kl m km I Int k • klmtta . a o . =g c , -g a . +g o . -IE U

iL>' PII iLII iLII /LII u, iLlJ

From (3 c) and (11)

k I, i.p 111 n, ~iL 2 (kl 11111 km In + kn It,.+ . klmll)U • U u. a = g g - g g g g IE

iL), PIJ

(11 a)

(11 b)

(12 a)

(12 b)k I,>.p m n, IJ; 2 (il nm km 1m + ht 1m . klmn)a , o a. a = g g - g g g g - I E

/L>' PIJ

From (4) and (12) the following well-known relations for any anti-symmetrictensor Fkl (Laporte and Uhlenbeck, 1931; Fierz and Pauli. 1939) areobtained immediately.

a>. BiLF =1:u If (E f.,+ E .. f )

mn m a~ >.p. >.p. af3

I" um,p.; u~ =4F+mn

p. lip

fP miL~ n 4 F -mn• U (]'.=

P. IJp

(13 a)

(13 b)

(13 c)

156

where

and

Harish-Chandra

. . .r:- ~ P k I,:lp _ ~ p- .i I, ~p. - z a. a - z a. ap. kI u.~ III p.:I

(14 a)

(14 b)

r ' =!(P, _ ip X)

kl NI N

-. xF = t (P +iP ')

kl kl 1:1

FX =t E r'"

kl klml!

It is yet to be proved from (2) that for every proper Lorentz trans­formation there exists a spinor transformation such that the two applied

together leave ak. unchanged. For this purpose it is sufficient to consider>.p.

infinitesimal transformations. An infinitesimal Lorentz transformation isgiven by

, IX =X +e x (e =- e-) (15)

k k I: I kl If;:

where ~ki is a real infinitesimal quantity. Similarly an infinitesimal spinortransformation

(16 a)

(16 b)

is characterised by an infinitesimal spinor '1J",", The symmetry of '1Jp.J1 in1£, v follows from the invariance of E".J" It is therefore sufficient to showthat for every elll there exists a '1JP.11 such that

III ~ ~ ~ keo-. + 1] 0-. + 1]. CT • = O.

1,4>' 4 SA >. 4P.

From (5) and (17) the solution is easily obtained.

kl ~ - III A1] =;} ea. CT 1] .. = 1] =! e 0- • a .

4f3 ~, a.\ I, {J 4~ 4{l N,a.\ I,~

The transformation matrix of (16) is therefore

~p+ ~ III >'/3o "4ECT.O-4 k,4>' I

~ kI •a. + ! E CT • u>.(!4 R, Q~ I

(17)

(18)

(19a)

(19b)

A Note on ike a-Symbols 157

It must now be shown that the transformation (19) considered as a repre­sentation of the Lorentz group satisfies the integrability conditions (seeVan der Waerden, 1932). In any representation of the Lorentz group thetransformation (15) is represented by

1+! ~/IkJ

where Ikl = - Ilk are the representative matrices for the infinitesimaltransformations (see Van der Waerden l.c.). Therefore in our case, from (19)

. .(I )~= 1(0' . ii3 -r- a . (J'Af3) (20)

kta k. aA I I. aA k

where a. and f3 on the left side are to be looked upon as matrix indices.The integrability conditions for the Lorentz group are well known and are

lkll"ln _ Imtt Ikl = _ gklll Iht + tnt Ih t + gk1t I'm _ g'n Tkm (21)

Now from (4) and (20)

(I )? i (I'n1t)~ i t1t 1;),.i3kl .. = -- 2 Eklmn a = - 4: EkimN a a~ a '

Therefore'Y 1 p"

(Ikl Im•t - ImN Ikl)« = - 16 (EkJpq Emnr ." - Emnpq EklrJ a aA aq

),.{) a;~ as'p.y

j k 'l.~P I' s. ~y s r ; ,'Ii q p,p.y= - -I';; Eklpq Emn r1 {a • a a. a - a • a u. a }

v aA pp. aA P/lo

J { 2 pI' q S. p.y + . pqrt s, /loy= - 16 EkJpfJ Emm l - g U. i7 lEU· U

ap. t, ap.

. .s ; rp 'I, "Y • s rqpt,\Y

+2ua"g U +'0'. E (T } from (11)CIA t

8 pr (I'IS Y • P'lrt rrS yEilpq Emnrs {- g ) + 4 I E \L) }

a tilTherefore

I I - I I =! / E I fJ' _!.. E E /,'1rt rkl 1ft" mn kl klq mnrs 4 kip" mnrs t

'II i r t,- t t= -! (;; ± gM! gl1' g,) I +"2 Emnrs (8k 8/-81 8k) Is

Where the sum is to be taken over all permutations of k, 1, q with + or- sign according as the permutation is even or odd. So

I I - I I =- 1- ( E ± g g g) I'll + ~ (E r' - E t)kl 11In 111ft kl klq km In 'Is 2 mnks t mnls k

=![-g I +g I +g I -g I Jkm /n 1m 1m 1m /m 1m kn

+ -1 (E / IPfJ- E l IP'l)

mnis Ipq m"i.t spq

= [-g I +g I +g I -g I JInn In 1m en en 1m In itm

Therefore (21) is fulfilled.

158 H arish-Chandra

It is now possible to consider finite Lorentz transformations. Let thetransformations (15) be denoted by 1 + ! Eel .leI. The transformation matrixof (15) is

Therefore

(23 a)

! (Ekl Jkl)::, = E~n (22)where m and n are the matrix indices of the transformation r: A finiteLorentz transformation L can be generated from the infinitesimal trans­formation by the following common device.

(Okl)" t 01<1 JI.;I

L = 1+ ! - JId = en Lim,,~oo

= 1+.1 Old J + (t Okl JId)2+ (t okl Jk/)3+ ...Z Itl 2! 3!

On account of (22) one can write symbolically02 03

L = eO = 1+°+ 2! + 31 +...where on is a matrix defined by induction as follows:

(fr+ 1)/ = (0")/ 0/It is easy to verify that° olm° O"p + 1 (fJ o"m) ° Olp _ (.1. fJ'm O'lY E )2 ~P - 0kl mt' "Z mft kf II lmnr I.; -

for any tensor Old whose only non-vanishing components are 003 = - 030

and fJ12 = - 021' However since every antisymmetrical tensor can be broughtto this form by a Lorentz transformation it follows that the above identityis valid for every O,el' Written in matrix form it becomes

(J4+ tOkl okl 02 - (i old 8mnEldmJ2 = 0

Making use of this characteristic equation of 8, it can be proved that

h . - h ._-] cosh ytI> t - cosh *- (02 .10 kI\L= t [cos ylP t + cos ,/lP_ + ----tI> _ q,---- +"4 kl O )+ -

+ t [Sinh -JiJff-+ SinhviP_]°ycfJ+ ytI>_

+ (Sinh _ytI> t _ sinh ytI>-) 1.. (fJ~+ .i, 0 Old. 0)y tP-i- ",liP:.. tI>+- tI>_ "4 kl

(24)

wheretI>' i: - __1 fJ okl -l-.1 -/ J_Okl Omit "+ (8k/ 0 2 (23 b)± - "4 kl ~"4 V t2 I:klmn k/

This is the expression for the most general proper Lorentz transformation.

The spinor transformation A associated to (23) is given by

A = (1 + t okllkl ) " = et 9 1<1 11<1

n Lim,,-'=

(1_ okl I )2=1+(tokII

kl)+ Z 21 kl + .....

Now

A Note on tlte (J-S')Imbols 159

Put

. .1 0 0 ( 2 km I n, WI +. klmp ft, p.y) f (11)= - - g (J. (J lEa. (J rom16 kl mil ap. p, ap.

= - tOll 8Y+ i 0 8 limp {8'~ 3Y- i E" (e)Y} from (2)kl a 16 kl fltlt P a prs II

= ( - t 8 Okl + l.- 8 (J limn) sY. kl 16 kl mn a

e = - t 8 II+ L 8 0 limn (25)kl 16 kl mil

Then from (24),

A = ( 1+ :!+ ~~+ ...) + ( 1+ :! + ~~+ ...) (1 III I k/)

= cosh v9+ s~te (! OklIkl) (26 a)

or

A IS = cosh v& 3P+ sinh :Ie t III (J .» (26b)a a vB k,o..\ I

Therefore the spinor transformation associated to any given Lorentz trans­formation is completely determined. As an example consider the case ofa spatial rotation about the X a axis. In this all components of 0kl are zeroexcept 812 = - 821 = 2r/J (say). Therefore

8=- ,p2and

. .p sin r/J 0 3>'/1 3 0, ).{l.

= cos r/J 8 +-.- t En03 «(J • (J - (J • (J )a 1 ll). ll"-

. .p . 0 3, .\ f1 3 e, )./1

= cos rP ~ + ism rP t \.(J • (J' - a .:T )II ll). a.\

\27 a)

Similarly for a Lorentz transformation along the .':3 axis the only non­vanishing component of Ok! is 803 = - 630 = 2ifJ (say), so that 9= r/J2 and

~27 b)

160 Harish-Chandra

In the usual representation 0° is the unit matrix and o3.a./3 = - o~f1' Thereforein matrix notation (27£1) and (27b) can be written as

A = cos eP - i sin eP 03

A = cosh eP - sinh epa3

where the matrix elements of ·r are l1~ft.

Till now only the proper Lorentz transformations have been discussed.Reflection can now be included in the following way. From (11)

0, ,\ k 0, p _ 2 0.': . 0 k _ H: k11 11. 11. - g ., .-I1.-g (1.

P, >.p v /l-V p,v p,/I(28)

(29)

Therefore I1~X is the reflection matrix, and by reflection the spinors a,i. andbp. go over into ap' and b,i. given by

0, ~ • 0,>. ba = 0 tl. , o. = 11 •P. P. .\ p. p. >.

The quantities ~;, remain unchanged for the simultaneous application ofreflection to the tensor index k as well as to the spinor indices p., v. In theusual representation (T~j, is the unit matrix and therefore the only non-

vanishing components of 11~' ~ are 11~' 2= - 11~' i = - 1. In this case there­fore (29) coincides with the usual rules for reflection.

If ~i< and 11~p. be 1\\0 different sets of I1'S satisfying ~l) and (2) then itfollows from (5) that

't k l11.=a 11 •

.\fIo I >.p.where

k 'k ~p.a = to' . 0'

t >.p. tso that a~ are real and

kl"m kla a g =gHZ n

Therefore a~ must be the coefficients of a Lorentz transformation apartfrom the fact that they may reverse the direction of time.

Since the equations (1) and (2) are in a proper covariant form theyremain valid for all real transformations of the tensor space and anyarbitrary transformations of the spin-space (qf. Infeld and Van der Waerden,1933). However it must be borne in mind that for this general case

Eotta=- v'-g

~12S= --2­v'-g

A Note on the a-Symbols 161

where g is the determinant of the general gig matrix. Also ~12 and ~u

are no longer 1 but are equal to 'Y and ~ respectively where 'Y is a spinor'Y

density of weight 1, i.e., on transformation it gets multiplied by the deter-minant of the transformation in the spin-space (Infeld and Van der Waerden,1933). All the results [e.g., equations (5), (11) and (12)] therefore hold alsofor the general case which is of importantce in the general theory ofrelativity.

In conclusion let us consider an interesting application of (11) to theDirac equation of a particle of spin t.Expressed in terms of spinors itsplits up into the following two equations:

(33 b)

(33 a)

(Ik ()k ax+ x2 ax= 0 (32)which follows from them. This becomes obvious if one looks upon (30a)as the definition of b", Therefore (30) can be replaced by

k kJ a. = X a.

A A

£)kl=-x ax.\

i £)a}. ax= xb: (30 a)i ()ai ba = Xai,. (30 b)

To these are to be added the corresponding conjugate-complex equations.Apart from numerical factors the charge-current-density spinor is given by

Sa~= aa ati+ ba b~ (31)where aa. and b~ are the complex-conjugates of ali and b~ respectively. Itis obvious that the charge-density given by (21) is positive definite if theusual representation of a'S is used since in this case aD is the unit matrix.For every other representation of a's the charge-density is therefore eitherpositive or negative definite according as this representation is obtainedfrom the usual one by a Lorentz transformation without or with thereversal of the direction of time. However only the definite character ofthe charge-density is of importance, the sign being immaterial.

Notice that the equations (30) are completely equivalent to the secondorder equation

so that

bel • CIA k=la a.

It A

(31) now becomes. .

i m A 1J. (1J. 1J.S . =a a. + a . a. a a • a = a )ap a. p CIA Bp. I 111 m m

162or

Harish-Chandra

1 .1:, ai~ + 1 [ 111 (Il k~+ ,i ,i'f.') /., Il! ~ {" ]= T g a a; 'i a. a a a a - cr. a aa t: I\,!L '1/1 ill All. III

• klmn i Il+ t € a a a . from (11) (34)I 1Il 11, A,u.

The equations (33) and (34) are completely equivalent to the usual formu­lation of the Dirac-equation in the force-free case. The definite characterof the charge-density is not quite obvious from (34). Equations (33) resemblevery much the corresponding equations for a particle of spin O.

However the equivalence of (30) and (33) holds only for the force-freecase. In case of interaction with an electromagnetic field (30) go over into

i 1Ta.l.. a;,. = X ba.

i 1Ta.j... ba. = X a>-

(35 a)

(35 b)

Or from (3)

where 1T";-' is the spinal' corresponding to 1Tk == 0k+ ie CPA!, rPk being theelectromagnetic potentials and e the charge of the particle. The secondorder equation derived from (35) is

1T . 1Ta.p. a .+ x2 a' = 0a.;\ Il A

k k I a~1T 1T a. + -} (1T 1T -1T 1T) a . a' a. + X2 Q. = 0

k A kllkaA Il A

which is not the same as that obtained by replacing Ok by 1Tk in (32). There­fore to take electromagnetic interaction into account it is not sufficient toreplace ci by 1T~ in (33). The correct generalisation of (33) in this case is

,~ k1T_a. = X a. (36 a)

- A ,\

where

k i' Il1T a. + - f. a. = - Xa·k.\ X A Il A

(36 b)

~ k la~f. = t fa. a and f = i) r/> - i) r!>A 'k! Au °kl l: I t :»

The extra term in (36 b) corresponds precisely to the magnetic moment .:X

of the electron in Dirac's theory. The expressions (31) and (34) for thecurrent vector remain unchanged. From (36) it follows that the Diracequation is completely equivalent to the second order equation

1T 1Tk a. + i ef: a. + X9a. = 0k A A {J. A

A Note on the a-Symbols 163

provided the current vector is defined by (34). Equations (36) emphasisethe fact that even in the simple case of spin t correct electromagneticinteraction cannot be introduced simply by replacing Uk by 17k in anyarbitrary formulation which is valid for the force-free case.

SUMMARY

The a-symbols are defined by means of the equations (1) and (2). Alltheir properties are deduced from their definition without making use ofany explicit representation. Certain interesting relations concerning theproduct of three or more a's are obtained. They are shown to be usefulin transforming tensors into spinors and vice versa.

Directly from (1) and (2) it is deduced that corresponding to everyproper Lorentz transformation there exists a spinor-transformation suchthat the two applied together leave the a's unchanged. The spinor trans­formation corresponding to the most general proper Lorentz transformationis explicitly given. Also the spinor-transformation corresponding to reflec­tion is obtained. It is pointed out that since the defining equations (1) and(2) and the relations deduced from them are already in a proper covariantform they can be taken over as such to the general theory of relativity.

Finally the Dirac equation for a particle of spin {- is discussed froma new angle. Here only one spinor together with its space-time derivatives(and nat two spinors) is used to describe the particle. It is shown that theDirac equation is completely equivalent to a second order equation for thissingle spinor. The expression for the charge-current density in tcrms ofthis single spinor and its derivatives is obtained. In the present formulationcorrect electromagnetic interaction can be introduced only by the additionof an extra term depending explicitly on the field. This additional term isthe one which corresponds to the magnetic moment of the electron.

REFERENCES

Fierz and Pauli

Infeld and Van der Waerden

Laporte and Uhlenbeck

Van del' Waerden

.. Proc, Roy. Soc., A, 1939, 173,21)-32.

.. Preuss. Akad. Bed. Ber., 1933, 9. 380-474.

.. Phrs. Rev., 1931,37,1380-97.

. . "Die gruppen theoretische methode in derQuantumrnechanik", Berlin, Springer.