ALGEBRA OF THE DIRAC-MATRICES

By HARISH-CHANDRA

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

Received April 30. 1945

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

§.l. The Dirac matrices Ya (a= 1, 2, 3, 4) are characterised by the'commutation rules

(2)

(1)

Ya

iYaYb (a =1= b)

iyayb"Yc (a, b, c all different)

YaYb+ Y~Y« = 2 Dab.

These four matrices give rise to a set of 16 quantities

1

Ij

which is closed under multiplication if we regard two quantities whichdiffer by a numerical factor - 1, i or - i as essentially the same. As iswell known these sixteen matrices are linearly independent and apart fromequivalence possess only one four-dimensional representation which isirreducible. Several important identities concerning these matrices, whichare independent of any particular representation have been established byPauli (1936) by making use of the well-known results following fromSchur's theorem. The object of the present paper is to point out that thecommutation rules of the above 15 quantities (all excluding 1) can beexpressed quite elegantly by one single formula and that the above­mentioned identities can therefore be derived directly from the commutationrules. It seems that perhaps the present method is more general andpowerful than that of Pauli. The identities are obtained in a form suchthat the five elements of any pentad (see Eddington, 1936) can be regardedas basal elements. The use of the matrix B of Pauli is avoided so that theidentities (341) and (345) of his paper can now be generalised to the caseep+ =1= .p+, ep =1= 1j;. This was not possible previously. Some new tensoridentities are also obtained.

Further the present method yields quite easily the matrix determinantof a quantity composed linearly from the above sixteen matrices. As is

30

AIg-ebra of the Dirac-Matrices 31

well known this determinant is independent of the representation used.Eddington (1936) has already calculated it by using the rather cumbersomemethod of employing a particular representation. As a physical applicationof the evaluation of this determinant we shall consider the case of a chargedparticle of spin -1 having an explicit spin interaction with the electro­magnetic field. It will be shown that apart from quantum effects andupto the first approximation the particle behaves as if it possessed a puremagnetic dipole moment, which points either along or opposite to themagnetic field in the rest system. This magnetic moment which arises fromthe explicit spin interaction is to be distinguished from the usual magneticmoment of the electron which is due purely to quantum effects, and whichwould therefore disappear if the non-commutability of the different operatorswere ignored.

(3)

(4)instead of Ya' Therefore

§ 2. For the purpose of this paper it 1S more convenient to multiplythe y's by i and use

The matrices E1> E3, E3, E4 anticornmute and their squares are equal to- 1. Put*

(5)

(6)

(7)

so thatE52= _ 1, EaE5= - E5Ea (0= 1, 2, 3.4).

Following Eddington we define

E,..p ~ E,..E> (fL, v = 1, 2, 3. 4, 5, f' =f= v)

Eo> := E> } ( 1 2_ V=, .···,5)E,o = - Eo,- E,

E", == 0 (v=O.I,···5).

Then the following equations hold (ef Eddington I.e.) for u, v= 0, 1,"·5

EIt> -= - E>fL (8a)

E,.,> 2 == E,.,> E,.,> = -- 1 (fL 7' v) (8b)

E E = E,'p (p., v, p all different) (Sc)p.> -,..p .EfL > Eq p = Ecr p E", ==j= iE'\T (fL, v, a, p, A, T all different) (&1)

£5 = iEl~E3E~ = iYIY2Y3Yl = irs

1s is the same as in Pauli's paper.

* The present definition of Es differs in sign from that of Eddington. The advantage is thatnow

corresponding to (3).A3

32 Harish-Chandra

In (8d) the positive or the negative sign is to be chosen according as(fL, v, a , p. A, 7) is an odd or even permutation of (0, 1, 2, 3, 4, 5). It iseasy to see that (8) is equivalent to the single equation

EA,u. t., = - 0AP D,u.p + 0,u.p 0AP + E AP 0,u.p - Ej.Lp DAp

(9)

Here o/-,p is the usual Kronecker's symbol

'" IlfL=vo _J

/-,P - lO fl- =/=- "

and EAu.vprrr is antisymmetric in all 6 indices and E012315 = 1. It is convenientto make the convention that the same index appearing once below and onceabove in the same term implies a summation. Thus for example

6

s, VEp" = E Epp Ep<Tv=o

while no summation is intended in the expression

Epp EVlT

In fact (9) can be looked upon as a tensor equation in a six-dimensionalspace whose metric tensor is 'ftj.LP' Equation (9) is invariant to any ortho­gonal transformation of this six-dimensional space, if we regard E.\j.L as anantisymmetric tensor. Also if FA,u. is any set of fifteen E-numberst, anti­symmetric in A, fl- and satisfying the same commutation rules as (9), thenit is not difficult to prove that

FAj.L = a A ro; PEvp

where aAv arc the coefficients of an orthogonal transformation of thesix-dimensional space, i.e.,

E a?lP aj.Lv = 0?l,u.y

I aA" 1=1where I aA" ! denotes the six-dimensional determinant of the transformation.Thus every matrix transformation

F1\1' =AE;\.u A- 1

is equivalent to an orthogonal transformation of the six-dimensional space.The converse is also true due to the equivalence of all four-dimensionalrepresentations.

t Any linear combination of the E', and 1 is called an E-number (cf. Eddington, I.e.).Every matrix with 4 rows and 4 columns is an E-number due to the linear independence ofthe E's and 1.

Algebra of the Dirac-jJ;fatrices 33

For future use we note the following relations which follow directlyfrom (9):-

E,\jL E.p+ E,\. EjLp""- D,\. DiLP- b,lp. Svp+ 2 biL• o,\p+ E,I. bfl.p + E,\I' o.p

- 2 E\p Sp.. T Ep-p 8,\. + E.p ,SAIL (lOa)

E,'" Epp = 5 Atrp - 4 Etrp (lOb)Also if

S = S + S'\iL EAIL (SAIL = - S,.,\) (1ta)

T= t+ t,lJ1. PJ1. (tAJ1. = - tiL),) (11b)

where s, S'\w f. t"Ap. are ordinary numbers it follows from (9) that

ST = st - 2 S,IiL {hP. + (st A,. + tS AIL - 4 S,\p r IL - ; 5"'{3 (1'0 Ea j3 yii,lp.) EA" (12)

so that

(16)

03a)

(15)

(14a)

(14b)

ST - TS = -8 s,\. tV~ EAIL

ST + TS = 2 st - 4 S'\,. tAIL + 2 (st;\IL + tsAp. - ; saj3 ty/j ~aj3I'8>'p.) P.IL (13b)

In particular on choosing S= E",j3' (13) gives

TEaj3 - Eaj3 T= 4 Ua,\ E"j3 - tj3,\ Pa)

TEap+ Eap T = - 4 tat'~ + 2 tEap - ity/j ~apY5}'p ppOn contracting with t\p- (lOa) yields

TE.p+ (hlL E,\. E/Lp = - 3 tvp+ 3 tv'\' E,Ip - tpA EA. + T8.p

+ (E. p - 0vp) t

Multiplying tl4a) by Ef3 y on the right and using (I5) we obtain

Eaj3 TEf1 y = T3ay+ 4 (~ay- Eay) t +- 4 Ua f3 Ep y r- !yf3 Ei3a) - Rta y

On contracting a, y (16) gives the well-known result

. 2 T - Eap TEa,s = 32 t.

Multiplying (l4a) by Ea,s on the right we get

Ea,s TEa,s = - T + 4 t1l.a P a + 4 tAP Pj3 - Stap Ea,s (a =1= ~) (17)

As is well known (cf. Pauli, 1936) it follows from the commutationrules (9) that the spur of EX/L is zero. Therefore for any four-rowed repre­sentation

sp (T)= 4 t

sp (E,IJ1. T) = - S t'\IL

Also since 1, E'\IL are 16 linearly independent matrices T in (lIb) can beany arbitrary matrix of 4 rows and 4 columns. Let if; and <P be any twomatrices with 4 rows and 1 column and 0/+ and eP+ with 1 row and 4

A3a

34 H arish-Chandra

columns. Then ep,p+, t/;t/;+, ,pt/;+ and fc/>~ are square matrices with 4 rowsand 4 columns and we can choose T equal to any of them. We notice thatfor T= ,pep+

t = *sp (T) = i ep+,p (18a)

(If.' = - t sp (E.l.Jl< T) = - 1- c/>+E~j.t cPo (18b)

Substituting these values in (lIb) and (14b) and multiplying by f+ on theleft and f on the right we obtain

:P+cP ',p+f= t ifJ+t/; ',p+¢ - t ifJ+ £All- f· ep+E),p.rP (19)

f+¢·q,+Ea.p¢J+ ¢J+Eai3ep·ep+ifJ= t ifJ+ifJ·ep+Eaf3cP+! f+EapifJ·¢+¢

i+ 8 c/>+EYS,p·ifJ+p·PifJ· !apyS.l.p

On choosing T= ifJ¢+ (16) gives in the same way

¢J+EapifJ·c/>+Ef3acP= ifJ+ifJ·ep+¢+ ifJ+c/>·¢-ft/;- ifJ+Eap¢·t/J+EPtlf

ifJ+Eapt/;·q,+EPy.p = - f+Eayt/J· ep+ifJ + ifJ+¢ ·¢-t-Eayr/J

-! ifJ+Eflyep',p+Ea{Jr/J+! IjJ+Efla,p·,p+E-lifJ, (a::l= y),

Similarly on putting T = q,,p+ (17) gives

¢J+Eap .p·c/>+Ea,p r/J = - ifJ+.p·¢+f - t ,p+ExQc/>'o/+Exa, ifJ

- t ,p+E),p¢·f+Exflf+ f+Eapf·.p+Etlp,p, (a =1= (J).

Now let a, b. c, d be indices which run from 1 to 4 only.following Pauli (1936) we put

ifJ+ifJ == iQo 1

f+EaaifJ= ifJ+Earf= iifJ+YaifJ == rs, \ifJ+Eair/J= - f+YaY5r/J= if+Yar/J :;; iSa Ir/J+Eabr/J=- r/J+YaYbr/J == Mao (a"* b) If

Maa=O

r/J+EOf>rp = bp+Y5ifJ == ta, IM

A

~ M cd Jao= :z- l':abcd

Here

(20)

(21)

(22)

(23)

Then

(24)

andA. i bed

Ya == lYaYs=j1 Eabcd I' I' 'Y

where the tensor !,,1>cd is anti symmetric in all the four indices and !123~ = 1.If the corresponding quantities constructed from .p+, ,p be distinguished by

3SAI/[ebra of the Dirac-Matrices

a dash, (19) can be written in the following form:

if1+rp'rp+!J1= - -.t DoDo' + t SaS'a + ! SaS'" -- t MabM'ab + t QsQr: (25)

Choosing a = 0, fJ = 5 in (20) we get

i!J1+cp'rp+ys!J1+ iifl+Y6rp'rp+ifl= - t QoQs' -! QsQo' + ~ MabMa/ (26)

Simlarly putting a= °in (21) we obtain

SaS'a + Qr,Qs'= - QoQ 0' + ifl+rp' rp+ifl- ifl+YaCP' 4>+Ya!J1- ifl"'-Yr/p· rp+Ysif1· (27)

On the other hand We get on taking a = 5

SQS'a + QsQs' = - QoQ~' + ifl+cp' 4>+ifl- lj;+..yacP" 4>+;alj; - f+yscp' CP+Y5lj;. (28)

On putting a = 0, Y= 5 (22) gives

- s)i'" = - iifl+ysrp' 4>+lj; + irp+ysif1· ifl+ep

+ t </>+Yaifl'ljJ+1r</> + t ljJ+Yarp .c/>+yQtf; (29)

Also if we put a = 0, fJ = 5 in (23) we get

- ljJ+yr,cP' </>'rslj; = - 4>-' ifl·ifl+</> + t s.s>+ t SaS'a (30}

which is the same as equation (44 P).* From (30) and (25) we get

- t/J+Ysrp·rp+Yslj;- t/J+rp.</>+t/J= t QoQo' + i MabM'ab- t DsQs' (31)

which corresponds to equation (43 P). Equations (26), (27), (28), (29) and(31) can be written in the following form:

*MabM'ab- ~ (QuQs' + Q6QO) = (t/J+rp'rp+YsljJ- 4>+rp'lj;+y~tf;)

+ (rp+t/J. t/J+ysrp - t/J+lj;.</>+Y6</» (32)

2 Sasa'= - 2 DoQ; - 2 Q6QS' - (t/J+YaCP'rp+yat/J- tf;+Yat/J'rp+ya</»

- (t/J+Y6rp· rp+Y6lj; - t/J+YsljJ· </>+YsCP) + (t/J+rb </>+ljJ- ifl+tf;rp+q,) (33)

2 SaS'a = - 2 QoDo' - 2 QiiQS' - (lj;"'Yarp·</>+yat/J-lj;+Yat/J·rp+ya</»

- (ljJ+ys</> "CPTY6ljJ- ifl+yst/J·</>+ys<P) + (t/J+</>'rp+lj;- o/'¢J'q,+</» (34)

- 2 SaS'a = i {</J+Ysifl·t/J+rp -lj;+ysep'rp"1"tj;}

+ t {ep+YaljJ'ljJ'y"ep + rp+Yaifl·tj;+y"</>- 2 rp+Yarp·ifi+yaiflj (35)

t MabM'ab= QoQo' - QsQs' - 2 (ifl+rp'ep+ifl- !fr+ljJ.rp+ep)

- 2 (ljJ+Y5rp·rp+YSlj;- q,+yo¢ljJ+Yilif;) (36)

* P refers to Pauli's paper.

36 H arish-Chand ra

Equations (32) to (36) are the generalisation of equations (344 P), (34) P),(34:3 P), (34 5 P) and (342 P) respectively. Equations (32) and (36) havealready been given by Pauli as equations (47 P) and (43 P). The otherswere not obtained by him. It is noteworthy that we have derived theidentities directly without employing the matrix B of Pauli. One moreinteresting identity can be derived by interchanging 1J+ and ifl+ in (23) andputting a= 0, f3= 5

t ep+Yaifl·ifl+y"c/;+! c/;i Yaifl ·ifl+:/ ep = - Q5Q 5' .: QoQo'·

However on putting eP+ = if;+ and ep = if; it degenerates merely into the sumof (341 P) and (34 3 P).

It may be mentioned here that it is possible to derive tensor identitiesin addition to the invariant identities given above, by choosing othersuitable sets of values for a, f3 and Y in the equations (20) to (22). Onputting 1> = ifl and 1>+ = ifl+ the following identities are obtained:

T - t = tAp.Pp.

(T- tF=- 2 t (,U.P_! ta{Jt yli € i> ~\ EApp.p 2 ai_ YUI\P

t(T- t)2+ 2 tp,ptP.p}2-11al'i(YJ€a{3YOApE\PCl'/y'li'ta'J3'tY'8'

i , r I , '\.

= 8- ea f3 f.l p fTT t I_EY8p. P fT T t , .t: ,e Q~' E"P1'-1' v. P. I' IT T at'YW'p

~ ~b

M{/bS - DoS" = 0 [a =~ 0, f3 = a in (20)J (37a)

M"z,So + DoS" -r-r; 0 [a = a, f3 = 5 in (20») (37b)

M ab£?5+ iDoMab -r- i (S~§,\ - Sz,S,,) = 0 [a = a, f3 = b in (20)J (37c)

MaoSo+ iQ5SI/ = 0 [a = 0, y = a in (22)] *(37d)

MabS/' - iQ'jS" = 0 [11 = a, y = 5 in (22)1 (37e)

S"Sb+ S"Sb+ M{/{Mcb = - 8"bQ 0

2['L= a, 13= b in (21) or (22)] (37f)

The generalised identities for the case ep =1= If and 1J \" :=/= If' can be obtainedby similar substitutions and they need not be given here explicitly.

§ 3. Now we shall calculate the matrix determinant of T. As is wellknown this determinant, which we denote by del. T is the same for all four­dimensional representations of Exw In fact it is equal to the independentterm in the characteristic equation of T. It is therefore sufficient todetermine the characteristic equation. For this purpose we make use of(12) and find tha t

'" This identity w[).~ mentioned by Prof. Bhabha in a lecture.

Algebra of the Dirac-Matrices 37

Now

i= 8 Eaf'.u.vrYTt"ptrYT 16 (tapti\p - ti\ptap - fa>.tpp) Fi\!,

= 2 iE'1.PJ-LVrYTtJ-LpttrT (tapIApE i\p - 2 ta;"tppFl.p)

= 'J i "a{3J-LvrY7"f at t_ (T - t) - 4 it, t (I €a.{3J-Lv rY7"t t TEi\p- Qt' -"p u. a~ t'P p.v a .

t t EQ tlJ.!-PrYT t t - 1. t ~a."p.·rY7"t t ta.\ f3p I I'. rYT .- 6 ;>'p~ I' at> jJ.P trT

(38)

(39)

(39) is easily verified for a 'tensor' tiL. whose only non-vanishing compo­nents are to!. t23 , 145, Since by a suitable orthogonal transformation everyantisymmetrical tensor tp. p can be brought into this form. it follows that theinvariant equation (39) holds for every tp.po Also

€apY8Ap €APQ' f1'/o' ta'P,tY'i' = 16 (teptY8 - fay!f18 - laS!Yfl)' (40)

Substituting (39) and (40) in (38) we get

{(T- t)2+ 2 tp..!JL·}2_ 8 {(ta{3tapp- 2 t aPt{3ytyotc;a}

- i i ta{3tYC;ti\p€a.pyClXp (T - t) == O. (41)

Equation (41) is the characteristic equation of T. Putting T= 0 on theleft side of (41) we obtain the term independent of T so that

det T = (2 tp..tJ.!-· + t 2)2_ 8 {(tapta{Y··· 2 (tQ f3 t {3 ytYOtc;(I) }

+ ii ta{3 t yc; t ;>. p €4{3Y8 XPt (42)

(44)

(43)

det T' = (t2+ 1- tp.ptP.P)2- t {(t a,B t a/3 )2_- 2 taf3tpl'tYOtSa}

i+ 0 ta,{ltyst >Op €"IHoi\Pt

so that

(42) agrees with the result given by Eddington* (1936). For the purposeof the following discussion it is convenient to replace T by T' where

T' = t+ 1- tx"ExjJ.

t45)

Let us now revert from E:l.1' to the original matrices E, and theirproducts. Obviously T' can be put in the form

T' t + t Fa+ 1. t Eilb + i E"EbE' .j= ,1 2 ab 6 Eabcd '5'

+ 1 E"EbE'E"24 S €Jbcd

Here a, b, c. d run from 1 to 4 only and fa and sa arc four-dimensionalvectors while s is a scalar. From (5) we have

.. Eddington has chosen Es = - jE1E~E8E4 and therefore in his case the sign of i in (42)is reversed, .

38 Barish-Chandra

';-4 EabedEaEbE'Ed = - iE:; = - iE06

1. EaElrre - 'E E - 'E"ij" €abcd . C -- I Ii d - - I ,is'

On comparing (43) and (45) and using (46) we find

tOa = ta

tali = Sa

tOl)=S,

For brevity the following notation for any vector Aa or aintroduced

Thent t/A~t/A"= tOatoa + I tab 12 + tOl)to5 + ta:;ta;,

= I ta 12 + I tab 1

2 + I Sa 12+ s2

(46a)

(46b)

I

~ (47)IJ

tensor Bab is

(48a)

(48b)

(49)Also put

so that(t2)00 = - S2 - I t; 1

2

(t 2)0.; = tOtab- SSb

(t 2)ab = - tatb+ t/teb - SaSe

(t2)a:; = - st; + te/S6

lt2)oo = taSa

(t 2)05 =- Sli - I Sa 12

(50a)

(SOb)

(SOc)

(50d)

(50e)

(50f)

(52)

Thus we havetal1tfJl'tl'stSa = (t2)a:yCt 2)a:l'

= (t 2)ol (2)00 + (t2)SS(t2)S5 + 2 (t 2)06( t 2)ll5 + 2 (t 2)oi t 2)oa

+ 2 (t2)a5(t2)a S+ (t2)ab( t2)ab

= (S2+ I ta 12)~+ (S2+ I Sa 12) 2+ 2 «(a Sa) 2 + 2 I ~t"b- SSb /2

+ 2 I t/Sb- st; 12 + 21 tatb+ S"Sb- t/teb 1

2 (51)

From (44), (47), (49) and (51) we have

det. T= (t 2+ I ta 12 + I lab 12 + I Sa 12 + 52)2_ 2 ,I t a 12 + I tab 12 + I Sa 12 + S2)2

+(S2+ I ta 12)2+ (52+ Is" 12)2 + 2 (r'sa)2+ 2 i (~tab-SSb 12

+ 21 I/Sb- st; 12 + 21 tatb+ SaSb- t"'(,,b 1

2

, abed 4 . abed+ I S tabtedE t+ I tatbcSd€ t

since

Algebra of the Dirac-Matrices 39

Instead of E, or Ya which satisfy (4) or (1) respectively it is convenient

to introduce the matrices all- (f.L= O. 1, 2, 3) whose commutation rules are

ap'a" + a"ap' = 2 gp.p (53)

where gp.p is the usual metric tensor of flat space-time (gp.p = 0, f.L '*' v.

gOO=-gl1=-g22=-g33= 1). For this purpose it is sufficient to choose(av a2. aJ = (E l , E2, E:;) and a o = - iE4= Y4' From now onwards thegreek indices run from 0 to 3. We make the convention that for anytensor AilD•••

rACO),. = ~4)" (54)

where Ac•ll.. denotes that among a, b.: .. there are n indices equal to 4. Thenew quantity Aco).. defined by (54) is obtained by replacing each index 4 byO. Since for raising the greek indices we use the tensor g/loP the followingrelation holds

A BiLl' · •• = (- 1)'" A Bob . . ./10' • • • ab •••

where m is the total number of indices a, b,'>: in each of the tensorsA&b' .. and Bab

• • • which are contracted together. Also from (54)

EOl 23 = - El 23 0 = i E1234 = i.Therefore we put

EIJ."UT s; i TJ lJ.I'uT

where '10133= 1. Thus (45) and (52) can be written as

T' = t - t aP, + 1_ t a.iLaV - 1- n a"'a."aUSTlAo -z p.p o "lp,pO"T

1- -- S'YI alJ.a"a!'aT24 "IP.PaT

(55)

det T'= «(2- I tp, 1%+ I tlJ." 12_ I 5p.1 2+ s2)2- 2 (- I tp, 12 + I tILl' 12

- I 5p.1 2 + 5 2) 2

+ (S2 - I til- 12) 2+ (S2 - I Sp. 12) 2+ 2 (tJLsp.) 2 - 2 I sSp, + r"tv,", 12

- 2 I «; +tp.PSp 12+ 2 I tptp+ Sp.Sp + t,../Tt lT" 12

- s tp.ptrrrTJp."UTt - 4 tp.tplT''IT 7/JJ. PIT'Tt . (56)

A notation similar to (48) has been employed in (56) for greek indices also.

The above result can immediately be applied to discuss the case ofa particle of spin -! having a charge- and dipole-interaction with an electro­magnetic (or meson) field. If Pp. be the energy-momentum vector of theparticle, cPp. the electromagnetic potentials and F,up the field-strengths thewave equation for the particle can be written as

(57)

40 Harish-Chandra

Here gl is the charge and g2 the dipole-strength and m the mass of theparticle. Our object is to determine the classical analogue of this particle.Hence we treat Pw fl!' Fl!v as numerical quantities commuting with eachother. This corresponds to the neglect of quantum effects. As is wellknown the condition for the existence of a solution of (57) is that

(58)

Here 1Tp. = PI! - gJfJL and we have written g instead of g2 for simplicity.(58) corresponds to the classical equation of motion of the particle.Comparing (58) and (55) we find

t=m

lJL=-1Tp.

tp.v = igF fJ-V

slJ-=O,s=O

so that on account of (56), (58) reduces to

(1TI'7T1J- - m2+ t g2FI'vFlJ-vp - 417p.1T1J- (t g2Fva-Po") - 2 (t g2FfJ-vFiLV) 2

- 4 g27T F,<LpF TT17+ g4F -F Fa-'TF p. = 0 (59)p. "a- p. VCT 'T .Now

Fp.pPa-F<TTF'T,<L = 2 (1 F/LVFfJ-I'P + (! F,<LVF*IJ-I')2

where F°p.v is the tensor dual to Fp.v and is defined by

(60)

(62)

to the

F°p.v= hP.I'CTT F17T

(61)

Equation (60) is easily verified for the particular Lorentz-frame in whichonly F Ol and P 23 are different from zero. Since it is a tensor equationit must hold for every other frame also. Thus we obtain from (59) and (60)

[1Tp.7T/L- m2 + t g2F.uvF/LPJ2 + 4 {(t g2FiLI'P°p.v)2 - TT.uTTfJ- (1g2PI'CTPVCT)

- g217p.PI!vFpaTTCT = O.

When the explicit spin interaction is absent g= 0 and (62) reducesusual classical equations of a point-charge

(63)In this case

(64)

where vI! is the classical four-velocity of the particle. Since we wish to retainonly terms of the lowest order in g in (62) we can substitute (64) in the termsof (62) containing g, so that on ignoring terms of the order g2. (62) becomes

("/LTTI" - m2 -l-- t g2FI!vFI"V) 2 = 4 m2g 2 (-1 FI"vFi''' - F,up,,) (65)

A 1gebra of the Dirac-Matrices 41

where F,.. = Fp..v·. Therefore

'lTJJ.'lTJJ.- m 2 =± 2mg ,It Fp..P·-.:.... Fp.P (66)

if the second and higher powers of g are neglected. Now

1 F p. - F FI-' = H2.. p.. p.where H is the magnetic field in the rest system of the particle. Therefore

'fTp.'lTI-'- m 2 =± 2 mg IHI (67)

For the non-relativistic case we find on putting Ti 0 = m + W that

?T2

W =2/n ± g I H I (68)

where 'IT = (Tit, 'lT2 , 'ITa). (68) shows that in the rest system and in a weakelectromagnetic field the particle manifests only a magnetic moment g suchthat the direction of the moment is either along or opposite to the magneticcomponent of the field. This is precisely what is to be expected of aparticle of spin !.

I am thankful to Prof. H. J. Bhabha for helpful comments and advice.

SUMMARY

The Dirac-matrices generate an algebra consisting of sixteen linearlyindependent elements. A formula is given for expressing the product ofany two elements as a linear combination of these sixteen. This determinesthe structure of the algebra completely. It is shown that certain knownidentities concerning these matrices can be obtained comparatively easilyby the present method. Some new identities are also deduced.

The characteristic equation of a general element of the algebra is derivedand from it an expression is obtained for the determinant of any four-dimen­sional matrix representing the element. This expression is used to discussthe case of a particle of spin t having an explicit spin interaction with theelectromagnetic field. It is shown that in the classical limit Ii -.+0 and uptothe first approximation in the interaction constant g the particle manifestsonly a magnetic-moment g in the rest system, the direction of the momentbeing either along or opposite to the magnetic field in the same system.

REFERENCES

1. Eddington

2. Pauli

. . Relativity theory oj electrons and protons, Cambridge UniversityPrLSS, 1936.

.. Aim. Inst, Poincare, 1936, 6 109-36.