The correspondence between the particle and the wave...

502 G. K . Batchelor

vortex filaments (Taylor 1938). The second terms on the right-hand sides represent 1 he decrease of vorticity due to viscous dissipation.

Further equations for the rates of change of parameters defining the basic velocity correlation could be derived by considering higher powers of r in the equations (6*27)-(6*30), but it is clear from the deductions already made that their complexity would limit their practical value. All terms derived from the double-velocity correlation are measurable, but there have as yet been no measurements of either triple-velocity or velocity-pressure correlations, and these terms must therefore be deduced indirectly from the equations given.

R e f e r e n c e s

Batchelor, G. K. 1945 Australian Council for Aeronautics, Report, A.C.A. 13. Karman, T. de & Howarth, L. 1938 Proc. Roy. Soc. A, 164, 192.Robertson, H. P. 1940 Proc. Camb. Phil. Soc. 36, 209.Taylor, G. I. 1935 Proc. Roy. Soc. A, 151, 421.Taylor, G. I. 1938 Proc. Roy. Soc. A, 164, 15.

T he correspondence between the partic le and the wave aspects of the meson and the photon

B y H a r ish -Ch a n d r a

J . H. Bhabha S t u d e n t , Tata Institute of Fundamental Research, Bombay

(Communicated by H. J . Bhabha, F.R.S.— Received 10 September 1945)

In the Kemmer formulation of the meson equation certain rj* matrices are introduced. They are determined completely by the condition that for a Lorentz transformation r%\[f transforms as a vector. The jT * ’s , together with another matrix J3 (defined in terms of them), turn out to be very convenient for discussing the properties of the meson matrices fik. With their help, a formula for the spur of a product of any number of /?fc’s is derived both for the 10- and the 5-row representations. Also they enable one to bring out, without making use of an explicit representation, the complete equivalence of the Kemmer formulation with the vector and scalar meson equations in the usual tensor form. Besides, the use of the T7*-matrices now permits the introduction of nuclear interaction in the Kemmer formulation in an elegant way. The extension of the ‘particle’ formulation to the case of a particle of zero rest-mass is discussed. Finally, a specially convenient representation of the s is given both for the case of spin 1 and of spin 0. The JT*’s and the /? are explicitly given for this particular representation.

1. Kemmer (1939) has given a formulation of the theory of the meson J based on the following equation:

i/1kdki/r + x f = 0.( 1)

J The term ‘meson’ is used in this paper to denote any particle with spin 1 or 0, having a non-vanishing rest-mass.

on June 2, 2018http://rspa.royalsocietypublishing.org/Downloaded from

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Here /?* ( Jc = 0,1,2,3) are matrices acting upon xjr which itself is a one-column matrix. The /?*’s satisfy the following commutation rules:

f ik p tp m + f im p ip k = g k lfim + (2 )

gkl being the usual metric tensor for flat space-time,

9 k i == 0, k = ¥ l ‘, 9 o O = ~ 9 — ^ 2 2 = 9 3 3 = 1 •

I t is well known that (1) is completely equivalent to the usual ‘wave’ formulation of the meson equations. However, until now this fact could be established only by making use of an explicit representation of the /?*’s. A method of proving it directly will be given in this paper. I t will be shown that the correspondence between the two formulations is indeed so close and complete that one can go over with ease from one formulation to the other a t any stage of the calculation.

Further, it was not possible until now to introduce in (1) the interaction between the meson and the heavy particles in an elegant way. In his attempt to take into account this nuclear interaction Wilson (1940) had to make explicit use of the cumbersome representation of the /^-matrices. This completely destroyed the beauty of the ‘ particle ’ formulation. However, as a result of the recognition of the close correspondence mentioned above the nuclear interaction can now be included in the matrix scheme without affecting its elegance.

In using the matrix method for the quantum-mechanical treatment of the physical processes involving mesons it is necessary to evaluate the spurs of the multiple products of the j3k s. The usual methods for their evaluation are extremely cumbersome (cf. Booth & Wilson 1940). In this paper a general formula for the spur of a product of any number of /?fc’s will be derived quite easily. I t is hoped that it will be of use in future calculations.

I t is well known that the Kemmer formulation can be applied only to a particle of non-vanishing rest-mass, i.e. x + 0. The derivation of the second order equation

dkdkft + x 2fr = 0 (3)from (1) depends directly on the explicit assumption y 4= 0. I t will be shown that by a slight modification the same formalism can be adapted to the case of zero rest-mass so that it can now be applied with equal elegance to the case of the photon or the electromagnetic field.

2. The Hermitian conjugate of (2) is+ j p m jp l jp k — gkl^fm _|_ gird^fk^

where /?tfc is the Hermitian conjugate of fik. Thus the /?|’s satisfy the same commutation rules as the /?fc’s. Therefore every irreducible representation of the is equivalent to the corresponding one of the /?£’s, because it has been shown by Kemmer that two inequivalent irreducible representations of the having the same numberof rows and columns do not exist. Therefore there exists a matrix A such that

A^A-1 = jPk.

Particle and wave aspects of meson and photon 503

(4)



504 Harish-Chandra

I t follows from (4) (see Pauli 1936) that A can always be chosen to be Hermitian, i.e.

A* = A. (5)The Hermitian conjugate of (1) is

- id k i/r*flk + x ^ = 0,

so that from (4) — idki/r^Aj3k + x ^ A = 0.

Put \]A.A = i/r*, (6)

then \Jf * satisfies the equation— idk\lr*j3k + x ^* = 0. (7)

Let Imn be the matrices representing the infinitesimal transformations of the Lorentz group, so that by an infinitesimal transformation

x 'k = x k + e kl°^ ( = ) (8 °)

yjr is transformed to ijr' — (14- |e w/ w) \]r. (86)

From the relativistic invariance of (7) it follows that

x]r*' = ^ * ( 1 - |% J W), (8c)

and from (8) and (6) one obtainsP MA + A Ikl = 0. (9)

Besides, I kl satisfy the following well-known commutation relations[J k l Jm n j __ _g k m jln _j_ glmj[kn g k n j lm _g ln jk m (10d)

[ f ik , I lm] = (106)

where [A,B] = A B — BA. I t is well known that in the present case

I kl = fikfil—fil/3k, (11)and (9) and (10) are satisfied.

Introduce a one-column vector-matrix 1 k such that transforms as a vector,so that from (8c) _ t 'T___ „ i

1kl-Lm ykm1l yim^k* (12)

e c+ II £ (13)

therefore from (12) and (9)r *Ilm = 9kir m-9km^7- (14)

From (12) and (14) it follows that= ^ k ’ (15a)

r * ki ki = 3thus from (15a) and (14)

(156)

3 = m r i m =




which immediately gives

©II1cr

and therefore Pjc^l — (16)

Also from (12) and (14) COiII£§gHci (17a)

\ m . .n l™ = - 3 / ? , (176)

so that r tw „ 1 = 0.But from (106)

= 3/?,+ 2/5*4; (18)

therefore 3 /^ A 7 „ - + = 0.

On contracting l, m one gets n p , T = 0.so that 3 /?A T » + 2 / ? A .7 |- 0 .

On interchanging l, m and subtracting it is found that

n fii 'im =so that = o. (19)

I t will now be shown that for a given representation of the the choice of 1 kis unique apart from a numerical factor. For this purpose transform (12) into the corresponding spinor equations

" -/t> P- aft ucl rf- vji "b Cva rJ uf)’ (20a)

= h(e/i/}rIaiJ + e^T a/j). (206)

The Greek alphabets will be used to denote spinor indices, the Latin aphabets being reserved for tensor indices. Here

fa- = Uklrfk (21a)

= cd’ v, (216)

^ aft r k^’a/2’ (21c)

The <r-symbols have the usual meaning. eSv and e are the fundamental antisymmetric spinors for raising and lowering the indices, e12 = e12 = 1. Similarly, e12 = = 1.Every finite irreducible representation of I is characterized by a positive number k which can assume only integral or half-integral values. The following relation holds for a particular irreducible representation

= (22

Similarly, a finite irreducible representation of is characterized by a positive number (integral or half-integral) given by

= W +l). ( 226)



As is well known (Dirac 1936; Fierz 1939) for any given values of and l the following relations hold:

Ifivik) — ^(A:) vv(k) —

Ifivd') = UfSJ') Vv(l) ~ >5 uM(k)uv(k-$) = v ^ k - ^ v ^ k ) = 0,

ufl(k)v'l(k) = 2k, vfl(k = 2&+ 1, vv(k + £) uv(k -H £) — uv(k) vP(k) = e"*",

uA(l)u^(l-^) = vA(l-i)v*(l) = 0,uA{l)vt{l) = 2 1 , v*(l )uA

v*(l + £) u*(l + - u*(l) v*(l) = €**,

where u^k) is a spinor matrix with 2k+1 rows and 2k columns, while v^k) has 2k rows and 2&+1 columns. Similarly for uA(l) and vA(l). Corresponding to definite values of k and l split up 7 aA in /I aA(k, l) (cf. Bhabha 1945). From (20) it is found that

~ W % a 7 ^ = \ 7 a+ (24a)l afi(246)

from which it follows that 7 a (k, l) = 0 unless

k(k+ 1) = f, 1) = §,

i.e. unless k = \ , l = \. Therefore (20) reduces to

7 a./} i^/ia (25a)

~ I/iv($) ^a# ~ \ e(iP + 7 a/i’ (256)

where 7 a#is written instead of 7 a/!j(£,i) f°r brevity. Multiply (25a) by v^(\) so that

506 Harish-Chandra

from (23)- f i>„(£) 7 aj = K ( i ) 7 4 + \evav^{\)

On interchanging v, a and adding it is found that

vv(\) 7 *$ = ~ vV{ \)7 v^

so that - vv(\)7 a/? = \ev a 7rf,

or, on multiplying by uv(%), 7 ^ = wa(l)7 + (26)

where 7^ = 7 Similarly, by substituting (26) in (256) one obtains

7 <x/l(h i) = wa(i) U (i) (27a)

where 7 = \v*(\) 7^ = |v a(£) v*{\) \). (27 6)

Similarly from (14) FJ*(£, \) = v*(\) F*, (27 c)

where f * = \r*^(\, V i u ^ v f t ).(27



In easel k, l, m, n form a complete system of commuting variables, 7 and r * are ordinary numbers. That k, l, m, n do actually form a complete system of commuting variables for the various irreducible representations of the follows from the dimensionality of the 5- and 10-row representations (cf. § 6) which are the only nontrivial irreducible representations as shown by Kemmer. Therefore and are determined completely except for a numerical constant. Incidentally (27) proves tha t non-vanishing rI k and r* satisfying (12) and (14) always exist whenever the irreducible representation k = f , l = \is contained in the representation of the /?fc’s. This condition is of course satisfied for the case of spin 1 and 0 (cf. § 6).

Notice that from (27)

so that unless r* or 7 is zero T * 7 fc4= 0. However, from (13) and the non-singularity of A it follows that if 7 = 0 then r* = 0 and conversely. Therefore

117**0,

unless r% = l k = 0. Thus if l k exists at all it can always be so normalized^ that

r $ l k = 4.

This determines l k completely except for a phase-factor of modulus unity. On introducing this normalization (16) becomes

( 28 )

In view of (27), (19) is easy to understand. I t has been shown by Bhabha (1945) that the only non-vanishing matrix elements of J3r are of the type

OMIAI k± \,l± % )

Since the only non-vanishing elements of r* and 7 Z are |) and 7 Z(^, ^), itfollows that a product of any odd number of /?/s multiplied by 7£ on the left and 7 Z on the right would give zero. Also notice that if 6 be any operator commuting with Ikl, then 01 k and r%0satisfy equations entirely similar to (12) and (14). I t therefore follows from the above result that 01 k and r* 0 can differ from 7 k and by a numerical factor only. One such operator 0 is Thus fimfiml k and r k fim/}m arenumerical multiples of l kand 71*. Due to (13) the multiplying factors are conjugate complex of each other.

I t may be mentioned here that all the results obtained so far depend only on the existence of I kl and A satisfying (9) and (10) and are completely independent of the commutation rules (2). They hold therefore for the matrices occurring in any relativistic wave equation of the type (1) (see Bhabha 1945). From now onwards (2) will be used explicitly.

% m and n are the eigenvalues of I 12 and 1“ respectively.§ If necessary the sign of A may be reversed without disturbing (4) and (5) to make r k*'Ik

positive.


Vol. 186. A. 33



508 Harish - Chandra

Notice that from (2) fikfikfiiJr fiifikfik = (29a)

and p M k = (296)

so that from (17) and (11)

= PnPn{Pm == -37* ,

i.e. ( 5 - l ) ( 5 - 3 ) 7 * = 0, (30)

where Bis a number given byfinfinTk = B 7 k;

therefore -B = 1 or 2? - 3. For the 10-row representation of the fik s fimfim has only 2 and 3 as eigenvalues, while for the 5-row representation the eigenvalues are 1 and 44 Therefore for the 10-row*representation 3 while for the 5-row one Now consider I f lImnrI r. From (12)

_ T m Jl -*■mn T 'T — n Tm rT Hn r v nr x l x m

WirierOn the other hand, from (11) and (29)

= PlPm0m0n- += + (3i)

therefore - B ) + g,nB } l r = g,nl r+ 2 ^ 7 , . (32)

For the 10-row representation = 3 and so

f in f i l l r — 9 n irI(33a)Multiplying (33a) by fin

= W - M J l r = 2A7r =

i.e. ft,7r = -ft/1,. (33 b)

J The easiest way to verify it is to calculate from the equation (100) of § 6. Noticethat from (29a) it follows that (/?w/?w — 5/2)2 commutes with the and therefore for any irreducible representation of the /?fc’s it is a multiple of the unit matrix. Thus for any irreducible representation 88 = /?w/?m satisfies the equation

(8 8 -B 1) ( 8 8 - B 2)^= 0,where B lyB 2 are numbers with B x + B 2 = 5. Also it is easy to prove from (2) that the characteristic equation for 88 is

88(88- 1) (88 — 2) (88 -3 ) ( 8 8 - 4) = 0; this, together with (29a), permits only two possibilities

B 1==l , B 2 = 4, = 2, Z?2 = 3,besides the trivial one B = 0 corresponding to /?* = 0.



}for spin 1,

for spin 0.

Thus for spin 1, /?/7r is antisymmetric in l, r.On the other hand, for the 5-row representation = 1, so that from (32)

PlPn^r 119 nr'Multiplying (34a) by

P P l P r P r = P n ^ - m ^ r = 4 & 7 r =

therefore p nl r = \gnrp lrJl = Prl n.

The corresponding equations for I are

P*PlPm = Pkfflm-P*P l=~P*Pk

Pje P i = r * P k — l9 k lP m P V

P* Pi Pm = 9klP*Now introduce a matrix p defined as follows:

p = Tfcr*fc.From (28) it follows that /?2 = /?,

P =

P^k ~also from (33) and (35)

PnPlP = 9nlP~ 1= PPnPl f°r Spin 1,

and similarly from (34) and (36)

PlPnP = 1 lP* = PPlPn for sPin °-

Therefore in both cases PnPiP~PPnPi ~

Now put PPk+PkP = ®k>

then 6k0l —

since PPkP — 0 from (19). Also from (40) and (38)

W r * = Pk= PkPlPmP + PPkPlPrrv

therefore dk6l0m + dm6l6k = {gkiPm + gmiPk) P + QmiPk)

~ 9 k »

Particle and wave aspects of meson and photon

(34a)

(346)

(35)

(36)

509

(37)

(38) (39a) (396)

(40)

(41)

(42)

thus the dk’s satisfy the same commutation rules as the s. Also from (19), (37) and (39)

k l ~ (PPk + PkP) T; = Pk k]r t o l = r tp i, I

(43 a)

33-2



and

510 Harish-Chandra

(436)eke,7m = e j , 7 m = = M f i 7 m = m ,7„,r t e ,e m = r t m »-

The representation of the #fc’s must be equivalent to that of the /?fc’s since from (43)

A/?‘ 7, = B7, =and the B values for two inequivalent representations are different, therefore

fik = seks-\ (44)where $ is a non-singular matrix. Notice that from (41)

Q k fi + f i 6 k = e io (45)so that from (44) and (45) /?*./?' + fi'j3k fik, (46)where /?' = S fiS -1 =Put 6kdl — 0l0k = Jkl. Then from (43)

■^kl^-m 9 km 9lm ^ k '

But Jkl — S ~lIkl S, and so^ k l^ ^ m 9 k m ^ r l~ ~ Qlm^'-^k’

Thus S 7 k satisfies an equation similar to (12) and therefore must be equal to c l k, where c is an ordinary number. Similarly, T* $ _1 is equal to c*Tk , where c* is another number. However,

cc*rz7t = r s s - ' s i t = =therefore cc* = 1 and

/?' = S 7 kr * ks~ l = cc*7kr * k = 7 kr * k = p,

so that f i k f i + f i f i k = fik- (47)Collecting all the equations together

= 7 kr * k, (48a)

fi2 = fi, (486)

f if ik “t fik fi ~ > (48c)

IIbTer (48d)

T'kfil'Im ~ (48e)

fikfil'^-m 9 k l^ rn 9km ^ l

I ! filf im = F k 9lm ~ FT 9km

= ~ f i l l k

fik fil ^ in — r k9lm

for spin 1, (49a)

nfiifim = 9 u n fil m~ \9lmPkIk

r* fit = l9 k in f im

for spin 0. (496)



511

I t will now be shown that the conditions (486) and (48c) are so restrictive that for any irreducible set of matrices (3k (which need not in general satisfy (2)) there exist either none or just two matrices (3satisfying them. Notice that if /? is a solution then 1 — /? is also one. Further /? + 1 — /?, since /?=(=! on account of (486). Now let (S' be any other solution. Then

p m - P ) + w - p )Pu ~ o,so that [ /U /? - /? ') 2] = [Pk, (P- P)]+ (P- IA> ~where [A, B]+ = A B + BA. Since the (3ks are irreducible it follows thatis a multiple of the unit matrix, i.e.

P4" (3' — (S(S' — (S'(S = (3+ (50)where c is a number. On multiplying (50) by or (S'

(S(S'/S={l-c)(S, (51a)/S'/S/3' = (l-c)jS'.(516)

From (48c) and (51) PP'PPk+PkPP'P = (1 -c ) / ik.

Now (PP'P,Pk]+ = W ifi , Pk]+-P (P

= PP’Pk -PPkP+PkP'P= PP'Pk+PkP'P = (1 - «) A . (52a)

since (3(3k(3 = 0 as follows from (48c). Similarly

[P'PP'.PA = P'PPk + PkPP = (1 (52 6)Subtracting (526) from (52a) one obtains

a P 'P -p n P k ] = o,therefore /3'/3 — /3/3' is a multiple of the unit matrix, i.e.

P P ~P P ' =d(53)where dis a number. Multiplying (53) by (3 from both sides

d(S — 0,therefore d = 0, since /? 0 from (48c). Thus

P P = PP- (54)Multiplying by /? on the right gives

PP = PPP={1 -c)P ,while multiplication by y S' on the left gives

PP = P P P ~ {1 -c)(Stherefore (1 — c) (3 = (1 — c) /?'.Thus either (3 — (S' ox c — 1, in which case (3(3' = (S'(3 = 0 and therefore

/ ? ' = ! - / ?


from (50).




Now consider the matrix y =

for the case of spin 1: y2 = \PkP

— 1^(5 ~ PkPk) PPi= 1 ^ ( 5

= W i = 7-Further 7 = $/?*/?/?* = i ( l - /?) A/?* = 3A)therefore from (29a) and (48 c)

yfih+PkV = Afc-From the result just proved above it follows that either 7 = /?or 7

*PkPky = P=

= 2 7,whilethereforeand

For spin 0 take ThenAlso 7 =thereforeand 7 = /? or 1 — /?. But

/?%/? = 3A7 = 1 - / ?

P+ W PPk = 1 for SPin h 7 = 1/4/?/^-

y» = A /? ( 5 - B ) M = W i = 7-

i /W * =yfik + Pk7 =

P%y = \Pk^-PiPl)PPl = 47,

(55)

1 — /?. But

(56)

while PiPiP = /?>therefore y = 1 — /?and A + ^ 1 for spin 0. (57 a)Also from (496) and (57a) it follows that

PkPPl = grw(l — /?) for spin 0. (57 6)

I t is interesting to observe that for spin 0

= gkl(l- fi) j3 m + Pk= g klf t m(] + ( i f ikg lm. (58)

The condition (58) is stronger than (2) and may be used to separate the 5-row representation from the 10-row one.

A few other interesting results can also be derived. For example, consider J

<0 = (59a)

J eklmn is a tensor antisymmetric in k , Z, m, n and e0123 = 1.



I t follows from (58) that o) = 0 for spin 0. Therefore only the 10-row representation need be considered. In this case it is easy to prove from (48) and (49) that

<■> = (59

so that - -& (P mne*rm'n'Pk'I ,r* p nPk.7l,r* .pn._ 1 cklmne m’n ’ f> r j o~ 8fc fcm« P k 1 l 1 m’P n ’

= - l p ki,r* * p ‘ = i P t’i .r v p *= i f i k P f i k = l - f i - (59c)

I t follows immediately from (596) and (59c) that

to3 = 0),

<*>2P k + P ka>

a)fir(o = 0 ,

Pr0JPs + Ps(jJPr = 0.Also from (596)

Pr Ps<*> + <*>Psfir = \ efmnp1(lm r* - 7r r * ) f i n

= y mnP lP (P m P r-P rP m )P n

= |(1 -P )e 8lmn(PlPmPrPn-PlPrPmPn)

= \{l~P)esmU (PrPlPmPn-PlPrPmPn + PlPfrom (2)

= 9srw3 = gsr0).Therefore, as has already been pointed out by Schrodinger (1943) and Kemmer (1943), (o satisfies the same commutation rules as any of the fi/Js. Thus if one puts o) — /?4 and gu = 1, g4k = 0 (£ + 4), then (2) would hold for = (0,1, 2, 3,4).

Particle and wave aspects meson and photon 513

3. The above results can now be applied to evaluate the spurs of any matrix composed of the /?fc’s and their products. Notice that

sp{P) = r*l™ 4, (60sp(Pk) = *P(Pfik+fikfi) = = 0, (606)

and MPkPi) = s p iP M + f ik M = r rn fit fit 7 ” + r * J kp, 7™,so that from (49) sP(PkPi) = f°r spin 1>1

spiPkPi) = 29ki for spin 0./Consider now the spin 1 case, so that

PkPlPm^n r t p kj k2.'./3k i kn+i =

(60 c)

(61a)Due to (39)



514 Harish- Chandra

In case n the number of factors ftk ,..., fikn is odd one finds on using (40) that

• • • fik2 n-ikzn~ ^k^kx • • • ~On the other hand, for an even number of factors one gets from (38), (39), (40), (37) and (61a)

kofiky • • • fikzn kin+y ~ fikx fik2f i f t h s P • • •

~VlkokxUk Vltik3]lkJ2] ‘ • VUn-iktn- lkinkin+ii' (®^C)Now SP(Pkx• • • Pkn) ~ sP(PPkx • • • Pkn + Afci • • • PkJ

= r* /iki... pkn i m+r*/3kiso that SPiPkx • • • Pk2n) ~ Vltxfej[fc2<21 VU2k3\ I k * 3* • • • VUn fc2n-X1 ^

+ VUX Vlt, kj [fc/31 — VUn 1 [k!l\ (62a)

# i - A J = o. (626)

For spin 0 r%fapml n = gklgmn,

therefore njkx• • • P k 2n-x k2n — (63a)

Wkx P k 2n ^k 2n+ x * * * @k2n * (636)

Thus Sp {f$ 'kx" * * Pk2n ) Q kx k2 Qk2 fc4 ''' g ki k2n k3 * * * Q k2jikx*(64a)

SP{Pkx'“Pk2n+1) = ° ‘ (646)

Multiplying (626) and (646) by arbitrary vectors i, A ^ * , ..., 4(2w>*2n5 one finds that

sp{(AW(5) (A<2>(S)... (A<2">P)}= A 'f ^ t 'A t f^ 1*... J ^ n - l ,2 n ) tx + j^{2,Z)U ^ S ) U j \® n ,\ ) txfor Spin (65a)

while for spin 0

spftAWp) (A ^p)... (A(2ri)P)}= (A(2)A(3)) (A(4)A(5)) ... (A(2n)A(1)) + (A(1)A(2)) ... (A<*»-«A<*»>), (656)

where (A<r>p) = A ^ kfik, (A rW s)) =

A%’8)1 = {AW A&) 8lk -A W A ($.

4. Let us now return to equation (1). First consider the spin 1 case. Multiplying (1) by r* from the left

iI? 0 kdkft + xT?& = °-Similarly, multiplying by r% fa,

^ k P i P m ^ + Xr k P A = 0,i(d{ r% \jr-dk r? \jf)+x r t fa f = o.

Q f = u k,

j F Z M = ° u -V

or from (49a) Put (б б а)

(б б б )



515

On account of the antisymmetry of .T j^ in 1c, l, is also antisymmetric. The aboveequations can now be written as *


dk U l- d l Uk = G 1d, (67

dk G ki + X 2 U l = 0 , (676)

which are precisely the usual equations of the wave formulation. To understand the equation

d if = dk ( 68)

which follows from (1), replace ftk by /?/?* + /?*/?. Then

fikM = PPkM + f i kP M =

= ( d t7 m SX

Similarly xjr = {P + \PPPk) f = 7 m Um + ±/lk7 m-?C

therefore from (68)

’I *s>Vk + ± p T * d , e mk = 7 t S,Vl‘- 7 l» D t + T ‘ St Gni. (69)

Multiplying by r* r one obtains dkUk = 0, (70a)

while on multiplying by r * rj3s it is found that

Grs + dsGlr + dr Gsl = 0. (706)

The equations (70) are completely equivalent to (69) or (68) because the latter follow from the former.

In the same way the energy-momentum tensor

T m = X & * ( f f k i - f i k 0 i - f i i P k ) & (71)can also be transformed

= f ' i ’n( n 9 k ,- r Z g m,)f+>/r*/}k7 mr»">/3lt

= 9k,U'"'Um-U 1 U k- ~ G l ,Q " ‘l X

and + |/?fc/?/?fe)

= JJ\ r n J J _ J _ Q V c m QU m 2y2 mk ^ u m ' 2X2 ^ Jm«)

therefore

Tu = \{CtlmaH + G}mG-'k + \gk, G ^ a m + x \V lU l+ Uf Uk- g a U'™Um).} (72) X



516 Harish-Chandra

The angular momentum tensor Mkl m is given by

l kl, m (d,</r*x'k - dk /?,„i// - f* ( x kd , + A A )

(73)I t would be interesting to transform the spin term to the usual wave formulation. I t is found th a t

thereforet * P M r n t = k a t n Ung,m- O l, U ,-U J Gkm + U*nGmngk \,

X

i= -[G tnUnglm-G }-U ngkm- U}Gkm+ U \G lm\.

Xip*flmIk l xjr is minus the complex conjugate of Therefore the spin terms ki,m is

S k l , m = 'ey? f t * (k l f im Am^fci)

= - 2 x { G t n u n 9 im - GtnUngkm- 2OlmUt + 2 Uk + complex conjugate}, (74)

so that for k and l not zero

Ski,0 = - I M . u i ~ Gl Uk) + complex conjugate].

The commutation rules for pf and ip*, when the second quantization is in troduced, have been given by Kemmer. They a re j

[ ( W D r A M 'U = I W * M ,,( /W J = (75)where, /t and v refer to the matrix elements of the 1-row and 1-column matrices

^*Ao and Ao^'> Ao ij/' respectively. Now

f*Ao = f*(AAi + AoA) = u fm rm /h + lv Gomr *m>X

X

^*Ao = ^*(AAo+AoAAo) = u 'm{rmg00- r 0gm0)+?-Gtmr * m/30X

= u * r t+ lG S k_r*i/30,X

K f = i k m + l,e„ 'n o „ k,X

J In Kemmer's paper [A, B ] denotes a Poisson bracket, so that in his case[A ,B]= i (A B-BA) .

Also on the right is replaced by (/?o)^ here so that the right side vanishes automatically for the matrix elements corresponding to the eigenvalue 0 of /?0. This artifice of replacing S^v by (/?o)p was pointed out to me by Professor Bhabha.



517Particle and wave aspects of meson and photon

where the underlined index k runs from 1 to 3 only. From (75)

( I? flop)] =

=

= f c G L = )’

which are the correct commutation rules of the usual wave theory.I t is now extremely easy to include nuclear interaction in the above scheme. The

total Lagrangian can be written as

L = L H + L M+ L E+ L J,

where L H is the Lagrangian of the field of the heavy particle, LM of the meson field and L E of the electromagnetic field. L H and L M include the interaction of the electromagnetic field with the heavy particle and the meson respectively. L 1 is the interaction term between the meson field and the heavy particles, and is given by

U = g1xP*ock(TPNr ^ + TNPi/r*7k) W +<72 W*cckaW. (76)

W and W* refer to the heavy particle. a fc,s are the Dirac matrices acting on W. P* = p i L,where Lis a matrix such that

a = L

tpn and r VP are the operators corresponding to the conversion of a neutron into a proton and vice versa. In fact (76) is exactly the same as the usual interaction term in the wave formulation of the meson theory. Obviously (76) can be extended to include pseudo-vector interaction by adding terms of the typej

i|l klmn{TPNr* nf + tnp ijr* l n) W

and | W*akaleklrnn(TPN + rNP 7™) W,

where eklmn is a tensor antisymmetric in all the four indices and e0123 = — 1.Let us now consider the case of scalar mesons. Multiplying (1) by r* 1 on the left

ir* l/3kdki/r + x r* lijf = 0.

% If rj = 1 — 2/31 is taken as the reflexion matrix then ^*7*. is a vector and not a pseudo - vector, since

V 'l* = ( ! " “ 2/?o) — ^k “ 2 00 k + %9ok 0 = %9ok ?0 ~ ^k-7j satisfies the relations

V2 = L yPk = ~~ PkV (* + 0), t]/30 =For the 5-row representation the reflexion matrix must be taken to be —1/ so that ^ * 7 fc may transform as a vector since in this case

k ^k o9ok*



518 Harish-Chandra

Similarly, multiplying by r * mftm and using (496),

4 idkr * krjr + x r * m/3mijr = 0.

Put n f = U k (77

and r t h f = - - g * V , (776)l

then (1) is equivalent todlu = u l, dkuk = o ,

which are again the usual equations for the scalar field. Also it is found that

* = Off + i h m = c/V

and = -Z tf /S „ 7 * U + 7*U„

so that (68) can be written as

7 lc9l U * - l / ikV ‘Z d,U = - +

or dk Ul — dl Uk = 0.

The expression for the current vector and the energy-momentum tensor can be transformed exactly as above. They agree with the usual expression of the wave formulation. Further, from (58) it is found that

= 7 {V '9 u U „ -U lg lmU},l

so that

Ski,m = - 2 I *(4zArn+ Am4 ) f 1 { ( ^ ~ ^ ^ ) + complex conjugate},

therefore $w>0 = 0 in this case.

Also = l u t r ^ f i t + Z u t r t ,

f* P l = u tr * + £ .w r

M = ytmT"U0-ji„ u ,

so that f f i , ” , = 7 U', f * « 7 0 = £7J

and -CoAoV = — j ^ , P == 4)>




therefore from (75)

[c /t, - ? [ / ' ] = [ | m , f j ] = - S ( x - x ') ,

which are the correct commutation rules of the usual scalar theory.The nuclear interaction can be introduced in the scalar theory by exactly the

same expression (76) as in the vector theory. However, on account of (496) and the fact that <xkcck = 4 the second term in (76) can be written as

The g2 term gives rise to the ‘ charge ’ interaction while the gx term brings about the ‘dipole’ interaction. This is precisely the reverse of what happens in the vector theory. The pseudo-scalar interaction can be introduced through the terms

^ xf ,*akafamelamn(TPNr * nt/f + W

and jy V*aka‘a,ma’‘eamn(TPNr*^,t/r + TNPf * ^ rT') V.

In the case of neutral mesons the additional condition

= r$i/r (78

has to be imposed in both the scalar and the vector case. I t then follows from the equations of motion that

= (786)

Also for this case tnp and tpn are both to be replaced byThe above method of treating neutral mesons is completely equivalent to the one

employed by Majorana (1937) and Belinfante (1939). Let a bar denote the complex conjugate and a curl the transposed of a matrix. On taking the transposed of (2) and reversing the sign it is obvious that the matrices — satisfy the same commutation relations as fik, so that for an irreducible representation of fik there exists a matrix 6 having the property that

- A - 6fikB~\ (79

From (79a) and the irreducibility of fik it is easy to prove that (cf. Pauli 1936)

0 = ±6. (796)

To decide between the two possibilities notice that from (11) and (79 a)

I kl0 + d l kl = 0 .

Put r k = i k.

Then Fk0 satisfies an equation similar to (14) and therefore



520 Harish -Chandra

where a is a number. Since 0is a non-singular matrix 4= 0, and therefore without disturbing (79) 0 can be replaced by 0a~x. In other words, can be so chosen consistently with (79) that

rke = n . (79c)

On taking the transposed of (79c) and using (796) it is found that

7* = ± 0 -1A*,

therefore I ' l l , = ± = ± /? 7 * = ±gu .

I t therefore follows on account of (28) that in (796) only the positive sign is permissible. (78) can now be written as

$ * l k = r % f = r kd}Jf = $0 1 k, ■ (80a)

= - W M = == (806)

so that from (80) and (56), (57)xjr* = $0. (81)

As a consequence of (81) the current density $*(3k$ vanishes since

$*P k$ = =Also the usual Lagrangian

L = (dk\Js*{3k\jf — r/r)-1- (82a)

can be written in the formL = ^i/r0j3kdki/r + xfrOiJf,(826)

%

where x[r is written instead of $ to emphasize the fact that now there is only one canonical variable \jr in the Lagrangian.

The connexion between 0 and A is obtained from the equation

- A = er>-iAftkA -W = A-'0/}k0-'A,

so that 0^~XA —

where c is a number. Due to 6 = Band A — A f the above equation can be written as

d~xA = cA-x0.Multiplying by T* on the left

r%o-xA cr%A-xo = c rko.

But T fB - 'A = TkA = r%,therefore from (79c), c = 1 and

o-xa = a -xo. (83 a)




The matrix ££ of Belinfante (1939) corresponds to A~xd.

Let it also be remarked that dfi = fid, (836)

since dfi = dlkrkd = fid.5. Equation (1) can now easily be adapted for the case of zero rest-mass. Instead

of (1) writeifikdkijr + y ft = 0, (84)

where y is a matrix satisfying the following conditions:

y 2 = y, (85

yfik +fiky = fik. (856)

In (84) y appears in place of the mass y of the particle. Multiplying (84) by (1 — on the left one obtains

ifikdk(yxfi) = 0. (86)

Thus yijr satisfies (1) with y = 0. Also on multiplying (84) by dmfimfil on the left it is found that

= (87)

therefore yxfi also satisfies (68). The second order wave equation

dkdk(yfr) = 0

follows from (86) and (87). Further, if <j) is any solution of the equation

0, (88)

then yfikdk</> = fikdk( 1 - y ) = 0.Also since y( 1 — y) = 0

(1 — y)(j)satisfies (84). Thus (1 — y)<ftis also a solution of (84). This corresponds to a gauge transformation of the usual theory.

Equation (84) can be derived from the Lagrangian

L = -[dkifi*fikyifi — ifi*yfikdkifi] + ijr*yijr. (89)%

y must satisfy the condition yt = AyA~x. (90)

In fact from the result that for any irreducible representation of the s there can exist only two y’s it follows that either (90) is fulfilled or

AyA~x — 1 — yf.

In our case y can be either fior 1 — fi,and since

Afi = A 7kr* k = A l kT kA = p A




(90) is satisfied. The Lagrangian (89) is gauge-invariant, i.e. it remains unchanged when is replaced by ijr + ( \ —y)<j) and ^ satisfies (88). This is why the expression (89) has been preferred to the following more obvious and simpler one,

L = -. {dk — xjf} + \Jr*y\jr,

which, though it is not gauge-invariant, yields the same equations of motion.The expression for the current vector derived from (89) is the same as in the

meson theorys k = ft*(ftk7 +yftk)ft = (91)

it is not gauge-invariant.The energy-momentum tensor derived from (89) is

Tki = 7 {3* ft* PiYft ~ ft dk } + 9ki ft*yft.

However, it can be verified that

TU = T 'kl~ \ i dm{ft*\-(9klfim- 9kmfti) y - y(9klftm ~ ft}’ (92)

where Tki = ^-[dkft*ftift-ft*fti dk ft] ■

T'kl has the same form as in the usual meson theory. Also it is found that due to (84)

n , = 9 a ~h- PiPM

where dkl = ft*(9ki~ ftkfti~ ftiftk)yft-(99)Therefore 6klwhich is symmetric in k and l may be taken to be the energy-momentum tensor instead of Tkl.(93) differs from (71) only in having y instead of y. Thus in the present theory y formally plays a part somewhat similar to that of the mass in the meson theory. On account of the factor y, 6kl is gauge-invariant.

The expression (73) of the angular-momentum tensor remains valid also in the present case. However, the spin is not gauge-invariant.

The Hamiltonian formulation of (84) may be obtained In the same way as for the meson theory (Kemmer 1939). Multiplying (84) by /?0(1 — y)

ftodo(yft)+ftoftkd-(yft) = o,while from (87) with 1 = 0

( l -ftl)do(yft)-ftkftod-(yft) °>therefore ^o(yft) + (ftoftk-ftkfto)d-(yft)= 0. (94)

This gives the equation of motion of yijf. No such equation exists for ( I —This corresponds to the well-known fact that the Hamiltonian formulation gives the equations of motion only of the gauge-invariant quantities. Obviously y^r is gauge-invariant while (1 — y) ^ is not.



In order to eliminate the ‘ longitudinal waves ’ ^ is to be chosen in such a way that it satisfies in addition to (87) the equation

= (95)

This can be done by a suitable gauge transformation.There are now two possibilities. Either y = 1 — Consider first the

case of the 10-row representation of the J3k so that now

7 = i - p =

I t can be verified that in this case (84) corresponds to the usual Maxwell equations. The Lagrangian (89) becomes

iF ^ F k)- (d k V lF ^ + F ^ d k U,),

where Fu = ! (9f>n)h

and Uk = r%i/r. (966)

From (69) it follows that the elimination of the longitudinal waves in this case corresponds to imposing the condition dkUk = 0. With this choice of y (84) may be used to describe the photon or the electromagnetic field. In this case the reality condition (78) or (81) has to be imposed. The interaction of the photon with the charged particles can be introduced in a way entirely similar to (76). For example, to introduce interaction with an electron of charge e the interaction term

ey*ukw r kdxir

has to be added to the sum of the independent Lagrangians of the electromagnetic and electron fields. On account of (81) and (83), (89) can now be written in the form

L' = \L = iijfdyfikdk + \\Jf6y\jf. (97)

For the other choice y — /? (84) is equivalent to

dkFlk = Ul, (98a)dk Ut — dl Uk 0, (98

Fkl and Ul being still defined by (96). I t follows from (98) that there exists a U

such that Ul = U and dldt U = dlUj = 0.

Equations (98) are thus formally equivalent to the equations of the usual scalar theory. The energy-monentum tensor turns out to be

UiU,+ UlUk- g u U*»Vm,

and therefore agrees with the usual expression. The current vector and the commutation rules for the second quantization are, however, different. A gauge transformation consists in adding to Fkl another antisymmetrical tensor F'kl satisfying

»F'U = 0.


Vol. 186. A. 34



524 Harish-Chandra

The elimination of the longitudinal waves corresponds to imposing the condition

+ dmFkl + diFmk = 0 .

Coming to the 5-row representation of the /?fc’s and taking 1 —/? = it is found that (78) is equivalent to

dmUm+U = 0, (99

dk U 0 , ( 9 9 6 )

where U and Um are defined by (77) with 1. The energy-momentum tensor comes out to be

- e H = gu V W .

Gauge transformation consists in adding to Um a tensor U'm such that

ZmU'm = 0.

(88) corresponds to imposing the condition TJl = dl Uk. The current vector has the same form as in the usual scalar theory

h u w k-utu}.0

Since from (99) Uis a constant the energy-momentum density is also constant and therefore the field is of no physical interest.

The case y = /? corresponds to the usual scalar theory, the equations being

dku = uk,The Lagrangian is TJ\ Uk - (dkW . Uk + UjcdkU).

Gauge transformation consists in adding to Uk a vector U'k satisfying

dk U'k = 0.

In this case (95) follows automatically from (87) since

'dkfikf>>lM/ — = dk/3k(l — ft) fth[f — dljh]f= dkfikfllij/ — dkgkl( 1 — /?)fjr — dljh]r from (57 6)= dkPkftl\Jf — dlrjr.

6. Till now the theory has been developed in an abstract way. I t is useful to know what the matrices l k, F*k and /? actually are for a particular representation. For this purpose it is best to choose the following representation. For the 10-row representation, xjr and fiax = fikcrkx are

[M 0,1)\ / 0 vau+ 0

& = I ftib2) j > AtA = V 2 1 - \U*Vt 0 - 2Vi\ ^ ( 1, 0 ) / \ 0 u + vx 0

(100)




where va = va(| ) and = va(l) and similarly for others. The splitting up of ijr corresponds to the decomposition of the whole representation with respect to the and l values so that the components of \jr are ijr(k,l) (cf. Bhabha 1945). For this representation

/0 0 0

, = I 0 1 0

\ o 0 0

(101a)

Also (1016)

(1016) is easily obtained from (596), (100) and (101a) if one notices that the spinor corresponding to eklmn is given by (cf. Harish-Chandra 1945)

2 ^k,<xk l,fipPm,yv^n,8p a."t (102a)

Wlere P(*p)(y8 = i(eayefl8 + e/lyeas) | an(l (kp)(vp) —

On the other hand, for the 5-row representation

0

W > ,0 /aA ^ 0 j

72 7«A p*X = 0, fi/ V a\ 1\ o r (1 o \

lo o)

(1026)

R e fer en c es

Belinfante 1939 Physica, 6, 849-887.Bhabha 1945 Rev. Mod. Phys. 17, 200-216.Booth & Wilson 1940 Proc. Roy. Soc. A, 175, 483-518. Dirac 1936 Proc. Roy. Soc. A, 155, 447-459.Fierz 1939 Helv. Phys. Acta, 12, 3-37.Harish-Chandra 1945 Proc. Ind. Acad. Sci. 23, 152-163. Kemmer 1939 Proc. Roy. Soc. A, 173, 91-116.Kemmer 1943 Proc. Camb. Phil. Soc. 39, 189-196. Majorana 1937 Nuovo'Cim. 14, 171.Pauli 1936 Ann. Inst. Poincare, 6, 109-136.Schrodinger 1943 Proc. Roy. Irish Acad. A, 48, 135. Wilson 1940 Proc. Camb. Phil. Soc. 36, 363-380.



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