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On the Rarita-Schwinger equation for the vector-spinor field
Nuclear Physlcs B29 (1971) 104-124. North-Holland Pubhshmg Company
ON THE RARITA-SCHWINGER EQUATION FOR THE VECTOR-SPINOR FIELD H i r a L. B A ] S Y A
School of Mathematics, Trinity College, Dubhn Received 15 October 1970 Abstract: It has been shown that the Rarlta-Schwlnger equation for the v e c t o r - s p m o r held with a r b i t r a r y p a r a m e t e r s has three solutions for some values of the p a r a m eters; for some other values it has two solutmns. It is also seen that the Umezawa-Vlscontl formula for the highest order of demvatlve in the reciprocal operator ~s not satlshed in the case of Rarita-Schwmger v e c t o r - s p m o r field. The electromagnetm interactmn of the v e c t o r - s p m o r field is discussed. By a new f o r m a h s m m quantlzatlon we have shown that ~(x) --t~(x/c0 holds even m case of RamtaSchwlnger held and the interacting spin ~2 held has been quantized. It is seen that the interaction Hamiltonian is not an infinite series of the coupling constant.
1. INTRODUCTION In 1967 M u n c z e k [1] d i s c u s s e d the R a r i t a - S c h w i n g e r e q u a t i o n with a r b i t r a r y p a r a m e t e r s showing that there a r e three solutions and each solution c o r r e s p o n d s to d i f f e r e n t m a s s s t a t e . We have s h o w n now t h a t in a d d i t i o n to t h r e e s o l u t i o n s , the e q u a t i o n a l s o h a s two s o l u t i o n s d e p e n d i n g on d i f f e r e n t v a l u e s of the p a r a m e t e r s . E a c h s o l u t i o n c o r r e s p o n d s to a d i f f e r e n t m a s s state and a different spin state. In M u n c z e k ' s f o r m a l i s m , he u s e d the c a n o n i c a l m e t h o d of q u a n t i z a t i o n . We f e e l t h a t t h e c a n o n i c a l m e t h o d of q u a n t i z a t i o n d o e s not a l w a y s give the c o r r e c t r e s u l t if not c a r e f u l l y u s e d . S i n c e the v e c t o r - s p i n o r f i e l d i s s e e n to b e a m u l t i - m a s s f i e l d in o u r f o r m a l i s m , we h a v e u s e d the m e t h o d of q u a n t i z a t i o n of the m u l t i - m a s s f i e l d a c c o r d i n g to T a k a h a s h i [2] i n o u r d i s c u s s i o n . T h i s m e t h o d i s the g e n e r a l i z a t i o n of the m e t h o d of q u a n t i z a t i o n of o n e - m a s s f i e l d by the s a m e a u t h o r a n d U m e z a w a [3]. In o u r d i s c u s s i o n we h a v e n o t i c e d t h a t the U m e z a w a - V i s c o n t i [4] f o r m u l a f o r the h i g h e s t d e r i v a t i v e i n the r e c i p r o c a l o p e r a t o r is not s a t i s f i e d . In the t h e o r y of i n t e r a c t i n g f i e l d s a c c o r d i n g to the a u t h o r s i n ref. [3], it w a s s t a t e d t h a t ~(x) = ~(x/cr) h o l d s only i n s i m p l e c a s e s l i k e s c a l a r o r s p i n o r f i e l d s . We have s h o w n now by a n e w f o r m a l i s m t h a t t h i s h o l d s e v e n i n c a s e of v e c t o r - s p i n o r field. In s e c t . 2 we d i s c u s s the s o l u t i o n s of the f r e e f i e l d e q u a t i o n a n d i n s e c t . 3 we q u a n t i z e the f r e e field. Sect. 4 i s d e v o t e d to the d i s c u s s i o n a n d q u a n t i z a t i o n of the i n t e r a c t i n g field. F i n a l l y , i n s e c t . 5 we p r e s e n t o u r c o n c l u s i o n s .
105
H. L . Baisya, R a r i t a - S c h w i n g e r equation
2. FREE FIELD EQUATION In this section we make a thorough analysis of the solutions of the f r e e field equation. The L a g r a n g i a n for the v e c t o r ° s p i n o r which contains both spin ~ and ½ fields can be written in the absence of i n t e r a c t i o n as = -tPtz(x) [(YX 6~v +a@~ 6~.v +V u 6X~) + b yUVXTv)aX + m 6pv+mCypYv]
g/v(X) ,
(2.1)
where
~p(x) = ~(x)n~,
(2.2)
~ U = Y4 g ~ p ,
(2.3)
and a, b and c a r e a r b i t r a r y p a r a m e t e r s . The quantity gcrp is taken as g~---
+1
for % p = 1 , 2 , 3 ,
-1
for cr, p = 4 ,
0
otherwise .
(2.4)
F r o m eq. (2.1) the equation of motion for ~z(x) is
A#v(a)C,v(x) = o,
(2.5)
where A~v(0) =
-(rx, pv ax+ m M p v )
,
rx, v.v = [~x our +a(% C~v+~v ~x,) + b~u~x~v],
(2.6)
(2.7) (2.8)
MI~ v = 61~v + c y l ~ y v .
Multiplying eq. (2.5) by Y~z and 0g on the left we have r e s p e c t i v e l y 0~qc~t(x) if4a+2
1
4 a + 2 [m(4c+ 1) + ( a + 4 b - 1 ) V 0 ] y ~ t ( x ) ,
(2.9)
¢0and
(a + 1)(ya)avC, v(X) + (a + b)[Y]yv¢/v(X) + m a , ¢ ~ u ( x ) + mc(yO)g/v¢,v(X) = 0 .
(2.10)
F r o m eq. (2.9) and eq. (2.10) we can have that y#~U(x)
=0 ,
(2.11)
if c =-(a+2b) ,
(2.12)
2 b = 3a 2 + 2 a + 1 .
(2.13)
106
H.L.Bazsya
R a m t a - S c h w i n g e r equatzon
U s i n g eq. (2.11) in eq. (2.9) we have 8~p(x)
=0 .
(2.14)
A g a i n u s i n g eqs. (2.11) and (2.14) both in eq. (2.5) we obtain (7X~X + m ) ~p(x) = 0 .
(2.15)
E q s . (2.12) and (2.13) will give the v a l u e s of b and c f o r any value of a. F o r e x a m p l e , if we take a = - 1 , then b = 1 and c = -1. In that c a s e , eq. (2.5) becomes [-(ya+m)bpv+(YpSv+Yv~,)-7,(7a-
rn)yv] ~v(X)
=
0 .
(2.16)
Eq. (2.16) is the usual Rarita-Schwinger equation for the spin 3 field. The parameter a in the Lagrangian is not fixed. This can be seen by using a point transformation [5] t
~p(x) = Tpv~v(x ) ,
(2.17)
T t~v = 5 t_Lv+ ¼PY p Y v "
(2.18)
where
p cannot have the v a l u e -1, in which c a s e the t r a n s f o r m a t i o n will be s i n g u lar. In the following we shall d i s c u s s the s o l u t i o n s of eq. (2.5) u n d e r the f o l lowing a s s u m p t i o n s : C a s e I. A s s u m e a = - 1 ; b and c a r b i t r a r y . C a s e II. In addition to a = - 1 , a s s u m e that (i) b - 1 = 0, and that (ii) 2 b + c - 1 ¢ 0 . C a s e III. In addition to a = - 1 , a s s u m e that (iii) b - 1 ¢ 0, and that (iv) 2 b + c - 1 = O. C a s e I. With a = - 1 , eq. (2.5) b e c o m e s A U v ( ~ ) ~ v ( X ) = [ A u v ( a ) - ( b - 1 ) y p ( y a ) y v - re(c+ 1)7Uyv] t~v(X ) = 0 , (2.19) where ' 8 ) = -(70+rn)61~v+(Yp~v+YvSlz)-yU(ySA~tv(
rn)y v .
(2.20)
Now we c a n have !
AUv(~)~ v = r n ( ~ u - y 0 y U ) ,
(2.21)
A p v ( 8 ) 7 v = 2(78)7~ + 3 r n y ~ - 2~/~ ,
(2.22)
t
a/~A/zv(8) = rn 8 v - rnYv(y8 ) , '
a
71jAI~u( ) = 2a v - 2(78)y/z + 3 r n y u ,
(2.23) (2.24)
A p v ( 8 ) ~ v = - m 8 # - ( b - I ) E ] 7 ~ - mcy/~(ya) ,
(2.25)
A/zv(~)-gv = 2 8 / z - ( 4 b - 2)y/z(y~ ) - (4c + l ) m Y/z ,
(2.26)
H. L. Bazsya, Rarita-Schwinger equatton
a,A,v(a)
= - m a v - (b -
1)[]~v-
mc(ra)v
107
(2.27)
v ,
v p A # v ( a ) = 2a v - 2 ( 2 b - 1 ) ( ~ ) ~ v - m(4c + 1)~ v .
(2.28)
In obtaining eqs. (2.25)-(2.28), u s e h a s b e e n m a d e of eqs. (2.21)-(2.24). F r o m eqs. (2.25) and (2.26) we h a v e 1 Ap.v( a )(av+-~mVv )
=
- y p [ ( b - 1)[~+m(2b+c- 1 ) ( y ~ ) + ½ m 2 ( 4 c + l ) ] = V p ( 1 - b)(vO+M1)('ya+M2) ,
(2.29)
w h e r e M 1 and M 2 a r e the s o l u t i o n s of ( b - 1)M 2 - (2b + c - 1)mM+½m2(4c+ 1) = 0 ,
(2.30)
with b ¢ 1; i.e. 1
M1,2-
[ ( 2 b + c - 1)~-(4b2+c2-4bc- 6 b + 6 c + 3 ) ~] .
m 2(b-l)
(2.31)
A g a i n f r o m eq. (2.27) and eq. (2.28) we h a v e (0# + ½ m v # ) A # v ( a ) = - [ ( b - 1 ) D + m ( 2 b + c - 1)(va) +½m2(4c + 1)]Tv
= (1-b)(ya+M1)(va+M2)v v .
(2.32)
F r o m eq. (2.31) we h a v e M 1 +M 2 = ~
( 2 b + c - 1) ,
(2.33)
m2 M1 M2 - 2(b - 1) (4c + 1) .
(2.34)
U s i n g eq. (2.34) in eq. (2.33) we obtain 1 b- 1 -
2 3m 2 (M 1 - 2 m ) ( M 2 - 2m) .
(2.35)
F r o m eqs. (2.19) and (2.32) we c a n w r i t e
(a N + ½ m r ~ ) A ~ v ( ~ ) ~ v v ( x )
= o,
or
[ ( b - 1)[~+m(2b+c- 1)(V0) + ½ m 2 ( 4 c + 1)]Vv, q/v(x) = 0 ,
(2.36)
( 1 - b)(yO + m l ) ( y a +M2)Yvt?v(x ) = 0 .
(2.36a)
or
H. L. Baisya , Rarzta-Schwznger equation
108 Therefore,
we have the following t h r e e c a s e s : (a) Yv~v(x) = 0 .
(2.37)
(b) (y~ +M1)Yvt~v(X) = 0 .
(2.38)
(c) (y0 +M2)'Yvt~v(X) = 0 .
(2.39)
We f i r s t c o n s i d e r the c a s e (a): If eq. (2.37) holds, then eq. (2.19) r e d u c e s to !
Avv(~)t~v(x) = A#v(~)q;v(X) = 0 .
(2.40)
H e n e e , the field ~ # ( x ) is the R a r i t a - S c h w i n g e r field f o r spin 3 s a t i s f y i n g
( ~ +m)q;u(x) = 0 ,
~ts@p(x) = 0 ,
yVqJp(x) = 0 .
(2.41)
We now c o n s i d e r the c a s e (b): F r o m eq. (2.28) and u s i n g eq. (2.29) we have
~v~v(X) = - M l ( 2 b - 1)Yv@v(X) + (2c + ½)mYvt~v(X ) .
(2.42)
F r o m eq. (2.19) and eq. (2.42) we obtain
@O+rn)@p(x) = [ - ( b - 1)Ml+(C+½)m]Yv~v~v(X)+~VYv~v(X)
.
(2.43)
F r o m eq. (2.42) we have ( ~ v - ~rn Yv) ~/v(x) = - Ml(2b - 1)Yv~v(X) + 2era Yv~v(X) . M u l t i p l y i n g eq. (2.44) by (0 F +½m y p ) on the left we obtain
(2.44)
(~/~ +½m y , ) ( ~ v - ½m Yv) ~v (x) = (ap + ½ m y , ) [ - ( 2 b - 1)M 1 + 2cm] yvt~v(X). (2.45) Again, m u l t i p l y i n g eq. (2.43) by (yO- m) on the left and s i m p l i f y i n g we h a v e
(~-m2)~p(x)
= (~p+½myla)[(1-2b)Ml+2Cm]Yv~v(X)
.
(2.46)
H e n c e , c o m p a r i n g eq. (2.45) and eq. (2.46) we have
(L~- m2) t~p(x) = ( ~ , + ½my p)( ~v-½mYv)~v(X) ,
(2.47)
which m e a n s that
q;p(x) = (~/~ +½m y/~)@(x) ,
(2.48)
with
+(x) _ ~ - 1m~ (%_ +m ?v)+v(X) _
1
M 2 - m2
(a v - ½m Yv) + v ( x )
(2.49)
S i m i l a r l y , f o r the e a s e (e) a l s o , the s o l u t i o n is s a m e with M 1 r e p l a c e d by
M 2. T h e r e f o r e , we c a n w r i t e the m o s t g e n e r a l s o l u t i o n of eq. (2.19) as tp/c(x) = ~ ( x )
+Nl(Obt +½my/l)@(x , 1) +N2(~:j +½m y~)qv(x, 2) ,
(2.50)
H.L. Baisya, Rarita-Schwinger equatton
109
where 3
A~v(2) ~V~(x) = 0 ,
(2.51)
(y2 +M1)@(x , 1) = 0 ,
(2.52)
(78 +M2) ~V(x, 2) = 0 ,
(2.53)
N 1 and N 2 a r e coefficients which will be d e t e r m i n e d a little later. C a s e II. In this case, under the assumptions (i) and (ii) eq. (2.36) becomes (c + 1)m [72 +M] 7 v ~ v ( X ) = 0 ,
(2.54)
4c+1 M - 2(c + 1) m .
(2.55)
(a) Yv@v(X) = 0 .
(2.56)
where we have defined
T h e r e f o r e , we have e i t h e r
or
(2.57)
(b) (yO+M)yv~v(x) = 0 . Case (a). If eq. (2.56) holds, then eq. (2.19) r e d u c e s to v
Auv(2),v(x) = 0.
(2.58)
That is, the field ~Vu(x) is the R a r i t a - S e h w i n g e r field for spin { d i s c u s s e d before. Case (b). F r o m eq. (2.28) we have 2 a v t~v(X ) - 2(y0)V v tPv(X ) - (4c + 1)m Yv ~Vv(X) = 0 .
(2.59)
Using eq. (2.57) in eq. (2.59) we have 2v • v ( x )
= cMyv¢v(x
(2.60)
) .
Now f r o m eq. (2.19) and using eq. (2.57) and eq. (2.60) we have
(72 + m)~/u(x) = [c(M- m) +m]~uTv%(x) + a~Tv,v(x).
(2.61)
F r o m eq. (2.60) we can write (2.62)
( a v - ½m Yv) * v (x) = ( c M - ½m)7 v * v (x) .
Then we obtain (2p +½m ~ # ) ( 2 v - ½m 7v)~Pv(x ) = (2# +{m ~ # ) ( e M - ½ m ) Y v ~ v ( x )
.
(2.63)
Multiplying eq. (2.61) by (T2- m) and simplifying we obtain (D- m2)~V#(x) = (a# +½m 7U) [ c M - ½m] 7 v ~ v ( X ) . In obtaining eq. (2.64) use has been made of eq. (2.55) and eq. (2.57).
(2.64)
110
H. L . Ba isya , Ramta-Schwinger
equation
H e n c e , c o m p a r i n g eqs. (2.63) and (2.64) we obtain (~ - m 2)~Vu(x) = (~ ~ + ½m y p)( 8 v - ½m Yv) g/v(x) ,
(2.65)
which m e a n s that the f i e l d ~V~(x) is ~ p ( x ) = (8 p + ½rn 7/z)~(x) ,
(2.66)
with ~(x)
1 -
rn 2
~.
- M2
1
(8 v- ½m 7v)~v(x)
m2
(8 v - ½m -rv)g~v(x)
(2.67)
We s e e that t h e r e a r e only two s o l u t i o n s to eq. (2.19) u n d e r the c a s e H. T h e m a s s a s s o c i a t e d with one of the field is g i v e n by eq. (2.55). C a s e III. In this c a s e , u n d e r the a s s u m p t i o n s (iii) and (iv) eq. (2.36) b e comes (c + 1) [ [ ? - M 2] ~vt~v(X) = 0 ,
(2.68)
w h e r e we h a v e defined M2
4 c + 1 rn 2 ---~1
(2.69)
M will be r e a l if 4 (4c + 1)/(c + 1) is r e a l and it will be i m a g i n a r y if + 1)/(c + 1) is i m a g i n a r y . T h e r e f o r e , f r o m eq. (2.68) we have the following c a s e s : e i t h e r
(a) 7v~v(x) = o
(2.70)
or
(b) ( ~ - M 2 ) y v ~ v ( x )
= 0 .
(2.71)
C a s e (a) i m p l i e s that the field ~/z(x) is the R a r i t a - S c h w i n g e r f i e l d f o r spin as discussed before. C a s e (b). F r o m eq. (2.27) and u s i n g (iv) and eq. (2.71) we obtain ~R~ 2 - ~ v ~ v ( X ) + ½(c + 1) ~ - ¥ v q V v ( X ) " e(V~)Vv~v(X) = 0 . (2.72) Again, f r o m eq. (2.28) we have 2 8 v ~ v ( x ) + 2 c ( y a ) Y v @ v ( x ) - m ( 4 c + 1)Yv~Vv(x ) = 0 .
(2.73)
F r o m eq. (2.19) we obtain
(~8 + m)~p(x) = 8btyvqvv(x)- m c ylcYv@v(X) +yhtav@v(X)- ½(1- c)ygt(yS)yvqvv(x ). (2.74)
U s i n g eq. (2.72) in eq. (2.74) we obtain
H. L .
Bazsya , Rarita-Schwinger equation
(Ta + m ) @ u ( x ) = ~ . y v ¢ / v ( X ) - [ r n c
111
+ I -4cc2 - ~ - 7P vvt~v(x) c+l + ~ - VuavC'v(X) .
(2.75)
M--~ ½(c + 1) - ½m - c(v,)l vvC, v(x) .
(2.76)
F r o m eq. (2.72) we can write (a v - -~m 1 ~ v) ~v(x) = T h e r e f o r e , we have
(~ ~, +½m 7~)(~v - ½m ~)~vCx) = ( ~ +½my u) IM~ ½(c + 1) - Zm 2 - c(7~)] 7v~v(x)
(2.77)
Using eq. (2.73) in eq. (2.75) we have
(~+m)~u(x)
~M2(c + 1) + m2 t = -c au~v%(X) + L 4~ ~Tv~v(x) + ½(c + 1 ) ( ~ a ) ~ p ~ v ~ v ( X ) .
(2.78)
Multiplying eq. (2.78) by (7~- m) and simplifying we obtain
( n - m 2 l % ( x ) = (0,+½m~,) IM~½(c+l)-½m-c(Ta)l
~v%(x).
(2.79)
Hence, comparing eq. (2.77) and eq. (2.79) we have (E]- m 2) ~u(x) = ( ~U +½m 7 9 ( ~v - ½m 7v) ~v (x) ,
(2.80)
which means, as before, that the field ¢,/~(x) can be expressed as
~u(x) = (a. +½m~.)~(x),
(2.81)
with 1 gJ(x) - [3- m 2 (av-½mTv)C'v(X) 1
- M2 - m2 (~v-½mYv)C'v(x) •
(2.82)
In this case also, there are two solutions to eq. (2.19) and the m a s s a s s o ciated with one of the field is given by eq. (2.69). 3. QUANTIZATION We now proceed to obtain the commutation relation for the fields satisfying eqs. (2.16) and (2.19). F o r this purpose we need the r e c i p r o c a l operator d(a) which will satisfy
112
H. L .Batsya,
A(a)d(a) With a = -1,
the derivative n,,(a)
Rartta-Schwrnger
= (o-
m2)(‘7
operator
= [$,(a)
equation
- M+
- ~4:) .
(3.1)
fil(c+ ~)Y~LY~],
(3.2)
eq. (2.6) is
- (b- ~)Q(Y~)Y,-
where A;,(a) is given by eq. (2.20). The reciprocal operator d(a) which will satisfy (eq. 3.1) can be written as
dvh(a)
= p-
hz+-
M&i&a)
+ q
_I- m2)(n-
- ~~(3
hif~)(ay+f~yy)(ya
- hqaA
+$mYA)
~2)(1-M~)(ay+BmrU)(ra-M2)(aX+gmYX)
,
(3.3)
where 1
d&a)
1 ry2)[bVx- gyVyA + rm (YA-
= -(+
YAaV) - -3;2
%A1
(3.4)
which satisfies $,(6)&h(a)
= (n- m2)6nX ,
(3.5)
?zl and n2 are given in the appendix. The commutation relation for the spin $ field satisfying written as
eq. (2.16) can be
expand the field QcL(x) in terms of the wave functions
ti (x) = c
Y
4
c J d30{u(Y)(x
ILP ’
Y
+?)(z)
P
+v(~) &’
u ti(x, z) and 6
. lr)+ (41 ’ “VP
(3.7)
The wave functions and the creation and annihilation operators are characterized by the momentum p and spin state Y corresponding to the mass state i. We normalize the wave functions as -i J doA(x)S:
z)l?,(a,
-i IdoA(z)$)(x,
z]r,(6,
-z)ur)(x, j) = •(~) bij
,
(3.8)
5(P - P’) ,
(3.9)
Byy, b(p- p’)
and
H. L . Baisya , Ramta-Schwznger equation
11 3
where E(m) =+I , F X,
E(M1) = - I ,
o'p( a, -~-) = -'),'k6o.p
(M2)
= +I ,
(3.10)
+ Tp 6X(~ + yo.5,kp - b 7(:rYXTp •
(3.11)
c
The operators a(~r)(i) and b(br)(i) satisfy the relations
)+(j) o(;) (r b )(i)b ')+(j)+bp, All
:~
~ ~(p-p') U rr'
(r) (j)bp (z) = ~ijSrr, 5 ( p - p ' ) .
(3.12)
other relations of this type vanish. We ean show that
id(a)hl+)(X-X')~?(i)-l = ~ f d3pu(r)(x, ,)5(r)(x,,,)E (i) ' r
(3.13)
id(a)A}-)(x-x'), (i)-I = ~ f d3pv(r)(x,i)v(r)(x',i)e (i) ,
(3.14)
where
n (i)
=
I-~ (m i j¢i
),
(i : 1 , 2 , 3 )
(3.15)
(i.e. rn l = m , m2=M1, m 3=M2). From eqs. (3.13) and (3.14) we have
f d3p{u(r)(x,i)u(r)(x',i)+v(pr)(x,i)v(r)(x',i)}E(i) = id(a) ~ ~(i)- 1 hi(x - X') .
(3.16)
i We now define the field quantity
~(x) - ~ ~ f d3pE(i){u-(pr)(x,z)a(r)+(i)+~(r)(x,i)b(r)(i)} i
.
(3.17)
r
We shall obtain the commutation relation between ~(x) and ~ (x) instead of between ~ (x) and ~ (x). The commutation relation between ~(x) and ~(x) is obtained to be {~(x),~(x')} = r~ fd3p{u(r)(x'Ou-(ff)(x"i)+v(r)P
( , ) i ~(r)tx'~ ,,ji~E (i)
= id(a) ~i 77(i)-1 Ai(X-X') = id(a) (m 2.M1)( m12 2_M~)2 ,,,m(x_x,) -
1
,) + AM.(X-X ,
,,
,,-1 ,,
,., AM2(X_X,)]| -id(a)A(x-x') (3.18) D
114
H . L . Ba zsya, R a r i t a - S c h w i n g e r equatzon
T h e r e l a t i o n eq. (3.18) can also be written as
{¢v(x), ~/x(x')} = id'vx(a)&m(X-x')
2 1 - i ~ ni(a u +~m yv)(Va - Mi) i=1
3
2
× (Ok+½m~X)AMi(X-X') = { ~ ' v ( x ) ' ~ (x')} + 2
1
l
{J/~(x,i),~'~(x',i)}.
(3.19)
i=1 The c a u s a l G r e e n ' s function [6] is
(o IT*(~(x), ~(x') I0)
: d(a) ~ •(i)- l&c(X _ x ' , i) i
1 1 : d(a) [ (m2_ M2)(m2 - M2 ) A c ( X - X ' , m ) - (m2_ M21)(M2_ M~) & c ( X - x " M 1 ) L
,
+ (m2_M2)(M2_M2)
A c ( X - x ' , M 2)
]
.
(3.20)
The e n e r g y - m o m e n t u m v e c t o r is
v
i
r
(3.21) T h i s s a t i s f i e s the H e i s e n b e r g equation [~Va(x) , P p] = -iO txYla(X) .
(3.22)
The total e n e r g y of the field which is p o s i t i v e is given by
i
y
where
wi(p) = +
2
2
+m.$ .
(3.24)
In our d i s c u s s i o n of the f r e e field we have noticed that the U m e z a w a Visconti [4] f o r m u l a for the highest o r d e r of d e r i v a t i v e in the r e c i p r o c a l o p e r a t o r is not s a t i s f i e d . The highest o r d e r of d e r i v a t i v e in our r e c i p r o c a l o p e r a t o r is s e v e n w h e r e a s the field has m a x i m u m spin I . Also, we have introduced the quantity ~(x) and obtained the c o m m u t a t i o n r e l a t i o n between ~(x) and ~/(x). As a r e s u l t of the introduction of ~(x), we have s e e n that the total e n e r g y of the field is positive. If we would have u s e d ~(x) instead of ~(x), the e n e r g y would not have been p o s i t i v e . We shall see in the d i s c u s sion of the i n t e r a c t i n g field that ~(x) is a v e r y useful quantity.
H. L Bazsya, Rarita-Schwinger equation
115
4. E L E C T R O M A G N E T I C I N T E R A C T I O N In this s e c t i o n we i n t r o d u c e the e l e c t r o m a g n e t i c i n t e r a c t i o n and q u a n t i z e the i n t e r a c t i n g field. We f i r s t w r i t e the r e c i p r o c a l o p e r a t o r d(0) given by eq. (3.3) in a d i f f e r ! ent f o r m . We w r i t e du~(O) a s
d'ux(O) = [do, u~.+dp,u~.OP+dpcr, u~-~pOcr+dt.t~p, u~.OpOcr~p ] m '
(4.1)
where 1
[do, u)t] m = m 5u)~- ~m 7uT)t , 1
i
[dp, uX]m = (-YllSuX +~YpYuY)t + ½YuSt.tX- 3YXSp u ) , 1
[d p~, uX]rn = (- 3 m Y ~ 7uOX~ + ~
1
Y P Y)t Su(~
2
3m 5P uSXa) '
2 [d p~p, uX] m - 3m 2 Ytz6ua6Xp •
(4.2)
We a r e p u t t i n g the s u b s c r i p t m in eq. (4.1) in o r d e r to i n d i c a t e that the m a s s p a r a m e t e r is m. A l s o , we can w r i t e
(0 v + ½m Vu)(Y pO U - M1)(0X +½m 7X) = [do, ux+d•,
u X O p + d p ~ , uX 8pO~+dtiap, uX OpOcrOp] M l , m
,
(4.3)
where
[do, v)d M1, m = - ~ m 2 M1YvYX , [ d u , vX] M1, m = ( ¼m2 YvY pYX - ~1 m M 1 YX6 p u - ~rn , M 1 Yu 5p X),
[d pff , v~t] M1, m = (½m 7 U~/X6 v a - M 1 5Uv6Xff + ½m yvy U 6Xcr) , [dump, vX] M1, m = 7 u 6ua6Xp •
(4.4)
A s i m i l a r e x p r e s s i o n c a n be w r i t t e n f o r (a v +½m Vu)(7 #~ p - M2) (0~, + ½rn YX) with M 2 in p l a c e of M 1 in eqs. (4.3) and (4.4). F o r c o n v e n i e n c e of s y m m e t r y , we s h a l l w r i t e M 3 f o r m f r o m now on. Substituting eqs. (4.1) and (4.3) in eq. (3.3) and r e a r r a n g i n g the t e r m s we c a n w r i t e d(O) a s
116
H.L. Baisya, Ramta-Schwinger equation
dvx(O)= d(o1,)vx+d(1) ~ +d (I) a a +d(1) p,v~t p pa, vX p a pap,vX +
+d
0 ~
p a p
d (2) []+d (2) [DO +d (2) 0 +d (2) OpOa~p O, v~. p, v~. p pa, vX [::]~p a pap, v~. {~ +d
~ + p pa, w
p ~
pap,
(4.5)
p a p'
where d(1) 2 2 [do, vX] M1, M 3 O, vX _=M~lM22[do, vX]M3 +7/1M3M2 -
2 2 7/2M3Ml[do, vX]M2,M 3 '
d(1) 2 2 2 2 p, v~t - M1M2 [d P, vX] M3 +7/1M3M2 [d p, vX] M 1 , M 3 2 2 - n2M3Ml[dp, v~t]M2,M 3 , 2 2 d(1) 2 2 pa, vX = M1M2 [d pa, v~t] M 3 +7/ 1M3M2 [d P a, vk ]1M1, M3
22 22 d (1) pap, vk - M1M 2 [d P~P, vX] M3 +7/1M3M2 [d pap, vX]M 1, M 3 - n2M~M~l [dpao, vXl M2,M3 ' d (2) - - ( M ~ 2 2 2 O, v~t +M2) [do, v~t]M3 - n l(M3 +M2) [do, vX] M 1, M8 2
2
+ 7/2(M3 +M 1) [do, v~t]M2,M 3 ,
d(2)
2
2
p, vX = "(M~I +~22 )[dp, vX] M3 " 7/1(M3 +M 2) [dp, v~t] M1 ' M3
+7/2( + )Eap, v1M2,
'
d(2)
pa, vX
+ 7/2(M3 +M ) [dpa ' vX]M2,M3 ,
H. L. Baisya, Rarita-Schwinger equation
] 17
d (2) I~ap, vX 2 2 + n2(M 3 + M 1) [d~ap, vX] M2, M S ' d (3) 0, vX - [do, vX] M 3 +n 1 [do, v3.] M1, M 3 - n2 [do, vX] M2, M 3 ' d (3) . --- [d
p, vA
d (S)
pa, vX
=- [d
p, vk]M3+n 1 [d p,v~t]M1,M 3 -n2[d p,v~]M2,M3 , ~,vX]M3+nl[dpa,
vX]M1,M3-n2[d~cr, v~lM2,M3 '
d(3)~p, vk - [d V~p, v X ] M 3 + n l [ d p ~ p , v ? J M 1 , M 3 - n 2 [d pap, v?JM 2,M 3
(4.6)
We n o t i c e that d (3)
]~ff, VX
d (3)
= 0
p crp , v ~t
--- 0
(4.7)
(4.8)
One can e a s i l y s e e that eqs. (4.7) and (4.8) a r e t r u e . T h e f o r m of d(0) given in eq. (4.5) is u s e f u l to w o r k when i n t e r a c t i o n is i n t r o d u c e d . We i n t r o d u c e the m i n i m a l e l e c t r o m a g n e t i c i n t e r a c t i o n by r e p l a c i n g ap by
D~ = a~z-ieA~z ,
(4.9)
in eq. (2.5) with a = -1. T h e equation of m o t i o n now b e c o m e s
( FU, ~taD P + mM~ta) ~a(x) = 0 , or
Axa(a) ¢a(x) : ix(X),
(4.10)
jx(x) = ie F U, ka ~Va(xJA • •
(4.11)
where
T h e H e i s e n b e r g o p e r a t o r ~(x) can b e w r i t t e n as
~(x) : ~(x/e)+½ ~ where
d4x'[d(~),¢(Xo-X'o)]AM M M ( x - x ' ) j ( x ' ) 1 2 3 '
(4.12)
] 18
H.L. Baisya, Ramta-Schwinger equatzon 1
AMIM2M3(X-x')
=- (M2_ M2)(M2 - M2 ) A M I ( X - x ' )
1
1
(M2_M2)(M22_M2) A M 2 ( X - X ' ) + (M2_M2)(M22_M2) A M 3 ( X - X ' ) .
(4.13)
T h e f i e l d @(x/a) s a t i s f i e s the f r e e f i e l d c o m m u t a t i o n r e l a t i o n : {tp(x/cr), q/(x'/a)} = id(O )n M1M2M3 (X - x') .
(4.14)
We shall now show that the s e c o n d t e r m of eq. (4.12) will v a n i s h . Now
½ [d(O),E(Xo-X'o)]AM1M2M3(X - x ' ) =
1
½ [d(O), E(x o - Xo) ] AMI(X - x')
1
½ [d(0) E(x o - Xo)]AM2(X - x ' )
1
+ (M2_M2)(M2_M2)½[d(a),
E(Xo-X'o)]AM3(X-X').
(4.15)
We e v a l u a t e e a c h t e r m of eq. (4.15) by u s i n g the f o r m u l a given by K a t a y a m a
[71. We define
Niz~p.. " - n p n ~ n p . . .
,
(4.16)
w h e r e n is the unit n o r m a l to the s u r f a c e or. Also, we define
-½[d(O),E(Xo-Xo)]A(x-x')
=- lid(a)]] K 5(4)(x- x ') ,
(4.17)
w h e r e K is the m a s s p a r a m e t e r . By the f o r m u l a we have when
(4.18)
d(a) = a ~ a e , when
d(a) = ~ pO(yOp ,
[[0 p 0~0p]] = (npn~Spc e + npnpScr a + ndnpSua
+ 2nLtn(~npna)a a =- Arp(ypa~ a ;
(4.19)
when d(a) : [B,
[[D]] = -1 ;
(4.20)
[[ff32]] : - ( [ ] + K2) ,"
(4.21)
when d(a) = if32 ,
H. L. Ba isya, Rarita-Schw roger equat ion
119
when d(8) = lisp ,
[[liSp]] : -ap ;
(4.22)
when d(a) = []apSa,
[[[]apSa]l:-SpSa+K2Npa
;
(4.23)
when d(a) = E]apSaa p ,
[[[~apSaap] ] = -apaaap+K2hTpapaaa;
(4.24)
when d(a) = O2ap ,
[[[]28p]] = -ap(O+ K2) .
(4.25)
Using eqs. (4.18)-(4.25) in eq. (4.15) we obt~,in ½ [d(8), E(x o - Xo)] AM1M2M3(X - x') = +d(2)+d(2)8 o
p
1 .r_d(1) N (M 2 - M2)(M 2 21 - M2) [ p a p a p
-d(2){-8 pc
a +M~N
p a
t
pa
_ d (1) N p~
8
p~aa
} - d (2) ~-8 8 8 +M211.~papaaa} pap-
/x a p
+ d(o3)(0+ M21) +d(3)8 p p ([3+ M~I)] 5 (4) (x-x') _
+d(2)+d(2)8 o
p
(U 2 p
1
2 2 M 2) I-d(1) N _M2)(M2_ pa pa
d(1)
N
pap
8 papa a
_d(2)~_SpaG+M22Npa}_dpapL(2)~-8p8 8p+M22~pGpaaa} pal a
+ d(o3) ([~+ M22) +d~)8 p ([]+ M22) ]6 (4) (x - x,) 1 I-d(1) N - d (1) .~ 8 + (M 21 - M3)(M 2 2_M32) L pa I~G pap papa a 2
+d(2)+d(2)~ -d(2){-8 8 + 4 N p a } - a p a p.(3) { o p t~ pG p a
-8
P 8 G8 p + M 32g^p a p a S a }
+ d(o3)(E;+ M~)+d~)8 p([~+ M~)] 5(4)(x-x'). Therefore, substituting eq. (4.26) in eq. (4.12) we obtain
(4.26)
H.L. Baisya, Rarita-Schwinger equation
120
1
I
1
- (M21 - M 2 ) ( M I - M2 ) [
]j(x) ,
(4.27)
where the s q u a r e b r a c k e t contains the quantities as in the p r e v i o u s e x p r e s sion. Now collecting the coefficients of d~,l)~Ypcr j(x),~v d!Z)Nlz
E _
' 1
1
~M~,_ M 2~.) ( M 2~._ M 3 ~.+(H-@(M~-@] ) +[ - [
-[
]d (1))q
p~p
] d (2) a
a j(x)-[ p~pa a
j(x)-[
l d (3) Da J p P
j
]d(2)j(x)
o
] d (2) a
0 j(x)
]d(o3)Dj(x)
] d (2) ~ ~ ~ j ( x ) - [ ~o'p p c r p - [
~(~)N '" ~ e tx(y :#t
j(x)
+[(M~,-@(M~-@-(H-@(~
~V(x) = ~ ( x / ~ ) .
(4.29)
~(x) -= ~(0 ~i(x),
(4.30)
~(x) = @ x / ~ ) .
(4.31)
Now defining
we have
H. L. Baisya, Rarita-Schw~nger equatzon
121
Hence the commutation relation between ~P(x) and ~(x) can be written as =
= id(O)AmMiM2(X- x')] XO=X,0 3
3
2
1
1
= (,r(x),~2(x')}Xo:X ° + 2 (d(x, Mi).~(x',Mi)}Xo=X ° • i=1 (4.32) Here, we have written m for M 3 again. T h e r e f o r e , the commutation relation of the interacting spin 3 field is _3 _3 {~V~(x), ~ 2 (x,)}Xo=X,° = id'( O)nm(X- X')]Xo=Xo, , (4.33) and it is clear that it does not contain any electromagnetic field. The interaction Hamiltonian can be obtained to be
H(x,n) = H(x) = ~(x) j(x) 3
1
3
3_
!
_3
!
!
1
_3
i
!
+ ~ ( x ) Ftz, k p ff/p(X, 1)- ~ ( x , 1)rtz, ~.p ~ ( x , 1)+ ~ ( x , 2)F ,kp ¢~p(X, 1) 3
1
i
i
I
I
+ ~(x)F.Xp~pp(X. 2 ) - ~ ( x , 1)Ftz, Xp~p(X. 2)+~(x, 2)F ~, kp ~ ( x , 2)]A t~ (4.34) The interaction Hamiltonian for the spin ~ field is hermitian as we show below:
[ie~(x)F ,Xp ~9~p(x)A ] * 3
3
3
3
t
= -ie ~v~t(x)7~gexy4 ~V2(x)A 3
3
= -ie ~ * (x)v4Y4y gy4 gax ~(x)A tp =-ie~(x)(-g.aya)~(x)ASt~ = ie ~ ( x ) y a ¢~e(x)Aa .
(4.35)
H. L. Bazsya , Rarita-Schw znger equation
122
S i m i l a r l y , the t e r m s 1
1
-ie fJ~(x, 1) F p , Xp tpp(X, 1)A P
1
and
1
+iv tp;(x, 2) F#, ~p ~p(X, 2)A/~,
in eq. (4.34) a r e also h e r m i t i a n but other t e r m s a r e not. Hence the total Hamiltonian is not hermitian. As pointed out in the Bhabha t h e o r y [8], the interaction Hamiltonian can be made h e r m i t i a n by m e a n s of an indefinite m e t r i c , but in that c a s e the positivity of the e n e r g y will be destroyed. We have noticed with other equations [9] also that in the m u l t i - m a s s t h e o r y this situation will prevail.
5. CONCLUSIONS By introducing the new quantity ~(x) we have shown that the total e n e r g y of the field is definitely positive when the field d e s c r i b e s p a r t i c l e s with many m a s s states. We have also noticed that the U m e z a w a - V i s c o n t i f o r mula is not satisfied in c a s e of v e c t o r - s p i n o r field. We feel that this f o r mula needs modification for m u l t i - m a s s fields. We intend to do this on some other occasion. In our d i s c u s s i o n of the i n t e r a c t i n g field, we have seen that our c o m m u tation relation does not contain any e l e c t r o m a g n e t i c field. This states that in our f o r m a l i s m , the type of i n c o n s i s t e n c i e s pointed out by Johnson and Sudarshan [5] do not s e e m to appear. By using the a n t i c o m m u t a t i o n r e l a t i o n between ~/(x) and ~(x), we have been able to obtain the H e i s e n b e r g o p e r a t o r in a closed f o r m and seen that %9(x) = tp(x/cr) holds. The interaction H a m i l tonian obtained in this f o r m a l i s m is not an infinite s e r i e s of the coupling constant. We do not know whether this will be true in case of higher (> 9) spin fields. We would like to mention that because of the n o n - h e r m i t i c i t y of the i n t e r a c t i o n Hamiltonian, the unitarity of the S - m a t r i x b r e a k s down as in the Bhabha theory [8]. The author would like to thank P r o f e s s o r Y. Takahashi for valuable suggestions.
APPENDIX
To obtain the coefficients N 1 and N2: In o r d e r to obtain the coefficients N 1 and N 2 we use the anticommutation 1
1
r e l a t i o n s satisfied by g/~(x, 1) and ~ ( x , 2). The anticommutation r e l a t i o n s can be obtained as follows. The D i r a c field s a t i s f i e s the a n t i c o m m u t a t i o n relation {~(x, 11, ~(x', 1/} = i(yO - M l l A ( x - x', M1) .
(A.1) ~k ~ t
1
Multiplying eq. (A.1) by Nl(alx +½my/~) f r o m the left and by N l ( a v - ~ m ~ v ) f r o m the right we have
123
H. L . Bais ya , Rarita-Schwinger equatzon --
,
e-~
1
N l ( a # +½m yp) {¢,(x, 1),~(x', 1)}Nl(a v - ~m Vv) = i IN 11 2 (a u +½m ~.)(~a - M1)A(x- x', M1)(~ v -'~m Yv) , or 1
1
{@~(x, 1), ~tPv(X', 1)} = -i IN1 ] 2(a tz + ½m y/~)(ya - M1)(O v +½m Yv)A(x- X', M 1) -: -inl(a p +½m yp)(ya - M1)(a v +½m y v ) n ( x - x', M1) = idtzv(a , 1 ) A ( x - x ' , M 1 )
,
(A.2)
where we have defined ,
(j = 1,2). 1
Similarly, the antieommutation relation satisfied by ~ ( x , 2) can be written as 1
1
2 X 2 {@g( , 2), @v(X , 2)} = -in2(a t + ½ m y t ~ ) ( y a - M 2 ) ( a v -
+½m y v ) n ( x - x', M2)
=-- i d p v ( a , 2)A(x- X', M2) •
(A.3)
Now Axp(a)d#v(a , 1) = -n 1 Axu(a)(a p +½myp)(ya - M1)(O v +½mYv) = (.~-M21)Sxv ,
(A.4)
where AX/I(a) is given by eq. (3.2). Now using eq. (2.29) in eq. (A.4) we have -nl(1 - b)y~t(ya +M2)([~- M~l)(a v +½m yv ) = ([~- M~I) Sky .
(A.5)
T h e r e f o r e , we shall have to have -n 1(1 - b)yx(ya + M2)(a v +½m Yv) = 5Xv •
(t.6)
Multiplying eq. (A.6) by (F4)va from the right we obtain -nl(1 - b)yX(ya + M2)(a v +½m yv)(r4)vz = (r4)xe.
(A.7)
Now
(a v +½m yv)(r4)v~ = -[a4v ~- acry4- ( M j - m)54G+ (m + b M j o 2bm)Y4y~] • (A.8) Using eq. (A.8) in eq. (A.7) and putting X = (r = 4 we obtain n l ( b - 1)(M 1 - 2m)@a + M2) - 1 = 0 .
(A.9)
Operating on @(x, 1) by the left-hand side of eq. (A.9) and using eq. (2.52) we have
124
H.L. Batsya, Rarita-Schwinger equation nl(b-
1)(M 1- 2m)(-Ml+
M2) = 1
or
2 M2-2m nI =
3m 2 M1 - M 2
-,
(A.10)
where we have used eq. (2.35). In a s i m i l a r m a n n e r w e c a n h a v e 2 n 2 --
M 1- 2m (A.11)
3m 2 M 1 -
RE FE RE NCE S [1] H M u n c z e k . P h y s . Rev. 164 (1967) 1794. [2] Y . T a k a h a s h l . R e l a t l v l s t m q u a n t l z a t l o n of m u l t i - m a s s f i e l d s , u n p u b h s h e d . [3] Y T a k a h a s h l and H. U m e z a w a . P r o g . T h e o r . P h y s . 9 (1953) 14, Nucl. P h y s . 51 (1964) 193. [4] H. U m e z a w a and A. V l s c o n t l . Nucl. P h y s . 1 (1956) 348. [5] K. J o h n s o n and E . C . G . S u d a r s h a n , Ann. of P h y s 13 (1961) 126. [6] Y T a k a h a s h l . I n t r o d u c t i o n to f m l d q u a n t l z a t l o n ( P e r g a m o n P r e s s , Oxford. 1969) 235 [7] Y . K a t a v a m a . P r o g . T h e o r . P h y s . 10 (1953} 31. [8] H. L. B a l s y a , Nucl. P h y s . B23 (1970) 633. [9] H L B a l s y a . On a m u l t i - m a s s e q u a t i o n and the q u a n t l z a t l o n of h i g h e r - s p i n f i e l d s , Nucl P h y s . . to be p u b h s h e d
ON THE RARITA-SCHWINGER EQUATION FOR THE VECTOR-SPINOR FIELD H i r a L. B A ] S Y A
School of Mathematics, Trinity College, Dubhn Received 15 October 1970 Abstract: It has been shown that the Rarlta-Schwlnger equation for the v e c t o r - s p m o r held with a r b i t r a r y p a r a m e t e r s has three solutions for some values of the p a r a m eters; for some other values it has two solutmns. It is also seen that the Umezawa-Vlscontl formula for the highest order of demvatlve in the reciprocal operator ~s not satlshed in the case of Rarita-Schwmger v e c t o r - s p m o r field. The electromagnetm interactmn of the v e c t o r - s p m o r field is discussed. By a new f o r m a h s m m quantlzatlon we have shown that ~(x) --t~(x/c0 holds even m case of RamtaSchwlnger held and the interacting spin ~2 held has been quantized. It is seen that the interaction Hamiltonian is not an infinite series of the coupling constant.
1. INTRODUCTION In 1967 M u n c z e k [1] d i s c u s s e d the R a r i t a - S c h w i n g e r e q u a t i o n with a r b i t r a r y p a r a m e t e r s showing that there a r e three solutions and each solution c o r r e s p o n d s to d i f f e r e n t m a s s s t a t e . We have s h o w n now t h a t in a d d i t i o n to t h r e e s o l u t i o n s , the e q u a t i o n a l s o h a s two s o l u t i o n s d e p e n d i n g on d i f f e r e n t v a l u e s of the p a r a m e t e r s . E a c h s o l u t i o n c o r r e s p o n d s to a d i f f e r e n t m a s s state and a different spin state. In M u n c z e k ' s f o r m a l i s m , he u s e d the c a n o n i c a l m e t h o d of q u a n t i z a t i o n . We f e e l t h a t t h e c a n o n i c a l m e t h o d of q u a n t i z a t i o n d o e s not a l w a y s give the c o r r e c t r e s u l t if not c a r e f u l l y u s e d . S i n c e the v e c t o r - s p i n o r f i e l d i s s e e n to b e a m u l t i - m a s s f i e l d in o u r f o r m a l i s m , we h a v e u s e d the m e t h o d of q u a n t i z a t i o n of the m u l t i - m a s s f i e l d a c c o r d i n g to T a k a h a s h i [2] i n o u r d i s c u s s i o n . T h i s m e t h o d i s the g e n e r a l i z a t i o n of the m e t h o d of q u a n t i z a t i o n of o n e - m a s s f i e l d by the s a m e a u t h o r a n d U m e z a w a [3]. In o u r d i s c u s s i o n we h a v e n o t i c e d t h a t the U m e z a w a - V i s c o n t i [4] f o r m u l a f o r the h i g h e s t d e r i v a t i v e i n the r e c i p r o c a l o p e r a t o r is not s a t i s f i e d . In the t h e o r y of i n t e r a c t i n g f i e l d s a c c o r d i n g to the a u t h o r s i n ref. [3], it w a s s t a t e d t h a t ~(x) = ~(x/cr) h o l d s only i n s i m p l e c a s e s l i k e s c a l a r o r s p i n o r f i e l d s . We have s h o w n now by a n e w f o r m a l i s m t h a t t h i s h o l d s e v e n i n c a s e of v e c t o r - s p i n o r field. In s e c t . 2 we d i s c u s s the s o l u t i o n s of the f r e e f i e l d e q u a t i o n a n d i n s e c t . 3 we q u a n t i z e the f r e e field. Sect. 4 i s d e v o t e d to the d i s c u s s i o n a n d q u a n t i z a t i o n of the i n t e r a c t i n g field. F i n a l l y , i n s e c t . 5 we p r e s e n t o u r c o n c l u s i o n s .
105
H. L . Baisya, R a r i t a - S c h w i n g e r equation
2. FREE FIELD EQUATION In this section we make a thorough analysis of the solutions of the f r e e field equation. The L a g r a n g i a n for the v e c t o r ° s p i n o r which contains both spin ~ and ½ fields can be written in the absence of i n t e r a c t i o n as = -tPtz(x) [(YX 6~v +a@~ 6~.v +V u 6X~) + b yUVXTv)aX + m 6pv+mCypYv]
g/v(X) ,
(2.1)
where
~p(x) = ~(x)n~,
(2.2)
~ U = Y4 g ~ p ,
(2.3)
and a, b and c a r e a r b i t r a r y p a r a m e t e r s . The quantity gcrp is taken as g~---
+1
for % p = 1 , 2 , 3 ,
-1
for cr, p = 4 ,
0
otherwise .
(2.4)
F r o m eq. (2.1) the equation of motion for ~z(x) is
A#v(a)C,v(x) = o,
(2.5)
where A~v(0) =
-(rx, pv ax+ m M p v )
,
rx, v.v = [~x our +a(% C~v+~v ~x,) + b~u~x~v],
(2.6)
(2.7) (2.8)
MI~ v = 61~v + c y l ~ y v .
Multiplying eq. (2.5) by Y~z and 0g on the left we have r e s p e c t i v e l y 0~qc~t(x) if4a+2
1
4 a + 2 [m(4c+ 1) + ( a + 4 b - 1 ) V 0 ] y ~ t ( x ) ,
(2.9)
¢0and
(a + 1)(ya)avC, v(X) + (a + b)[Y]yv¢/v(X) + m a , ¢ ~ u ( x ) + mc(yO)g/v¢,v(X) = 0 .
(2.10)
F r o m eq. (2.9) and eq. (2.10) we can have that y#~U(x)
=0 ,
(2.11)
if c =-(a+2b) ,
(2.12)
2 b = 3a 2 + 2 a + 1 .
(2.13)
106
H.L.Bazsya
R a m t a - S c h w i n g e r equatzon
U s i n g eq. (2.11) in eq. (2.9) we have 8~p(x)
=0 .
(2.14)
A g a i n u s i n g eqs. (2.11) and (2.14) both in eq. (2.5) we obtain (7X~X + m ) ~p(x) = 0 .
(2.15)
E q s . (2.12) and (2.13) will give the v a l u e s of b and c f o r any value of a. F o r e x a m p l e , if we take a = - 1 , then b = 1 and c = -1. In that c a s e , eq. (2.5) becomes [-(ya+m)bpv+(YpSv+Yv~,)-7,(7a-
rn)yv] ~v(X)
=
0 .
(2.16)
Eq. (2.16) is the usual Rarita-Schwinger equation for the spin 3 field. The parameter a in the Lagrangian is not fixed. This can be seen by using a point transformation [5] t
~p(x) = Tpv~v(x ) ,
(2.17)
T t~v = 5 t_Lv+ ¼PY p Y v "
(2.18)
where
p cannot have the v a l u e -1, in which c a s e the t r a n s f o r m a t i o n will be s i n g u lar. In the following we shall d i s c u s s the s o l u t i o n s of eq. (2.5) u n d e r the f o l lowing a s s u m p t i o n s : C a s e I. A s s u m e a = - 1 ; b and c a r b i t r a r y . C a s e II. In addition to a = - 1 , a s s u m e that (i) b - 1 = 0, and that (ii) 2 b + c - 1 ¢ 0 . C a s e III. In addition to a = - 1 , a s s u m e that (iii) b - 1 ¢ 0, and that (iv) 2 b + c - 1 = O. C a s e I. With a = - 1 , eq. (2.5) b e c o m e s A U v ( ~ ) ~ v ( X ) = [ A u v ( a ) - ( b - 1 ) y p ( y a ) y v - re(c+ 1)7Uyv] t~v(X ) = 0 , (2.19) where ' 8 ) = -(70+rn)61~v+(Yp~v+YvSlz)-yU(ySA~tv(
rn)y v .
(2.20)
Now we c a n have !
AUv(~)~ v = r n ( ~ u - y 0 y U ) ,
(2.21)
A p v ( 8 ) 7 v = 2(78)7~ + 3 r n y ~ - 2~/~ ,
(2.22)
t
a/~A/zv(8) = rn 8 v - rnYv(y8 ) , '
a
71jAI~u( ) = 2a v - 2(78)y/z + 3 r n y u ,
(2.23) (2.24)
A p v ( 8 ) ~ v = - m 8 # - ( b - I ) E ] 7 ~ - mcy/~(ya) ,
(2.25)
A/zv(~)-gv = 2 8 / z - ( 4 b - 2)y/z(y~ ) - (4c + l ) m Y/z ,
(2.26)
H. L. Bazsya, Rarita-Schwinger equatton
a,A,v(a)
= - m a v - (b -
1)[]~v-
mc(ra)v
107
(2.27)
v ,
v p A # v ( a ) = 2a v - 2 ( 2 b - 1 ) ( ~ ) ~ v - m(4c + 1)~ v .
(2.28)
In obtaining eqs. (2.25)-(2.28), u s e h a s b e e n m a d e of eqs. (2.21)-(2.24). F r o m eqs. (2.25) and (2.26) we h a v e 1 Ap.v( a )(av+-~mVv )
=
- y p [ ( b - 1)[~+m(2b+c- 1 ) ( y ~ ) + ½ m 2 ( 4 c + l ) ] = V p ( 1 - b)(vO+M1)('ya+M2) ,
(2.29)
w h e r e M 1 and M 2 a r e the s o l u t i o n s of ( b - 1)M 2 - (2b + c - 1)mM+½m2(4c+ 1) = 0 ,
(2.30)
with b ¢ 1; i.e. 1
M1,2-
[ ( 2 b + c - 1)~-(4b2+c2-4bc- 6 b + 6 c + 3 ) ~] .
m 2(b-l)
(2.31)
A g a i n f r o m eq. (2.27) and eq. (2.28) we h a v e (0# + ½ m v # ) A # v ( a ) = - [ ( b - 1 ) D + m ( 2 b + c - 1)(va) +½m2(4c + 1)]Tv
= (1-b)(ya+M1)(va+M2)v v .
(2.32)
F r o m eq. (2.31) we h a v e M 1 +M 2 = ~
( 2 b + c - 1) ,
(2.33)
m2 M1 M2 - 2(b - 1) (4c + 1) .
(2.34)
U s i n g eq. (2.34) in eq. (2.33) we obtain 1 b- 1 -
2 3m 2 (M 1 - 2 m ) ( M 2 - 2m) .
(2.35)
F r o m eqs. (2.19) and (2.32) we c a n w r i t e
(a N + ½ m r ~ ) A ~ v ( ~ ) ~ v v ( x )
= o,
or
[ ( b - 1)[~+m(2b+c- 1)(V0) + ½ m 2 ( 4 c + 1)]Vv, q/v(x) = 0 ,
(2.36)
( 1 - b)(yO + m l ) ( y a +M2)Yvt?v(x ) = 0 .
(2.36a)
or
H. L. Baisya , Rarzta-Schwznger equation
108 Therefore,
we have the following t h r e e c a s e s : (a) Yv~v(x) = 0 .
(2.37)
(b) (y~ +M1)Yvt~v(X) = 0 .
(2.38)
(c) (y0 +M2)'Yvt~v(X) = 0 .
(2.39)
We f i r s t c o n s i d e r the c a s e (a): If eq. (2.37) holds, then eq. (2.19) r e d u c e s to !
Avv(~)t~v(x) = A#v(~)q;v(X) = 0 .
(2.40)
H e n e e , the field ~ # ( x ) is the R a r i t a - S c h w i n g e r field f o r spin 3 s a t i s f y i n g
( ~ +m)q;u(x) = 0 ,
~ts@p(x) = 0 ,
yVqJp(x) = 0 .
(2.41)
We now c o n s i d e r the c a s e (b): F r o m eq. (2.28) and u s i n g eq. (2.29) we have
~v~v(X) = - M l ( 2 b - 1)Yv@v(X) + (2c + ½)mYvt~v(X ) .
(2.42)
F r o m eq. (2.19) and eq. (2.42) we obtain
@O+rn)@p(x) = [ - ( b - 1)Ml+(C+½)m]Yv~v~v(X)+~VYv~v(X)
.
(2.43)
F r o m eq. (2.42) we have ( ~ v - ~rn Yv) ~/v(x) = - Ml(2b - 1)Yv~v(X) + 2era Yv~v(X) . M u l t i p l y i n g eq. (2.44) by (0 F +½m y p ) on the left we obtain
(2.44)
(~/~ +½m y , ) ( ~ v - ½m Yv) ~v (x) = (ap + ½ m y , ) [ - ( 2 b - 1)M 1 + 2cm] yvt~v(X). (2.45) Again, m u l t i p l y i n g eq. (2.43) by (yO- m) on the left and s i m p l i f y i n g we h a v e
(~-m2)~p(x)
= (~p+½myla)[(1-2b)Ml+2Cm]Yv~v(X)
.
(2.46)
H e n c e , c o m p a r i n g eq. (2.45) and eq. (2.46) we have
(L~- m2) t~p(x) = ( ~ , + ½my p)( ~v-½mYv)~v(X) ,
(2.47)
which m e a n s that
q;p(x) = (~/~ +½m y/~)@(x) ,
(2.48)
with
+(x) _ ~ - 1m~ (%_ +m ?v)+v(X) _
1
M 2 - m2
(a v - ½m Yv) + v ( x )
(2.49)
S i m i l a r l y , f o r the e a s e (e) a l s o , the s o l u t i o n is s a m e with M 1 r e p l a c e d by
M 2. T h e r e f o r e , we c a n w r i t e the m o s t g e n e r a l s o l u t i o n of eq. (2.19) as tp/c(x) = ~ ( x )
+Nl(Obt +½my/l)@(x , 1) +N2(~:j +½m y~)qv(x, 2) ,
(2.50)
H.L. Baisya, Rarita-Schwinger equatton
109
where 3
A~v(2) ~V~(x) = 0 ,
(2.51)
(y2 +M1)@(x , 1) = 0 ,
(2.52)
(78 +M2) ~V(x, 2) = 0 ,
(2.53)
N 1 and N 2 a r e coefficients which will be d e t e r m i n e d a little later. C a s e II. In this case, under the assumptions (i) and (ii) eq. (2.36) becomes (c + 1)m [72 +M] 7 v ~ v ( X ) = 0 ,
(2.54)
4c+1 M - 2(c + 1) m .
(2.55)
(a) Yv@v(X) = 0 .
(2.56)
where we have defined
T h e r e f o r e , we have e i t h e r
or
(2.57)
(b) (yO+M)yv~v(x) = 0 . Case (a). If eq. (2.56) holds, then eq. (2.19) r e d u c e s to v
Auv(2),v(x) = 0.
(2.58)
That is, the field ~Vu(x) is the R a r i t a - S e h w i n g e r field for spin { d i s c u s s e d before. Case (b). F r o m eq. (2.28) we have 2 a v t~v(X ) - 2(y0)V v tPv(X ) - (4c + 1)m Yv ~Vv(X) = 0 .
(2.59)
Using eq. (2.57) in eq. (2.59) we have 2v • v ( x )
= cMyv¢v(x
(2.60)
) .
Now f r o m eq. (2.19) and using eq. (2.57) and eq. (2.60) we have
(72 + m)~/u(x) = [c(M- m) +m]~uTv%(x) + a~Tv,v(x).
(2.61)
F r o m eq. (2.60) we can write (2.62)
( a v - ½m Yv) * v (x) = ( c M - ½m)7 v * v (x) .
Then we obtain (2p +½m ~ # ) ( 2 v - ½m 7v)~Pv(x ) = (2# +{m ~ # ) ( e M - ½ m ) Y v ~ v ( x )
.
(2.63)
Multiplying eq. (2.61) by (T2- m) and simplifying we obtain (D- m2)~V#(x) = (a# +½m 7U) [ c M - ½m] 7 v ~ v ( X ) . In obtaining eq. (2.64) use has been made of eq. (2.55) and eq. (2.57).
(2.64)
110
H. L . Ba isya , Ramta-Schwinger
equation
H e n c e , c o m p a r i n g eqs. (2.63) and (2.64) we obtain (~ - m 2)~Vu(x) = (~ ~ + ½m y p)( 8 v - ½m Yv) g/v(x) ,
(2.65)
which m e a n s that the f i e l d ~V~(x) is ~ p ( x ) = (8 p + ½rn 7/z)~(x) ,
(2.66)
with ~(x)
1 -
rn 2
~.
- M2
1
(8 v- ½m 7v)~v(x)
m2
(8 v - ½m -rv)g~v(x)
(2.67)
We s e e that t h e r e a r e only two s o l u t i o n s to eq. (2.19) u n d e r the c a s e H. T h e m a s s a s s o c i a t e d with one of the field is g i v e n by eq. (2.55). C a s e III. In this c a s e , u n d e r the a s s u m p t i o n s (iii) and (iv) eq. (2.36) b e comes (c + 1) [ [ ? - M 2] ~vt~v(X) = 0 ,
(2.68)
w h e r e we h a v e defined M2
4 c + 1 rn 2 ---~1
(2.69)
M will be r e a l if 4 (4c + 1)/(c + 1) is r e a l and it will be i m a g i n a r y if + 1)/(c + 1) is i m a g i n a r y . T h e r e f o r e , f r o m eq. (2.68) we have the following c a s e s : e i t h e r
(a) 7v~v(x) = o
(2.70)
or
(b) ( ~ - M 2 ) y v ~ v ( x )
= 0 .
(2.71)
C a s e (a) i m p l i e s that the field ~/z(x) is the R a r i t a - S c h w i n g e r f i e l d f o r spin as discussed before. C a s e (b). F r o m eq. (2.27) and u s i n g (iv) and eq. (2.71) we obtain ~R~ 2 - ~ v ~ v ( X ) + ½(c + 1) ~ - ¥ v q V v ( X ) " e(V~)Vv~v(X) = 0 . (2.72) Again, f r o m eq. (2.28) we have 2 8 v ~ v ( x ) + 2 c ( y a ) Y v @ v ( x ) - m ( 4 c + 1)Yv~Vv(x ) = 0 .
(2.73)
F r o m eq. (2.19) we obtain
(~8 + m)~p(x) = 8btyvqvv(x)- m c ylcYv@v(X) +yhtav@v(X)- ½(1- c)ygt(yS)yvqvv(x ). (2.74)
U s i n g eq. (2.72) in eq. (2.74) we obtain
H. L .
Bazsya , Rarita-Schwinger equation
(Ta + m ) @ u ( x ) = ~ . y v ¢ / v ( X ) - [ r n c
111
+ I -4cc2 - ~ - 7P vvt~v(x) c+l + ~ - VuavC'v(X) .
(2.75)
M--~ ½(c + 1) - ½m - c(v,)l vvC, v(x) .
(2.76)
F r o m eq. (2.72) we can write (a v - -~m 1 ~ v) ~v(x) = T h e r e f o r e , we have
(~ ~, +½m 7~)(~v - ½m ~)~vCx) = ( ~ +½my u) IM~ ½(c + 1) - Zm 2 - c(7~)] 7v~v(x)
(2.77)
Using eq. (2.73) in eq. (2.75) we have
(~+m)~u(x)
~M2(c + 1) + m2 t = -c au~v%(X) + L 4~ ~Tv~v(x) + ½(c + 1 ) ( ~ a ) ~ p ~ v ~ v ( X ) .
(2.78)
Multiplying eq. (2.78) by (7~- m) and simplifying we obtain
( n - m 2 l % ( x ) = (0,+½m~,) IM~½(c+l)-½m-c(Ta)l
~v%(x).
(2.79)
Hence, comparing eq. (2.77) and eq. (2.79) we have (E]- m 2) ~u(x) = ( ~U +½m 7 9 ( ~v - ½m 7v) ~v (x) ,
(2.80)
which means, as before, that the field ¢,/~(x) can be expressed as
~u(x) = (a. +½m~.)~(x),
(2.81)
with 1 gJ(x) - [3- m 2 (av-½mTv)C'v(X) 1
- M2 - m2 (~v-½mYv)C'v(x) •
(2.82)
In this case also, there are two solutions to eq. (2.19) and the m a s s a s s o ciated with one of the field is given by eq. (2.69). 3. QUANTIZATION We now proceed to obtain the commutation relation for the fields satisfying eqs. (2.16) and (2.19). F o r this purpose we need the r e c i p r o c a l operator d(a) which will satisfy
112
H. L .Batsya,
A(a)d(a) With a = -1,
the derivative n,,(a)
Rartta-Schwrnger
= (o-
m2)(‘7
operator
= [$,(a)
equation
- M+
- ~4:) .
(3.1)
fil(c+ ~)Y~LY~],
(3.2)
eq. (2.6) is
- (b- ~)Q(Y~)Y,-
where A;,(a) is given by eq. (2.20). The reciprocal operator d(a) which will satisfy (eq. 3.1) can be written as
dvh(a)
= p-
hz+-
M&i&a)
+ q
_I- m2)(n-
- ~~(3
hif~)(ay+f~yy)(ya
- hqaA
+$mYA)
~2)(1-M~)(ay+BmrU)(ra-M2)(aX+gmYX)
,
(3.3)
where 1
d&a)
1 ry2)[bVx- gyVyA + rm (YA-
= -(+
YAaV) - -3;2
%A1
(3.4)
which satisfies $,(6)&h(a)
= (n- m2)6nX ,
(3.5)
?zl and n2 are given in the appendix. The commutation relation for the spin $ field satisfying written as
eq. (2.16) can be
expand the field QcL(x) in terms of the wave functions
ti (x) = c
Y
4
c J d30{u(Y)(x
ILP ’
Y
+?)(z)
P
+v(~) &’
u ti(x, z) and 6
. lr)+ (41 ’ “VP
(3.7)
The wave functions and the creation and annihilation operators are characterized by the momentum p and spin state Y corresponding to the mass state i. We normalize the wave functions as -i J doA(x)S:
z)l?,(a,
-i IdoA(z)$)(x,
z]r,(6,
-z)ur)(x, j) = •(~) bij
,
(3.8)
5(P - P’) ,
(3.9)
Byy, b(p- p’)
and
H. L . Baisya , Ramta-Schwznger equation
11 3
where E(m) =+I , F X,
E(M1) = - I ,
o'p( a, -~-) = -'),'k6o.p
(M2)
= +I ,
(3.10)
+ Tp 6X(~ + yo.5,kp - b 7(:rYXTp •
(3.11)
c
The operators a(~r)(i) and b(br)(i) satisfy the relations
)+(j) o(;) (r b )(i)b ')+(j)+bp, All
:~
~ ~(p-p') U rr'
(r) (j)bp (z) = ~ijSrr, 5 ( p - p ' ) .
(3.12)
other relations of this type vanish. We ean show that
id(a)hl+)(X-X')~?(i)-l = ~ f d3pu(r)(x, ,)5(r)(x,,,)E (i) ' r
(3.13)
id(a)A}-)(x-x'), (i)-I = ~ f d3pv(r)(x,i)v(r)(x',i)e (i) ,
(3.14)
where
n (i)
=
I-~ (m i j¢i
),
(i : 1 , 2 , 3 )
(3.15)
(i.e. rn l = m , m2=M1, m 3=M2). From eqs. (3.13) and (3.14) we have
f d3p{u(r)(x,i)u(r)(x',i)+v(pr)(x,i)v(r)(x',i)}E(i) = id(a) ~ ~(i)- 1 hi(x - X') .
(3.16)
i We now define the field quantity
~(x) - ~ ~ f d3pE(i){u-(pr)(x,z)a(r)+(i)+~(r)(x,i)b(r)(i)} i
.
(3.17)
r
We shall obtain the commutation relation between ~(x) and ~ (x) instead of between ~ (x) and ~ (x). The commutation relation between ~(x) and ~(x) is obtained to be {~(x),~(x')} = r~ fd3p{u(r)(x'Ou-(ff)(x"i)+v(r)P
( , ) i ~(r)tx'~ ,,ji~E (i)
= id(a) ~i 77(i)-1 Ai(X-X') = id(a) (m 2.M1)( m12 2_M~)2 ,,,m(x_x,) -
1
,) + AM.(X-X ,
,,
,,-1 ,,
,., AM2(X_X,)]| -id(a)A(x-x') (3.18) D
114
H . L . Ba zsya, R a r i t a - S c h w i n g e r equatzon
T h e r e l a t i o n eq. (3.18) can also be written as
{¢v(x), ~/x(x')} = id'vx(a)&m(X-x')
2 1 - i ~ ni(a u +~m yv)(Va - Mi) i=1
3
2
× (Ok+½m~X)AMi(X-X') = { ~ ' v ( x ) ' ~ (x')} + 2
1
l
{J/~(x,i),~'~(x',i)}.
(3.19)
i=1 The c a u s a l G r e e n ' s function [6] is
(o IT*(~(x), ~(x') I0)
: d(a) ~ •(i)- l&c(X _ x ' , i) i
1 1 : d(a) [ (m2_ M2)(m2 - M2 ) A c ( X - X ' , m ) - (m2_ M21)(M2_ M~) & c ( X - x " M 1 ) L
,
+ (m2_M2)(M2_M2)
A c ( X - x ' , M 2)
]
.
(3.20)
The e n e r g y - m o m e n t u m v e c t o r is
v
i
r
(3.21) T h i s s a t i s f i e s the H e i s e n b e r g equation [~Va(x) , P p] = -iO txYla(X) .
(3.22)
The total e n e r g y of the field which is p o s i t i v e is given by
i
y
where
wi(p) = +
2
2
+m.$ .
(3.24)
In our d i s c u s s i o n of the f r e e field we have noticed that the U m e z a w a Visconti [4] f o r m u l a for the highest o r d e r of d e r i v a t i v e in the r e c i p r o c a l o p e r a t o r is not s a t i s f i e d . The highest o r d e r of d e r i v a t i v e in our r e c i p r o c a l o p e r a t o r is s e v e n w h e r e a s the field has m a x i m u m spin I . Also, we have introduced the quantity ~(x) and obtained the c o m m u t a t i o n r e l a t i o n between ~(x) and ~/(x). As a r e s u l t of the introduction of ~(x), we have s e e n that the total e n e r g y of the field is positive. If we would have u s e d ~(x) instead of ~(x), the e n e r g y would not have been p o s i t i v e . We shall see in the d i s c u s sion of the i n t e r a c t i n g field that ~(x) is a v e r y useful quantity.
H. L Bazsya, Rarita-Schwinger equation
115
4. E L E C T R O M A G N E T I C I N T E R A C T I O N In this s e c t i o n we i n t r o d u c e the e l e c t r o m a g n e t i c i n t e r a c t i o n and q u a n t i z e the i n t e r a c t i n g field. We f i r s t w r i t e the r e c i p r o c a l o p e r a t o r d(0) given by eq. (3.3) in a d i f f e r ! ent f o r m . We w r i t e du~(O) a s
d'ux(O) = [do, u~.+dp,u~.OP+dpcr, u~-~pOcr+dt.t~p, u~.OpOcr~p ] m '
(4.1)
where 1
[do, u)t] m = m 5u)~- ~m 7uT)t , 1
i
[dp, uX]m = (-YllSuX +~YpYuY)t + ½YuSt.tX- 3YXSp u ) , 1
[d p~, uX]rn = (- 3 m Y ~ 7uOX~ + ~
1
Y P Y)t Su(~
2
3m 5P uSXa) '
2 [d p~p, uX] m - 3m 2 Ytz6ua6Xp •
(4.2)
We a r e p u t t i n g the s u b s c r i p t m in eq. (4.1) in o r d e r to i n d i c a t e that the m a s s p a r a m e t e r is m. A l s o , we can w r i t e
(0 v + ½m Vu)(Y pO U - M1)(0X +½m 7X) = [do, ux+d•,
u X O p + d p ~ , uX 8pO~+dtiap, uX OpOcrOp] M l , m
,
(4.3)
where
[do, v)d M1, m = - ~ m 2 M1YvYX , [ d u , vX] M1, m = ( ¼m2 YvY pYX - ~1 m M 1 YX6 p u - ~rn , M 1 Yu 5p X),
[d pff , v~t] M1, m = (½m 7 U~/X6 v a - M 1 5Uv6Xff + ½m yvy U 6Xcr) , [dump, vX] M1, m = 7 u 6ua6Xp •
(4.4)
A s i m i l a r e x p r e s s i o n c a n be w r i t t e n f o r (a v +½m Vu)(7 #~ p - M2) (0~, + ½rn YX) with M 2 in p l a c e of M 1 in eqs. (4.3) and (4.4). F o r c o n v e n i e n c e of s y m m e t r y , we s h a l l w r i t e M 3 f o r m f r o m now on. Substituting eqs. (4.1) and (4.3) in eq. (3.3) and r e a r r a n g i n g the t e r m s we c a n w r i t e d(O) a s
116
H.L. Baisya, Ramta-Schwinger equation
dvx(O)= d(o1,)vx+d(1) ~ +d (I) a a +d(1) p,v~t p pa, vX p a pap,vX +
+d
0 ~
p a p
d (2) []+d (2) [DO +d (2) 0 +d (2) OpOa~p O, v~. p, v~. p pa, vX [::]~p a pap, v~. {~ +d
~ + p pa, w
p ~
pap,
(4.5)
p a p'
where d(1) 2 2 [do, vX] M1, M 3 O, vX _=M~lM22[do, vX]M3 +7/1M3M2 -
2 2 7/2M3Ml[do, vX]M2,M 3 '
d(1) 2 2 2 2 p, v~t - M1M2 [d P, vX] M3 +7/1M3M2 [d p, vX] M 1 , M 3 2 2 - n2M3Ml[dp, v~t]M2,M 3 , 2 2 d(1) 2 2 pa, vX = M1M2 [d pa, v~t] M 3 +7/ 1M3M2 [d P a, vk ]1M1, M3
22 22 d (1) pap, vk - M1M 2 [d P~P, vX] M3 +7/1M3M2 [d pap, vX]M 1, M 3 - n2M~M~l [dpao, vXl M2,M3 ' d (2) - - ( M ~ 2 2 2 O, v~t +M2) [do, v~t]M3 - n l(M3 +M2) [do, vX] M 1, M8 2
2
+ 7/2(M3 +M 1) [do, v~t]M2,M 3 ,
d(2)
2
2
p, vX = "(M~I +~22 )[dp, vX] M3 " 7/1(M3 +M 2) [dp, v~t] M1 ' M3
+7/2( + )Eap, v1M2,
'
d(2)
pa, vX
+ 7/2(M3 +M ) [dpa ' vX]M2,M3 ,
H. L. Baisya, Rarita-Schwinger equation
] 17
d (2) I~ap, vX 2 2 + n2(M 3 + M 1) [d~ap, vX] M2, M S ' d (3) 0, vX - [do, vX] M 3 +n 1 [do, v3.] M1, M 3 - n2 [do, vX] M2, M 3 ' d (3) . --- [d
p, vA
d (S)
pa, vX
=- [d
p, vk]M3+n 1 [d p,v~t]M1,M 3 -n2[d p,v~]M2,M3 , ~,vX]M3+nl[dpa,
vX]M1,M3-n2[d~cr, v~lM2,M3 '
d(3)~p, vk - [d V~p, v X ] M 3 + n l [ d p ~ p , v ? J M 1 , M 3 - n 2 [d pap, v?JM 2,M 3
(4.6)
We n o t i c e that d (3)
]~ff, VX
d (3)
= 0
p crp , v ~t
--- 0
(4.7)
(4.8)
One can e a s i l y s e e that eqs. (4.7) and (4.8) a r e t r u e . T h e f o r m of d(0) given in eq. (4.5) is u s e f u l to w o r k when i n t e r a c t i o n is i n t r o d u c e d . We i n t r o d u c e the m i n i m a l e l e c t r o m a g n e t i c i n t e r a c t i o n by r e p l a c i n g ap by
D~ = a~z-ieA~z ,
(4.9)
in eq. (2.5) with a = -1. T h e equation of m o t i o n now b e c o m e s
( FU, ~taD P + mM~ta) ~a(x) = 0 , or
Axa(a) ¢a(x) : ix(X),
(4.10)
jx(x) = ie F U, ka ~Va(xJA • •
(4.11)
where
T h e H e i s e n b e r g o p e r a t o r ~(x) can b e w r i t t e n as
~(x) : ~(x/e)+½ ~ where
d4x'[d(~),¢(Xo-X'o)]AM M M ( x - x ' ) j ( x ' ) 1 2 3 '
(4.12)
] 18
H.L. Baisya, Ramta-Schwinger equatzon 1
AMIM2M3(X-x')
=- (M2_ M2)(M2 - M2 ) A M I ( X - x ' )
1
1
(M2_M2)(M22_M2) A M 2 ( X - X ' ) + (M2_M2)(M22_M2) A M 3 ( X - X ' ) .
(4.13)
T h e f i e l d @(x/a) s a t i s f i e s the f r e e f i e l d c o m m u t a t i o n r e l a t i o n : {tp(x/cr), q/(x'/a)} = id(O )n M1M2M3 (X - x') .
(4.14)
We shall now show that the s e c o n d t e r m of eq. (4.12) will v a n i s h . Now
½ [d(O),E(Xo-X'o)]AM1M2M3(X - x ' ) =
1
½ [d(O), E(x o - Xo) ] AMI(X - x')
1
½ [d(0) E(x o - Xo)]AM2(X - x ' )
1
+ (M2_M2)(M2_M2)½[d(a),
E(Xo-X'o)]AM3(X-X').
(4.15)
We e v a l u a t e e a c h t e r m of eq. (4.15) by u s i n g the f o r m u l a given by K a t a y a m a
[71. We define
Niz~p.. " - n p n ~ n p . . .
,
(4.16)
w h e r e n is the unit n o r m a l to the s u r f a c e or. Also, we define
-½[d(O),E(Xo-Xo)]A(x-x')
=- lid(a)]] K 5(4)(x- x ') ,
(4.17)
w h e r e K is the m a s s p a r a m e t e r . By the f o r m u l a we have when
(4.18)
d(a) = a ~ a e , when
d(a) = ~ pO(yOp ,
[[0 p 0~0p]] = (npn~Spc e + npnpScr a + ndnpSua
+ 2nLtn(~npna)a a =- Arp(ypa~ a ;
(4.19)
when d(a) : [B,
[[D]] = -1 ;
(4.20)
[[ff32]] : - ( [ ] + K2) ,"
(4.21)
when d(a) = if32 ,
H. L. Ba isya, Rarita-Schw roger equat ion
119
when d(8) = lisp ,
[[liSp]] : -ap ;
(4.22)
when d(a) = []apSa,
[[[]apSa]l:-SpSa+K2Npa
;
(4.23)
when d(a) = E]apSaa p ,
[[[~apSaap] ] = -apaaap+K2hTpapaaa;
(4.24)
when d(a) = O2ap ,
[[[]28p]] = -ap(O+ K2) .
(4.25)
Using eqs. (4.18)-(4.25) in eq. (4.15) we obt~,in ½ [d(8), E(x o - Xo)] AM1M2M3(X - x') = +d(2)+d(2)8 o
p
1 .r_d(1) N (M 2 - M2)(M 2 21 - M2) [ p a p a p
-d(2){-8 pc
a +M~N
p a
t
pa
_ d (1) N p~
8
p~aa
} - d (2) ~-8 8 8 +M211.~papaaa} pap-
/x a p
+ d(o3)(0+ M21) +d(3)8 p p ([3+ M~I)] 5 (4) (x-x') _
+d(2)+d(2)8 o
p
(U 2 p
1
2 2 M 2) I-d(1) N _M2)(M2_ pa pa
d(1)
N
pap
8 papa a
_d(2)~_SpaG+M22Npa}_dpapL(2)~-8p8 8p+M22~pGpaaa} pal a
+ d(o3) ([~+ M22) +d~)8 p ([]+ M22) ]6 (4) (x - x,) 1 I-d(1) N - d (1) .~ 8 + (M 21 - M3)(M 2 2_M32) L pa I~G pap papa a 2
+d(2)+d(2)~ -d(2){-8 8 + 4 N p a } - a p a p.(3) { o p t~ pG p a
-8
P 8 G8 p + M 32g^p a p a S a }
+ d(o3)(E;+ M~)+d~)8 p([~+ M~)] 5(4)(x-x'). Therefore, substituting eq. (4.26) in eq. (4.12) we obtain
(4.26)
H.L. Baisya, Rarita-Schwinger equation
120
1
I
1
- (M21 - M 2 ) ( M I - M2 ) [
]j(x) ,
(4.27)
where the s q u a r e b r a c k e t contains the quantities as in the p r e v i o u s e x p r e s sion. Now collecting the coefficients of d~,l)~Ypcr j(x),~v d!Z)Nlz
E _
' 1
1
~M~,_ M 2~.) ( M 2~._ M 3 ~.+(H-@(M~-@] ) +[ - [
-[
]d (1))q
p~p
] d (2) a
a j(x)-[ p~pa a
j(x)-[
l d (3) Da J p P
j
]d(2)j(x)
o
] d (2) a
0 j(x)
]d(o3)Dj(x)
] d (2) ~ ~ ~ j ( x ) - [ ~o'p p c r p - [
~(~)N '" ~ e tx(y :#t
j(x)
+[(M~,-@(M~-@-(H-@(~
~V(x) = ~ ( x / ~ ) .
(4.29)
~(x) -= ~(0 ~i(x),
(4.30)
~(x) = @ x / ~ ) .
(4.31)
Now defining
we have
H. L. Baisya, Rarita-Schw~nger equatzon
121
Hence the commutation relation between ~P(x) and ~(x) can be written as =
= id(O)AmMiM2(X- x')] XO=X,0 3
3
2
1
1
= (,r(x),~2(x')}Xo:X ° + 2 (d(x, Mi).~(x',Mi)}Xo=X ° • i=1 (4.32) Here, we have written m for M 3 again. T h e r e f o r e , the commutation relation of the interacting spin 3 field is _3 _3 {~V~(x), ~ 2 (x,)}Xo=X,° = id'( O)nm(X- X')]Xo=Xo, , (4.33) and it is clear that it does not contain any electromagnetic field. The interaction Hamiltonian can be obtained to be
H(x,n) = H(x) = ~(x) j(x) 3
1
3
3_
!
_3
!
!
1
_3
i
!
+ ~ ( x ) Ftz, k p ff/p(X, 1)- ~ ( x , 1)rtz, ~.p ~ ( x , 1)+ ~ ( x , 2)F ,kp ¢~p(X, 1) 3
1
i
i
I
I
+ ~(x)F.Xp~pp(X. 2 ) - ~ ( x , 1)Ftz, Xp~p(X. 2)+~(x, 2)F ~, kp ~ ( x , 2)]A t~ (4.34) The interaction Hamiltonian for the spin ~ field is hermitian as we show below:
[ie~(x)F ,Xp ~9~p(x)A ] * 3
3
3
3
t
= -ie ~v~t(x)7~gexy4 ~V2(x)A 3
3
= -ie ~ * (x)v4Y4y gy4 gax ~(x)A tp =-ie~(x)(-g.aya)~(x)ASt~ = ie ~ ( x ) y a ¢~e(x)Aa .
(4.35)
H. L. Bazsya , Rarita-Schw znger equation
122
S i m i l a r l y , the t e r m s 1
1
-ie fJ~(x, 1) F p , Xp tpp(X, 1)A P
1
and
1
+iv tp;(x, 2) F#, ~p ~p(X, 2)A/~,
in eq. (4.34) a r e also h e r m i t i a n but other t e r m s a r e not. Hence the total Hamiltonian is not hermitian. As pointed out in the Bhabha t h e o r y [8], the interaction Hamiltonian can be made h e r m i t i a n by m e a n s of an indefinite m e t r i c , but in that c a s e the positivity of the e n e r g y will be destroyed. We have noticed with other equations [9] also that in the m u l t i - m a s s t h e o r y this situation will prevail.
5. CONCLUSIONS By introducing the new quantity ~(x) we have shown that the total e n e r g y of the field is definitely positive when the field d e s c r i b e s p a r t i c l e s with many m a s s states. We have also noticed that the U m e z a w a - V i s c o n t i f o r mula is not satisfied in c a s e of v e c t o r - s p i n o r field. We feel that this f o r mula needs modification for m u l t i - m a s s fields. We intend to do this on some other occasion. In our d i s c u s s i o n of the i n t e r a c t i n g field, we have seen that our c o m m u tation relation does not contain any e l e c t r o m a g n e t i c field. This states that in our f o r m a l i s m , the type of i n c o n s i s t e n c i e s pointed out by Johnson and Sudarshan [5] do not s e e m to appear. By using the a n t i c o m m u t a t i o n r e l a t i o n between ~/(x) and ~(x), we have been able to obtain the H e i s e n b e r g o p e r a t o r in a closed f o r m and seen that %9(x) = tp(x/cr) holds. The interaction H a m i l tonian obtained in this f o r m a l i s m is not an infinite s e r i e s of the coupling constant. We do not know whether this will be true in case of higher (> 9) spin fields. We would like to mention that because of the n o n - h e r m i t i c i t y of the i n t e r a c t i o n Hamiltonian, the unitarity of the S - m a t r i x b r e a k s down as in the Bhabha theory [8]. The author would like to thank P r o f e s s o r Y. Takahashi for valuable suggestions.
APPENDIX
To obtain the coefficients N 1 and N2: In o r d e r to obtain the coefficients N 1 and N 2 we use the anticommutation 1
1
r e l a t i o n s satisfied by g/~(x, 1) and ~ ( x , 2). The anticommutation r e l a t i o n s can be obtained as follows. The D i r a c field s a t i s f i e s the a n t i c o m m u t a t i o n relation {~(x, 11, ~(x', 1/} = i(yO - M l l A ( x - x', M1) .
(A.1) ~k ~ t
1
Multiplying eq. (A.1) by Nl(alx +½my/~) f r o m the left and by N l ( a v - ~ m ~ v ) f r o m the right we have
123
H. L . Bais ya , Rarita-Schwinger equatzon --
,
e-~
1
N l ( a # +½m yp) {¢,(x, 1),~(x', 1)}Nl(a v - ~m Vv) = i IN 11 2 (a u +½m ~.)(~a - M1)A(x- x', M1)(~ v -'~m Yv) , or 1
1
{@~(x, 1), ~tPv(X', 1)} = -i IN1 ] 2(a tz + ½m y/~)(ya - M1)(O v +½m Yv)A(x- X', M 1) -: -inl(a p +½m yp)(ya - M1)(a v +½m y v ) n ( x - x', M1) = idtzv(a , 1 ) A ( x - x ' , M 1 )
,
(A.2)
where we have defined ,
(j = 1,2). 1
Similarly, the antieommutation relation satisfied by ~ ( x , 2) can be written as 1
1
2 X 2 {@g( , 2), @v(X , 2)} = -in2(a t + ½ m y t ~ ) ( y a - M 2 ) ( a v -
+½m y v ) n ( x - x', M2)
=-- i d p v ( a , 2)A(x- X', M2) •
(A.3)
Now Axp(a)d#v(a , 1) = -n 1 Axu(a)(a p +½myp)(ya - M1)(O v +½mYv) = (.~-M21)Sxv ,
(A.4)
where AX/I(a) is given by eq. (3.2). Now using eq. (2.29) in eq. (A.4) we have -nl(1 - b)y~t(ya +M2)([~- M~l)(a v +½m yv ) = ([~- M~I) Sky .
(A.5)
T h e r e f o r e , we shall have to have -n 1(1 - b)yx(ya + M2)(a v +½m Yv) = 5Xv •
(t.6)
Multiplying eq. (A.6) by (F4)va from the right we obtain -nl(1 - b)yX(ya + M2)(a v +½m yv)(r4)vz = (r4)xe.
(A.7)
Now
(a v +½m yv)(r4)v~ = -[a4v ~- acry4- ( M j - m)54G+ (m + b M j o 2bm)Y4y~] • (A.8) Using eq. (A.8) in eq. (A.7) and putting X = (r = 4 we obtain n l ( b - 1)(M 1 - 2m)@a + M2) - 1 = 0 .
(A.9)
Operating on @(x, 1) by the left-hand side of eq. (A.9) and using eq. (2.52) we have
124
H.L. Batsya, Rarita-Schwinger equation nl(b-
1)(M 1- 2m)(-Ml+
M2) = 1
or
2 M2-2m nI =
3m 2 M1 - M 2
-,
(A.10)
where we have used eq. (2.35). In a s i m i l a r m a n n e r w e c a n h a v e 2 n 2 --
M 1- 2m (A.11)
3m 2 M 1 -
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