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Causality of propagation of the Bhabha-Gupta field coupled to external electromagnetic and gravitational fields
Nuclear Physics B127 (1977) 537- 547 © North-Holland Publishing Company
CAUSALITY OF PROPAGATION OF THE BHABHA-GUPTA FIELD COUPLED TO EXTERNAL ELECTROMAGNETIC AND GRAVITATIONAL FIELDS J. P R A B H A K A R A N
Department of Physics. Loyola College, Madras-600034, India T.R. G O V I N D A R A J A N
and M. S E E T H A R A M A N
Department of Theoretical Physics, University of Madras. Madras-600025, India Received 16 November 1976
• 3 . 1 Recent investigations lmve revealed that the Bhabha-Gupta (mixed spm-~ spm-~) theory is free of pathologies like a~lusality of propagation in the prc~nce of minimal electromagnetic coupling as well as a trilinear coupling to a spinor field and scalar field, provided the arbitrary parameters occurring in the Bhabha-Gupta Lagrangian are suitably restricted. Wc extend the treatment in this paper to show that causality is preserved even when external electromagnetic and gravitational fields are simultaneously pre.~nt.
1. I n t r o d u c t i o n
In recent papers [1 ], we investigated certain c-number field theories for spin -3 particles wherein causality of field propagation is retained in the presence of interactions with external fields of different types. We observed in particular that unlike the familiar Rarita-Schwinger formalism [2], the Bhabha-Gupta theory [3] shares with the Fisk-Tait theory [41 the advantage of being causal, provided a certain restriction is placed on the free parameters present in the Bhabha-Gupta Lagrangian. Ii1 the present paper, we extend this result by showing that the Bhabha-Gupta theory (with the same restriction on the free parameters) remains causal even when an externalgravitathmal interaction is present simultaneously with minimal electromagnetic interaction. This result is of particular interest against the background of recent similar investigations on the Rarita-Schwinger theory by Madore [5]. He has shown that the anomalous characteristic surfaces which develop when minimal electromagnetic interaction is introduced into the Rarita-Schwinger equation get eliminated if interaction with the gravitational field is also included, provided we consider only a "linear approximation" of the gravitational field, and also the charge e and the mass m are related by m = e / ~ . In the case of the BhabhaGupta theory, however, it turns out that causality could bc achieved, without 537
538
J. t'rabhakaran et al. / Propagation of the Bhabha-Gupta field
having to impose any such restriction on e and m, the only requirement being a specific choice of the arbitrary parameters in the theory * However, since the free total charge is of indefinite sign, the introduction of an indefinite metric in the quantization of the free field would become inevitable in such a theory. Our calculations leading to the above-mentioned result are presented in sect. 2. We then inquire, in sect. 3, into the significance of the specific choice of parameters m the Bhabha-Gupta theory, which always achieves causality for different types of interactions. We show that this choice renders all the constraints (i.e. relations among the field components not revolving any time-derivatives) in the theory primary, which means that all the constraints simply follow as a consequence of the singular nature of the matrix a ° which multiplies the time-derivative of the field components in the equations of motion. This happens because a ° becomes diagonalizable for the choice of parameters, thereby satisfying the sufficiency criterion laid down by Amar and Dozzio [6] for causal propagation. Sect. 4 is devoted to a brief discussion of these results, taken in conjunction with our results presented in earlier papers.
2. Causal propagation of the interacting Bhabha-Gupta field The Lagrangian density tor the Bhabha-Gupta field (having two different mass states and spins 3 and ½) in simultaneous interaction with the electromagnetic and gravitational field is given by .C = ~ ' ( i ~ , - -~t(~
• ~
- m) f,,
• v)(i,"~
- ~,) + ( i S
• "Q))('Y " ~ ) ]
1 5+ ~(~ " 7)(i7 "c/) + m) 7 " V) + a~(i7 " 9 - Xm)O
+ d(i~ c-/) . ~b + iS " ,'7)q5).
(2. I )
Here ~b~' is a vector-spinor and ¢ is a Dirac spinor. The 7 u arc space-time-dependent matrices obeying ,yuyv + 7v,,/~ = 2owav ,
(2.2)
where g~" is tile metric tensor associated with the gravitational field, and ~u = ~ t ~ , 0 . = qst,y°" Further, c-Do= V .
ieAu ,
* If, however, we decide not to make such a choice, then cau~lity could still be achieved by a procedure similar to that of Madore. Btlt the relation between m and e in this case will be slightly different. See appendix for details.
539
J. Prabhakaran et aL / Propagation o f the Bhabha-(;upta fieM
where V~, is the covariant derivative acting on spinor fields and A u is the electromagnetic potential. The gravitational field enters solely through V~,. The constants a, X and d are real and arbitrary. The following relations involving commutators of ,7)~, and the Riemann curvature tensor R ~ v p X will be used in the calculations of this paper [71: . . . .
R a . . qJp .
(2.3)
where f i x is a vector-spinor and C ~ v = - -4I R p x ~
7P@ - iel"~..
(2.4)
Fq. (2.3) has the consequence that ('r " CO)2 _ :l)2 ....
1 ~'"~r"~..
(2.5)
= ±4 R - ~ e o" F ,
(2.6)
where o • F - o " V t " u v and o u~ = ½ i [ 7 u, Tv] . It may also be noted that in view of the symmetry properties of the RiemannChristoffel tensor and eq. (2.2), we have
RuupX ~'IJ'yPTh = 2
R u . "7u ,
R . . p x 7"7~7°y x = - 2 R .
(2.7)
Returning now to the Lagrangian density (2.1) we note that the equations of motion following from it are ( i 7 " c/)
m) f t ,
+ ~7.(iy
-- ~(i3',CO " t~ + i CO,7 " if)
" CO + rn) 7 " ~ + id~D~,O = 0 ,
a( i7 " CO - X m )O + id co . (a = 0 .
(2.8) (2.9)
These are the basic equations of the Bhabha-Gupta field in interaction with gravitational and electromagnetic fields. They are not t r u e equations of motion however. By conabining them initially one can deduce constraint relations which do not involve time derivatives of ff~ or 0. We shall now proceed to obtain the constraints, using basically the same kind of technique as in the non-interacting case. As a first step, contracting (2.8) successively with 3,~ and c-/)~ we get, respectively 2 i c o " qJ + ~ m T " q/ + i d T " COO = 0 ,
(2.10)
~ i y • CO CO" ~ + ~ m 7 • cO,/ " ~
-i(½7" R " ~ - ieT " F " (u) - mCD " ' • J + 3ffaR - ½ e o " b ) 3' " ~ + i d c o 2 0 = 0 .
(2.1 I)
J. Prabhakaran et aL / Propagatk~n of the Bhabha-Gupta fieM
540
In derivhlg (2.11) we have used relations (2.3)-(2.7). The derivatives of flu and occurring in the first two and the last terms of (2.11) can now be eliminated by oper. ating on (2.10) with (7 " c/)) and subtracting from (2.1 I). The resulting equation yields c/) . qj in terms of non-derivative quantities: i~
(2.12)
" ~ = --m~ ,
where l
(-~.,,, • R - ~ + '-.,,. - C • e )
O' = - - 2 n ' l 2
3
2 ie
d R - ~(~---
I ie
1
2 e o • F) ¢
(2.13)
In the above, y • R " ~ =- 7 U R u v ( U v and similarly for 7 " F " ¢'- The tensor/S,ut, is the L" dual o f FUV: / , , z , = ~1 ~Ut,,oh ,'px, and Gut, is the Einstein tensor: i Gut, = Ruv .... ~gut,R •
(2.14)
The term containing/~enters eq. (2.13) through the use of the idcntity ~(o • F)(~, - ~) = i 7 " F ' ~ - i757 • P ' ~ .
(2.15)
If we now substitute for c/) • ff from (2.12) into (2.9) and (2.10) and then eliminate (7 " cZ))¢ between the resulting equations, we immediately get a c o n s t r a i n t relation (not involving derivatives of flu or ¢). It is 7'~
=(2
3d2-1C~a / -3dX¢'
(2.16)
The field equations (2.8) and (2.9) together with the constraint relation will now be used to check whether tile propagation of the Bhabha-Gupta field is causal. The shock-wave formalism of Madore and Tait [8] will be used for this purpose. This formalism exploits the fact that a surface which supports discontinuities in highest derivatives occurring in the wave equation is a characteristic surface. For eqs. (2.8) and (2.9), a characteristic surface o is one on which 3uqA, and 0u¢ can have discontinuities while flu and 4~have to be continuous. Using square brackets to denote the magnitude of the discontinuities, we have, on o, I~.l
= 1~1 = o ,
but [auOv] = ~ukt, ,
[0u¢] = ~juK,
(2.17)
J. Prabhakaran et al. / Propagation o f the Bhabha-Gupta fieM
541
where ~Ju is the vector normal to the surface o *. Note that K is a Dirac spinor and k u is a vector spinor; these can be non-zero only if ~u is normal to a characteristic surface. Our problem now reduces to the following question. Does there exist a space-like characteristic surface o, i.e. a surface o for which the normal ~u is timelike and on which K or k v is non-zero? To determine the answer to this question we take the discontinuities of the field eqs. (2.8) and (2.{)) and the constraint (2.16), i.e. we replace O, ~ v ",rod Ou¢, ~ u ~ v in these equations by their discontinuities:
D],
[~,.l,
[~,~<~],
[Su~.] ,
given by eqs. (2.17). We then get y'~ku
~Tu~'k
~7"k
1
(2.18)
+ 37u 7 " ~ 7 " k + d ~ u K = 0 ,
(2.19)
a7 " ~K + d~ - k = O ,
"k = 0 .
(2.20)
(In writing tile last equation we have used tile fact that c~, eq. (2.13), is free of derivatives.) On c o m b i n i n g eqs. (2.19) and (2.20) we get 3' " ~K = 0, so that ~2K = 0, and hence K=0,
(2.21)
since ~2 5/:_0. Substituting (2.20) and ( 2 . 2 1 ) i n (2.18), we obtain
7 " ~jk. -- ~ J u T " k + -~YuT' ~ 7 " k = O.
(2.22)
This equation enables us to write ks = (~u
7u 7 ' ~J)X,
X-
17.~7-k 3 ~2
(2.23)
Finally we have to see whether a non-null k u of this form is consistent with the constraint equation. Differentiating b o t h sides of (2.16), taking discontinuities, and using the fact that K = 0, we get
where = c~(tb, ~ k u , 4~ -~ K ) .
~' The forms of the discontinuities exhibited here are the most general ones, for the discontinuity in the derivative of each spinor component should be along the outward normal to o and hence be parallel to ~.. For a detailcd discussion of the forms of the discontinuity formulae, see Madorc and Tait, ref. [8].
J. Prabhakaran et al. /Propagation of the Bhabha-Gupta field
542
Feeding (2.23) into (2.24) we get
~t×--
1--~-jL~V'G't-3m2~S~"P"
×.
.
.
.
This equation leads to
{ [( <12el t2 +
1 - 2 a / 3m2J
}
(17"~)2 X = 0 ,
(2.26)
when only the electromagnetic field is present. That this equation allows timelike t u, thus giving rise to acausal propagation has been already noted [ I ] . On tile other hand, when the gravitational field alone is present, eq. (2.25) gives 7"tX =
( 3.2 1
1 - 2a ] 3 m 2 7 " G ' t X '
(2.27)
Operating by 3, • ~ on both sides, we get ~:X=
( 3elI
~13m2[
(7.G't)7.~+2t'a-tlX.
(2.28)
Using (2.27) once again on the right-hand side of (2.28), we finally get [t 2 + b2(G • t) 2 -- 2b~ • G • t]X = 0 ,
(2.29)
where we have called [1 - (3d2/2a)]/3nfl = 6. We shall now show that (2.20) admits timelike ~u. If we choose the symmetric second-rank tensor field Guy to have only the components G0: = G:0 non-vanisliing, with all the other components zero, and also ~ = ( t o , t i , O, 0), then we get from (2.29), ~(l
. t,~c,~:) - ~:} × = o ,
and this corresponds to a characteristic surface for which to -- -+1~,1(1 - bZG2o2) -~/~ .
(2.30)
It is immediately evident that i f b 4: O, and
G~o2 < i/t, ~ ,
(2.31)
one has ~o > [~l, i.e. ~u is timelike and the characteristic surface is spacelike. On the other hand, if Go: is such that the inequality in (2.31) is reversed, then to becomes complex and there is no propagation of the waves. The theory is thus clearly unsatisfactory. Evidently, therefore, there is only one way to avoid these difficulties and to safeguard causality. It is by setting b = 0. Since b = [I -- (3d:/2a)]/3m 2 , this means that a > O, and we choose the arbitrary parameter d such that d:--Ta2 .
(2.32)
J. Prabhakaran et at / Propagation of the Bhabha-Gupta fieM
543
Provided this choice is made, eqs. (2.26) and (2.29) b o t h give ~2x=O, which yields X = 0 and therefore ku = O. More generally, even when both the electromagnetic and gravitational fields are simultaneously present, eq. (2.25) gives, in view of (2.32),
v .~x:0, attd hence ~2X=0:~X---O~ku=O
,
so that the discontinuities k , and K b o t h vanish for ~2 4= O, i.e. there can be no characteristic surface with ~2 ~ 0 * This completes our d e m o n s t r a t i o n that the propagation is causal even in the presence of b o t h the electromagnetic and gravitational fields, provided we choose d 2 = 2a.
3. Significance of the choice 3d 2 = 2a We have shown in sect. 2 that the propagation of the Bhabha-Gupta field is causal even when external electromagnetic and gravitational fields are simultaneousl~ present, provided 3d 2 = 2a. (For the same choice of parameters, we had shown in earlier papers [1] that causality is preserved with electromagnetic field alone, and also with a trilinear type of coupling with a spinor field and a scalar field.) This at once raises the interesting question: What is the speciality of the choice 3d 2 = 2a for which causality is preserved no matter what the interaction is? In other words, is it possible to understand the causal behaviour of the field in terms of the structure of the Bhabha-Gupta field equations? We address ourselves, in this section, to this i m p o r t a n t question. Following Gupta [31 we first write eqs. (8) and (9) in the form (ie~ • :b
m)u"- q~, = 0 .
(3.1)
I
The range of the underlined indices is from 0 to 4; ,lp,, - ~9,, and q"4 -= 4~. (a'°)~ u- (P = 0, 1,2, 3) are 20 X 20 matrices. We are interested in the structure of the matr]-x ~0, which consists of fot, r blocks (u°)u~', (o~°)u 4, (~°)4t' and (0~°)44. These * Madore [5I has shown recently that the acausal modes of propagation of the Rarita-Schwinger lield with minimal electromagnetic coupling, could be eliminated if interaction with an external gravitational field is included, provided the charge e and tile mass rn are related by m = e/3x,/~. This result is true only under the assumption of a linear approximation, Even in the case of the Bhabha-(;upt~ field, a similar type of result may be obtained under the same approximation, if v,,e decide not to make the choice 3d 2 = 2a. We show this explicitly in the appendix.
J. IYabhakaran et al. / tS"opagation ~.l'the Bhabha-Gupta fieM
544
arc respectively 16 X 16, 16 × 4, 4 X 16 and 4 X 4 blocks which are of the form (°~°)n
v = ")'()gn u - T u g °''
1 t.,' .,0
I .~, TO~t,
37 gn + ~ n
•
(ot°)u 4 = d(gu ° - 7 u T °) ,
d
1~,o
(d~)44 =77
"
The matrix a ° is a singular matrix (irrespective of the values of a and d), with eigenvalues 0, -+1 and -+I/X. A set of four constraints results from the singularity of o~°. and these could be called p r i m a r y constraints following Johnson and Sudarshan [9]. They have the form (IH
") " ~17" c/1)7°~° =iCO " ~312 +i)k 7 . COO,
(3.2)
where I ~,312 = ~'k + ?~'k~ " *-
These constraints are seen to relate the dependent components ~o to the dynamically independent components 0 3/2 and O. But the other dependent conrponents "t • ~ d o not occur in (3.2) at all. This implies that there are further (secondary) constraints, which are to be obtained by differentiating (3.2) and using the equatkms of motion. We have in fact obtained these already in covariant form (2.1 6). For the choice 2a = 3d 2, a peculiar thing happens. It turns out thal the set of constraints (2.1 6) follows straightaway f!ona the equations of motion, withot,l use of derivatives of the primary constraints. This can be seen as follows. Contracting (2.8) with "/~ we get (2.10) and then, on elinfinating :-/3 • ~ between (2.9) and (2.10) we end up with (2.16) provided 3d 2 = 2a. This means essentially that eq. (2.16) follows simply from the equations of nlotion, by virtue of the singularity of a o i.e., the rank of the matrix a °. which was originally 16, gets further reduced to 12, when 3d 2 = 2a. Tht, s we see that the choice 3d 2 = 2a renders all the constraints in the theory primary. It is the absence of secondary constraints that seems to be responsible for ridding the B.G. theory of acausality troubles *. One can understand this situation in terms of the sufficiency criterion for causal propagation given by Amar and l)ozzio. In the case of a mr, hi-mass
* We m a y n o t e i n c i d e n t a l l y t h a t tile g e n e r a l t h e o r e m dlle to .Iohn:~on a n d S u d a r s h a n asserts t|la! the a b s e n c e o f s e c o n d a r y c o n s l r a i t l t s in half-integFal hi,~hcr spin t h e o r i e s ,.viH~ s :-, 3 / 2 has the c o n s e q u e n c e t h a ! q u a n t i z a t i o n has to involw.' an i n d e f i n i t e m e t r i c . T h i s i,c seen to bc c o n q s t c n t v.'llh t h e fac! t h a t the c h o i c e 3 d ) = 2.~ m a k e s the t o t a l char,uc in the !'roe field s i t u a t i o n o f i n d e f i n i t e xi,~ n.
J. Prabhakaran et aL / Propagation o.f the Bhabha-Gupta field
545
theory, this condition states that the minimal equation satisfied by ~o is oe° ,~FI [(c~°) z
a 2] r i " = O ,
i
where -+a~. +-a2 .... arc tile non-zero eigenvalues of ~o. Ill other words, in the normal Jordan form of ~0-, the block of the nt,tl eigenvalues is required to be diagonalisable " For the Bhabha-(;upta theory, the non-zero eigenvalues arc -+1, -+1/), and it is easy to verify that 0~° satisfies the minimal equation o~O[(o~O) 2 -
l]I(o~O)2-
~2] =0
provided 3d 2 = 2a.
4. Discussion
We have shown that unlike the Rarita-Schwinger theory, the Bhabha-Gupta theory preserves its causal character when minimal interaction with both an external electromagnetic field and a gravitational field is present, provided the arbitrary parameters in the BG Lagrangian are suitably restricted. The significance of this particular choice of parameters is seen to be traceable to the fact that the matrix o~° becomes diagonalizable with this choice and that the number of constraints in tile theory equals the multiplicity of ttle zero eigenvalues of oe°, i.e. a l l the constraints are primary. It is tile absence of secondary constraints that seems to be responsible for ridding the theory of acausality troubles. The choice of parameters also has the cousequance that the sufficient condition for causality of propagation, given by Amar and I)ozzio is seen to be satisfied. This nice feature of the Bhabha-Gupta theory which renders it causal in the presence of minimal electromagnetic and gravitational interactions, taken in conjt, nction with our earlier results (namely, causality with other types of interactions, as well as the absence of complex energy eigenvalues in the spectrum even t'ot arbitrarily high magnetic field strengths) leads one to speculate whether the search for consistent half-integer spbl theories would lead one to theories in which the free total charge is indefinite and more than one mass/spin are present. The example of the Fisk-Tait equation [41 for spin -3 particles lends support to this conjecture, flowever, a deeper understanding is essential before any general criterion for the consistency of interacting higher-spin theories could be laid down. The authors are grateful to Professor P.M. Mathews ["or Iris encouragement, keen interest in the work and useful suggestions. One of us (J.P.) thanks Dr. V. * A m a r and Dozzio have s h o w n that their condition is sufficient for causality, with m i n i m a l e l e c t r o m a g n e t i c coupling i.e. v~hcn ~u -" i~u W.4 u. We have verified that the s a m e criterion holds even w h e n minimal gravitational cot, piing is present, i.e. w h e n an , "~),u = Vta leA;a.
546
z Prabhakaran et al. / l~'olmgation o f the Bhabha-Gupta field
Srinivasan for a discussion, and another of us (T.R.G.) acknowledges with gratitude the tinancial support of the University Grants Commission, India, in the form of a Junior Fellowship.
Appendix
We now demonstrate that causality of propagation could be achieved even though we do not make the choice 3d 2 = 2a, provided e and m are related by [ 1 . - ( 3 d 2 / 2 a ) ] ( e 2 / 3 m 2) = G, G being the gravitational constant. This is an approximatc result and is thc extension e r a similar conclusion due to Madorc [5] for the case of the Rarita-Schwinger particles. We start from eq. (2.25), which may be rewritten as 7 " ~X = b 7 • G • ~X
2 i e b 7 s 3' /2-. ~X,
(A.I)
where b=(1
3d2 t - -L-,
1
Multiplying both sides of (A.I) by 7 ' ,~ and using (A.I) once agahl, we get 2b(~ " G " ~) + 4 e 2 b 2 ( t 2'' ~)21×
[~2 + b 2 ( G . ~)2
(A.2)
..... 4 i e b 2 7 S ( l ?" ~) • ( G • ~)X .
This equation leads to (A 2
(A.3)
B2)X = 0,
where A = ~2 + b 2 ( G .
~)2 _ 2b(~" G • ~) + 4 e 2 b 2 ( l :'' ~)2 ,
(A.4) (A.5)
B = 4 e b 2 ( G • ~)" ( I ~'' ~ ) .
Following Madore, wc now look upon the spin -3 field as a test field which does not enter into the Einstein equations. These equations are therefore G ~ v = --8zrG r~v ,
(A.6)
where rt, v is the Maxwell stress-energy tensor: l
v
p
1
v,
o
ruv = 7 ~ [ /:p/:v +zg~ F p o l m ] . Using (A.6) and (A.7), we may write A as A =
aav ~
,
(A.7)
J. Prabhakaran et al. / Propagation of the Bhabha-Gupta fieM
547
where a,~t3 = (1 - b 2 e2 F u v F U V ) g ~ + 167rb(G - be2)rc~ + (87rGb )2 r u a ~ • If we n o w d e c i d e to neglect t e r m s involving G 2, Ge t h e n to t h a t a p p r o x i m a t i o n , a~ O c~ ga~, if we c h o o s e G = be 2 . ( A . 3 ) will t h e n lead to a c h a r a c t e r i s t i c surface for w h i c h ~2 = 0, so t h a t t h e p r o p a g a t i o n is causal.
References [ 1 ] J. Prabhakaran, M. Seetharaman and P.M. Mathews, J. Phys. A8 (1975) 560; Phys. Rev. D I 2 (1975) 3191. [2] W. Rarita and J. Schwinger, Phys. Rev. 60 (1941) 61. [3] H.J. Bhabha, Phil. Mag. 43 (1952) 33; K.K. Gupta, Proc. Roy. Soc. A222 (1954) 118. [4] C. Fisk and W. Tait, J. Phys. A6 (1973) 383. [5 ] J. Madore, Plays. Letters 55B (1975) 217. [6] V. Amar and U. Dozzio, Nuovo Cimento Lett. 12 (1975) 659. [7] tt.A. Cohen, Nuovo Cimento 52A (1967) 1242. [8] J. Madore and W. Tait, Comm. Math. Phys. 30 (1973) 201. [9] K. Johnson and E.C.G. Sudarshan, Ann. of Phys. 13 (1961) 126.
CAUSALITY OF PROPAGATION OF THE BHABHA-GUPTA FIELD COUPLED TO EXTERNAL ELECTROMAGNETIC AND GRAVITATIONAL FIELDS J. P R A B H A K A R A N
Department of Physics. Loyola College, Madras-600034, India T.R. G O V I N D A R A J A N
and M. S E E T H A R A M A N
Department of Theoretical Physics, University of Madras. Madras-600025, India Received 16 November 1976
• 3 . 1 Recent investigations lmve revealed that the Bhabha-Gupta (mixed spm-~ spm-~) theory is free of pathologies like a~lusality of propagation in the prc~nce of minimal electromagnetic coupling as well as a trilinear coupling to a spinor field and scalar field, provided the arbitrary parameters occurring in the Bhabha-Gupta Lagrangian are suitably restricted. Wc extend the treatment in this paper to show that causality is preserved even when external electromagnetic and gravitational fields are simultaneously pre.~nt.
1. I n t r o d u c t i o n
In recent papers [1 ], we investigated certain c-number field theories for spin -3 particles wherein causality of field propagation is retained in the presence of interactions with external fields of different types. We observed in particular that unlike the familiar Rarita-Schwinger formalism [2], the Bhabha-Gupta theory [3] shares with the Fisk-Tait theory [41 the advantage of being causal, provided a certain restriction is placed on the free parameters present in the Bhabha-Gupta Lagrangian. Ii1 the present paper, we extend this result by showing that the Bhabha-Gupta theory (with the same restriction on the free parameters) remains causal even when an externalgravitathmal interaction is present simultaneously with minimal electromagnetic interaction. This result is of particular interest against the background of recent similar investigations on the Rarita-Schwinger theory by Madore [5]. He has shown that the anomalous characteristic surfaces which develop when minimal electromagnetic interaction is introduced into the Rarita-Schwinger equation get eliminated if interaction with the gravitational field is also included, provided we consider only a "linear approximation" of the gravitational field, and also the charge e and the mass m are related by m = e / ~ . In the case of the BhabhaGupta theory, however, it turns out that causality could bc achieved, without 537
538
J. t'rabhakaran et al. / Propagation of the Bhabha-Gupta field
having to impose any such restriction on e and m, the only requirement being a specific choice of the arbitrary parameters in the theory * However, since the free total charge is of indefinite sign, the introduction of an indefinite metric in the quantization of the free field would become inevitable in such a theory. Our calculations leading to the above-mentioned result are presented in sect. 2. We then inquire, in sect. 3, into the significance of the specific choice of parameters m the Bhabha-Gupta theory, which always achieves causality for different types of interactions. We show that this choice renders all the constraints (i.e. relations among the field components not revolving any time-derivatives) in the theory primary, which means that all the constraints simply follow as a consequence of the singular nature of the matrix a ° which multiplies the time-derivative of the field components in the equations of motion. This happens because a ° becomes diagonalizable for the choice of parameters, thereby satisfying the sufficiency criterion laid down by Amar and Dozzio [6] for causal propagation. Sect. 4 is devoted to a brief discussion of these results, taken in conjunction with our results presented in earlier papers.
2. Causal propagation of the interacting Bhabha-Gupta field The Lagrangian density tor the Bhabha-Gupta field (having two different mass states and spins 3 and ½) in simultaneous interaction with the electromagnetic and gravitational field is given by .C = ~ ' ( i ~ , - -~t(~
• ~
- m) f,,
• v)(i,"~
- ~,) + ( i S
• "Q))('Y " ~ ) ]
1 5+ ~(~ " 7)(i7 "c/) + m) 7 " V) + a~(i7 " 9 - Xm)O
+ d(i~ c-/) . ~b + iS " ,'7)q5).
(2. I )
Here ~b~' is a vector-spinor and ¢ is a Dirac spinor. The 7 u arc space-time-dependent matrices obeying ,yuyv + 7v,,/~ = 2owav ,
(2.2)
where g~" is tile metric tensor associated with the gravitational field, and ~u = ~ t ~ , 0 . = qst,y°" Further, c-Do= V .
ieAu ,
* If, however, we decide not to make such a choice, then cau~lity could still be achieved by a procedure similar to that of Madore. Btlt the relation between m and e in this case will be slightly different. See appendix for details.
539
J. Prabhakaran et aL / Propagation o f the Bhabha-(;upta fieM
where V~, is the covariant derivative acting on spinor fields and A u is the electromagnetic potential. The gravitational field enters solely through V~,. The constants a, X and d are real and arbitrary. The following relations involving commutators of ,7)~, and the Riemann curvature tensor R ~ v p X will be used in the calculations of this paper [71: . . . .
R a . . qJp .
(2.3)
where f i x is a vector-spinor and C ~ v = - -4I R p x ~
7P@ - iel"~..
(2.4)
Fq. (2.3) has the consequence that ('r " CO)2 _ :l)2 ....
1 ~'"~r"~..
(2.5)
= ±4 R - ~ e o" F ,
(2.6)
where o • F - o " V t " u v and o u~ = ½ i [ 7 u, Tv] . It may also be noted that in view of the symmetry properties of the RiemannChristoffel tensor and eq. (2.2), we have
RuupX ~'IJ'yPTh = 2
R u . "7u ,
R . . p x 7"7~7°y x = - 2 R .
(2.7)
Returning now to the Lagrangian density (2.1) we note that the equations of motion following from it are ( i 7 " c/)
m) f t ,
+ ~7.(iy
-- ~(i3',CO " t~ + i CO,7 " if)
" CO + rn) 7 " ~ + id~D~,O = 0 ,
a( i7 " CO - X m )O + id co . (a = 0 .
(2.8) (2.9)
These are the basic equations of the Bhabha-Gupta field in interaction with gravitational and electromagnetic fields. They are not t r u e equations of motion however. By conabining them initially one can deduce constraint relations which do not involve time derivatives of ff~ or 0. We shall now proceed to obtain the constraints, using basically the same kind of technique as in the non-interacting case. As a first step, contracting (2.8) successively with 3,~ and c-/)~ we get, respectively 2 i c o " qJ + ~ m T " q/ + i d T " COO = 0 ,
(2.10)
~ i y • CO CO" ~ + ~ m 7 • cO,/ " ~
-i(½7" R " ~ - ieT " F " (u) - mCD " ' • J + 3ffaR - ½ e o " b ) 3' " ~ + i d c o 2 0 = 0 .
(2.1 I)
J. Prabhakaran et aL / Propagatk~n of the Bhabha-Gupta fieM
540
In derivhlg (2.11) we have used relations (2.3)-(2.7). The derivatives of flu and occurring in the first two and the last terms of (2.11) can now be eliminated by oper. ating on (2.10) with (7 " c/)) and subtracting from (2.1 I). The resulting equation yields c/) . qj in terms of non-derivative quantities: i~
(2.12)
" ~ = --m~ ,
where l
(-~.,,, • R - ~ + '-.,,. - C • e )
O' = - - 2 n ' l 2
3
2 ie
d R - ~(~---
I ie
1
2 e o • F) ¢
(2.13)
In the above, y • R " ~ =- 7 U R u v ( U v and similarly for 7 " F " ¢'- The tensor/S,ut, is the L" dual o f FUV: / , , z , = ~1 ~Ut,,oh ,'px, and Gut, is the Einstein tensor: i Gut, = Ruv .... ~gut,R •
(2.14)
The term containing/~enters eq. (2.13) through the use of the idcntity ~(o • F)(~, - ~) = i 7 " F ' ~ - i757 • P ' ~ .
(2.15)
If we now substitute for c/) • ff from (2.12) into (2.9) and (2.10) and then eliminate (7 " cZ))¢ between the resulting equations, we immediately get a c o n s t r a i n t relation (not involving derivatives of flu or ¢). It is 7'~
=(2
3d2-1C~a / -3dX¢'
(2.16)
The field equations (2.8) and (2.9) together with the constraint relation will now be used to check whether tile propagation of the Bhabha-Gupta field is causal. The shock-wave formalism of Madore and Tait [8] will be used for this purpose. This formalism exploits the fact that a surface which supports discontinuities in highest derivatives occurring in the wave equation is a characteristic surface. For eqs. (2.8) and (2.9), a characteristic surface o is one on which 3uqA, and 0u¢ can have discontinuities while flu and 4~have to be continuous. Using square brackets to denote the magnitude of the discontinuities, we have, on o, I~.l
= 1~1 = o ,
but [auOv] = ~ukt, ,
[0u¢] = ~juK,
(2.17)
J. Prabhakaran et al. / Propagation o f the Bhabha-Gupta fieM
541
where ~Ju is the vector normal to the surface o *. Note that K is a Dirac spinor and k u is a vector spinor; these can be non-zero only if ~u is normal to a characteristic surface. Our problem now reduces to the following question. Does there exist a space-like characteristic surface o, i.e. a surface o for which the normal ~u is timelike and on which K or k v is non-zero? To determine the answer to this question we take the discontinuities of the field eqs. (2.8) and (2.{)) and the constraint (2.16), i.e. we replace O, ~ v ",rod Ou¢, ~ u ~ v in these equations by their discontinuities:
D],
[~,.l,
[~,~<~],
[Su~.] ,
given by eqs. (2.17). We then get y'~ku
~Tu~'k
~7"k
1
(2.18)
+ 37u 7 " ~ 7 " k + d ~ u K = 0 ,
(2.19)
a7 " ~K + d~ - k = O ,
"k = 0 .
(2.20)
(In writing tile last equation we have used tile fact that c~, eq. (2.13), is free of derivatives.) On c o m b i n i n g eqs. (2.19) and (2.20) we get 3' " ~K = 0, so that ~2K = 0, and hence K=0,
(2.21)
since ~2 5/:_0. Substituting (2.20) and ( 2 . 2 1 ) i n (2.18), we obtain
7 " ~jk. -- ~ J u T " k + -~YuT' ~ 7 " k = O.
(2.22)
This equation enables us to write ks = (~u
7u 7 ' ~J)X,
X-
17.~7-k 3 ~2
(2.23)
Finally we have to see whether a non-null k u of this form is consistent with the constraint equation. Differentiating b o t h sides of (2.16), taking discontinuities, and using the fact that K = 0, we get
where = c~(tb, ~ k u , 4~ -~ K ) .
~' The forms of the discontinuities exhibited here are the most general ones, for the discontinuity in the derivative of each spinor component should be along the outward normal to o and hence be parallel to ~.. For a detailcd discussion of the forms of the discontinuity formulae, see Madorc and Tait, ref. [8].
J. Prabhakaran et al. /Propagation of the Bhabha-Gupta field
542
Feeding (2.23) into (2.24) we get
~t×--
1--~-jL~V'G't-3m2~S~"P"
×.
.
.
.
This equation leads to
{ [( <12el t2 +
1 - 2 a / 3m2J
}
(17"~)2 X = 0 ,
(2.26)
when only the electromagnetic field is present. That this equation allows timelike t u, thus giving rise to acausal propagation has been already noted [ I ] . On tile other hand, when the gravitational field alone is present, eq. (2.25) gives 7"tX =
( 3.2 1
1 - 2a ] 3 m 2 7 " G ' t X '
(2.27)
Operating by 3, • ~ on both sides, we get ~:X=
( 3elI
~13m2[
(7.G't)7.~+2t'a-tlX.
(2.28)
Using (2.27) once again on the right-hand side of (2.28), we finally get [t 2 + b2(G • t) 2 -- 2b~ • G • t]X = 0 ,
(2.29)
where we have called [1 - (3d2/2a)]/3nfl = 6. We shall now show that (2.20) admits timelike ~u. If we choose the symmetric second-rank tensor field Guy to have only the components G0: = G:0 non-vanisliing, with all the other components zero, and also ~ = ( t o , t i , O, 0), then we get from (2.29), ~(l
. t,~c,~:) - ~:} × = o ,
and this corresponds to a characteristic surface for which to -- -+1~,1(1 - bZG2o2) -~/~ .
(2.30)
It is immediately evident that i f b 4: O, and
G~o2 < i/t, ~ ,
(2.31)
one has ~o > [~l, i.e. ~u is timelike and the characteristic surface is spacelike. On the other hand, if Go: is such that the inequality in (2.31) is reversed, then to becomes complex and there is no propagation of the waves. The theory is thus clearly unsatisfactory. Evidently, therefore, there is only one way to avoid these difficulties and to safeguard causality. It is by setting b = 0. Since b = [I -- (3d:/2a)]/3m 2 , this means that a > O, and we choose the arbitrary parameter d such that d:--Ta2 .
(2.32)
J. Prabhakaran et at / Propagation of the Bhabha-Gupta fieM
543
Provided this choice is made, eqs. (2.26) and (2.29) b o t h give ~2x=O, which yields X = 0 and therefore ku = O. More generally, even when both the electromagnetic and gravitational fields are simultaneously present, eq. (2.25) gives, in view of (2.32),
v .~x:0, attd hence ~2X=0:~X---O~ku=O
,
so that the discontinuities k , and K b o t h vanish for ~2 4= O, i.e. there can be no characteristic surface with ~2 ~ 0 * This completes our d e m o n s t r a t i o n that the propagation is causal even in the presence of b o t h the electromagnetic and gravitational fields, provided we choose d 2 = 2a.
3. Significance of the choice 3d 2 = 2a We have shown in sect. 2 that the propagation of the Bhabha-Gupta field is causal even when external electromagnetic and gravitational fields are simultaneousl~ present, provided 3d 2 = 2a. (For the same choice of parameters, we had shown in earlier papers [1] that causality is preserved with electromagnetic field alone, and also with a trilinear type of coupling with a spinor field and a scalar field.) This at once raises the interesting question: What is the speciality of the choice 3d 2 = 2a for which causality is preserved no matter what the interaction is? In other words, is it possible to understand the causal behaviour of the field in terms of the structure of the Bhabha-Gupta field equations? We address ourselves, in this section, to this i m p o r t a n t question. Following Gupta [31 we first write eqs. (8) and (9) in the form (ie~ • :b
m)u"- q~, = 0 .
(3.1)
I
The range of the underlined indices is from 0 to 4; ,lp,, - ~9,, and q"4 -= 4~. (a'°)~ u- (P = 0, 1,2, 3) are 20 X 20 matrices. We are interested in the structure of the matr]-x ~0, which consists of fot, r blocks (u°)u~', (o~°)u 4, (~°)4t' and (0~°)44. These * Madore [5I has shown recently that the acausal modes of propagation of the Rarita-Schwinger lield with minimal electromagnetic coupling, could be eliminated if interaction with an external gravitational field is included, provided the charge e and tile mass rn are related by m = e/3x,/~. This result is true only under the assumption of a linear approximation, Even in the case of the Bhabha-(;upt~ field, a similar type of result may be obtained under the same approximation, if v,,e decide not to make the choice 3d 2 = 2a. We show this explicitly in the appendix.
J. IYabhakaran et al. / tS"opagation ~.l'the Bhabha-Gupta fieM
544
arc respectively 16 X 16, 16 × 4, 4 X 16 and 4 X 4 blocks which are of the form (°~°)n
v = ")'()gn u - T u g °''
1 t.,' .,0
I .~, TO~t,
37 gn + ~ n
•
(ot°)u 4 = d(gu ° - 7 u T °) ,
d
1~,o
(d~)44 =77
"
The matrix a ° is a singular matrix (irrespective of the values of a and d), with eigenvalues 0, -+1 and -+I/X. A set of four constraints results from the singularity of o~°. and these could be called p r i m a r y constraints following Johnson and Sudarshan [9]. They have the form (IH
") " ~17" c/1)7°~° =iCO " ~312 +i)k 7 . COO,
(3.2)
where I ~,312 = ~'k + ?~'k~ " *-
These constraints are seen to relate the dependent components ~o to the dynamically independent components 0 3/2 and O. But the other dependent conrponents "t • ~ d o not occur in (3.2) at all. This implies that there are further (secondary) constraints, which are to be obtained by differentiating (3.2) and using the equatkms of motion. We have in fact obtained these already in covariant form (2.1 6). For the choice 2a = 3d 2, a peculiar thing happens. It turns out thal the set of constraints (2.1 6) follows straightaway f!ona the equations of motion, withot,l use of derivatives of the primary constraints. This can be seen as follows. Contracting (2.8) with "/~ we get (2.10) and then, on elinfinating :-/3 • ~ between (2.9) and (2.10) we end up with (2.16) provided 3d 2 = 2a. This means essentially that eq. (2.16) follows simply from the equations of nlotion, by virtue of the singularity of a o i.e., the rank of the matrix a °. which was originally 16, gets further reduced to 12, when 3d 2 = 2a. Tht, s we see that the choice 3d 2 = 2a renders all the constraints in the theory primary. It is the absence of secondary constraints that seems to be responsible for ridding the B.G. theory of acausality troubles *. One can understand this situation in terms of the sufficiency criterion for causal propagation given by Amar and l)ozzio. In the case of a mr, hi-mass
* We m a y n o t e i n c i d e n t a l l y t h a t tile g e n e r a l t h e o r e m dlle to .Iohn:~on a n d S u d a r s h a n asserts t|la! the a b s e n c e o f s e c o n d a r y c o n s l r a i t l t s in half-integFal hi,~hcr spin t h e o r i e s ,.viH~ s :-, 3 / 2 has the c o n s e q u e n c e t h a ! q u a n t i z a t i o n has to involw.' an i n d e f i n i t e m e t r i c . T h i s i,c seen to bc c o n q s t c n t v.'llh t h e fac! t h a t the c h o i c e 3 d ) = 2.~ m a k e s the t o t a l char,uc in the !'roe field s i t u a t i o n o f i n d e f i n i t e xi,~ n.
J. Prabhakaran et aL / Propagation o.f the Bhabha-Gupta field
545
theory, this condition states that the minimal equation satisfied by ~o is oe° ,~FI [(c~°) z
a 2] r i " = O ,
i
where -+a~. +-a2 .... arc tile non-zero eigenvalues of ~o. Ill other words, in the normal Jordan form of ~0-, the block of the nt,tl eigenvalues is required to be diagonalisable " For the Bhabha-(;upta theory, the non-zero eigenvalues arc -+1, -+1/), and it is easy to verify that 0~° satisfies the minimal equation o~O[(o~O) 2 -
l]I(o~O)2-
~2] =0
provided 3d 2 = 2a.
4. Discussion
We have shown that unlike the Rarita-Schwinger theory, the Bhabha-Gupta theory preserves its causal character when minimal interaction with both an external electromagnetic field and a gravitational field is present, provided the arbitrary parameters in the BG Lagrangian are suitably restricted. The significance of this particular choice of parameters is seen to be traceable to the fact that the matrix o~° becomes diagonalizable with this choice and that the number of constraints in tile theory equals the multiplicity of ttle zero eigenvalues of oe°, i.e. a l l the constraints are primary. It is tile absence of secondary constraints that seems to be responsible for ridding the theory of acausality troubles. The choice of parameters also has the cousequance that the sufficient condition for causality of propagation, given by Amar and I)ozzio is seen to be satisfied. This nice feature of the Bhabha-Gupta theory which renders it causal in the presence of minimal electromagnetic and gravitational interactions, taken in conjt, nction with our earlier results (namely, causality with other types of interactions, as well as the absence of complex energy eigenvalues in the spectrum even t'ot arbitrarily high magnetic field strengths) leads one to speculate whether the search for consistent half-integer spbl theories would lead one to theories in which the free total charge is indefinite and more than one mass/spin are present. The example of the Fisk-Tait equation [41 for spin -3 particles lends support to this conjecture, flowever, a deeper understanding is essential before any general criterion for the consistency of interacting higher-spin theories could be laid down. The authors are grateful to Professor P.M. Mathews ["or Iris encouragement, keen interest in the work and useful suggestions. One of us (J.P.) thanks Dr. V. * A m a r and Dozzio have s h o w n that their condition is sufficient for causality, with m i n i m a l e l e c t r o m a g n e t i c coupling i.e. v~hcn ~u -" i~u W.4 u. We have verified that the s a m e criterion holds even w h e n minimal gravitational cot, piing is present, i.e. w h e n an , "~),u = Vta leA;a.
546
z Prabhakaran et al. / l~'olmgation o f the Bhabha-Gupta field
Srinivasan for a discussion, and another of us (T.R.G.) acknowledges with gratitude the tinancial support of the University Grants Commission, India, in the form of a Junior Fellowship.
Appendix
We now demonstrate that causality of propagation could be achieved even though we do not make the choice 3d 2 = 2a, provided e and m are related by [ 1 . - ( 3 d 2 / 2 a ) ] ( e 2 / 3 m 2) = G, G being the gravitational constant. This is an approximatc result and is thc extension e r a similar conclusion due to Madorc [5] for the case of the Rarita-Schwinger particles. We start from eq. (2.25), which may be rewritten as 7 " ~X = b 7 • G • ~X
2 i e b 7 s 3' /2-. ~X,
(A.I)
where b=(1
3d2 t - -L-,
1
Multiplying both sides of (A.I) by 7 ' ,~ and using (A.I) once agahl, we get 2b(~ " G " ~) + 4 e 2 b 2 ( t 2'' ~)21×
[~2 + b 2 ( G . ~)2
(A.2)
..... 4 i e b 2 7 S ( l ?" ~) • ( G • ~)X .
This equation leads to (A 2
(A.3)
B2)X = 0,
where A = ~2 + b 2 ( G .
~)2 _ 2b(~" G • ~) + 4 e 2 b 2 ( l :'' ~)2 ,
(A.4) (A.5)
B = 4 e b 2 ( G • ~)" ( I ~'' ~ ) .
Following Madore, wc now look upon the spin -3 field as a test field which does not enter into the Einstein equations. These equations are therefore G ~ v = --8zrG r~v ,
(A.6)
where rt, v is the Maxwell stress-energy tensor: l
v
p
1
v,
o
ruv = 7 ~ [ /:p/:v +zg~ F p o l m ] . Using (A.6) and (A.7), we may write A as A =
aav ~
,
(A.7)
J. Prabhakaran et al. / Propagation of the Bhabha-Gupta fieM
547
where a,~t3 = (1 - b 2 e2 F u v F U V ) g ~ + 167rb(G - be2)rc~ + (87rGb )2 r u a ~ • If we n o w d e c i d e to neglect t e r m s involving G 2, Ge t h e n to t h a t a p p r o x i m a t i o n , a~ O c~ ga~, if we c h o o s e G = be 2 . ( A . 3 ) will t h e n lead to a c h a r a c t e r i s t i c surface for w h i c h ~2 = 0, so t h a t t h e p r o p a g a t i o n is causal.
References [ 1 ] J. Prabhakaran, M. Seetharaman and P.M. Mathews, J. Phys. A8 (1975) 560; Phys. Rev. D I 2 (1975) 3191. [2] W. Rarita and J. Schwinger, Phys. Rev. 60 (1941) 61. [3] H.J. Bhabha, Phil. Mag. 43 (1952) 33; K.K. Gupta, Proc. Roy. Soc. A222 (1954) 118. [4] C. Fisk and W. Tait, J. Phys. A6 (1973) 383. [5 ] J. Madore, Plays. Letters 55B (1975) 217. [6] V. Amar and U. Dozzio, Nuovo Cimento Lett. 12 (1975) 659. [7] tt.A. Cohen, Nuovo Cimento 52A (1967) 1242. [8] J. Madore and W. Tait, Comm. Math. Phys. 30 (1973) 201. [9] K. Johnson and E.C.G. Sudarshan, Ann. of Phys. 13 (1961) 126.