Field equations in twistors.pdf

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1/8

Field equations in twistorsAsghar Qadir

Citation: Journal of Mathematical Physics 21 , 514 (1980); doi: 10.1063/1.524449 View online: http://dx.doi.org/10.1063/1.524449 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/21/3?ver=pdfcov Published by the AIP Publishing

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2/8

Field equations in twistorsAsghar Qadir a)

International Centre for Theoretical Physics, Trieste, Italy(Received 1 December 1978)

As part of the twistor program for the quantization of relativity the physical fieldequations have been translated into twistors.

I. INTRODUCTION

There have been numerous programs for trying to combine quantum mechanics and the general theory of relativityat various levels, I from the attempt to try to develop a quantum theory of gravity to the attempt to quantize the underlying space-time. A particularly hopeful approach of he lattertype is the twistor program for the quantization of relativity I-J of Penrose. It is based on a search for a theory whichyields quantum theory and relativity theory as special limitsof a more general theory.4.5 A particularly hopeful feature oftwistors is that they appear naturally in the solution of thezero mass field equations. 6 There is further reason to hopethat they will prove to be satisfactory for the purpose, as itappears that the quantum electrodynamic divergences maybe avoided in this formalism when using Penrose graphs. 7-10

In this paper, as part of the twistor quantization program, we shall be translating the field equations of physicsfrom spinors into twistors. For this purpose we shall be usinga set of four twistors, chosen in a particular way, which wecall the twistor quadrad. The equations are written as scalar equations with the fields themselves being expressed as

scalar functionsof

the twistor quadrad. These functions arerequired to be of the type which are eigenfunctions of theCasimir operators of the Lie algebra ofU(2,2).8,11

I t s hoped that the solutions ofthese equations will leadto a deeper, and more thorough, understanding of Penrosegraphs.9 In particular, they should lead to a more rigorousbasis for the apparently ad hoc (at present) rules for writing and calculating these graphs. There remains, a t present,some ambiguity regarding the choice of contours for calculating these graphs, which may be eliminated by a betterunderstanding of the fields represented by these graphs. Weshall not be studying the solutions ofthe field equations here,however.

II. THE FIELD EQUATIONS IN SPINORS

I t has been shown 6 that the source-free, zero rest-massfield equation for a spin nl2 field (n*O) can be written as

VAA CPA,N = 0 (2.1)

A , .. ,N being n indices. For a spin-zero field we have theKlein-Gordon equation

l n leave of absence from Mathematics Department, Quaid-i-Azam University. Islamabad. Pakistan. Present address: Physics Department. University of Texas. Austin. Texas.

V AA VAA cp = 0. (2.2)

Thus, for n = 1 Eq. (2.1) becomes the Dirac equation formassless particles, for n = 2 it becomes the source-free Maxwell equation, and for n = 4 the linearized Einstein free-fieldequations.

Contour integral solutions of these equations werefound by Penrose,, 6 He showed 6 7 that they could be expressed as contour integrals of a linear combination of twotwistors Ucr and va of the form

(2.3)

where O and L are the spin or basis and e cr., eA cr. are basisspinstors 8 , 12 (injection and projection operators from thespinor space to the twistor space). Then Eq. (2.1) becomes

aCPr aCPr+ 1

av a aucr (2.4)

where CPr is the component of P A .. N have r I s and n - rO's.

The massive field equations in spinors are more complicated. Here the Klein-Gordon equation is

(VA A V AA + m 2)cp = 0. (2.5)The Dirac equations are a pair of coupled equations

VAA,a A = ( m N Z ) PA VAA,p A = m / V Z a A.(2.6)

The Maxwell equations are expressed in terms of the spinorfield cpAB and its dual field cj5AB

VAACPAB = 21TJA B, VAA,cj5AB = 0. (2.7)I t should be noted that the solutions of these equations lie in

a complex space-time, in general (see, for example, Ref. 13).The linearized gravitational field equations can similarly bewritten. However, they do not give any additional insight atthis stage, and so they are not considered here.

The solutions of the massive field equations, withsource terms present, generally have both unprimed andprimed spinor indices---


3/8

formal transformation in spinors, we need to transform cABand CA 'B ' separately by multiplying by the square root of } 2.Takingf} 2 to be complex, we would multiply CAB and cA 'B ' byX and i , respectively, such that x i = f } 2 and i=l=ingeneral.

To translate the fields cpA ...NA...M( xPP',x,i) into twistors, we need to remove the spinor indices. One way of doingthis is to convert the fields into scalars by contracting in twospinors SQ tQ so as to saturate the indices fiQ =llQ . ingeneral), obtaining

F (xPP'SQiQ.,x,i) = cp A .NA .. L S ~, ..tM (2.8)where SA tA' are variable spinor fields. Thisfunction will behomogeneous of degree nand m in S and 5Q', respectively,there being n unprimed and m primed indices. If the fieldsare to be dealt with under conformal transformations, theymust have a conformal weight related to the degree ofhomogeneity of the field in X and i. Thus we h l lrequirethat the functions be homogeneous in X and X.

The field equations in terms of he spinor fields (cp s) canbe converted to field equations in terms of Fby differentiating it (F) with respect to S and tQ (Since cp is symmetric inthe un primed indices and in the primed indices, only onedifferentiation is required.) Thus if

VAP CP ANA .M = O

we shall now have

aVA p F = O .

aS ASimilarly, if we had

V V A ... NA...Q - 0AP QA CP - ,we shall now have

a aVAP,V QA , _ F = OaS A aS A

(2.9a)

(2.9b)

2. lOa)

2. lOb)

We are now in a position to set up the twistor formalismfor writing the field equations by counting the number ofvariables involved

III THE TWISTOR QUADRAD

The fields are functions of the position, x PP consistingoffour independent complex variables (as xPP =I=x PP ' in general), the spinors SQ and t Q , consisting of four independentcomplex variables (two for each spinor) and the conformal

functions, as we shall call them, X and i , consisting of twocomplex variables (one for each function). Thus the fieldsare functions often independent complex variables. Wewant to write these fields in terms o ftwistors in a naturalway.

Noting that 10 = 4 3 2 1, we could try writingthe fields as functions of four twistors wa, X a , ya, and Z a,with Z contributing four variables, Y three, X two, and Wone. This may be done by restricting the functions so that Yonly occurs skewed with Z, X only occurs skewed with YandZ, and Wonly occurs skewed with X, Y, and Z. We shall callthis set of four twistors the twistorquadrad. We shall write itas

515 J. Math Phys., Vol. 21. No 3 March 1980

7]aa = ( W a , xa ,ya ,za ) (a = 0,1,2,3). (3.1)

n terms of he space of all twistors we may think of Z asa given point, Yas any point on a given line through Z, X as apoint in a given plane passing through the line containing Yand Z, and Was a point in a given 3-hypersurface containingthe plane which contains X, Y, and Z. Thus there are fourindependent complex variables specifying Z. Of the four independent complex variables specifying Y, one is not relevant as it picks out a particular point on the given line.Similarly, two of the four independent complex variablesspecifying X are not relevant as they pick a point out of aplane, and three of the four independent complex variablesspecifying Ware not relevant as they pick a point out of a 3-space. Thus six ofthe 16 independent complex variables contained in the twistor quadrad are irrelevant. We thus haveten independent relevant complex variables in the twistorquadrad. This being the number of independent complexvariables on which the fields depend, we should be able towrite the fields as functions of the twistor quadrad subject tothe above-mentioned restrictions.

To see what we are doing in terms of a complex Minkowski space, we define the dual twistor quadrad ij au theinverse of 7]aa, i.e.,

ijaa 7]ba = o ~

or, equivalently,

ij au 7]a(3= o ~.We shall call the twistor quadrad real if

ijaa = iiaawhich would be written explicitly as

(3.2a)

(3.2b)

(3.3a)

t a = Za Yo = Ya, Xa = Xa , Wa = Wo (3.3b)

The real twistor quadrad may be thought of as four nullrays in a real Minkowski space, such that each ray intersectstwo of the other three but not the third. Thus, Z and W donot intersect and X and Y do not intersect. In the twistorpicture all four twistors of the real quadrad lie in N, the spaceof null twistors. The lines joining Z and Y, Z and X, Yand W,andXand WlieinN, but the lines joining Zand Wand YandX do not lie in N.

For the general twistor quadrad there is no clear picturein the twistor space because every pair of twistors can beconnected by a line in the twistor space. However, we canvisualize it more easily in a complex Minkowski space asmerely a complexification of the real twistor quadrad.

We shall require that the functions of the twistor quadrad, representing the fields, be homogeneous in all fourtwistors, so as to ensure homogeneity of the fields in thespinors 5Q and t Q nd in the conformal functions Xand i (aswe shall see later). Writing the fields as functions oftwistorsasf(r(Y), it can be shown 8 that they must satisfy theconditions

7]aaabJ(7]C}) = 0 for a > b, (3.4)

hJ(7]CY) for a = b,

where ha is the degree of homogeneity off in the ath twistor.Generally 11 for a set of n complex, n-dimensional vectors,

Asghar Qadir 515



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TJaa , these functions give representations ofthe Lie algebra ofGL(n,q, which ca n be regarded as finite-dimensional if theha's are positive integers and infinite-dimensional otherwise.For twistors, due to their dimension and their conjugationproperties, the representation will bell of the Lie algebra ofU(2,2). Using Eq. (3.4), we shall reduce t he number of variables from 16 to ten, as Eq. (3.4) gives six additionalconstraints.

IV TRANSLATION OF SPINOR VARIABLES INTO

TWISTORS

We shall use basis spinstors,8.12 eAa, eA a, ~ a , 'a' towrite twistors in terms of their component spinors and viceversa. They are defined in component form by

o 0o

oo 0

~ ,

01 .

(4.1a)

(4,lb)

We define the four twisto rs of the quadrad in terms of spin

ors by

wa = 1 T A ~ a+ u A'eA a,X a = P A ~ a+ 0'e A,a,y a A A ~ a+ f1A A,a,z a = S A ~ a+ TJA'eA a

and the four twistors of the dual quadrad by

i a = t A ~'a + JAeAa'Ya = / C , ~ a+ jiAeAa,

Xa = P A , ~ a+ yAeAa,tva = i i A ~'a + ? e Aa,

(4,2a)

(4,2b)

where SA and tA' are the twistors referred to earlier, [Theinter-relationships of the component spinors become clearerin the subsequent calculations from Eq. (4.11) on. For a geometric interpretation of the quadrad, see Ref. 8.]

The position X AA is defined by the point of interactionof Yand Z [being the position vector of the point relative tothe origin which was chosen for the expansions given by Eq,(4.2)]. Thus we have

TJA = _ i ~ A S A (4.3a)

f1A' = - i ~ A A A (4.3b)

Multiplying Eqs. (4.3a) and (4.3b) by AB and SB' respectively, and subtracting

i(AASB - A ~ A ) X A A= SBf1 A' - AB TJA . (4.4)By using the spinstors for projection, Eq. (4,2a) gives

AA = YCleAa, f1A' = y a ~ aSA = z a eAa, TJA = z a ~ a

Using Eq. (4.5) and Eq. (4.4) gives

(4,Sa)

(4,Sb)

AASB A ~ A= ya(eAaeBf3 - eBaeAf3)Zf3, (4.6)

Using the fact that any expression skew in two spin or indices

516 J. Math. Phys., Vol. 21, No.3 March 1980

can be written as the same expression with the indices contracted times AB' and the definition 2,8of the infinity twisto r

e A a ~ B e B { 3= I a{3 (4,7)

we see that Eq, (4.6) becomes

AASB- A ~ A= y al a;zf3AB' (4.8)

Putting Eqs (4,S) and (4.8) into Eqs, (4.4), we obtain

~ y ~ y[az{3]I~ A = 2i a {3y (4,9a)y PI ~ 1 T

Similarly, from the dual quadrad we could obtain

-:AA' _ . ~ a Y [ 7 { 3 l a { 3X-. - - 2 _ _ (4,9b)

y J P TZ TThen, as z a = 0 [by the definition of the dual quadrad,see Eqs. (3.2)]

(4.10)

Now we note that A is inversely proportional toy PI p1TZ 1T.Thus Y PI p1TZ T is a convenient choice of the

conformal function (which should compactify the Minkowski space), Similarly, as?A' is inversely propo rtiona l toYJ p1Ti , YJ p1Ti1Twould serve as the other conformalfunction, Thus we define

(4,11)

Clearly, for a real quadrad, i = i and XAA' = ,iAA' = XAA',Thus Eq. (4,10) will be trivially satisfied,

We can evaluate the twistors of the dual quadrad interms of the quadrad by

tv = J.t P1TW XVy PID,

XP = t I'VP1TWI'X Z1TID,

Y = t W Y PZ TIDV J-lVP1T

i = t x v y PZ TID~ J-lvprr

where

D = t J.tVP1TWJ.tX Y PZ ,

Now, since

and

I ~ f 3 y b= 21 yba f ~

x YJ p1Ti ,putting Eq. (4.12) into Eq. (4, IS), we see that

i = t .tVP1TYJ.tz 1 p1TI D,which, by Eq, (4.14), gives

X = i I D .

Thus we see that

x l i = D

(4.12)

(4,13)

(4,14)

(4.1S)

(4,16)

(4.17)

(4,18)

It is comparatively easy to writeJIJ'TJaa (which we shalldenote by J aa) in terms of derivatives with respect to thespin or variables, However, because of the constraints on thefields [contained in Eq. (3.4)] it is not so easy to invert this soas to be able to write the sp inor field equa tions in terms of the

Asghar Qadir 516



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twistor quadrad . Let us first translate the derivatives a a intospinor derivatives.

V THE TR NSL TION OF THE DERIV TIVES

To translate the derivatives, we shall need a relationbetween aaa and jaa (==a lafJaa)' For this we prove the following theorem.

Theorem: The derivates with respect to the quadrad andthe dual quadrad are related by

(5.la)

(5.lb)

Proof (i) Differentiating Eq. (3.2a) with respect to thequadrad, we see that

lO:c lO:a - baa - bu y 7aa cy77aa = . (5.2)Multiplying both sides by fJb{3using Eq. (3.2b), and simplifying, we obtain

acyfJa{3= - fJc{3fJar (5.3)

Now the chain rule can be written as_ - -b{3aaa - (aaa77b{3)a . (5.4)

Putting Eq. (5.3) into Eq. (5.4) gives Eq. (5.la).(ii) Similarly, differentiating Eq. (3.2a) with respect to

the dual quadrad and following the same procedure, we derive Eq. (5.l b). Hence the theorem.

For simplicity, we shall first take the fields to be conformally invariant, so that a j aX and a j a i give zero. Thus wedo no t need to consider t he conformal function derivatives inour translations as yet. We shall include them later by directextension of he earlier results. We can then write the derivatives with respect to the quadrad in terms of the derivativeswith respect to SA' ~ A ,and tA' as

a aS A a ~ A atA'

a w a a w a aw a

a aSA a ~ A

a x a ax a ax a

a aSA a ~ Aaya aya aya

a aSA a ~ Aaza aza aza

ax a

atA'aya

atA'aza

a

5.5)

We must calculate this transformation mat rix using the expressions for the spinor variables in terms of wistors derivedin the previous section.

Now, from Eq. (4.2a) we can evaluate the first columnof the matrix, as SA can be written in terms oftwistors asgiven in Eq. (4.5b):

aSA aSA aSA= - = - =0 , (5.6a)a w a a x a aya

aS A= 0 .

aza(5.6b)

Similarly, to obtain t he last column of the matrix, weuse Eq. (4.2b) to enable us to write

517 J. Math . Phys., Vol. 21, No.3 March 1980

tA' =ZaeA,a. 5.7)

Thus we have

atA' a= e A

aZa(5.8a)

atA' atA' atA'_ = _ = --=- =0.

aYa aXa aWa(5.8b)

Using Eq. (5.1) with Eq. (5.8), we obtain

atA' - - {= - Z ~ { 3 e A,

aw a

atA' - - {= - z Y{3eA a x a a

atA'

To obtain the middle column of the matrix, we use Eq.

(4.9a), which givesa ~ A a ~ A

= =0 , (5. lOa)a w a a x a

J ,AA' eAyeA'I Z[{3 8[plyplI Z1T_ X _ . _ = 4i p {3y a P 1 T (5. lOb)aya (yvlva Zo )2

a ~ A eAyeA' YPI Z[ 8[ply{311= 4i { p1T a py. (5.lOc)

aza (yvlva z a )2

Equation (5.5) may be symbolically written as

(aaa) = (Aij)(aA,A')' (5.11)where a aa ) represents the matrix of the derivatives withrespect to the quadrad, A ij) the transformation matrix, and

a A A , the matrix of derivatives with respect to the spinorvariables. We want to invert the matrix (Aij) subject to theweight constraints contained in Eq. (3.4) and subject to t hefact that aj aX = a j a i = 0, to obtain

(aA,A ,) = (Bij)(aaa) 5.12)

We now want to determine the components of (Bij).We can invert the transformation matrix 8 (Aij) by

explicitly writing out its components a i j and requiring thatthe components of its inverse matrix satisfy the simultaneous equations

(5.13)

where 8 i,k is the Kronecker delta. Writing Eq. (5.12) as

aa aw a

aS A

[ ~qAa ~

~ Ja

a ax aPAA,a qAA'

a rAA,a SAA,aa ~ A aa

pA'a qA'a ~ a ~ aaya

atA' aaza

Asghar Qadir 517



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it is found that Eq. (5.13) can be satisfied by 8

pAa = qAo = ,-Ao = 0, (5.15a)

Y l y za l JSAo = 2 ~ f J fJy,

Y pZ J P(5.15b)

a aw Jp AA = leA 04 ,. ax fJq 404 = IeAJ eA ,

(5.16)

Y ~ P(5.17a)

eA'7f3W U JfJy~ a = ----.:r'c--'-' _ _Y f , i ~P

(5.17b)

r4' = ~ a= 0 (5.17c)

Thus we get the translation of the spinor derivativesinto the twistor derivatives by putting Eqs. (5.15)-(5.17) into

Eq. (5.14) to obtaina y1ozY1J a

= 2eA f3 f3y ,aS A ylpZ l J az uP

(5.18)

a _. f3L aax AA' - IeAaeA f3 (5.19)

- - f3_ _ YlaZy/ y

- - 2 ~ -fJ - - - ,aS A' Y17,r/ p Taz o

(5.20)

where L 0 f3is the usual canonica l generato r for the Lie algebra of U(2,2) (the group for twistors) being given b y8.11

L fJ = r(aaafJ' (5.21)

The dependence on the conformal functions X and Xcan be included by considering the additional terms in thederivatives of X and i with respect to the quadrad twistors

- az X - -a aX' (5.22a)

- _ a- YaX ax ' (5.22b)

(where the + subscript signifies that the conformal functions are also varying). Multiplying the complete expressions for a l ax by Xu and a;aya by ya, we obtain a laxand a ; ax. The complete expressions for the derivatives withrespect to the spinor variables are

a 1 X u (5.23)ax Y ~ P 1 T ax a '

= 2 Y l a ~ Y ( r l a f JL - WYYI)-L-) , (5.24)as A' y p Z ~P az)' aYI)

518 J. Math. Phys., Vol. 21, No.3, March 1980

a . fl L aa ~ A = leAaeA, fl' (5.25)

a y a ~ .ax y PZ J T aya

(5.27)

Equations (5.23)-(5.25) may be verified by operat ing on thespin or variables with them.

I t is interesting to note that

(5.28a)

(5.29a)

(5.30a)

(5.31a)

where w, x, y, and z are the degrees of homogeneity of thefield in W a, xa, ya, and za, respectively. Considering thedual quadrad

- i a - aZ = ~ A - - - X - - = - w, (5.28b)as A' ax

_ _ ay = x ax = - x , (5.29b)

X = - X ~= - y , (5.30b)ax

a aW =

- 5 0 4 - x

= - z . (5.31b)aS A ax

VI THE FIELD EQUATIONS IN TWISTOR S

We have succeeded in writing the partial derivativeswith respect to the spin or variables in terms of wistors of thequadrad. However, we have not taken into account the factthat the spinors, SA and tA and the conformal functions, Xand X are themselves functions of position, i.e., they arefields themselves. Thi s may be understoo d as follows. Wetake a fibre, defined by (SA tA X, andx on thecompactifiedcomplexified Minkowsk i space with points specified by X AA '

to obtain in the fibre bundl e with variables ( ~ A ,SA tA X,X). We take a section of his fibre bundle wi th variablesyAA ,such that

(6.1)

So, whereas t he position X AA' and the fields (SA tA X, and.f are independent variables, they are all functions of position, A which happens to be equal to A . We must dealwith derivatives with respect to A and not X AA ' only. Bythe chain rule we obtain that

Asghar Qadir 518



7/8

(6.2)

Even now we have not taken into account the covariantderivative for the fields, but only the partial derivative. Toobtain the covariant derivative with respect to yAA',YVAA"we note that 8

YV - (Y ';,:;'1) _ aXAAx - AA' + 1


8/8

iJ = D Z aZf3 [ y ya 3 (9 a 3) X(Yf3 a - U9f3 a) 1~

(6.21)

= 0 (6.23)

.i =.1, (6.24)4 = ~ (6.25)0 = 0 = 0 .

- (6.26)

Equation (6.12) together with Eqs. (6.14)-(6.18) and(6.20)-(6.26) comprise the field equations in twistors.

VII CONCLUSION

We have seen that we can write the massive field equations, with source present, in terms oftwis tors, using a twistor quadrad (set offour twistors) by imposing six constra intson the fields. This was done by writing the twistor derivatives in terms of the spinor derivatives by a transformation

matrix. The matrix being nonsquare did not have a uniqueinverse. The choice of the inverse was made by requiring consistency, internally, and with the six constraints. Wethus have a unique inverse, in the sense that any furtherpermissible change of the inverse matrix cannot change thetransformation formulas.

It still remains to solve the twistor field equations. Somegeneral points can be mentioned about the solutions. It iseasily verified that the zero rest mass, source-free, equationsare solved by fields of the type

f Wa , x f3yyZO) = A a 3y .. ,'jI- vp,,,,,X waw f3xy .x y 1-'. . . y v Z pZ11', (7.1)

where

A a ... 3y ... i I-''''V 1 11' = A (a . f3)( 1 .. i)( 1 ' ' ' ' \ ' ) ( 1 11')' (7.2a)

A ( a .. 3y) ... j I V P ' 1 T = A (a .. 3 11 . 61 1- ) ... 1''''11'

= A ( a . f3ly ,s 1- ... vl p) '

= Aa ... 3(y .. s 1- ) .. p ..= A a . 3 (y .,sII-' .vl p) '11'= A a .. 3y .. s( I-''''V 1') ..11 = 0, (7.2b)

with

I a, . ,p}l.;;; l{y ,D}I.;;; l{ p , .. ,v}l.;;; l{ p .. ,1T}I (7.2c)I t would be interesting to find the most general conditionsfor solutions of the ze ro rest-mass field equations .

520 J. Math. Phys., Vol. 21, No.3, March 1980

A field of the above type, which is only a function of Z,will beaspin-z/2 field. Ifit also depends on y, it will bea spin(z - y)/2, conformal weight y field. I t would be interestingto determine what all the degrees of homogeneity refer to. I tis hoped that this will lead to a better understanding of Penrose graphs. Y

Of immense importance is the solution of the massivefield equa tions with source terms present. In particul ar, we

could write the complete gravitationa l field equations (i.e.,Einstein equations) in twistors. 12 This is of prime importance in the program for the quantization of relativity using twistors.

By regarding the fields as functions giving us a representation of the Lie algebra ofU(2,2), the Casimir operatorscan be evaluated in terms of the degrees of homogeneity ofthe fields in the twistor variables. 8 . 11 This would have somebearing on the significance of the degrees of homogeneity.Conversely, if the significance is already understood , itwould be interesting to consider the conserved quantitiescontained in the Casimir operators due to the requirement ofinvariance under U(2,2). I t would also be of interest 14 toconsider the cohomology classes of the functions representing the fields.

I t should be mentioned that by bringing in th e infinitytwistor we are breaking con formal invariance, as infinity is

being located in some sense. However, Poincare invariance is being maintained . That would be broken by the introduction of he origin twistor. Thus our fields cannot have theorigin twistor in them. This places a severe restriction on thefields. It is to be expected that massive fields will include theinfinity twistor, as masses break conformal invariance, butthe massless fields need not do so.

ACKNOWLEDGMENTS

I would like to express my gratitude to professor R.Penrose, the father of twistors, for suggesting the problemand for his encouragement and advice. Thanks are also dueto Professor Abdus Salam, the Intern ational Atomic EnergyAgency, and UNESCO for hospitality at the InternationalCentre for Theoretical Physics, Trieste.

Quantum Gravity, edited by C.J. Isham, R. Penrose, and D. W. Sciama(Clarendon, Oxford, 1975).

'R. Penrose, J. Math. Phys. 8, 34S (1967).

JR. Penrose, Int. J. Theoret. Phys. I, 61 (1968).'R. Penrose, An Analysis o f he Structure of Space- Time (Adams PrizeEssay. 1967).

'R. Penrose, Structure of Space-Time, in Battelle Rencontres, edited byB. S. DeWitt and J. A. Wheeler (Benjamin, New York, 1968).

'R. Penrose, J. Math. Phys. 10, 38 (1969).'R. Penrose and M. A. H. MacCallum, Phys. Rep. C 6,243 (1973).SA. Qadir. Twistor fields, Ph.D. Thesis, Lond on (1971).'A. Qadir, Phys. Rep. C 39,133 (1978).

lOG Sparling, in Ref. I, pp. 408-Sao.A. Qadir. Int. J. Theoret. Phys. 16. 25 (1976).K. Dighton, Int. J. Theoret. Phys. 11, 31 (1974).F. A. E. Pirani. in Brandeis Summer School 1964, Vol. I: Lectures onGeneral Relativity. edited by F. A. E. Pi rani, H. Bondi, and A. Trautmann(Prentice-Hall, Englewood Cliffs, N. J., \965).

14R. Penrose, private communication.

Asghar Qadir 520

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