CENTRAL RESEARCH INSTITUTE FOR PHYSICSBUDAPEST

K F K l - 1 9 8 1 - 8 9

B. L U K Á C S E.T. N E W M A N Z. P E R J É S J. P O R T E R Á, S E B E S T Y É N

STRUCTURE OF THREE-TWISTOR PARTICLES

‘

H ungarian ‘Academ y o f ‘ Sciences

CENTRAL RESEARCH

INSTITUTE FOR PHYSICS

BUDAPEST

2017

KFKI-1981-89

STRUCTURE OF THREE-TW ISTOR PARTIC LES*

B. Lukács, Z. Perjés and Á. Sebestyén Central Research Institute for Physics H-1525 Budapest 114, P.O.B. 49, Hungary

E.T. Newman and J. Porter

University of Pittsburgh, Pittsburgh, Pa. 15260, U.S.A.

HU ISSN 0368 5330 ISBN 963 371 871 6

*Work s u p p o r t e d by a Na t i o n a l S c i e n c e F o u n d a t i o n c o - o p e r a t i o n grant

L L U

L _ n

- о,

и -

j

L I D •

R E F E R E N C E S

1. Our notation is in agreement with R. Penrose and M.A.H.

MacCallum, Physics Reports 6C, 241, (1973) 2. R. Penrose, Int. J. Theor. Phys. JL, 61 (1968)

3. K.P.Tod and Z. Perjés, Gen. Rel. and Grav. 1_, 903 (1976) 4. K.P.Tod, Reps. Math. Phys. ЗЛ., 339 (1977)

5. Z. Perjés, Phys. Rev. D l l , 2031 (1975) 6. Z. Perjés, Phys. Rev. D 2 0 , 1857 (1979)

7. The coincidence of internal and space-time C a s i m i r invariants has been observed in Penrose: Twis t o r s and Particles; an

Outline, in Quantum Gra v i t y and the Structures of Time and S p a c e , Eds. L. Castell, M. Drieschner, and C.F. von

Weizsäcker, Carl Hanser, 1975

8. L.P. Hughston: Twistors and Particles, Lecture Notes in Physics, Vol. 97, Springer, 1979.

9. E.T. Newman and J. Winicour, J. Math. Phys. 15;, 1113 (1974) 10. R. Penrose and G.A.J. Sparling: A ' N o t e on the n-Twistor

Internal Symmetry Group, in Advan c e s in T w i s t o r Theory, Eds. L.P. Hughston and R.S. Ward, Pitman, 1979

11. T e r Haar: Elements of Hamiltonian Mechanics, N o r t - H o l l a n d , 1968

12. H.S.M. Coxeter: Regular C omplex P o l y t o p e s , Chap. 8, Cambridge U.P., 1974

13. E.P. Wigner, Ann. of Math. 40, 149 (1939)

14. R. Penrose: Appli c a t i o n s of Nega t i v e D i m e n s i o n a l Tensors, in Combinatorial Mathematics, Ed. D.J.A. Welsh, Academic, 1971

ABSTRACT

The simplest physical system to have a non-trivial intrinsic structure in Minkowski space-time is a three-twistor particle. We investigate this structure and the two pictures of the particle as an extended object in space time and as a point in unitary space. We consider the effect of twistor trans lations on the mass triangle defined by the partial centre of mass points in space-time. Finally we consider the connections between twistor rotations and spin and we establish the spin deficiency formula.

АННОТАЦИЯ

В пространстве-времени Минковски простейшей физической системой является частица, состоящая из т рёх твисторов. Нами изучается эта частица, являющаяся обширной системой в пространстве-времени и точкой в унитарном пространстве.

В прострастве-времени массовый треугольник определяют парциальные массовые центры. Записав эффект твисторных трансляций на массовый треугольник, о п р е ­ делив связь между спином и ротациями твис т о р о в , нами выведена формула с п и н о ­ вого дефекта.

KIVONAT

Minkowski téridőben a legegyszerűbb, nemtriviális belső szerkezetű fizi­

kai rendszer a háromtvisztor-részecske. Megvizsgáljuk ezt a részecskét, amely a téridőben kiterjedt és az unitér térben pontszerű rendszer. A téridőben a parciális tömegközéppontok tömegháromszöget definiálnak. Felirjuk a tvisztor- transzlációk hatását a tömegháromszögre. Megállapítjuk a spin és a tvisztor- elforgatások kapcsolatát és levezetjük a spinhiány-formulát,

a бa

b b

f ЗГ f

The notation converts identities of the ki n d U 6^ = U ^ into trivial partitions of some index line.

S ymmetrization and skewing in like indices is denoted according to the scheme

The dimension n of the tensor system is g i ven by the loop n =

In twistor t h e o r y one is interested in d i m e nsion n=4. T a k i n g twistor complex conjugates is an involution that has the effect of tur n i n g the s ymbols upside down. Th u s the blobs of twistors Z^, > Z3 and °f their complex conjugates are drawn

The s k e w unit t w i s t o r and the infinity twistor are denoted, respectively

e»e-r6 . U J J and x“ 3 . u .

It is useful in computations to keep in min d some of their algebraic properties in the blob n o t a t i o n such as

1.

I N T R O D U C T I O N

In Penrose's theory of twistors, zero-mass objects are a ssign­

ed a fundamental role. T h e zero-mass particles are represented (classically) b y a single twistor w h i l e the mas s i v e particles are represented by several twistors . The basic idea of twistor p a r t i c l e theory is that the kinematic variables, e.g. m o m e n t u m and angular momentum, associated w i t h the m a s s i v e particles can be expressed in terms of two or m o r e twistors. (This description

is v i a the so-called kinematic t w i s t o r ) . On the other h a n d the internal structure of the particle does depend critically on the n u m b e r of twistors used in the description. (Frequently in twistor

6__ Q

t h e o r y leptons are described by t w o - t w i s t o r systems w h i l e

had r o n s are d e s c ribed by three-twistor systems.) The linear t r ans­

formations among the two or three (or more) twistors w h i c h

p r e s e r v e the k i n e matic twistor (or variables) are referred to as

7

internal symmetry transformations , (1ST) . T h e ttoo-twistor p a r ­ ticles have the simplest space-time description. They can be t h o u g h t of as e i t h e r a rea l center of mass wo r l d - l i n e w i t h an associated m o m e n t u m and spi n or as a complex center of mas s world-line.

T h r e e - twistor particles are the simplest systems w h i c h do po s s e s s an e x t e n d e d structure in co m p l e x M i n k o w s k i - s p a c e . A three twistor particle with twistors Xa , Y a , Za can be thought of as b e ing (in some sense) compo s e d of the three p a i r s (X01, Y a ) ,

(Yu , Za ) and (Za , Xa ) of twistors^ and thus it would have a sub­

structure of t h r e e two-twistor m a s s i v e sub-systems. These s t ruc­

ture are however not disjoint since any pair of them has a twistor in common. Furthermore the world-lines, masses, spins, etc. of the parts make u p those of the entire system. A point to be

I К ' ф - : К ' í р

^(7.8)

of the zero-mass constituents, w e obtain the spin definiency formula in the blob notation

(7.9)

A P P E N D I X : T H E B L O B N O T A T I O N

The blob notation of abst r a c t tensor systems has been first introduced by Penrose^. Its advantages over the mor e conventional formalisms of Ricci, Levi- C i v i t a and E i n s t e i n are b e s t understood probably in terms of the p h y siology of h u m a n perception.

A tensor is drawn in the d i agrammatic notation as a blob w i t h arms and legs depicting the upper and lower indices:

a b f

, aD

cde U

gh

д p £

The outer product of tensors T c^ c and U ^ is the juxtaposition of blobs

Tcde gh

To contract a pair of indices, one connects the corresponding arm and leg,

T ag u f = 1cde gh

A Kronecker symbol 6^ is represented b y a line segment:

2

emphasized is that the subsystems and their k i n e m a t i c properties are not invariant under the 1ST.

The present work is an attempt to study the geometry of this twistor decomposition of t h r e e - t w i s t o r systems. T h e two m a i n questions investigated are (a) what are the c hanges in the s u b ­ systems caused by the action of the 1ST, and (b) what is the

relationship of the total s ystem v a r i ables to t h o s e of the related sub-system variables. For peda g o g i c a l reasons w e have c hosen a purely classical approach, m o s t of the results e a sily surviving quantization.

In Section 2 the g eneral theory^ ® of m a s s i v e n - t w istor par­

ticles is reviewed. At the center of the f o r m alism lies the

kinematic t wistor in terms of which the momentum, angular momentum and center of mass line of the part i c l e are expressed. A s m e n ­ tioned before, the 1ST of the system leaves the kinematic twistor invariant.

In Section 3 we specialize to thr e e - t w i s t o r systems and study the 1ST w h i c h turns out to be the inhomogeneous SU(3) group

(I S U (3)). T h ree spaces pla y a critical role h e r e (1)( t w i s t o r ot a a cl

space T on w h i c h we choose three p o i n t s Z^ = (X , Y , Z ) , or alternately three copies of T, i.e. T x T x T, (2) complex

M i n k o w s k i space which has the n a t u r a l l y chosen triple of points x a , y a , za , ea c h being the intersections of the twistor pairs

(Ya , Za ) , (Za , X a ) , and (Xa , Ya ) and (3) unitary space, a three- c omplex dime n s i o n a l affine space on which the e l e m entary representa­

tion of the I S U (3) group acts as the isometries. It can be viewed as the homo g e n e o u s space ISU (3)/ S U (3) .

In Section 4 we investigate the translation subgroup of the I S U (3) group and show that a translation along a given axis in u nitary space leaves the cor r e s p o n d i n g twistor unchanged while shifting the other two in complex space-time a l ong their subsystem or partial c enter of mass line. We also find a unique c o r r e s p o n ­ dence between the time d e v elopment of the s y s t e m in Minkowski space and a special translation in unitary space.

In S ection 5 we study the SU(3) subgroup of ISU(3) and show h o w from its generators w e can find a unique "complex internal c enter of mass" world line in uni t a r y space. T h i s is in analogy w i t h the use of the homogeneous Lo r e n t z group generators to find

we o b t a i n

2 ^ - (£ - I)(L p ГЬ - L fП ) + 2 J jJ + 2 I Ll .

(7.4) We now c ompute the spin of a t h ree-twistor particle

similarly. Inserting the rest-mass square,

i

^m2

= П T J + П + П t J

and the kinematical twistor +

(7.5)

in Eq. (7.2) w e have

(7.6) U s i n g a judicious amount of identities of the form (7.3)

for various subsystems, w e obtain for the spin of the three- -twistor particle;

(7.7) When w e compare this expression with the spin twistors of the massive subsystems (Cf. Eq. (7.4)) and w i t h the spin twistors

3

the center of m a s s 1 . W e further discuss the internal or unitary spin (which is analogous to the P auli-Lubanski vector) and show its relationship to the space-time spin. S e c t i o n 6 deals with the mass triangle d efined by the partial c e n t e r of ma s s points w h i l e Section 7 presents the spin deficiency formula, i.e. the relati o n s h i p between the total spin and the constituent spins.

A c c o r d i n g to this formula the total spin is the sum of the massive subsystem spins minus the spins of the three twistors.

In the concluding sections we use the P enrose blo b notation

14

to facilitate lengthy algebraic computation. A n introduction to the blob notation is g i ven in an Appendix.

2. THE T W I S T O R C O N S T I T U E N T S

Consider a massive particle in Minkowski space-time as a s y s t e m of n >

2

massless constituent twistors Z A ,ot i=l,...n. The particle has the kinematical twistor

д аВ = 2 z ^ ) y ^

V (2.1)

ct ß

w h e r e I is the infinity twistor which breaks the conformal invariance. T h e summation convention holds for Roman twistor

"flavor" indices (Flavor indices share the p r o p e r t y of Greek twistor indices that they are raised and lowered by c omplex con-

a ” i

jugation, (Z ^ ) = (Z^). E a c h term on the right of Eq. (2.1) for a f ixed value of i is the kinematical twistor of one of the massless c o n s t i t u e n t s .

The kinematical twi s t o r is decomposed into the s p i n o r parts

(2.2)

Her e у AB is the total angular m o m e n t u m spinor (symmetric in its

Д

indices) and the Hermitian spinor p ß , is the four-momentum. The c enter-of-mass line of the system consists of the points of real

Thus, typically, in tensor notation c 2 =

‘^ Ф с в ' (^ 5 ?l76) (Z2 Z3 V > (Z? Zp 3lXP)

X {

(z'fzi

n I ) Ú ]z 2r

( z ^ z 3

IT a ) Z ^ Z3 3 1 к л 2 lí a J T 3 a)

- (zK z \ r] I ) zi;]z 3 z 3 iTOzf z2,z“ }

3 1 к л 2 £ T | a w J 3

where antisymmetrization in indices in the brackets is understood.

7. THE S P I N D E F I C I E N C Y F O R M U L A

In this section we prove the spin def i c i e n c y theorem. The content of this theorem is that joining three two-twistor par­

ticles by identifying their constituents pairwise g i ves a total spin which is the sum of the spins diminished by the spins of the twistor constituents,

A

S 12k + S 23k + S 31k

(7.1)

5 and using the identity

- \ p Пгк + L* Ь П + l-n - LJ . j L 7.

(7.3) Equation (2.1) takes the form for a t w o -twistor particle in the blob notation: L-J = ^ . Inserting this in the spin twistor (2.11) wr i t t e n in terms of blobs as

4

M i n k o w s k i space-time

AA', , d e f -2,A B A' , - A 'В ' A . , -1 A A ' ...

x (t) === ш (m pB + у pB ,) + тш p (2.3) where m^ ==í pB ,p® is squared rest-mass and the real p a r ­ ameter T is the p r o p e r time. It does n o t appear to be possible to e x p r e s s the c e n t e r of m a s s in terms of the kinematical q u a n t i ­ ties in a m a n i festly twistor invariant form, due to the fact that the con c e p t of the center of mass in n o t invariant with respect to translations. However, u s i n g the constituent twistors directly,

g one can define a center of m a s s point twistor :

Raß dg| 2m 2zot z3 M ik^

i к (2.4)

The quantities

M j, äsi l“ !» I .

lk l к aß (2.5)

are c a lled mass amplitudes and for n > 2 are the partial mas s

amplitudes of the two-component subsystem labeled by i and k, and are such that the m a s s - s q u a r e d of the system m a y be wri t t e n as a sum of partial mass-squares:

2 .. r:ik m = M., M

lk ⁽2.6)

The p o int twistor (2.4) decomposes according to

,a0

1 R R ' A B . A

2 r R R ' r

t r*

l r B '

. В

1 Г А ' £ A ' B '

(2.7)

A A '

w h e r e r , the center of mass p o i n t , is a p o i n t of the complex A A '

M i n k o w s k i space-time. T h e point r lies on the complex center of m a s s line of the system. The co m p l e x center of mass line is the set. of p oints9

AA' -2, AB A' - A 1В ' A . , . _AA ' -2 , , AA' -1 z = m (у pB + у Рв') + lS m + Ap m

(2.8)

T a k i n g the sum, and adding the lengths of imaginary parts [Eg. (6.6)],

/ »2 _ lrDaß -A aß -В

(XA -XB ) 4 lRA Räß ' RB Raß

- <Rf +

(6.10)

For calculational convenience and to illustrate its usefulness, Penrose's blob notation w i l l be used to write the lengths in terms of the constituent twistors /Cf. Appendix/.

Let us write

Z^a 1

F r o m equation (2.8) , and A = (13), В = (23),

П T - i

о Ф

A v

(6.11)

(6.12)

w h e r e the mas s amplitudes are

M _A

= A

^{M- =} _{L J} ^(6.13)

Proceeding w i t h the evaluation of terms in Eq. (6.10) w e obtain the expressions for the sides of the mass triangle:

a -2

:_ _ _ _ _ _ _ t ?

* £ i t i а о n - i

о О

Д • - _____ _i_________

m

* П t - T П T J [ H i

- - - i - - - L ? "

20 T J П T J i i

(6.14)

5

where X is complex and S ^ , denotes the Pauli-Lubanski spinor

SA A ' " i(yA B PA ' ” * V b ' PA )- (2.9) In the rest-frame of the parti c l e def i n e d by the form of the

A A 1 1 П оП

four-momentum [p ] = — , the P auli-Lubanski spinor becomes /2 ip ⁱ_

proportional to the nonrelativistic spin. Thus the spin jA A i is given

-1 о

:A A ' m SA A ' ‘ (2.10)

T h e Pauli-Lubanski s p i n o r is the sole nonvan i s h i n g part of the spi n twistor5

S3 = \ (Ä Ap3 + \ m2 Ő 3)

a 2 ao 2 a ⁽2.11)

according to

О

- s A ’B О

(2.12)

T h e internal symmetry transf o r m a t i o n s of the kinematical twistor are10

z az i ⁼ u ’* ( z “ + 1 к ^ k £

-j-aß z £Z ß Л •

z 1a = ü?; ( z K + к a

дк£

^aß z ß Z £

(2.13)

where [ ] js an n x n unitary m atrix and [ A ^ j is skew. The t r a n s i ­ tion Z? -*■ Z? amounts to selecting a n e w set of m a s s l e s s c o n s t i ­ tuents for which the kinematical twistor of the s y s t e m remains unchanged. Thus the kinematical variables discussed remain u n ­ changed under internal symmetry transformations, w i t h the e x c e p ­ tion of the center-of-mass twistor. The center-of-mass point is defined directly in terms of constituents and it has been shown

aß = _ 1 R «3 e

r

Y6 = , A Ä _ A Ä 1 _ 2 R 1 R 2aß 2 R 1 a ß y S R 2 ( 1 2 ’ ( A Ä rA Ä

The length of the imaginary part of vector r АЛ is

(6.5)

_aß - . A Ä it- г \

R RaS ' 4y yAA- (6-61

The partial mass centers are null separated in complex M i n k owski space-time since any pair of them lie on a common twistor. Let us however consider the real mass triangle. Fro m Eq. (6.5), the condition of null separation for the arbitrary subsystems A and В is

-RA RBaS " vXA “ XB )" " (У/ yB ) + 2i(xA - x B ).(yA - yB )=0. (6.7) Hence using Eq. (2.8), we can express the side c connecting

AÄ 2 A Ä the m a s s points A and В in terms of the spin vectors S =m у as

„2 def , v2 _ , -2C _ -2C ,2 , , - (xA " B } " m A SA В SB ) ‘ (6.8) It is quite surprising that the spin difference appears in a side length of the mass triangle. Fr o m Eq. (6.7) we further have that each side of the mas s triangle is o r t hogonal to the difference of the spins at the endpoints of the side.

T o complete the analysis of the mas s triangle, we now ask how the lengths of the sides of the mass triangle depend directly on the constituent twistors. A direct substitution into Eq.

(6.8) yields u n w i e l d y results. Instead we consider the variants of Eq. (6.7)

“Räß R~ ~ 'XA-XB' “ (УА~УВ^ “ 21(XA~XB* **УА~УВ*

“RA Räß = *XA_XB* “ (yA+yB)" + 21 *XA_XB* ’ (УА+УВ;

“R- RBaß ~ ^XA _XB^ " ^УА + У В1 ” 2 1 ^XA _XB ; * ^УА +УВ ) ' (6.9)

6 о

by Hughston that internal transformations (2.13) move the com­

plex center of mass point over the entirety of the complex center- of-mass line.

The central d o g m a of twistor particle theory asserts that the state of the s y s t e m at any instant of time is completely described by the v a l u e s of the constituent twistors Z? and by their co m p l e x conjugates. The space of n-twistors TxTx...xT

ct — i

admits a naturally defined symplectic form d Z ^ d Z a . In the sense of H a m iltonian dynamics z“ and Z^ together play the role of c a ­ nonically conjugate variables. Accordingly, any function of the form f(Z?,Z^) is c a l l e d a dynamical q u a n t i t y ^ .

We introduce the Poisson bracket of dynamical quantities f(Z,Z) and g(Z,Z)

[ f ,gl ^{' Эf} 3g _ _3f_ 3g \ ,3Za 3Zr 3Zr 3Z°V

r a a r

(2.14)

The P o i s s o n b r a c k e t is antisymmetric in f and g and is imaginary when b o t h f and g are real. F r o m (2.14) w e identify the general

i cx

coordinates q a n d canonically conjugate m o m e n t a p. as follows,

Ct 1

(2.15) The tw i s t o r v a r i a b l e s have the Poisson brackets

[Z«, z£] ^{fk ,a}

6i V (2.16)

A t r a n s f o r m a t i o n g e n e rated by a dynamical quan t i t y f is given

в?; = z“ z1

к k a (2.18)

6z“ = i[z“ , f]

(2.17)

5K = i [ ^ci' f b

л ct к ct

The un i t a r y t r a n s f ormations Z. = U. Z, are generated by the func- 1 X К.

tions

6. THE M A S S T R I A N G L E

The subsystem of twistors Z^ and z“ has the squared rest mas s

2 -12 3 -

m 12 = 2M12 M = 2d d 3*

T h e center of mass point twistor (2.4) of the subsystem may be written

Rotß _

12 M

12

(z“ Z^ - z“ zj) .

^(6.1)

Using Eq. (6.1) and the center of mass twistors of the remaining massive subsystems, we obtain the center of mass of the three-

-twistor particle as the linear combination

H " 8 - > 1 2 *12 m

M j^2 3 aß

2 3 23

«11

31 M ^{R a ß)}К З Г ⁽6.2) The non-diagonal spinor part of (6.2) w h i c h is linear in the position vectors [Cf. Eq. (2.7)] is

AÁ _{= m} - 2

,

_(m., ² ₂ ^AÁ

:12

2 AÁ m 23 Г 2 3

2 AÄ.

m 31 Г 3 1 } (6.3) Thus the center-of-mass point r AÁ of the particle is a w e i g h t e d mean of the partial center of masses. Hence the four mass center points lie in a plane. The center of m a s s r AÄ is in the barycenter of the triangle f o r m e d by the partial m a s s centers. Note h owever that the weights are mass-squares rather than masses. We now compute the sides of the triangle.

The invariant distance of two complex points in Minkowski sp a c e - t i m e,

AÁ A Ä , . AÁ r - x + ry

1 (6.4)

AÁ A Á , . A Á r2 “ x 2 + ^ 2

can be expressed in terms of the point twistors and R ^ 'as^

7

forming a Hermitian matrix, a n d the transformations

Z? = Z? + Л,, Ia ^ Zß are g e n e r a t e d by t h e mass amplitudes

1 i ÍK p

M ik - z i z? r.e a n d " ik " Za ze 1<,e- (2-1 9 >

The Л-transfo r m a t i o n s commute and are c a l l e d internal t r a n s l a ­ tions .

As long as w e are concerned with a massive part i c l e in free motion, the decomposition int o allowed twistor const i t u e n t s is immaterial for the motion of the particle. This reflects the

invariance of the kinematical twistor w i t h respect to the internal transformations (2.13). The idea is, however, that the b e h a v i o u r of the particle in interactions should depend on the substructures present in a twistor decomposition.

T h e n-twistor particle w h e r e n > 3, possesses massive parts.

Such a substructure consists of two o r more null constituents.

Clearly, a two-twistor part i c l e has n o massive subsystems. The simplest place to study m a s s i v e subsystems is a t h r e e - t w i s t o r particle. In the n e x t section we discuss some features u n i q u e to three-twistor particles.

3. S T R U C T U R E OF T H R E E - T W I S T O R P A R T I C L E S

T h e internal structure of a m a s s i v e particle described by three twistors c a n be exami n e d in terms of the t h r e e two-twistor subsystems obtained by considering t h e three twist o r s pairwise.

Each s u c h two-twistor s u b s y s t e m defines a massive particle in space-time with w e l l - d e f i n e d (real a n d complex) cent e r - o f - m a s s line, spin and cente r - o f - m a s s point. T h e s e phys i c a l properties of the subsystems combine to yield the properties of the entire system in unexpected and interesting ways, given an ordered

triple of twistors (Z^, Z^/ Z^) 6 T. (It is sometimes preferable to t h ink of the triple as a point in TxTxT.) A n y one of these, Z ? , h a s its kinematical twistor, A?®, and any p a i r of these,

Ot OL OLß

(Z ^ , Zj), i < j, has its associated k i n ematical twistor A^j.

While a single twistor d e s c ribes a m a s s l e s s s y s t e m in M i n k o w s k i

w h ere w e introduce the real parameter j which can take any real value for a classical system (and it w i l l take the values 0, 5,

3 i

1, ... after quantization). This common m a g n i t u d e is a Cas i m i r invariant of both the Poincaré and of the inhomogeneous SU(3) groups, the second common C a s i m i r invariant being the mass square:

d1 dL = A A1 AA'

m 2

There is a further p r o p e r t y that connects the space-time spin and (the negative of) unitary spin: The projections of the spin

twiscor (2.11) ont o the constituent twistors are the negative of the components of the u nitary spin,

s“

Z? ZK

ß i a ^(5.12)

This result (which can be proved by direct computation) gener- 5

alizes a relation holding for two-twistor particle spins . For a two-twistor particle, however, the unitary spin is replaced by the conformally invariant quadratic expressions

Z? a.1 Zk , i, к = i, 2, i -k a

w h ere [o£] are the Pauli matrices.

8

space-time, a pair of twistors describes a massive system. The internal symmetry t r a n s f o r m a t i o n s change t h e kinematical twistors of the one and two tw i s t o r subsystems w h i l e the kinematical

twistor of the entire system is unchanged and in fact this con-

ex В ос В

strains the changes in and A ^ since

.aß 1

2

Z

i/j i< j

Aaß

ij = Z Aaß

= Z i/j

.aß

ij - Z Aaß

The internal s t a t e s will b e used in the description of interactions and the manner in w h i c h the various concepts are linked is of importance. To p r o c e e d f urther the internal symm e t r y group must b e examined more closely. The internal t r anslations are g e n e rated by the partial m a s s amplitudes w h i c h can be g i v e n e q uivalently by

d - J G ^ M jk and d i 2 e ijk ^ * (3.1)

2 i-

The mass squ a r e d is e a s i l y g i v e n by m = 2d d^ and is positive.

The u nitary transformations are generated b y В£ w h i c h satisfy

d. bJ d k = 0. (3.2)

1 К

Writing B,a as a trace-free part, A £, plus a trace r esults in B* = A.1 + 6.1 В

к к к (3 .3 )

X 2Г

where В = - В^,. Fro m (3.2), the trace m a y be written as

„ _ -2 3 ,i ,k В = 2m d . A, d .

x к (3.4)

Thus the t r a c e of t h e generators of the uni t a r y t r a n s f o r m a t i o n s can be w r i t t e n in t e r m s of the remaining 14 internal symmetry generators and w i t h o u t loss of gen e r a l i t y the u nitary t r a n s f o r m a ­

tions will be re s t r i c t e d to SU(3) in the remainder o f the paper.

As in (3.1) introducing a l t ernative translation parameters

[ s * , s k ] =

(dk d. - Ő k d r d r )sj - (d1 d p - < s j d r d r )Sk

(5.9) which are the S U (2) Poisson brackets. When the transformations generated by S'!' are referred to arbitrary t hree-twistor systems, then they do not b e l o n g to ISU(3) since S. is not a linear com-

i I i I 3

bination of the A_. s and d s . We can consider, however, the restriction of these transformations to systems with a fixed

momentum d1 = D 1 . Then, S^ are ISU(3) generators. We thus have the theorem:

The unitary spin S t generates the SU(2) subgroup of 1SU(3) leaving the mass amplitudes invariant [Cf. Eq. (5.7)].

The mass amplitudes determine the scalar products of the momentum parts тг^д) of constituent twistors by = тт^д, ^ •A ' Hence the effect of transformations g e n e r a t e d by is a rigid rotation of the m o m e n t u m spinors тг. , (together with the frame defined b y any pair of them) about the total four-momentum p д | .A For systems with a fixed value of the uni t a r y momentum, d1=D , we may choose coordinates D “*" = 6^. In this coordinate system,

/2 -3

the unitary spin has the component form, w i t h J real, Э.

J 3 J1 + iJ2 0

sk ]

1 2

= 2 m J l-iJ2 J 3 0

0 ^r\ 0

Thus the Poisson brac k e t s (5.9) , restricted to systems with a fixed momentum, may be written

J ,J, ] — is . j

a b abc c a, b, c, = 1, 2, 3 [5.10) 7

It has been k n o w n for some time that the m a g n itudes of the space-time spin and of the uni t a r y spin are equal:

1 2 i к _

r. in S1 S,

2 к 1 - S-. , S A A '

AA 1 = j (j + l)m^{- 2} (5.11)

9

t1 = i e ^ k A

2 E j k ‘

^(3.5)

the internal symmetry transf o r m a t i o n s take the form Л

z

^a

i u k (z“

^{l aß -m,}

ek£m 1 V

(3.6)

v о 3

w i t h U;T an e l e m e n t of SU(3) and t an element Of C . T h i s trans­

formation is r e p resented b y the p a i r

(y,t)

a n d the g r o u p product structure follows from the composition of t w o successive trans­

formations. T h u s (U,t) followed b y ( Ц ',t ' ) g i ves after a short calculation

( y ' ,t') . (U,t) = ( y ' y , t + U + t') (3.7)

w h e r e

y +

is the Hermitian adjoint of

y.

Equat i o n (3.7) defines the 14-parameter group d enoted by ISU(3) and called the inhomo­

geneous S U (3) group of internal symmetry trans f o r m a t i o n s (1ST).

3 i

Thi s group acts on C w i t h coordinates z , i = l , 2, 3 a s a group of point t ransformations where

(y,t)

gives

Z1 =

u£(zk

+ tk ) . (3.8) T h a t is to s a y the ISU(3) is r e a l i z e d as the isometry g r oup on

3 i - 3

C of the H e r m i t i a n line element dz d z . a n d this gives C the

3 1

structure of a unitary space, U . A n alternate point of view is 3

to consider U as the homogeneous space I S U (3)/ S U (3). W e will 3

gi v e later y e t another m ethod of obtaining U . 3

The transformations in (3.6) which act on T constitute the twistor realization of ISU(3). T h e same elements of ISU(3) act on U J via (3.8). We n e e d not b o t h e r to co m p u t e the generators of the I S U (3) group in the isometry representation s i nce they are already available in the twistor r e a lization [cf. eqs. (2.18),

(3.3) , and (3.1) ]:

wit h

S* = S(X1 X . - Y1 Y .)

3 3 3

s\ X^ = s x 1 , S* Y^ = - SY1 , S > 0

D 3

x1 d. = Y1 d. = О, x1 X. = Y1 Y, = 1,

l i i i

(5.8)

The vector S1 = /S X1 contains all the information in S ^ . From (5.4) and the first t e r m in (5.6) we have

В = В ^.m

1 3 (т t ) .

U p to the choice of origin В can be identified with the imaginary part of T . There exists a real line which is imbedded in the

complex line and parametrized by the real par t of т d e f i n e d by В = 0.

To reiterate the material of this section, we h a v e shown that a point in three twistor space selects an origin in unitary space. Aj(Z) represent the eight generators of SU(3) rotations about this point while d1 are the generators of the t h ree complex

л i ot i

translations. The t ensor field A.( Z . , t ) on the uni t a r y space 3

represents the SU(3) generators about the point z = t . Assuming that the three twistors z“ are h e l d fixed (i.e. we h a v e a given internal structure) then simply from the algebraic structure of

i A i

А^ and A.. one is led to the complex line and internal spin-tensor.

A t the p resent we m a k e no attempt at a physical interpretation of the " o r i g i n " , the internal c enter of mass line and internal spin tensor o t her than to say that the y are to represent the internal structure of the three twistor particle. A different choice of the three twistors obtained from the 1ST w o u l d r e p r esent a d i f ­ ferent particle h aving a d i f f erent origin, w o r ld-line and spin tensor but with the same k i n e m a t i c values.

I

are S U (3) generators.

This is true in particular on the center of mass world-line.

However an examination of (5.5) shows that t1 is a function of

ct —i л i ct *

the Z^ and Z^ and the functional dependence of A., on the Z s

is changed. From (5.7) we see that the St generate t r a n s f ormations which k e e p the mass line fixed and by d i r e c t calculation we have

For a fixed numerical value of t1 , At

10

zr

a d

d ,

1 e ^ k Za I . zl

2 3 aß к

\ e . V Ia0 Zk

2 i]k a ß

(3.9)

We identify the translation generators d1 and cL as the components of the complex internal m o m e n t u m of a particle in the u n i t a r y space. The generators A, i of SU(3) rotations constitute the total

к «

internal angular momentum of the particle. U n d e r a translation

^ i i i

z = z + t , the total ang u l a r m o m e n t u m and В changes as [ C f . Eq. (3.7)]

ii . i A. =

к к + d1 t,

к + t1 d. - к

1, ,r

3 (d 4 + * Г a r )5k' (3.10)

/S

В = В

+ § < d l 4 + t1

a i> (3.11)

and the complex momentum remains invariant. The b e h a viour of dynamical quantities with respect to SU(3) rotations is implicit in our tensor notation.

For future use we w i s h to spell out the m e a n i n g of the trans­

formation (3.10); the A^ are the generating functions of the isometries in unitary space with the origin as a fixed p o i n t

Л A

while the A, are the generating functions of isometries keeping

1 i ^ i

the p o int z = t fixed. In this m a n n e r A^ can b e thought of as a t ensor field on unitary space with t* = z1 .

T o summarize this section, there are three spaces w h i c h play fundamental roles here. T h e first is twistor space T on w h i c h we take three points (Z^, Z^, Z^) to spe c i f y our m a s s i v e system.

(An a l t ernative and sometimes n e c e s s a r y point of view w o u l d be to choose a twistor from each of three different copies of T ) . The IST, i.e. ISU (3) , acts on these three points preserving the kinematic twistor. Since p a i r s of twistors d e f i n e points of com­

plex M i n k owski space, the three twistors define three p o i n t s in complex M i n k o w s k i space w h i c h are m o v e d about b y the 1ST. The

3

third space is the u n i t a r y space U h aving the

isometry group ISU(3). T h e generators of ISU(3) define a

A1 i i - i - C . = C . + d t . + t d ..

3 3 1 3 ^(5.2)

Cj can be d e c o mposed into the four parts

C* = a d1 d. + a1 d. + d1 a. - S1

3 3 3 3 m 2 3 ^(5.3)

w h ere

S1 d11 = s1 d. = sj

3 3 ^{i i}

ai

^i = d1

I — j_

’k " V

(5.4)

a = (-%)2 C1 d 3 d = A j B, a1 = C1 d 3 - a d 1 ,

in 3 m m ^

a . = C1 d . - ad . .

3 2 3 i 3

J m J

“*i к (Note that the Hermitian adjoint is def i n e d by = S ^ ) .

If we now insert (5.3) into (5.2) with . i i , . ji r

t = - a + i t d , t. a . -- ix d

l : (5.5)

we obtain

Cj = (a i(r - x) ) d1 d - — S1

3 m 3

(5.6) Thus along the internal c enter of m a s s line d e f i n e d by (5.5), Л .

ct has only the first and last terms of its canonical decomposition.

The Hermitian t ensor S1 called the internal spin-tensor can be

3 -í I дЬ p l p — d

explicitly s olved for and written = e e jcd ^A a+2B ^ a ^ b ^ * Fr o m its derivation or by direct calculation it is seen that is

(essentially) the invariant part of c]: under translations i.e.

[Sj,d3 ] = 0. [sj,djj = C (5.7)

and that it has a canonical decomposition

11

i i

v e c t o r field d and t e n s o r field A ^ on the uni t a r y space in a

m a n n e r analogous to the w a y the Lorentz g r oup defines the momentum and angular m o m e n t u m fields on M i n k owski space. In section 5 we w i l l show h o w Aj and d* define an internal c e n t e r of mass line

and internal spin in ana l o g y to the way angular m o m e n t u m and m o m e n t u m determine a c e n t e r of m a s s and spin.

4. I N T E R N A L T R A N S L A T I O N S

We now explore the effect of internal translations on the structure of the three-twistor particle. C o n s i d e r first a trans­

l ation ,

A iz i

z + ti

w i t h t1 = (0,0,Л), by the complex amount A in the z J direction of the u nitary space. In the t wistor realization, Eq. (3.7), the m o m e n t u m p a r t s of the c o n stituent twistors [tt^a , where

Z i = ^ чa ') ] remain uneffected,

and

71 iA' 71 iA' (4.1)

(4.2)

W e compare the change in the w p a rts with the effect of a trans­

lation in Minkowski space-time

~AA ’ AA' A A ’

x = x + a

(4.3)

T h i s latter gives

ш~A = 0) + iáA A ’

I t ^{= TT}_{A ’} ^(4.4)

In addition to its M i n k owski space structure (momentum, mass, angular momentum, center of mass, etc) a t h ree twistor particle has an associated u nitary space structure, n a m e l y a p o i n t (or origin) in unitary space and a complex "internal c enter of mass"

w o r ld-line also in unitary space with a r e l a t e d internal spin tensor (which is the unitary space analogue of the Pauli-Lubanski spin v e c t o r ) .

To see the point structure w e note tha t three t w i s t o r space ct ß has 24 real dimensions (3x4x2) w h ile the k i n e m a t i c twi s t o r A has ten real components (momentum and ang u l a r momentum) and thus

ot ß

the kinematic subspace defined b y A c o n s t a n t is 14 real dimen­

sional. The equivalence classes of points in this s p ace (eight dimensional) defined as those points c o n n e c t e d by SU(3) trans­

formations, i.e. Z'?

= и?

Zj, U

e

SU(3), can be identified wi t h the points of unitary space. The equivalence classes can be

3

p a r a m e t r i z e d by points in C (6 real dimensions) i.e. by the translations Z ' , = z f + A 4^ I Z^, from some arbitrarily chosen

"origin"

z“ .

(Note that by associating this arbitrarily chosen origin wi t h the group identity element, the k i n e m a t i c subspace can be considered as the ISU(3) group space. Note further t h a t if we had considered originally the g r oup I U ( 3 ) , the U(l) part would have an action on the ISU(3) m a n i f o l d w h i c h would not be the action of an ISU(3) element. Neveifheless locally one could du­

plicate the U(l) action by an ISU(3) element. This explains from a group theoretical point of v i e w the r e lationship (3.4) between the U (1) generator and the ISU(3) generator).

In order to u n d e rstand and see the internal c e n t e r of mass line and internal spin tensor we define

and obtain from (3.10) and (3.11) the tran s f o r m a t i o n law under (5.1)

translations z i

+ ti z

12

for any twistor. Consider, in particular, the pai r Z® and Z^.

Choose

A A ’ , ,-A 1A5 -A 2A1.

a = A (7T1 и + it2 ТГ ) (4.5)

w here X is complex. Fro m E q s . (4.4) ,

or using (2.5) ,

~A A .

iX -2A A"

W 1 = + TT TT? 711A'

-A A

iX -1A A' Ш 2 = Ы 2 - TT

4 712A ‘

-A W I =

A j.

Ш д + iX M 12

-A

*2 -A

Ш 1 = A

“ i " iX M 12

-A

*1

(4.6)

C h o o s i n g the parameter X to be

'iA/M12 (4.7)

we have the following result:

The special internal translation with (t1 ) = (0,0,Л) shifts the twistors z” and p a r allelly along the time-like c enter of mass line of the massive two-twistor subsystem they represent

and leaves the twistor Z^ invariant. A similar result is obtained for translations along the other two axes.

Consider nex t the space-time translation in the direction of the total fou r - m o m e n t u m of the system:

A A1

a ,-lA

T (тг

^-2ATT -ЗА

TT A ' .

71

3 }

^(4.8)

Is this possibly an internal translation? From Eqs. (4.4) we obtain

~ A A

i x ( M i2 -2A

M 13 - З А

Ш 1 = U l + IT -f TT

~ A A

i x ( M21 -1A

M 23 - З А

W 2 = Ш 2 + IT + IT

~ A ы з

= A

3 + i x ( M 3 i -1A

TT -f

M 32 -2A TT

or, in matrix f o r m using (3.1),

This defines an internal translation wit h (t1 ) = ix(d1,d2,d3 ) .

(4.9)

(4.11)

What w e have, is a translation in the direction of the unitary m o m e n t u m d'*' by the amount т . The significance of this result lies

in the fact that it establishes a m a p from the time development of t h e system in space-time to the d e v e l o p m e n t in unitary space p a r a l l e l to the unitary momentum.»

T o conclude this s e c t i o n we o b s e r v e that translations of the form (4.11) exhaust t h e unitary t r anslations which can be pictured equivalently as space-time translations. The reason for this is that space-time translations not along the centre-of-mass line of the s y s t e m alter t h e angular momentum. However the angular m o m e n t u m is p r e s e r v e d by all internal transformations since these prese r v e the k i n e matic twistor.

Kiadja a Központi Fizikai Kutató Intézet Felelős kiadó: Szegő Károly

Szakmai lektor: Tóth Kálmán Nyelvi lektor: Révai János Gépelte: Balezer Györgyné

Példányszám: 375 Törzsszám: 81-619 Készült a KFKI sokszorosító üzemében Felelős vezető: Nagy Károly

Budapest, 1981. november hó