I1+ NI,;~)\:~) ('IMI.:NT<~ VOL. J, l l A. N. 3 1') ])ic+.ml>rt ' l.()t;7

The de Sitter Model for Elementary Particles with Nonstatic Frame.

R. PRASAD (*)

I ~ s t i l ~ e t H e n r i P o i n c a r d - P a r i s

(ricevuto il 13 Ottobre 1967)

DIRAC (1) has shown that, in a similar manner to the Lorentz groul~, the two de Sitter groups, S0,.I and S03.2, have a spin angular momentum which must be added to the orbital angular momentum to obtain a constant of the motion.

The de Sitter group with positive curvature is usually taken to represent the external space, and in fact (2) represents, with a nonstatic frame, the steady-state expanding universe. SC~raSDL'~G~.R (a) extensively investigated this space, and pointed out that the static frame leads to serious difficulties of interpretation.

The de Sitter group with negative curvature has been proposed (4) as flm internal spacc of elementary particles, and when combined with the external space, this provides a mass formula and classification scheme. The static frame for the internal space, however, also obscures several important features, the most important of which is the oscillating character of this space.

The nonstatic frame, on the other hand, clearly illustrates the basic difference between the two spaces pointed out by WIG~mR (5), namely, that the space with positive curvature is infinite in the t ime dimension and finite in the space dimension, whereas the space with negative curvature is infinite in the space dimension and finite in the time dimension.

Instead of associating the total energy with the time-dependence of the external space (which is equivalent with the mass in the rest frame of the particle), the more natural definition of mass as the eigenvalue of the total Laplace-Beltrami operator is adopted (e). This, however, introduces a new quantum number associated with the external t ime-dependence and which should be related to the spin in a similar manner to the relation between the hypercharge and the isospin.

I t will be found that the factorization method has extensive application in fhe solution of the differential equations, which are all special cases of the Legendre or

(*) A d d r e s s for c o r r e s p o n d e n c e : 70, C h e t w y n d Rot~d, L o n d o n N .W. 5. ( ' ) I ' . A. M. DIRAC: Attn. of ~faih., 36, 657 (1945). (,) j . L . SYNOE: Relativity: The General Theory, Ch. 8 ( A m s t e r d a m , 1964), p. 321. (s) E. SCIIRSDIh'~ER" Expaneling Universes, Ch. 1 ( C a m b r i d g e , 1957), p. 2.% (~) I~. ])RA.~AD: Nuovo Cimenlo, 44 A, 299 (1966). (a) ]~]. p . WIOh'ER: Prec. Nat. Acad. Set., 36, 184: (1950). ( ') E . SCIIRbDI+~'<~ER. Expanding Universes, Ch. 4 ( C a m b r i d g e , 1957) p. 7•.

T I l l ' : I)|'~ S l ' l " l ' E l ~ ) , I (~ ] ) l ' : l , IZ()]~, I " , ] , ] "MI ' )N ' I ' .kH, Y l ' , k I ~ , ' l ' l C l , l " ~ VV[ ' ] ' t [ N I ) \ ' ~ ' l ' . k ' [ ' I ( ! I"[~.A'~[I'i ~)7.*{

(:h.g(,nl)a.u(,r (.(l(talions. SCI[[r162 in Ll.('l (7) w(~rk,d oul the: basic t heo ry of thq~ fm~lotizatioiJ m[,,thod 1o deal w i th very s i in i la r equation.% a n d ]NFELI) a n d H U L l , (8)

h a v e g iven t he so lu t ions to all e q u a t i o n s of t h e fac/ :or izable type . The, p r o b l e m of u. h y d r o g e n a t o m in a h y p c r s p h e r i c a l space c o n s i d e r e d by SCIIRODIN~EIr (a), a n d t h e Keple, r p r o b l e m in a space of c o n s t a n t n e g a t i v e c u r v a t u r e c o n s i d e r e d by INFELD a n d SOKILD 0~ p rov ide r a d i a l e q u a t i o n s w h i c h a re ful ly e q u i v a l e n t w i t h t h o s e d e r i v e d he re w h e n t h e C o u l o m b force is p u t equa l to zero.

In the case of t he i n t e r n a l r ad i a l e q u a t i o n a s i t u a t i o n ar i ses w h i c h is s imi l a r to t h a t of Di rac ' s r e l a t iv i s t i c e q u a t i o n s for t h e e l ec t ron in a s p h e r i c a l field 0~). T h a t is, t h e r e is no so lu t ion wh ich is q u a d r a t i c a l l y i n t c g r a b l e ove r t h e whole r a n g e of t he v a r i a b l e Z (s inh Z----r/R), n a m e l y (0, oo), b u t t h e so lu t i ons w h i c h a re L2(0, 1) a n d L2(1, oo) c an be used to d e t e r m i n e the c igcnva lues of t h e p r o b l e m a c c o r d i n g to t h e m e t h o d of T i t c h m a r s h 0~).

The e x t e r n a l space has t he q u a d r a t i c f o r m

�9 o ,! (~) ~ = ~i + ~, + ~ - ~ i + ~

a n d can be p a r a m e t r i z e d w i t h n o n s t a t i c c o - o r d i n a t e s aM

x~ = R + . s i n g . s i n 0 . cos ~ . c o s h t ,

x 2 = R + ' s i n y . s i n 0 . s in ~ . c o s h t ,

(2) x s = B + . s i n g ' c o s 0 . c o s h t ,

x~ ---- R+" s inh t ,

x 5 -- 1~ ~. cos g. cosh t ,

w h i c h l eads to t he m e t r i c f o rm

(3 ) ds~. = R~. c o s h 2 t ( d z 2 A- s in * z(d0 * + s in 2 0 dq~ ~} - - I ~ 2 dt 2 -i-

Simi la r ly the i n t e r n a l space sat isf ies

2 2 2 X 2 (4 ) - - R ~_ = x~ § x2 -~- x3 - - x4 - -

which can also be p a r a m e t r i z e d

(5)

w i t h n o n s t a t i c (or osc i l la t ing) c o - o r d i n a t e s as fol lows:

x~ ~ R _ ' s i n h Z sin 0 cos qv cos t ,

x 2 = R _ . s i n h ~ s in 0 s in q~ cos t ,

x3 = R _ . s i n h Z cos 0 cos t ,

x 4 ---- R_ . s in t ,

x e ~ R _ . c o s h Z cos t ,

(') •. SCHRODINGER. l~rOC. Roy. Irish Acad., A 4 6 , 9 (1940); A 4 6 , 183 (1941); A 4 7 , 53 (1941). (I) T~ INFELD and T. E. H~TLL" ]~eV. llfod. Phys., 23, 21 (1951). ( ') E. SCnRSDI~OER. Prec. l~oy. Irish Acad., A 46, 183 (1941).

(lo) L. [I~FELD and A. SC~ILD. Phys. Rev., 67, 121 (19~5). (,1) p . A. M. DIR~.C: Prec. Roy. See., A l l 7 , 610 (1928). ( " ) :E. C. TITCHMARSI[: Eigen]uncliort Expansions, P a r t 2, Ch. 19 (Oxford, 1958), p. 247.

974 R. I'RASAD

and this leads to the m(qric form

(6) ds 2_ = R 2_ cos 2 t (d f f + sinh 2 z(d02 + sin 2 0 dcp2))-- R 2_ dt 2 .

Construct ing now the invar iant Laplaee-Bel t rami operator act ing on these two manifolds

1 (7) V -.q ~ , g~ ~ - g ~ + ~,2w= o

gives two equat ions for the external space

(8) { ~ ,,,+:,} sin 21 Z ~Z~ sin2 Z ~ - - -sin2 ~v,(r) = a2 ~r

1 ~ ~ } (9) - - cosh a t -- - - / ,+ /~+ cosh 2 t ~.(t) = at hu,.(t),

~t ~t

where a2 is the separat ion constant:, and two equat ions for the internal space

{ 1 cq ~ / ( / + ~!} sinh2 Z ~ . . . . . ~gi(r) = f12 ~,(r) ,

(10) sinh2 Z Z X sinh2

- - cos 3 t - - - - / , _ R _ cos 2 t ~ ( t ) = f12 ~g~(t), ( l l ) cos i Ot ~t

where f12 is again a separat ion constant . These four equat ions can be reduced to the normal form of the Sturm-Liouvil le

equat ion by means of the following substi tut ions. For tile external equations, by writ ins

1 (12) ~o(r) = ~~

sin Z

1 (13) W,(t) = (cosh t)! ~ , ( t ) ,

one obta ins the result

(14) z ( l + 1) ~ } ( 1) 2 s i n ~ g ~-Z2 ~ , ( r ) = a + - ~ ~ , ( r ) ,

(15) e o s h 2 t

where the no ta t ion adopted is

(16) ~t I= 1 - - ( a + �89

{ Z ; R~ = (b + �89 0/4.

T I I E 1)1'1 ~SI'F'I'J"Ir M'()I)] , ; I . I".~R l,S].l , . '3ll ,]N'l ' . 'd{Y I ' . ' t l r ~,VI'I'II N O N S ' I ' . k T I C J"I 'AMI'] .~)77~

Similarly f,,r the internal st)ae,,, making lh,. subs.ilutions

1 (17) W,(r) = sini~ ~ (~,(r),

1 (18) ~t~(t) - - (cos t) ! r

gives the results

/~(l + 1) ~, = _ (~ +

0} ( ' ) cos ~ ~ ~-~ ~,(t) = d +-~ ~ ( t ) , (20)

where the notation

(2U 2

p_R_ 4

has been used. That the correct form for the eigenvalues has been chosen can be seen by making

the substitutions which transform the equations from the ~cm-changing ~> type to the ~/-changing ~ type. The transformations

(22)

and

(23)

1 sin X = cosh~ '

I .H~ ~.( r ) = (cosh z)i

1 eosh t ~ - - - ,

sin y

1 �9 ~ = isiny)i

give for the external equations

- - . I I . ( t )

(24) { a ( a + 1) ?2 = _ ( i + 1 , cosh* z ~z~}//*(r) ~) //.(r) ,

tb(b-}- I ) a e = (a + (25) [ sin~y ~2}/ / ' ( t ) -~)~//.(t)

and the transformations

1 s inh X = s-inllz '

1 q~(r) -- (sinh z)--i .//~(r)

(26)

(JT(i R. I'I~A~AD

and

(27)

1 eos t

eosh y '

1 q~(t) = (cosh y't) .ll~(t)

g ive for the in t e rna l equa t ions

/c(c+l) e~ i ( ~)2 (28) ~ s inh 2 z ~z2) H,(r) = - - l + 11,0"),

(29) [--cosh' y end-//-I,(t)----- + .

I t is necessary, therefore , to de t e rmine va lues for t lm paramete r s a, b, c, d, bu t first the ques t ion of spin mus t be considered. The values of the orbi ta l angular m o m e n t u m , l, in the rad ia l equa t ions must , of course, be pos i t ive in tegra l inc luding zero. I t wil l be more conven ien t , however , to ut i l ize Di rae ' s j q u a n t u m number , charac te r ized here by p and q for tlle ex t e rna l and in te rna l equa t ions respec t ive ly . I f J is the to t a l angular m o m e n t u m then

{ J = l + � 8 9

(30) J2~o = J(J + 1) ~o = {(ej)z-- �88 ~o,

so tha~

(31) (~j?= (J-4- i-)',

where e is Dirac 's p a r i t y ope ra to r

(32)

and

I e = -t- 1 for s ta tes of pa r i ty ( - - 1) :+t ,

[ e = - 1 for s ta tes of pa r i ty ( - - 1 ) : - t ,

(33) e 2 = 1 .

Ttm resul t ob t a ined is t h a t if

(34)

and if

(35)

{;= j = - - l - - 1

~ j=~_�89 �9

~ j = z ,

so tha t in b o t h cases we can replace l(l+ 1) by j( j + 1) where j can take posi t ive and nega t i ve in tegra l va lues bu t not zero.

T I l E I)l~ SIT'I'I,:P~ 3 [OI )EL [.'{)l~ ELI':MEN'F.kR.Y I*AICI'ICL]ZS *.VITII N4)NSTATIC I. 'RAM] ~: 977

W]lP v~l]ll( 'S f o r I l l ( . pl l l '~i l l lCt(q 's (I. b. c . r a t ( ' ( h ~ t t ~ r n l i l w d b y compar ison wi1]l l lw Log(,ndrc and Gegcnbau r equat ions . Tlw L(,g(.ndr(. cquq | i (m is

d 2 d m2 / u (36) (1 - - t ,2) ~ __ 2/* ~ + n ( n + 1) - - 1 - -p2 ) = 0 .

By mak ing the subs t i tu t ions

(37) I p ~ c ~ [ u = (sin Z ) - i v ,

(38) / p* = - - p ,2, / * '= s inh t ,

t u = (cosh t ) - t v ,

f /* = c o s h z , (39) / u ~ (sinh Z ) - I v ,

]" /* = s i n t , (40)

I u = (cos t ) - t v

in eq. (36), gives respec t ive ly

( s i n 2 z ~ '~ V = n-~- v ,

{ ( (42) c~fa-* t ~tq

(43) (sinh* Z ~-Z* v = - - n q- v ,

(cos't ~ v = n+-~ v

and i t is seen tha t these equa t ions have the same form as eqs. (14), (15), (19}, (20), respect ively. This enables the ident i f ica t ion of m and n for each of these equa t ions .

However , the subs t i tu t ion

(45) u = (/**-- 1)~no~

could also be made in eq. (36) and would resu l t in t he Gegenbauer equa t i on

- - - - 2 ( m + 1)/* + ( n - - m . ) ( n + m + 1) c o = 0 (46) (1 ~ )d~* - - ~

I t is known tha t this equa t ion has only solut ions ana ly t i c at the po in ts /*---- • 1 when ---- n - - m is pos i t ive in tegra l inc luding zero. This condi t ion de te rmines t he four para-

62 - II Nuavo Cimemto A.

978 1~. PRASAI)

meters a, b, c, d, since p and q are known to be positive or negative integral excluding zero. It is now possible to make the identifications of the parameters with the quantum numbers 1, Y, J , X. The following scheme is adopted:

(47)

(48)

(p-}- ~)= • m = - - J or J +1 I .'. a is �89

a = n or - - n - - l = X /

(a + � 8 9 ~ m = X + [ / ... b is integral , b - - ~ or - - n - - 1 J

(49) (q + �89 • or I + 1

e = n or - - n - - l - - - - Y / �89 . ' . e is

(5o) (c + � 8 9 1 7 7 Y + � 8 9

d = n or - - n - - 1 d is integral .

i .

These results could have been obtained in the case of the internal and external time equations and the external radial equation by applying the factorization method directly.

The linearized equations are

(51) (m + �89 ctg Z - - O(m, n, Z)

{(m + ~) ctg z + ~---~} q~(m + 1, n, Z)

=V(n + �89 (~ + �89 ~(., + L n, z),

=V(~ + �89 (m + �89 q)(~, n, z),

(52)

{ ~ ( m + � 8 9 tght--~t- ~ O ( m , m + 1, t) =V/(m+~)~--(n+�89

(m+ �89 t g h t + ~ ~(m+ ~,n+ Ltl = V ( m + �89 + �89 + Lt) ,

(53)

f f / l ( m + � 8 9 c t g h Z - - d - d } ~ ( m , ~ t + 1, Z) =~,/(m+ �89189 j -

( . , + �89 c tghz+ ~ ~(m+L,~+~,z/= V ( ~ + �89 + �89 n + L Z),

(54)

]{ d/ - - ( m + �89 t g t - - ~ q)(m,n,t)

- (m + �89 t,g t + a~ ~ ( ~ + a, n, t)

-- ~ / (n + ~ ) 2 _ (m + ~)~ r + 1, n, t) ,

= V ( n + �89 (m + �89 ,~(.~, n, t).

I t should be noted here that if in place of tile eigenvalue

% /a z__ b 2 = %/ia ~b)ia-+--~

g

TIlE I)E ~I'I"FI~;Ir Mi)l)l.:l, I.'(tb~ F|,I'i~,II,iN'F.',,I.IY I"klUi'ICI,It.,s v,.'lq'li" N )Ns'l'. 't 'l 'l~' I,'I,~AME !'t7[)

lhc first (,qualion of each pair wcr(, u ritlcn with tlm('ig~,nwduc (a b) and (he s('('ond equation of t'a(.h pair w(.re wrilten with lh(' (,i~envalue (a-~ b). then (.a('h pair of equations would bc dir(,etly analogous to 1)irac's lin(~arization of tile I Iamil lonian in the case of the electron in a splwrically symmetric lield (11). It shouhl also be stated that, because of the symnletries occurring in the original equatfons, it is not necessary to use negative values for the parametcrs directly in the linearized equations. It can be seen in eqs. (41). (42), (43), (44). that the replacement

m - , - - - m ,

n - ~ - - n - - I

leaves these equations unchanged. Except in the case of Dirac's ?' quan tum number it would be expected that negative values for the parameters refer to antiparticles.

Equations (51), (52), (54) can be solved directly by the factorization method, and the wave functions are written out explicitly in the l i terature (is). In the case of eq. (53), this could be solved utilizing Darwin's method for solving the Dirac equations (~4), but it is easier to use the general method of Titchmarsh (~2), who has also applied it to this particular equation (15), or the known solutions of the Laplace-Beltrami equation in toroidal co-ordinates (re).

The main interest centres on the eigenvalue spectrum rather than the wave func- tions themselves, so the latter will not be given here. The eigenvalue spectrum for each of the Legcndre or Gegenbauer equations is

(55) 2,,,,, = ( n - - m ) ( n + m -F- 1) ,

where

(56) :r n - - m = O, 1 ,2 .3 . . . . .

Writ ing this out for each equation, with the identification given in eqs. (47), (48), (49), (50), one obtains

(57) ~ = X ( X + 1 ) - - J ( J q - 1),

(58) 2 t = ()iS+ 1) 2 - - ( b + � 8 9

(59) 2 ~ = I ( I ' - 1) - - Y ( Y + 1),

(60) Z, = (d + �89 - ( r + ~ ) ~ .

Since the definition of the total internal and external wave functions is given by

(61) f ~o(r, t) = ~ , ( r ) . ~ g ~ ( t ) .

I ~Pi(r. t) = ~ ( r ) . ~ ( t ) .

( 'D The two ex t e rna l e q u a t i o n s a re comple te ly so lved in 1'. M. 3,IoR~E t~l).d i | . FEe3IiB.X-CH: Melhor o/ Theoretical Physics, Ch. 6 (New York , 1953), p. 732. The i n t e r n a l t i m e e q u a t i o n has so lu t ions of the s a m e fo rm as the e x t e r n a l r ad ia l equa t ion , see E. ( ' . TITCHMARSII: Eigen/unction Expansions, P a r t I ,

Ch. 4 (Oxford, 1958), p. 79. (~) C. G. DARWIN: Prec. Roy. Noc., A 118, 65t (1928). (tJ) E. C. TITCHMARSH: Eigen/unclion Expansions, P a r t I . Ch. 4 (Oxford , 1958), p. 100, also P. 103. (~') P. M. MORSE a n d H. FF, SHBXCI[: 3lethods o! Theorelical Physics, P a r t 2, Ch.10 (New York , 1953),

p. 1302.

980 R. PRASAD

the total eigenvalue spectrum for the external wave field is

9 (62) 2 ~2 2"='~'-- 2"=2'X-- ~'~-- "X-- '~ ' - -J 'J- - l'--~+ + 4

and that for the internal wave field is

(63) 2 2 ' 9 ~ = ~: + ~ = - - ~ ( r § 1 ) ~ - ( r + 1) + ~(~ + ~ t - , _ R_ + - .

4

Previously (4) the connection between the internal and external fields was established by equating the eigenvalue of the total Laplace-Beltrami operator across the boundary, but here the connection is established by requiring that the external eigenvalue spectrum be the same as the internal one, that is

(64)

o r

(65) 2 2 2 2

9 = 2 ( X ~ - �89 (X + 1)--J(J + 1 ) + 2 ( Y + t)a + (Y + t ) - - I (1+ 1 ) - - -

2

I t would seem desirable, therefore, to look for a new external quantum number, X, which would distinguish between particles of given (I, I 7, J), and such that, if at tention is confined to the (J, X)-plane, relations like (47), (56) should be observed; it should also be possible to construct configurations like those observed in the (I, Y)-plane

I should like to thank Prof. J.-P. VIGIER for several helpful suggestions made in connection with this work.