PHYSICAL REVIEW D VOLUME 13, NUMBER 2

Sca&r effective Lagran~a~ in tie Sitter space*

15 JANUARY 1976

J. S. Docker and Raymond CritchleyDepartment of Theoretical Physics, The University, Manchester, England

(Received 10 July 1975)

Scalar field theory is considered in a de Sitter-space background and the effective Lagrangian calculated

exactly using the proper-time regularization technique. The imaginary part vanishes and this result is shown

to hold for all spins on quite general grounds.

I. INTRODUCTION

In this paper we calculate the effective Lagran-gian due to scalar particles in a de Sitter space-time. It is shown to be real both by explicit cal-culation and by general eigenvalue arguments.A similar result is proved to hold for higher-spin particles.

Our general method is the proper-time regular-ization technique employed by Schwinger' togetherwith a contour way of avoiding singularities. Othermethods are brief 1.y mentioned.

The interest in the present system lies in itsexactly solvable nature, and it could provide aplace to try out the various renormalization tech-niques.

Section II sets out the basic formalism and Sec.III deals with the propagator and Green's functionon de Sitter space. These are calculated in arapid and, we hope, illuminating way. In Sec. IVwe evaluate the effective Lagrangian Z('~ and showthat its imaginary part vanishes. The real partis computed and three infinite terms extractedleaving the radiative correction contribution asa principal-part integral.

Section V contains a discussion of why we ex-pect, mathematically, ImZ" l to be zero (for anyspin} while in Sec. VI we explore briefly the Fock-space content of the theory and again indicatewhy we obtain ImS(i) 0

II. EFFECTIVE LAGRANGIANS

According to Schwinger' the probability densityof pair production in an external field is deter-mined by the imaginary part of the effective La-grangian density, 2 ', or, equivalently, by thereal part of the singl. e-loop vacuum-to-vacuumamplitude, as discussed first by Feynman. ' Stan-dard theory' ' yields the formal expression (werestrict ourselves to scalar particles} for W '"= fZ"'dx,

= ——ln DetGF2

Z= ——Tr lnG2 F7

where GF is the complete Feynman Green's func-tion. We use in (1}the conventional formalityof treating GF as a space-time matrix.

In his elegant paper' Schwinger calculates W' "for the case of a constant and uniform electro-magnetic field. Now, strictly speaking, thissituation violates the boundary conditions usuallyassumed when Eq. (1) is derived, but not so asto invalidate the final result. This is broughtout in the calculations of Nikishov' and Narozhnyiand Nikishov' in which the boundary conditionsare explicitly considered.

For a constant electric field we can choosegauges such that either A is time-dependent and

Ao is zero or A is zero and&0 is space-dependent.In the first case pair creation would be consideredas a time-dependent perturbation effect, whereasin the second it would be thought of as barrierpenetration.

This choice of gauges in electromagnetism cor-responds to a choice of coordinates in generalrelativity. Thus for de Sitter space we can findboth static and nonstatic coordinate systems. Inthe former there should exist a natural quanti-zation procedure, ' but the present paper will usethe nonstatic, Robertson-Walker form of themetric. It would be interesting to do the calcu-lation using the static description. Nikishov' hasdiscussed pair creation in the static gauge forthe constant electric fiel.d. We should, however,be careful to remember that a particular coordi-nate system may not cover the whole manifold.

The fact that in the nonstatic gauge of the con-stant el.ectric field a clear-cut division into posi-tive- and negative-frequency states is not pos-sible, and that the field does not vanish at in-finity, but yet the quantity calculated using (1}is usually taken to be meaningful gives us hopein applying (1) to the case of de Sitter space.

We choose de Sitter space because it is wellstudied, has some physical relevance, and isone analog of a constant electromagnetic field.At one level R»p, is the analog of E„„and so+p &~ p

= 0 trans lates into A p pp& (~,——0, which means

that the space-time manifold, St, is a symmetricspace, of which de Sitter space is a particularly

13 224

SCALAR EFFECTIVE LAGRANGIAN IN DE SITTER SPACE 225

well-known example.At this level of analogy, spin, in general rel-

ativity, plays the passive role that charge playsin electromagnetism in the sense that it is thespin-curvature coupling that knocks a particleoff a geodesic. Further, a constant electric fieldpossesses a direction whereas a space of constantcurvature is isotropic, and so de Sitter spacemay not be a good physical analog of a constantelectric field, although we might expect to findan exact solution even if " turns out to be real.

Since we are now dealing with a curved space-time some slight formal changes are necessaryto the standard theory of effective Lagrangians.These have been given by DeWitt" and we re-state them in order to fix our notation and con-ventions which differ a little from his.

The Green's function G satisfies

(Cl"'+ m2)G(x", x') = G(x", x')„„." + m G(x", x')

= 5(x", x'), (2)

where 5(x", x') is the scalar & function

5(x", x') =(-g") ~'5(x" -x')(-g') ~'.

Following Schwinger and DeWitt we now write

G(x", x') =(x"j Gi x')

with covariant orthogonality,

(x"~x') =5(x",x'),

so that (2) reads

(H+ m2)G = 1

with

(3)

G, (»-, *)=, I d.&*-,.~*,0) 8--",0

(4)

where ( x", r( x', 0) is a "quantum-mechanical"propagator which satisfies the "Schrddinger equa-tion"

~ af —&x", r~x, o& Cr"&x", ~~x', 0&=f5(~)5(x",x')

and has the covariant initial condition

(5)

(x"iHi x'& =Z,"5(x",x'),

where &," is the scalar Laplace-Beltrami operator

&,w-=Alai„"=(-g) 'e„[(-g) 'g"'e. 9]Hence, inverting (3) with m2-m2 —i0 and using

the fifth parameter formalism,

( x", Oi x', 0) = 5(x", x').

We assume that it is expression (4) that goes intothe relevant formula for ".

The action, Wi'~, equals f~Z~'~ dx where dxis the invariant volume element on Ãt so that, forus, &"'(x) is a scalar function. Standard theory"'then gives

2"'(x) = —— —(x 7~x 0) e 'i di t NE2T

2 0

where the coincidence limit, lim, ~ „(x",7~ x', 0)is defined as the average of the limits obtainedby taking x" —x' to be timelike and spacelike inturn. (This may necessitate choosing differentT contours in the two cases in order to ensureconvergence. )

Schwinger's calculation amounts to deriving aclosed form for ( x", r

~

x', 0) and then substitutinginto (6). This is the path we shall follow.

III. de SITTER SPACE

It is clear that Eq. (5) will be solvable exactlymost easily when ~ possesses some symmetry.The more the better, so to speak. Particularlyimportant are homogeneous spaces of which sym-metric spaces form a subset. de Sitter spaceis a homogeneous space, SO(1, 4)/SO(1, 3), andthe solution of (5) reduces to a group-theoreticalproblem, essentially that of finding the spectrumof those representations of SO(1, 4) that havenonzero Plancherel measure. However, in thepresent work we shal. l not make use of any ex-plicit group theory ideas, but will just solve thedifferential equations "as they come. " In factwe can simply make use of the considerableamount of work that has already been performedon de Sitter space. The calculations of Gutzwiller, 'Tagirov, ' GOheniau and Schlombl. ond, "and ofBorner and Durr" are particularly relevant. Afew basic facts follow.

The geodesic distance between x" and x' isdenoted by s(x", x') and can be real, zero, orimaginary, corresponding to timelike, lightlike,and spacelike intervals, respectively. Consideringde Sitter space as a four-dimensional sphere $4of radius -a' in a five-dimensional pseudo-Eu-clidean space R', (signature 1, —1, —1, —1, —1)leads to the introduction of p= cosh(s/a) as a moreconvenient variable. Denoting the coordinatesof the R', by $, (a = 0, 1, 2, 3, 4), we have the fol-lowing relations:

226 J. S. DOWKER AND RAYMOND CRITCHLE Y 13

p(x", x') = 1+ ~g n"($." —h!)(h&' —hn), n=

1= —1+ ~, n"(5!'+(!)(&l'+$h)

Locally, the curvature of S', is given by A„,~= ~»R(g„g„, -g„~g„,) with R = —12/a'. Furtherreferences on the geometry of de Sitter spacecan be found in the above-mentioned papers. Weshould also refer to the book of Hawking andEllis" and that of Schrodinger. '~

dx —5(p- ]).dp

To obtain ( x", ~l x', 0) we can continue this ex-pression to P& 1 and multiply by i.

More precisely, what we are saying is that theGreen's function GP($", $') and propagator( (", 7'l $', 0) are the boundary values, on the pos-

(7)

A. The propagator &x',r g',0) and GF

Because the Hamiltonian is invariant under the

group of motions of de Sitter space, the biscalarpropagator ( x", el x', 0) will depend on x" andx' through only the SO(1, 4) invariants constructedfrom x" and x'. There is only one such invariant,P(x", x'), S', being a harmonic space. The solutionof (5) is formally given by

&x", rlx', 0}=s-"&" 5(x", x')

in which Q"2 can be replaced by its radial part,Q/I 2

0

1 d2 d(p' —1) + 4p-a' dp' dp

if we give 5(x", x') as a function of p. Life ismade a little complicated by the pseudo-Euclideanmetric on Bg, but in our case we can proceed byusing a continuation from the Eucl, idean spaceORE = S„ the four-dimensional sphere, corres-ponding to BR. For this space the invariant 6

function takes the form

&.(x",x')= 4~. —„&(P-l), IPI-. 11 d

and so the Euclidean quantum-mechanical prop-agator is

(x", Tl x', 0)s= ~ exp ——2 (p —1) +4p—1

itive real axis, of functions, G, (z)0, ago, $", $')and (z)0, $", rl s$,', $', 0) „which are regularin the lower-half z plane. We are here continuingfrom the Euclidean signatured embedding space8, to the Lorentzian signatured space R', by con-tinuing in the "timelike" coordinate $0, just asin ordinary quantum field theory.

On the negative real axis they are the boundaryvalues of functions regular in the upper-half zplane. This is equivalent to requiring G~, as afunction of P, to be the boundary value of an an-alytic function from the lower-half P plane.

Further comments on this continuation procedureand why it produces what it does will be givenlater.

The structure of (7} shows the same propertyas the corresponding one in flat space, "that byapplication of the operator d/dp (- —d/sds) wecan step up the dimension of SEE by 2. We needonly solve for S' in order to find the propagatoron S2". This follows more precisely from therelation

d2 d d(p —1) ~ +Np

d2 d= —(p'-1) +(Ã-2) p ——(N-2) .dp dp dP

We here recognize the differential operatorfor Legendre functions (of course) and so we ex-pand 5(P —1) as

5(P-1) = g (n+-') &.(P).n=0

Incidentally the same calculation can be per-formed for anti-de Sitter space, H~~ =SO(2, 3)/SO(2, 2). Then we would continue from the neg-ative-definite space H„ for which 5(P- 1}is givenby

h)P —l)=f dhhth hPP (P), hP-).0

SCALAR EFFECTIVE LAGRANGIAN IN DE SITTER SPACE 227

From (7) we have

Ks(P T) =( x", T(x', 9)@—4~ q d

(n+ q}exP i —2[&4 —(n+ ~) ] P„(P).4m'a4 dp a' (9)

There are two ways of proceeding, both of which are instructive. Firstly and more rapidly, if we aresimply interested in Ge(x", x') we can immediately employ (4) to write

c

Ge(x", x') = ~ 427 d 1, 1 2dTe ' ' — (n+ &) exp —i —,(n+-, )' P„(p}

dp 0 ~ a

1 d ~ (- 1)"(2n+ 1)Sv'a' dp ~ (n+-')'+m'a'

ccc'ecch( ) C( c"' ' cc') '

Dougall's formula has been used in the last step.If we multiply by i and set P- P -i0 as our continuation beyond the branch point at P = 1 we find

Tagirov's" expression for G~(x", x'),

a'

cosh(mrna

j dP 2'™a

8)(a'cosh(mrna) dp—8(p —1)[ie™P ~„, , (p) —(2/w—) cosh(mva)Q ~„,.—,(p)]] .

The second method is more longwinded but has the advantage of providing an insight into the expressionfor Ke(p, ~). The expansion (9) is transformed into its "dual" sum involving classical paths. To thisend a Mehler-Dirichlet formula is brought in. This gives

~ ((' ')= —~() ~ .gc"C —*(c—("~ l)') — ((-cocC') ' —coc( ~ l)C c(C,

e

with p=cos6 (a6) is the geodesic distance on S ). Then

i9~ -u'2Ke(p, r) = —,exp, — (p —cosy) ' —8, —,—,di().

Bm a4 4a' dp 8+ 27'' M

A 6)-function transformation takes us to the classical paths form

(2vi)~' y 2 1 d "" " ia'Ke(p, T) = — e'" @ ', — (p —cosy) ' g (-I)'(y+2vk) exp —(y+2vk)' dy,

which is, basically, an expression for the propagator on S'=SO(5)/SO(4) in terms of the propagator onSO(5). The latter is given exactly by its (Iuasiclassical expression" and the integration in (ll) corres-ponds to removing the SO(4) divisor.

The relation between (9}and (11) is a duality one and corresponds to the well-known relation between(9 and ( functions. " We refer the interested reader to the work of McKean" for comments on this topic.

For completeness we give the expression for the propagator on H',

(- 2xi )~', ,2 1 d 2a 2

Kz(p, r) = e'( )' ' —„I (coshi —p) ~~ exp i fdf .16@a v+2 dp Jy, 47 (12)

Expressions like (11) and (12) have been derivedby Perrin 0 and McKean for the cases of S andH', respectively. They are well suited to ex-pansions valid for small T If, indeed, Kz(.1, 7)

is so expanded we have checked that the resultscoincide with the general ones of DeWitt. '

Returning to the Green's function, Ge(x", x'),(11) is substituted into (4) and the r integration per-

228 J. S. DO%'KE R AND RA YMOND C HITCH LE Y

formed to yield a MacDonald-Kelvin function

Kg, (ma~ y+2wk~ ). The summation over k canbe done and what emerges is just a Mehler-Dirichlet integral for P g„, , (-—P),

p g„,„,(--p)

(p —cost )' cosh[ma(w —t }]dt,

and the final answer is as before.

s 2. d7lim +lim

~K, v'e

p j. p~j. 0 T

Then we find from (9)

(13)

~ (n+-,')n(n+1)e "' "'l"'~"32v'a';

)—

2~ dVxeT

(14)

Already, in this form, we can easily see thatIm ' is zero. For convergence reasons the 7contour runs just beneath the real axis and socan be rotated to along the negative imaginaryaxis 7- —i&, whence

—e 'Q n(n+-,')(n+1}"d7 -2,

327 Q

X8 -(7/a2)(n+g2)2

(15)

IV. THE EFFECTIVE LAGRANGIAN, RENORMALIZATION

Without more ado we now substitute (x", 7'( x', 0)=i ( x", 7'1 x', 0)e into Eq. (6) for 2 ' . This willgive us the coincidence limit arrived at from thespacelike direction. The timelike contributionwill be calculated later. It was not a Priori ob-vious to the authors that the two limits would givethe same contribution, and so we separate them

formally,

g(i) g(x) + g (i)

which is real. This fact can be traced back tothe compactness of the Euclidean space S,.

Expression (15) contains divergences which areremoved by a renormalization process. To in-vestigate this the classical paths form, (11), isnecessary, since it is the behavior of K(l, 7)near 7 =0 that is important.

If (11) times a factor of i is substituted into

(13) for 8"' it can be seen that a technical dif-ficulty arises when the limit P-1 is taken. Takendirectly this limit produces ~ —. Such behavioris exactly that encountered by Hadamard (Ref. 16,p.134). Hadamard suggests the use of a contour, and

this is the course we shall adopt for simplicity in ourcalculation. We firstly write (11)in contour form andthen perform the differentiation to obtain for K, =iK~,

(p )(2wx), (9)4),to2 164m'a

sa(t:»O' P) 0 «P P ) 4'P

(16)

where the contour C runs just above the real axis.The signature is such that (cos&p —P)'i'

i(P —-cos&p)'t' on the top side of the cut from6I to 2m —e.

An alternative procedure of extracting the "fin-ite part" is that due to Riesz. ' This amountsto replacing the —,

' power in (16) by (v —I)/2 and

corresponds to continuing in the dimension of themanifold 0R or VR~. Since we have the basic in-gredients we might as well make this statementmore precise now, before carrying on with (16}.

The ~ function on S„ is

d (~-2)/25e =(2wa') "t' („,)„6(P-1).

If the same path that led to (11) from ('l) is pur-sued we find for the Euclidean quantum-mechan-ical propagator on S„

~pi a e ''t" 'l ' 4' ZQ

Ke(p, 7, n) =- (,„„&,Pf, (p —cosy)h "' 'g(- I)'(q+2wk) exp (y+2wk)' dp,2wa' "" 47

(17)

where the symbol Pf stands for Hadamard's "Partie finie. "" Riesz's method of obtaining the Pf is toconsider a function Ke(P, 7, v) defined by (17) in a region of the v plane where the integral is finite(Rev& 2) and then to continue this function analytically to the desired values of v, =n.

Further considerations along these lines would ultimately bring us to the dimensional regularizationtechnique. This has been used by Candelas and Raine~ in a parallel calculation. It will be considered,along with other methods of renormalization, in a further paper, where we shall also deal with the re-normalization of the energy-momentum tensor.

Returning to (16), we set p= 1 and substitute in (13) for C"' to get

SCALAR EFFECTIVE LAGRANGIAN IN DE SITTER SPACE 229

(, ) ~i~ " d7 y exp[ —i(m'~ —(p'a '/4r)]2wa ao T sin'(-,' q ) (18)

The divergences arise from the ('p-0 part of the integral, and so we expand ((l/sin'(~(p) as

(psin '(2(p) = —,+S((p),

where

&(W)=g d. W*"=& ~24

S*~ &(W)240

and is nonsingular at p = 0. Then we find

X//2

Jt(psin '(2y)e'~ ' "dp= —8a + S((p)e'~ ' 4'd(p,

C T C(19)

where the contour C, on the right, now consists of the real p axis apart from small clockwise indentationsaround ('p = 2nw (n = +1, s2, ~ ~ ~ ), where T((p) has poles.

We substitute (19) into (18) to get

vsm g2 2

p (l) e-'lilt 8' S (q) v a /4T dq29 3 y5/2

The term involving & ((p) will yield a finite answer and so we perform the p integration explicitly for thefirst two terms of the expansion of S (('p) using

+2n sP a /rdl+aP SQ SQ

This gives for '

(y) 1 dT j 27 S dT ' 2 17 "dT -'=2Z —e '+ —e ' —e '64/ T 2 P g T2 240/ 27P2g4

v'g p 2 2Z'((p) e( & / d(p2'm'g I T5/2 (20)

The formula

d' 1san'z dz' sxnz

allows us to find the explicit form of &((p),

. +8 1 - 17

(y —2nw) ((p —2nw) 240

= 0((('),

where the P' means that the term n = 0 is not included. Then we can use

( 1)~+'(9) —2nw+ie) ~ = ~, , P —iw8((p —2nw)

) dj' ' y —2nn

to see that the i)-function contributions to the final term in (20) cancel, positive n with negative n, due to the sym-metry of the integrand. This is, of course, just our previous result that the imaginary part of " is zero. Thereal part, apart from the renormalization terms, comes from the principal-part contribution. Since p isreal for this, we can perform the & integration to give a Hankel function K,/, (mai()) and we finally obtainthe expression for Z ',

230 J. S. DOWKER AND RAYMOND CRITCHLEY

1"" m' m' 17

64m „' u 4g'u' 480a &

m" d' 1 1 P 7, 1 1

p 1+4 1+ e64(('a', dp' 2 sin(~ p} C( 24 2' & 360 p may (21}

where the first integral contains three terms which correspond more or less to the three renormalizationterms of DeWitt."This is seen by noting that R 1/a', and e.g. , R„„(„R""~'-I/a'.

We now wish to show that the limit taken from the timelike side produces the same answer as (21). Inlieu of arapid way of proving this we have to resort to actual computation. Unfortunately this is not quiteas straightforward as the spacelike calculation just given. What we have to do is to invert E(I. (4} for(&",&~ x', 0), —= K(t(, 7'}, with Gr given by (10), and then substitute into (13) for &(". Thus, firstly, we have

'o-fCi 8 ((()K(P, o) = — Ge(p, t(') e'" '

du, ', c)0271' -~- fc

(22}

and then, quite generally,

1 . . ] 2 2lim + lim —

i

— G (p t('}e4 p + 277 „ a) &-~-$C

(23)

The 0 integration is now performed using the well-known generalized function Fourier transform,

e""dx= i((e(y)f 1«OQ X

(24)

We find

~ «((c(«5C

2m -fc

dp, G~, p). (25)

There is a slight difficulty in that Gr(P, m') can have poles on the negative m' axis and a branch point atm'= 0, but expression (25} can be justified by splitting the complex t(' integration into a pole + principal-part formperfor, ming the integration by (24), and then recombinina the terms into contour integrals.

We can now rotate the m —ic to -~ —ic contour into coincidence with the m —ic to ~ —ic one and we ob-tain the general formula

p(0

lim+ lim G (P t(')dt('4 P z P 1+

(26)

We did not use this formula to calculate Z"' earlier, but it reproduces expression (21) if the formula forthe P (1 region of Gr from (10) is substituted. In fact we do not need it for one part of the expression forG~ for P & 1. Thus, for P & 1, we have

Gr(t(, m2) = ~~ —Q, ~2„.—(p)+ 8, (1 —tanhm(((() P,(„( (—p) . -i d 1 d

The first term here can be directly written in the form of (6) using the formula

(27)

i ' ' " d&,'" t exp[- i(m'&+ t'a'/4&)]@ (t2+((((~ ' 8(T r' ' (cosht —P)((2(pl=a— dt

which is easily proved by doing the r integration to give a Hankel function H, &,(mat) The result is .just astandard Mehler-Dirichlet-type integral for Q y/2

Then the contribution to ~' from this particular term in G~ is found to be

~ 00 4 2 ~ OO

mat(28)

SCALAR E FFECTIVE LAGRANGIAN IN DE SITTER SPACE 231

The final term has an imaginary part coming fromthe poles at t = 2»»n (n =1,2, . . .) which is easilyevaluated using a previously given formula for(t+ie) ~ (the t contour runs just above the realaxis). We find

1 dxx(1+4x')(I+ tanh]»x) .2 ga (29)

The real part of (28) is then identical to 2»») of(21).

The imaginary part, (29}, is, in fact, canceledby the contribution of the second term in Gz, (27).This contribution is finite and so the limit p-1can be taken immediately in (26), for this term.(In general this is not possible and some regular-ization method has to be employed before the limitis taken. The method we have used in the body ofthis paper is that of Schwinger. ') The calculationis rapid and yields precisely the negative of ex-pression (29}, as required. Presumably this canbe proved more directly.

The conclusion is that the effective Lagrangian2[») is real and is given by twice expression (21).

U. EIGENFUNCTION EXPANSIONS, EIGENUALUES

Tagirov'o obtained expression (10) for theGreen's function G~ using an eigenfunction expan-sion and an addition formula for Legendre func-tions. In this section we wish to present an eigen-function approach to the calculation of K and TrK.

We intend to work on S, (we could treat S,„) andwe can write the propagator in the general form

ds' =a'(cos7') '(dr' —do') ——~ T -—2 2

= dt ' —a' cosh'(t/a) dc', —~ & t « (32)

where d02 is the interval on the unit, three-dimen-sional sphere, S'. The time coordinates are re-lated by

where the extra factor of i comes from the con-tinuation from%ad to 3L Equation (15) is an ex-ample of this formula with d, =

8 l(f+ I)(2l + 1).(This result can easily be generalized to n dimen-sions. } Clearly the validity of this continuationprocedure will place some restrictions, as yetunspecified, on the manifolds that we can treatin this way.

It is easy to see from (31) that we can rotatethe T contour by -»»/2, and hence prove that 2 ']

is real, provided that Ap+~&0. This will certain-ly be the case if JpE is compact because from verygeneral theorems t].o~ 0. (Remember we have,somewhat inconveniently, chosen 3' to have anegative-definite metric. )

Thus, for those spaces for which it is correctto continue in the way we have from compact, orclosed, spaces, the effective Lagrangian is real.W'e defer a discussion of the continuation process.

It is instructive to rederive Eq. (9}for Kz on S,using the eigenfunction expansion. However, wedo not wish to repeat Tagirov's calculation and soproceed in the following way.

A coordinate system that covers the entirede Sitter space is

K,(x",x', r}= P e-"»' 4»(x"){[»'(x'), (30) c [ oshc(t/ c)o] s', tso( ——)

=e

d,e -'"r'

where l now labels the eigenvalue rather than theeigenfunction and d, is the corresponding degen-eracy factor.

Thus for g '] we will have, from (6),

g(1) 1 d e -j}).&T -jm TdT 2 32

&t 8&2a4, (31)

where the A.; and the t|)j are the eigenvalues andnormalized eigenfunctions of Uz' on Ifz (here S,),

UE g; =A.;P.UE' is formally self-adjoint on%E and we canassume the real eigenvalues A, j to be arranged in

nondecreasing order, Ap «A. , «A2 « ~ ~ ~ «Aj « ~ ~ ~,the equal signs allowing for degeneracies.

Taking the trace of KE we arrive at the textbookquantity

Zz(r) -=(TrKz}(r) = g e

tan r = sinh(t/a) .

The eigenfunctions of the Klein-Gordon equationin de Sitter space have been discussed by manypeople. Similarly the eigenfunction problem onspheres, S„, has been virtually exhausted. "'"To add something new we therefore consider notjust the scalar equation but an appropriate gen-eralization to arbitrary spin.

If we take the standard Dirac-Fierz-Paulihigher-spin equations and replace ordinary deriv-atives by covariant ones then inconsistenciesarise unless, as shown first by Buchdahl, thespace has constant curvature. Since, for the mo-ment, we are concerned only with de Sitter spacewe can ignore this particular inconsistencyquestion.

We can describe the propagation of a massivespin-j field by a (2j+1)-component function tt»)

which satisfies a second-order differential equa-tion obtained, if one wishes, by iterating theDirac-Fierz-Pauli first-order equations. This

232 J. S. DOWKER AND RAYMOND CRITCHLEY

is, in the case of $4,

[g" V» V„+m' ——,' 8(j + 1)]cp = 0, ll = ——,

(33)

where V'„ is the covariant derivative operator. Inthe present instance it is very convenient to usethe method of Cartan moving frames: Then (33)reads

~ )——~—(L (-, —'A)(] m ——,)(()+1)ImA d 1~t2 A t A'

=0

where A(t) = z a cosh(t/a). The differential opera-tor L is just the (right) angular momentum opera-tor on S' and the j are the usual angular momentummatrices.

The theory on $4 is obtained from that on $4 bythe replacement t ™it We ca.n see from (32} thatthis produces the metric on S4,

is given by r=t+xa/2.The eigenvalue problem that we have to solve is

then

s' A(it) s 1, +3A(, t} ]|t —A, (,t} [L+(-,'-A(it)) j]'

The angular wave functions are easy to find. Theyare simply the spinor hyperspherical harmonicsthat we have used elsewhere,

Y"~~(g) =(m, g( JLj NM)

(2L + 1) (2J + 1)21r3 &»» (g)(-'» ~}

[ge SU(2)] .

Here J=L+j is a "total angular momentum" andthe Q are the usual SU(2) representation matrices.We thus write

dse' --—[dt ' +a' cos2(t/a) d(7'],

which is virtually in standard geodesic-polar co-ordinate form. The geodesic distance, r, on S,

(4,) = g e„","(t)]]JLjNM)JINN

and obtain the equation for 4,

(34)

d' A d A . 2 1 ~ ~~ 1~2 1 ~2 ~, A 1JNdt2 +3————j' ——(L'+J' —2 j') —+(J' —L') —4 (t) = —&(@»L (t)

with

It +»I=2L+1,

( t - »)=2J+1,

](., = l (l + 1) —j (j + 1}—2,and

l & max(] g/, /] f) .

(35)

Square integrability requires that I be half an inte-ger while the periodicity in the angular variableshad already been ensured through (34). The con-dition on the minimum value of l shows that ~, is

with now A =-,' a cos(t/a). The M and N labels canbe dropped. The solution to this equation isstraightforward and is effected by making thetransformation to x= sin(t/a) = cos(r/a), (0 & r~ wa), which is the natural variable on S„andchanging the normalization of 4 by '4 = cos(t/a)4The resulting equation is identified with the dif-ferential equation for the representation functionsI'~, of SU(2) (e.g. , Vilenkin, 26 p. 138) related toJacobi functions. We find

(l + &)i /2

cosset/'a

j

non-negative and hence we can conclude that theeffective Lagrangian is real. This of course pre-supposes that the higher-spin field theory makessense. For spin one-half there are no problemsand our approach is equivalent to that via the Di-rac equation.

We could now compute the spinor analog of theformula (9) for Ks, using, this time, the eigen-function expansion (30), with the eigenfunctionsgiven by (34), with (35). For example, in dyadicform, the spinor propagator is given by

Ks(t", t'; r) = g e ' ('4 ~~(t"))JLj NM)(JLjNM~4~~(t').tJNNL

We do not wish to take this expression any fur-ther. If we wanted the real part of 2"' we couldproceed as in the earlier sections. That is, thematrix trace would be taken, the point t' would bechosen as the origin, t,', of coordinates on $4(to= &a /2), and the renormalization effected by asuitable Mehler-Dirichlet formula for the P'„, folmm

lowed by an expansion in powers of T.

Since in essence there is nothing in this calcula-tion different from the spin-zero one we do notgive it here. In any case thexe are more efficientmays of proceeding.

13 SCALAR EFFECTIVE LAQRANQIAN IN DE SITTER SPACE 233

G (x", x') =i (O~T(&p(x" )y (x )) (0), (36)

where the vacuum, ~0), is the one appropriate to theparticular mode decomposition,

s,a(Csac sa + sa9 sa)i

where the cp(,',) are as given by Tagirov, for exam-ple, and are considered to be the analogs of planewaves because as a —~ (the flat-space limit) theytend to positive- and negative-frequency exponen-tials. This is the criterion that Chernikov andTagirov used to select the particular vacuum ~0)

from amongst all those invariant under the de Sit-ter group. It seems that analyticity and covariancealso select this vacuum, up to a nonmixing Bogo-liubov transformation.

Equation (36) shows why we find IrnZ'" to bezero. ImZ'" calculated from GF will give the cre-ation rate for particles which are defined as exci-tations corresponding to y",,'. This rate is clearlyzero, and this would be true for a GF constructedas in (36) from a given mode decomposition in anyreasonable (i.e. , Cauchy complete) Riemannianspace. As a general statement, this is, therefore,not very significant. These "particles" may, ormay not, be physical particles which are usually

VI. DISCUSSION

In this paper we are not so much interested in thereal part of 2 ' as in its imaginary part and wehave found this to be zero. The basic mathemati-cal reason for this is that we have obtained de Sit-ter space, and its Qreen's functions, by continua-tion from the Euclidean sphere S,. A similar nullresult will also be true for manifolds SR which canbe obtained by continuation from a compact, orclosed, Euclidean space XE. A sufficient, and

probably necessary, condition for this continuationto be possible is that % should possess a global,timelike Killing vector, i.e. , that there shouldexist a static coordinate system, for then the sim-ple replacement t —it leads to no obvious difficul-ties, at least not in the continuation of the metric.Unfortunately we do not have a general theory ofthe continuation process. In flat space the analy-ticity of the Qreen's functions, Wightman functions,etc. depends on the Poincare invariance of thetheo-ry. In a curved space one would therefore expectthe symmetry group to play a similar role.

It seems reasonable that if the coefficients in adifferential equation are analytic functions of t then

the solution should be also.We now turn to another aspect of this work —its

Fock-space content. The discussions of Chernikovand Tagirov" are relevant here. Specifically the

Gr of (10) is given by the vacuum expectation val-ue

only defined in asymptotic regions.The effective Lagrangian as computed using the

Feynman Qreen's function

. (Oout~T(y(x" )y (x'))~Oin)(0 out/ in)

(37)

yields the creation rate for physical particles.The asymptotic regions, for which Pout) and

~Gin) are the vacuum states, are usually such thatphysical particle states can be recognized as timeexponentials. If no such regions exist then someother criterion must be found for identifying phy-sical particles. In de Sitter space the criterionemployed by Chernikov and Tagirov (see alsoNachtmann") is that of the flat-space limit a- ~mentioned earlier. Apparently there is only oneset of basis functions that tends to the standardexponential form as a tends to infinity —the cp",,'.In this case we have no option but to identify the"in" and "out" Pock spaces so that ~Oout) is pro-portional to ~0 in). This means G„ is the same as

30

It is interesting to compare this situation with

that of the constant electric field E in the non-static gauge. ' There one can find different "in"and "out" bases, both of which tend to the (same)noninteracting exponential basis as E tends tozero. The G„calculated according to (37) agreeswith the standard expressions and the pair-crea-tion rate evaluated via the Bogoliubov transforma-tion amongst the parabolic cylinder functions alsoagrees with Schwinger's expression.

Of course it is necessary here also to have acriterion for selecting physical particles. Thelimit E-0 does not tell us which particular com-binations of parabolic cylinder functions to choosefor the "in" and "out" bases. These are decidedby an appeal to the WKB approximation at t =a ~which in no way involves making E small. ' ltshould be noted that it is G„ that results fromsolving the Qreen's function differential equation.The G„of (36) never enter the theory at all.

An alternative way of finding the appropriate "in"and "out" bases is to solve an auxiliary problemwhich approaches the constant-E case as someadiabatic parameter vanishes and for which Etends to zero at t =~ ~. Such an exactly solublefield exists' and corresponds to replacing a lineartime variation of the potential by a tanh one. Itwould be useful if an exactly soluble gravitationalcase could be found which becomes static at t =+~and which tends to the de Sitter space as some param-eter is made to vanish. However, it ought not tobe necessary to use the flat-space limit. . It shouldbe possible to do everything in the actual curvedspace. As an alternative to the modes y(,", we

might think of using those combinations that tend

234 J. S. DO%'KE R AND RA YMQND C RITC H LE Y 13

to time exponentials at t =+~ as being more appro-priate for asymptotic real particles. In fact theBogoliubov transformation for these modes is givenby Gutzwiller [Ref. 9, Eq. (3.11)]and we find therelative pair-creation probability to be mode-in-dependent and equal to

u = sech'ma~.

A possible difficulty with this mode decom-

position is that lmC~&, calculated from(26), turns outto be negative. This disagrees with the Bogoliubovtransformation result and so is probably due tosome technical error in our application of the gen-eral formulas. For example, it may now be neces-sary to reinstate De%itt's noncausal loops. 4 Onewould like to justify this mode decomposition be-fore investigating such questions in any depth.

*This paper replaces an earlier one in which the mainemphasis was placed on using the mode decompositionmentioned in Sec. VI as giving a negative Imp( ~ ) Thecalculational methods were the same as those of thepresent paper but a sign mistake led us to believe thatImg{ ) was positive. The actual negative value isstill a puzzle to us ~ The Feynman Green's functionwas calculated by eigenfunction summation to be

-[87t'4 cosh{m ax)]

d'd &@-&]2.)sr p+10) '@-&gg, &

—.(-p-& 0)]

and does not possess the usual analyticity properties.The real part is nonzero for spacelike separations and

is also infinite at p = -1 al.though the Green's functionstill appears to satisfy Eq. (2) on S&.

~J. Schwinger, Phys. Rev. 82, 664 (1951).R. P. Feynman, Phys. Rev. 76, 749 (1949); SO, 440(1950).

J. Schwinger, Phys. Rev. S3, 615 (1954). See alsoA. Sal.am and P. T. Matthews, ibid. 90, 690 (1S53).

4B. S. DeWitt, Dynamical Theory of Groups and Fields(Gordon and Breach, New York, 1965).

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~~S. W. Hawking and G. F. R. Ellis, The Large ScaleStructure of Space-Time (Cambridge Univ. Press,New York, 1973).

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S. Minakshishundaram and A. Pleijel, Can. J. Math. 1,320 (1949); A. Selberg, J. Indian Math. Soc. 20, 47(1956); H. Acyl, Bull. Am. Math. Soc. 56, 115 (1950).

9H. P. McKean, Commun. Pure and Appl. Math. 25, 225(1972); H. P. McKean and I. M. Singer, J. Diff. Geome-try 1, 45 (1967).F. Perrin, Ann. Ec. Norm. Super. 45, 1 (1928). Seealso the references in the article by J. S. Dowker, inthe Proceedings of the Conference on Functional Inte-gration and Its Applications, London, 1974, edited byA. M. Arthurs (Clarendon, Oxford, 1975).

2~M. Riesz, Acta Mathematica 81, 1 (1949).+L. Schwartz, Theoric des Distributions (Hermann,

Paris, 1966).2~P. Candelas and D. J. Raine, Phys. Rev. D 12, 965

(1975),L. Robin, Fonctions Spheriques de Legendre et Fonc-tions SPheroidales (Gauthier-Villars, Paris, 1957,1958, 1959), Vol.s. I, II, and III.P. Appell and J. Kamph de Feriet, Fonctions Hyper-geometriques et HypersphA. iques (Gauthier-Villars,Paris, 1926) ~

26N. J. Vilenkin, SPecial Functions and the Theory ofGroup Representations (Am. Math. Soc., Providence,Rhode Island, 1968) ~

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%'e might say that "adiabatic particl. es" [L. Parker andS. A. Ful, ling, Phys. Rev. D 9, 341 (1974)] are the onlyparticles we have.