Fortschritte der Physik / Progress of Physics: Band 28, Heft 4 1980 [Reprint 2021 ed.] 9783112522868, 9783112522851

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FORTSCHRITTE DER PHYSIK H E R A U S G E G E B E N IM AUFTRAGE D E R P H Y S I K A L I S C H E N

GESELLSCHAFT

DER DEUTSCHEN D E M O K R A T I S C H E N

REPUBLIK

VON F. KASCHLUHN. A. LÖSCHE, R. R I T S C H L UND R. ROMPE

H E F T 4 • 1980 • 13 A N D 28

A K A D E M I E

- V E R L A G

EVP 1 0 , - M 31728

•

B E R L I N

B E Z U G S M Ü G L1C11K KIT E N Bestellungen sind zu richten — in der D D R an den Poslzeitungsvcrtrieb, an eine Buchhandlung oder an den A K A D E M I E - V E R L A G , DDK - 1 0 0 0 Berlin, Leipziger Straße 3—4 — im sozialistischen Ausland an eine Buchhandlung für fremdsprachige Literatur oder an den zuständigen Postzeitungsvei trieb — in der B R D und Westberlin au eine Buchhandlung oder an die Ausiieferungsstelle K U N S T U N D WISSEN, Erich Bieber, 7000 Stuttgart 1, Wilhelmstraße '1—6 — in Österreich an den Globus-Buchvertricb, 1201 Wien, Höchstädlplatz 3 — in den übrigen westeuropäischen Ländern an eine Buchhandlung oder an die Aualieferungsstelle K U N S T U N D W I S S E N , Erich Bieber GmbH, CH - 8008 Zürich/Schweiz, Dufourstraße 51 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandcl; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, D D R - 7010 Leipzig, Postfach 160. oder an den A K A D E M I E - V E R L A G , D D R - 1080 Berlin, Leipziger Straße 3—4

Zeitschrift „Fortschritte der P h y s i k " Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. Artur LAsche, Prof. Dr. Rudolf Kitsehl, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 1080 Berlin, Leipziger Straße 3 - 4 ; F e m r u f : 2236221 und 2236229;Telex-Nr. 114420; B a n k : Staatsbank der D D R , Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektiou Physik der Humboldt-Universität zu Berlin, DDR - 1040 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: V F B Druckhaus „Maxim Gorki", DDR - 7400 Alteohurg, Carl-von-Osaietzky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik" erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je Band 1 8 0 , - M zuzüglich Versandspeien (Preis für die D D K : 120,— M). Preis je Heft 1 5 , - M (Preis für die D D R : 1 0 , - M). Bestellnummer dieses Heftes: 1027/28/4. (c) 1980 by Akademie-Verlag Berlin. Printed in the German Democratio Republio. AN (EDV) 57618

ISSN 0 0 1 5 - 8 2 0 8 Fortsehritte der Physik 28, 173—199 (1980)

Vacuum Stress-Energy Tensor and Particle Creation in Isotropie Cosmological Models A . A . GRIB, S. G. MAMAYEV, V . M .

MOSTEPANENKO

Department of Theoretical Physics, Leningrad Institute of Precise Mechanics and Optics, Leningrad, 197101, USSR

Summary Quantum theory of quantized scalar and spinor fields in homogeneous isotropic space-time is considered. Particle interpretation of quantized field is given which is based on diagonalization of the Hamiltonian constructed by means of the metrical stress-energy tensor (SET). This interpretation allows to define a normal ordering prescript which respects the conservation property of SET. With the help of Zeldovich-Starobinsky regularization procedure [3] this allows to obtain the total renormalized vacuum expectation values of SET which include both nonlocal terms corresponding to the particle creation and local ones describing the vacuum polarization. Estimates for the total SET of scalar and spinor field on various stages of the evolution for realistic cosmological models are obtained. Also the effect of spontaneous breakdown of gauge symmetry for selfinteracting' scalar field in a hyperbolic Friedman model is considered.

Contents Summary

173

1. Introduction

174

2. Classical solutions of spin 0 and 1/2 field equations in isotropic universes 2.1. The metric 2.2. Spin 0 field 2.3. Spin 1/2 field 3. Quantization and particle interpretation in non-stationary metric 3.1. 3.2. 3.3. 3.4.

Field quantization Hamiltonian diagonalization: scalar field Hamiltonian diagonalization: spinor field Equations for the parameters describing thé creation process

4. Vacuum expectation values of the stress-energy tensor 4.1. Normal ordering 4.2. Renormalization 4.3. Massless fields . 12

Zeitschrift „Fortschritte der Physik", Heft 4

175 175 176 178 179 179 180 182 183 184 184 185 188

174

A. A. GBIB, S . G. MAMAYEV, V. M. MOSTEPANENKO

5. The density of created particles and the total S E T of massive fields in the Friedman cosmological models 189 5.1. Spin 0 field 5.2. Spin 1/2 field 5.3. Comparison with the background

190 194 195

6. Spontaneous symmetry breakdown in hyperbolic space-time

196

7. References

198

1. Introduction The theory of quantized fields in curved space-time has received a great deal of attention in recent years. Hawking-Penrose theorems on singularities as well as observational evidence for the hot cosmological model and existence of black holes suggest that very strong gravitational fields must occur in our Universe. In such fields the quantum effects of other fields must be essential, namely creation of particles and antiparticles, vacuum polarization and spontaneous breakdown of symmetry. The particle creation from vacuum by a classical gravitational field has been under intensive investigation after the appearance of L. PARKER'S papers [ i ] — see e.g., [2—13]. As is well known, the primary difficulty of this problem is the construction of the Fock space for the quantized field, i.e. its interpretation in terms of particles. The IS'-matrix picture, which is quite natural in electrodynamics, here is possible only in asymptotically flat space-time and is of no use in realistic cosmological models. In refs. [1, 2, 8, 14, 16] various approaches to the definition of particles in the presence of nonstationary gravitational field were proposed. One of them is to define particles in such a way that the instantaneous Hamiltonian of the quantized field, contracted via metrical ("new improved") [14, 15] stress-energy tensor, is diagonal in terms of creation and annihilation operators [2, 5, #]. Mathematical aspects of this approach were elaborated in [17—19]. On the basis of this method in [5, 7, 10] the stress-energy tensor of created scalar and spinor particles was calculated for various stages of evolution of realistic cosmological models. In these models such an interpretation is physically natural since the Hamiltonian mentioned above here is the generator of scale transformations which are observable as the expansion of the Universe (the importance of this fact has been pointed out in [JJ]). Independently of the study of particle creation, a great number of papers appeared recently (see, e.g., [20—32]) in which vacuum expectation values of the stress energy tensor (SET) operator in curved space-times are investigated. The principal problem here is to obtain renormalized values of originally divergent quantities. That is achieved by means of various regularization schemes, e.g. adiabatic regularization [20, 21, 26], covariant point-splitting [22—25], dimensional regularization [27—29], ^-function regularization [30—3J]. All these schemes are based more or less on the Fock-Schwinger-De Witt proper time method [12, 33] and do not rely on any particle interpretation. Instead an existence of a stationary in-region is postulated to provide a well defined initial vacuum state |0ln). In this state expectation values of S E T are computed with the assumption that they depend on the metric in a local geometrical manner. This S E T may be called vacuum polarization tensor. Up to the present time there was a troublesome lack of contact between the two a b o v # mentioned directions of research. The main reason is that the created particles arfe described by generally nonlocal terms in vacuum expectations of S E T , while vacuum polarization is accounted for by local geometrical terms. The latter were computed in

Vacuum Stress-Energy Tensor and Particle Creation

175

[20—32] either for massless field, when no particle creation occurs, or in the adiabatic limit, when it may be neglected. The main purpose of the present paper is to calculate the total vacuum expectations of S E T in homogeneous isotropic space-times, containing both local terms, which are due to the vacuum polarization, and nonlocal ones, corresponding to the created particles. This is achieved by means of a general method based on the diagonalization of the Hamiltonian and subsequent use of a Pauli-Villars type of regularization procedure proposed in [3]. In the case when there is no particle creation or it is negligible, our results coincide with those of refs. [22, 25, 26, 32] (for isotropic Universe). I n Sec. 2 we present complete orthonormal sets of solutions of conformally coupled Klein-Gordon-Fock and Dirac equations in homogeneous isotropic space-times and introduce the necessary notation. I n Sec. 3 the scheme of quantization of scalar and spinor fields is outlined and particle interpretation of quantized fields is constructed. The physical relevance of this method has sometimes been questioned due to a number of confusions. For example, in [34] this method led to an infinite number of created particles per unit volume of 3-space. However, in this paper the canonical Hamiltonian, constructed via canonical SET of the field was used. But, in dealing with gravitational interaction one must use the metrical SET, i.e. t h a t obtained by variation of action with respect to gik and not the fields [14]. The use of the corresponding "metrical" Hamiltonian gives finite results for the number of created particles per unit volume in isotropic space-times. I t is shown in this Section t h a t the widely used anzatz (see Eqs. (3.11), (3.22)) for calculation of the SET of created particles [3, 4, 9] is exactly equivalent to the diagonalization of the metrical Hamiltonian. I n Sec. 4 the general method of regularization of vacuum expectations of SET is given. This method respects axioms 1 — 4 of ref. [32] thus giving a unique result up to the usual renormalization ambiguities. In principle it is applicable to any space-time admitting a mode decomposition of the field. Its main advantage is that it provides a constructive way to obtain the total S E T of the quantized fields, including vacuum polarization as well as the contribution from the created particles. I n Sec. 5 particle creation and vacuum polarization are calculated for the realistic cosmological models of the Friedman type. The numerical estimates of the effect are obtained for various stages of the evolution of the Universe. Sec. 6 is devoted to the effect of the spontaneous symmetry breakdown in the hyperbolic isotropic space-time for the self-interacting scalar field. The negative sign of the 3-space curvature plays here the same role as the negative square of the mass in the Goldstone model and leads to the spontaneous breakdown of gauge and discrete symmetries. Throughout the paper units h = c = 1 and sign convention (—) of [36] are used.

2. Classical Solutions of Spin 0 and 1/2 Field Equations in Isotropic Universes 2.1. The metric In co-moving coordinates the metric of a homogeneous isotropic space-time may be written as ds2 = gik dxi dxk = a2{tj) {drf — dl2) (2.1) lere a(rf) is the scale factor and dl2 is the metric of a 3-space of constant curvature = —1, 0 or + 1 : dl2 = yap dx« dxi = dr2 + f2(r) {dd2 + sin 2 6 d

ftiiVo) = -Mvo)

ftikno) •

(2.27)

Comparison of Eq. (2.26) with its scalar analog (2.15) shows t h a t in the adiabatic regime 1) the imaginary p a r t of the frequency in (2.26) may be neglected and (h/m = á/ma2 there will be no essential difference between spin 1/2 and spin 0 cases. In antiadiabatic expansion, when h¡m^> 1, the imaginary part imá is dominant for X < ma and the creation of nonrelativistic particles with spins 1/2 and 0 will be quite different [6]. This may be considered as a consequence of the Pauli principle. 3. Quantization and Particle Interpretation in Non-Stationary Metric 3.1. Field quantization Operator of the quantized field r¡0 and hence in the instability of the vacuum |0). Objects defined as particles at r¡ = r¡0 have no such sense at later times, and in order to define particle observables one must introduce some concept of particles valid at any V ^ Volt seems fairly evident t h a t no such concept with a t least some physical relevance is possible in arbitrary curved space-time. Particle interpretation of the field is intimately connected with the space-time symmetry, and the general Riemannian manifold has no symmetry at all. In our space-time (2.1), however, there is a 6-parameter in variance group of the 3-space rj = const, and a group of conformal transformations (dilatations) generated by ^-translations according to (2.3). This allows one to introduce the concept of particles as such objects, whose creation and annihilation operators diagonalize the field Hamiltonian (2.10) or (2.21) (which are the generators of scale transformations in the metric (2.1)). This definition of particles is generally dependent on time — t h a t is

180

A . A . GRIB, S . G . MAMAYEV, V . M . MOSTEPANENKO

why these particles should, perhaps, more properly be called "quasiparticles". As it will be shown below, if a(r/) becomes constant at some time rj > TJx, particles defined in this way coincide with what is usually called particles in stationary space-time. 3.2. Hamiltonian diagonalization: scalar field Inserting the expansion (3.1) into Eq. (2.10) with the use of (2.6), (2.7), (2.12), (2.17) and the properties of the functions Oj{r, 6, {rj) {EM

&(+>(*,(-> + ajaj< +>)

+ Fj(v) + Fj*(v)

Sj«-)«/-)}.

(3.4)

Here J = {A, I, —m) denotes the quantum numbers of the complex conjugate function, 0j = 0 j * , and Ej(rj) and Fj(rj) are expressed in terms of solutions of Eq. (2.15) as =

+

=

+

(3-5)

An identity holds E j 2 — \Fj\ 2 = 1. Note that the quantity Ej(rj) is an adiabatic invariant of the classical oscillator (2.15). Initial conditions (2.16) give Ej(rj0) = 1, Fj(rj0) = 0. However, if a(rj) 4= const., at any rj > rj0 we have Ej(rf) > 1, F j ( r j ) 4= 0 (the parametric excitation of the oscillator takes place). The particle interpretation of the field at r] > rj0 is given by creation and annihilation operators 6 / ( ± ) , in terms of which the Hamiltonian (3.4) is diagonal. Diagonalization is achieved by means of a time-dependent Bogoliubov transformation

= *j*(v) W'Hv) - ( - 1 ) mPAi) £/ = *j*(ri) &(»?) - ( - 1 ) - PAv) &V+>(*?),

(3.6)

I«./]2 - Ifrl 2 = 1 with coefficients (»?)) •

Thus we arrive at the time-dependent particle concept in non-stationary space-time.

(3-9)

Vacuum Stress-Energy Tensor and Particle Creation

181

With the help of (3.7) and (2.15) a system of first-order differential equations for a ; and /J; may be obtained [7] &l

- H^L BW ~~ 2a)2 P x

'

«.c-«e

B,* - — ~ 2a>2

Px

'

n 0 = f (o(r¡') dr¡'. lo

(3.10)

The solutions gi{rf) of (2.15) and their derivatives may be expressed in terms of on and Pi as gx(v) = a r ^ f o ) [«/(»?) e*9™ + pfo) c -fa)

Cj +

(3.15)

we find the average number of pairs created by the moment rj in the Heisenberg state |0) which was vacuum at rj = rj0 to be njW = = |frMI 2

(3.16)

where @>.{'>]) is the coefficient of the Bogoliubov transformation (3.6). Its dependence on the momentum X only is explained by the spherical symmetry. The density of created

182

A . A . G R I B , S . G . M A M A Y E V , V . M . MOSTEPANENKO

pairs (number of pairs in the unit volume of 3-space) is found to be [1, 2] oo 0 Eqs. (3.10) show that if a(tj) = const, for rj > rj1 > rj0 (or d(rj) -> 0 as rj oo), the quantities 0 as rj —oo) we arrive at the asymptotically — static situation treated in [1,1S\. Eqs. (3.11) immediately show that in such a case the method of the Hamiltonian diagonalization gives precisely the same results as the usual in- and owi-formalism. 3.3. Hamiltonian diagonalization : spinor field Now we shall briefly sketch the spin 1/2 version of the construction of the preceding subsection. Inserting the expansion of the spinor field operator yi in terms of creation and annihilation operators into the Hamiltonian (2.21) and making use of the properties of the spinor eigenfunctions tpj{±)(x), listed in [5, 38] we find ¿fi 1 ' 2 )^) to have the form of (3.4) with Ej and Fj defined by EAv) = -CON 1

- \M2) - A Re

(n+h-)], (3.18)

FAv)

=

^ mafi+fi- - j

(/!+ - /!_)

Here fx± = f[t} are the solutions of Eqs. (2.26) with initial conditions (2.27). Obviously, Ej(Vo) = h FJ(VO) = 0 and E/ + ]Fj\2 = 1 for all V. The Bogoliubov transformation, diagonalizing H(s)ux,

vx

=

2mux,

—

2mvx

(3.26) Ux =

»("(I

±

2sx)

with 6)

wm =

_ = CO

m?aa

,,,„,

mdX

W(H2) =

COi

0)i

.

(3.27)

Since |8(?70) = 0, the initial conditions for (3.26) are *foo) =

«foo)

= «fa0 ) = 0.

(3.28)

In order to find the large A behavior of Sx, ux and Vx use the system of the Volterra-type integral equations, equivalent to (3.26) — (3.28): v Ux +

ivx

=

J

W (S) (V) [ 1 =h

exp

2sx(r¡')]

f

oj{V")

dr¡"J

dr¡',

lo V ^

j J

rf dr¡'w^{r¡')

i J m

dr¡"w^\r¡")

[1

±

2sx{r¡")]

cos 2 / cofe) L n'

drh

184

A. A. Gkib, S. G. Mamayev, V. M. Mostepanenko

For A ma we may put a> ~ A, w(0) ~ m2«4/A2, mi!1/2) ~ rad/A. If, moreover, A/ma then 1 and we may use the first iteration of (3.29): ux + iv,j ~ / w(s)(»?') 1o

a I a.

e2Uir>-i"> d r j ' ,

(3.30)

V

If a ( y f ) is smooth enough in the interval (r/0, r/), then for s = 0 (3.30) gives .s, ex A-6, Ux oc A-4, vx oc A-3, while for s = 1/2 we find oc A-4, ut oc A-3, oc A-2. Thus of (3.17) and (3.24) is finite in both cases. 4. Vacuum Expectation Values of the Stress-Energy Tensor

As it was pointed in Sec. 1, the quantities of prime interest in the theory of quantized fields intracting with external gravitational field are vacuum expectation values of the stress-energy tensor (SET) operator (0] Tik |0). The vacuum state |0) here may be either the in-vacuum (defined at r\ ->• —oo) in ¿'-matrix-type problem or the initial vacuum state at rj = rj0 constructed in the preceding section. We shall assume that A{rj) and a sufficient number of its higher derivatives go to zero as rj —oo (or rj -> rj0). In this section the expressions for the total SET including both the contribution of the created particles and the vacuum polarization will be obtained. 4.1. Normal ordering It is well known that even in flat space-time vacuum expectation values of SET diverge. This divergence is ascribed to the contribution of the zero-point oscillations of the quantized fields and is usually excluded by normal ordering of the SET operator. It has been generally accepted that no feasible normal ordering procedure can be defined in a general curved space-time since there is no invariant definition of creation and annihilation operators for interpolating field. In our case of homogeneous isotropic space-time, however, there is a natural construction of instantaneous vacuum state and creation and annihilation operators for "quasiparticles" Cj(lt)(ry). Thus we can define a time-dependent prescription of normal ordering with respect to these operators, which is in fact equivalent to substraction of the divergent contribution of the zero-point oscillations of the instantaneous vacuum |0,) defined by (3.8) [7] : Nn{Tik)

= Tikr-{%\Tik

|0>.

(4.1)

Writing SET for the scalar field (2.6) in terms of the fieisenberg operators (3.12) and taking the average of (4.1) in the state |0) we find after a lengthy but straightforward calculation for the models with k = 0, — 1 00

m2a2

2(o Ux

Vacuum Stress-Energy Tensor and Particle Creation

185

where ux are defined in (3.25). In derivation of (4.2) Eqs. (3.14), (3.6), (3.2), (3.3) and the properties of the eigenfunctions 0j{x) given in [37] were used. For the spherical model (x = + 1 ) integration in (4.2) must be replaced by summation over A = 1, 2, ... I t is easy to cheek that (0| N n {TfJ) |0) obey (modewise) the conservation equation Vk(0\ N„{T,(")*) |0> = 0

(4.3)

which reduces here formally to the first equation in (3.26). In fact both terms of the right hand side of (4.1) are separately conserved. This may be considered as a proof of the correctness of our normal ordering: any other prescription would give a nonconserved result. Using the same procedure for the SET of the spin 1/2 field (2.20) we obtain for x = 0, — 1 oo d?

co; its behavior for d = X\m}~ia(s > 1 is

and for d

1 (0)(«5) = - I 2«"2 r2 (—¡f^j0-1

•

W

For radiation-dominated Universe (q = 1/2) the exact form of n(°)(6) may be obtained /l

(0)(S) = JL. e -*i'/2 ]/2á

n

. ó 2 \" 2

( T

+

' T )

ó* + 2

¡3 R

. and P 0 ( 0 ) we now must use expressions (5.9) for sx, ux. The total SET in this case has components e =

2K^mi{mt)'3o

Ft o) P~2 for the expansion law (5.2). The method of calculation of ii(1/2>( 0)

(1.1)

for —> oo and similarly for the other two variables going to infinity. In addition we consider the more restricted class of amplitudes going down asymptotically like F(s, t) = 0(|«]—)

(|f| < 4, e > 0)

(1.2)

for |s| - > oo. In this latter case one may write fixed t dispersion relations of the form OO

^=4/(7 4

— s

-s', + s + t — 4/ 7 I A (vs '> t)'

ds

'-

(1.3)

204

F. SCHWABZ

In the general case (1.1) it is more complicated due to the subtraction term at s = F(s, t) = F(s0, t) x

i I 4

-

s0:

i — s0) (s + s0 + t n (2s' + t - 4) A(.s', t) ds' s) (s' -s0)(s' + s + t - 4) (s' + «0 + t - 4)'

(L4)

Due to s—«-crossing symmetry there is only one subtraction constant necessary. The partial wave series in the «-channel physical region is written as F(s, za) =

(s) Z

(21 + 1) /,(«) Pt{zs)

(1.5)

1=0,2...

where Q(S) = [(,s — 4) s] 1 ' 2 and zs — 1 + 2T/(s — 4) is the scattering angle in the schannel. In our normalization unitarity is expressed as a t ( s ) ^ |/,(«)|* = n»^) + 0!»(a).

(1.6)

The partial wave projection reads +i m

=

/ ns. —1

o *.) p,%) dzs =

_^)a]1/>

[(a

/ F(s, t) p, ( i + -JLJj 4—8

dt.

(1.7)

In the unphysical region 0 5S s Si 4 a different normalization is used which is obtained from (1.5) and (1.7) by setting o = 1. The scattering lengths a t are defined by

=

(L8)

2. Construction of Amplitudes: No Subtractions Are Required

In this chapter we describe how our amplitudes are obtained if the asymptotic behaviour is such that dispersion relations without subtractions may be written. This makes the construction somewhat simpler as compared to the more general case where the amplitudes are allowed to increase like (1.1) such] that a dispersion relation needs two subtractions. t Assume that the variable t is fixed at some value of t in the range |i[ < 4. In a first step a new variable z = (2s + t — 4)/2 is introduced. Correspondingly a function G(z, t) = F(s, t) is defined. Crossing symmetry under s—«-exchange means symmetry in 2 of 0(z, t) = 6(—z, t). Asymptotically it behaves as 0(]z|~e). Its lowest branch points are at z = ± (i + 4)/2. Its analytic structure allows a conformal transformation of each complex z-plane onto the unit circle in a new variable w(t) defined by

i _ ¿ y ( s _ 4)/( S + t)

(2.D

Further a function H(w, t) is defined by H(w, t) = G(z, t). The symmetry in z implies that H(w, t) is an even function of w, i.e., H(w, t) = H(—w, t). The asymptotic behaviour for |s| oo means in this new variable lim H(w, t) = 0. If we approximate

An Approximation Scheme for Constructing 7t°Tu° Amplitudes

205

H(w, t) by a polynomial there occur only even powers: H(w,t) =

£

cn(t)w«(t).

(2.2)

»=0,2...

For the real and the imaginary part of the amplitude in the physical region it follows from this expansion t) = £ C»W c o s »?(«» 0 (2.3a) 71=0,2...

and

t) = £

c

u{t) Sin mp(s, t)

(2.3b)

n = 2.4...

respectively where the angle i is ensured if they obey the relation £

( - l ) " c „ ( 0 = O.

(2.4)

n=0,2...

Now we choose a finite number of values of t and make for each of them an ansatz like (2.2). As already mentioned earlier we choose for t the five values tx = 8/3, t2 = 4/3, t3 = 0, f4 = —4/3 and t5 = —8/3 and write H{w,h) =

Z

n=0.2...

Cn(ti)wn(ti).

(2.5)

The upper summation limit N will be fixed later on. The expansion coefficients c„(i;) are not independent. First of all they are restricted by t—«-crossing symmetry. These constraints may be read off from Fig. 1. The amplitude has the same values at those pairs of dots which are symmetric with respect to the line 2s -)- t — 4 = 0. It leads to thirteen linear constraints for the cn{t{), i = 1, ..., 5. Writing Eq. (2.4) for each value of leads to five additional equations. To find the constraints for the cn(ti) which lead to a unitary amplitude we have to define partial waves. To this end consider the amplitude at a fixed value of s > 4. Our analyticity assumptions imply that the lowest branch points for the real part occur at t = 4 and u = 4. For the imaginary part the lowest singularity occurs at the boundary of the double spectral functions which is for 4 < s < 2 0 at u = t = 16s/(s — 4) and if s > 20 at u = t = 4s/(s — 16). Correspondingly we define two expansions R(s, t)= and M*> *)=

£ fi„(s) wrn{s, t) »=0,2...8 £

«.(«) W(s,

n = 0.2...8

t)

(2.6a) (2.6b)

for the real and the imaginary part respectively where „ 14

M

^

- f E H E M 1 + V(4 - t)/(s + t)

Zeitsclirift „Fortschritte der PhyBik", Heft 4

(0 < s)

(2.7)

206

F . SCHWARZ

and 1 _ y[i6s - t(s - 4)]/[(* + 4) 2 + t(s - 4)] wa(s, t) =

1 + y[16s - t(s - 4)]/[(« + 4) 2 + t(s - 4)]

(4 < s ^ 20)

1 - V[4s - t(s -

16)]/[4^ + (t + g + 4) (s -

16)]

1 + }/[4s - t(s -

16)]/[4s + (i + s + 4) {s -

16)]

(2.8)

(20
t) = (t + 4)^/2 (S _ 4)n=0,l... 2 1 (» - 1) c2n(t) _ 1 (t + 4)1/2 (s _ 4)2 ^ („ _ 1) [I6(n - 1)2 - 19] c2n(t) D »=0,1... + 0[(« - 4)3] (s - 4)i/2 2j(a>

(A.5)

* ( í + 4)2 _ 4)1/2 g c2n(i) 4 n=0,l... - (t + 4) (8 - 4)3/2 2J [2(» - 1)2 - 1] c , ( 0 + 0[(« - 4) 5 ' 2 ]. n=0,l...

(A.6)

To obtain the behaviour of the partial waves near threshold, one has to know in addition the expansion of the matrices NaM^ 1 and N f M f 1 for s 4. To this end the elements of each factor are exanded separately into powers of s — 4. Although in principle this is straightforward it turns out to be rather tedious. For this kind of problems the programming system REDUCE turned out to be very useful. It menables one to do algebraic manipulations on a computer. In fact, many of the expansions given in Appendix A and B could not have been obtained without applying REDUCE. The non-vanishing matrix elements in the decomposition of the matrix NaM0-1 into powers of s — 4 which are needed for the further calculations are given in Table 1. The decomposition of N f M , - 1 has the same structure, i.e., there are zeros in the same places as in the decomposition of NaMa~ 1. Only the non-vanishing elements are numerically different ant it is not given explicitly. Combining these decompositions with the expansions (A.3)... (A.6) of the amplitudes we have the following result. The threshold behaviour is the same for both the unsubtracted and the subtracted case. The real part of the partial waves behaves like r¡(s) = 0[(s — 4)'+1/2],

(A.7)

223

An Approximation Scheme for Constructing rfin 0 Amplitudes

The imaginary part of the 5-wave goes linearly to zero whereas the higher waves go to zero like (s — 4)'. So we add two linear constraints for the expansion coefficients c„(i{) to make the leading term of a 2 and at vanish and have then for all waves o < («) = 0 [ ( « - 4 ) ^ ] .

(A.8)

B . The behaviour for large s To determine the behaviour of the partial waves for large values of s one has to know the behaviour of the angular functions sin 2np(s, t) for s going to infinity. The leading terms which are relevant for our calculations are

vfi it -I- 41 2 and

c o s 2n