Existence of the S-matrix in quantum field theory in curved space-time

ANNALS

OF PHYSICS

118,

490-510

Existence

(1979)

of the S-Matrix in Quantum in Curved Space-Time*

Field Theory

ROBERT M. WALD+ Enrico

Fermi

Institute,

University

of Chicago,

Chicago,

Illinois

60637

Received August 21, 1978

The existence of the s-matrix is proven for particle creation by an external gravitational field of compact support. No infrared divergences occur even for massless quantum fields; in particular, a localized gravitational field always produces a finite expected total number of particles. The results of this paper apply to both boson and fennion fields as well as to more general linear, external potential interactions.

1. INTRODUCTION The theory of the spontaneous creation of particles by quantum processesin a strong gravitational field has been studied by many authors recently in the approximation where the gravitational field is treated as a classical, externally prescribed field. This theory is closely analogous to other external field problems such as a Dirac field in an external electromagnetic field or a scalarfield in an external scalar potential. While particle creation by a gravitational field is certainly of no importance for ordinary laboratory physics, it may be of considerable importance in the early universe and near black holes. Indeed, the most striking application of the theory is Hawking’s demonstration that particle creation near a Schwarzschild black hole results in a flux of escapingparticles with an exactly thermal spectrum [ 1, 21. The nature of the theory of particle creation by an external field is as follows: One assumesthat the states of the quantum field can be characterized as particle (i.e., Fock space) states both in the distant past (“in states”) and distant future (“out states”). The quantum field operator is assumedin the distant past to reduce to the standard free-field expression in terms of the “in” annihilation and creation operators and in the distant future to reduce to the corresponding expression in terms of the “out” Fock space operators. Thus, the field operator interpolates between the “in” and “out” annihilation and creation operators and yields a relation between them. From this relation, an expression for the S-matrix can be derived. In [2] a derivation of the S-matrix was given for the caseof a real scalar field. In a straightforward manner, the relations between the “in” and “out” annihilation and * Supportedin part by NSF grant PHY 76-81102 A01 and by the Alfred P. SloanFoundation. +SloanFoundationFellow. 490 00034916/79/040490-21$05.00/O Copyright All rights

0 1979 by Academic Press, Inc. of reproduction in any form reserved.

CURVED SPACE S-MATRIX

491

creation operators yield a formula for the S-matrix in terms of operators describing the scattering of ordinary (“c-number”) classical solutions. However, in order that the expression derived be mathematically meaningful, certain conditions must be satisfied by these operators; in particular, one of them must satisfy a Hilbert-Schmidt condition. If these conditions are not satisfied, then, strictly speaking, the S-matrix does not exist, though, of course it remains possible that meaningful physical predictions can still be made (as is the case, for example, in the “infrared catastrophe”). The existence of the S-matrix has been shown for particle creation by a number of other types of external fields, such as a scalar field in an external scalar potential [3] and a Dirac field in an external electromagnetic field [4-61. In this paper, we shall present a proof of the existence of the S-matrix for particle creation by an external gravitational field of compact support. The proof relies on a recent result of Fulling, Sweeny, and Wald [7] on the short-distance singularity structure of the two-point function in quantum field theory in curved space-time. The method of proof is quite general and applies to both boson and fermion fields as well as more general types of external field interactions. Aside from the restriction to space-time curvature of compact support, the only further requirement is that the space-time be globally hyperbolic [S]. In particular, no restriction is made on the strength of the gravitational field. While the proofs given here apply only to the compact support case, there is no reason to believe that the results would not remain valid if the asymptotic fall-off requirements were suitably weakened. An interesting consequence of the results proven below is that no infrared divergences occur in particle creation by an external field of compact support. The expected total number of massless particles created by an arbitrary gravitational field of finite extent and duration is always finite; similarly, the total number of massless charged particles (if such existed) created by an electromagnetic field of compact support is also finite. In Section 2 we derive the expression for the quantum S-matrix in terms of operators describing the classical scattering of “c-number” solutions for the three fields which will be treated in the paper: the neutral scalar field, the charged scalar field, and the Dirac field. The analysis follows closely that of [2], where only the neutral scalar field was considered. The uniqueness of the s-matrix follows very simply from the derivation. We show that a sufficient condition (and also a necessary condition if the operators describing classical scattering are everywhere defined) for the existence of the s-matrix in all cases is that the operator B, defined below, satisfy the HilbertSchmidt condition tr B+B < co. In Section 3, we prove that this condition is satisfied for an external field of compact support.

2. DERIVATION

OF THE S-MATRIX

We wish to consider particle creation of a linear quantum field by an external classical field. We assume that the states of the quantum field can be characterized both as vectors in the Fock space &&z?~,J constructed from the one-particle Hilbert

492

ROBERT

M.

WALD

space, #in , of incoming states and as vectors in 3&(.%&t). The S-matrix yields the relation between these two ways of characterizing the states. The quantum field is an operator valued distribution-defined on all C” test functions of compact supportsatisfying the wave equation appropriate to the field. Because the equation is linear (the quantum field is assumed to interact only with the external field and not with itself), no mathematical difficulties arise in defining the field operator. We assume that in the past the field operator asymptotically approaches the standard free-field expression in terms of the annihilation and creation operators of Ff,(&‘i,). Similarly, we assume in the future that it approaches the free-field expression for ~&~t(X~~t). The fact that the quantum field operator satisfies the field equation everywhere allows us to relate the “in” and “out” annihilation and creation operators and solve for the S-matrix. We now shall carry out this program for the real scalar field, the charged scalar field, and the Dirac field. 2.1. Real Scalar Field 4

An analysis of the real scalar field 4 was given in [2] and we will follow that analysis closely here. The equation for $ is. (V,VJJ - m3 4 -t V+ = 0,

12.1)

where V, denotes the covariant derivative and V encompasses a possible nonminimal coupling to the curvature as well as a possible additional external real scalar potential. We consider only space-times which are globally hyperbolic [8]. For two classical solutions I$~ and d2 of Eq. (2.1), the Klein-Gordon inner product

is conserved, i.e., is independent of the choice of Cauchy surface C on which the integral is evaluated. For the free field (flat space-time, V = 0) the quantum field theory is constructed as follows: The one-particle Hilbert space Z is taken to consist of all positive frequency solutions with finite Klein-Gordon norm. (The Klein-Gordon inner product Eq. (2.2) is positive definite on the space of positive frequency solutions.) The Hilbert space of states is taken to be the symmetric Fock space constructed from &‘, ~~(~)=co~~(~09~)o.--,

(2.3)

where 0, denotes the symmetrized tensor product. It will be convenient to work explicitly with the dual Hilbert space 2 of the oneparticle Hilbert space .%‘, so that all maps will be linear (rather than antilinear). J? is in antilinear correspondence with ti and since complex conjugation is also an antilinear map taking &@(positive frequency solutions) into negative frequency solutions we may linearly identify fl with the space of negative frequency solutions with the inner product given by minus the Klein-Gordon product (so that norms will be

CURVED

SPACE

S-MATRIX

493

positive definite). We will use a bar to denote the natural map between 2 and 2, i.e., if u E X’, then 5 is the element of 2 whose action on Z is the same as taking the inner product with (T. Annihilation and creation operators, a and u+ are defined on YS(X) as follows. For a state (2.4) y = (c, 71 7 72 9 773 Y.> where qi E &,Y X, we define for all one particle states u E X 46) Y = (6 . 71.) (2)l’2 a . 72 ) (3)“2 a . r/3 ,... ), a’(u) Y = (0, co, (2)1’2 u 0,171 ) (3)‘/” u 0, 772)... ), where the “e” in Eq. (2.5) denotes scalar product and 0, in Eq. (2.6) denotes symmetrized tensor product. Finally, the quantum field operator 4 is given by, for all test functionsf, Kf>

= -4v)

+ ai(

0.7)

where iuf+ is the one-particle state obtained by taking the positive frequency part of the advanced minus the retarded solution of the wave equation with sourcef, while ia,is the vector in 2 associated with the negative frequency part of this solution. Equation (2.7) is equivalent to the more familiar mode sum, (2.8) where {Fi} ranges over a complete orthonormal set of positive frequency solutions and a, and LZ~+are the annihilation and creation operators for the ith state. To define “in” and “out” states in quantum field theory in curved space-time we need a notion of “positive frequency” in the asymptotic past and future (or, equivalently, past and future complex structures [9]). We are primarily concerned here with the case of curvature of compact support, where the space-time actually becomes flat in the past and future. In this case we can define the “past positive frequency” part of a solution by looking at its data on a Cauchy surface in the flat region in the past and decomposing it by the flat space formula. For space-times which are only asymptotically flat, we can for massless fields, perform this decomposition at past null infinity 9-, provided that LJ- is a good initial data surface for the field [lo]. Similar asymptotic prescriptions will also undoubtedly work for massive fields if the curvature is required to fall off sufficiently rapidly in the past. For nonasymptotically flat spacetimes several prescriptions for defining “past and future positive frequency” (or, equivalently, the Feynman propagator) have been given [ll-131 but for general space-times which do not approach flatness in the past-in particular, space-times with initial singularities (as are of interest in cosmology)-there are difficulties in obtaining a natural notion of “past positive frequency.” This corresponds to the physical ambiguity in defining the notion of an incoming particle if the field is strong in the past. Below, we will explicitly deal with the case of curvature of compact

494

ROBERT

M.

WALD

support, where these difficulties are not present. However, the mathematical constructions of this section will work in general cases with any definition of “past positive frequency” provided that (i) the Klein-Gordon inner product is positive definite on the subspace of past positive frequency solutions, (ii) any solution of Eq. (2.1) with initial data of compact support can be expressed as the sum of a past positive frequency solution and the complex conjugate of a past positive frequency solution (- a negative frequency solution), both of which have finite Klein-Gordon norm, (iii) each negative frequency solution has vanishing Klein-Gordon inner product with any positive frequency solution, and (iv) the past positive frequency solutions obtained by decomposing solutions with data of compact support are dense in the Hilbert space of all past positive frequency solutions. Given the notion of “past positive frequency,” the one-particle “in” Hilbert space %‘rn is taken to be the past positive frequency solutions with finite Klein-Gordon norm. The Hilbert space Fin of “in” states is taken to be ~XA?W). The requirement that the field operator reduce to the free-field expression on sin in the “distant past” (i.e., outside the future of the support of the curvature in the compact support case) implies that for a test function h with support in the “distant past” we have #z) = -ff(uh”-) + a+(u;+),

(2.9)

where a and a+ are the annihilation and creation operators for sin and ia:+ and iuhpare the past positive and negative frequency parts of the advanced minus retarded solution of Eq. (2.1) with source h. But the fact that #I satisfies Eq. (2.1) (in the distributional sense) implies that Eq. (2.9) holds for all test functionsfof compact support. Namely, we write an arbitrary test functionfof compact support as the sum of a test function h with support in the distant past plus a test function k of the form (V,V@ m2 + V) g, where g is a test function [7]. By Eq. (2.1) we have 4(k) = 0 so $(f) = $@). Furthermore, CT~= 0 so crf” = ui- and oFf”’ = c$+. Since Eq. (2.9) holds for #‘z), we have 4(f) = --a(~,“-) + a’(c7;‘). (2.10) In an identical manner, we obtain the analogous equation for # in terms of the “out” annihilation and creation operators, b, b+, and the future positive and negative frequency parts of the advanced minus retarded solution with source f, 4(f)

= -b(cr;-)

+ b+(atFf).

(2.11)

Equations (2.10) and (2.11) give the relation between a, a+ and b, b’ -da;->

+ a+(~;+) = -b(ufF-)

+ b+(c$+)

(2.12)

which will allow us to relate 4,, to * out, i.e., to determine the S-matrix. This relation is most conveniently reexpressed in terms of operators A, B, C, D [2] acting on the oneparticle Hilbert spaces Xrn and Xout which describe the classical scattering of positive frequency waves. These operators are defined as follows:

CURVED

SPACE

495

S-MATRIX

Consider a classical solution of Eq. (2.1) which is purely positive frequency in the future (and thus may be associated with a vector u E %&). This solution may also be decomposed into its past positive and negative frequency parts. With its past positive frequency part, we may associate a vector T E sl,, ; with its past negative frequency part we may (linearly) associate a vector T’ E J& where 4, is the dual Hilbert space of si, . We define A: &‘&,t + Xi,, B : X&t -+ .T&, by, Au=T,

Ba = 7’.

We define the operators C : tii, + XOut and D : %r, + manner with “past” and “future” interchanged. We assume (as is easy to prove for the case of curvature the operators A, B, C, D are everywhere defined, i.e., for and T’ have finite Klein-Gordon norm and the analogous D. As shown in [2], invariance of the Klein-Gordon inner the following relations among these operators: A+A - BiB = I, B+A = A+& C+C - D+D = I, A+ = C,

D+e = C+D, B+ = -jj+.

(2.13) Y&

in exactly the same

of compact support) that all normalized u E &‘&t , 7 property holds for C and product, Eq. (2.2), implies (2.14a) (2.14b) (2.14c) (2.14d) (2.14e)

Note that A+A = CA is an everywhere defined self-adjoint operator and hence, by the Hellinger-Toepolitz theorem [14], is bounded. This implies A is bounded. Similarly B, C, and D are bounded. Equations (2.14a) and (2.14c), along with (2.14e) also imply that A-l and C-l exist as everywhere defined bounded operators. Equation (2.12) holds for all test functionsf. But, given any state u E 21, , we can find a test function f such that 11uf”- - 0 jl and I/ $‘(I , are as small as we like. This implies that for all u E Xi, , a(6) = b(Ca) - b+(Du).

(2.15)

As we shall now show, Eq. (2.15) gives enough information to uniquely determine the S-matrix (up to an arbitrary phase factor). The main result of this section will be that the necessary and sufficient condition (given that A, B, C, and D are everywhere defined) for the existence of the S-matrix (and thus for a completely satisfactory theory of particle creation) is that the operator I3 satisfy the Hilbert-Schmidt condition, tr B+B < co. In Section 3 we will show that this condition is satisfied in the case of curvature of compact support. It is worth remarking that when the S-matrix exists, tr B+B is equal to the expected total number of particles created from the vacuum [2], so this latter quantity is always finite when the S-matrix exists. Let !I’,, denote the “in” vacuum state. We wish to express YO as a vector in Z&t . Physically, this will tell us all the amplitudes for particle creation from the vacuum in the given spacetime. The correspondence S : &‘n + .FOut between all states of 4, with states of Z&t is the S-matrix.

496

ROBERT

M.

WALD

We may write out the vector SY,, in 5%,t as SYLl

=

cc,

71

3 72

(2.16)

7 r/3 ,...).

Since Y,, satisfies a(G) Y,, = 0 for all u E A?& , we have {NW

(2.17)

- ~+(wS(c, 71, 72 ,...I = 0.

Equation (2.17) yields relations which determine the rn . Writing DC-l we find that for all T E &&t

T = Ca and E = (2.18a) (2.18b)

? * 771= 0, p/2- T . y2 = cE?, 31i27 * r], = 2’/“(E?) 0, q1 , 4112~ . 7)a = 3112( ET )0,7?2?

(2.18~)

(2.18d)

etc. The first equation, (2.18a), implies ql = 0 and the other odd equations imply by induction that qn = 0 for all odd n. In other words, particles are always created in pairs. Equation (2.18b) states that the action of v2 E X&t 0, A?&, when viewed as a map from 2% into &&t, must be identical to the map E. This is possible if and only if (1) E is symmetric, i?+ = E, (since 72 must be) and (2) tr E+E < co since 11v2 /I2 must be finite. Condition (1) is implied by Eq. (2.14d). Condition (2) will hold if and only if tr B’B < co; namely, if B+B is trace class, so is D+D (indeed, tr DtD = tr B+B) and since C-l and (C-l)+ = A-l are bounded, so is E+E; conversely, since C and C+ are bounded, if E’E is trace class, so is B+B. Thus, if tr B+B = co, no solution for SY,, exists. We shall now assume that tr B+B < co and will show that the S-matrix exists. In Section 3, we will prove tr B+B < co for curvature of compact support. If tr E+E < 00 we can view E as a vector in &&t 0, A&t . Let Edenote this vector. We can express E in a useful form as follows. Since E+E is trace class, it is compact, and since it is self-adjoint, by the Hilbert-Schmidt theorem [14] we can find a complete orthonormal set of eigenvedtors {yi} in Z&t with eigenvalues ci2 such that E+E = c c$($

, -),

where any degeneracy except possibly ci = 0 is finite. For ci f 0 let pi Then,

(2.19) EyilCi v (2.20)

We have E = C Cipi(yi 7 a) 1

(2.21)

or in terms of the two-particle state E, we have E =

c

cipi I

@ yi .

(2.22)

CURVED

Furthermore,

SPACE

S-MATRIX

497

the fact that E is symmetric, Et = E, implies

(2.23)

Thus, if ci # cj , then pi is othogonal to yi . This means that E leaves invariant each finite-dimensional subspace spanned by all p’s and y’s associated with the same eigenvalue ci > 0. Using again the fact that E is symmetric, we can diagonalize E on each subspace to obtain the final form for the two-particle state E: There exists a complete orthonormal basis of one-particle states hi such that E = 1 kihi @ Ai . z

(2.24)

Since for all u E &7&t , (E6, EC?) = (E+Ee, CT) = (a, (C--L)+ D+Dc-lu) = (a, (c-l)+

[C+C - I] C-10)

= 11u 112- II C-la 112 < II u /I29

(2.25)

we have I ki / < 1 for all i Returning to Eqs. (2.18) we find by induction that for even n we have (2.26)

or, in more suggestive notation,

S I W = c exp [ 6 C k&+(U ~+GQ]I hd. i

(2.27)

Our next task is to show that C,, 11vn II2 < c&so that we can adjust c to make Ij 5’?P0II = 1. Then we must extend the definition of S to all states in 9& and prove that it is unitary. We now outline these steps.

498

ROBERT

M.

WALD

To show C Ij qn II2 > cc we note this sum is bounded by a constant times C /I E Iln so if /I E II2 = tr E+E < 1 the sum clearly converges. If /I E iI2 > 1 (but still finite, of course) we can break the sum, Eq. (2.24), for E into two pieces, E = 5 ki& @ hi + i=l

f

k,h, @ Aj ,

(2.28)

j=N+l

where the first sum is finite and the second sum has norm less than one. If we do the partial sum over all terms which do not contain any Xi with i < N, we will, by the above argument, get a finite result. But, by calculations along the lines of Appendix B of [2], the full sum is simply &, [c neven(n!/2”(n/2)!2)l ki I”] times this partial sum, and this factor is finite since each ) ki I < 1. This proves that jl SY, II is finite. To complete the definition of the S-matrix, we define its action on simple n-particle states by (2.29) (SY, lies in the domain of powers of b and b+ so the right-hand side is well defined. Some properties of this state vector, which physically describes stimulate emission, were discussed in [15].) Since the span of all simple n-particle states is dense in Fin , this defines S on a dense domain. To show that S preserves inner products on this domain, it suffices to show that it does so for two vectors of the form Eq. (2.29). However, the inner products in both 4n and F&t can be expressed, respectively, in- terms of -commutators involving a(p) and a+(o) in 4n and the corresponding (b(Cp) b+(Dp)) and (b+(Ca) - b(Da)) in F&t . Hence, inner products will be preserved if the corresponding commutators are equal. An easy calculation [2], using Eqs. (2.14~) and (2.14d), shows that this is the case. We may then extend the definition of S to all of 4, by continuity. It is not difficult to show that the range of S is dense in Z&t . This proves that S is unitary. To summarize, we have derived, from physical considerations, equations for the S-matrix. If the operators A, B, C, D are everywhere defined, these equations have a solution for a unitary operator S if and only if tr B+B < co. This solution is unique up to an overall phase factor. It is worth noting that the state vector, s‘y, , describing particle creation from the vacuum always has the “factorized form,” Eq. (2.27), showing independent creation of pairs of particles in “eigenmodes” hi . 2.2. The Charged Scalar Field X

We consider a charged (complex) scalar field satisfying (V, - ieA,)(Vw - ieA@) X - m2X + VX = 0 which generalizes Eq. (2.1) to include the possibility

(2.30)

of an external electromagnetic

CURVED

SPACE

499

S-MATRIX

field described by a vector potential A, . For two classical solutions x1, xz, of Eq. (2.30) the Klein-Gordon inner product,

(XIpX~)KG = i Ir

MV,

-

ie4)

x2 - x2(Vu + ieA,) n,> dxu,

(2.31)

is independent of choice of Cauchy surface C. For the free field, the one-particle Hilbert space 2 is taken to be the direct sum of S+ , the Hilbert space of positive frequency solutions of the Klein-Gordon equation, and s’?, where &‘L is the Hilbert space of negative frequency solutions (with inner product given by minus the Klein-Gordon formula to make it possitive definite) and the bar again denotes the dual operation 2 = a?+ @ 2F .

(2.32)

A vector in %I+ is called a particle state, while a vector in fl is called an antiparticle state. The Hilbert space of free-field states is again taken to be SS(S), the symmetric Fock space constructed from the one-particle space 2. Annihilation and creation operators are defined as before, Eqs. (2.5) and (2.6). However, the quantum field operator x is now given by the formula

X(f) = -a&-)

+ a+(‘+),

(2.33)

where iKf- E Z? C 2 is the vector in S< associated with the negative frequency part of the advanced minus retarded solution of the complex conjugate of the KleinGordon equation (which, of course, is the same as the Klein-Gordon equation here, where A, = 0) with sourcefand itcf+ E sS? C S is the vector in fl corresponding to the positive frequency part of this solution. In exactly the same manner as in the case of a real field, in curved space-time the “in” and “out” annihilation and creation operators must satisfy for all test functionsf -a(K:“-)

+ a+(Kr”+) = --b(K:-)

+ b+(KfF+).

(2.34)

Operators A, B, C, D can be introduced as follows: We define A, : ~?+~“t + X+in and B, : .X+out + X’ -in exactly as in the real case by taking the past positive and negative frequency parts of solutions which are purely positive frequency in the future. In an analogous manner, we define A- : &?,,“t + Kin and BP : &!&t + S&n by taking the past positive and negative frequency parts of the future negative frequency solutions. We define A : S&t -+ Zi, and B : X&t -+
+ A-P-, + B-P-,

(2.35a) (2.35b)

where P, : X&t ---f X&t is the projection operator onto X+oUt and P- is the projection operator onto ZoUt . Again, we define C : Xin -+ S&t and D : Xi, -+ sout in the “time-reversed” manner.

500

ROBERT

M.

WALD

With these definitions of A, B, C, D, it may be verified that all the relations, Eqs. (2.14), satisfied by the corresponding operators in the real case hold here as well. Furthermore, Eq. (2.34) again implies that for all crE *in a(G) = b(G)

l

- b+(Do).

(2.36)

Since Eqs. (2.36) and (2.14) were the only things used in deriving the S-matrix in the real case, all the discussion applies here as well. Specifically, if A, B, C, D are everywhere defined, the S-matrix exists if and only if tr B+B < co. It should be pointed out that although the formula, Eq. (2.24), for the twoparticle amplitude E holds here, it is not a very convenient formula since Xi will, in general, be a superposition of particle and antiparticle states. A more physically relevant formula can be obtained as follows. The operator E+E can be written as the sum of an operator which maps g+out into 2 +OUtand an operator which maps X&t into Z&t . Performing the analysis on the first of these operators separately, we obtain Eq. (2.19) where each yi lies wholly in H+out . The corresponding pi will lie in *o”t . Using the symmetry property of the full operator E, Et = E, it follows that we can express E in the form (2.37)

where (ri} form an orthonormal basis of particle states (ti+,& and {pi} form an orthonormal basis of antiparticle states (X?Out). This displays the two-patticle amplitude as a sum of amplitudes for producing pi , yi particle pairs. The full state vector still has the “factorized form” S I Od

= c exp [i C cib+(pi) b+(yi)] I but). z

(2.38)

2.3. The Dirac Field I/I

A Dirac field consists of a pair of 2-component spinor fields 5” and vA’ satisfying (in curved spacetime and with an external electromagnetic field given by the vector potential AAA,), cvAA’

-

idAA’)

(VA,.,’ - ie/fAA

5”

+

&

?A’

=

0,

(2.39a) (2.39b)

where standard spinor notation [M] has been used. For two solutions Z& = (tlA, ~~8) and & = (fZA, r],,,) of the Dirac equation, the Dirac inner product

CURVED

SPACE

501

S-MATRIX

is independent of choice of Cauchy surface C. Unlike the Klein-Gordon inner product, the Dirac inner product is positive definite on the space of all solutions (not just the positive frequency solutions). The free-field theory is constructed as follows: As for the complex scalar field, the one-particle Hilbert space X is taken to be &+ @ Y? , where %+ and XL are, respectively, the Hilbert spaces of positive and negative frequency solutions of the Dirac equation with finite Dirac norm. However, the Hilbert space of all states is now taken to be the antisymmetric Fock space constructed from X,

where OR denotes the antisymmetric tensor product. For all u E 2, annihilation creation operators are defined on FA(X) by a(6) Y = (u . 71 ) 21&7 . 72, 31&J . ys,...), a’(u) Y = (0, cu, 21&J @A 71 ) 31k7 @A 72 ,...),

and (2.42) (2.43)

where y

=

cc,

71

> 72

9 r/3

,-.I.

Since the field operator 4 is a Dirac spinor, the test “functions”fon will be dual Dirac spinors, f = (at, , p”‘). The formula for #Jis WI

= 4w)

+ at&J,+),

(2.44)

which it acts (2.45)

where wf- E XQ C 2 is the vector in Xl associated with the negative frequency part of the advanced minus retarded solution of the complex conjugate Dirac equation with source f, and We+ E &? C X is the vector in &? , corresponding to the positive frequency part of this solution. The discussion of the S-matrix for a Dirac field in curved space-time parallels that of the complex scalar field with a few sign changes and with antisymmetric tensor products replacing symmetric ones. The “in” and “out” annihilation and creation operators are related by a(w,“-) + a+(w;+) = b(&)

+ b+(w;“).

(2.46)

The operators A, B, C, D are defined exactly as in the case of a charged scalar field. Since the Dirac product is positive definite on negative frequency solutions, the relations they satisfy have several important sign changes, A+A + B+B = f,

B+,j = -A+jj, CC+ A+ =C,

D+D = I, D+C = -C+& B+ = D.

(2.47a) (2.47b) (2.47~)

(2.47d) (2.47e)

502

ROBERT

In terms of these operators,

hf. WALD

we have a(6) = b(Cu) + b+(Du)

(2.48)

and thus [b(Eu) + b+(Da)] s!Po = 0.

(2.49)

We will now outline the derivation of the S-matrix under the assumption that tr B+B < co. An important technical difference which occurs here is that Eq. (2.47c), because of the sign change, no longer guarantees that C-l exists as a bounded operator. Therefore, we divide the discussion into two cases: (i) ker C = 0. First, we note that ker C = 0 if and only if ker A = 0; Namely, if Cy = 0 for y # 0, then Eq. (2.47~) implies- Dy # 0 and Eqs. (2.47d) and (2.47e) imply that A(Dy) = C+ijg = D'Cy = 0, so Dy E ker A. Thus, if C has a nontrivial kernel, so does A. Conversely, if A has a nontrivial kernel, Eqs. (2.47a), (2.47b), and (2.47e) imply that C does also. Since tr B+B < co by the Hilbert-Schmidt theorem the spectrum of BiB and hence (by virtue of Eq. (2.47a)) the spectrum of A+A is pure point. Since ker A = 0, we have ker AtA = 0, so zero is not in the spectrum of A+A. Hence (AtA)-r = (CC-l exists as a bounded operator. But the bounded operator C+(CC+)-l is a right inverse of C, CC+(CC+)-l = I. Furthermore, C+(CC+))l could fail to be a left inverse of C only if ker C f 0 which is contrary to the hypothesis. Thus, C-l exists as a bounded operator and we may proceed --with the derivation of the S-matrix as in the scalar case. Defining E = --DC-l and paralleling the derivation of SF0 beginning with Eq. (2.49) above, we again obtain Eqs. (2.18) with aa replacing OS and with the following two additional differences: (i) the states rln are now antisymmetric and (ii) the operator E is now antisymmetric, Et = -E, by virtue of Eq. (2.47d). These two facts combine to make a consistent solution again possible for rln, provided, of course, tr B+B < 60. The same derivation as led to Eq. (2.37) now shows that the two-particle amplitude E has the form 6 = T cipi @A Yi = T MPi where {n> form an orthonormal &. The solution for 7n is rln = 0

0 Yi - Yi 0 PiA

basis of Z+ and {pi} form an orthonormal (n odd)

(n!)l/Z n/2 vn= c p/2(42)! 0x AE

(2.50) basis of (2Sla)

(n even)

(2.51 b)

and again we have the factorized form

S I Qd = c exp [B C cib+(pi)h’(y,)] 1oout).

(2.52)

CURVED

SPACE

503

S-MATRIX

An argument similar to the one given in the scalar case shows that Ij SF,, I/ is finite. The extension of the definition of S to all of 4n and the proof that S is unitary also parallel the scalar case, with anticommutators of annihilation and creation operators replacing commutators in the latter argument. (ii) ker C # 0. This case has been discussed by LaBonte [5, 171. Since tr D+D = tr B+B < co, inview of Eq. (2.47~) ker C must be finite dimensional. Let C denote the restriction of C to a map from (ker C)’ to (ran C) = (ker C+)’ = (ker A)‘. Then by the same - argument as above, C has a bounded inverse C-l. Letting T = Ca and E = -DC-‘, Eq. (2.49) again leads to Eqs. (2.18), with the further difference that equations now hold only for T E (ker A)l, not all 7 E &‘&t . On the other hand, if crE ker C, we have by Eq. (2.49) b+(Da) SF,, = 0. (2.53) Note that Eq. (2.47d) implies that A% = 0, i.e., Da E ker A. Conversely, if h E ker A, then by Eqs. (2.47e) and (2.47a) DBh = B+Bh = h with Bh E ker C by Eq. (2.47b).

Thus Eq. (2.53) can be rewritten as b+(T) SYo = 0

(2.54)

for all 7 E ker A. Equation (2.54) says that SY,, contains the state 7 with certainty. Let a1 ,..., %n be an orthonormal basis for ker A. Then the solution of Eqs. (2.18) (with 7 E (ker A)l) and Eq. (2.54) (with T E ker A) is c = 0,

(2.55a) (2.55b)

n cm, %a = 0, I = 1) 3, 5)... ) %n+z = 0,

rlm+z

=

k

(m

+

112

O!l’”

y,“(Q)!

"1

@A

.**

@A

%I

@A

EY

I = 0, 2, 4 ,... .

(2.55~)

The full definition of the S-matrix and the proof of unitarity go through as before. The state vector, Eq. (2.55), shows the creation of the particles, 01~,..., (Y, , with certainty along with amplitudes (given by c) for creating ordinary particle-antiparticle pairs. The states 01~,..., OL, may each be taken to lie wholly within JF+,,~~ (particles) or fioUt (antiparticles). If the external vector potential A, vanishes, charge conjugation symmetry of the Dirac equation shows that the dimension of ker A in %+ou+ equals the dimension of ker A in flout . In other words, Slu, will contain equal numbers of particles and antiparticles. However, if A, # 0 (or, say, if we were to consider the two-component neutrino equation), then this equality of dimension does not obviously follow. LaBonte [17] quotes Wightman as having shown that ker A = 0 for an external potential which vanishes sufficiently rapidly in the past and future, so this phenomenon probably does not occur in cases where one has a clear, unambiguous physical interpretation of the results. In cases where it does occur and particle-antiparticle equality fails, it is presumably attributable to bad definitions of “in” and “out” particle states rather than lepton and charge nonconservation.

504

ROBERT

M.

WALD

Finally, it is worth emphasizing that in constructing the quantum field theory in curved space-time, we assumed that the particle states were symmetric in the scalar case and antisymmetric in the Dirac case. Had we assumed the “wrong” statistics in either case, we would have met an immediate inconsistency: in Eq. (2.18b) r/z and E would have different symmetries and no solution for Q would exist. More generally, we have the following “spin-statistics theorem”: No reasonable quantum field theory in curved space-time (or in an external potential) with symmetric statistics exists if the conserved inner product for c-number solutions is positive definite for all solutions; no theory exists with antisymmetric statistics if the inner product is positive definite only on positive frequency solutions.

3. PROOF OF trB+B < co

We turn now to the proof that in the case of curvature (or external potential) of compact support, the conditions needed for the derivations of the previous section are indeed satisfied, namely, the operators A, B, C, D are everywhere defined and tr B+B < co. We will treat only the case of the real scalar field; with the minor modifications mentioned at the end of this section, the proof also goes through in the charged scalar and Dirac cases. We consider a globally hyperbolic [S] Cm space-time which is flat outside a compact region in the sense that the space-time outside this region is isometric to Minkowski space-time with a compact set removed. All external potentials are also assumed to vanish outside the compact region. We first show that the operators A, B, C, D are everywhere defined. Let (J E:.#&t , i.e., u is purely positive frequency in the future and has finite KleinGordon norm. To show that A is defined on (7we must show that if we propagate u into the past and take its positive frequency part, its Klein-Gordon norm will be finite. But the standard theorems [8] on Cauchy evolution assure us that in the past u differs from the Minkowski space solution (which, of course, has finite norm) by merely a smooth solution with data of compact support on a Cauchy surface in the past. It is easy to show that the positive frequency part of any such solution has finite norm. Thus A is defined for all u E &&t . Similarly, B, C, and D are everywhere defined. We turn now to the proof that tr B+B < co. We introduce the two-point distributions G and H as follows: For two test functions f and g we define G(f, g) = -(a;-,

a;+) - (c,“-, CT,“‘),

(3.1)

H(f, g) = -(c$-,

cr,“) -

(3.2)

(6;-,

a?),

where uf”, etc., were defined in Section 2. Formally, G is the “in” vacuum expectation value of 6(f) 4(g) + 4(g) 4(f), while H is the “out” vacuum. expectation value of this quantity; this identification is “formal” since it is contingent on the existence of a

CURVED’

SPACE

505

S-MkTRIX

consistent theory, which is what we are trying to prove. In any case G and H are well defined by Eqs. (3.1) and (3.2): Both G and Hare solutions of the wave equation in each variable. Furthermore for support offand g outside the future of the support of the curvature, G is a Hadamard distribution [7] since it is equal to the free-field vacuum anticommutator function which is well known [18] to be of the Hadamard form. Similarly H is of the Hadamard form outside the past of the support of the curvature. Therefore, by the theorem of Fulling, Sweeny, and Wald [7] both G and Hare Hadamard distributions everywhere. Hence, if we define the distribution S by S(vK d = W; d - H(f, g),

(3.3)

then S can be realized as a smooth function S(x, x’) S(f, g) = j S(x, x’)f(x) g(x’) dV dV’.

(3.4)

Since S(x, x’) is a solution of the wave equation in each variable we may rewrite Eq. (3.4) as

stid = - s,,,q(x)

+(x,

x’) ~u~ug(xr) d,D dZu’,

(3.5)

where C is a Cauchy surface, which, for convenience, we shall take to be a plane in the flat region in the future. For all test functions f and g, we have

s

q(x) EX.?Z

x

a

Sk

4

z

a

%7(x')

= (CT;-, of+)+ (CT;-, a;+>- (c$, CT,“) -
(3.6)

Let A, p E A&t . We can find a test function f such that jl uf”+ - h 11and /I uf” 11are arbitrarily small. Similarly, we can find a test function g such that 11u,“’ - p 11and /I uf- j/ are arbitrarily small. Hence, Eq. (3.6) implies that for all A, p E ZOut tt

-u’) g p(x’) =(BA, Ap) +(Bp, Ah) s w ; S(x, ZXT

= (0% 4)

+ (Ah, Bp)

= (X, @+A + A+B] p) = 2(X, A+&),

(3.7)

where Eq. (2.14b) was used in the last step. In a similar manner, we find that for all Ji E =%ut , p E xxlt ,

(3.8)

where Eq. (2.14a) was used in the last step.

506

ROBERT

M.

WALD

We now decompose the solution S(X, x’) into its future positive and negative frequency parts in each variable. As we shall show below, the Fourier transforms of S and its time derivatives with respect to the spatial variables x and x’ on C exist. In terms of these Fourier transforms, the positive and negative frequency parts are given by (3.9a) (3.9b) (3.9c)

(3.9d) where the first & denotes the positive or negative frequency part in x and the second * in x’ and where w = (k2 + mz)lJ2. Equation (3.7) implies that the operator A+B is given in terms of the function S- by 2A+Bp(x)

Similarly,

= i Jz S--(x,

F

x’) atl

p(x’) d%‘.

(3.10)

B+B is given by Y 2B+Bp(x) = --i s, S+-(x, x’) arl p(x’) d3x’.

(3.11)

Hence, the trace of (A+B)+(A+B) will be finite if and only if S-- E &&t @ sOut ; we have Y

+i

4 tr[(A+B)+ (A+B)] = - LXX s--(x, x’) at w

S--(x, x’).

(3.12)

Similarly, 5

2

4 tr[(B+B)+ (B+B)] = I,,, s+-(x, x’) at atl P-(x,

x’).

(3.13)

But, we have (A+B)+(/i+B) - (B+B)+(B+B) = B+(/iA+ - BB+) B -= B+(C+C - a+@ B = B+B,

(3.14)

where Eqs. (2.14e) and (2.14~~) were used. Thus, tr B+B will be finite if the right-hand sides of Eqs. (3.12) and (3.13) are finite.

CURVED

SPACE

S-MATRIX

507

We now shall prove tr B+B < cc and thus complete the proof of existence of the S matrix by the following steps: By simple estimates, we establish that all the Fourier transforms on the right-hand side of Eq. (3.9) exist and are square integrable functions of k and k’. This shows S-- and S+- are quare integrable on Z x Z. Similar arguments establish that the time derivatives of these functions appearing in the Klein-Gordon inner product in Eqs. (3.12) and (3.13) are also square integrable. This proves finiteness of the right-hand sides of Eqs. (3.12) and (3.13) since these expressions are simply sums of L2-inner products of square integrable functions. Outside the past of the support of the curvature, His equal to the flat space value of the anticommutator, d, of the free-field operator; while outside the future of the support of the curvature, G is equal to d. Let %?denote the (compact) region of the hyperplane 2 given by the intersection of Z with the future of the support of the curvature. Then by the above remark, S = G - H vanishes if both x and x’ lie in Z outside %. Furthermore, if x’ E V we may bound the behavior of S(x, x’) on Z for large 1x / as follows: The free-field anticommutator d(x, x’) decreases exponentially with the squared geodesic distance u between spacelike separated x and x’ if m # 0; in the massless case it is equal to l/g [18]. Thus, on Z, H(x, x’) is at worst 0(1/j x 1”) as 1x 1+ co. On the other hand, G(x, x’) is the solution of the (curved space) wave equation in x’ with initial data d(x, JI’) on a Cauchy hyperplane 2 in the past of the curvature (where x lies on Z and y’ lies on 2). Hence, as proven, for example, in Hawking and Ellis [8] the nth Sobolev space norm of G(x, x’) (viewed as a function of x’ on Z with x fixed) on the subset V of .X is bounded by the nth Sobolev space norm of d(x, JJ’) (viewed as a function of JI’) on %?,where @’is the intersection of the past of q with 2. From the form of A, this latter norm is (0(1/l x 1”) as j x 1+ co. Thus, there exist constants c, such that (3.15) where V’j denotes a combination of j space and time derivatives with respect to the primed variable and the sum is taken over all such combinations. But, according to Lemma 7.4.1 of [8], the absolute value of a function on % is bounded by its nth Sobolev norm if IZ > 4 dim %?= 3. (More precisely, it is actually bounded by its n > 4 Sobolev norm where only spatial derivatives are taken, but this norm is obviously bounded by the norm defined by Eq. (3.15).) Thus, there exists a constant K such that for all x E 2 and x’ E e we have

I e, .e < WI x 12.

(3.16)

Similarly, since the nth Sobolev norm of I derivatives of S with respect to the primed variable is bounded by the (n + I)th Sobolev norm of S, the same bound applies to (space or time) derivatives of S with respect to the primed variable. In the massive case, we have much stronger bounds on the asymptotic behavior of A and the righthand side of Eq. (3.16) could be replaced by a constant times exp(--m2 / x I”). 595/118/z-18

508

ROBERT

M.

WALD

We can repeat the above argument for aS/at. as/at is simply -ad/at plus the solution in x’ obtained by Cauchy evolving (aA/at)(x,f). However, since in the massless case A(t, x; t’, y’) = [-(t - t’)a + (x - Y’)~]-~, we have aA = 2(t - t’)[-(t

at

- t’)2 + (x - y’)“]-” = 0(1/l x 14)

and we obtain correspondingly stronger bounds on the Sobolev norms. The final result is that there exists a constant I? such that for all x E 2, x’ E %’ we have

This bound also applies to derivatives of as/at with respect to the primed variable. Again much stronger bounds occur if m # 0. Consider, now, the integral

I

/ S(x, x’)12 d3x d3x’.

ZXZ

We can break up this integral into a sum of four pieces as follows: s,,, = Lx,

+ Lx(r-w

+ Lnxr

+ LX~%‘g) *

(3.19)

The first integral clearly is finite since V is compact. The second integral vanishes since S(x, x’) = 0 when both x and x’ lie on 22 outside g. The third integral is finite because of the bound given by Eq. (3.16) since

s

I S(x, x’)12 d3x d3x’ < K”(vol5Q (Z-WXYP

/z--v &

d3x < co.

(3.20)

Finally, the fourth integral is finite by virtue of the analogous bound to Eq. (3.16) with x and x’ interchanged. Thus, S(x, x’) is square integrable on Z x 2. Hence, its Fourier transform $ exists and is also square integrable. Similarly, using the bounds (3.16) and (3.18) it follows that the Fourier transforms of a,!?/&, &S/at’, and iPS/at at’ exist and are square integrable. If m # 0 this implies that S+- and S--, defined by Eq. (3.9), are square integrable. However, if m = 0 the danger of an infrared divergence remains: if the Fourier transforms of %‘/i?t, B/at’, or a2S/at at’ are badly behaved near k = 0 or k’ = 0 (though still square integrable, of course), then further division by w = / k 1 or U’ could make the square integral of s+- and $-- diverge. To show this cannot happen, we note that Eq. (3.18) implies that &T/i% is an L1 function of x. Hence, its Fourier transform with respect to x is bounded, and, in particular, is finite at k = 0. Thus, at worst ~2 &/at diverges ask/j k 1 as 1k 1-+ 0 and thus is square integrable. Similarly, (w’)-l X/at and (ww’)-l a2S/at at’ are square integrable, and hence so are S- and S-, even if m = 0. Now, as+-/& is given by the same formula, Eq. (3.9b), as for S+-, except that on the right-hand side S is replaced everywhere by W/at. Exactly the same argument as

509

CURVED SPACE S-MATRIX

given above proves that %+-/at is square integrable. Similarly, all other derivatives of S+- and S-- appearing in Eqs. (3.12) and (3.13) are square integrable. Hence, the right-hand sidesof Eqs. (3.21) and (3.13) are finite, which proves that tr B+B < co. For the caseof the charged scalar field x, we define the distribution G by G(f,

2)

=

(I?,“-,

=

-(ET-,

f?f-)

+

0;‘)

(K;+,

-

((ip-,

K;+)

K,“‘),

(3.21)

where, as defined in Section 2, ids is the advanced minus retarded solution of the complex conjugate equation, while iof is, again, the advanced minus retarded solution of the original equation. G is, formally, the “in” vacuum expectation value of x(f) x+(g) + x+(g) x(f) and is a Hadamard distribution outside the future of the support of the curvature. The theorem of Fulling, Sweeny, and Waid [7] was proven explicitly only for the case of the real scalar field, but the proof depends only on the good “c-number” Cauchy evolution properties of the field equation and the ability to construct a local Hadamard parametrix, both of which hold here as well. Thus, G is a Hadamard distribution everywhere. Hence, if we define H by the sameformula with “future positive frequency” replacing “past positive frequency” and define S = G - H, the distribution S again will be realized as a smooth function. The argument given above for the case of the real scalar field may then be paralleled to prove that tr B++B+ < co where, as defined in Section 2, B, is B restricted to %+,,t . Repetition of the argument using S++ and S--‘- instead of S-- and ,S’ - shows that tr Bet BP < co also, and thus tr BtB = tr B++B, + tr B-+ BP < co. For the Dirac field we make a sign change in the definition of G, G(f, g) = (w;-, c$-) - (co;‘-, w:“‘)

(3.22)

and similarly in the definition of H, so that they are (formally) the “in” and “out” vacuum expectation values of the commutator #(f) $+(g) - #+(g) g(f). Again the results of Fulling, Sweeny, and Wald [7] hold, and the proof that tr B+B < co then parallels the charged scalar case. This completes the proof of the existence of the S-matrix for the neutral scalar, charged scalar, and Dirac fields. ACKNOWLEDGMENT I wish to thank Pong Soo Jang for helpful discussions on fermion fields at an early stage of this work. REFERENCES 1. S. W. HAWKING, Comm. Math. Phys. 43 (1973, 199. 2. R. M. WALD, Comm. Math. Phys. 45 (1973, 9. 3. R. SEILER, in “Troubles in the External Field Problem

for Invariant Wave Equations” Wightman, Ed,), Goidon and Breach, New York, 1971, and references cited therein.

(A. S.

510

ROBERT

M.

WALD

4. A. 2. CAPRI, .I. Mafhernaticul Phys. 10 (1969), 575. 5. G. LABONTE, Cunad. J. Phys. 53 (1975), 1533. 6. R. SEILER, 1977 Erice Lecture. 7. S. A. FULLING, M. SWEENY, AND R. M. WALD, Comm. Math. Phys. 63 (1978), 257. 8. S. W. HAWKING AND G. F. R. ELLIS, “The Large Scale Structure of Spacetime” Cambridge Univ. Press, London/New York, 1973. “A non-perturbative approach to quantum field 9. A. A~HTEKAR AND A. MAGNON-ASHTEKAR, theory I: interactions with external potentials,” to be published. 10. R. GEROCH, J. Mathematical Phys. 19 (1978), 1300. 11. J. HARTLE AND S. W. HAWKING, Phys. Rev. D 13 (1976), 2188. 12. H. RUMPF, Phys. Lett. 61B (1976), 272. 13. P. CANDELAS AND D. J. RAINE, Phys. Rev. D 15 (1977), 1494. 14. M. REED AND B. SIMON, “Functional Analysis,” Academic Press, New York, 1972. 15. R. M. WALD, Phys. Rev. D 13 (1976), 3176. 16. R. PENROSE, in Battelle Rencontres (C. M. Dewitt and J. A. Wheeler, Eds.), Benjamin, New York, 1968, 17. G. LABONTE, Comm. Math. Phys. 36 (1974), 59. 18. J. D. BJORKEN AND S. D. DRELL, “Relativistic Quantum Fields,” McGraw-Hill, New York, 1965, Appendix C.