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Renormalization of semiclassical field theories
Volume 91B, number 3,4
PHYSICS LETTERS
21 April 1980
RENORMALIZATION OF SEMICLASSICAL FIELD THEORIES S. RANDJBAR-DAEMI, Bernard S. KAY and T.W.B. KIBBLE Blackett Laboratory, Imperial College, London S W 7 2BZ, UK
Received 28 January 1980
We discuss a simple model which shares some of the essential features of the semiclassical theory of gravitation. In this model, a classical scalar field v interacts with a quantized field ~. We describe a consistent and unambiguous renormalization scheme for the model.
There has recently been a revival of interest in the semiclassical form of general relativity, where a classical gravitational field is coupled to a quantized matter field, with a suitably defined expectation value as the source of the gravitational field: Guy -- K2(~ lT~uvl~).
(1)
(See for example refs. [ 1 - 3 ] .) One can view this theory either as an approximation to the true quantum gravity, or conceivably as an alternative theory in its own right [4]. The model is also of interest in that it involves a nonorthodox quantum mechanics which is intrinsically nonlinear. See, in this context the recent article of Penrose [5]. Two of us [3] (here referred to as I) recently introduced an action principle from which both eq. (1) and the Schr6dinger equation governing the time development of the quantum state can be derived. Using this action principle as a guide we have begun a new approach to renormalizing this model and calculating its physical implications. This approach should at least clarify the expected difficulties. Many of these difficulties are not specific to gravity but would arise in any semiclassical model. The aims of the present note are: (i) to resolve s o m e of these general problems by applying our method to a simpler model, a renormalizable flat-space-time theory; (ii) to demonstrate the advantages of our approach as compared to possible alternatives, again in the context of this simple model; and
(iii) to describe some preliminary results for the gravitational case itself. The simple model, which we may call the v~ 2 model, consists of a classical scalar field o coupled to a quantized scalar field ~. Their interaction may be described by the action integral I = I v + I ~ + lin t ,
(2)
where
zo =fd4x (½0uvO"o I¢
~-m202) ,
=fdt[Im(O/lf)-(4Jl/tol¢)-~((flff)-l)],
lin t
=
- -
fd4x ½Xo(~ 1~21¢).
(3a)
(3b) (3c)
Here t = x 0 a n d / t 0 is the hamiltonian operator for a free scalar field of mass la. The corresponding EulerLagrange equations are i(d/dt) l¢(t)) =/4o(t)[ff(t)) - a(t)lff(t)),
(4a)
where
no(o=& + Xfd xo(x) 2(x), ([] + m 2) v ( x ) = - ½ X ( ~ ( t ) 1 5 2 ( x ) l ¢ ( t ) )
(4b)
(a is needed to give the constraint (~1¢) = 1). Like the semiclassical theory of gravitation discussed in I, this model exhibits nonlinear quantum effects (see also ref. [6]). Although the Schr6dinger equation (4a) is superficially linear, it involves the field o which ac417
Volume 91B, number 3,4
PHYSICS LETTERS
cording to eq. (4b) depends nonlinearly on the quantum state. Thus the time evolution does not respect linear superposition. However, the linearity of eq. (4a) alone suffices to permit a transformation to the Heisenberg picture, in which the field equations read [[] + U2 + Xu(x)] ~ x ) = 0,
(5a)
([-1 + m 2) u(x) = -½X(ff [62(x)lff) -- -XOff ( x ) ,
(5b)
say. Of course the operator 62 requires renormalization. The cancellation of infinities may be achieved by adding suitable counter terms to the action integral, namely
= - f d 4 x [½6m202(x) + XScbv(x)] ,
(6)
where 6m 2 and 6q~ are constants. Then the right-hand side of eq. (5b) is replaced by --XqbrCen(X)= --Xqbqj (X) -- 8m2v(x) -- X6cb.
(7)
The essence of our approach is to treat the full closed system described by the pair of coupled equations (5) consistently order by order in perturbation theory. This has the advantage, as we shall show, that ambiguities in the handling of divergences may be resolved in each order by appealing to physical renormalization conditions on the full system. This is to be contrasted with the alternative and more conventional approach (see for example refs. [1, 2]) in which the problem is split into two parts: (a) One studies the quantum 6 field in an arbitrary fixed u-field background, described by eq. (5a), and attempts to define the operator 62(x) in this background. This is the analogue of "quantization in a curved background space-time". (b) Having defined 62 one substitutes it into the right-hand side of eq. (5b) and looks for consistency. This is the "back-reaction problem". A fundamental difficulty with this latter approach occurs in defining 62. Using, for example, a point-splitting regularization technique one obtains for reasonable states I ~ ),
<~ 16(x) 6(x + e) l¢>
(8)
= -(4rr2) -1 {e- 2 - ~ [02 + Xv(x)] ln(e2/v)
+ % (x, ~)), where v is some constant with dimension of length squared and the function e, which contains the only 418
21 April 1980
state dependence, is finite as e ~ 0. To obtain a welldefined expectation value qbren, one must discard the infinite part of eq. (8). This can be achieved by choosing
6m2div = X2(16n2)- 1 In (e2/v) ,
(9a)
and 8~di v = (47r2) -1 [e -2 -- ~-/l2 ln(e2/v)] .
(9b)
However, the finite parts remain undetermined, and we have a residual ambiguity represented by the freedom to make transformations qbren(X) ~ qbren(X) + a + flu(x),
(10)
where a and/3 are arbitrary constants. (A term in (6(x)) might also appear if not excluded by symmetry. Its presence would not change our argument.) The important point is this: There is in general no way of removing this ambiguity and fixing ~ and/3 within the context of quantum theory in a fixed ubackground. A closely related though more complex ambiguity occurs for the energy-momentum tensor (.see refs. [7-9]). For quantum theories in a curved background space-time, the analogue of eq. (10)is
]'uv-+ f"uv+aguv +[3Guv+71Huv+82Huv,
(11)
in the usual notation (see e.g. refs. [9,10]). Similarly for a quantized field in a classical background electromagnetic field, we have
iu ~ ]. + aOvFuU .
(12)
Our approach bypasses this ambiguity. In each order of perturbation theory the arbitrary factors such as a and/3 in (10) are fixed by physical renormalization conditions applied to the complete theory, exactly as in conventional renormalization theory. The renormalized expectation value qbren(X) is defined by an unambiguous prescription. (See eqs. (21) and (25) below.) We could at least formally seek to solve the coupled equations (4) in a perturbation series in X, given suitable initial data at some time to, which, with the usual adiabatic switching hypothesis, may be taken to be t O = _oo. It is more convenient, however, to work with the Heisenberg-picture equations (5). We assume a zeroth-order solution, comprising a solution Vo(X) of the free KleinGordon equation together with a state Iff>in the Fock space of a free scalar field 60. We seek solutions v and 6 of the coupled equations (5) tending asymptotically to v0 and 60 as t ~ _oo.
Volume 91B, number 3,4
PHYSICS LETTERS
The expectation value qS(x) = ½(~ l~2(x)l~) appearing on the fight-hand side of eq. (Sb) may be regarded as (half) the coincidence value of the function
¢b~(x,y)= ½(@l[O(x) ¢(y)+&(y)¢(x)][~) .
(13)
21 April 1980
(For technical reasons we find it more convenient to associate the powers of the coupling constant X with the "propagators" A R rather than with the vertices.) Eq. (5b) may be converted similarly to an integral equation,
It will also be useful to define the related functions • ~(x, y) = ½(~ [[~0(X) ¢(V) + ¢(y) ¢0(X)] 1~),
(14a)
v(x)=vO(x)_xfdnx'AR(x_x';m)
~ ,) , qSren(X
and
q~Cren(X)= ½~O(x,x) + x-i6m2v(:x)+6~ .
~ o ( X , Y ) = ½@[[¢0(x) ¢0Cv) + q~0(Y)q~0(x)] 1¢).
In terms of diagrams, we have •
qbq~(x,y):
where the additional symbols are defined by:
×
~
v,
×-
%(x,y):
--y,
....
y .
Note the symbol indicating dependence on the quantum state ft. We can convert eq. (5a) to an integral equation incorporating the assumed initial conditions. For the functions defined in eqs. (13) and (14) we obtain
• *(x, y) = %*(x, y) - X f d 4 x ' AR(x - x "
p) v(x') ~ ( x ' , y ) ,
(15a)
and
• o(x,y) = % ( x , y ) - X f d 4 y ' A g ( y - y ; l ~, ) v ( y ' ) ~ o ( X¢, y ) , , (15b) where A R is of course the retarded Green's function. Diagrammatically these equations become
,=
o
@ +
+,/~
+~ .
v0(x):
o
,
)t-Qim2 :
[]
,
6q5:
®
,
--~AR(x--y, m2):
× ~ y
(16a) .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
A
(16b) where we have introduced the additional notation: -X/Xg(x -y;
~): ×
.
y,
(17)
~
,(20)
.
By combining eq. (20) with eqs. (16a) and (16b), one easily obtains an order by order solution for v(x). In our approach the counter terms 6m 2 and 6q5 (not merely their infinite parts) are determined (in each order) by imposing renormalization conditions on the full interacting system. First, we require that when v0 = 0 and Iff) is the vacuum state [~2) of the quantum field q50 then nothing happens, i.e. v(x) = 0 and q] = q~0"This requires qS~an(X) = 0 or equivalently
6q~ = -~q~oo(X,X) .
(21)
Diagrammatically, we have = - '/2 ~.~-.~"--'~ a .
.
(19)
(14b)
We shall represent these functions diagrammatically as follows:
v(x):
(18)
with, as in eq. (7),
(22)
This implies that all tadpole contributions without factors of v cancel. The second condition is essentially a mass renormalization condition for the classical field v. We again choose I ~ ) = I~2) but take v0 to be a nonzero solution of the Klein-Gordon equation. Now ~a~n no longer vanishes, so that v is not equal to v 0. Its Fourier transform g ( p ) = f d g x eip'Xv(x),
(23) 419
Volume 91B, number 3,4
PHYSICS LETTERS
is not confined, as ~0 is, to the mass shell. However, we may require that the only value o f p 2 for which has a delta-function singularity, i.e. a term proportional to 6(p 2 -- m2), is p2 = m 2. It is not difficult to see that the necessary condition for this is the cancellation o f the loop containing a single o insertion, i.e. ~2 ~-~q~o
= - -~.~
? .
(24)
(There are two symmetry-related terms on the right, making equal contributions, and thereby cancelling the factor o f ½ above.) This condition may be written
6m 2 =
-X2fd4yAR(x - - y ; / a ) @~0(X, y ) e - i p . 0 ' - x ) , (25)
for all p satisfying p2 = m 2. These renormalization conditions suffice to render any physical quantity finite and unambiguous. We may for example consider an initial state in which u vanishes asymptotically as t ~ _oo while I@) is some normalized many-particle state I~), and ask for the probability that the system will be found in some designated set o f states in the distant future. This is a meaningful question provided that o also vanishes in some suitable sense as t ~ +co, so that free " o u t " states exist (an assumption which can and o f course must be checked). It is straightforward to compute the "scattering amplitude" (X, out [~, in) for any designated process. Note however, that although such amplitudes have the usual linear dependence on final-state variables, X, they depend nonlinearly on the initial state ft. This is inevitable in a semiclassical theory (see refs. [6,11]). We could now proceed to apply this method to other more interesting models. Semiclassical electrodynamics would require only a single counter term, which may be fixed by a charge renormalization condition. The semiclassical theory o f gravity, however, presents obvious difficulties. To eliminate the divergences in the expectation value o f the stress tensor we have to add four counter terms,
= f d 4 x x/S~[~A + ~(1/K2) R(x) + 6ARZ(x) + 6BRuv(x ) RU"(x)] ,
(26)
corresponding to the four arbitrary terms in (11). The 420
21 April 1980
second o f these represents a renormahzation of the gravitational constant, and can be fixed in the usual way. The others, however, present more o f a problem. In particular the last two if present in a classical lagrangian would yield field equations o f the fourth order which have many unacceptable solutions. It is possible, however, in the context o f our approach that these terms can be constrained to vanish in some suitable sense. The first term would also pose problems. It represents a cosmological term, and on observational grounds must be required to have an extremely small value (see ref. [12]), essentially zero. To try to understand this situation better and to determine whether some satisfactory renormalization scheme for this model can be found, we have begun a detailed study o f the low orders o f perturbation theory using as expansion parameter a dimensionless ratio ~2/ot2 where a is some characteristic length defined by the quantum state. It is not difficult to show that no serious difficulties arise in the lowest nontrivial order. Whether the same techniques will be effective in higher orders is now under active investigation. We wish to thank C.J. Isham and M.J. Duff for helpful discussions. B.S. Kay wishes to thank the Science Research Council for contract no. G R / A / 61463.
References [ 1 ] G.T. Horowitz and R.M. Wald, Phys. Rev. D17 (1978) 414. [2] G.T. Horowitz, Semiclassical relativity, the weak field limit, Chicago preprint. [3] T.W.B. Kibble and S. Randjbar-Daemi, J. Phys. A13 (1980) 141. [4] L. Rosenfeld, Nucl. Phys. 40 (1963) 353. [5] R. Penrose, in: General relativity, an Einstein centenary survey, eds. S.W. Hawking and W. Israel (Cambridge U.P., Cambridge, 1979). [6] T.W.B. Kibble, Commun. Math. Phys. 64 (1978) 73. [7] R.M. Wald, Commun. Math. Phys. 54 (1977) 1. [8] P.C.W. Davies, in: GRG Einstein centennial volume, eds. A. Held and P. Bergman (Plenum, New York), to be published. [9] B.S. Kay, Casimir Effect in QFT, Phys. Rev. D, to be published. [10] S.M. Christensen, Phys. Rev. D17 (1978) 946. [11 ] R. Haag and U. Bannier, Commun. Math. Phys. 60 (1978) 1 [12] S.M. Christensen and M.J. Duff, preprint ITP-79-01 Santa Barbara.
PHYSICS LETTERS
21 April 1980
RENORMALIZATION OF SEMICLASSICAL FIELD THEORIES S. RANDJBAR-DAEMI, Bernard S. KAY and T.W.B. KIBBLE Blackett Laboratory, Imperial College, London S W 7 2BZ, UK
Received 28 January 1980
We discuss a simple model which shares some of the essential features of the semiclassical theory of gravitation. In this model, a classical scalar field v interacts with a quantized field ~. We describe a consistent and unambiguous renormalization scheme for the model.
There has recently been a revival of interest in the semiclassical form of general relativity, where a classical gravitational field is coupled to a quantized matter field, with a suitably defined expectation value as the source of the gravitational field: Guy -- K2(~ lT~uvl~).
(1)
(See for example refs. [ 1 - 3 ] .) One can view this theory either as an approximation to the true quantum gravity, or conceivably as an alternative theory in its own right [4]. The model is also of interest in that it involves a nonorthodox quantum mechanics which is intrinsically nonlinear. See, in this context the recent article of Penrose [5]. Two of us [3] (here referred to as I) recently introduced an action principle from which both eq. (1) and the Schr6dinger equation governing the time development of the quantum state can be derived. Using this action principle as a guide we have begun a new approach to renormalizing this model and calculating its physical implications. This approach should at least clarify the expected difficulties. Many of these difficulties are not specific to gravity but would arise in any semiclassical model. The aims of the present note are: (i) to resolve s o m e of these general problems by applying our method to a simpler model, a renormalizable flat-space-time theory; (ii) to demonstrate the advantages of our approach as compared to possible alternatives, again in the context of this simple model; and
(iii) to describe some preliminary results for the gravitational case itself. The simple model, which we may call the v~ 2 model, consists of a classical scalar field o coupled to a quantized scalar field ~. Their interaction may be described by the action integral I = I v + I ~ + lin t ,
(2)
where
zo =fd4x (½0uvO"o I¢
~-m202) ,
=fdt[Im(O/lf)-(4Jl/tol¢)-~((flff)-l)],
lin t
=
- -
fd4x ½Xo(~ 1~21¢).
(3a)
(3b) (3c)
Here t = x 0 a n d / t 0 is the hamiltonian operator for a free scalar field of mass la. The corresponding EulerLagrange equations are i(d/dt) l¢(t)) =/4o(t)[ff(t)) - a(t)lff(t)),
(4a)
where
no(o=& + Xfd xo(x) 2(x), ([] + m 2) v ( x ) = - ½ X ( ~ ( t ) 1 5 2 ( x ) l ¢ ( t ) )
(4b)
(a is needed to give the constraint (~1¢) = 1). Like the semiclassical theory of gravitation discussed in I, this model exhibits nonlinear quantum effects (see also ref. [6]). Although the Schr6dinger equation (4a) is superficially linear, it involves the field o which ac417
Volume 91B, number 3,4
PHYSICS LETTERS
cording to eq. (4b) depends nonlinearly on the quantum state. Thus the time evolution does not respect linear superposition. However, the linearity of eq. (4a) alone suffices to permit a transformation to the Heisenberg picture, in which the field equations read [[] + U2 + Xu(x)] ~ x ) = 0,
(5a)
([-1 + m 2) u(x) = -½X(ff [62(x)lff) -- -XOff ( x ) ,
(5b)
say. Of course the operator 62 requires renormalization. The cancellation of infinities may be achieved by adding suitable counter terms to the action integral, namely
= - f d 4 x [½6m202(x) + XScbv(x)] ,
(6)
where 6m 2 and 6q~ are constants. Then the right-hand side of eq. (5b) is replaced by --XqbrCen(X)= --Xqbqj (X) -- 8m2v(x) -- X6cb.
(7)
The essence of our approach is to treat the full closed system described by the pair of coupled equations (5) consistently order by order in perturbation theory. This has the advantage, as we shall show, that ambiguities in the handling of divergences may be resolved in each order by appealing to physical renormalization conditions on the full system. This is to be contrasted with the alternative and more conventional approach (see for example refs. [1, 2]) in which the problem is split into two parts: (a) One studies the quantum 6 field in an arbitrary fixed u-field background, described by eq. (5a), and attempts to define the operator 62(x) in this background. This is the analogue of "quantization in a curved background space-time". (b) Having defined 62 one substitutes it into the right-hand side of eq. (5b) and looks for consistency. This is the "back-reaction problem". A fundamental difficulty with this latter approach occurs in defining 62. Using, for example, a point-splitting regularization technique one obtains for reasonable states I ~ ),
<~ 16(x) 6(x + e) l¢>
(8)
= -(4rr2) -1 {e- 2 - ~ [02 + Xv(x)] ln(e2/v)
+ % (x, ~)), where v is some constant with dimension of length squared and the function e, which contains the only 418
21 April 1980
state dependence, is finite as e ~ 0. To obtain a welldefined expectation value qbren, one must discard the infinite part of eq. (8). This can be achieved by choosing
6m2div = X2(16n2)- 1 In (e2/v) ,
(9a)
and 8~di v = (47r2) -1 [e -2 -- ~-/l2 ln(e2/v)] .
(9b)
However, the finite parts remain undetermined, and we have a residual ambiguity represented by the freedom to make transformations qbren(X) ~ qbren(X) + a + flu(x),
(10)
where a and/3 are arbitrary constants. (A term in (6(x)) might also appear if not excluded by symmetry. Its presence would not change our argument.) The important point is this: There is in general no way of removing this ambiguity and fixing ~ and/3 within the context of quantum theory in a fixed ubackground. A closely related though more complex ambiguity occurs for the energy-momentum tensor (.see refs. [7-9]). For quantum theories in a curved background space-time, the analogue of eq. (10)is
]'uv-+ f"uv+aguv +[3Guv+71Huv+82Huv,
(11)
in the usual notation (see e.g. refs. [9,10]). Similarly for a quantized field in a classical background electromagnetic field, we have
iu ~ ]. + aOvFuU .
(12)
Our approach bypasses this ambiguity. In each order of perturbation theory the arbitrary factors such as a and/3 in (10) are fixed by physical renormalization conditions applied to the complete theory, exactly as in conventional renormalization theory. The renormalized expectation value qbren(X) is defined by an unambiguous prescription. (See eqs. (21) and (25) below.) We could at least formally seek to solve the coupled equations (4) in a perturbation series in X, given suitable initial data at some time to, which, with the usual adiabatic switching hypothesis, may be taken to be t O = _oo. It is more convenient, however, to work with the Heisenberg-picture equations (5). We assume a zeroth-order solution, comprising a solution Vo(X) of the free KleinGordon equation together with a state Iff>in the Fock space of a free scalar field 60. We seek solutions v and 6 of the coupled equations (5) tending asymptotically to v0 and 60 as t ~ _oo.
Volume 91B, number 3,4
PHYSICS LETTERS
The expectation value qS(x) = ½(~ l~2(x)l~) appearing on the fight-hand side of eq. (Sb) may be regarded as (half) the coincidence value of the function
¢b~(x,y)= ½(@l[O(x) ¢(y)+&(y)¢(x)][~) .
(13)
21 April 1980
(For technical reasons we find it more convenient to associate the powers of the coupling constant X with the "propagators" A R rather than with the vertices.) Eq. (5b) may be converted similarly to an integral equation,
It will also be useful to define the related functions • ~(x, y) = ½(~ [[~0(X) ¢(V) + ¢(y) ¢0(X)] 1~),
(14a)
v(x)=vO(x)_xfdnx'AR(x_x';m)
~ ,) , qSren(X
and
q~Cren(X)= ½~O(x,x) + x-i6m2v(:x)+6~ .
~ o ( X , Y ) = ½@[[¢0(x) ¢0Cv) + q~0(Y)q~0(x)] 1¢).
In terms of diagrams, we have •
qbq~(x,y):
where the additional symbols are defined by:
×
~
v,
×-
%(x,y):
--y,
....
y .
Note the symbol indicating dependence on the quantum state ft. We can convert eq. (5a) to an integral equation incorporating the assumed initial conditions. For the functions defined in eqs. (13) and (14) we obtain
• *(x, y) = %*(x, y) - X f d 4 x ' AR(x - x "
p) v(x') ~ ( x ' , y ) ,
(15a)
and
• o(x,y) = % ( x , y ) - X f d 4 y ' A g ( y - y ; l ~, ) v ( y ' ) ~ o ( X¢, y ) , , (15b) where A R is of course the retarded Green's function. Diagrammatically these equations become
,=
o
@ +
+,/~
+~ .
v0(x):
o
,
)t-Qim2 :
[]
,
6q5:
®
,
--~AR(x--y, m2):
× ~ y
(16a) .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
A
(16b) where we have introduced the additional notation: -X/Xg(x -y;
~): ×
.
y,
(17)
~
,(20)
.
By combining eq. (20) with eqs. (16a) and (16b), one easily obtains an order by order solution for v(x). In our approach the counter terms 6m 2 and 6q5 (not merely their infinite parts) are determined (in each order) by imposing renormalization conditions on the full interacting system. First, we require that when v0 = 0 and Iff) is the vacuum state [~2) of the quantum field q50 then nothing happens, i.e. v(x) = 0 and q] = q~0"This requires qS~an(X) = 0 or equivalently
6q~ = -~q~oo(X,X) .
(21)
Diagrammatically, we have = - '/2 ~.~-.~"--'~ a .
.
(19)
(14b)
We shall represent these functions diagrammatically as follows:
v(x):
(18)
with, as in eq. (7),
(22)
This implies that all tadpole contributions without factors of v cancel. The second condition is essentially a mass renormalization condition for the classical field v. We again choose I ~ ) = I~2) but take v0 to be a nonzero solution of the Klein-Gordon equation. Now ~a~n no longer vanishes, so that v is not equal to v 0. Its Fourier transform g ( p ) = f d g x eip'Xv(x),
(23) 419
Volume 91B, number 3,4
PHYSICS LETTERS
is not confined, as ~0 is, to the mass shell. However, we may require that the only value o f p 2 for which has a delta-function singularity, i.e. a term proportional to 6(p 2 -- m2), is p2 = m 2. It is not difficult to see that the necessary condition for this is the cancellation o f the loop containing a single o insertion, i.e. ~2 ~-~q~o
= - -~.~
? .
(24)
(There are two symmetry-related terms on the right, making equal contributions, and thereby cancelling the factor o f ½ above.) This condition may be written
6m 2 =
-X2fd4yAR(x - - y ; / a ) @~0(X, y ) e - i p . 0 ' - x ) , (25)
for all p satisfying p2 = m 2. These renormalization conditions suffice to render any physical quantity finite and unambiguous. We may for example consider an initial state in which u vanishes asymptotically as t ~ _oo while I@) is some normalized many-particle state I~), and ask for the probability that the system will be found in some designated set o f states in the distant future. This is a meaningful question provided that o also vanishes in some suitable sense as t ~ +co, so that free " o u t " states exist (an assumption which can and o f course must be checked). It is straightforward to compute the "scattering amplitude" (X, out [~, in) for any designated process. Note however, that although such amplitudes have the usual linear dependence on final-state variables, X, they depend nonlinearly on the initial state ft. This is inevitable in a semiclassical theory (see refs. [6,11]). We could now proceed to apply this method to other more interesting models. Semiclassical electrodynamics would require only a single counter term, which may be fixed by a charge renormalization condition. The semiclassical theory o f gravity, however, presents obvious difficulties. To eliminate the divergences in the expectation value o f the stress tensor we have to add four counter terms,
= f d 4 x x/S~[~A + ~(1/K2) R(x) + 6ARZ(x) + 6BRuv(x ) RU"(x)] ,
(26)
corresponding to the four arbitrary terms in (11). The 420
21 April 1980
second o f these represents a renormahzation of the gravitational constant, and can be fixed in the usual way. The others, however, present more o f a problem. In particular the last two if present in a classical lagrangian would yield field equations o f the fourth order which have many unacceptable solutions. It is possible, however, in the context o f our approach that these terms can be constrained to vanish in some suitable sense. The first term would also pose problems. It represents a cosmological term, and on observational grounds must be required to have an extremely small value (see ref. [12]), essentially zero. To try to understand this situation better and to determine whether some satisfactory renormalization scheme for this model can be found, we have begun a detailed study o f the low orders o f perturbation theory using as expansion parameter a dimensionless ratio ~2/ot2 where a is some characteristic length defined by the quantum state. It is not difficult to show that no serious difficulties arise in the lowest nontrivial order. Whether the same techniques will be effective in higher orders is now under active investigation. We wish to thank C.J. Isham and M.J. Duff for helpful discussions. B.S. Kay wishes to thank the Science Research Council for contract no. G R / A / 61463.
References [ 1 ] G.T. Horowitz and R.M. Wald, Phys. Rev. D17 (1978) 414. [2] G.T. Horowitz, Semiclassical relativity, the weak field limit, Chicago preprint. [3] T.W.B. Kibble and S. Randjbar-Daemi, J. Phys. A13 (1980) 141. [4] L. Rosenfeld, Nucl. Phys. 40 (1963) 353. [5] R. Penrose, in: General relativity, an Einstein centenary survey, eds. S.W. Hawking and W. Israel (Cambridge U.P., Cambridge, 1979). [6] T.W.B. Kibble, Commun. Math. Phys. 64 (1978) 73. [7] R.M. Wald, Commun. Math. Phys. 54 (1977) 1. [8] P.C.W. Davies, in: GRG Einstein centennial volume, eds. A. Held and P. Bergman (Plenum, New York), to be published. [9] B.S. Kay, Casimir Effect in QFT, Phys. Rev. D, to be published. [10] S.M. Christensen, Phys. Rev. D17 (1978) 946. [11 ] R. Haag and U. Bannier, Commun. Math. Phys. 60 (1978) 1 [12] S.M. Christensen and M.J. Duff, preprint ITP-79-01 Santa Barbara.