PHYSICAL REVIEW D VOLUME 45, NUMBER 8 15 APRIL 1992

Higher derivatives and renormalization in quantum cosmology

Francisco D. Mazzitelli* International Centre for Theoretical Physics, P.O. Box 586, 34100 Trieste, Italy

(Received 15 October 1991)

In the framework of the canonical quantization of general relativity, quantum field theory on a fixed background formally arises in an expansion in powers of the Planck length. In order to renormalize the theory, quadratic terms in the curvature must be included in the gravitational action from the beginning. These terms contain higher derivatives which change completely the Hamiltonian structure of the theory, not making clear the relation between the renormalized theory and the original one. We show that it is possible to avoid this problem. We replace the higher-derivative theory by a second-order one. The classical solutions of the latter are also solutions of the former. We quantize the theory, renormalize the infinities, and show that there is a smooth limit between the classical and the renormalized theories. We work in a Robertson-Walker minisuperspace with a quantum scalar field.

PACS number(s): 04.60. + n, 03.70. + k

I. INTRODUCTION

The problem of finding a satisfactory quantum theory of the gravitational interaction has been addressed from very different points of view. There are efforts directed to obtaining a well-defined quantum theory up to and beyond the Planck scale, such as superstrings and discre- tized gravity. Many works have also been done in the (old) canonical quantization of general relativity, al- though it is well known that due to renormalizability problems some modifications must be introduced at high energies. As it stands, the perturbative theory can only be considered a low-energy effective theory of a more fun- damental one.

The canonical approach [ I ] is still of interest for different reasons. On the one hand, there is some hope of giving a nonperturbative meaning to the theory through the use of Ashtekar's variables [2]. On the other hand, this theory is a convenient framework for investigating conceptual problems that arise when trying to build up a quantum theory of spacetime. In fact, in the last few years there has been considerable progress in understand- ing the semiclassical limit of the theory [3-41. It is now widely believed that, expanding the theory in powers of the Planck length, classical general relativity arises to lowest order. To next order it is possible to obtain quan- tum field theory on a f ixed background (in the Schrodinger picture).

In discussing the classical or semiclassical limit of the theory, there are still some points which need clarification. In order to analyze decoherence and corre- lations [ 5 ] (the two important ingredients of classical be- havior), it is necessary to introduce reduced density ma- trices and Wigner's functions [6]. Up to now it has not been clear which is the correct definition of these objects

*Electronic address: mazzitef@itsictp.bitnet.

45

[7]. This is undoubtedly due to the fact that we do not know how to interpret the wave function in the Universe. When obtaining quantum field theory on curved spaces form the full quantum theory, the formal derivations ig- nore the infinities which arise due to the Dresence of quantum matter fields [3,8]. I t is the aim of this paper to discuss the regularization and renormalization of these infinities in a particular model of cosmological interest. We will work within a minisuperspace describing a Robertson-Walker universe plus a quantum scalar field.

The covariant renormalitation program for quantum fields on curved spaces in the Heisenberg picture is well known [9]. At first sight, it may appear that the problem we want to address here is the Schrodinger picture coun- terpart of it. This is only partially true. There are addi- tional complications which take place because, to absorb the infinities, quadratic terms in the curvature are needed PI.

The presence of higher derivatives changes completely the Hamiltonian structure of the theory. At the classical level, a theory with higher derivatives contains more de- grees of freedom, since more initial data are needed to solve the dynamical equations of motion. At the quan- tum level, the difference is even more dramatic. Non- commuting variables in the lower-derivative theory, such as positions and velocities, become commuting in the higher-derivative theory. Although the higher-derivative terms were supposed to be only small corrections (as they will be in our case), it is clear that the transition between both theories cannot be smooth. This situation also arises in other contexts. An example is Wheeler- Feynman electrodynamics [lo- 121. There, the elec- tromagnetic interaction between two nonrelativistic par- ticles can be described by a nonlocal Lagrangian or, alter- natively, by an expansion of the form

where the term proportional to l /c2" may contain up to n derivatives of the coordinates of the particles. The

2814 @ 1992 The American Physical Society

52 HIGHER DERIVATIVES AND RENORMALIZATION IN . . . 2815

higher-order Lagrangian (1) contains unphysical "runa- way" solutions. These are the solutions that cannot be expanded in powers of 1 /c. A quantum theory based on this Lagrangian will reproduce neither the results of the nonlocal Lagrangian nor those of the fundamental theory, quantum electrodynamics. However, there is a way of avoiding the undesired properties of the higher derivatives. If we are only interested in classical solu- tions which can be expanded in powers of 1 /c, the theory can be alternatively described as a second-order Hamil- tonian system [ll-131. This system can be quantized with the usual methods and, as long as the particles are nonrelativistic, each successive term in Eq. (1) only intro- duces a small correction to the previous one.

In the context of quantum cosmology, the Robertson- Walker minisuperspace has also been studied with quad- ratic terms in the curvature [14- 161. The Lagrangian

contains first and second derivatives of the scale factor. As we mentioned before, it is not possible to take the a-0 limit in the Hamiltonian version of this theory. The Lagrangian (2) also contains solutions which cannot be expanded in powers of a. These solutions imply, for instance, the instability of Minkowski spacetime 116,171. The Starobinski program of R inflation is also based on this type of solutions [18]. At this point we cannot say if they are unphysical or not. In the case of Wheeler- Feynman electrodynamics, we know the fundamental theory. As a consequence, we have a criterion for decid- ing if a solution is physically acceptable. In the case of gravity we do not know the fundamental theory. We only know that general relativity is a good classical theory and that quadratic terms in the curvature must be included in a variety of situations as small quantum corrections. The conservative attitude then should be to consider only perturbative solutions, replace the theory by an approximated second-order one, and quantize this reduced theory. In this paper we will assume this point of view, and we will show that in this context it is possi- ble to handle the field-theoretical infinities that occur when considering a quantum scalar field.

The paper is organized as follows. Section I1 contains a brief review of the higher-order theories and the method of reduction developed in Ref. [13]. In Sec. I11 we derive the gravitational action in d dimensions for a flat Robertson-Walker minisuperspace with R terms and a matter field. We also reduce this higher-derivative theory to a second-order one. In Sec. IV we quantize the reduced theory. We solve the Wheeler-DeWitt equation using a WKB approximation and show that the infinities can be absorbed in the bare constants of the theory. We discuss our results in Sec. V. The appendixes contain some details of the calculations.

11. HIGHER-DERIVATIVE THEORIES AND THEIR REDUCTION

Consider a Lagrangian depending on a single variable q ( t ) and its first N derivatives:

the Euler-Largrange equation is

If the Lagrangian is nondegenerate, this is a differential equation of order 2N.

The canonical formulation of higher-order theories was developed by Ostrogradski [19] a long time ago. One considers q , q , . . . , q ' N p " as independent variables and defines the conjungate momenta as

The Hamiltonian is defined, as usual, through the Legen- dre transformation

and the canonical equations of motion are

An alternative procedure, which has been used in the context of general relativity [20] and quantum cosmology [14], is to substitute for the Euler-Lagrange equation (4) a set of N differential equations of order 2. I t is then possi- ble to write a Lagrangian for this set of equations. The new Lagrangian depends on N variables and their first derivatives. The usual canonical formalism can then be applied. This method is particularly useful in quantum cosmology because one can choose as independent vari- ables quantities such as the radius and the scalar curva- ture of the spacetime [14- 161.

In any case, we see that the phase space of the theory has 2N dimensions. This can be traced back to the fact that 2N initial values are needed to solve the Euler- Lagrange equations. If we quantize the theory, using, for example, Ostrogradski's method, the wave function will depend on q and its N - 1 first derivatives:

Note that q and q, noncommuting variables in conven- tional theories, become commuting if higher derivatives appear in the Lagrangian.

Consider now the case in which the higher derivatives occur as small corrections. To be specific, assume that the Lagrangian is of the form

where a is a small parameter. Higher powers in a imply the presence of higher and higher derivatives. The num- ber of dimensions of the phase space is then dependent on the order considered. Let us further assume that we are

2816 FRANCISCO D. MAZZITELLI - 45

only interested in classical solutions which can be ex- panded in powers of a, that is,

The Euler-Lagrange equation can be written, order by or- der in a, as

= 1% [ $ I - [ I ] q = q O , etc.

To solve this set of second-order differential equations, only two initial data are needed: q ( 0 ) and q ( 0 ) [21]. The phase space of the perturbative theory is then two dimen- sional. As a consequence, it should be possible to find a perturbative (in a ) canonical formalism involving a single variable and its conjugate momentum. The problem is to find the perturbative Hamiltonian and the perturbative definition of the momentum associated with q. This was in fact done in Ref. [13].

The Hamiltonian of our higher-order theory can be written as [cf. Eqs. ( 5 ) and (6)]

N n - l m

(12) n = 1 m=O

and of course is conserved. This energy depends in prin- ciple on the 2N initial data. However, substituting a per- turbative solution in Eq. (12), we get a perturbative Ham- iltonian that is only a function of two initial data. Let us denote it by Hpert . In the perturbative formalism we are looking for, H,,,, must be that generator of time evolu- tion, that is,

where the curly brackets denote the Poisson brackets in the perturbative theory. Equation (13) can be rewritten as

As a consequence, the momentum P(q,Q) and the pertur- bative Lagrangian L,,,, must be given by

It is easy to check that the Euler-Lagrange equation asso- ciated with L,,,, agrees with Eq. (1 1). The phase space is

always two dimensional, and the limit a-0 can be taken at any stage of the calculation.

I t is now straightforward to quantize the theory. In contrast to Eq. (81, the wave function will be only a func- tion of q. The inclusion of higher orders in a (and as a consequence of higher derivatives) does not modify this fact.

111. THE CLASSICAL ACTION

Let us now turn to quantum cosmology. The more general gravitational action containing up to quadratic terms in the Riemann tensor is

where we have included a cosmological constant. We work in d + l dimensions in order to regularize the theory. The generalized Einstein equations associated with this classical action are

G ( R , ~ - - ~ R ~ , ~ + A ~ , ~ ) + ~ H ~ , ' + ~ H ; ~ + ~ H , ~ = O ,

(1 8)

where

We will consider a flat Robertson-Walker universe with the line element

Here t is the conformal time and a = e Z 6 is the scale fac- tor. As this metric is conformally flat, without loss of generality we can set y=O in the classical action. Indeed, one can prove that, for conformally flat space- times (see Appendix A),

In d = 3 dimensions we have an additional linear relation between these tensors, the generalized Gauss-Bonnet theorem, so H:',' is proportional to H E . However, as we are working in d dimensions we must keep B f 0.

For the Robertson-Walker metric, the gravitational ac- tion can be written as

HIGHER DERIVATIVES A N D RENORMALIZATION IN

where we omitted a global volume factor. As expected, the classical action depends on the first two derivatives of the scale factor. According to Eq. (12), the Hamiltonian associated with this action is

This Hamiltonian is proportional to the left-hand side (LHS) of the temporal-temporal component of the Ein- stein equations (18) for the Robertson-Walker metric. Indeed, using the geometric identities of Appendix B one can show that

H =O is a classical constraint, which we would obtain by including a lapse function in Eq. (22) and varying the ac- tion with respect to this function.

As anticipated, we will treat the higher-derivative terms as small perturbations. The constants a/G and P/G in the action (24) are our small parameters. All of the formalism reviewed in the previous section can be ap- plied to the Robertson-Walker Lagrangian without change. The perturbative energy can be obtained from Eq. (25). As we will work up to the first order in a/G and P/G, all we have to do is replace the higher deriva- tives appearing in the exact Hamiltonian using the zeroth-order dynamical equation. This is given by

and, of course, it coincides with the trace of the ordinary Einstein equations for a Robertson-Walker universe. From Eqs. (25) and (27) we get

where the functions F,(QI) are linear in a,B and are given by

Note that in exactly three spatial dimensions F, vanishes, and only a particular combination of a and fl appears in

the Hamiltonian. This is because the relation = 3 3 ~ ' ~ ) i ,, s valid at d = 3 .

The canonical momentum conjugated to 4 is, accord- ing to Eqs. (15) and (28),

In terms of the canonical variables 4 and P4, the pertur- bative Hamiltonian reads

We stress again that we are neglecting terms quadratic in a/G and B/G. The time reparametrization invariance of the theory is restored by imposing Hpert =O.

Before going to the quantum version of this constraint, let us add matter to our model. For simplicity we will consider a free scalar field, minimally or conformally cou- pled to the curvature. The action is

where c = O (minimal coupling) or (= 1 (conformal cou- pling). These two values simplify the canonical structure of the theory [22]. Indeed, after the substitution q+exp[ - +( d - 1 )4+]q, the action becomes

As there is no mixing between the temporal derivatives of p and 4, the matter canonical Hamiltonian is proportion- al to the temporal-temporal component of the energy- momentum tensor. It is given by

2818 FRANCISCO D. MAZZITELLI

H~ = + e - ~ ( ~ - I J ! I - C J defined by

where II, is the momentum conjugated to p. It is convenient to work with the Fourier modes The matter Hamiltonian can then be written as

where nk is the momentum conjugated to the Fourier mode pk .

Our classical theory is defined by the perturbative gravitational part plus the matter contribution (which of course does not contain higher derivatives). The classical constraint associated with the reparametrization invari- ance of the complete theory is then

In principle, the presence of the matter field modifies Eq. (27). However, as we are considering the scalar field as a quantum field, the corrections introduced in (27) will be of the same order as those associated with the higher- derivative terms. For this reason we are not including them. Strictly speaking, this is equivalent to reducing the theory taking G - ' as the parameter of expansion [the analogue of a in Eq. (9)].

IV. RENORMALIZATION IN THE WHEELER-DEWITT EQUATION

According to the Dirac's quantization procedure, at the quantum level the classical constraint (37) becomes an equation on the physical states. That is, the wave func- tion \V($,pk ), which describes the state of the Universe, must satisfy

This is the so-called Wheeler-DeWitt equation (WDWE). The operator H,,,, + HM is the classical Hamiltonian with the usual substitutions

beginning. We will look for WKB solutions of Eq. (38):

where

The splitting in Eq. (40) is ambiguous. For example, siO' can be absorbed into X. To fix these ambiguities we will impose the additional conditions

The first condition is necessary in order to obtain a Schrodinger equation for the matter wave function X. The second one fixes the Berry's phase between the fast (matter) and the slow (gravitational) degrees of freedom [23-251.

Inserting the ansatz (40) into Eq. (38) and using the conditions (42) we find, up to O ( G - ' ) terms,

We will choose the simplest factor ordering, placing all and the derivatives on the right.

In previous works, where the action (24) has been e-d!d- lJ d S - i 0 ax-

quantized without reduction, the wave function has an 2 d ( d - 1 ) d 4 a4 ( H , - ( H M ) ) x . (44)

additional dependence on derivatives of 4. Moreover, the coefficients in the WDWE are singular in the limit The interpretation of the above equations is clear and a,P+O. Instead, in our case the only gravitational vari- has been discussed at length in the literature. If we iden- able in the wave function is 4. In the limit a,P-0, the tify dReS/d+ with the momentum Pd, Eq. (43) is the WDWE (38) is well defined and agrees with the one de- classical constraint (31) modified by the back reaction of rived without including the higher-order terms from the the quantum field [26]. Moreover, introducing a time

45 - HIGHER DERIVATIVES AND RENORMALIZATION IN . . .

variable through the above-mentioned identification,

Eq. (43) becomes the temporal-temporal Einstein equa- tion

Here, the higher derivatives are replaced by lower deriva- tives as in Eq. (28) .

Using the definition (45) , Eq. (44) becomes

which is the Schrodinger equation for the matter wave function.

As long as the back reaction is neglected, no renormal- ization is needed. However, when the back reaction is considered, it is necessary to handle the infinities that ap- pear in the mean value ( H M ) for d - 3 . We expect the divergences to be absorbed into the bare constants of the theory. We set

where the subindex R stands for "renormalized." As the bare constants enter into the phase of the wave function we will also have to define a renormalized phase.

To proceed, we must isolate the infinities in ( HM ) . As we are working in d dimensions, our theory is already regularized. To renormalize it, we will use the adiabatic subtraction [27] . This method of renormalization has been extensively used in the context of quantum field theory on curved spaces. It has been described in the Schrodinger picture in Ref. [28] .

Using the Gaussian ansatz

the Schrodinger equation (47) becomes

where

o $ = k 2 + m 2 e 2 <

It is convenient to define

Xexp [ - i l ~ , ( t ) d t ] / d ~ ) . (53)

With these definitions, Eq. (50) takes the form

This equation can be solved iteratively as

Note that each iteration contains two time derivatives more than the previous one. The maximum number of time derivatives of the scale factor in a given iteration is called the adiabatic order.

From Eqs. (49) , (52) , and (53) , it is easy to find an ex- pression for ( HM ) in terms of W k ( t ). The result is

The renormalization prescription is the following. One computes ( HM ) using the iterative solution (55) and Eq. (56) up to the fourth adiabatic order. The renormalized mean value is defined as

The computation of ( HM is long but ~trai~ht'forward. We refer the reader to Ref. [28] for the details. The re- sult is

( H M ) a d 4 = ( ~ M ) : g + ( H M )!:4 , (58)

where

2820 FRANCISCO D. MAZZITELLI - 45

and

In Eq. (59), p is an arbitrary mass scale induced by the di- mensional regularization. The last term in Eq. (60) can be written in terms of H&' and H%' using the linear rela- tion (23). It is finite due to the identity

valid in conformally flat spacetimes. The important point for us is that (H, )ad4 can be

written as

where the coefficients a, b, c, and e are finite in the limit d-3. From Eqs. (461, (57), and (62) we see that the infinities in ( H M )can be absorbed into the bare con- stants. The renormalized equation is

2 e 2 4 ( ~ R ~ o o + ~ R ~ R g O O + a R ~ & ) + ~ R ~ g ' )

where the constants GR, A, , a, , and pR are functions of the scale p introduced by the regularization.

We define the renormalized phase of the wave function through

S, = G , S , + R ~ S \ ~ ' + ~ , R ~ S ~ ~ ) + B ~ R ~ S \ ~ ' . (64)

From Eq. (45) we obtain, for d = 3 ,

As a consequence, Eq. (63) can then be written as

+(H,),,,=o . (66)

This is the renormalized version of the constraint (43).

V. CONCLUSIONS

Let us summarize the results obtained in this paper. We analyzed the flat Robertson-Walker minisuperspace with a quantum scalar field, including quadratic terms in the curvature. In contrast to previous works [14-171, we reduced this higher-order theory before quantization. We obtained the new Wheeler-DeWitt equation and solved it using the WKB approximation. As usual, in this approximation the phase of the wave function satisfies the classical Hamilton-Jacobi equation modified by the back reaction of the quantum matter field. The back-reaction term is divergent. We showed that the divergences can be absorbed into the bare constants of the theory.

It is worth noting that it is possible to set a , =OR =O in our final equation (66). In this case we have

We see that there is no trace of the higher-order terms. Equations such as (67) have been studied in Refs. [29,30]. Here we derived it within the context of quantum gravi- ty. The reduction of the higher derivatives was crucial for the derivation. Otherwise, the phase of the wave function would be a function of 4 and R , even in the limit a, ,pR +O. Moreover, this would be a singular limit in the Hamilton-Jacobi equation.

T o regularize the theory we worked in d spatial dimen- sions. We then defined the renormalized value of ( HM ) by using an adiabatic subtraction. This method produces covariant results. For example, it gives the correct answer for the trace anomaly. For this reason, it is more adequate than noncovariant methods based on the inser- tion of cutoffs in the sum over modes of the quantum field. This was also pointed out in Refs. [29,30].

In our treatment we have considered only pure WKB states. As we have mentioned r261. in order to discuss . *.

properly correlations and decoherence, the analysis should be based on the use of Wigner's functions and re- duced densitv matrices. The Wheeler-DeWitt eauation should be then replaced by a master equation for the re- duced density matrix. It would be interesting to see if the infinities that appear in the master equation and in the Wigner's function can be handled using the techniques developed here.

45 HIGHER DERIVATIVES AND RENORMALIZATION IN . . .

Let us finally discuss possible generalizations of this work. Our main tools have been the reduction mecha- nism and the renormalization prescription. The reduc- tion mechanism presented in Refs. [11,13] can be applied to general situations. In principle, there should be no obstacles to applying it to the action (17) without freezing out any degree of freedom. The extension of the renor- malization prescription is, however, more complicated. Of course, it should be straightforward to extend our re- sults for interacting fields on flat or closed [31] Robertson-Walker universes. Moreover, it should also be possible to consider some Bianchi spacetimes. However, the generalization to nonhomogeneous spaces seems non- trivial. What is needed is the analogue of the Schwinger- DeWitt expansion of the two-point function in the Schrodinger picture. In this context, the use of embed- ding variables [32] may be useful for making explicit the covariance of the theory.

ACKNOWLEDGMENTS

I would like to thank Professor Abduos Salam, IAEA, and UNESCO for financial support.

APPENDIX A

Here we prove Eq. (23). The square of the Riemann tensor can be written as

where C,,, is the Weyl tensor defined by

The Weyl tensor vanishes if and only if the spacetime is conformally flat. From Eq. (Al ) and definitions (19), (201, and (2 1) we obtain

The tensor A,, comes from the variation of the first term in Eq. (Al). As this term is quadratic in the Weyl tensor, its variation vanishes for conformally flat spacetimes. Thus, Eq. (23) follows.

APPENDIX B

For the Robertson-Walker line element (22) we

~ , = d $ ,

The explicit expressions for the tensors H:;, H:;, and H:: defined in Eqs. (1 9), (201, and (6 1 ) are

For the Robertson-Walker spacetime we have

The tensor HI: can be obtained from the relation

which follows from combining Eqs. (23) and (61).

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2822 FRANCISCO D. MAZZITELLI 45

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