On the quantization of gravity

Volume 94B, number 4

PHYSICS LETTERS

25 August 1980

ON THE QUANTIZATION OF GRAVITY M. NOURI-MOGHADAM 1 and J.G. TAYLOR 2

Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA Received 2 June 1980

We analyse the construction of quantum states for the case of gravity and matter using the solution of the 2-submanifold boundary value problem and "sum over paths" quantisation. This leads to a specification of such states in terms of a complete commuting set of operators. The "sum over topologies" definition is obtained only by a very ad hoc assumption which is made precise. The problem of the arbitrariness of the background metric is discussed and resolved by analogy with QED.

There have been many attempts in the past to quantize the gravitational field [1 ]; none have yet been successful. We may isolate at least three possible reasons for this: (i) The quantized self-interacting gravitational field theory is perturbatively non-renormalisable [ 1 ]. (ii) It is difficult to obtain non-perturbative results for quantum gravity, due to unsolved mathematical problems concerning the classification o f solutions to Einstein's equations and related questions [2]. (iii) It is not clear precisely which objects should be calculated in the first place, since the very fabric o f s p a c e - t i m e is being allowed to fluctuate [3]. Recent developments [4,5] have led some to hope that supergravity may produce a perturbatively renormalisable quantum field theory or at least a theory with unambiguous S-matrix elements. Others [2] have proposed that non-perturbative calculations may save the day. Be that as it may, it would seem that even prior to, or at least hand in hand with, such detailed analyses as are required to construct such a renormalisable quantum gravity we must also attempt to understand what it is that should be calculated. In other words we must try to clarify (iii) above. Indeed it may be that such a study may lead us to a very different 1 Research supported in part by Fulbright Senior Fellowship Grant. On sabbatical leave from Department of Mathematics, Tehran University of Technology, Tehran, Iran. 2 Permanent address: Department of Mathematics, King's College, London, UK.

program from those currently being studied. At the very least we will be able t o expose the detailed assumptions which have gone into developing these and any other possible programs for quantising gravity. That is the purpose o f this paper. We start by quantising the gravitational field following the lines of ref. [3]. There it was assumed that (a) quantum mechanics as the basis o f dynamics does not allow any preferred classical fields, either matter or metric and (b) dynamics must be expressed in a coordinate independent fashion. Clearly (a) intially avoids the thorny problem o f which preferred classical field to choose; (b) is surely acceptable to all. From this we .conclude that an action can only be written down in terms o f the antisymmetric tensors available; if these have values in a Lie algebra we may obtain [6] the Einstein-Caftan theory o f gravity interacting minimally with matter in Yang-Mills form. This initially is to be written in first-order form, and can be expressed most elegantly in the language o f forms. Thus E i n s t e i n Cartan theory has as Lie algebra that o f Sp(4) and action density dA = Tr 75 [DWl ^ DCOl] ,

(1)

where co 1 = dxU(eff'), a +Buabaab), Dco 1 = dco 1 + ico 1 A COl, eua and Bu ab being the vierbein and spin connection, respectively. The quantisation of such a classical system on an arbitrary background manifold appears to be best achieved [3] b y introduction o f states defined by the eigenval487

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ues of the appropriate antisymmetric tensor fields. For definiteness let us first discuss the pure gravitational field case with action density (1). Then we expect states of the system to be given as eigenstates of a complete orthonormal set of observables on a 3dimensional submanifold o. These cannot include all of the components of eua and B# ab, since these will be non-commuting as quantised variables. We argue by analogy with dynamical systems of a finite number of degrees of freedom or for matter fields. In these cases we can determine the complete anticommuting set by principle of uniqueness: choose those observables for which the 2-point or 2-submanifold boundary value problem (for finite numbers o f degrees of freedom or the field case, respectively) has a unique solution. By the 2-point boundary value problem we denote the choice of boundary values x (a 1) = x 1, x2(a2) = x 2 for the single degree of freedom x(t) regarded as a function of time. We may justify this specification of a complete commuting set of observables for a general dynamical system as follows. Suppose the total set of dynamical variables for the system (when expressed in first order form) are denoted by 3`. Then states I , t) at time t are defined by the overlap functions (neglecting spacetime curvature effects) ( ,t21 , t 1)=

[3']exp

A[3']

.

(2)

Eq. (2) is not yet complete, however, since the labels on the 1.h.s. and their limits of integration on the r.h.s. have not been given. To do that we may subdivide the set 3' into 2 subsets a,/3 with the property that specification of the values of the variables in a at t 1 and t 2 gives a unique solution of the equation of motion

a state at any time t, and being an instantaneous specification the corresponding quantized operators defined by (2) will commute. (See ref. [3] for further details of the definition of quantised operators associated with (2).) In the gravitational case the 2-submanifold boundary value problem (usually called the "sandwich problem") is not completely clear, especially since we may have to deal with manifolds with singularities. However, the complete commuting set of dynamical variables on a given 3-submanifold is expected to be effectively the conformal 3-geometry 3g and the trace of the extrinsic curvature Tr K on the surface [7] *1 Using the result we may therefore define the Hilbert space of states [3g, Tr K, a) by the path integrals (3g 2 , T r K 2 , o213g 1, T r K 1, a 1)

= f [Ide ~ exp(~/'

i -- 1 , 2 .

3 r

g lai

We suppose that the set a is taken as a maximal set with the above properties, though it may not be unique. Then the states I , t) are made precise by inserting the labels o f the observables in a at time t; the definition (2) is completed by requiring that the integral is to be taken over the functions a, t3 wRh ~(ti) = ~i. It is appropriate to call the set a o f variables a complete commuting set, since they give a maximal specification of 488

=

3g i ,

TrKIoi=TrK i .

(5)

If matter fields are present then further integration over them must be included in (4) with similar restrictions as in (5). There should also be extra terms in the measure to ensure general coordinate invariance; these are ignored in (4) for simplicity but their complete form is given in ref. [3]. From the uniqueness of the classical solution ecl, Bcl of Einstein's equations interpolating the boundary values (5) we may write the expression (2) as (3g 2, T r K 2, o213gl , T r K 1, 01 )

= (3)

(4)

where the integration is over all values of e and B with

•

with boundary values

dA ) ,

Ol

5A/~3" = 0 ,

a(ti) = a i ,

25 August 1980

f [ I d h exp

0"2

)

[dA(h + g c l ) - A ( g e l ) ]

,(6)

al

where h +gel stands for e and B in (6). Thus (6) is the standard expression for quantum fluctuations on a given background space-time, and the matrix (6) can be de*1 York [8] has obtained the outline of a proof that such is the case for suitably modified variables closely related to ag and Tr K on 3-submanifolds a without boundary, in the case of thin-sandwich problems. Further boundary conditions on o must be specified if be ~ 4~.

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composed into a set of matrices, one for each gcl, with appropriately varying values of Ol, a 2 and boundary conditions (5). Each of these matrices can be evaluated by perturbation theory about gcl in the standard fashion, though with the complication of finite times in general. We now see that we still have to face the problem raised earlier of choice Ofgcl. Though we live in a particular choice Ofgcl there seems to be no reason why this should be the case. Moreover this particular solution in our case has a singularity in it, so making difficult the extension of (6) to the case. It may be [9] that singularities are avoided in a recently developed theory of gravitation which contains an antisymmetric tensor field [10]. Other methods, such as complexification [2] or retrodiction from final states [11] may be possible. However, these approaches still do not begin to answer the question of why this particular gcl has to be used. There appear to be at least 2 possible ways to resolve this difficulty. The first of these is to sum (6) over all gcl' using the extra factor exp[(i/h) fg~ dA(gcl)] as the appropriate measure. This had been cancelled from (4) to produce (6), so indeed is correctly to be inserted before such summation is attempted. However, the summation would not be possible in (6) even after insertion of the extra factor since it would violate the boundary conditions which depend on a particular gcl" Nor does it seem possible to avoid this problem by taking o l and a 2 to be at time _~o and +co respectively and define a vacuum transition with hlai = O, since gcl has still to be chosen. It may be claimed that due to ignorance of the particular value Ofgcl , summation over gcl should indeed be performed by the method remarked above, leading to (after cancellation of an infinite factor) f d [ h ] e x P ( h f d A [h]) .

(7)

It is this expression which has been used in ref. [2]. However, we see that the choice of weighting factor in the summation over gcl is quite unjustified. In the 1-dimensional case if the values o f x 1 and x 2 are uncertain we can define

(~2t21~it,) =fdx 1

,l(x )fax:

X/d[x] exp(~A[x])

,

x(ti)=x

~,~%) i .

(8)

25 August 1980

However, ffl and if2 in (8) are arbitrary; their choice in a particular situation is obtained by other criteria such as given experimental situations. The similar expression in ref. [3] gives a very different expression to (7); the expected replacement of (7) should be @ 2 a 2 l ~ l a 1) = f d 3 g 1 d T r K 1 ~l(3gl , TrK1)

×

fd3g 2 d T r K 2 ff~(3g2, TrK2)

×fdedBexp(~f dA(e,B)). 02

(9)

Ol

Thus further advance when using (9) requires the development of a theory specifying the wave functions ~01, ~2; this would indicate why a certain region of the space of classical solutions is weighted more heavily than others. But by analogy with the one-dimensional problem of eq. (8) such a theory would be a radical departure; for example, it would require a theory constraining the way experimentalists set up their apparatus in ordinary quantum mechanics. It is presently unavailable. By contrast the other approach relies on argument by analogy with quantum electrodynamics. There background field configurations including charge distributions or point charges are not considered in the fully interacting theory. These distributions are expected to arise only taking account of the charged particle fields. By principle of parsimony the background field is chosen to be zero (except when a constant value may produce a lower energy vacuum, as corresponds to the case of spontaneous symmetry breaking). We might suppose that the same happens in quantum gravity. In that case we take the background as the flat one of Minkowski. The question as to why we observe a nonflat background in the universe is to be answered by appeal to the effect of non-trivial matter vacuum and configuration fluctuations about flat space-time. Progress has been made recently [12] in the analysis of such effects in gauge theories on a lattice, for example to achieve confinement in simplified models; we propose that similar effects in general relativity may produce a non-trivial effective background metric. Such a program has answered the question of which gcl by the answer: the trivial one as bare background, but a presently unknown though expectedly non-trivial one as effective background. Such a background could have confining properties, so would then be compact. We 489

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hope to report on the more detailed analysis of such a possibility elsewhere. In summary we see that we may construct a quantisation theory o f the gravitational field by careful analogy with other quantum theories using the uniqueness o f the 2-subboundary value problem. Such an approach leads automatically to the choice of a preferred metric on which the effects o f fluctuations o f the metric and other fields is to be considered. By analogy with QED we argue that this background is flat; and that we must search for a non-trivial effective background b y considering the effect of metric fluctuations producing a phase transition as in gauge theories. We would like to thank Princeton University for its hospitality while this work was completed and Professor York for a helpful discussion and communicating his results prior to publication.

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25 August 1980

References [1] See, for example, Isham, Penrose and Sciama, eds., Quantum gravity (Clarendon, Oxford, 1974). [2] S.W. Hawking, Phys. Rev. D18 (1978) 1747; Nucl. Phys. B144 (1978) 349. [3] J.G. Taylor, Phys. Rev. D19 (1979) 2336. [4] S. Christensen, M. Duff, G. Gibbons and M. Ro~ek, Vanishing one-loop 13function in gauged N > 4 supergravity, ITP-80-14, Santa Barbara preprint (March 1980), unpublished. [5] J.G. Taylor, On the ultraviolet-divergence of superfield supergravity, Proc. EPS High energy physics conf. (CERN, Geneva, 1979). [6] J.G. Taylor, Phys. Rev. D18 (1978) 3544. [7] C. Misner, K. Thorne and J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973). [8] J. York, private communication. [9] J.W. Moffatt and J.G. Taylor, in preparation. [10] J.W. Moffatt, Phys. Rev. D19 (1979) 3554; 3562. [11] B. Kay, Commun. Math. Phys. 71 (1980) 29. [ 12] J. Kogut, Progress in lattice gauge theories, Les Houches Conf. on Common trends in particle and condensed matter physics (Feb. 1980).