General Relativity and Gravitation, VoL 14, No. 3, 1982

Is Quantum Gravity Deterministic and/or Time Symmetric?

DON N. PAGE

Department of Physics, The Pennsylvania State University,

University Park, Pennsylvania 16802

Received May 25, 1981

Abstract

S. W. Hawking suggests that quantum gravity introduces a new level of uncertainty into physics by turning pure states into mixed states. Although the evidence for this informa- tion loss is based upon a semiclassical approximation and hence is not conclusive, it is interesting to examine the implications. As originally formulated, Hawking's proposal violates CPT invariance by singling out one direction of time in which pure states turn into mixed states. An alternate hypothesis is suggested whereby the theory could be time symmetric and yet allow a loss of information. In this model the universe as a whole would be an open system, and even its density matrix would not have a deterministic evolution. The question remains of how much uncertainty there actually is in quantum gravity.

In classical mechanics the state of a system is completely described by the precise values of all dynamical variables. The theory is deterministic in the sense

that for an isolated system the values at one time completely determine them for all other times, In quantum mechanics the most complete description possible is a density matr ix P, which gives the expectat ion values of all observables. Only a complete set o f commuting observables may have precise values, and that occurs only for pure states, which have t r (p 2) = 1. [States with t r (p 2) < 1 are mixed states, which do not have precise values for any such complete set.] Quantum mechanics is deterministic in the l imited sense that the density matr ix o f an iso- lated system at one time completely determines it at all other times. This evolu- t ion may be described by a linear superscattering operator $ which maps a den- sity matr ix Pi at t ime ti to a density matr ix Of a time tf. In the ordinary theory,

Pf=~(Pi) =SpiS* (1)

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where the S matrix is unitary, so the evolution preserves tr(p 2) and keeps pure states pure.

S. W. Hawking suggests [1] that black hole formation and evaporation may produce a fundamental loss of information by reducing tr(p 2). Gravitational col- lapse splits a system into a blacl~hole subsystem and a subsystem outside, each of which would be in a mixed state even if the whole system were in a pure state. As the hole evaporates away, that subsystem would be lost into a singularity, leaving the entire remaining system in a mixed state, according to Hawking. He describes the evolution by an irreversible superscattering operator ~; which does not have the form (1).

If this is so, it represents one of the most remarkable conceptual develop- ments in physics since quantum mechanics. It would say that even if an exter- nally isolated system (e.g., the whole universe) had precise values for a complete set of commuting observables at one time, after any black holes form and evapo- rate there would be no complete set with precise values, unlike the case in ordinary quantum mechanics.

Unfortunately, the evidence for this information loss is not conclusive, since it is based upon the semiclassical approximation, which can break down notice- ably even for large black holes [2]. Other arguments for information loss [3, 4] are also based upon the classical concept of a sharp distinction between the in- side and the outside of a blackhole, which may be invalid in quantum gravity [2].

But if there does turn out to be a new level of uncertainty in quantum grav- ity, that will raise questions about the time symmetry (CPT invariance) of the theory. The irreversible superscattering operator that Hawking proposes breaks CPT invariance [2-4], because tr(p 2) decreases only in the future direction of time. R. Penrose [5, 6] argues that quantum gravity should violate CPT invari- ance in order to produce the observed time asymmetry of the universe (e.g., the second law of thermodynamics). However, there is little indication that simply having an irreversible superscattering operator would solve the mystery of the apparently highly regular initial conditions of the universe, or even that there is any solution in terms of dynamical laws. Thus one might place greater emphasis on the c F r invariance of quantum field theory in flat space-time and of classical general relativity to suggest that quantum gravity should be time symmetric. R. M. Wald argues [3, 4] that, except possibly for measurements made in strong gravitational fields, the time asymmetry introduced by an irreversible super- scattering operator might be merely due to the description in terms of density matrices. But eliminating the CPT invariance this way requires giving up a view of density matrices as basic, even though they are the most complete description available in quantum mechanics and underlie the probabilities of all possible experimental outcomes.

An alternate hypothesis that can restore CPT invariance and yet retain den- sity matrices as fundamental is to abandon the assumption of a superscattering

IS QUANTUM GRAVITY DETERMINISTIC AND/OR TIME SYMMETRIC? 301

operator, if quantum gravity indeed proves not to give S-matrix evolution. Ex- cluding the possibilities of nonlinearly evolving density matrices or of persisting black holes or naked singularities, this would mean that quantum gravity would be even less deterministic than Hawking suggested, for it would not even predict a definite mixed state.

Hawking originally obtained a superscattering matrix [1 ] by assuming an S matrix from H1 to H2 | H3, where H1, / /2 , and Ha are Hilbert spaces of states on an initial hypersurface, hidden hypersurface (enclosing any holes that form and disappear), and final hypersurface, respectively. Evolving an arbitrary mixed state on H1 to/ /2 @ H3 and summing over the unobserved states on//2 gives a state on H3 and hence the superscattering matrix, which is irreversible and time asymmetric if the states on H2 are nontrivial [2]. A time-symmetric hypothesis would be to assume an S matrix from Ho | H1 to H2 | Ha, where now Ho rep- resents the space of unpredictable states that can enter our universe from a hole or singularity, and//2 represents the space of states that can be forever lost from our universe into a hole or singularity. One would not have a superscattering matrix from H1 to H3, because the density matrix pO) on H1 would not deter- mine the full density matrix p(O, 1) on H0 @ H1 that would evolve to p(2, 3) on //2 | H3, from which p(3) on Ha is obtained by contraction over//2. In this hypothesis the universe as a whole would be an open system, with unpredictable information entering from Ho and other information being lost to / /2 .

One can apply this description to Hawking's more recent formalism [7]. This uses a path integral over geometries to calculate an n-point Green's function, which is interpreted as giving the expectation values of fields in some state. To get a superscattering operator, Hawking applies annihilation and creation opera- tors on H~ to change the ingoing state to an arbitrary p(I) and uses other op- erators on Ha to determine p(3) by means of the n-point function. The resulting time asymmetry comes from preparing the state on H~ and measuring it on/ /3 . This is analogous to choosing p(O, 1) = p(O) | p(1) with fixed p(O) for an arbitrary p(1), but allowing 19 (2, 3) to have correlations between p(2) and 19 (3). One could get a time-reversed superscattering operator by preparing a state on//3 and mea- suring it on H~, which would be analogous to choosing an uncorrelated 19 (2, 3) with p(2) fixed but allowing p(o, 1) to be correlated. If there is really no S matrix from H~ to / /3 , the time-reversed superscattering operator would not be the in- verse of the original one. Other methods of preparing and measuring the state would give still different correspondences between p(1) and p(3), showing that fundamentally there would be no unique superscattering operator.

Although in principle anything could come out of rio, so that even the final density matrix p(3) could not be predicted, one might wish to make a statistical prediction of what would occur in repeated experiments with the same p(1). Even this could not be done without some additional assumptions, but one might assume an ensemble of p(o, 1), s in repeated experiments has some mean

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(p(O)) and no mean correlations between p(0) and pO), as is clone in the thermo- dynamics of systems coupled to heat baths. Then although the precise p(3) would vary from experiment to experiment with the same pO) but varying p(o, 1), one would get a definite mean (p(3)). For example, with the appropriate assump- tions (p(3)) might be given by Hawking's time-asymmetric superscattering opera- tor applied to pO). Since repeated measurements cannot distinguish between identical density matrices and a statistical ensemble of density matrices with the same mean, the sequence of experimental outcomes would be statistically the same as if there were a unique superscattering operator and final density matrix.

It would be interesting to know how much uncertainty there is in the Final density matrix for certain given initial density matrices, such as those that form black holes. This could be described by the effective dimensionality of rio and //2, or how much information can enter and leave the universe. The semiclassical approximation would give In dim//2 at least as large as the maximum entropy of the hole but would suggest that Ho arises only from the final momentarily naked singularity and hence might have a dimensionality of order unity. In a time- symmetric view one would expect both dimensionalities to be equal. Thus one might ask whether the semiclassical approximation is correctly estimating the size of H0 or of H2. The answer is not yet clear from the path integral formalism. The most conservative possiblity still remains open [2], that Ho and//2 are trivial and that full quantum gravity gives an S matrix which is both deterministic and time symmetric.

Acknowledgments

The ideas in this paper were sharpened by the author's participation in Nuf- field conferences on quantum gravity at Cambridge and Oxford and by discus- sions with S. W. Hawking, R. Penrose, and R. M. Wald. This work was supported in part by NSF Grant No. PHY 79-18430.

References

1. Hawking, S. W. (1976). Phys. Rev. D, 14, 2460-2473. 2. Page, D. N. (1980). Phys. Rev. Lett., 44, 301-304. 3. Wald, R. M. (1980). Phys. Rev. D, 21, 2742-2755. 4. Wald, R. M. (1981). To appear in Quantum Gravity 2: A Second Oxford Symposium,

eds. lsham, C. J., Penrose, R., and Sciama, D. W., Clarendon Press, Oxford. 5. Penrose, R. (1979). In GeneralRelativity: An Einstein Centenary Survey, Chap. 12, eds.

Hawking, S. W., and Israel, W. Cambridge University Press, Cambridge. 6. Penrose, R. (1981). To appear in Quantum Gravity 2: A Second Oxford Symposium,

eds. lsham, C. J., Penrose, R., and Sciama, D. W., Clarendon Press, Oxford. 7. Hawking, S. W. (1981). To appear in Quantum Gravity 2: A Second Oxford Symposium,

eds. Isham, C. J., Penrose, R., and Sciama, D. W., Clarendon Press, Oxford.