Quantum gravity and Brownian motion

Volume 139, number 5,6

PHYSICS LETTERS A

7 August 1989

Q U A N T U M GRAVITY A N D B R O W N I A N M O T I O N ~r Claus K I E F E R

Instttutfur TheorettschePhystk, UmversttatHetdelberg,Phtlosophenweg19, D-6900Hetdelberg,FRG Received 22 May 1989; accepted for pubhcatmn 12 June 1989 Communicated by J.P. Vlgler

The emergence of classical properties in quantum gravity can be treated analogously to the theory of Brownlan motion. The local scale factor x/h of a three geometry, which plays the role of an lntnnsic time variable in the Wheeler-DeWltt equation corresponds to the coordinate of a Browman part,tie, while the conformal part of the three geometry corresponds to the heat reservom It ~sshown that for small values of the scale factor, where the strong couphng hmlt G--,ooapphes, there is no interaction between intrinsic time and conformal degrees of freedom As the interaction is switched on, the conformal degrees of freedom lead to a suppression of interferences for different scale factors, and intrinsic Ume emerges as a classical quantity. A master equation for the reduced density mamx similar to Brownian motmn ts derived and discussed.

Since the advent of q u a n t u m field theory in the twenties, the construction o f a consistent q u a n t u m theory o f gravity remains elusive. Despite this failure, there has been some improvement since DeWitt [1] published his pioneering work on canonical q u a n t u m gravity. This approach is especially suited for studying conceptual problems and cpsmological applications. What are the characteristic features o f the canonical theory? The fundamental kinematical concept is a wave functional 7' on the space o f all spacelike three geometries which can be represented by the three metrics hab(X). It obeys the Wheeler-DeWitt equation HI ~ ) = 0 which is a second order partial differential equation o f the hyperbolic type. This becomes explicit if one chooses the scale part ~ hab --v/h and the conformally invarlant part ~ab- h-~/3h,b as coordinates [ 1 ],

(6nGh 2 C4

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16nGh 2

N/Fh ( oN/ Irl) -~

c2

16~G x/~(3)R

- -

--

c ' x / h hachbd~

) ku[x/~, h~bl = 0 .

~2 ab cd

(I)

¢, Selected for Honorable Mention by the Gravity Research Foundation (1989)

Here (3)R is the Ricci curvature of the three geometry. The cosmological constant was set equal to zero. Although there is no external time parameter, an "initial" value problem with respect to an intrtnstc timelike variable (which for this choice o f variables is given by x/~) may be formulated. This novel concept o f time has important consequences for the behaviour o f wave packets in q u a n t u m cosmology [2,3]. Furthermore, the issue o f boundary condittons plays a central role in the theory. Although there have been appealing proposals in recent years to fix them [4,5 ], the final decision about this point remains elusive. It is clear, however, that no valuable theory will result without having an idea o f how to impose sensible boundary conditions on ( 1 ). The canonical approach is able to provide a profound understanding o f how classical properties emerge from q u a n t u m gravity [ 6-8 ]. The configuration space may be divided into a relevant part (a " s y s t e m " ) and a less relevant part (an "environm e n t " ) whose interaction forces the system to become classical by suppressing interferences between different values o f the quantity being "observed". This is called continuous measurement and has been proven fruitful in q u a n t u m mechanics [ 9 ].

0 3 7 5 - 9 6 0 1 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )

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The physical content of this mechanism can be illuminated by drawing an analogy between quantum gravity and the theory of Browman motion. The local scale factor x/h of the three geometry - which is the relevant part for cosmology [ 10,11 ] - may be compared to the coordinate x(t) of the Brownian partlcle while the conformal invariant part hab may be compared to the heat reservoir wherein the particle is immersed. The coupling of the particle to the heat bath corresponds to the coupling of x/hto/~a~ through (3)R in the Wheeler-DeWitt equation (1). An ~mportant property of this potential term m ( 1 ) is that it vanishes as x / h ~ 0 . This can easily be estimated by comparing the curvature term with the kinetic terms. L being a typical length scale, one has (3)Roe1/L 2 and x/hoe L 3. Therefore the ratio of the potential and kinetic term becomes less than one for L < h~c 3 which is the Planck scale. We are thus left with a free theory for L ~ 0 . This is referred to as the strong couphng hmtt of gravity which corresponds to choosing either G--.oo or c ~ 0 (light cones shnnking to straight lines in the four dimensional theory). As there is no coupling between different space points in this limit, one is confronted with an ultralocal theory which may be formulated by taking independent Kasner universes at every space point [ 12 ]. One cannot solve the relevant equation ( 1 ) in full generality, but one can study an approximation of a closed Friedmann universe with small perturbations on it [ 13 ]. The intrinsic time variable becomes the usual scale factor a which interacts with higher multipoles through the Riccl tensor in ( 1 ). The kind of couphng determines which quantities can become classical and which not. It was shown in refs. [7,8] that the modes force a to become classical, as expressed by the reduced density matrix p(a, a') which is obtained from the total wave function by tracing out all multipoles x,, Xn

(a-a')2). e x p ( - ~N3

(2)

Here N is a cut-off in the number of modes, and q/o is a solution of the mlmsuperspace background,

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O _a4)gto(a ) = 0 . Hoq/o(a)=-(a f-~ a-~a

(3)

Unimportant phase factors have been neglected in (2). One can immediately recognize from (2) that interferences between different scale factors are exponentially suppressed. The remaining width becomes zero in the limit of an infinite number of modes, but a cut-offat the Planck scale ( N ~ a) leads to a finite width which decreases with increasing a

t8]. The reduced density matrix does no longer obey a unitary von Neumann equation. As is known from irreversible thermodynamics, it has to obey a master equation like Boltzmann's stosszahlansatz. This master equation can be obtained by differentiating (2) with respect to a and a'. A short calculation yields, using (3), 2 0 2p

a Oa--~ - a

,2 0 2p

~

Op

(9,0

+a~a - a ' ~

N 3 a2-a '20p + 3 a' Oa

-a4p+a'4p

N 3 a'2-a 20p 3 a aa' = 0 .

(4)

The first line is the von Neumann equation [p, Ho ], the second hne expresses the influence of the conformal degrees of freedom. One can simplify (4) further by taking a ~ a' because of the exponential suppression in (2):

2N 3 [p, Ho] + ~

Op ( a - a ' ) O--a

2N 3 ( a ' - a )

--y-

~0p= 0 .

(5)

This can be compared with the master equation of a Brownian particle with coordinate x (and unit mass) in a heat bath with linear coupling ("linear response theory") which reads [ 14 ]

Op(x, x' ) 1 Op Ot - i[p' H] - ~ , ( x - x ' ) ~x

p(a, a' ) = T r a c e I ~v) ( ~uI ~o(a)~(a')

7 August 1989

+y(x-x')

0_.__p_p- 2 y k B T ( x - x ' )2p.

Ox'

(6)

Here y is the damping coefficient of the particle (divided by its mass), T is the temperature of the heat bath and kB is Boltzmann's constant. Companng (5) with (6) shows that in the gravitational case the

Volume 139, number 5,6

PHYSICS LETTERS A

damping is given by the number o f interacting modes, 7= 2N3/3, which vanishes for vanishing coupling. The "temperature" attributed to the conformal part of the three metric vanishes, a fact that may be understood by remembering that the " e n v i r o n m e n t " is in a pure state. It is also important to remember that the multipoles leading to the special form (2) were in their ground state [8 ]. This was a consequence o f the Hartle-Hawking path integral proposal and may change by using another boundary condition which leads, for example, to a coherent state. A suitable method to pick out a sensible boundary condition m a y be to demand that it lead to classical properties in accordance with our experience. The total wave function ~ determines which quanttttes become classtcal and which do not. This point of view differs from previous proposals [4,5 ] where it is only demanded that they should lead to appealing classical trajectories, for example to those containing an inflationary phase. Q u a n t u m theory in its usual form, however, does not predict probabilities for classical paths [ 2 ]. The issue o f boundary condition is also connected with the problem o f the origin o f the arrow of time [ 15 ]. A boundary condition o f low entropy at small scale factor would enable one to derive the second law o f thermodynamics [ 15 ]. As argued abOve, for small scales the interaction via (3)R in (1) is effectively switched off and no classical properties emerge if a separating initial condition is imposed. At the Planck scale, therefore, q u a n t u m effects o f gravity are essential, as is usually argued in a more intuitive way. The total density matrix therefore reads in this limit Ptot =P,/~®P~ab •

(7)

When the interaction in ( 1 ) is switched on, it may

7 August 1989

be treated as a perturbation. The usual scattering picture [ 16,17 ], however, is not suitable, because it preassumes an external time parameter. Instead one has to deal consistently with the wave equation ( 1 ) as was done in the investigation o f wave packets [ 3 ]. In the q u a n t u m theory o f gravity, concepts o f thermodynamxcs, q u a n t u m theory and space ( - t i m e ) physics, the central issues o f Einstein's famous 1905 papers, emerge in a unified fashion and deserve further study. I would like to thank H. Dieter Zeh for critical comments.

References [ 1] B.S. DeWltt, Phys. Rev. 160 (1967) 1113. [2] H.D. Zeh, Phys. Lett. A 126 (1988) 311. [3] C. Klefer, Phys. Rev. D 38 (1988) 1761. [4] J.B. Hattie and SW. Hawking, Phys. Rev. D 28 (1983) 2960. [5] A. Vdenkm, Phys. Rev. D 37 (1988) 888. [ 6 ] H.D. Zeh, Phys. Lett. A 116 ( 1986 ) 9. [7] C. IOefer, Class. Quantum Gray. 4 (1987) 1369 [8] C. Kaefer, Class. Quantum Grav. 6 (1989) 561. [ 9 ] E. Joos and H.D. Zeh, Z. Phys. B 59 (1985 ) 223; H.D. Zeh, Found. Phys. 1 (1970) 69. [10] R. Brout, F. Englert and E. Gunzlg, Ann. Phys. (NY) 115 (1978) 78. [ 11 ] F. Giirsey, Ann. Phys. (NY) 24 ( 1963 ) 211. [ 12] M. Pilate, m: Quantum structure of space-ume, eds. M. Duff and C.J. Isham (Cambridge Umv. Press, Cambridge, 1982). [ 13] J.J. Halhwell and S.W. Hawking, Phys. Rev. D 31 (1985) 1777. [ 14] A.O. Caldeira and A.J Leggett, Physlea A 121 ( 1983) 587. [ 15 ] H.D. Zeh, The physical basis of the direction of time (Spnnger, Berlin, 1989) [16] M.P. Ryan, Hamdtoman cosmology (Springer, Berhn, 1972). [ 17] M. Pdatl, Phys. Rev. D 28 (1983) 729

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