On the possibility of interpreting quantum mechanics in terms of stochastic metric fluctuations

Volume 137, number 1,2

PHYSICS LETTERS A

1 May 1989

ON THE POSSIBILITY OF INTERPRETING QUANTUM MECHANICS IN T E R M S O F S T O C H A S T I C M E T R I C F L U C T U A T I O N S S. B E R G I A ~, F. C A N N A T A ' a n d A. PASINI Dipartimento di Fisica, Universita di Bologna, Via Irnerio 46, 40126 Bologna, Italy

Received 17 October 1988; accepted for publication 27 January 1989 Communicated by J.P. Vigier

A model based on five-dimensional gravity for stochastic metric fluctuations taken as a possible subquantum background leads to a multiplicative stochastic differential equation for the geodesic displacement between pairs of test particles. An approximate solution for very short correlation time of the fluctuations is compared with quantum mechanics.

1. Introduction

After the discovery made by Nelson [ 1 ] that the evolution of a quantum mechanical system can be described in terms o f a stochastic process, there have been many attempts to reach a physical interpretation of this process. Nelson himself has wondered whether " q u a n t u m fluctuations are just ar real as thermal fluctuations and arise from certain interactions" [2]. This possibility has been recently examined in detail by Smolin [3]. Metric fluctuations, a priori, seem to be a good candidate because of their universality. In this respect they look more promising than the electromagnetic fluctuations o f stochastic electrodynamics [4]. The interest for the subject was recently revided (for a list of early references, see ref. [5] ). A paper by Frederick [ 6 ] suggested an approach in terms of stochastic perturbations of the geodesic equation, and indicated that the r a n d o m motions induced by the metric fluctuations occur at the velocity of light. The latter idea was implemented in a modified version of the model of causal interpretation of q u a n t u m mechanics in terms o f a fluid with irregular fluctuations (developed by Bohm and Vigier [ 7 ] ) proposed by Vigier [ 8 ], from which it was shown that the K l e i n - G o r d o n equation can arise from a stoAlso at INFN, Sezione di Bologna.

chastic process [9] (see also Guerra and Ruggiero [ 10 ] ). Vigier and coworkers therefore argue that a quantum-relativistic evolution equation can be derived by stochastic fluctuations o f the spacetime metric. Fluctuations of the spacetime metric as possible source o f stochasticity have also been considered by Diosi [ 11 ], Diosi and Lucacs [ 12 ], Roy Choudhury and Roy [13 ], but in a different context and with the aim o f studying the behaviour of quantum mechanical systems affected by random external forces, a problem that must be kept distinguished from the one discussed here. In this note we do not aim at providing a global theory; nor do we intend to give an introduction to the subject which would supersede previous attempts. Rather, we intend to call attentions to some aspects which, in our view, should be taken into account in any attempt to pursue the programme outlined above, and to point out difficulties which at the present stage seem hard to avoid. Tho this end we present an explicit model in which: (i) the metric fluctuations are conformal, i.e. they preserve locally the causal structure of spacetime and survive in the local inertial system of free-falling test particles (see section 2); (ii) the geodesic displacement between two test bodies (see section 3 ) is considered the most natural observable affected by these fluctuations. We obtain for the geodesic displacement a stochastic differential equations similar to the

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multiplicative stochastic differential equations analysed by van Kampen [14]. The equation is then solved under simplifying hypotheses on the driving stochastic process (section 4). Our results can be summarized as follows: (i) Ehrenfest's theorem of quantum mechanics is violated, in as much as the expectation value of the geodesic displacement does not obey the classical equation; the kinetic energy of test particles is not conserved; (ii) there is no way to obtain the inverse dependence on the mass of the test bodies of the diffusion coefficient, as a consequence of the fact that the equation of the theory of gravitation are at this stage independent of the mass of the test bodies. These difficulties, though made evident by the use of a specific model, seem hard to avoid in any attempt at interpreting stochastic quantum mechanics in terms of metric fluctuations. In section 5 we make a preliminary analysis of the modifications which should be introduced in an attempt at overcoming the above difficulties. One of the features that should be properly understood is the role of the fluctuation-dissipation theorem [ 15 ], in view of the non-dissipative character of the quantum fluctuations that should also be obtained. These aspects will be dealt with in more detail in a subsequent publication, in which quantitative aspects of our model will also be presented.

2. A model for conformal metric fluctuations

1 May 1989

function on manifold:

the

four-dimensional

L(x") = ~ y~/sZ(x") ds =~'/2(xU) ~ ds=~/2(xJ')L ,

to

Z ~ ( x ) = q~(x) -~/3

(g*u~(x) + A,,(x)Ap(x)~(x) X \ A~(x)~(x)

A,Ax)~(x)'] • (x) / ' (3)

where g ~ and q~ are defined as

g*u~=~J/2g,,,, ,

(4)

@=~3/2.

(5)

This reparametrization ensures that no extra factor of qb multiplies the four-dimensional curvature scalar R in the dimensionally reduced theory [ 17 ]. As is usual in modern approaches, the extra dimension is here taken seriously, namely non-zero modes (i.e. depending on the fifth coordinate) in the Fourier decomposition over the compact extradimension are retained both for the metric components as well as for matter fields (if any). One starts by considering a background solution, for instance (~/p~ is the Minkowski metric + - - - ):

g~,,-rl~,

~(x)

q~, ~g*u~ = qu~ + h~,~, ]h~,~[ << 1,

A~,=O,

qb=~c=COnst=t.

(6)

Next one considers a weak-field approximation for the actual field, namely (7)

and

A~,(x)~(x)'] ~(x) .1'

q ~ c ~ q , = q,c + qr = 1 + q r

(1) where x = ( x u, xS), /~=0, 1, 2, 3. Au(x) can be interpreted as the electromagnetic potential, ~(x) is a scalar field which, if it can be taken independent of x 5, governs locally the size of the fifth dimension. Indeed, the length L(x ~) of the fiber becomes a 22

(2)

where L is the length of the Kaluza-Klein theory (see, e.g., lectures by Orzalesi [ 16 ] ). A change of variables is often considered, leading

Conformal fluctuations of the spacetime metric arise naturally in a Kaluza-Klein [ 16 ] approach. In the five-dimensional case, the metric tensor YABcan always be parametrized in the form [ 17 ]

= (gu~(x)+Au (x)A~(x)~(x) \ A~(x)~(x)

spacetime

(8)

(A u is still assumed zero for simplicity). Eqs. ( 3 ) - (8) form the basis of the customary approach to five-dimensional quantum gravity [ 17 ], in which the corrections in (7) and (8) represent quantum fields on a classical background. In our case we develop a classical weak-field approximation to

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relation times of the noise applying the procedure discussed by van Kampen [ 14,27 ]. We have already mentioned at least one physical reason, namely universality, which characterizes our approach with respect to stochastic electrodynamics. it should now also be stressed that it leads naturally to a distinct analytical set-up, namely a multiplicative rather than an additive stochastic differential equation. The next question is why bother to keep all the terms in expression (13) for R'ook throughout, instead of expression ( 15 ) for flat spacetimes (after all ordinary quantum mechanics generally neglects the effects of gravitation). The main reason for this is that for the perturbative solution of eq. (18) a nontrivial reference behaviour for the system in the absence of the stochastic perturbation is needed. Since there is no geodesic displacement in a flat spacetime (as a matter of fact in the absence of non-uniform gravitational fields), eq. (18) trivializes. The behaviour of our solution for a flat spacetime obtains as a limiting case from the general solution. This feature enables us in principle to analyse, in a definite framework, an interesting view recently expressed by Smolin [ 3 ] with respect to the non-dissipative character of the quantum fluctuation, which appear to elude the fluctuation-dissipation theorem. According to Smolin, quantum fluctuations are ordinary statistical fluctuations: the typical non-dissipative character of quantum fluctuations would emerge only for a selected class of reference frames, namely local inertial systems. To any other observer they will display the character of ordinary thermal fluctuations. This possibility is related to the Unruh effect [28]; accelerated observers would find for any background field a thermal spectrum or, in other words, everything would go as if their detector was in a thermal bath. The interplay between fluctuation and dissipation is transparent in the case of an oscillating forced electric dipole, for which the macroscopic friction force in the equation of motion of the source system requires the existence of a dissipative system acting on it with a stochastic force [ 15 ]. The fact that our gravitational detector, when it responds oscillator-wise (like, for instance, when hit by a gravitational wave [29 ] ), is not damped, led us to speculate [ 30 ] that the fluctuation-dissipation theorem would in principle be eluded in our ap-

1 May 1989

proach, or that, in some way, we would have nondissipative fluctuations. Furthermore, supposing this condition realized in the rest system of the "oscillator", accelerated observer would find a thermal spectrum, so that Smolin's argument would be fulfilled. Note in fact that the Unruh effect is not necessarily a quantum field theory effect, and would in principle be exhibited by our stochastic background field 0 [31 ]. In our framework the transition from the general to the non-dissipative case, or from finite to zero temperature, corresponds to taking the limit of our solution as hu~--,0. As we will see in the next section things do not work in this way. Possible ways of improving the situation are indicated in the final section.

4. Aspects of our approximate solution We will limit ourselves to determine (in an approximation valid for short correlation times of the external noise) the expectation value ( u t ) of the process ut and of its first moments from eq. (18). We extract for book-keeping purposes from A 1 in eq. (18) a parameter a measuring the intensity of the fluctuations. In the hypotheses made (A~, is a stationary process), (A~) is time-independent. We also assume, for simplicity, stationary gravitational fields (Ao also time-independent). Then we redefine in eq. (18)

Ao +o~(A~ ) =A'o-,Ao,

(21)

with constant Ao. If ~Tc<< 1 (To is the autocorrelation time of the external noise), the expectation value ( u , ) obeys, to second order in o~Tc [ 14 ], the equation d(U,)dt

-(A°+a2i(A~'exp(zA°)Al('-T) o

×exp( -

zAo)) d r )

(u,) .

(22)

The essential features of the solution for ( u , ) can be grasped in a one-dimensional simplified version of the problem, namely with R (°) and R,(O) in eqs. (19) and (20) expressed by a constant ro and a stochastic process rt(¢). From eq. (22) 25

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PHYSICS LETTERS A

d (<~>

1 May 1989

The analogue of eq. (23) is now

=[(~o O)+a2(c,0(t)

d<(2~) - 2 ( ¢ ( ) , dt

'~q(<~.>'~

0 c2(t)J/k<(># '

(23,

dt

-<

)+O12[g21(t)((2)+g22(t)(~)]

with c, ( t ) =

d(~2)

---7 s i n h ( ~ o r) c o s h ( ~ o z)

=a2[ga,(l)(f2)"l-g33(t)(~2)]

(32)

dt

0 X/f°

where g2, (t), g22(/), g31 (t), g33(/) are given by × d r ,

(24) g2~ ( t ) = ; 2r d r ,

c2(t) = - ~ 1 sinh: ( x ~ ° r) dr. O ro

0

0

(25) g22(/) =g33(/) = -- i 2z2(rt(O)rt-~(~) ) d r ,

Eq. (23) can be re-expressed as

0

ol2c2(t) d<~> _ [ro+tX2c~(t)]<~>=O

d2<~> dt 2

~

(26) If u obeys the differential equation (18), its moments UaU b obey d(uaub) dt

Y Ao&cd(UcUd)

-

(27)

c.2

with (28)

Aab,cd -~ Aacfbd + Abdt~ac .

In the one-dimensional case. eqs. (27) can be cast in the form [ 14 ] ~

= (Bo +o~2B,)

,

(29)

with Bo =

B,=

0 2ro r( )

0 2r(0)

,

(30)

,

(31)

where again a factor o/2 has been extracted from BI for convenience. To eq. (29) the same procedure applies [ 14 ] that was used for eq. (18). In this case we shall limit ourselves to the free-particle case ro=0. 26

g3, (t) = i 2 d r .

(33)

0

We shall consider the case in which the coefficients C1( t ) , C2(/), g21 ( l ) , g22(t) = g 3 3 ( / ) and g31 (l) can be

considered as slowly varying functions of t to be taken at a certain constant value. Let us first of all observe that eq. (26) describes a damped motion, with a drag force determined by c2. This is by no means unexpected in the theory of multiplicative stochastic differential equations. Van Kampen, for instance [ 14 ], stresses that "if one has a non-dissipative system described by Ao... the additional term, due to the fluctuations, is usually dissipative" (see, e.g., the case of a stochastically perturbed harmonic oscillator [14]). Note, however, that this result does not mean that we are dealing with a dissipative system in the sense of having velocitydependent forces: in fact there is no explicit friction term in the stochastic differential equation. The "transfer of motion" between the system of test particles and the medium is confirmed by the third of eqs. (32), which shows that the kinetic energy ½m ( 2 of a test particle relative to the other (neglecting the reciprocal influence! ) is not conserved. This means that we do not have thermal equilibrium, and that the thermal bath picture advocated in connection with Smolin's mechanism in our preliminary analysis cannot be implemented in our scheme at this stage.

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137, number

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relation times of the noise applying the procedure discussed by van Kampen [ 14,271. We have already mentioned at least one physical reason, namely universality, which characterizes our approach with respect to stochastic electrodynamics. it should now also be stressed that it leads naturally to a distinct analytical set-up, namely a multiplicative rather than an additive stochastic differential equation. The next question is why bother to keep all the terms in expression (13) for R’OOkthroughout, instead of expression ( 15 ) for flat spacetimes (after all ordinary quantum mechanics generally neglects the effects of gravitation). The main reason for this is that for the perturbative solution of eq. ( 18 ) a nontrivial reference behaviour for the system in the absence of the stochastic perturbation is needed. Since there is no geodesic displacement in a flat spacetime (as a matter of fact in the absence of non-uniform gravitational fields), eq. ( 18 ) trivializes. The behaviour of our solution for a flat spacetime obtains as a limiting case from the general solution. This feature enables us in principle to analyse, in a definite framework, an interesting view recently expressed by Smolin [ 3 ] with respect to the non-dissipative character of the quantum fluctuation, which appear to elude the fluctuation-dissipation theorem. According to Smolin, quantum fluctuations are ordinary statistical fluctuations: the typical non-dissipative character of quantum fluctuations would emerge only for a selected class of reference frames, namely local inertial systems. To any other observer they will display the character of ordinary thermal fluctuations. This possibility is related to the Unruh effect [ 28 ] ; accelerated observers would find for any background field a thermal spectrum or, in other words, everything would go as if their detector was in a thermal bath. The interplay between fluctuation and dissipation is transparent in the case of an oscillating forced electric dipole, for which the macroscopic friction force in the equation of motion of the source system requires the existence of a dissipative system acting on it with a stochastic force [ 15 1. The fact that our gravitational detector, when it responds oscillator-wise (like, for instance, when hit by a gravitational wave [ 29 ] ), is not damped, led US to speculate [ 301 that the fluctuation-dissipation theorem would in principle be eluded in our ap-

LETTERS

1 May 1989

A

preach, or that, in some way, we would have nondissipative fluctuations. Furthermore, supposing this condition realized in the rest system of the “oscillator”, accelerated observer would find a thermal spectrum, so that Smolin’s argument would be fulfilled. Note in fact that the Unruh effect is not necessarily a quantum field theory effect, and would in principle be exhibited by our stochastic background field @ [ 3 11. In our framework the transition from the general to the non-dissipative case, or from finite to zero temperature, corresponds to taking the limit of our solution as h,,-+O. As we will see in the next section things do not work in this way. Possible ways of improving the situation are indicated in the final section.

4. Aspects of our approximate solution We will limit ourselves to determine (in an approximation valid for short correlation times of the external noise) the expectation value (u,) of the process uI and of its first moments from eq. ( 18 ). We extract for book-keeping purposes from A, in eq. ( 18) a parameter LIImeasuring the intensity of the fluctuations. In the hypotheses made (A,, is a stationary process), (A,) is time-independent. We also assume, for simplicity, stationary gravitational fields (A, also time-independent). Then we redefine in eq. (18) &+a(A,

> =Ab-tAo,

(21)

with constant Ao_ If err, K 1 (2, is the autocorrelation time of the external noise), the expectation value (u,) obeys, to second order in c17~ [ 141, the equation co

(A,, ew(~Ao)Al(,-., 0

Xexp(-TAO))

dr

(u,)

.

(22)

The essential features of the solution for (u,) can be grasped in a one-dimensional simplified version of the problem, namely with R(O) and R,(@) in eqs. ( 19) and (20) expressed by a constant r. and a stochastic process r,( @). From eq. (22) 25

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d2~ v

dr 2 = R~ook~k '

(14)

with R'ook given by eq. (12). In the case of a conformally flat space-time (hu~=0), one gets 02¢ R,OOk= 1 ( 1 + ¢ ) -~ OxkOx/

i#k,

R'ooi=0.

(15)

The symmetry between space and time has disappeared from eq. (14); it comes then all the more natural to restrict to non-relativistic motions (in eq. (14) the proper time r becomes then the coordinate time), with the aim of reproducing non-relativistic quantum-mechanical results. We will focus our attention on the local behaviour of a pair of test particles, considering eq. (14) at a given point in space, and studying the evolution of the solution ~i in time. Up to this point the fields ¢ ' and ¢ have been considered as purely classical and the Riemann tensor components in eqs. (13) and (15) are thought of as being computed accordingly. We now come to the stochastic properties of ¢. Note, first of all, that its stochastic character arises naturally in a five-dimensional framework read along the lines first indicated by Einstein and Berg~ann [ 16 ]: with the extra dimensions taken seriously, as in eqs. ( 1 ), (3), one is bound to consider fluctuations of the metric components about the average, which is read as the background value. We therefore assume that, at a given instant of time, ¢ is a random variable on a sample space ~ (wef~ labeling the elementary outcomes [23]), such that a stochastic process ¢,(09) arises when the random variable 0 is used to describe the state of the environment at reach instant of time (see ref. [23], p. 40). It seems also physically plausible to attribute to 0(09) (a particular samplepoint is picked, letting t vary over a certain interval) continuous sample paths [23 ]. More specific requirements to be imposed on 0 are Gaussian character and stationarity, in particular 0 and its functions appearing in eqs. ( 13 ), ( 15 ) will be assumed to have expectation values constant in time and such that their correlation functions depend only on the time difference [23 ]. An additional requirement has to do with the correlation times of 0. It is clear that a wide separation exists between the time scale of the atomic or nuclear phenomena and the time scale at which fluctuations of 0 take place. The 24

1 May 1989

latter is roughly fixed at Planck's unit of time for the same reason the scale of distance is roughly fixed at Planck's length (see section 1 ). This situation is often idealized in the assumption that the environment is a white noise. However, analyses of the quantum conformal noise [19,24] indicate a natural ultraviolet cut-off determined by a "residual" length. Due to the formal analogy between quantum and stochastic fields, it it plausible that this feature would manifest itself also in our approach. Now, if the environment is not a white noise, as seems to be suggested by physical reasons, the Markovian character of the temporal evolution of the system interacting with it is a priori lost [23]. Already at this stage, therefore, our formal scheme does not coincide with that of Nelson [1 ]: it seems however worth investigating how for this set-up can go in reproducing quantum mechanical features. R'0ok in eq. (13) can now be split in two terms, R'ook= R ~O~iook+ Riook( O ) ,

(16)

with R ~O~,oo~ independent of ¢. If a vector u with components, u"= ~',

a = 1, 2, 3,

=~,,

a = 4 , 5, 6,

(17)

is introduced, eg. (14) can be rewritten in the form /J=Au= (A 0 +AI )u,

(18)

with Ao=

(0 '0) R¢O~

(19)

and

Both Ao and A~ are functions of time, denoted, according to the convention previously introduced, as Ao=Ao(t), A~ =A~,(¢). Eq. (18) is a multiplicative stochastic linear differential equation, driven by the "certain" matrix Ao and the "random" matrix A~ [ 14 ]. Under the above assumptions for ¢, difficulties associated with the white character of the noise [25,26] can be avoided [27], and it will be legitimate to look for a solution valid for very short cor-

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For a ~ 0 , eqs. (23) and (32) represent a simplified version of the classical equations of motion of a system with one degree of freedom, q - ( , and conjugate momentum p = m(. Indeed, in the free-particle case we are considering ( t o = 0 ) , one has dq2/ dt=2qp/m, d(qp)/dt=p2/m, which coincide respectively with the first and the second of eqs. (32). The third equation expresses, with a = 0 , the conservation of the kinetic energy. Note that we kept ro~0 in eqs. (23), and of course ro~0 implies curvature, hence non-vanishing acceleration in the geodesic displacement. Our expectation values are to be compared with quantum mechanical expectation values, which (Ehrenfest theorem) obey the classical equation of motion. Our expectation values violate, in general, the Ehrenfest theorem, whereas one would expect (for flat spacetime) a vanishing acceleration for the expectation value while having a non-vanishing stochastic acceleration. This illustrates a very peculiar feature that fluctuations capable of reproducing quantum mechanics should manifest.

5. Concluding remarks On physical grounds one finds reasons for the lack of equilibrium expressed by eqs. (23) and (32). The point is that in our approach there is, to this stage, no room for a reaction of the test particles on the field. In these conditions there must be a net transfer on energy from the medium to the pair of test particles and equilibrium is impossible. Similar indications arise from the general result on the violation of the classical equation of motions by the expectation values. Agreement with the Ehrenfest theorem would have to be achieved by means of a compensating mechanism. We already know that our stochastic equation should be modified to account for the reaction of the test particles on the medium. This extra contribution should be fine-tuned to the multiplicative noise in such a way as to exactly cancel modifications of the classical equation of motion. This might seem hopeless, but is looks unavoidable in any attempt at reproducing quantum mechanics in terms of metric fluctuations. The necessity of an additional term describing the reaction of the test particles on the field is evident also for another reason: we want to reproduce a dif-

1 May 1989

fusion coefficient in h/m (m = mass of the test particles). This cannot be achieved in the framework of eq. (18), which is independent of the mass. This dependence could however arise if the back reaction of the test particles on the metric is taken into account. Indeed the geodesic postulate can be used only for the motion of test particles [ 32 ]: for (a pair of) extended massive particles whose retroaction on spacetime is not neglected, the geodesic (displacement) equation should be modified. Under a Lorentz boost a spherical distribution becomes ellipsoidal and acquires a quadrupole moment, thus becoming capable of emitting gravitational waves. The power emitted is larger the higher the mass of the particle. One thus expects that as consequence of this mechanism, metric fluctuations would give rise to larger average displacement in a given time the smaller the mass of the particle, which could give rise in principle to a diffusion coefficient inversely proportional to the mass. Apart from these qualitative arguments, a modification of our scheme in agreement with a rigorous treatment of radiation reaction in G R [33] must be worked out. The last aspect discussed entails another unavoidable conclusion (this point has been stressed in general terms by Ghirardi, Omero, Rimini and Weber [ 34 ] ); the hypothetical background medium of stochastic mechanics cannot be independent of the test particles: it must be, in some way, "prepared" by them. The notion of a random medium to some extent prepared by particles travelling through it has also been considered by Garbaczewski [35], although this author considers the idea only applicable to material media, where "physical reasons of randomness can be explicitly identified". Here we have indicated a physical mechanism for the general (nonMarkovian) case.

Acknowledgement We thank F. Guerra for advice and helpful comments and R. Bergamini and A. Barletta for useful discussions.

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References

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