Nonlinear corrections to quantum mechanics from quantum gravity

Volume 154, number 5,6

PHYSICS LETFERS A

8 April 1991

Nonlinear corrections to quantum mechanics from quantum gravity Orfeu Bertolami Centro de Fisica da Matéria Condensada, Av. Prof Gama Pinto 2, 1699 Lisbon Codex, Portugal Received 3 October 1990; revised manuscript received 25 January 1991; accepted for publication 6 February 1991 Communicated by J.P. Vigier

Nonlinear corrections to the Schrodinger equation are shown to arise in minisuperspace models within quantum cosmology. These corrections are at nuclear physics scale many orders of magnitude below the recently established bounds. They could be observed ifthe present capability to measure the detuning ofresonant transitions were extended to the study of very heavy bound states with energy of about 102 TeV.

Indirect experimental evidence suggests that likewise the other known interactions, gravity must be quantized [1,21. Unfortunately, quantum effects become competitive with the classical field theory of gravity only at the Planck scale M~= 1.2 x 1 019 GeY. Within the canonical Hamiltonian formalism to quantize gravity [3,4], an important consequence is that the Schrodinger equation (in general, the Tomonaga—Schwinger equation [51) emerges as the evolution equation of small subsystems of a universe whose size is sufficiently large. This is achieved by adopting the so-called minisuperspace approximation and a probabilistic interpretation of the wave function of the universe [3,6]. The Schrodinger equation arises then in the semiclassical approximation to the Wheeler—DeWitt equation. Nevertheless, the resulting equation contains higher-order nonlinear corrections which are of course suppressed by powers of the Planck mass, leaving again little hope that they may have any observational implication. However, since the subject of nonlinearities

equation have been repeatedly suggested in the literature [10—20].In most of these generalizations the main aim is to incorporate, within the framework of QM, observers and their associated consciousness processes [11], in a way that is independent of the interpretation of the QM formalism, as well as to obtam a measurement theory automatically from the QM fundamental equation. The hope is to avoid the problematical features associated with the collapse of the state vector [16,171. On the other hand, complex nonlinearities were considered as they have implications for the cosmological constant problem [21] and for the reduction ofthe wave function [22]. Let us now turn to the derivation of the Schrödinger equation and its nonlinear corrections within quantum cosmology. Our starting point is the Wheeler—DeWitt~ equation and the minisuperspace approximation which turns the superspace finite dimensional. In particular, we are interested in the homogeneous minisuperspace models in which the three-metric h0 and matter fields, generically de-

in ordinary quantum mechanics (QM) has been recently revived [71, stimulating a series of precision tests aiming to establish the upper limits ofthese corrections [8,9], we believe that it is not completely devoid of interest to discuss nonlineanties in the Schrodinger equation arising from quantum cosmology. Real nonlinear generalizations of the Schrödinger

noted by 0, are independent of the position. As we shall see, the Schrodinger equation arises by considering inhomogeneous perturbations about a homogeneous minisuperspace background.

0375-9601/91/S 03.50 © 1991

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We shall not consider nonlinear and nonlocal terms in the Wheeler—DeWitt equation itself originated from string field theory arguments [23,24].

Elsevier Science Publishers B.V. (North-Holland)

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The action for the homogeneous models is the following: 1 1’ / awa S ~ j dt (~~Ha —N(t) [g~HaHp+ U(w)])~ —~-—

(1) where w’~denotes the superspace variables such as h~,0 and the time t, Ha is the conjugate momentum to w’~,N(t) is the lapse function, 2= l6itMj2 and ~ is the superspace metric (signature (+, —)). The so-called superpotential U(w) is given by _,

U( w)

= \/~ {

V( 0)

(3

~R] ,

...,

(2)

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variables are semiclassical, i.e. some are quantum variables, denoted by q Actually, the latter vanables must be introduced since it is inconsistent with the uncertainty principle to set the inhomogeneous modes of the fields to zero. Moreover, considering quantum variables is an important step to understand the connection between the minisuperspace approximation and the full theory and also to describe the field fluctuations which lead to gravitational waves and density fluctuations [251. In ref. [25], the inhomogeneous modes of the three-metric h~and of a scalar field have been considered. It is found that, by introducing the inhomogeneous modes ~‘.

where h = I det h~I, V(0) is the potential of the matter fields which may include a cosmological constant and ~3~Ris the curvature is the curvature induced by the three-metric h 1~. For the action (1) the Wheeler—DeWitt equation reduces to

ofthe fields, one has to add to eq. (3) a Hamiltonian operator H~which depends only on the quantum variables. The Wheeler—DeWitt equation then

(~~.v2_ ~ U(w))W(wa)=0,

(3a)

where the Laplacian V~is built with the metric g~°~ (w), such that the full superspace metric is, expanded in powers of 2,

(3b)

gap(w,q)=g~(w)+O(A). (6) The wave function of the universe is written in terms of classical and quantum variables as in gen-

such that V 2YJ’=VaV a ~

vg

9Wa and Va is the superwhere covariant g= Idetgapl, ôaä/t space derivative. As argued in refs. [3,6], a probabilistic interpretation of the wave function w( Wa), similar to the one of QM, can be achieved in the situation where some variables Wa are semiclassical. This means that it is the size of the universe that determines the accuracy of quantum measurements. The Schrodinger equation is then the evolution equation of subsystems that are much smaller than the universe. Thus it follows that the latter is described semiclassically by a wave

function of the form ~ji(w)=A(w)

exp[i(S0/2+S1 + ...)]

(4)

By substituting lowest order in 2,this S into eq. (3) one finds that at 0 satisfies the Hamilton—Jacobi equation. At order 2, one obtains an equation for the amplitude A (Wa). The next order reveals that A (Wa) is harmonic. Aiming to get the evolution equation for the subsystems, one has to consider that not all superspace 226

becomes

(2v~_~u(w)_Hq)w(w,q)o,

(5)

—

eral one considers a superposition of terms ~(w, q)=çií(w)~(w,q) ,

—

(7)

where ~i( w) is given by eq. (4) which of course satisfies eq. (3a). In order to obtain the evolution equation for the wave function x ( w, q) we have to remember that the velocity at W’~ on the classical trajectory in superspace is a wa/at = 2N( t )V’~S~ since the momentum is given by H~= (1/2 )VaS 0. Thus, by proceeding as previously one obtains, considering terms to 0(22), in the gauge N( t) = 1, the following modified Schrödinger equation, 2p ox V H~x &~. x i (8) p where e is the typical energy scale of Hq and p ( w,

a-,

q)=A(w)~(w, q).

We see then that the Schrodinger equation is implemented by a nonlinear correction. Formally this

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correction is similar to the so-called quantum potential which is obtained in the pilot-wave or hidden-variable approach to QM by splitting the usual Schrodinger equation in its real and imaginary parts for a wave function like the one given by eq. (4) [26]. Closer to our result is the discussion of ref. [20] where a nonlinear correction which resembles the one in eq. (8) emerges by exploiting similarities between QM and stochastic variable theories see also ref. [27J. We shall simplify eq. (8) by assuming that the rate of change ofA ( w) in superspace, i.e. V’~A( w), is not significant on the time scales which are relevant when studying the evolution of the subsystems described by the wave function x( w, q). The resulting equation is then formally linear, however it contains several sources of nonlineanities. Nonlinearities may arise, as discussed in refs. [2,16,191, via the dependence of the parameters of the potential V( 0) in H~on the state of the system through expectation values of local operators and/or via the quantum evolution of the subsystem since the four-metric depends on the quantum state of the system through the equations of the gravitational field. Furthermore, nonlinearities appear implicitly in eq. (8) as the wave function x( w, q) depends on the superspace variables which in their turn are functions of the subsystem state. We shall further simplify the correction we have obtamed to the Schrödinger equation by disregarding all dependence of x( q) on the superspace variables except time. This means that among the su—

~

perspace variables only the time is relevant in the evolution of the subsystems. This assumption linearizes the correction to the Schrodinger 2~/Ot2 seeequation, the defiwhich explicitly is given by ô nition ofthe Laplace—Beltrami operator in eq. (3b). This correction shifts the energy E of a given level of the subsystem from the corresponding eigenvalue of Hg by an amount that is proportional to the squared energy of the level in question if one considers as is usual that the time evolution ofthe subsystem is given by exp( iEt). Let us now discuss if there is any scale at which the nonlinear correction in eq. (8) in its simplified form can be of any relevance by considering the recently established bounds to nonlinear terms in the Schrödinger equation [8,91. It has been thought for some time that the most —

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effective way to study the effect of nonlinearities in QM was via neutron interferometry [28,29]. However it has been pointed out [71, that the measuring of the detuning of resonant transitions could give much more stringent bounds on the presence ofnonlinearities in QM. Indeed very recent results set an upper limit on ilbnlinearities of 10—26 to 4 X 10_27 on the fraction of the binding energy per nucleon of the studied systems [8,91. Aiming to estimate the effect of the correction in eq. (8), we consider a subsystem which is composed of two particles with reduced mass ~uand which is bound by a Coulombian or by a harmonic force. It follows then from our previous discussion that due to the nonlinear correction the splitting between the m from the n orbital levels is given by 2f,

mn~61td(.Em/Mp)

(9)

where ö= 1— (En/Em) 2~ For a system bound by a Coulombian force we find that to saturate the limits of refs. [8,91 it is required that ö ~‘2Em ~ 3 x 102 TeV; e is taken to be the binding energy per particle of our bound system. Assuming that the theory which gives origin to the binding of the system becomes strongly coupled (a 1) at the energy scale that saturates the bounds of refs. [8,91, one obtains for the reduced mass ~ 4 TeV (ö~1) for m being the ground state. In the case of a system bound by a harmonic force the limit for the energy Em is the same and this implies for the ground state that u~2 X 102 TeV (ö~1) if we assume that the fundamental frequency w 0 is of the same order as ~u. Thus we have seen that the canonical Hamiltonian formalism in the minisuperspace approximation plies that the Schrodinger equation necessarily imhas nonlinear corrections. We conjecture that these nonlinearities could be observed if well understood frequency measurement techniques in nuclear physics were extended to much more massive bound states. Of course, this possibility is not very realistic. At this point however, we feel that we should confront our proposal with other suggested measurable effects of quantum gravity. Most of the suggestions concerning possible measurable effects of quantum gravity were based on the assumption that quantum gravity might induce the breaking of quantum coherence [30]. This possibility, which is inspired in the physics of black holes, 227

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would have rich phenomenological implications [311. However, the idea that quantum gravity would allow pure states to evolve into mixed ones has been severely criticized on the grounds that it would imply the violation of locality or energy—momentum conservation [321 and the loss of predictability as well as of the connection between symmetry primciples and conservation laws [331. Nevertheless, Hawking argued passionately in favour of the idea that topology changing transitions in quantum graYity would imply the loss ofquantum coherence [34]. More recently, the realization that the contribution of topological fluctuations to the functional action could be computed through the classical solutions of the Euclideanized equations of motion has brought again much interest in quantum gravity and the related question of loss of quantum coherence [35]. Indeed, gravitational instantons or wormhole-type solutions were thought to have some bearing in the collapse of the state vector [36] and in the loss of coherence of superconducting quantum interference devices (SQUIDs) on timescales of about 1 0~seconds [371. Although these proposals are of course interesting, they are based on assumptions which cannot always be convincingly justified. For instance, the conjectured effect of wormholes on SQUIDs is based on the hypothesis that the strength with which the former couples to macroscopic objects depends on the ratio of the electron mass to the Planck mass linearly [37], being therefore a much more striking effect than the one we have found. However, the most important objection to these possibilities was advanced by Coleman, who persuasively argued that wormho!es do not lead to any observable loss of quantum coherence at all [38]. Assuming this position, the sole effect of wormholetype solutions is to shift the values of the bare coupling constants emerging from the very high energy “theory of everything”. Thus, the observable phenomenological constants of nature would have an inherent contribution from wormhole physics [37,38]. This was though to provide a way to solve the cosmo!ogical constant problem [39]. Moreover, it has been pointed out that the shifting of coupling constants by wormholes would spoil the finiteness or the renormalizability of the theory of everything [40]; however, this consequence has been shown to be un228

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founded by the “big fix” mechanism of Coleman [39]. To conclude, we could say that effects of quantum gravity are very elusive if one believes as we do, that the loss of quantum coherence is not among the possible implications of quantum gravity. Indeed, we have seen that although the presence of nonlinear corrections to the Schrodinger equations is a fairly straightforward implication of the canonical Hamiltonian formalism to quantize gravity and of the minisuperspace approximation, we expect no significant effect of these corrections. Our estimates show that they could be observed only for extremely heavy bound states with energies of about 102 TeV, if the present techniques of measuring resonant frequencies could be used beyond the nuclear realm. It is a pleasure to thank Dr. M.C. Bento, Dr. B.J. Hiley and Dr. J.M. Mourão for helpful comments. Note added. After submitting this work we became aware of ref. [41], which contains results similar to ours. In studying the corrections to quantum mechanics due to quantum gravity, the author of ref. [411 has constructed a model oftwo interacting particles and argued that the resulting equations contain features that could have been obtained from the Wheeler—DeWitt equation. Our from approach is somewhat different as we have started the more gemera! framework of the homogeneous minisuperspace models. In ref. [411 no connection between the theoretical corrections to quantum mechanics and the recently obtained experimental limits to these corrections [8,9] was established.

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