Non-linear Schrödinger equation coming from the action of the particle's gravitational field on the quantum potential

PhysicsLettersA 166 (1992) 111—115 PHYSICS LETTERS A North-Holland Non-linear Schrodinger equation coming from the action of the particle’s gravitational field on the quantum potential J.L. Rosales and J.L. Sánchez-Gdmez Department ofTheoretical Physics, UniversidadAutónoma de Madrid, Canto Blanco, Madrid 34, Spain Received 30 December 1991; revisedmanuscript received 26 February 1992; acceptedfor publication 15 April 1992 Communicated by J.P. Vigier The effect of the gravitational field of a particle on its own wave function is analyzed by obtaining the “stochastic” metric induced by the quantum potential when such aneffect is taken into account. Upon going to the non-relativistic limit, a non-linear Schrodinger equation is derived in which the non-linear term is related to gravity. This term is completely negligible for “micro- scopic” systems, yet some arguments are given which suggest that it must be important for macroscopic systems; in the latter case it would produce an appreciable deviation from the strict quantum behaviour, in the sense that one would get classical trajectories for the motion of the system’s centre of mass. The biggest fundamental problems faced by the era!, they have been introduced more or less ad hoc. contemporary quantum theory are, surely, the con- Here, the non-linear term comes from “first prin- struction of a consistent theory ofgravity, and the — ciples”, and no free parameter is left so that its small- consistent as well — derivation of classical properties ness (whichguarantees that QM correctly applies for (trajectories and the like) in the case of macroscopic microscopic systems) is not “put on by hand” but systems (admittedly, the second one is commonly arises naturally in the theory. In such a sense, we regarded, especially for the “pragmatists”, as a pseu- claim the present non-linear Schrödingerequation to doproblem; obviously we do not agree with such a be universal. narrow vision of science). Both problems have been We will consider the simplest quantum system — traditionally considered as belonging to very differ- a neutral, scalar particle — and introduce a Lagrang- ent domains: the first is thought of as relevant just ian density describing the corresponding quantum at the “Planck scale”, that is at distances of lO—~~ (Klein—Gordon) fluid [8,91, cm, whilst the second would be important for mac- / roscopic distances. The feeling is however gradually £9=p -~— + ~— (~,3~S8~S+~ ä,~p (1)increasing that both problems are in fact strongly re- X m P lated to each other [1—6],in the sense that gravity where p is the fluiddensity and S represents a certain is deeply involved in the “quantum” behaviour of action for the fluid (r stands for the proper time). macroscopic systems, a factwhich would clearly have Upon making — after Feynman — the assumption important implications for the so-called reduction postulate that gives rise to the infamous “measure- p(x, t) =p(x) , (2a) ment problem” in QM. S(x, r)=S(x) — imc2t, (2b) In this Letter, we derive a non-linear Schrödinger . and by introducing !P=exp(P+iS/1l), where P= equation which describes the time evolution of neu- . . ~lnp, one easily arrives at the Klein—Gordon tral scalar particles when their own gravitational field is included in the quantum potential. Indeed, non- equation linear modifications ofthe Schrodingerequation have o w+ w— o 3 beentreated in a variety of contexts [7] but, in gen- 2 — ( ) 0375-9601/92/s 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved. 111 Volume 166, number 2 PHYSICS LETTERS A 15 June 1992 Next we compute the energy—momentumtensor cor- Moreover, in spite of its similarity with the corre- responding to this stochastic K—G fluid, which is ob- sponding classical term, T(c) depends in fact on the tamed through wave function through p. As we are here concerned with the effect of gravity on the quantum potential, _______ 82’T~L~~_ 8 S+ 8(8) ~VpgPV~f. (4) we, as already said, shall take the “background” met- — 8(8~S) nc to be just Minkowskian (see further comments Upon using the Euler—Lagrange equation corre- below). sponding to the variation of p (and taking into ac- Now if we assume count (2b)), i.e. (12) h28~S8~S=m2c2+— Dp— —~ ~9~pc9Mp where 2ith2G A~m~ we get ~ mc~ m (13) (5) (A~and mp being the Planck length and mass, re- where spectively), and neglect terms oforder e2 and higher, we easily arrive at T~)=mpvMvP, (6a) — ~ —g,~~Dy). (14) T~)=mp(u~u1+VQg’~~’). (6b) In order to solve (14) we make the following ap-We have defined proximation that we believeto be quite justified from ~�4= _!_a’~s (drift velocity), (7a) a physical point of view: we will neglect in (5) the m “kinetic” term, i.e. mpu,~up.This amounts to assum- ing that the “trajectories” contributing mostly to ~ h8~p h = — — — — 0 ~P (osmotic velocity) (7b) are the “semi-classical” ones, that is those for which 2m P — m 8,~pis small. This approximation being done, one and VQ is the “quantum potential” easily obtains VQ(. (8) ~pgppDy=gp’~1~~, (15) \2in) ~° and by contracting indices Next we shall obtain the metric from this energy— —3D ~ — 40~+o ~2) , (16) momentum tensor by solving the corresponding Einstein equations which has the obvious solution (9) (17) Notice that then, from (15), To do this, let us set PJI; = ~ Dp, (18) R —R~°~+ ~ (lOa) in agreement with the assumption of isotropy of thepV— PP g,~_—g~ 2°)+ 8g,~,, (lOb) K—G fluid. Now recall that g~~=~( I + q’), so that we shall have where ~ satisfies (‘ 1 8xGh28~tG ~ (11) ~ (19) This equation shows the deviation from the Mm- We set g~= i~ (Minkowski) for simplicity; yet we kowskian metric due to the effect of the particle’s are aware that, due to T(c), this is not exactly so. gravitational field on the quantum potential. 112 Volume 166, number 2 PHYSICS LETTERS A 15 June 1992 The K—G equation corresponding to the modified first term on the r.h.s. of (20) (the “kinetic” term) metric must now be written taking into account that shouldbe !12/2mR2 while the gravitational term is of the order of y/R ‘. Now recall that y — given after ~ g= —det(g~~) (21) — is of the order of m~c24,where mp and A~V /4t We are primarily concerned with the non-relativistic are, respectively, the Planck mass and Planck length; (low velocity) case, so that by making the usual ap- therefore we conclude that, for the gravitational term proximation to go from the K—G to the Schrödinger to be relevant in the case of “elementary” particles, equation we obtain the particle mass, m, must be of the order of 8!!’ h2 h2R/m~c2A~~,which for a — very optimistic — local-ih — = - — i~ Ot 2m I !!~I2!1~, (20) ization, R= 1 fm, gives about l0~°GeY/c2. Indeed this means that the gravitational effects are quite ir- which is the non-linear Schrodinger equation we relevant insofar the quantum behaviour of “elemen- mentioned. tary” particles (or, in general, of microscopic sys- The non-linear term contributes — practically — tems) is concerned — as already pointed out. nothing to “microscopic” observables, i.e., those of In dealing with compound (macroscopic) sys- elementary particles, atoms, etc. (for instance, its tems, one has the difficulty that (20) does not admit contribution to the Lamb shift in hydrogen is less separate solutions, even if the constituents do not in- than l0~°eV), hence QM remains unchanged at teract with each other; this being one of the facts that the microscopic level. Things, however, could be dif- motivated introducing a non-linear logarithmic ferent in the macroscopic case; the gravitational ef- Schrodinger equation (see the paper by Bialynicki- fect should greatly increase with the mass (it is non- Birula and Mycielski [7]; nevertheless, we again re- linear). Moreover, one knows that the non-linear mark that here we have notpostulated the non-linear Schrodinger equation has solitonic solutions. For in- equation; we have derived it). This non-separability stance, in the — admittedly unrealistic case — of one renders, in practice, almost impossible obtaining an dimension an explicit solution of (20) reads exact solution in the case ofsuch compound systems /2a)’ /2 even after making some simplifying assumptions.!P(x, t)=( — exp(~[mvx— (E_a)t]) Therefore we are driven to try finding approximate solutions and in this sense we think it worth justi- 2 (x— Vt)), (21) fying, at least qualitatively, why such a — technicallyxsech(~ — complicated task should be performed. In other where y= 4~tGh2/c2, E= ~my2, and a is an integra- words, we shall argue that gravitational effects are relevant for compound systems — in the sense pointedtion constant (notice the dependence on x — yt). Much less is known in the three-dimensional case, out above. Admittedly, we shall not present any which we are primarily interested in here. Some gen- mathematically rigorous argument, however we think eralproperties are know that canbe useful in several the following ones to be at least of some indicative problems [10], but no solution has — to our know!- value. edge — been found which “modulates” a quantum In order to compute the density, p, appearing in wave packet (that is reducing itself to the said wave- (19) let us assume that one can use a wave function packet when the non-linear gravitational effects are which factorizes into a product of relative and CM neglected). In any case, we shall produce here some wave functions (we also assume the relative wave semi-quantitative arguments in order to show that function itself to be a product of one-body wave such gravitational effects can be rather important as functions, i.e., some kind of mean field approxi- far as the quantum behaviour of the CM (centre of mation; this is not strictly necessary but simplifies mass) of a macroscopic systemis concerned. But first appreciably the argument). Thisassumption amounts let us, for a while, return to the microscopic case, in factto neglecting the gravitational effects overany We note that for a particle initially localized in a single constituent, which seems to be rather a sen- region of (lineal) size R, i.e. volume about R ~, the sible approximation (in spite of the fact that, as re- 113 Volume 166, number 2 PHYSICS LETTERS A 15 June 1992 marked already, such a function cannot be an exact given above for the microscopic case (i.e. compare solution of (20)). Thus we put kinetic and gravitational terms), thus finding that the non-linear term should be relevant when the X~, 1) number of constituents satisfies =0~(xi —X)...ON(xN—X)w(X, t) (22) h2 yN2 where the 0’s represent stationary relative wave 2mND2 (24) functions and ~(X, t) is the CM wave function. In- that is deed the 0’s should account for the fact that the con- stituents (of a solid body, the case which is discussed N3> 2mc2y’ (25) here) havealmost fixed positions with respect to the CM, that is Ø~(x~— X) must be peaked around some where D is now the dispersion of the CM wave packet. value a~,with a dispersion ofthe order of I A, which Taking for definiteness m to be the proton mass and would represent the oscillation around the “equilib- D= 1000 A, (25) yields N> 1O’~(that is M> l0~ rium” point. On the other hand, the CM initial dis- g). Of course these figures must be considered only persion depends just on the preparation of the cor- as indicative, although the order of magnitude must responding wave packet, being then much larger be practically right. One should realize the impor- (~1000 A in a macroscopic system usually) than tance that the system be a compound one; indeed, as any typical dispersion of constituents. This taken into shown above, a singleparticle ofmass 1 g, say, should account, and upon using (22), one can find the fol- not be affected by its own gravitational field (at dis- lowing (approximate) expression for the density, tances larger than the Planck length), hence it must p(x, t)~N~qi(x,1)12, (23) not be regarded as macroscopic, at least insofar lo- calization and trajectories are concerned. However where N is the number of constituents. To obtain this we see that a system ofthe same mass, butwith about equation, the CM coordinates have been considered 1023 atoms, behaves — according to the preceeding independent from — i.e. uncorrelated with — the con- estimate — as classical as regards the motion of its stituent coordinates (that is, to get p(x, t) we inte- CM, since the non-linear term, coming from its own grate overall the x,’s andX, disregarding the obvious gravitational field, is quite important and — presum- constraint ~ x./N=X). This is thought to be a harm- ably — would cause the collapse of the CM wave less approximation provided N>> 1. function. Now we assume that the system “moves” in a space—time with a metric given by (19) and (23), This work has been supported in part by CICYT i.e., we consider the system as a “test body”, then (Spain) under contract No. PB. 88-0173. We thank analyzing the quantum evolution of its CM under S. Bergia and F. Cannata for some useful discus- the gravitational potential derived from that metric. sions. One of us (JLR) is indebted to ProfessorJ.P. It goes without saying that this is, again, a rough ap- Vigier for hospitality at the Institut Henri Poincaré, proximation; nevertheless it should give an estimate Paris, where part of this work was done. We are also of the relevance of the non-linear term in macro- indebted to the referee, whose criticism has helped scopic systems. We then consider (20) as the non- to clarify some points in this work. linear Schrodinger equation for the CM wave func- tion, ~(X, 1), where now M=Nm is the system’s mass, which is also to be used in deriving the gray- References itational potential from the metric. Hence we shall have [11R. Penrose, in: Quantum concepts in space and time, eds. R. Penrose and C. Isham (Clarendon, Oxford, 1986). = — ~~-_ ~ wI2w. (20’) [2] F. Károlyházy, in: 62 years of uncertainty, ed. A. Miller (Plenum, New York, 1990); A. Frenkel, Found. Phys. 20 (1980) 159. Now we can follow almost literally the reasoning [3] L. Diosi, Phys. Rev. A 40 (1989)1165. 114 Volume 166, number 2 PHYSICS LETTERS A 15 June 1992 [4]J. Ellis, S. Mohanty and D.V. Nanopoulos, Phys. Lett. B S. Weinberg, Phys. Rev. Lett. 62 (1989) 485; Ann. Phys. 221 (1989) 113. (NY) 194 (1989) 336. [5] E.J. Squires, Found. Phys. Lett. 3 (1990) 190. [8] A. Kyprianidis, Phys. Rep. 155 (1987) 1. [6] J. Unturbe and J.L. Sánchez-Gómez, Nuovo Cimento B, in [9] J.P. Vigier, Found. Phys. 21(1991) 125. press. [10] B.J. LeMesurier, G. Papanicolaou, C. Sulem and P.L. Sulem, [7] I. Bialynicki-Birula and J. Mycielski, Ann. Phys. (NY) 100 Physica D 31(1988) 78. (1976) 62; 115