A geometric interpretation of de Broglie wave-particle model
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A geometric interpretation of de Broglie wave-particle modelThis article has been downloaded from IOPscience. Please scroll down to see the full text article.2002 Europhys. Lett. 58 169(http://iopscience.iop.org/0295-5075/58/2/169)Download details:IP Address: 18.104.22.168The article was downloaded on 31/08/2013 at 11:33Please note that terms and conditions apply.View the table of contents for this issue, or go to the journal homepage for moreHome Search Collections Journals About Contact us My IOPscienceEurophys. Lett., 58 (2), pp. 169175 (2002)EUROPHYSICS LETTERS 15 April 2002A geometric interpretation of de Brogliewave-particle modelA. FeoliDipartimento di Ingegneria, Universita` del Sannio, Palazzo Bosco LucarelliCorso Garibaldi 107, I-82100 Benevento, Italy andINFN, Sezione di Napoli, Gruppo Collegato di Salerno - Salerno, Italy(received 15 October 2001; accepted in nal form 24 January 2002)PACS. 04.30.-w Gravitational waves: theory.PACS. 03.75.-b Matter waves.Abstract. We present a solution of the linearized eld equations of Einstein GeneralRelativity that can be interpreted as a model of a de Broglie quantum object formed by a waveand a particle. In the absence of external elds the particle travels at a constant velocity withits energy density localized in a spacetime region of the order of its Compton wavelength. Thede Broglie wave packet, moving with the particle, acquires a clear geometric meaning beinginterpreted as a special kind of gravitational wave.Introduction. In this letter we present a model of a quantum object formed by a waveand a particle which are both solutions of the Einstein equations in the weak-eld approx-imation. In this way, a geometric interpretation of de Broglie waves is possible. Althoughnowadays there are many interesting attempts to quantize gravity, we will take the oppositeway. Actually, the idea of a geometrization of quantum mechanics proved to be very fertile: alot of sophisticated models were published, increasing the number of dimensions as in Kaluza-Klein approach or using Weyl geometry, Finsler spaces, scalar-tensor theories of gravity orother possible generalizations of Einstein Relativity. We do not intend to propose in this lettera complete theory of quantum gravity, but we just want to develop the suggestion given in aprevious paper  that the de Broglie waves of one-particle quantum mechanics, in the absenceof other external eld, could be oscillations of spacetime. Our approach tries to accomplishthe original de Broglie ideas in the standard framework of the classical General Relativity. In1924 de Broglie  proposed his relation P = K between the classical constant momentumP = mcu = m(c,v) = (E/c, p) of a free particle and the wave number K (/c,k) of theassociated plane wave. Generalizing this concept, in 1927, he formulated the theory of thedouble solution  as an alternative to the standard interpretation of quantum mechanics.He not only believed that the wave eld is a mathematical tool to compute probabilities, butalso that all particles are accompanied in spacetime by a real pilot wave = R(t, x, y, z) exp[iS(t, x, y, z)](1)c EDP Sciences170 EUROPHYSICS LETTERSin such a way that the guiding formulaP = S (2)holds. This idea was improved in 1952 by Bohms hidden variable theory  and then by Vigierand his co-workers in the Stochastic Interpretation of Quantum Mechanics . According totheir interpretation, the waves represent collective motions on the top of a chaotic medium(the covariant subquantal Diracs aether) which induces random stochastic jumps at thevelocity of light. These real waves can be considered as describing the particles immediateneighborhood, i.e. the aethers reaction to the particle motion. In a dierent context Vigierand Synge  showed also that for a Klein-Gordon particle, the equation of motion can bewritten as a geodesic equation in the metricg =[1 +(mc)2RR] , (3)where = diag(1,1,1,1). This geometric model has been recently studied by Shojaiet al.  but it had some problems as pointed out by Holland : there is no well-denedsource of the metrical eld in the physical sense that a localized mass may generate agravitational eld; the metrical coecients are independent of the wave intensity and do not ingeneral diminish in magnitude with distance in what is otherwise empty space. Furthermorethe relativistic generalization of de Broglie-Bohm theory of motion in the case of Klein-Gordonequation is aected by some troubles : the zero component of the probability current hasan indenite sign so it cannot be treated as a probability density and the four-velocity of thequantum particle is not generically time-like.We consider the case of a free particle moving with a constant velocity v0 and we want toconstruct the associated de Broglie wave packet. We start from a metric tensorg = + h(x), (4)where |h | 1 and we neglect any terms of order |h |2 and the back-reaction of the metricperturbation on the motion of the source. We will show that our geometrical model solvesall the problems listed by Holland with the advantage of deducing a de Broglie-like metricdirectly from the Einstein equations and not from the Klein-Gordon equation.The vacuum solution. It is well known that the linearized Einstein eld equations invacuum areh = 0, (5)if the Lanczos conditions(h 12 h)= = 0 (6)are satised. The classical solution is a plane wave with wave number such that KK = 0.We start fromh = e = eR(t, x, y, z) exp[iF (t, x, y, z)], (7)where e is a constant polarization tensor, R and F are functions to be determined and thereal part of h is the physical gravitational eld.Note that a solution of this kind of eq. (5) has the corresponding that solves = 0. (8)A. Feoli: A de Broglie wave-particle model 171From (7) in (5) we have = 0 that can be split into its real and imaginary part:(F )(F ) =RR, (9)(R2F ) = 0. (10)Substituting (7) in (6) we have(e 12 e)R = 0,(e 12 e)F = 0. (11)Searching for de Broglie waves, we consider as interesting solutions of (9), (10) and (11) thosesatisfying the constraintRR= 2, (12)where is a positive constant.The phase of the wave eld will beF = Kx = kx t, (13)where K is a constant wave number such that KK = 2 that represents a gravitationalwave propagating along the x-axis with a phase velocityVph = c1 + (k)2. (14)The amplitude R(x, y, z, t) must satisfy (10) and (12) and simultaneously lead to a non-trivial expression of e from eqs. (11). We were able to nd  a solution of this kind, in theform of the Bessel function of the 0th-order :R = AJ0(y2 + z2/), (15)where A is a constant such that A 1 in order to have |h | 1.From (11), using (13) and (15), the non-vanishing components of the polarization tensorare:e00 = e11 = e10(K1K0+K0K1), (16)e22 = e33 = e10(K1K0 K0K1). (17)The de Broglie waves. A particular case of the previous solution is obtained by putting =mc, (18)the Compton wavelength of a particle of rest mass m.Writing F = S/, as in the de Broglie-Bohm interpretation of Quantum Mechanics (seeeq. (2)), eq. (9) becomes a Hamilton-Jacobi equation(S)(S) = m2c2 (19)172 EUROPHYSICS LETTERSand we can interpret S as a momentum P according to eq. (3).Furthermore eq. (13) becomesS = Px = px Et (20)typical of a de Broglie wave.Expressing in the usual way the momentum as a function of v, the non-vanishing compo-nents of the polarization tensor become:e00 = e11 = (v2 + c2)/c2,e22 = e33 = (c2 v2)/c2,e01 = e10 = 2v/c. (21)The nal solution acquires the form of a wave packet of cylindrical symmetry around thepropagation direction:h = e(v)AJ0 cos(Et px). (22)So we have shown that it is possible to obtain a de Broglie wave from the same equations asthe standard gravitational waves.As, in principle, the parameter v can vary from c to c, we can construct a wave packetsuperposing waves with slightly dierent values of v with respect to the velocity of the sourcev0, considering v0 v v v0 + v and v v0. Integrating on p = p/mc, the generalsolution becomes the real part ofh = AJ0 p0+pp0pe(p) exp[i(Et px)/] dp. (23)Assuming that E varies slowly enough with p to use the expansionE = E0 +(dEdp)p=p0(p p0) + O[(p p0)2] (24)and neglecting any terms of order (p p0)2 or higher, we obtainh AJ0e(p0) + O[A2] + O[Ap], (25)where = 2sin([dEdp t x]p/)[dEdp t x]cos(E0t p0x). (26)In that caseVG (dEdp)p=p0= v0 (27)the velocity of the quantum particle corresponds to the group velocity of the associated wavepacket. In the standard theory of gravitational waves eqs. (5) and (6) do not uniquely deter-mine the reference frame. In our model, we x a well-dened reference frame, the frame inwhich the wave packet and the particle move at velocity v0.If our solution is to have a geometric meaning, it is necessary to consider the role of gaugetransformations. In fact by a suitable coordinate transformation of the form x = x + (x)and such that (x) = 0, it is possible thath = h = 0. (28)A. Feoli: A de Broglie wave-particle model 173According to Einstein : If a h-eld can be made to vanish by the addition of terms likethose in (28), i.e. by means of an innitesimal transformation, then the gravitational eld beingdescribed is only an apparent eld. So far it has been impossible to nd a transformation(hence a choice of the reference frame) that can make our h-eld vanish, so it seems a realphysical eld. Surely we can say that it acquires non-trivial values in the physical interestingframes in which the particle moves at velocity v0 and in all the frames connected by meansof a Lorentz transformation.A model of quantum particle. We propose now a description of the particle movingwith the de Broglie wave packet. Following the usual approach, we must substitute eq. (5)with = 2kT , (29)where k = 8G/c4 and a non-vanishing energy-momentum tensor appears:T = uu . (30)In the standard quantum mechanics the function of eq. (26) represents the de Brogliewave packet in one dimension, associated with a particle of momentum p0. In our case thepresence of the amplitude J0(y, z) suggests the existence of a spherical extended particle ofradius less than the rst zero of the cylindrical Bessel function, i.e. rmax 2.4, moving alongthe x-axis. It is possible to suggest one particular solution of eqs. (6) and (29) that allows todescribe the details of the source according to the above requirements. We wish in fact thatthe energy density of the particle (i.e. a nucleon) is localized in a spacetime region of radiusrp = /g < rmax, where g = 2/, and that it moves with the same group velocity as theassociated de Broglie wave packet i.e. v0. So we begin with the ansatz = 0sin grgr(r rp) and = 0 (r rp) (31)and = NSSm2c2, (32)where is a function to be determined, N is a positive dimensionless constant and 0 has thedimension of an energy density. Equation (6) becomes equivalent to the conservation of theenergy-momentum tensor if we assume that the particles four-velocity has the same constantvalue as the wave packet (25):Smc= u = (1,v0c). (33)Hence eq. (29) becomes = b2 sin grgr, (34)whereb2 =2k0N. (35)Starting from this assumptions and imposing thatb = g =2mc, (36)174 EUROPHYSICS LETTERSeqs. (6) and (34) are satised by the solution found rst by de Broglie  and then wellstudied by L. Mackinnon : =sin grgr+ D, (37)where D is a constant and r =2(x v0t)2 + y2 + z2 is the distance from the particlescenter.The energy density is non-vanishing just in a zone of radius rp = /g, so solution (37)holds for r < rp and the mass of the particle must be entirely contained in this spacetimeregion. In the rest frame of the source, we obtain the relationmc2 = /g0d3x =420g3(38)that, together with (35) and (36), gives the values of the constantsN =8Gm2c=8m2m2pl(39)(where mpl is the Planck mass) and0 =mc24r3p=2m4c523, (40)which represents the maximum of the energy density in the particles center. Finally we obtain = (D +8m2m2plsin grgr)uu (r rp). (41)For r rp and v c the classical Newtonian term has to be recovered so the constant D canbe determined using this condition for r = rp. Considering 1 and u (1, 0), we have:00 = 2m2m2pl(1 + 4sin grgr)(r rp) (42)and00 = Gmrc2(r rp) (43)from which we can easily compute (h)nh. As | | 1, the metric holds for m mpl, acondition that is always satised for quantum particles. Finally note that the tensor iscontinuous in r = rp, but its rst derivative has a jump on the spherical shell of the particlewhere in fact (r rp) = 4(r rp).The more general solution of the linearized eld equations is a sum of the particularsolution (h)nh of the non-homogeneous equation (29) and of the wave-like one (25) of thehomogeneous associated (8). So we have shown that Einstein Relativity is compatible withthe idea indicated by de Broglie in 1926 when he answered the conundrum wave or particle?,by wave and particle .A. Feoli: A de Broglie wave-particle model 175Conclusions. Louis de Broglie believed that the wave eld was the solution of theSchroedinger or Klein-Gordon linear equations and a small singular region represented theparticle. In 1970 he explicitly said that it seemed too early to try describing the internalstructure of the singular region and that, on the other side, it was probable that this de-scription should introduce very complicated non-linear equations . In this letter we havepresented an attempt to describe the structure of the source as localized in a spacetime re-gion of the order of its Compton wavelength and associated to a de Broglie gravitational wavepacket. In our approach, in fact, the wave and the particle are, respectively, the solutions of thehomogeneous (8) and non-homogeneous (29) linearized Einstein equations. The advantages ofour model are that it makes General Relativity compatible with de Broglie ideas, allows a geo-metric interpretation of de Broglie waves without any generalization of Riemannian spacetimeand is not aected by the same problems as Synge, Vigier and Holland approaches [6, 8].The author is grateful to G. Scarpetta, G. Papini and G. Lambiase for useful sugges-tions.REFERENCES Feoli A. and Scarpetta G., Found. Phys. Lett., 11 (1998) 395. de Broglie L., Philos. Mag., 47 (1924) 446; Ph.D. Thesis on Recherches sur la theorie desquanta, also published in Ann. Phys. (Paris), 3 (1925) 22. de Broglie L., J. Phys. (Paris), 8 (1927) 225. Bohm D., Phys. Rev., 85 (1952) 166; 85 (1952) 180; de Broglie L., Nonlinear Wave Mechanics(Elsevier, Amsterdam) 1960; see also Vigier J. P., Found. Phys., 21 (1991) 125. Bohm D. and Vigier J. P., Phys. Rev., 96 (1954) 208; Nelson E., Phys. Rev., 150 (1966)1079; Lehr W. and Park J., J. Math. Phys., 18 (1977) 1235; Vigier J. P., Lett. NuovoCimento, 24 (1979) 258; 265. Synge J. L., Geometrical Mechanics and de Broglie Waves (University Press, Cambridge) 1954;Vigier J. P., C. R. Acad. Sci. Paris, 266 (1968) 598. Shojai F. and Golshani M., Int. J. Mod. Phys. A, 13 (1998) 677; Shojai F., Shojai A. andGolshani M., Mod. Phys. Lett. A, 13 (1998) 2725; 2915; 2965; Shojai A., Int. J. Mod. Phys.A, 15 (2000) 1757. Holland P. R., Found. Phys., 17 (1987) 345. Holland P. R., Phys. Rep., 224 (1993) 95. Einstein A. and Rosen N., J. Franklin Inst., 223 (1937) 43. de Broglie L., C. R. Acad. Sci., 180 (1925) 498. Mackinnon L., Found. Phys., 8 (1978) 157; Lett. Nuovo Cimento, 31 (1981) 37; 32 (1981) 311. Bell J. S., Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press,Cambridge) 1993, p. 171. de Broglie L., Linterpretation de la mecanique ondulatoire par la theorie de la double solutionin Foundations of Quantum Mechanics, edited by dEspagnat B. (Academic Press, New York)1971, p. 345; the English translation is in Ann. Fond. Louis de Broglie, 12 (1987) 399.
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