A geometric interpretation of de Broglie wave-particle model

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  • A geometric interpretation of de Broglie wave-particle model

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    2002 Europhys. Lett. 58 169


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  • Europhys. Lett., 58 (2), pp. 169175 (2002)


    A geometric interpretation of de Brogliewave-particle model

    A. Feoli

    Dipartimento 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 [1] 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 [2] 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 [3] 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[i

    S(t, x, y, z)


    c EDP Sciences


    in such a way that the guiding formula

    P = S (2)

    holds. This idea was improved in 1952 by Bohms hidden variable theory [4] and then by Vigierand his co-workers in the Stochastic Interpretation of Quantum Mechanics [5]. 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 [6] showed also that for a Klein-Gordon particle, the equation of motion can bewritten as a geodesic equation in the metric

    g =[1 +




    ] , (3)

    where = diag(1,1,1,1). This geometric model has been recently studied by Shojaiet al. [7] but it had some problems as pointed out by Holland [8]: 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 [9]: 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 tensor

    g = + 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 are

    h = 0, (5)if the Lanczos conditions


    12 h

    )= = 0 (6)

    are satised. The classical solution is a plane wave with wave number such that KK = 0.We start from

    h = 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 171

    From (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,


    12 e

    )F = 0. (11)

    Searching for de Broglie waves, we consider as interesting solutions of (9), (10) and (11) thosesatisfying the constraint


    = 2, (12)

    where is a positive constant.The phase of the wave eld will be

    F = 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 velocity

    Vph = c

    1 + (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 [1] 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 tensor


    e00 = e11 = e10



    ), (16)

    e22 = e33 = e10



    ). (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 (see

    eq. (2)), eq. (9) becomes a Hamilton-Jacobi equation

    (S)(S) = m2c2 (19)


    and we can interpret S as a momentum P according to eq. (3).Furthermore eq. (13) becomes

    S = 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 of

    h = AJ0 p0+p

    p0pe(p) exp[i(Et px)/] dp. (23)

    Assuming that E varies slowly enough with p to use the expansion

    E = E0 +(dEdp


    (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)


    = 2sin

    ([dEdp t x


    )[dEdp t x]


    E0t p0x

    ). (26)

    In that case

    VG (dEdp


    = 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 that

    h = h = 0. (28)

  • A. Feoli: A de Broglie wave-particle model 173

    According to Einstein [10]: 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 e


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