A geometric interpretation of de Broglie wave-particle model

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Europhys. Lett., 58 (2), pp. 169–175 (2002)

EUROPHYSICS LETTERS 15 April 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 final 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 field 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 fields 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-field 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 field, 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 ν = �Kν between the classical constant momentumP ν = mcuν = mγ(c,�v) = (E/c, �p) of a free particle and the wave number Kν ≡ (ω/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 field is a mathematical tool to compute probabilities, butalso that all particles are accompanied in spacetime by a real “pilot wave”

φ = R(t, x, y, z) exp[

i

�S(t, x, y, z)

](1)

c© EDP Sciences

170 EUROPHYSICS LETTERS

in such a way that the guiding formula

Pµ = −∂µS (2)

holds. This idea was improved in 1952 by Bohm’s 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 Dirac’s “aether”) which induces random stochastic jumps at thevelocity of light. These real waves can be considered as describing the particle’s immediateneighborhood, i.e. the aether’s reaction to the particle motion. In a different 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

gµν =[1 +

(�

mc

)2 �R

R

]ηµν , (3)

where ηµν = 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-defined“source” of the metrical field in the physical sense that a localized mass may generate agravitational field; the metrical coefficients 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 affected by some troubles [9]: the zero component of the probability current hasan indefinite 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

gµν = ηµν + hµν(x), (4)

where |hµν | � 1 and we neglect any terms of order |hµν |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 field equations invacuum are

�hµν = 0, (5)

if the Lanczos conditions

∂µ

(hµ

ν − 12δµν h

)= ∂µσµ

ν = 0 (6)

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

hµν = eµνφ = eµνR(t, x, y, z) exp[iF (t, x, y, z)

], (7)

where eµν is a constant polarization tensor, R and F are functions to be determined and thereal part of hµν is the physical gravitational field.

Note that a solution of this kind of eq. (5) has the corresponding σµν that solves

�σµν = 0. (8)

A. Feoli: A de Broglie wave-particle model 171

From (7) in (5) we have �φ = 0 that can be split into its real and imaginary part:

ηαβ(∂αF )(∂βF ) =�R

R, (9)

∂µ(R2∂µF ) = 0. (10)

Substituting (7) in (6) we have(

eµν − 1

2δµν e

)∂µR = 0,

(eµν − 1

2δµν e

)∂µF = 0. (11)

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

�R

R= λ−2, (12)

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

F = −Kµxµ = kx − ωt, (13)

where Kµ is a constant wave number such that KµKµ = λ−2 that represents a gravitationalwave propagating along the x-axis with a phase velocity

Vph = c√

1 + (λk)−2. (14)

The amplitude R(x, y, z, t) must satisfy (10) and (12) and simultaneously lead to a non-trivial expression of eµν from eqs. (11). We were able to find [1] a solution of this kind, in theform of the Bessel function of the 0th-order :

R = AJ0(√

y2 + z2/λ), (15)

where A is a constant such that A � 1 in order to have |hµν | � 1.From (11), using (13) and (15), the non-vanishing components of the polarization tensor

are:

e00 = e11 = e10

(K1

K0+

K0

K1

), (16)

e22 = e33 = e10

(K1

K0− K0

K1

). (17)

The de Broglie waves. – A particular case of the previous solution is obtained by putting

λ =�

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

ηαβ(∂αS)(∂βS) = m2c2 (19)

172 EUROPHYSICS LETTERS

and we can interpret −∂µS as a momentum Pµ according to eq. (3).Furthermore eq. (13) becomes

S = −Pµxµ = 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 final solution acquires the form of a wave packet of cylindrical symmetry around thepropagation direction:

hµν = eµν(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 different 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

hµν = AJ0

∫ p0+∆p

p0−∆p

eµν(p) exp[−i(Et − px)/�] dp. (23)

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

E = E0 +(

dE

dp

)p=p0

(p − p0) + O[(p − p0)2] (24)

and neglecting any terms of order (p − p0)2 or higher, we obtain

hµν � AJ0eµν(p0)ψ + O[Aλ2] + O[Aλ∆p], (25)

where

ψ = 2λsin

([dEdp t − x

]∆p/�

)[dEdp t − x]

cos(

E0t − p0x

�

). (26)

In that case

VG ≡(

dE

dp

)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 fix a well-defined 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 xµ′ = xµ + ξµ(x)and such that �ξµ(x) = 0, it is possible that

h′µν = hµν − ∂νξµ − ∂µξν = 0. (28)

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

According to Einstein [10]: “If a h-field can be made to vanish by the addition of terms likethose in (28), i.e. by means of an infinitesimal transformation, then the gravitational field beingdescribed is only an apparent field”. So far it has been impossible to find a transformation(hence a choice of the reference frame) that can make our h-field vanish, so it seems a realphysical field. 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

�σµν = −2kTµν , (29)

where k = 8πG/c4 and a non-vanishing energy-momentum tensor appears:

Tµν = ρuµuν . (30)

In the standard quantum mechanics the function ψ 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 first zero of the cylindrical Bessel function, i.e. rmax � 2.4λ, 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 = π/g < rmax, where g = 2/λ, and that it moves with the same group velocity as theassociated de Broglie wave packet i.e. �v0. So we begin with the ansatz

ρ = ρ0sin gr

gr(r ≤ rp) and ρ = 0 (r ≥ rp) (31)

and

σµν = −Nξ∂µS∂νS

m2c2, (32)

where ξ is a function to be determined, N is a positive dimensionless constant and ρ0 has thedimension of an energy density. Equation (6) becomes equivalent to the conservation of theenergy-momentum tensor if we assume that the particle’s four-velocity has the same constantvalue as the wave packet (25):

−∂µS

mc= uµ = γ

(1,

�v0

c

). (33)

Hence eq. (29) becomes

�ξ = b2 sin gr

gr, (34)

where

b2 =2kρ0

N. (35)

Starting from this assumptions and imposing that

b = g =2mc

�, (36)

174 EUROPHYSICS LETTERS

eqs. (6) and (34) are satisfied by the solution found first by de Broglie [11] and then wellstudied by L. Mackinnon [12]:

ξ =sin gr

gr+ D, (37)

where D is a constant and r =√

γ2(x − v0t)2 + y2 + z2 is the distance from the particle’scenter.

The energy density is non-vanishing just in a zone of radius rp = π/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 relation

mc2 =∫ π/g

0

ρd3x =4π2ρ0

g3(38)

that, together with (35) and (36), gives the values of the constants

N =8Gm2

π�c=

8m2

πm2pl

(39)

(where mpl is the Planck mass) and

ρ0 =mc2π

4r3p

=2m4c5

π2�3, (40)

which represents the maximum of the energy density in the particle’s center. Finally we obtain

σµν = −(

D +8m2

πm2pl

sin gr

gr

)uµuν (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 γ � 1 and uµ � (1, 0), we have:

σ00 = − 2m2

πm2pl

(1 + 4

sin gr

gr

)(r ≤ rp) (42)

and

σ00 = −Gm

rc2(r ≥ rp) (43)

from which we can easily compute (hµν)nh. As |σµν | � 1, the metric holds for m � mpl, acondition that is always satisfied for quantum particles. Finally note that the tensor σµν iscontinuous in r = rp, but its first derivative has a jump on the spherical shell of the particlewhere in fact ∂ασµν(r ≤ rp) = 4∂ασµν(r ≥ rp).

The more general solution of the linearized field equations is a sum of the particularsolution (hµν)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 [13]”.

A. Feoli: A de Broglie wave-particle model 175

Conclusions. – Louis de Broglie believed that the wave field 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 [14]. 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 affected 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

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