CHAPTER 5

DUALISM WITHIN DUALISM

OPEN QUESTIONS

PHILIPPE GUERET

The general point of view of wave-particle duality proposed by Albert Einstein and Louis de Broglie represents all atomic objects such as photons, electrons, protons, etc. as consisting of the physical association of two entities: (1) a wave­packet, devoid of energy and momentum but nevertheless objectively real and propagating in space and time; (2) energetic corpuscles always localized inside the wave-packets.

Moreover, within this wave-particle duality exists another which appeared early in classical physics for the explanation of diffraction phenomena by means of the Huygens-Fresnel principle. This" dualism within dualism" retains all of its interest in the modern theory of light and serves as a basis to interpret the wave­mechanics in terms of the "double solution" hypothesis.

Our purpose is to expose this problem and draw from it some experimental and mathematical inferences coming into sight of open questions for all that.

Let us consider a pointlike source S of a wave phenomenon in a homogeneous isotropic medium and let I be a spherical wave-surface at a given time t (Figure 1). According to the Huygens hypothesis, each point M of I is regarded as the source of a secondary spherical wavelet the radius of which is vat at a forthcoming instant t + at, V being the wave propagation velocity (phase velocity). At the same time t + at, the wavefront becomes I I with radius Vet + at) and behaves as the envelope of all the wavelets emitted by the elements of I. Therefore, Huygens evidenced a progressive propagation mechanism of the wave phenomenon from a point M of space to another.

The Huygens construction has been completed by the Fresnel hypothesis of

PHILIPPE GuERET • Institut de Mathematiques Pures et Appliquees, Universite Pierre et Marie Curie, F-75230 Paris Cedex OS, France.

Wave-Particle Duality, edited by Franco Selleri. Plenum Press, New York, 1992.

97

F. Selleri (ed.), Wave-Particle Duality© Plenum Press, New York 1992

98 PHn.JpPE GuERET

s.....::~ ______ I--.....

:1:'

FIGURE 1. I is a spherical wave emitted by the source S. According to Huygen's hypothesis, each point M of I emits at its turn a spherical wavelet.

possible interferences between the wavelets themselves. Then, it is natural to admit that the secondary sources distributed on the I surface could have the same phase as the oscillating state of the wave front I. A more detailed investigation of this situation shows that actually the wavelets have a phase advance of -rr/2 before the I oscillations.

The Huygens-Fresnel principle does not only give a wave front propagating forward, but also a wave front I" propagating in the opposite direction and which is another envelope of the wavelets (Figure 2). A mathematical analysis of this fact enables us to justify the Huygens-Fresnel principle and to eliminate the undesirable I" wave inconsistent with empirical evidence. This was performed by Kirchhoff for scalar waves as acoustic waves in fluids. Kirchhoff built up an accurate formula including two terms adding or canceling each other according to whether one considers a point of I' or a point of I". The electromagnetic wave problem is more intricate because of the vectorial nature of the field. The expres­sion of the Huygens-Fresnel principle requires three assumptions:

1. In a point M of space, the field must be the sum of all the fields sent by the elements of the wave front I. This holds also for the first derivatives of the fields.

2. The integral expression of the field, taken on I, must be equal to zero in each point located inside the spherical surface I (no back-waves).

3. The secondary wavelets are real electromagnetic waves.

DuALISM WITHIN DuALISM 99

s ___ .... __ +-_+-~

FIGURE 2. The wavelets have two envelopes: I', which propagates forward, and I", propagating backward.

On this basis, one can deduce different formulas, equivalent to one another, but adjusted to the applications: in optics or for the transmission techniques of short waves, for instance.

The emission mechanism of light vibrations is not yet fully understood. However, it is known from the pioneer works of Einstein and Bohr that the energy exchanges between matter and electromagnetic radiation are carried out by quanta with energy hv. Particularly, when one atom emits one photon with energy hv, an electron of this atom passes from an energetic level E2 to another of lesser energy E1' the height difference of these levels being precisely equal to hv = E2 - E1•

Then, in optics, it is reasonable to think that

1. The electromagnetic radiation is not emitted continuously but by limited wave-trains (or wave-packets) containing photons and coming from the different atoms from which the course is composed.

2. The energy density of these waves inside a small volume measures the probability of finding photons in this space region.

3. Electromagnetic waves surrounding photons emitted by different atoms can very likely interfere as shown by the Brown-Twiss or the Pfteegor­Mandel experiments.

Thus, the following representation occurs: the light source S being made of a very great number of atoms, the electromagnetic wave emitted at a given instant t, is the sum of individual wave-packets due to the different atoms. These emit only

100 PHILIPPE GuERET

for a time T corresponding to the mean life of their excited levels. After a fairly long time, the wave-packets emitted at the instant t will be kept away and new atoms take the relay by emitting new waves with new amplitudes and phases: there is a loss of coherence between successive emissions of radiation. A source is coherent with itself only providing that we compare waves emitted at very near instants, the time interval being clearly below T.

The successive atomic emissions of the source S are made of wave trains limited in the propagation direction. The phase velocity V of the waves inside the wave train is related to the wavelength 'A and the frequency v by V = 'A v and the wave-train velocity v itself (the group velocity) is given by the Rayleigh formula

lIv = a(lI'A)/av = a(v/V)av (1)

The wave-train length L = VT (coherence length) is a measurable size by well­known methods of the interference techniques: for instance, by disappearing of the interferences by interposing transparent plates or by separating sufficiently the mirrors of a Michelson interferometer.

We can define T as the "coherence time" of a wave train by remarking that at a given point M, it behaves as a monochromatic plane wave of the same length L as the wave train. Such an approximation meets its justification by applications which can be made to the calculations of interferences and diffraction phenomena.

We have experimental numerical data on the coherence length (and conse­quently on the coherence time T) of the light wave-packets: for ordinary visible light of mean wavelength 'A = 0.5 J.Lm, L is equal to about 1 m which corresponds to a coherence duration T of about 10-8 s. Moreover, the time constant T is connected to the spectral ray width of the light: one can easily devise that a long wave train is sufficiently near that of a sinusoidal wave to have a well-defined frequency whereas a short wave train admits a more spread spectrum.

These considerations about the structure of light emissions bring up more physical consistence to the Huygens-Fresnel principle. Thus, the wave-surface I corresponds to the geometrical distribution of the wave trains in space surrounding the source S. The wave-packets take the place of the Huygens wavelets. For a source of ordinary intensity, the great number of atomic emissions gives to the wave front an apparent "materiality" in the sense that the I wave can be regarded itself as responsible for the interference phenomena. On the contrary, for a source of very low intensity, emitting photons one by one, for instance, I is reduced to its statistical signification of geometrical function of repartition of wave-packets around the source and can be useful for calculations of probabilities.

Let us recall that Einstein propounded such a model for atomic emissions of radiation under the name of NadeLstrahlung. In the case of light, this model exhibits a wave-wave dualism with the wave-particle dualism.

The idea of a wave associated with a particle of matter occurred first to de Broglie(1) and was embodied in his well-known doctoral thesis of 1924.(2) His

DuALISM WITHIN DuAUSM 101

assumptions were that the particle has an internal vibration (the "de Broglie clock") and moves in the space-time of special relativity, where at every moment it is localized.

A free particle at rest has mass mo and energy E = mOc2 . If this energy is taken to equal one quantum hvo' the internal particle vibration frequency is defined by Vo = moc21h and a wave function '1'0 can be associated with the vibration so that

(2)

T being the proper time of the particle. '1'0 is uniform throughout space and does not serve in any way to locate the particle.

Now, if it is supposed that the same particle is moving freely with uniform velocity v in the + x direction, a Lorentz transform gives

T = (t - ~xlc)Y (1 - ~2)

(~ = vic) and, for an observer at rest, the wave function '1'0 becomes

'I'(x,t) = aexp{[21Tivo(t - ~xlc)]Y(1-~2)}

or, on taking vofY (1 - ~2) = v and c2/v = V

'I'(x,t) = a exp [21Tiv(t - xiV)]

(3)

(4)

(5)

Note that, under the Lorentz transformation, the wave frequency becomes v = vofY (1 - ~2), while the particle frequency becomes v = Vo Y (1 - ~2) according to the clock slowdown formula. In order to remove this contradiction, de Broglie lays down his "phase concordance principle": the internal particle vibration and its associated wave remain in phase where the particle is located. de Broglie waves consequently act as a guide for the particle motion.

According to Eq. (5), for an observer at rest, the particle motion is associated with the propagation of a plane wave of frequency v and phase velocity V > c. On defining, as usual, the wavelength by A = Vlv, one gets for this phase wave

A = c2hlEv = hlp (6)

E and p = Ipl respectively representing the energy and relativistic momentum of the moving particle.

On the basis of classical reasoning in wave theory, one can consider a wave­packet made up of a superposition of plane wave with closely related frequencies. Using k = vIV, such a packet can be written as

<I>(x,t) = fM {aexp [21Ti(vt - kx)]}dk (7)

102 PHILIPPE GuERET

To apply this formula to matter waves, one puts k = p/h and E = hvobtaining

ct>(x,t) = I~ {aexp [*(Et - px) ]}dP (8)

Let Po be a central value in the wave-packet and assume that E varies slowly enough with p to justify the Taylor expansion

(9)

Substitution of Eq. (9) into Eq. (8) yields

In Eq. (10) the exponential term outside the integral represents a plane wave moving with constant velocity, and the integral behaves like a wave-packet when

dE -t - x = 0 dp

One thus obtains for the wave-packet velocity U (group velocity)

U=dE=~ dp t

and one notes that

(11)

(12)

1. In the nonrelativistic case (SchrOdinger waves): E = p2/2m, p = mvand thus

dE d U = - = -(p2/2m) = v

dp dp (13)

2. In the relativistic case (de Broglie waves): E2 = p2C2 + mijc4 , p = mov/V (1 - ~2) and thus

dE d VI - ~2)m vc2 U = - = - (p2C2 + m2c4)112 = pc2/E = 0 = v (14)

dp dp 0 V (1 - ~2)moc2

In both cases the identification of the group velocity U of a wave-packet with particle velocity v holds. This coincidence is a common source of confusion between the relativistic de Broglie waves (phase velocity V = c2/v, frequency

DuAUSM WITHIN DuALISM 103

v = E/h) and the nonrelativistic SchrOdinger waves (phase velocity V = v/2, frequency v = mv2/2h). Moreover, the SchrOdinger waves which are statistical featured waves without local concentration of energy (as the Born waves) cannot be likened to real physical waves. For this reason, de Broglie(3) expressed his "double solution hypothesis": to each solution 'It = a exp (is) of the propagation equation, there must correspond a solution Uo = fexp(iS) with the same phase S, but one whose amplitude f exhibits a singularity moving with the particle velocity. The U 0 corresponds to an extended wave phenomenon centered on a very small region standing, strictly speaking, for the particle. The Uo wave would be the solution of a yet unknown nonlinear equation and the qr wave the solution of its linear approximation, at least outside the singularity.

If we take 'It = a exp(iS), the linear Klein-Gordon equation

splits into

(C): V(a2VS) = 0

(J): (VS)2 + mac2 = IiDala

(C) is a continuity equation and (1) a Jacobi equation.

(15)

(15)

(16)

In the right-hand side of Eq. (16,J) one recognizes the relativistic generation of the quantum potential introduced by de Broglie, which expresses the reaction of the wave deformed in the presence of obstacles to its propagation. In particular, the explicit calculation of this quantum potential in the two slit situation(4) shows how to obtain interference without the need to abandon the notion of well-defined trajectories.

Now, let us consider a wave-packet of arbitrary shape surrounding the particle and defined by

u(x,t) = R(~~k) exp (is) (17)

with ~k = melli, ~ = x - vt, S = (mc2/Ii)t - (mvlli)x, m = molY/(1 - (32). This wave-packet is the solution of a nonlinear two-dimensional Klein-Gordon equation

oVp Du - x2u = --u

Vp (18)

with 0 = 13""13,... = a2/ax2 - a2/c2at2, x2 = moc211i2, Vp = (UU)1I2 = lui. This equation is the relativistic extension of the nonlinear Hasse equation(5) (a Schro­dinger equation with a nonlinear term of "quantum potential" type) and describes propagation of kinks and solitons of arbitrary shape in the two-dimensional space­time.

104 PHILIPPE GuERET

Equation (18) stems from the Lagrangian

From this Lagrangian can be derived the same current as in the linear case, that is a condition required by the double solution theory. (3) We have

and for the energy-momentum tensor

T ... v = [(ulu)(<J ... u)(<Jvu) + (Ulu)(<J ... u)(<Jvu) - (<J ... u)(<JvU) - (<J ... U)(<Jvu)] - 8 ... v;£

(20)

(21)

In four-dimensional space-time, instead of (17), the nondispersive wave­packet associated with a particle of rest mass mo and traveling in the + x direction are defined by

u(r,t) = R(r) exp (is) (22)

with r = 1;1 = [(x - vt)2/Y(1 - ~2) + y2 + z2]1I2. They are solutions of

DU - -x.2u = (Rrr + 2R/r) exp (is) (23)

But a simple calculation shows immediately that, for u given by Eq. (22),

DvP vP U = (R)exp(iS) = (Rrr + 2R/r)exp(iS) (24)

so that Eqs. (23) and (18) have the same form. Among the solutions (22) there are solutions for which arise simultaneously

DvP Du = 0 and mij + C/i2/c2) vP = 0 (25)

i.e., the Eulerian differential equation

Rrr + 2R/r + 'X.2R = 0 (26)

which admits the general solution

[ sin 'X.r cos 'X.r ] u(r,t) = A-- + B-- exp(iS)

'X.r 'X.r (27)

where A and B are constants.

DuAUSM WITHIN DuALISM 105

As Mackinnon has shown, the sine solutions express the relativistic covari­ance of the phase concordance principle. (6) These solutions are also well known to represent the superposition of two spherically symmetrical waves, one converging and the other diverging. (3) In electromagnetism, they behave as waves in a phase­locked cavity similar to those analyzed by Jennison(7) and have the inertial properties of classical particles.

The cosine solutions, unbounded at the center, can also be retained since, in a realistic scheme, particles are not pointlike and have a radius ro very small but different from zero. These solutions can express the very high value of the amplitude near the particle.

The previous elementary calculations give a schematic model of V.o waves consistent with experimental data on the interferences of particles. Furthermore, as in the optical case, the coherence length L of matter waves is a measurable quantity: the electron longitudinal coherence length L used in interference experi­ments (Moellenstedt, Faget) is about 10-6 cm with a velocity on the order of 109 cm/s; for neutrons, L varies between 10-5 and 10-3 with a velocity on the order of 105 cm/s. It is important to underline the fact that an interference pattern is not dependent on the observer may be explained by de Broglie waves, but not by Schr6dinger ones, (8) even in the nonrelativistic case.

Thus, a dualism between the 'I' wave and the Vo waves appears within the wave-particle dualism for matter, exactly in the same way for light and the Nadelstrahlung model occurs again.

A satisfactory manner to avoid this problem would be to ascribe all the physical properties to the wave-packets and the statistical ones only to the a wave, in the Born way. de Broglie discussed this important problem in a very interesting but practically unknown article published in the Cahiers de Physique. Analyzing the Einstein model, de Broglie did not retain the above proposition. Indeed, he reminded the reader that, about 1920, in order to put the Nadelstrahlung to the test, Schr6dinger suggested the following experiment: "with the help of mirrors, to have a try at doing interfere the light beams emitted in nearly opposite directions" and de Broglie wrote: "The experiment was performed and has given a positive result. As the photons were emitted one after the other by the atoms of the source, this result seems to give an evidence, contrary to the Nadelstrahlung hypothesis, that just when the emission occurs, the wave going out the atom and carrying a photon is a classical spherical wave,"(9) i.e., a I wave.

This experiment seems to have made a great impression on de Broglie's mind and led him to abandon Einstein's idea. But nowadays, one knows that it is not simple to build a source emitting photons one after the other, so that the Schr6dinger experiment needs confirmation. If new experiments would negate the results of the former and uphold the Nadelstrahlung, a space to new inquiries about individual atomic processes would be created. The wave functions Vo should describe a real phenomenon propagating in space-time as in the original de Broglie wave mechanics, the wave functions '1', elements of an abstract functional space, retaining all their quantum mechanical assignments. This view implies that,

106 PHILIPPE GuERET

mathematically and experimentally, a few questions remaining open, as, for instance:

1. The transversal size of a wave-packet, i.e., the transversal coherence length.

2. The physical nature of the Uo waves (in other words, what is waving?). For light, it appears clearly that Maxwell electromagnetic waves are associ­ated with photons, in the case of massive photons introduced by de Broglie,(IO) or for conventional zero-mass photons as shown, particularly, by the Majorana equation. (11) But the problem remains unsolved for the other particles: for instance, are protons and neutrons associated with the same "nucleonic" wave? A solution of this question is very likely tied to a realistic reinterpretation of the quantum field theory.

3. The "empty wave-packet" problem, i.e., the examination of the proper­ties of wave-packets deprived of their particles. Empty waves are neces­sary to explain one particle interference with itself. The theoretical study of their possible intervention in the domain of neutronic interferometry(l2) and their eventual ability to induce stimulated emissions of radiation(13) have already been examined. But other problems remain unsolved: for instance, are the empty waves absolutely devoid of energy? A negative answer should lead to a concrete description of how a photon grows old by repeated diffractions and, particularly, by taking in account this consid­eration, it is possible to obtain an available alternative explanation of the redshift of astronomical objects. (9) Another problem is: why an empty wave separated from an electrically charged particle, would remain sensitive to an electrostatic field?

4. The wave-wave dualism within the wave-particle dualism involves the existence of two kinds of quantum potential: one 0 a/a built from the wave function 'II" = a exp(iS.) gives the trajectories of the wave-packet center,(4) the other, built from the wave function Uo' is connected with the position fluctuations of the particle inside the wave-packet. This scheme is consis­tent with stochastic theories, but without the need for a recourse to a hypothetical chaotic medium as the Dirac ether.

S. An important problem is the spin dependence of the U 0 waves. From its solution depends, namely, space-time specification of the polarization measurements as it occurs peculiarly in the EPR experiments.

REFERENCES

1. L. DE BROGLIE, in: Wave Mechanics: The First Fifty Years, Butterworths, London (1973). 2. L. DE BROGLIE, Ann. Phys. (Paris) 3, 22 (1925). 3. L. DE BROGLIE, Nonlinear Wave Mechanics, Elsevier, Amsterdam (1960).

DuALISM WITHIN DuALISM 107

4. C. PHILIPPIDIS, C. DEWDNEv, and B. 1. HILEv, Nuovo Cimento B 52,1 (1979). 5. R. w. HASSE, Z. Phys. B 37,83 (1980). 6. L. MACKINNON, Found. Phys. 8, 157 (1978). 7. R. C. JENNISON, 1. Phys. A Gen. Phys. 11, 1525 (1978). 8. L. DE BROGLIE, Cah. Phys. 147, 1 (1962). 9. L. DE BROGLIE, Ondes electromagnetiques et photons, Gauthier-Villars, Paris (1968).

10. R. MIGNANI, E. RECAMI, and M. BALDO, Lett. Nuovo Cimento 11, 568 (1974). 11. C. DEWDNEV, P. GuERET, A. KVPRIANIDIS, and 1. P. VIGIER, Phys. Lett. A 102 (7), 291 (1984). 12. E SELLER!, Found. Phys. 17(8), 739 (1987); A. GARUCCIO, P. GuERET, and E SELLER!, Found.

Phys. Lett. 1(2), 139 (1988).