the extended wave-particle duality

Volume 96A, number 4 PHYSICS LETTERS 27 June 1983

THE EXTENDED WAVE-PARTICLE DUALITY

R. HORODECKI Institute of Theoretical Physics and Astrophysics, University of Gdatlsk, 80-952 Gda~isk, Poland

Received 26 October 1982 Revised manuscript received 28 March 1983

The extension of the de Broglie wave hypothesis for particles for which the absolute value of the wave four-vector is not zero is proposed in terms of the so-called 3 (2 In I + 1) and 21 n I + 1 waves hypothesis. Some theoretical and experimen- tal consequences are discussed.

1. Introduction. It has been shown [1 ] that the fundamental de Broglie postulate on the wave-particle duality implies a model of the extended particle in terms of the three waves hypothesis (TWIT). Recently, Gueret and Vigier presented a model of the double solution theory [2] where the role of the de Broglie singular wave [3] is played by an intrinsic nondisper- sire Compton wave (C-wave) [ 1,4,5 ]. On the other hand, it has been shown that there is a subluminal wave (D'-wave) dual to the de Broglie wave (B-wave) in a strict sense, i.e. the group velocity of the D'-wave equals the phase velocity of the B.wave [6]. Thus, it is seems rather natural to think that the de Broglie energy- frequency relationship is only approximately correct and that other frequencies and wavelengths can be associated with the extended particle for the same value of the energy. Whence, one can expect that the TWH is a reflection of any wave phenomenon governed by a nonlinear law that provides a singular velocity-depen- dent spectrum of frequencies in which the role of fundamental frequency is played by the de Broglie frequency 6o 0:

~0 = moc2h- l (1 - 32) -1/2 , (1)

where o is the velocity of the particle,/3 = o]c, m o is the rest mass of the particle, h = 2rr~ is Planck's con- stant and c is the velocity of light.

The purpose of the present letter is to present an extension of the TWH in terms of the so-called 3 (21nl

+ 1)waves hypothesis [3(2 In I+ 1)-WH] and 2 In I+ 1 waves hypothesis [(2 In[ + 1)-WH] and discuss its possible significance.

2. 3(2 In I ÷ 1)- WH and (2 In I + 1)- WH. We now suppose that the particle can be represented by a nonlinear wave-packet that interacts in a hidden form with any subquantum medium that in the linear-interaction limit can be identified with the Dirac covariant extended ether [7-9] . As a consequence there will be various harmonics and wavelengths associated with the particle when it moves in any Lorentz frame with velocity o. Of course, there are many possible ways of choosing such a spectrum since we do not know of any universal nonlinear wave equation that governs the dynamics of the extended particle. Our fundamental condition is that a hypothetical spectrum of the extended particle must be unique and compatible with the TWH.

The possible spectrum that satisfies the above condition can be defined as follows:

E n=~¢o n , n =0,+--1,+2 . . . . , (2)

with

~n = t~n~0, (3)

whereE n and 6o n denote nth energy and nth de Broglie frequency respectively. By analogy to the TWH we define the 3(2 Inl + 1)momenta as follows

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P . = a ~ n / o r = r ~ , , , I k,, t = 2~r~.n 1 , (4)

Pn =~it°n/V=fikn , I k ~ i = 2 r r ( X ~ ) - I , (5)

Pn = ~i60n/c = f ikn, kn = 27r~n -1 , (6)

where the momentum Pn, its dual momentum Pn and the intrinsic momentum i~ n are connected by the relation

• • ( 7 ) Pn Pn •

According to eq. (1), in addition to the fundamental frequency 600, there are two lateral "bands" of the frequency; the upper and the lower for negative and positive value of n respectively. Now, for each fixed frequency COn there are three waves. Thus, we have the the extension of the TWH into the 3 (2 1 n I + 1)-WH.

It is characteristic that there is also a possibility of a complementary extension of the TWH in terms of the (2In[ + 1)-WH. For this purpose we define 2In[ + I wave-vectors

Pn' = h k ' n , n = O , + l , + 2 , (8)

where

f n t k' n =~ k 0 (9)

and

k' 0 = mbc~ -1 (1 - ~2)-1/2 , (10)

where k~ has the direction of the three-velocity and does not vanish in any Lorentz frame.

Now 3(2[n l + 1) frequencies are defined by the relations

co n - k n - ~ f , (11)

ton* - ' (12) - k n • ~ ,

~ ' - ' , ( 1 3 ) 60n - kn " c

t where the frequency 60n and its associated frequency 60n* are connected by

t , * Oan60 n = (~'n) 2 (14)

It is easy to see from eqs. (3)--(6) and (9 ) - (13 ) that the following relations are fulfilled:

~ t t t: k 60n = 60n , 60n = 60n-1 , Wn = 60n+1 , (15)

k n = k ' n _ l , kn=k 'n+ 1 . (16)

It must be noted here that the above relations are

unique for 0 < ~ < 1 since the de Broglie fundamental frequency 600 as well as the phase-group velocity inter- relation oof = c 2 are unique. As a consequence, there is a unique relationship between the (2In[ + 1)- and 3(2 In [ + 1)-WH and they can be treated as two differ- ent complementary ways of the descriptio n of the extended particle.

3. Some special cases. Notice that eqs. ( 3 ) - (6 ) or (9) - (1 3)contain some known characteristic frequencies and wavelengths of the wave-part icle duality. In particular, putting n = 1 in eqs. (9 ) - (13) we see that in fact the B-wave is associated with the three frequencies, (i) fundamental 600, (ii) first harmonic w 1 and (iii) second harmonic 6o 2. It was pointed out by de Broglie in his original paper [10] that the frequency 60 2 and the wavelength of the B-wave are characteristic o f the atomic phenomena if a stationary orbit condition is supposed. On the other hand, as shown by Kemble [1 1 ], the time-independent SchriSdinger equation is a nonrelativistic approximation to the equiva- lent o f the B-wave, associated with the fundamental frequency w 0. Moreover, recently it was pointed out [6] that 601 is the frequency of the dual wave to the B- wave for which a space4ike wave four-vector can be formed. Using the duality principle [I ,6] and the de Broglie phase connection principle one can construct a superluminal, soliton4ike dual wave to the C-wave (D c- wave) in a strict sense [1 2], which for the particle trav- eling in the x direction with velocity v relative to the laboratory system takes the form

sin [A60 0 (t -- x /of)] ~ ( t , x ) = A600 ( t_ x / v f ) e x p ( i [ k ~ x - 6 0 1 ( k l ) t ] } ,

(17) where

A60 0 = (mOc2 /•)(1 -- /32)-1/2 .

This singular wave "locates" a particle in the range of time defined by this wave. It is consistent with the minkowskian interpretation of special relativity according to which time must be "space-like" [13].

There is still another characteristic frequency 60-1 associated with the D-wave that appears in the dispersion relation for the D-wave. Now, we discuss briefly the B- and D-wave.

(A) B-wave. According to (1 1) and (12) the wave equations for the monochromatic superluminal and subluminal B-wave can be written as follows:

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V2~0 -- O~2a2@o/Ot2 = 0 , (18)

72 ~1 -- 0-202 ~1 / Ot2 = 0 . (19)

When the B.wave is coupled to the Coulomb field then by the SchrOdinger "ansatz" [14] for the phase velocities of and o the time-independent Schr6dinger equation and its associated equation can be obtained. The Schr6dinger case is well known, thus we discuss briefly only the second one.

In the nonrelativistic approximation the C-wave can be neglected [4]. Consequently a point-like charged particle is assumed to be associated with the B-wave that is coupled to the Coulomb field V. Then, the phase velocity o is given by

v = m~ 1 [2m0(E - V)] 1/2, (20)

where E is the energy of the particle. Now the frequency 6o2, which in the nonrelativistic

limit is given by

6o(0) = m0o2h_l , (21)

is the momentary frequency, related to a subluminal B.wave that is coupled to the Coulomb field. In particular, for the radial s-states a stationary solution of (19) can be assumed in the form

ffl (r, t) = ql (r) sin co~0)t. (22)

Then, substituting (20) and (22) into eq. (19) we obtain the equation

d2~°l + 2 dql+ 2m0 E 2 dr2 r - ~ - - ~/2 E ~ tpl = 0 , (23)

which was first obtained by Rodimov [15] who postu- lated eqs. (11) and (12) (n = 1 case) as a fundamental for massive particles. The assumption, that ~1 acts as a quantum potential is' not in coritradiction with the Bohm interpretation of the SchriSdinger equation in terms of the extended Hamilton-Jacobi equation [16]. In particular, it is not hard to see that a quantum force defined by

F = - a ~l(r, t)/ar (24)

satisfies the Gryzifiski stationary-orbit conditions (eqs. (19a, b)in ref. [17]) where he interpreted the Sehr6dinger equation in terms of the extended Newton's second law.

(B) D-wave. Putting n = 0 in eqs. (2) and (4)-(6) we get the TWH. In particular, in ref. [1] it was point-

ed out that the dispersion relation for the D-wave can be written in an implicit form:

"5 2 2 2 (25) = c co_ 1 6oc ,

where 6oe =cke = m0c2fl-1. Using eq. (1) the above equation can be written di-

rectly in terms of k~:

6o4 _ c2(k~)2 6o2 + c 2co2(k;)2 = 0 . (26)

Since the discriminant of (26) is positive, there are two stable modes:

(1) a low-frequency branch

6o21 = (c2/2)(k~)) 2 (1 - [1 -(k~)m)2/(k~)2]l/2), (27)

(2) a high-frequency branch

6o22 = (c2/2)(k~) 2 (1 + [1 - (k~m)2/(k~)2]l[2), (28)

where k ls the rmmmal value of k at the critical om 0 velocity c/21/2 which corresponds to the de Broglie frequency ¢O0m [2]. Thus, the dispersion relation (26) permits a natural minimum of the wave number k~ when the wavelength of the B-wave coincides with the Compton wavelength. We see that both dispersions (27) and (28) are normal if we restrict the wave vector k~ by the tachyon condition

k~ = Ik~l ~> k~m • ( 2 9 )

Then, using the usual definition of the group velocity we obtain the four group velocities Vg0?, 2 and v~),4 :

o(1) = + c (1 - a2)1/2 + a2/2 - 1 (30) g1,2 -21] 2 ( l - - a2 ) l ] 2 [1 - (1 - a2 )1 ]2 ]1 ] 2 '

(2) = ± c (1 --a2) 1/2 -- a2/2 + 1 , (31) g3,4 21/2 (1 -- a2) 1/2 [1 + (1 -- a2)1/2] 112

where a = k~m/ko, the signs "+" and " - " are related to the positive and negative frequencies, respectively. From eq. (30) it follows that the group velocity v (11) is negative for co01 > 0. However, o 4 ) appears to~e always superluminal and positive detinite for the high- frequency branch and co02 > 0, i.e.

c<~v (2 )<~ g3

when oo>k~/>k~m and ¢o02>0 .

Thus we have a singular superluminosity associated with the D-wave in the sense that there does not exist a unique one-to-one relationship between the momenta

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of the particle and the group velocities. This type of superluminosity may be interpreted in terms of a nonlinear self-interaction o f the extended particle.

4. Discussion. The above (nonlinear) extension of the de Broglie wave hypothesis constraints allow for the possibility of considering other waves than the straightforward B-wave. In particular, there is the possibility o f the existence o f singular waves associated with the internal spectrum (3) of the extended particle that may play a fundamental role at a subquantum level. For instance, it is not excluded that the singular superluminal Dc-wave carries a "quantum" of information the counterpart of which at a classical level may be Gabor's elementary cell of information [18]. It was mentioned by de B~oglie [ 19] that the identical singular waves may partially overlap and form a multi- soliton-like wave. Notice, that the C-wave as well as its dual wave can be neglected at a nonrelativistic level [4,12]. Thus, it seems rather reasonable to suppose that such singular waves play an essential role at high- energy interactions. According to (3) the more ener- getic particle is more "particle like".

In the low-energy limit the nonlinear oscillations o f the extended particle may be considered approximately as a superposition of the oscillations at frequencies given by (3), so the various wavelengths may be observable. In particular, the 3(21nl + 1)- and (2Jnl + 1)-WH predict the possibility of the detection of superluminal waves in experiments of the type propo. sed by Vigier et al. [20] (assuming that a photon has nonzero mass).

5. Concluding remarks. The 3(21n + 1)- and (2lnl + 1)-WH seem to be consistent with the idea that the particle is an intrinsically spatially as well as temporal- ly extended wave phenomenon governed by a nonlinear law [19]. As a consequence there are various

harmonics and wavelengths associated with the particle which have natural counterparts at various interpretation levels o f quantum phenomena. Moreover, the 3 (2 In I + 1)- and (2 [ n I + 1)-WH predict the existence ~ of real physical correlations inside a quantum-mechan- ical system.

The author would like to thank Dr. A.J. Horodecki and Mr. J. Walczyriski for useful discussions. He is also grateful to a referee for his very helpful criticism.

References

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(1982) 256,260. [3] L. de BrogUe, Tentative interpr6tation causale et non

lin~aire de la mechanique ondulatoire (Gauthier-Villats, Paris, 1956).

[4] L. Mackinnon, Found. Phys. 8 (1978) 157. [5] L. Mackinnon, Lett. Nuovo Cimento 31 (1981) 37; 32

(1981) 311. [6] R. Horodecki, Phys. Lett. 91A (1982) 269. [7] P.A.M. Dirac, Nature 168 (1951) 906. [8] K.P. Sinha, C. Sivaram and E.C.G. Sudarshan, Found.

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[10] L. de Broglie, C.R. Acad. SCI. 117 (1923) 507. [ 11 ] E.C. Kemble, Fundamental principles of quantum

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the extended wave-particle duality

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