De broglie wave and its dual wave
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Volume 87A, number 3 PHYSICS LETTERS 28 December 1981DE BROGLIE WAVE AND ITS DUAL WAVER. HORODECKIInstitute of Physics, University of Gdask, 80-952 Gda/isk,PolandReceived 16 October 1980Revised manuscript received 30 October 1981A wave new, the dual wave to the de Broglie wave is introduced with a group velocity of the de Broglie wave. Somephysical consequences of the hypothesis are discussed.In spite ofgreatnumber of physical results concerning Thus, the de Brogue frequency w as well as the phasede Broglie waves (B-wave) their physical interpretation velocity u are unique. The dispersion law (2) holdsstill remains an open question [1,2]. For explaining equally for all particles i.e., for massless and massivethe waveparticle duality the existence of a sub- bosons and fermions. In particular, for massless par-quantum medium (aether) was postulated [3,4,5]. tides, the phase velocity of the B-wave is equal to theOn the other hand it was pointed out  that the group velocity of the particle which is, of course thedispersion law for B-waves is unique if the fundamental velocity of light.postulate of de Broglie is assumed, which can be for- Further, it will be shown that for massive particlesmulated as follows: The wave four-vector (kB, iw/c) the phase velocity of the B-wave can also be a groupof the phase wave of de Brogue  is parallel to the velocity, provided a new wave is introduced whosemomentum four-vector (PB, iE/c) of the particle in phase velocity is equal to the speed of the particle.the four-space and the constant of the proportional- Such a possibility follows directly from the de Brogueity between the components of the four-vectors is postulate of the waveparticle duality. It must bePlancks constant, emphasized here, that de Broglie relations (1) areAs a consequence, de Broglie relations between not the results from the special relativity theory, butenergy E and angular frequency w and between mo- are implied by particlewave duality plus special re-mentumpB and wave vectorkB take the form of lativity theory.In this paper we assume that in the Lorentz frame,E=llw, P~l~~,IIBH2~B . (I) where the particle is at rest, it can formally be asso-The dispersion law for B-waves is then given by ciated withan intrinsic nondispersive C-wave (Comp-= +~ 2 2 2 4 _2)1/2 ( ~ ton-wave). In the Lorentz frame, in which the par-ticle moves with velocity u, it is assumed to be asso-where m0 is the rest mass of the particle and c is the ciated with the three waves: (i) transformed C-wave,velocity of light. (ii) superluminal B-wave, (in) subluminal wave, whoseDe Broglie has shown that the group velocity Ug phase velocity is the particle velocity u (D-wave).of the B-wave is equal to the velocity u of the part- The relations between all three waves can be easilyide, and that there exists a unique relationship be- derived from eqs. (1), (2) and (3). First, the expres-tween group and phase velocity Uf of the B-wave and sion for the energy E can be written in the formthe velocity of light:E-11wmu-uf. (4) 2 (3\ From (1) and (4) we find that the de Broglie frequencyg Uf C ~ w can be written as0 031-9163/81/00000000/s 02.75 1981 North-Holland 95Volume 87A, number 3 PHYSICS LETTERS 28 December 1981= ui-- kB or w = U kD (5) w = (c2ikDI2 m~c4 2132)h/2 (13)where The group velocity ~ of the D-wave is given byXB~21r~BI h/mv,2D = 2irlkDj~ h/mvf,(6) ~ = VkDW =Uf (14)are the wavelengths of the B and D-wave respectively, Thus, the group velocity of the D-wave appears to bem is the intertial mass of the particle and v = ui; Vf = equal to the phase velocity of the B-wave. Therefore,lUf I. The dual momentum PD associated with the wave the D-wave is called the dual wave.vector kD is defined as follows It is important to note that the group velocity of thePD ~0)/0=~IkD (7) D-wave is greater than the velocity of light, however,in contradistinction to the tachyon problem  theIt is easy to see from eqs. (3), (4), (5) and (6) that the dispersion law (13) is normal. On the other hand, it isfollowing relations are fulfilled anomalous in this sense, that iF/c and PD do not forma four-vector. In consequence, the superluminal mo-= ~ k~=kBkD (8) tion of the D-wave packet is not tachyon-like.XD/XB = j~2 xC/XD = XBk = ~, (9) The above result is rather hard to understand with-out an additional assumption. Therefore, we supposewhere 2irk~= Xc = ~oc(l j32)l/2 is wavelength of that the waveparticle duality is due to the existence ofthe transformed Compton-wave X~and j3 = v/C. the particleaether hidden interaction. It was pointedIt must be noticed here that the wavelength of the out, that the extended subquantum model of Dirac-D-wave aether predicts the realexistence of B-waves [1,10,11].= hv/mc2Volume 87A, number 3 PHYSICS LETTERS 28 December 1981The author would like to thank L. Kostro Ph.D.  P.A.M. Dirac, Nature 168 (1951) 906.for the conversations which provided the basis for  L. de Broglie, La thermodynamique de la particulethis work and Dr. AJ. Horodecki for useful discus- isoie (Gauthier Villar,Paris, 1964). W. Mott and I. Sneddon, Wave mechanics and its appli-sions. The author would also like to express his sincere cations (Clarendon, Oxford, 1948).thanks to a referee, since due to correspondence with  L. de Broglie, Recherches sur la thkrie des quantahim the idea of this paper was made real. (doctoral thesis, Univ. Paris, 1924). English translation:J.W. Haslett, Am. J. Phys. 40 (1972) 1315.References  L. de Broglie, Comptes Rendus 117 (1923) 507.. L. Robinet, Phys. Rev. D18 (1978) 3610. K.P. Sinha, C. Sivaram and E.C.G. Sudarshan, Found. J.P. Vigier, Lett. Nuovo Cimento 29 (1980) 467. Phys. 6 (1976) 65. M. Gryziiiski, Phys. Lett. 76A (1980) 28.  A. Garuccio and J.P. Vigier, Found. Phys. 10(1980) D. Bohm and J.P. Vigier, Phys. Rev. 96 (1954) 208. 797.97
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