Volume87A, number3 PHYSICSLETTERS 28 December1981

DE BROGLIE WAVE AND ITS DUAL WAVE

R. HORODECKIInstituteofPhysics,Universityof Gdañsk,80-952Gda/isk,Poland

Received16 October1980Revisedmanuscriptreceived30 October1981

A wavenew,the dualwaveto thede Brogliewaveis introducedwith a groupvelocityof thedeBrogliewave.Somephysicalconsequencesof thehypothesisarediscussed.

In spiteofgreatnumberof physicalresultsconcerning Thus,the de Broguefrequencyw aswell asthe phasede Broglie waves(B-wave)their physicalinterpretation velocity u areunique.The dispersionlaw (2) holdsstill remainsan openquestion[1,2]. For explaining equallyfor all particlesi.e., for masslessand massivethewave—particleduality theexistenceof a sub- bosonsand fermions. In particular,for masslesspar-quantummedium(“aether”)waspostulated[3,4,5]. tides,the phasevelocity of the B-waveis equalto theOn the otherhandit waspointedout [6] that the groupvelocity of the particlewhichis, of coursethedispersionlaw for B-wavesis unique if the fundamental velocity of light.postulateof de Broglie is assumed,which canbe for- Further,it will be shownthat for massiveparticlesmulatedas follows: Thewave four-vector(kB, iw/c) thephasevelocity of the B-wave canalso be a groupof the “phasewave” of deBrogue [7] is parallelto the velocity,provideda newwaveis introducedwhosemomentumfour-vector(PB, iE/c) of the particlein phasevelocity is equalto the speedof theparticle.the four-spaceand theconstantof the proportional- Sucha possibilityfollows directly from thede Brogueity betweenthecomponentsof thefour-vectorsis postulateof thewave—particleduality. It mustbePlanck’sconstant, emphasizedhere,that de Broglie relations(1) are

As a consequence,deBroglie relationsbetween not theresultsfrom thespecialrelativity theory,butenergyE andangularfrequencyw andbetweenmo- areimpliedby particle—waveduality plusspecialre-mentumpBandwavevectorkB taketheform of lativity theory.

In thispaperwe assumethat in the Lorentz frame,E=llw, P~”l’~~,II’BH2~B . (I) wheretheparticleis at rest,it canformally be asso-The dispersionlaw for B-wavesis thengivenby ciatedwithan intrinsicnondispersiveC-wave(Comp-

= +~ 2 2 2 4 _2)1/2 ( ~ ton-wave).In the Lorentz frame,in which thepar-ticle moveswith velocity u, it is assumedto be asso-

wherem0 is therest massof the particleand c is the ciatedwith the threewaves:(i) transformedC-wave,

velocity of light. (ii) superluminalB-wave,(in) subluminalwave,whoseDe Broglie hasshownthat thegroup velocity Ug phasevelocityis theparticlevelocity u (D-wave).

of the B-waveis equalto thevelocity u of thepart- The relationsbetweenall threewavescanbe easilyide,andthat thereexistsa uniquerelationshipbe- derivedfrom eqs.(1), (2) and(3). First, theexpres-tweengroupandphasevelocity Uf of the B-waveand sionfor theenergyE canbe written in theformthe velocity of light:

E-11wmu-uf. (4)

— 2 (3\ From(1) and (4) we find that the de Broglie frequency

g Uf — C ‘ ~‘ w canbewritten as

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Volume 87A, number 3 PHYSICS LETTERS 28 December1981

= ui-- kB or w = U kD ‘ (5) w = ±(c2ikDI2— m~c4 2132)h/2 (13)

where The group velocity ~ of theD-wave is given by

XB~21r~BI’ h/mv,2’D = 2irlkDj~ h/mvf,(6) ~ = VkDW =Uf (14)

are thewavelengthsof the B and D-wave respectively, Thus,thegroup velocity of the D-waveappearsto bem is the intertial massof the particleandv = ui; Vf = equalto thephasevelocity of theB-wave.Therefore,lUf I. The dual momentumPD associatedwith thewave the D-waveis calledthe “dual wave”.vectorkD is definedasfollows It is importantto note that the group velocity of the

PD —~0)/0=~IkD (7) D-wave is greaterthan thevelocity of light, however,in contradistinctionto thetachyonproblem[9] the

It is easyto see from eqs.(3), (4), (5) and (6) that the dispersionlaw (13) is normal.On the otherhand,it isfollowing relationsare fulfilled anomalousin this sense,that iF/c andPD do not form

a four-vector.In consequence,thesuperluminalmo-= ~ k~=kB’kD (8) tion of the D-wave packetis not tachyon-like.

XD/XB = j~2 xC/XD = XB’k = ~, (9) The aboveresult is ratherhardto understandwith-out an additionalassumption.Therefore,we suppose

where2irk~= Xc = ~oc(l — j32)l/2 is wavelengthof that the wave—particleduality is due to the existenceofthe transformedCompton-waveX~andj3 = v/C. theparticle—aetherhiddeninteraction.It was pointed

It mustbe noticedhere that thewavelengthof the out,that the extendedsubquantummodel of Dirac-D-wave aetherpredictsthe realexistenceof B-waves[1,10,11].

= hv/mc2<h/m0c = Xoc , (10) Thus,thedispersionlaw (2) canbe explainedby inter-

preting the energyE asthe energyof an elementarydoesnot exceedtheComptonwavelengthX~.In quasi-particleexcitationfrom the underlyingvacuumaddition, the“natural” maximumwavelengthof XD at stateof the Dirac-aetherif the latteris regardedasavelocity v” = c/-~.,/~exists,givenby superfluidstateof particle—antiparticlepairs [101.

X = h/2m0c= ~X0~ (11) However,contraryto eq.(2) the dispersion(13) cannotDm be usedto impart massto theexcitation,since iE/c andThiscorrespondsto the de Broglie frequency PD do not form a four-vector.Then,it may be possible,

= ~hm0c2/ii - (12) to interpretresult (14)by the assumptionthat the D-

wavepacketis associatedwith a collectivemotion whichIn particular,it is interestingto note that it is possible is carriedby the Dirac-aether.Thereare collectiveexcita-to give a definition of theComptonwavelengthin terms tions (similar to the Goldstonemodes)possessingtheof the B-wave.Let L’ be a Lorentzframe in which the dispersionlaw givenby (13) whereE is energyandPDparticleis at rest.Thenaccordingto the de Broguepos- is themomentumof excitation.Notice,however,onetulate [8] the de Broglie frequencyis given by WO = oddity: the dispersionlaw permits thenaturalminimumm

0c2mandtheB-wave vectorby kB = 0. However, of the momentumexcitation ‘POrn I = 2m

0cat velocityif L’ moveswith respectto the laboratoryLorentz frame c/~,/~whenthe wavelengthof B-wave coincideswithL with the velocity u” = (c/v’~u/v,thenfor theob- theComptonwavelengthof theexcitation.serverassociatedwith L, thereexistsa B-wave,with fre- The importantconsequenceof the aboveinterpreta-quencyand wave vector = V’~.’0andkB = (m0c/ tion is the possibilityof explaining the nonlocalsuper-h)u/v, respectively.Thus,the reciprocalof the length luminal interactionsresultingfrom the Einstein---of the latter is simply the Comptonwavelength. Podoisky—Rosenparadox.

In order to know the relationshipbetweenthe Althoughthe consequencesof the D-wavehypothe-group andthephasevelocitiesof both B and D-wave, sis still needto be investigatedfurther,we believethatone would haveto know the dispersionlaw for the D- the aboveconsiderationsunderlinethe role of thewave—wave,This canbe derivedfrom eq.(2) by usingeqs. particleduality. Furtheranalysisof the aboveresultsand(4), (7) and(9). The result is: in particular,the presentationof the D-wave-equation

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Volume 87A, number 3 PHYSICS LETTERS 28 December1981

The author would like to thankL. KostroPh.D. [4] P.A.M. Dirac, Nature 168 (1951)906.

for the conversationswhichprovidedthebasisfor [5] L. de Broglie, La thermodynamique de la particulethis work andDr. AJ. Horodeckifor usefuldiscus- isoiée(GauthierVillar,Paris, 1964).

[6] W. Mott and I. Sneddon,Wavemechanicsandits appli-sions.The authorwould alsolike to expresshis sincere cations(Clarendon,Oxford, 1948).thanksto a referee,since due to correspondencewith [7] L. de Broglie, Recherchessur la thkrie desquanta

him the ideaof this paperwas madereal. (doctoralthesis,Univ. Paris,1924).Englishtranslation:J.W. Haslett,Am. J. Phys.40 (1972) 1315.

References [8] L. deBroglie, ComptesRendus117 (1923)507..[9] L. Robinet,Phys.Rev.D18 (1978)3610.[10] K.P. Sinha,C. SivaramandE.C.G.Sudarshan,Found.

[1] J.P. Vigier, Lett. Nuovo Cimento 29 (1980)467. Phys.6 (1976)65.[2] M. Gryziiiski, Phys.Lett. 76A (1980)28. [11] A. Garuccio andJ.P. Vigier, Found. Phys.10(1980)[3] D. Bohmand J.P. Vigier, Phys.Rev. 96 (1954)208. 797.

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