IL NUOVO CIMENTO VOL. 80B, N. 2 11 Aprile 1984

Wave-Particle Duality and Extended Special Relativity.

R. HORODECKX

II~stitute o/ Theoretical Physics' and Astrophysics, Unicersity o/ Gdai~sk 80-952 Gdafisk, Poland

(rieevuto il 25 Agosto 1983)

Summary. - It is shown th~{, the wa.ve-pa.rbicle duality iml)lies the extension of special relativity to superluminal frames. The re,~l (me- dimensional SUl)erluminal Lorentz tranMormations are obtained as a spe(;ial case of the Parker-Antippa transformations. Some interrelations between extended special relativity and the man.v-wave hypothesis are discussed.

PACS. 03 .65 . Quantum theory; quantum mechanics.

As one knows, special re la t iv i ty (SR) serves very well as an essential fac tor in the in t roduc t ion of m a t t e r waves (~.3). On the o ther hand, it has been

shown (4) t h a t the de Broglie pos tu la te on the wave-par t ic le dual i ty implies

a model of the ex tended par t ic le in terms of the (, three wave )~ hypothes is

t h a t involves subluminal as well as super luminal waves (5,s). Recen t ly , the nonl inear extens ion of the de Brogl ie w a v e hypothes i s has been presented in terms of the 2 1 n [ + 1 and 3 ( 2 [ n [ - ! - 1 ) wave hypothes is (7) (called here

m a n y - w a v e hypothes i s (MWII)) . The aim of the present pape r is to show thai , if one accepts the pos tu la te

on the wave-par t i c le dual i ty as fundamen ta l , then it implies the extension

(1) L. DE BROGLIE: Recherches sur la th~orie d~.s quu~tta (Doct;or:tl 1,hesis, Unive~.~ity of Paris, 1924). English translation: J . W . HASLETT: Am.. J. Phys., 40, 1315 (1972). (2) E. MAC KIYNON: A~n. J. Phys., 44, 1047 (1976). (a) L. MXCKIN.~ON: Found. Phys., 8, 157 (1978). (4) R. HORODFCKI: Phys. Lett. A, 8"7, 95 (1981). (a) R. [{ORODECKI: Phys. Lett. A, 91, 269 (1982). is) R. IIo,tODECKI: Lett. ,Yuoro Cimento, 36, 509 (1983). (7) R. HORODECKI: Phys. Left. A, 96, 175 (1983).

217

218 ~t. }[ORODECKI

of SR to superluminal frames. For this we adop t the Minkowskian interpre- ta t ion of SR, according to which the invar iant space-t ime interval is an ob- server- independent theoret ical quant i ty (8).

I f we take the assumpt ion of the wave-par t ic le duali ty as the foundat ion of our argument , it does not seem too unreasonable to identify this interval with the rest wave-length of the Compton wave (C-wave) (3.,). As a consequence, the Minkowski space- t ime relation

(l) c,(At) : - ( a t ) ' = ).~

can be interpreted as a law of nature ra ther t h a n a logical construct ion upon spat ia l and tempora l intervals.

Pos tu la t ing (1), we assume tha t the ordinary (external) space-t ime of the part icle can be extended to an intrinsic space-t ime, avai lable to the particle, if its mot ion is assumed to be observer independent (3). I t is compat ib le with five-dimensional re la t iv i ty (9) if 2~ ~ is identified with the constant curva tu re of the de Sit ter microuniverse (~o). I f we assume now tha t eq. (1) is our funda- men t a l postulate, then it implies the existence of the Lorentz t ransformat ions . As one knows, by using these latter, observer-dependent t ime can be always <~ projected out)> f rom eq. (1)(i,). I t provides the familiar clock slow-down formula

(2) At : (Arc)(1 - - fl")--~,

where At c ---- c - ' ).c ~ -~ ~,~' is the proper t ime corresponding to a rest mass me, v is the velocity of the particle, fl = v/c, h ~ 2zth is P lanek 's cons tant and c is the velocity of light. I n part icular , pu t t ing in the above formula fl ~ 0, we get A t - ~ h /moc ' -= Ate, thus eq. (1) (called here Minkowski law (~IL)) implies the de Broglie quan t um law hv c ---- m oc 2 (~2). As one knows, the above relat ion can be wri t ten in the form

(3) (At) -1 ~-~ re2 ~ re - - va,

where vo and r2 are the fundamenta l frequency and second harmonics of the in ternal spec t rum of the particle, respectively (7). Thus the MI~ is compat ib le with the MWH, since it involves some characteris t ic frequencies associated with the internal spec t rum of the particle. On the other hand, it is clear f rom lhe point of view of the M~vVH, tha t the ML is only approx imate ly correct, thus

(8) F. CIIRISTENSEN: Br. J. Philos. Sci., 32, 37 (1981). (9) G. ARCIDIACONO: Gen. Rel. Gray., 13, 703 (1981). (~o) T. CHACKO: Int. J. Theor. Phys., 12, 95 (1975). (") J .D . JACKSON: Classical Electrody~mmics (J. Wiley & Soils, Inc., New York, N. Y., 1975). (l~) L. DE BI~O(;LIE: C. It. Acad. Sci., 117, 507 (1923).

WAVE-PARTICLE DUALITY AND EXTENDED SPECIAL RELATIVITY 219

it mus t be extendcd. In this pape r we restr ict our considerations only to the two-dimensional case (~3) assuming t h a t the ML can be extended as follows:

(4)

(5)

c , ( A t ) 0- _ ( A x ) '~ = ) .~

c ' ( A t ' ) , - - ( A x ' ) , = - - ; ~

for t imelike separated events,

for spacelike separated events.

:Note t h a t the extension of Minkowski space-t ime to the pseudo-Eu- clidean space- t ime is well known in the physical l i terature, in the context of ex tended SR (~3-2o), however, it was not considered in the context of the wave-par t ic le duali ty. In part icular , f rom eq. (5) the superluminal transfor-

mat ions can be deduced

(6) x = P(x '+ v,r),

(7) t = S C t ' + v , z ' / c q

with

= (fi/ lf l l)(/~- 1)*, v, = ~-, = e f t ,

which are a special case of these Parkcr -Ant ippa t ransformat ions (~4.~a,). :Note t h a t the Ant ippa factor is necessary here if we want to have supcrluminal

velocity v, compat ib le with causali ty (~6). Let us now consider the C-wave in its rest frame. In the two-dimensional

case it is given by (3)

(s) ~ . (x , t)

where Ak c = r%c/h.

sin (2~Akcx) 2.'~ Akcx

exp [i(2,~rc t)] ,

I f we now t ransform this wave into a Lorentz f rame moving with the relative

veloci ty/~ > 1, the wave function of eq. (8) becomes

(9) ~ (x ' , t ') oc sin [2~ Avo(t' -i- x'[v,)]

2UAVo(r + x'/v,) cxp [i[2~(ktx' + v~t')]] ,

where A v e = (mock/h)(1- fl~)-t, v~ and k~ are the frequency and the w,~vc

number of the D' -wave, respectively (~).

(ta) L. MARCHILDON and A.F . ANTIPPA: CaN. J. Phys., 61, 256 (1983). (t4) L. P&RKER: Phys. Rev., 188, 2287 (1969). (15) V.S. OLKItOVSKY and E. RECAI~II: Lett. ~NUOVO Cimcnto, l , 165 (1971). (t6) A . F . ANTIPPA and A. E. EVERETT: Phys. Rev. D, 4, 2198 (1971); 8, 2352 (1973). (t:) E. RECAM[ and R. MI(;NAyI: Lett..~uovo Cimento, 4, 144 (1972); 8, 110 (1973). (is) A .F . ANTIPPA: Nuovo Ci'mento A, 10, 389 (t972). (19) A . F . ANTIPPA: Phys. Rec. D, 11, 724 (1975). (~o) G.D. MACCA~RONE and E. RECAMI: Le~t. Nuovo Cime,do, 34, 251 (1982),

2 2 0 1r II()RODECK I

In particular, if we app ly the t r~seendcnt t ransformat ions

00) = (/7/lfil) c r , t = (fitlDil) x'/c

in eq. (8), we get for fi > 1 the stai~ionary De-wave

sin (2n A~,c t') (11) ~o(X', t ')oc 2~A~,c t, cxp I i (2nkc:x ')] ,

which was obtained in a recent paper (s). However , in contrast with the D c- wave t ha t was obtained in ref. (6) by apply ing Lorentz t ransformat ions in eq. (11), the above De-wave appears to be compat ib le with two-dimen- sional extended SR. As one knows, the superluminal De-wave locates the par- ticle in the t ime range defined by this wave. I t is consistent with the Minkow- skian interpreta t ion of SR according to which t ime mus t be (, spacelike ~)(8). I t is also consistent with the result obtained by RAJPUT etal . (zJ) for t achyon localizability in t ime (2~.). :Notice that , as ia the case of the/?-wave (lZ), the fre-

quency paradox appears here. In fact, we see t ha t under the superluminal t ransformat ions (6), (7) the wave frequency becomes

f l = (ti/IDtl)tT,,~(1 - / 7~ ) -~ .

I t seem~ to contradict the superluminal clock formula

(la) At, , - !ftl ( ~ ' - ' ~ .~r,

which suggcsts an internal frequency

tha t differs by ~ factor - - f l - 2 ( 1 - flz) from (12). The abow~, paradox can be (, resolved ~ by the extension of the de Broglie

phase connection principle (':) to superlumin~l frames. In part icular, it is not hard to see that , if the SUl)erluminal clock and the De-wave are assumed to be in phase at a t ime t ' = 0, they also remain in phase for t' > 0. Thus it is clear t ha t the two-dimensional dual model of the double-solution theory (2s.z~), in which the role of the de Broglie singular wave is p layed by the De-wave , may

(21) B.S. RAJPUT, P. JOSHI and O. P. S. NEGI: Lett . .Nuovo Cimento. 35, 147 (1982). (22) See also A.O. BArn:T, G.D..~IACCARRONE and E. REc.s,.m: Nuovo Cimento .I, 71, 509 (1982). (2a) L. DE BrtO(;L[~: Tet~tative d'iulerpr~tatio~l causale el .non Ib~aire de la mdcha~dquc ondulatoire (I'aris, 1956). (~4) Pit. Gt,r:m:T and ,l. P. VI(;IER: Lett. ]~Tuoro CimenSo, 35, 256. 260 (1982). (?5) PH. GI;EHZT and J .P . VIC.IER: 1,'oand. Phys., 11, 1057 (1982).

W A V E - P A R T I C L E D U A L I T Y A N D ] ' ~ X T E N D E D S F I r F ~ , E L A T I V I T Y 221

be ob ta ined (~). ~Notice t h a t re la t ion (1.4) (,all be also wr i t ten in the form

(15) ~ (tit!/TI)( ~ I ---- ~)--11 = - - Y--1 - - Y l ) "

We see t h a t the above ex tens ion of the ML appears to be compat ib le with the M W H , since it involves the ha rmonics ~_~ and ~ of the i~ternaI spec t rum of the

ex tended par t ic le (~). Final ly , we conclude tha t , if one accepts the wave-par t ic le dual i ty as

f undamen ta l law of nature , then it implies the ex tens ion of SR to faster-

than- l igh t f rames. However , this implies t h a t the de Broglie ex tended par t ic le

m u s t no t necessari ly be described in a convent iona l Lo ren t z - cova r i an t manner . We see t h a t the M-WIt involves t imelike ,~s well as sp~eelike q u a n t u m -

mechanica l s ta tes (7). Thus the a s sumpt ion t h a t the E P ] t pa radox (~6) fol-

lows f rom the fac t t h a t q u a n t u m - m e c h a n i c a l predict ions do no t bear on a

pair of spacelike separa ted events (2~) leads us to the (-onclusion t h a t the eonvent ion ,d q u a n t u m - m e c h a n i c a l descr ipt ion of the physica l real i ty is no t

comple te (:6). Thus it seems r a the r reasonable to suppose t h a t the f u n d a m e n t a l

exper iments on the dual na tu re of the e lementa ry part icles can be in te rpre ted

in te rms of the ex tended wave-par t ic le dual i ty t h a t involves a spec t rum of a q u a n t u m waves (7,.0s,2,). ] n par t icular , it implies t h a t the E P R - t y p e experi-

ments (3o.3~) can be in te rpre ted causal ly (:3..~..~3) in t e rms of ~ causal evolut ion

of spacel ike q u a n t u m - m e c h a n i c a l states.

* $ $

The a u t h o r wouhl like to t h a n k Dr. A. J . IIoROI)ECKI for useful discussions.

(~-6) A. EINSTEIN, B. PODOLSKY and iN. ROSEN: l'hy,~. Rev., 47, 177 (1935). (-~:) M. BITBOI,: Phys. Lett. A, 96, (i(i (1983). (2a) F. SELL~RI: Found. Phys., 12, 1131 (1982). (29) j . ANDRADE E SILVA, F. SELLERI and J. 1 ). VIGIER: Lett. ,u Cime~do, 36. 51)3 (1983). (3o) A. (31) A. (32) A. (19Sl). (33) N.

ASPECT, ]). (~rRANGIER and G. ROGER: Phys. Rev. Lett., 49, 91 (1982). Gaacuc(:Io and V.A. RAPISARDA: .NUOVO Cimento A, 65, 269 (1981). (}ARYCCI(), V.A. RAPISARDA and J . P . \'IC, lElt: Lett. ,Yuovo Cimen#o, 32, 451

CUFARO PETRONI and J. P. VI(;IER: Phys. Lett. A, 93, 383 (1983).

�9 ; R I A S S U N T O (')

Si mostra che la dualit'L onda-particella implic'l l'estensione delle rel'divit~ ristretta a sistcmi di riferimento superhmfinali. Si ottengono le trasformazioni di Lorentz superluminali monodimcnsion'di reali come ml case speciale dclle trasformazioni di Parker-Antipp~. Si discutono a.lcun(~ relazioni intcrcorrenti tra la relativitk ristretta estesa e l'ipotesi delle molte ondc.

(') Trad.uzio~le a cura della Redazione.

222 a. "HORODECK I

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