A dynamical basis for the de Broglie phase wave

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  • LETTI~R]~ AL NIJOVO CIM:EI~TO u 44, ~q. 8A 16 Dicembre 1985

    A Dynamical Basis for the de Broglie Phase Wave.

    M. H. MAC Gt~EGO~

    Lawrence L ivermore Nat iona l Laboratory (*) - L ivermore , Cal . 94550

    (riccvuto il 18 Luglio 1985; manoscr i tto revisionato r icevuto il 4 Dicembre 1985)

    PACS. 03.65 ~ Quantum theory; quantum mechanics.

    Summary . - Using the kinematic equations of perturbat ive special re lat iv i ty (PSR), which are val id in the l imit of small momentum transfers, we derive the part ic le-wave velocity equation vw = e 2, thereby providing a dynamical basis for the de Brogl ie phase wave. This samc PSR calculation also demonstrates that 1) the phase wave is accurately planar, and 2) the quanta of the phase wave are tachyons.

    The de BrogUe wave equation 2 ---- h/t~ was deduced on the basis of two postu la tes : 1) the extended Planck energy equat ion me 2 = hv; 2) the part ic le-wave velocity equa- t ion vw- - - -e 2. The de Broglie wave-length 2--~ w/v follows directly from these two postulates. The de Broglie phase wave ~(2, w, v), whose effect have been verified in a large number of experiments, is the only known phenomenon in physics that involves transmission in a vacuum at superluminal (w > e) velocities. Since this phase wave is produced by a part icle that travels at subluminal (v < c) velocities, the phase wave pro- duction process ties together the separate relat ivist ic domains of bradyons (sub- luminal particles) and tachyons (supcrluminal (~ particles )~). However, in spite of ex- tensive studies on this problem (~), the precise nature of the relat ivist ic l inking between bradyons and tachyons has not yet been delineated. As we demonstrate in the present paper, this l inking appears to be provided by the kinematic equations of perturbat ive special re lat iv i ty (PSR), which apply in the l imit where the momentum transfers from the particle to the phase wave are very smali.

    The de Broglie equation is customari ly wri t ten in the three-vector form

    (1) (l/~)n =p/h,

    (*) "Work performed under the auspices of the U. S. Department of Energy by the Lawrence Liver- more National Laboratory under contract 1~o. W-7405-ENG-48. (t) See, for example, E. ]:~ECAMI and W. I:~ODRIGUES Jr.** Tachyons: may they have a role in ele- mentary particle physics, Progress in Particle and Nuclear Physics, Vol. 15, edited by A. FAESSLER (Pergamon, Ox[ord, 1985, ill press), and references contained therein; G. D. ~r and E. I~E- CAMI: NUOVO Cimento A, 37, 85 (1980); E. RI~CAMI and 1%. ~I(~'N.~NI: I~iv. Nuovo Cimento, 4, 209 (1974); 1~. ]:[ORODECKI** NUovo Cimento B, 80, 217 (1984); ]K. C. CORt3]~N: Nuovo Cimento A, 29, 415 (1975); O. M. P. BILAN1UK, V* K. DI~SHPANDE and E. C. G. SUDAI~SHAN: .Am. J . Phys., 30, 718 (1962).

    697

  • 698 ~. H. MAC Glen~OIr

    where p = my is the relativisl ic three-momentum of a moving particle, and 2 and n are the wave-length and directional unit vector, respectively, of the corresponding planar de Broglie phase wave, of()., w, v). However, this equation can also be writtm~ in the cow~riant four-vector form (~)

    (2) b = hL ,

    where

    (3) P ~ (E/c, p) = re(c, v)

    is the e~tergy-.~vo~r~et~ltem four-vector of a massive particle, and

    (4) L :~- ( l /) ,)(w/c, n) = v(1/c, n/w)

    is the/reql,er~cy four-vector of a plane wave (a). As nlentioned above, dc Broglie deduced eqs. (1) and (2) by postulat ing the existence of the equations

    (5) hv = mc "~

    add

    (6) vw = c a

    Equat ion (5) represents an extension of the Planck electromagnetic equation E = by, which was originally applied to massless particles, to include massive particles as well, where , corresponds to an unspeeiiicd

  • A DYNAMICAL BAS IS FOIr TtI]~ D]~ BROGLl l~ PHAS]~ WAV~ 69~

    The production process we use to invest igate the kinematics of phase wave pro- duction is the general excitat ion diagram shown in fig. I. A particle m accelerates a spatial perturbat ion n at angle ~, which causes the part icle to recoil at angle 0. I f we characterize these particles as four-vectors, then we obtain a covariant set of energy- momentum equations by writ ing ~ is i = ~P l , where the sums can be over any number of init ial-state and final-state particles (:), including in part icular the diagram shown in fig. 1, The results that are of interest here occur only in the perturbat ive l imit

    (7) n

  • 700 3i. H. MAC G~JdGO~

    Equat ions (8)-(13) lead to the three-momentum equat ion

    (14) uw = c( ,~ _ ~o) ~2 9 _ c[ (m I _ ~)2 q~] ' , .

    In the PSR l imit ~b o[ ( .bn_ )//)2 _ ?,bo]89 ~ C(q?b~ - - ~2)+[1 - - .b lU / (m ~ - - ~b02)] ,

    where we have discarded terms of order n 2 and higher. Thus we have

    (16) w ~ c ,h / (m ~ -- m~) = cU/vl ,

    which for v~ ~ v 2 ~ v is just eq. (6). Hence in the PSR l imit of eq. (7), where the re lat iv ist ic mass n of the exci tat ion quantum is much smal ler than the relat iv ist ic mass m of the inc ident part icle, the excitation quantum, independently o] the value o~ its (small) relativistic ~ass , is scattered ]orward at the de Broglie phase velocity w = c~/v This is a pure ly kinet ic effect, and it explains why de Broglie waves arc produced in the same manner by all mov ing part ic les (provided that they induce spat ia l excitat ions).

    In order to complete this analysis, we should extend these results to include non- forward scatter ing angles, as shown in fig. 1. To accoinpl ish this, we replace the four- vectors of eqs. (11)-(13) by the fol lowing

    (17) /51 = (Oral,p1, O, Oj ,

    (18j /5 = (e.ffbz, P2 cos 0, P2 sin 0, 0) = (cm~, p~, p~, 0) ,

    (19) (~ = (on, q cos ~, q sin ~0, 0) = (on, qf, qt, O) ,

    where f and t denote forward and t ransverse momentmn components , respectively. Inser t ing eqs. (17)-(19) into eq. (8), we obta in ~he set of equat ions

    p: = p~ __ qt,

    0 = p~ ~ qt , (20)

    tg ~ == qt/qf ,

    p~ = (p~)~+ (p~)~.

    ~Ve then specify the input parameters m 0. ~h, ~', and % which are sufficient to determine the le f t -hand quant i t ies in eqs. (20). Thu~ we can solve for p~,p~, qf, and qt. The solut ion for q~ is

    (21) q f= [P l - {P~ - - (P~ - - P22) SOt2 m}~Jtsec ~ ~,

    which is exact. We now write the PSR l imit of eq. (7) in the equiva lent form

    (22) Ap ~=p~- -p2

  • A DYNAMICAL BASIS FOR T I rE DE BROGLI :E PHASE WAVE 701

    Thus in the PSR l imit of small momentum transfers, the angular dependence o] the scat- tered ]orward momentum q~ cancels out. Hence we can immediate ly take over the for- ward-scatter ing results derived above. In the case of forward scattering in the PSR l imit, we had q = nw = Ap ~ nc2/v (eqs. (10), (14), (16)). For the case of nonforward scattering, we now set

    (24) q~ ~ nw ~ ,

    which from eq. (23) then gives nw ~ ~_ Ap ~ ne2/v, so that we have

    (25) w ~ ~ c2/v ,

    which can also be written in the form

    (26) v ' w ~-- 02 .

    Equat ion (26), or its equivalent, eq. (25), presents us with a powerful new result. I t says that in the PSR l imit of small momentum transfers, the ]orward velocity w ~ o] a scattered excitat ion quantum depends only on the velocity v o] the inc ident mass ive pa~'ticle, and not on the relativist ic mass n o] the qnantnm or its scattering angle ~. Thus an en- semble of s imultaneously scattered quanta n will t ravel forward at the de Brogl ie velocity c2/v in the form of an accurately planar wave. The planar nature of this scat- tering process also has other implications. I t suggests that the spatial excitat ion quanta can be in the form of two-dimensional (( strings )~ or three-dimensional , sheets )) rather than discrete localized particles. The kinematics will be the same in all of these cases. When we use the part ic le-wave momentum transfers to devise a part ic le-wave steering mechanism (s), and when we study the problem of energy conservation in the wave packet, these higher-dimensional excitations appear to offer important advantages.

    As a means of del ineating the perturbat ive nature of these results, the exact equa- t ions were coded on a computer, using a 100 KeV electron as a typical incident particle, and the perturbat ive l imit defined in eq. (7) was approached numerically. The results of these calculations are displayed in figs. 2 and 3. F igure 2 shows values for the velocity w of an elastically scattered quantum n at various scattering angles ?, using units where c = 1. The velocities w are superluminal, and they approach constant values as the perturbat ion parameter n/m~ approaches the value 10 -5 from above. F igure 3 shows corresponding values for the forward velocity component w~= w cos ~. The velocities w ~ at all scattering angles converge to the de Broglie phase velocity c2/v as the perturbat ive l imit n/m~ < 10 -5 is reached.

    We have obtained these results without referring to the rest mass % of the scat- tered spatial excitat ion quantum. The value for n o can be obtained from the four- vector equation

    (27) ~'~ ----- (P l - /52) ' (/51 - - t52) = ng e2 ,

    where fJ1, P2, and 0 are defined in eqs. (9), (10), and (17)-(20). Since

    (28) P I ' P1 P2" ~2 ~ 2 2 = m 0 c :,

    (8) l~. H. MAC GREOOR: UCRL 93654, A Particle-Wave Steering Mechanism, Nov. 22 (1985).

  • 702 3[. }{. MAC GREGOR

    9

    w

    7 -

    t ~L ! I 10 < 10- 2

    /--300

    /

    .I I 10 5 10--';

    n//7] 1

    10 "~

    F ig . 2. - The ve loc i ty w of an e las t i ca l ly -scat tered quantum n (see fig. 1), p lo t ted as a funct ion of the per turbat ion p&rametcr n/mr (eq. (7)) for var ious va lues of the scat tered ang le ~f, and us ing un i ts where c = 1. The inc ident par t i c le ~ is a 100 keV e lectron. As can be seen, the va lues for w are super lnmina l (greater than c), and they become asymptot ica l ly constant ( independent of n) in the per tm'bat ion l im i t n[m~ ~ 10-L

    2 . 0 - -

    w #

    1.9

    i !

    1 . 8 - -

    30 ~ 0o / 45 ~ 60 ~ 75 ~

    1 ! 1.08 1.06

    1.o~

    1.02

    1.00

    ~__ j __j4 10 " 10 -~ 10 -3 i0-':' 10 5

    D/m 1

    F ig . 3. - The same p lo t as in fi~ 2, but for the fo rward ve loc i ty component w~= w cos q% rather than w. As can be seen, the ve loc i t ies w f for a l l scat ter ing ang les cfl approach the de Brogl ie phase ve loc i ty 1/v~ at t i le per turbat ive l im i t *~/m, ~10 -~.

    we ] l&ve

    (29)

    Sett ing m,a = mr - - ~, going to the PSl~ limi~ n

  • A DYNAMICAL BASIS FOR THE DE BROGLIE PHASE WAVE ~0~

    and keeping terms through order n ~, we obtain

    ~ _ n2(~ + 2 tg2 ~)/(~ _ ~) . (31) no -- ~1 Since nc 2 represents a real relat ivist ic energy, we see from eq. (31) that q~ is negative, and hence the rest mass n o is imaginary. Thus the superluminal excitat ion quanta are tachyon states (as they must be in this relat ivist ic formalism). We can write eq. (31) in the equivalent form

    (32) n o "~ in[ (v ~ sec 'z q~/v~) - - 1]89

    where vl is the velocity of the incident massive particle. As we have just demonstrated, if a massive part ic le moving at velocity v excites

    a spatial quantum and accelerates it kinematical ly, then in the PSR l imit this excitat ion quantum will t ravel forward as a tachyon state at the de Brogl ie phase velocity c2/v. Suppose we now consider the case where this moving tachyon excitat ion interacts and exchanges energy and momentum with a second massive particle. I f the tachyon rep- resents a component of a de Brogl ie phase wave, then we require that its velocity be unchanged by this interaction. That is, in order to prevent the break-up of the phase wave, we must have dw/dn = 0, where w and n are the forward velocity and relativist ic mass, respectively, of the tachyon. What requirement does this put on the velocity of the second massive particle ? In order to obtain the rather surprising answer to this question, we again go through a four-vector analysis of the scattering, and we again take the PSR l imit of very smal l energy and momentum transfers. We demonstrate the solution here only for the case of forward scattering, but the results we obtain can be shown to apply also for the case of nonzero incident angles. The four-vector equation for this scattering process is

    (33) Q1 +/51 = Q2 q-/52,

    where

    (34) r = (cnzr nz~wk, O, O)

    and

    (k = 1, 2 )

    (35) /5 k = (cml~, m~vk, 0, 0) , (k = 1, 2)

    are the four-vectors for the tachyon and the second massive particle, respectively. The PSR l imit for this case is the l imit

    (36) n l _n 2 and m 2~_m 1.

    There is a problem we must now deal with. For the tachyon, we have the equation

    (37) n = no/(1 -- w2/c2)~.

    Thus if the tachyon rest mass n o is held constant, we wil l obtain

    (us) dwldn l , , = (c2- - w2) lnw # O .

    Hence if we require dw/dn : 0 for the tachyon interaction, the rest mass n o must be

  • 704 )t . K. MAC GREGOR

    allowed to change during the interaction. In fact, we will have

    (39) dn0/dn[, ~ = (1 -- w2/c2)~ = no/n.

    Allowing this freedoIn for variat ions in n o, we insert eqs. (34) and (35) into eq. (33) and take the derivat ive dw/(b~ in the limi~ of eq. (36), which gives

    (40) dw/d't~ ~__ (v - w)/n @ (m/n)dc/dm

    I f the second massive particle possesses an invariant rest mass, we then have

    (4l) d~'/dm -- (c 2 - v~)/~v.

    Insert ing eq. (41) into eq. (40), we finally obtain

    (42) dw/dt~ ~ (c'2/c -- w) /n.

    Thus the der ivat ive dw/dn wufishes only if vw = c2; that is, only if the second massive particle is movi~g at the same velocity as the first massive part icle! The significance of this result is that if the de Broglie phase wave is a real physical entity, as the present results suggest, and not merely a superposition of wave packet components, then the particle wave packet itself must be a separate ent i ty that is similarly quantized, and eq. (42) indicates that the phase wave in the PSR l imit of eq. (36) can function to syn- chronize not only the phase but also the velocity of this accompanying wave packet.

    The present analysis suggests that we should accord physical real i ty to the de Broglie phase wave, and hence also to the constituent taehyon states (~). This in turn indicates that a physical basis should be provided for the ~ internal particle fre- quency ~> , (eq. (5)) that is t ransmitted to the phase wave. Such a topic is beyond the scope of the present paper, but we want to point out that there is a model, the rela- t ivist ical ly spinning sphere (6), which correlates the de Broglie frequency v with the rotat ional frequency of the sphere, which correctly reproduces the spin angular mo- mentum and magnetic moment of a spin-~ particle, and which properly transforms these quantit ies and the particle mass under Lorentz transformations.

    (9) A l though it is general ly bel ieved that energy cannot be t ransmi t ted at super lnminM velocities, a rguments to ttw contrary have appeared in the l i terature. Wee for example G. FEINBERG: Phys. Rer., 159, 1089 (1967); T. ALV:(C;ER and 3[. N. I~RFISLER: Phys. I?ev., 171, 1357 (1968).