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~

L a w r e n c e L i v e r m o r e N a t i o n a l L a b o r a t o r y (*) - L i v e r m o r e , Ca l . 94550

(r iccvuto il 18 Lugl io 1985; manosc r i t to rev is iona to r icevuto il 4 Dicembre 1985)

PACS. 03.65 ~ Quan tum theo ry ; q u a n t u m mechanics .

S u m m a r y . - Using the k inemat ic equa t ions of pe r tu rba t ive special r e l a t iv i ty (PSR), which are va l id in the l imi t of smal l m o m e n t u m transfers , we der ive t he par t ic le -wave ve loc i ty equa t ion v w = e 2, t he r eby p rov id ing a dynamica l basis for the de Brogl ie phase wave. This samc P S R calcula t ion also demons t ra t e s t h a t 1) t he phase w a v e is accura te ly planar , and 2) the quan ta of the phase wave are tachyons .

The de BrogUe wave equa t ion 2 ---- h/t~ was deduced on the basis of two p o s t u l a t e s :

1) t he ex tended P lanck energy equa t ion m e 2 = hv ; 2) t he par t ic le -wave ve loc i ty equa- t ion v w - - - - e 2. The de Brogl ie wave- leng th 2--~ w / v follows d i rec t ly f rom these two postula tes . The de Brogl ie phase wave ~(2, w, v), whose effect have been verif ied in a large number of exper iments , is the only known phenomenon in physics t ha t involves t ransmiss ion in a v a c u u m at super luminal (w > e) velocit ies. Since this phase w a v e is p roduced by a par t ic le t h a t t r ave l s at subluminal (v < c) velocit ies, the phase w a v e pro- duc t ion process t ies toge the r the separa te re la t iv i s t i c domains of bradyons (sub- lumina l part icles) and tachyons (supcr luminal (~ par t ic les )~). However , in spite of ex- tens ive studies on this p rob lem (~), the precise na tu re of the re la t iv i s t ic l inking be tween bradyons and tachyons has no t ye t been del ineated. As we demons t ra t e in the present paper , this l inking appears to be p rov ided by the k inemat ic equa t ions of pe r tu rba t ive special r e l a t iv i ty (PSR), which app ly in t he l imi t where the m o m e n t u m transfers f rom the par t ic le to the phase w a v e are v e r y smali .

The de Brogl ie equa t ion is cus tomar i ly wr i t t en in t he t h ree -vec tor fo rm

(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 i n ele- mentary particle physics, Progress in Part icle 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

6 9 8 ~ . H. MAC Glen~OIr

where p = m y is the re la t iv is l ic t h r e e - m o m e n t u m of a moving part icle, and 2 and n are the wave- length and direct ional uni t vector , respect ively , of the corresponding p lanar de Brogl ie phase wave, of()., w, v). However , this equa t ion can also be writtm~ in the cow~riant four-vector form (~)

(2) b = h L ,

w h e r e

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

is the e~tergy-.~vo~r~et~ltem four -vec tor of a mass ive part icle, and

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

is the/reql,er~cy four -vec tor of a p lane wave (a). As n len t ioned above, dc Brogl ie deduced eqs. (1) and (2) by pos tu la t ing the exis tence of the equa t ions

(5) hv = mc "~

a d d

(6) vw = c a

Equa t i on (5) represents an extens ion of the P lanck e lec t romagnet ic equa t ion E = by, which was or iginal ly appl ied to massless part icles, to include massive par t ic les as well , where , corresponds to an unspeeii icd <(internal f requency ~) of the mass ive par t ic le t ha t is ~ransmit ted to the de Brogl ie phase wave. Equa t ion (6) is the par t i c le -wave ve loc i ty re la t ionship tha t is r equ i red in order to ob ta in re la t iv is t ic invar ianee (4), as is apparen t f rom a:t inspect ion of cqs. (2)-(4)(5) .Nei ther eq. (5) nor eq. (6) has here tofore been prov ided wi th a dynamica l basis; t hey have remained as pos tu la tes of the systemat ics . In the present paper we use ~he k inemat ic P S R equa t ions to der ive cq. (6), and hence provide this missing dynamica l basis. We also ment ion a. model- dependen t resul t (% tha t can be used to ident i fy the (~internal par t ic le f rcql lency ~) v of eq. (5), and thus assign it physical signitieance.

The de Brogl ie phase wave has two i m p o r t a n t k inemat ic fea tures : 1) i t is p roduce4 by all types of mov ing pa r t i c l e - -e l ec t rons , neutrons , a toms, molecules, e i c . - - s o t h a t i t is not part icle-specif ic; 2) the ve loc i ty of the phase wave is not a constant , "(s we would expect for wave propaga t ion th rough ~ med ium, but instead is a funct ion of the ve loc i ty of the par t ic le tha t p roduced it. These features indicaie t ha t lhe phase wave is p roduced as a consequence of the m, otion of the part icle. Hence they suggest t ha t the produc t ion process is k inemat ic . We now subs tan t ia te this suggestion, and we also demons t ra t e t ha t the k inemat ic accelerat ion process is accura te ly planar .

(2) See \V. ]~I\~D~,~:R: Essential l?ehdivity, 2nd ed. (Spr inger-Ver lag , Heide lberg , 1977). p. 91. Tiffs f en r -vee t o r fol'mali~n~ was con ta ined in L. de Bro~lie 's thesis. (a) See "~V. I{INI)LI';R: Esse~d/al ]?elalicily, 2nd ed. (Spr inger-Verla~, t t e ide lbe rg , 1977), p. 72. ( ' ) See C, ~[OLLER: T]t6 Theory of t?elaticity (Oxford, London , 1952), p. 6, 7, 51, 52, and_ 56-58. (a) \ \ ' l w n we fo rm the fou r -vec to r /3_ h~ f r o m eq. (2), the first c m n p o n e n t van i shes iden t i ca l ly in all f r ames of reference because of eq. (5). Us ing the fou r -vec to r zeeo-compm~ertt lemma (ref. ("), p. 79), it t hen follows t h a t t]w o ther eont!aonent.s also vanis t t idenbica.tly, which es tabl ishes b o t h t h e sca lar ~-elr)eiio, r ( ' l a t ienship of eq. ((;), and also the l )ar l ic le-w~ve col l inear i iy coodi t ion nee v in a,11 frames of l'r

(") 31. L T, [~[AC ~tr Loll. A'~lOt:O Ci~c~to, 43, 49 (1985).

A D Y N A M I C A L B A S I S FOIr TtI]~ D]~ B R O G L l l ~ PHAS]~ W A V ~ 6 9 ~

The produc t ion process we use to inves t iga te the k inemat ics of phase w a v e pro- duct ion is the genera l exc i ta t ion d iagram shown in fig. I. A par t ic le m accelerates a spat ia l pe r tu rba t ion n at angle ~, which causes the par t ic le to recoi l at angle 0. I f we charac ter ize these par t ic les as four-vectors , t hen we obta in a covar ian t set of energy- m o m e n t u m equa t ions by wr i t ing ~ i s i = ~ P l , where t he sums can be over any n u m b e r of in i t ia l -s ta te and f inal-s tate par t ic les (:), inc luding in par t i cu la r the d iagram shown in fig. 1, The resul ts t h a t are of in te res t here occur only in the pe r tu rba t ive l imi t

(7) n << m ,

F i g . 1. - A n e n e r g y - m o m e n t u m d i a g r a m f o r t h e a c c e l e r a t i o n of a s p a t i a l e x c i t a t i o n q u a n t u m n b y a m o v i n g p a r t i c l e m . I f t h e r e s t m a s s of t h e p a r t i c l e m r e m a i n s i n v a r i a n t , t h i s i s f o r m a l l y a n e l a s t i c - c o l l i s i o n p r o c e s s . A s s h o w n i n t h e t e x t , t h e s c a t t e r e d q u a n t u m n i s a t a e h y o n s t a t e .

where m and n are the re la t iv is t ic masses of the par t ic le and spat ia l exc i ta t ion quan- t u m , respec t ive ly . This is a l imi t t h a t does no t seem to h a v e been prev ious ly inves t i - gated. The P S R l imi t can be obta ined numer ica l ly f rom the exac t solut ions of t he k inemat ic equat ions . I t can also be approached analyt ica l ly , as we now demons t r a t e for the case of forward (0 ~ sca t ter ing

The e n e r g y - m o m e n t u m four -vec tor equa t ion for the sca t ter ing process shown in fig. 1 is

(s) p~ _- p~ + Q,

w h e r e / 5 and Q charac ter ize t he s ta tes m and n, respec t ive ly . In wr i t ing out the four- vec to rs for th is equa t ion , i t is of ten convenien t to wr i te t he magn i tude of the par t ic le t h r e e - m o m e n t u m as

(9) p = o(m~- m~)~.

The magn i tude of the exc i ta t ion q u a n t u m t h r e e - m o m e n t u m is

(10) q = n w ,

where wc do not ye t specify i ts rest mass. In the case of ]orward scattering, t he four- vec tors in eq. (8) can be wr i t t en as fo l lows:

( 1 1 )

(is)

(13)

k l = [eml, ~(m~ - m~)~, 0, 0] ,

P~ = [~%, c ( ~ - ~g)~, 0, 03,

O = [on, nw, O, 0] .

(~) See ~V. RINDLEtr Essential Relativity, 2 n d ed . ( S p r i n g e r - V e r l a g , H e i d e l b e r g , 1977) , 19. 80.

700 3i. H. MAC G~JdGO~

E q u a t i o n s (8)-(13) l ead to the t h r e e - m o m e n t u m e q u a t i o n

( 1 4 ) u w = c ( , ~ _ ~ o ) ~ 2 �9 _ c [ ( m I _ ~ ) 2 q ~ ] ' , .

In t h e P S R l imi t ~b << m l , t he las t t e r m in eq. (14) call be e x p a n d e d to give

(15> o [ ( . b n _ )//)2 _ ?,bo]�89 ~ C(q?b~ - - ~ 2 ) + [ 1 - - . b l U / ( m ~ - - ~b02)] ,

where we h a v e d i sca rded t e r m s of o rder n 2 a n d h igher . T h u s we h a v e

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

which for v~ ~ v 2 ~ v is j u s t eq. (6). Hence in t h e P S R l imi t of eq. (7), w h e r e t he r e l a t iv i s t i c mass n of the exc i t a t i on q u a n t u m is m u c h smal l e r t h a n t h e r e l a t iv i s t i c mass m of t he i nc iden t par t ic le , the excitation quantum, independent ly o] the value o~ its (small) relativistic ~ a s s , is scattered ]orward at the de Broglie phase velocity w = c~/v Thi s is a p u r e l y k ine t ic effect, a n d it exp la ins w hy de Brogl ie waves arc p r o d u c e d in t he s ame m a n n e r b y all m o v i n g par t i c les (p rov ided t h a t t h e y induce spa t i a l exc i ta t ions ) .

In o rde r to comple t e th i s ana lys i s , we should e x t e n d the se resu l t s to inc lude non- fo rward s c a t t e r i n g angles, as shown in fig. 1. To accoinpl i sh th is , we replace t h e four- vec to r s of eqs. (11)-(13) b y t h e fo l 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 s in ~0, 0) = (on, qf, qt, O) ,

where f a n d t d e n o t e f o r w a r d a n d t r a n s v e r s e m o m e n t m n c o m p o n e n t s , r espec t ive ly . I n s e r t i n g eqs. (17)-(19) in to eq. (8), we o b t a i n ~he set of e q u a t i o n s

p : = p~ __ qt ,

0 = p~ ~ q t , (20)

tg ~ == qt/qf ,

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

~Ve t h e n specify t he i n p u t p a r a m e t e r s m 0. ~ h , ~', a n d % which are sufficient to d e t e r m i n e t h e l e f t - h a n d q u a n t i t i e s in eqs. (20). Thu~ we can solve for p~ ,p~ , qf, a n d qt. The so lu t ion for q~ is

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

which is exac t . We now wr i te the P S R l i m i t of eq. (7) in the e q u i v a l e n t f o rm

(22) Ap ~ = p ~ - - p 2 < < p ~ .

Nex t wc set p~ = p~ - - Ap in eq. (21), e x p a n d the square root , a n d d iscard t e r m s in (Ap) 2. Wc t h e n d i scover t h a t t he s e c : ~ f ac to r s cancel out , a n d eq. (21) is r educed to t h e a p p r o x i m a t e P S R equa l i t y

(23) q~ ~ Ap .

A D Y N A M I C A L B A S I S FOR T I r E D E B R O G L I : E P H A S E W A V E 701

Thus in the P S R l imi t of smal l m o m e n t u m transfers , the angu la r dependence o] the scat- tered ]orward m o m e n t u m q~ cancels out. Hence we can i m m e d i a t e l y t ake over the for- ward-sca t t e r ing resul ts de r ived above. In the case of fo rward sca t te r ing in t he P S R l imi t , we had q = n w = A p ~ nc2/v (eqs. (10), (14), (16)). F o r t he case of nonforward scat ter ing, we now set

(24) q~ ~ n w ~ ,

which f rom eq. (23) t hen gives n w ~ ~_ A p ~ ne2/v, so t h a t we have

(25) w ~ ~ c2/v ,

which can also be wr i t t en in the fo rm

(26) v ' w ~-- 02 .

E q u a t i o n (26), or i ts equ iva len t , eq. (25), presents us w i th a powerful new resul t . I t says tha t in the P S R l imi t of smal l m o m e n t u m t ransfers , the ]orward veloci ty w ~ o] a scattered exc i ta t ion q u a n t u m depends on ly on the veloci ty v o] the i n c i d e n t m a s s i v e pa~'ticle, a n d no t on the re la t iv is t ic m a s s n o] the q n a n t n m or i t s scat ter ing angle ~. Thus an en- semble of s imul taneous ly sca t te red quan ta n will t r ave l fo rward at the de Brogl ie ve loc i ty c2/v in the form of an accura te ly p lanar wave. The p lanar na tu re of th is scat- te r ing process also has o ther implicat ions. I t suggests t h a t the spat ia l exc i ta t ion q u a n t a can be in t he fo rm of two-d imens iona l (( s tr ings )~ or th ree-d imens iona l , sheets )) r a the r t h a n discrete local ized part icles . The k inemat ics will be the same in all of these cases. When we use the par t i c le -wave m o m e n t u m transfers to devise a pa r t i c le -wave s teer ing mechan i sm (s), and when we s tudy the p rob lem of energy conserva t ion in t he w a v e packet , these h igher -d imens ional exci ta t ions appear to offer i m p o r t a n t advantages .

As a means of de l inea t ing the pe r tu rba t ive na tu re of these results , the exac t equa- t ions were coded on a computer , using a 100 KeV e lec t ron as a typ ica l inc ident par t ic le , and the pe r tu rba t ive l imi t defined in eq. (7) was approached numerical ly . The resul ts of these calcula t ions are d isplayed in figs. 2 and 3. F igure 2 shows values for t he ve loc i ty w of an elast ical ly sca t tered q u a n t u m n at var ious sca t ter ing angles ?, using uni ts where c = 1. The veloci t ies w are super luminal , and t h e y approach cons tan t va lues as t he pe r tu rba t ion p a r a m e t e r n/m~ approaches the va lue 10 -5 f rom above. F igure 3 shows corresponding values for the fo rward ve loc i ty componen t w ~ = w cos ~. The veloci t ies w ~ at all sca t ter ing angles converge to the de Brogl ie phase ve loc i ty c2/v as the pe r tu rba t ive l imi t n/m~ < 10 -5 is r eached .

We have obta ined these resul ts wi thou t refer r ing to the rest mass % of the scat- tered spat ia l exc i ta t ion quan tum. The va lue for n o can be obta ined f rom the four- vec tor equa t ion

( 2 7 ) ~ ' ~ ----- ( P l - / 5 2 ) ' (/51 - - t52) = ng e2 ,

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

(28) P I ' P 1 P 2 " ~ 2 ~ 2 2 = m 0 c :,

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

702 3[. }{. MAC G R E G O R

9

w

7 -

t ~L ! I 10 < 10- 2

/--300

/

.I I 10 5 10--';

n//7] 1

10 "~

F i g . 2. - The v e l o c i t y w of a n e l a s t i c a l l y - s c a t t e r e d q u a n t u m n (see f ig. 1), p l o t t e d as a f u n c t i o n of t h e p e r t u r b a t i o n p & r a m e t c r n /mr (eq . (7)) for v a r i o u s v a l u e s of t h e s c a t t e r e d a n g l e ~f, a n d u s i n g u n i t s w h e r e c = 1. The i n c i d e n t p a r t i c l e ~ is a 100 k e V e l e c t r o n . As can be seen, t h e v a l u e s for w are s u p e r l n m i n a l ( g r e a t e r t h a n c), a n d t h e y b e c o m e a s y m p t o t i c a l l y c o n s t a n t ( i n d e p e n d e n t of n) i n t h e p e r t m ' b a t i o n l i m i t n[m~ ~ 1 0 - L

2 . 0 - -

w #

1.9

i !

1 . 8 - -

30 ~ 0 o / 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 i g . 3. - The s a m e p l o t as in fi~ 2, b u t for t h e f o r w a r d v e l o c i t y c o m p o n e n t w~= w cos q% r a t h e r t h a n w. As c a n be seen , t he v e l o c i t i e s w f fo r a l l s c a t t e r i n g a n g l e s cfl a p p r o a c h t h e de Brog l i e p h a s e v e l o c i t y 1/v~ a t t i le p e r t u r b a t i v e l i m i t *~/m, ~ 1 0 -~.

w e ] l & v e

(29)

Se t t i ng m,a = m r - - ~, go ing to t he PSl~ limi~ n << m 1, i n v o k i n g t he squa re - roo t expan- s ion s h o w n in eq. (15), u s ing t h e P S R ang le r e l a t i o n s h i p (see eqs. (17)-(20))

(30) s in 0 ~_ - - m l n t g r - - m ~ ) ,

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

and keeping t e rms th rough order n ~, we obta in

~ _ n 2 ( ~ + 2 tg2 ~)/(~ _ ~ ) . (31) no - - ~1

Since nc 2 represents a real re la t iv i s t i c energy, we see f rom eq. (31) t h a t q~ is negat ive , and hence the res t mass n o is imaginary . Thus the super lumina l exc i ta t ion quan ta are t achyon s ta tes (as t h e y must be in th is re la t iv i s t ic formal ism) . W e can wri te eq. (31) in t he equ iva len t fo rm

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

where vl is t he ve loc i ty of t he incident mass ive par t ic le . As we have jus t demons t ra ted , if a mass ive par t ic le m o v i n g at ve loc i ty v exci tes

a spat ia l q u a n t u m and accelerates i t k inemat ica l ly , t hen in the P S R l imi t th is exc i ta t ion q u a n t u m wil l t r a v e l forward as a t achyon s ta te at the de Brogl ie phase ve loc i ty c2/v.

Suppose we now consider t he case where th is m o v i n g t achyon exc i ta t ion in terac ts and exchanges energy and m o m e n t u m wi th a second mass ive part icle . I f the t achyon rep- resents a componen t of a de Brogl ie phase wave , t hen we requi re t h a t i ts ve loc i ty be unchanged by this in terac t ion . T h a t is, in order to p r even t the break-up of the phase wave, we mus t have d w / d n = 0, where w and n are t he fo rward ve loc i ty and re la t iv is t ic mass, respect ive ly , of the tachyon. W h a t r equ i r emen t does th is p u t on the ve loc i ty of t he second mass ive par t ic le ? In order to ob ta in t he r a the r surpr is ing answer to th is quest ion, we again go th rough a four -vec tor analysis of the scat ter ing, and we again t ake the P S R l imi t of v e r y smal l energy and m o m e n t u m transfers . W e demons t r a t e t he solut ion here only for t he case of fo rward scat ter ing, bu t the resul ts we obta in can be shown to app ly also for t he case of nonzero incident angles. The four -vec tor equa t ion for th is sca t te r ing 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 t he four -vec tors for the t achyon and the second mass ive par t ic le , respec t ive ly . The P S R l imi t for th is case is t he l imi t

(36) n l _ n 2 and m 2 ~ _ m 1.

There is a p rob lem we mus t now deal with. F o r t he t achyon , we h a v e the equa t ion

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

Thus if the t achyon res t mass n o is he ld constant , we wil l obta in

(us) d w l d n l , , = ( c 2 - - w 2 ) l n w # O .

Hence if we requi re d w / d n : 0 for t he t achyon in te rac t ion , the res t mass n o mus t be

704 ) t . K. MAC GREGOR

al lowed to change dur ing the in terac t ion . In fact , we will have

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

Allowing this freedoIn for var ia t ions in n o, we insert eqs. (34) and (35) into eq. (33) and take the der iva t 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 par t ic le possesses an invar ian t res t mass, we then have

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

Inser t ing eq. (41) into eq. (40), we finally ob ta in

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

Thus the de r iva t ive dw/dn wufishes only if vw = c2; tha t is, only if the second massive par t ic le is movi~g at the same ve loc i ty as the first mass ive par t ic le ! The significance of this resul t is t ha t if the de Brogl ie phase wave is a real physical en t i ty , as the present resul ts suggest, and not mere ly a superposi t ion of wave packe t components , then the par t ic le wave packe t i tself must be a separa te en t i ty tha t is s imi lar ly quant ized, and eq. (42) indicates tha t the phase wave in the P S R l imi t of eq. (36) can funct ion to syn- chronize not only the phase but also the ve loc i ty of th is accompanying wave packet .

The present analysis suggests tha t we should accord physical r ea l i ty to the de Brogl ie phase wave, and hence also to the cons t i tuent t aehyon states (~). This in t u rn indicates t h a t a phys ica l basis should be p rov ided for the ~ in terna l par t ic le fre- quency ~> , (eq. (5)) t ha t is t r a n s m i t t e d to the phase wave. Such a topic is beyond the scope of the present paper , bu t we want to poin t out t ha t there is a model , the rela- t iv is t ica l ly spinning sphere (6), which correlates the de Broglie f requency v wi th the ro ta t iona l f requency of the sphere, which correct ly reproduces the spin angular mo- m e n t u m and magnet ic m o m e n t of a spin-~ part icle, and which p roper ly t ransforms these quant i t ies and the par t ic le mass under Loren tz t rans format ions .

(9) A l t h o u g h it is genera l ly be l ieved t h a t ene rgy canno t be t r a n s m i t t e d a t supe r lnminM veloci t ies , a r g u m e n t s to t tw c o n t r a r y have a p p e a r e d in the l i t e r a tu re . Wee for e x a m p l e G. FEINBERG: Phys. Rer., 159, 1089 (1967); T. ALV:(C;ER a nd 3[. N. I~RFISLER: Phys. I?ev., 171, 1357 (1968).