The de Broglie Pilot Wave Theory and the Further Development of New Insights Arising Out of It D. J. Bohm 1 and B. J. Hiley ~

8/7/2019 The de Broglie Pilot Wave Theory and the Further Development of New Insights Arising Out of It D. J. Bohm 1 and B

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Foundations o f Physies, V ol. 12, No . 10, 1982

The de Brogl ie P i lo t Wave Theoryand the Fur the r Deve lopmento f N ew Insigh t s Ar is ing O ut o f It

D . J . B o h m 1 a n d B . J . H i l e y ~

Received April 16, 1982

W e briefly review the history o f de Broglie's notion o f the "double solution" ando f the ideas w hich developed fr om this. W e then go on to an extension o f theseideas to the many-body system, and bring out the nonloeality implied in sueh anextension. Finally, we sum ma rize furth er developments that have stemme d fro mde Broglie' s suggestions.

1 . I N T R O D U C T I O N

Orig ina l ly, Schr6dinger regarded the wave in tens i ty, g t*q j , as the ac tua ldens i t y o f e lec t r ic cha rge , and t h is gav e an image o f t he e l ec t ron a s a un iqueand i ndependen t r ea l i t y. Howeve r, Sch r6d inge r ' s equa t i on i t s e l f , because o fi t s l i nea r i t y, imp l i ed t ha t t h i s a s sumed cha rge dens i t y wou ld sp read ou twi thout l imi t in a shor t t ime; and ye t , the par t i c le i s a lways found in a smal lr eg ion o f space . T o r e so lve t h is d i f f i cu lt y, Bo rn p ropo sed t ha t t he wavein t ens i t y i s t he p robab i l i t y dens i t yo f f i n d i n g a par t ic le . H ow eve r, th i s l e ftopen t he ques t i on o f whe the r t he l oca l i zed pa r t i c l e ex i st s i ndependen t ly, o rwh ether i t i s in some sense pro du ced or a t l eas t loca l ized in the ac t o f obser-va t i on ( a s i s indeed imp l i ed i n He i senbe rg ' s ana ly s i s o f t he measu rem en t o fp o s i t i o n a n d m o m e n t u m ) .

I t is we l l -know n tha t i n the u sua l i n t e rp re t a t i on o f t he qua n tum theo ry,the la t t e r po in t o f v iew is adop ted . Th e m os t cons i s ten t vers ion of th i s in te r-pre ta t ion i s tha t g iven b y Boh r, (1) in which n o m eaning can b e ascr ibed toprec i se ly def ined par t i c le p roper t ies beyond the l imi t s spec i f ied by the

1Birkbeck College (University of Londo n), Malet Street, Lond on W C1E 7 HX .

1001

0015-9018/82/lO00-1o01503.00/o 1982 PlenumPublishingCorporation


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Hei senbe rg unce r t a in ty p r i nc ip l e . Th i s l a ck o f mean ing goes beyo ndregard ing these p roper t i es as independent ly ex i s ten t , bu t uncer ta int o u sbecause o f lim i ts i n ou r ab i l it y t o ob t a in know ledge o f t hem th roug h

m easurem ents . Ra ther, in B ohr ' s v iew, the un iverse is bas ica l ly anunan a lyzab l e who le , i n wh ich t he no t i on o f the s epa ra t enes s o f pa r t i c l e andenv i ronmen t i s an abs t r ac t i on t ha t ha s no con t en t , excep t a s an app rox -ima t ion t h a t m ay be app l i ed w i th in t he l im i t o f He i s enbe rg ' s p r i nc ip l e .

Mos t phys i c i s t s have adop t ed v i ews s im i l a r i n key ways t o t hosedesc r i bed above , t hough t he de t a i l s va ry cons ide r ab ly. Howeve r, de Brog l i e ,E in s t e in , Sch r6d inge r, and o the r s d i s ag reed w i th t h i s app roach , because t heyfe l t t ha t t he r e i s a un ique ly de f i ned r ea l i t y, wh ich can be g r a sped i n t hough tand is ye t i nd ependen t o f t hough t . W i thou t co ns ide r i ng th i s r e a l i ty, s c i ence is

reduce d to a se t o f fo rm ulas and rec ipes fo r p red ic t ing the resu lt s o fexpe r imen t s . Indeed , a l a rge num ber o f mod e rn phys i c i s ts have s i nce t hen , a tl eas t t ac i t ly, com e to ad op t such a po in t o f v iew, perha ps b ecause i t is pa r to f the p r a gm a t i c sp ir it o f the age . Ho wev e r, in ou r v i ew, t h is p r agm a t i capproach i s no t the on ly poss ib le one , nor i s i t even the bes t . For one can seetha t co ncep t s t ha t do n o t g ive imm ed ia t e new expe r imen t a l p r ed i c t ions m ays t i l l be va luab le , in tha t they permi t new ins igh t and unders tand ing ( f romw h i c h n e w p r e d i c t i o n s m a y u l t i m a t e l y e m e rg e ) . A n a p p r o a c h t h a td i scou rages th i s k ind of ins igh t wi ll thus t e nd to p reve n t c rea t ive new

percep t ions , such as those o f E ins te in , de Brog l ie , e tc .Th e p i lo t w ave the ory of de Brog l ie ~2) was indeed a s ign i f i can t an d

f ru i t fu l example o f imagina t ive concep ts tha t he lp l ead to new ins igh ts . Webeg in w i th a b r i e f h i s t o r ic a l su m m ary o f t h is i dea and i ts deve lopmen t . O fcou r se , i n such a sho r t space , we canno t hope t o g iven a comp le t e a ccoun t ,bu t ra ther, we se lec t a few po in t s tha t a re re levan t to our d i scuss ion .

In essence , de Brog l ie assumed tha t the re i s a phys ica l ly rea l wavesa t i s fy ing Schr6d inger ' s equa t ion , a t l eas t as a l inear approx imat ion , a longwi th a pa r t i c le f o l lowing a we l l de f i ned t r a j e c to ry. The m om en tum o f th i spa r t i c l e was r e l a t ed t o t he wave t h rough t he equa t i on

p = h V ~ ( 1 )

whe re ~ i s the phase o f the wav e func t ion . Th e abo ve re la t ionsh ip sug ges t st ha t t he p a r t i c l e i s be ing "gu ided" by t he backg round wave , and fo r t h i sr ea son , de Brog l i e c a l l ed t he l a t t e r a "p i l o t wave" . (One may he r e cons ide rt he ana log y o f an a i rp l ane gu ided by r ada r w aves , wh ich ca r ry i n fo rm a t ionabou t t he who le env i ronmen t ) .

D e Brog l i e gave a pa r t i cu l a r l y beau t i f u l exp l ana t i on o f how the p i l o twave wou l d ac tua l l y gu ide t he pa r t i c l e . He p roposed t ha t i n s ide t he pa r t i c l ewas a pe r i od i c p roce s s t ha t was equ iva l en t t o a c l ock . I n t he r e s t f r ame , t he


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de Broglie Pilot Wave Theory and the Further Developme nt 1003

clock would have a frequency, coo = moc2/h. By considering how this clockwould behave in other frames, he derived what is in essence the Bohr-Som-merfeld relationship

which is the condition for the clock to remain in phase with the pilot wave.One ma y here think o f the phase locking of synchronous motors

through a nonlinear interaction. A similarly nonlinear interaction would ofcourse b e needed to lock the ph ase of the clock to that o f the pilot wave.This sort of interaction was indeed implied in de Broglie 's further

developm ent of the model . Wh at he suggested in addi t ion was thatnonlin earity o f the play ed a crucial role, in that i t mad e possible a singularsolution, representing the particle, which had to fi t smoothly onto a weakbackground f ie ld , obeying Schr6dinger 's equat ion as a l inear approximat ion.This requirement of a smooth connect ion of the two members of the "doublesolution" was, of course, what explained the guidance condition. I t issignificant to note here that his model provides at least a conceptualconnect ion between quantum mechanics and Einste in 's a t tempt a t a unif iedfield theory, in which the particle was also treated as a nonlinear singularity

that merges with a l inear background field (modern soli ton theory is alsoclosely related in concept to this approach).

The fa te of the theory was decided at the Solvay Conference, wherePauli ~3~ ma de imp orta nt objections to i t , based on the fact tha t the theo ry didnot provid e a consis tent acc ount o f the man y b ody system. ( in par t icular, hediscussed a general two-body scattering process.) While de Broglie felt thathe had at least the germ of an answer, this was ap paren tly not app reciatedby the others who were present. This, along with the fact that not evenEinstein spoke up for the theory, led to a definite rejection, which was indeedso decisive that de Broglie himself gave up work on the theory.

Ho wev er, in 1952, one of us (Bohm (4'5)) published two p apers on thesubject.2 In the first of these, the conse quenc es o f wha t was in essence deBroglie 's theory were worked out in considerable detail , and shown toprovide a general ly consis tent account of quantum mechanics for the one-body system. In the second paper, th is work was extended to the many-bodysystem, in a way that answered Pauli 's objections, and made possible certainkey new insights into the meaning of the quantum mechanics . Thisenc oura ged de B roglie to take up his idea again, ult imately to propose atheory, in which the par t ic le is assumed to be in a " thermal" bath, provided

2 See also the Appendix to present paper.


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1004 Bohm and Hiley

by a backgr ound of vacuu m f luctuat ions . Since then, a number of others(some of which will be discussed in Section 3) have fo llowed along theselines, though, of course, it must be admitted that this approach is still not

acce pted by mo st phys icists (see for exa mp le Bo hm and Vigier, (6) Vigier, (v)Ne lson , (8) de la Pe na (9) and R oy . ~1)

2. T R A J E C T O R Y I N T E R P R E TAT I O N F O R A M A N Y - B O D Y S Y ST E M

We now give an account of how the m any- bod y system is to beunde rstoo d in terms of an extension of the ideas o f de Broglie 's , which were

discussed in the previous section.The basic start ing point is to write the N body wave function, as

q/(x 1 . . . x~ )= R(x 1 . . . x~) exp[ iS(s I . . . x . ) /h l

and to def ined the mom entum of the nth par t ic le as

p. = V. S (2)

(which is evidently a generalization of de Broglie 's relation (1)) . Bysubsti tut ing (2) in to the m any -bod y Schr6dinger equat ion, we then obtain theconservat ion equat ion in conf igurat ion space

8P t_~ .~ Vn. __ (P V'S ) - - 0 (3)8t n m

(where P = T * T is the probabil i ty d ensity in this space), and the mod ifiedHami l ton-Jacobi equa t ion

8S ~ (V"S )Z + V(x~ ... x. ) + Q(x~ ... XN) = 0 (4)8~- + 2 ~

From this, i t follows that each particle will be acted on, not only by theclassical potential , V, but also by an addit ional quantum potential

_ h 2Q (5)

R

In this interpre tation, the new features of qua ntum me chanic s are seento arise basically from Q. To i l lustrate how this comes about, let us begin by


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cons ide r i ng t he spec ia l c a se o f a two -bod y sy s t em, w i th a p rod uc t wavefunc t i on

~e(x l , x 9 = O~(x~) O~(x~) (6 )

De f in ing

OA(x) = R A (x ) e ~s ~( ~)/ ~

and (7 )

#B(x) = R~ (x) e ~ s~ /~

we can s ee immed ia t e ly t ha t

- h 2 v ~ R A x l)Q = 2 m R A (x l)

h ~ v ~ R . ( x ~ )

2 m R s ( x 2 )(8 )

equa t i on r educes t o two s epa ra t e pa r t s

~ s ~ ( v , s ~ ) ~ h 2 v ~ R ~ (x ~ )-t- + VA (x,) = 0 (9 a)g t 2m 2 RA(XI)

~ s~ ( v ~ s J h 2 V ~ R .(x ,)8 t ~" 2 ~ ~ -Vu(x 2) 2 m R ~ ( x z ) - 0 ( 9 b )

Ev iden t l y, t he conse rva t i on equa t i on a l so sp l i t s i n to two i ndependen t pa r t s .W e see t hen t ha t t he one -b ody equa t i on ( as t r e a t ed by de Brog t i e ) a r is e s

a s an abs t r ac t i on and a s im p l i c i c a ti on o f t ha t o f t he two -body sy s t em , andeven tua l l y o f t he N-bo dy sy s t em. ( I t is c l e a r m oreo ve r t han u l t ima t e ly t heseN-bod i e s mus t be ex t ended t o i nc lude t he who le un ive r s e . )

One o f the m os t im por t an t w ays i n wh ich t h i s i n t e rp r e ta t i on g ive s newins igh t i s t ha t i t enab l e s u s t o exp re s s quan tum mechan i c s and c l a s s i c a lm echan ics in t e rms of the same language , so tha t w e can see the i r s imi la r it i e sand th e i r d i ffe rences m ore c lea r ly than is poss ib le in the usua l app roac h , inwh ich t hey a r e t r e a t ed i n t e rms o f ve ry d i ff e r en t m odes o f de scr i p t ion .

Th e f i rs t ma in d i f f e rence can be s een by no t i ng t ha t t he quan tu mpoten t ia l , Q , i s no t a l t e red when the wave func t ion i s mul t ip l i ed by acons tan t , so tha t i t does no t fa l l to ze ro a t long d i s tances , where the wavein t ens i t y becom es neg l ig ib le . How eve r, t he c l a ss i c a l no t i on o f ana lyza b i l i t yo f a sy s t em in to i ndependen t pa r t s depend s c r i t ic a l l y on t he a s sumpt ion t ha twhen eve r the pa r t s a r e su ff i c ien t l y f a r r emov ed f rom each o the r, t hey do no t

The above i s the sum o f two i ndependen t f unc t i ons . I f t he c la s s ic a l po t en t ia lV is l ikewise a sum , [~ (x l ) + VB(x2) , i t fo l lows tha t the Ha m i l to n- Ja co bi


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s i g n i fi c a n t ly i n t e r a c t. T h i s m e a n s t h a t th e q u a n t u m t h e o r y i m p l i e s a n e wk i n d o f w h o l e n e s s , i n w h i c h t h e b e h a v i o r o f a p a r t i c le m a y d e p e n d s ig n i fi -c a n t l y o n d i s t a n t f e a t u r e s o f t h e o v e r - a l l e n v i r o n m e n t . T h i s d e p e n d e n c e

p r o d u c e s c o n s e q u e n c e s s im i la r t o t h o s e i m p l i e d b y ' B o h r ' s n o t io n o fu n a n a l y z a b l e w h o l e n e s s , b u t d i f fe r e n t in t h a t t h e u n i v e r s e c a n b e u n d e r s t o o da s a u n i q u e a n d i n p r i n c i p l e w e l l d e f i n e d r e a l i ty.

T o i l l u s tr a t e i n m o r e d e t a i l w h a t i s m e a n t h e r e , w e c o n s i d e r a n i n te r -f e re n c e e x p e r i m e n t , i n w h i c h a b e a m o f e l e c tr o n s o f d e f in i te m o m e n t u m iss e n t t h r o u g h a t w o s li t s y s t e m . I n F i g . 1 , w e s h o w t h e r e s u l t s o f ac o m p u t a t i o n o f t i le q u a n t u m p o t e n t i a l " l ) ; a n d in F ig . 2 , w e s h o w t h e t ra j e c -t o r i e s r e s u l t i n g f r o m t h e p o t e n t i a l .

Fig. 1. Qu antum potential fo r a pair of G auss ian slits. Th e slits can be seen in thebackground. The fringes are formed in the foreground, the dark bands coinciding with thevalleys of the quantum potential.


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O

O

,<

=

Fig. 2. The particle trajectories emanating from the Gaussian slits on the left-hand side of the figure. The fringes on the right result from

the bunching of the trajectories.


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1008 Bohm and Hi ley

What is especial ly significant in Fig. 1 is that the quantum potentialremains large at long distances from the sl i ts , taking the form of a set ofvalleys and high ridges, which lat ter gradually f lat ten out into broad

plateaux. In Fig. 2, one sees how the trajectories are ult imately bunched intothese plate aux by the ov er al l effect of the potential , and th at this bringsabou t the interference pattern. (So that , for example, i f one of the sl i ts hadbeen c losed, the quantum potent ia l would have been a smooth parabol icfunct ion, w hich would produ ce no pat tern of f r inges). The fac t tha t thequa ntu m potentia l does not in general fall off with the distance is thus whatexplains interference and diffraction patterns, and this is clearly also whatimpl ies the k ind of wholeness of par t ic le and environm ent to which we havereferred above.

One m ay re turn here to the analo gy of the a i rp lane guided by radarwaves . Evident ly, i t is not a case of mec hanica l p ressure of these waves onthe a i rp lane , but ra ther, informat ion concerning the whole environment i senfolded by the waves , and carr ied in to each region of space . The a i rp lanethus responds ac t ively to the f o r m of the wav es, an d this fo rm is not al teredas the intensity fal ls of with the distance. A similar respo nse to thef o r m ofthe qu antum potent ia l i s seen to be character is t ic o f the behavior of theelectron. This m eans that in the microw orld , the concep t of ac t ive infor-ma tion is relevant (see Bo hm and Hile y C12~ for m ore detai l).

W hat has been sa id thus far about the new kind of wholeness impl iedby the quantum theory for the one-body sys tem is fur ther s t rengthened by aconsidera t ion of the man y-b od y sys tem. Fo r here , one f inds that when thewave funct ion is no longer separab le as a product of funct ions o f the coor-dinates of each par tic le , the q uantu m potent ia l leads to a s t rong in teract ionbetween al l the part icles of the system , that does no t in general fal l off tozero when the par t ic les are d is tant f rom each other. There i s evident ly anextension of the depende nce of the par t ic le on i ts over a l l environment thatcharacter izes the one-body sys tem. But in addi t ion, there i s a yet morethoroughg oing brea kdow n of the poss ibi li ty of analys is , because the forceacting on each part icle is no longer expressible as a predetermined functionof the posi t ions of the o ther par t ic les . Rather, the funct ional form of theforce depends on the who le set of condit ions in which the wa ve function isdef ined and determined. (So that , for example , the form changes wheneverth is qu antum s ta te of the whole changes . )

Let us take , as an exa mple , the hypothet ica l exp er iment of Eins te in ,Pod olsky , and Ros en. ~13) W e conside r there the o riginal form of theexper iment , in which we s tar t wi th a quantum s ta te of a two-par t ic le sys temin which x I -- x 2 and P1 + P2 are both determine d. This is given by

7t(xl x2) = f (x l -- x2 -- a) = ~ C k exp[i k( x I - x 2 - a)] (10)k


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de Broglie Pilot Wave Theory and the Further Development 1009

where f ( x 1 - x 2 - a ) i s a p acket funct ion, sharply peak ed a t x ~ - x 2 = a ,while C k is i ts F ou rier coefficient . Evid ently, in this state,p~ + P2 = 0, whilex l - x2 can be ma de as wel l def ined as we please.

In th i s exper iment , one can measure x~ and immedia te ly know tha tx 2 = x 1 + a ( to an arbi t rar i ly h igh degree o f accuracy ) . Al ternat ively, we canmeasure p~ and immed ia te ly know tha t P2= - - P l " In both cases, the f irstpar t ic le i s d is turbed in the process o f me asu rem ent and, of course the d is tur-bances can account for the Heisenberg uncer ta in ty re la t ions as appl ied to thepar t ic le Ap~ zl x I ~ h. But s ince the second par t ic le i s assumed not to in teractwith the f irst in any way at al l , i t fol lows that we are able to f ind i tsp roper t i e s wi thou t i t s hav ing undergone any d i s tu rbance wha t soever.Never theless , according to the quantum theory, the uncer ta in ty pr inciple ,

Ap2 zlx 2 ~ h mu st s t il l apply. So Heis enb erg 's explanat io n of th is uncer ta in tyas due to a d is turbance resul t ing f rom measurement can no longer be used. I twas th is wh ich indeed led Eins te in , P odo lsky, and Ro sen ~3) to argue thats ince both x 2 and P2 were in pr inciple me asu rable to arb i t rary acc ura cywi thou t a d i s tu rbance ,v they mus t have a l ready ex i s t ed independen t ly inpar t ic le 2 as ' e lem ents ' of rea l i ty w i th wel l def ined values before themeasurement took p lace . And so , they conc luded tha t quan tum mechan ics i san abs t rac t ion g iv ing on ly an incomp le te and f rag men ta ry desc rip t ion o f theunder ly ing rea l i ty (as insurace s ta t i s t ics are abs t rac t ions that s imi lar ly y ie ld

an incom ple te and f rag men ta ry desc r ip t ion o f the peop le to who m they a reapplied).

As is wel l -known, Bohr ~4) answered th is a rgum ent by m eans of afur ther develo pm ent of h is not ion that the me asu rem ent process i s anunanalyzable whole , which led in th is case to the conclus ion that there i s nomean ing to the a t temp t to g iven a deta i led descr ip t ion of how corre la t ions ofpos i t ion and m om ent um a re car r ied a long by the mov emen ts o f the pa r t s o fa many-body sys tem. I t i s in teres t ing, however, to go careful ly in to how thet ra jec tory in terpre ta t ion di ffers f rom that of Bohr, and yet com es to a s imi larno t ion o f unan a lyzab le wholeness , though , o f course, in ano the r way. Forth i s case , wr i t ingf = R e is/h, we ob ta in fo r the quan tum po ten t i a l

--h2 { 32R ~2R )/.R -h2 o2R (zJx--a)/R(Ax--a)( 1 1 )Q = 2m \ ~x~ + ~ m OAx 2

with Ax-- - -x1 - - X 2 . This funct ion evident ly remain s large , even when thedis tance , a , separa t ing the par t ic les i s not smal l . Therefore , when theproper t ies o f the f irs t par t ic le are measu red, the qu antu m potent ia l br ingsabo ut a corresp ond ing dis turb ance of the second par tic le . An d f ro m this , i tcan be show n ~5) that in a s ta t i s tica l ensem ble o f s imi lar m easu reme nts ,He i senberg ' s uncer ta in ty so lu t ions ,Ap2 A x 2 ~ h will s t i l l be obtained.

825/12/10 7


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I f f o l lows t hen t ha t w i th t he a id o f t he quan tum po t en t ia l , som e th ingl ike H e i senbe rg ' s o r i g ina l exp l ana t i on o f t he unce r t a in ty p r inc ip l e can bemain ta ined . T he un cer ta in t ies in the prop er t ies o f par t i c les a re indeed now

seen t o fo l l ow f rom d i s tu rbances p roduced by t he quan tum po t en t i a l , whosee ff ec t s a r e moveove r unp red i c t ab l e and uncon t ro l l ab l e , because t he cu r r en tfo rm o f t he t heo ry i s on ly ab l e t o p rov ide i n fo rma t ion conce rn ing s t a t i s t i c a ld i s t r ibu t ions over the in i t i a l condi t ions of a l l the par t i c les concerned . (Sotha t we cann o t i n p r ac t i c e mak e o bse rva t i ons o f i nd iv idua l pa r ti c l es go ingbeyond He i senbe rg ' s p r i nc ip l e . )

This , however, ra i ses the in te res t ing ques t ion as to whether quantumuncer ta in t ies in genera l represen t no th ing more than our inab i l i ty to ob ta incomp le t e i n fo rma t ion abou t w ha t is i n i t se l f an i n p r inc ip l e de t e rmina t e

ind iv idua l sys tem, or wh ether there is no t som e k ind of fu r ther inde te rmin ismtha t is inheren t in the ind iv idua l sys tem itself. T he an swer to th i s que s t ion i sra ther sub t le , and requi res tha t we go more care fu l ly in to the fea ture o funana lyzab l e who lenes s t o wh ich we have a l r eady ca l l ed a t t en t i on . I n t hecase o f t he EP R expe r imen t , t h is who lenes s is shown by t he f ac t tha ti n t e r ac t ion be tween t he two pa r t i c le s depends on t he w ave func t i on o f t hewhole sys tem f rom which , as we have seen , i t fo l lows tha t in d i ffe ren tquantum s ta tes , th i s in te rac t ion may be d i ffe ren t . In p r inc ip le , th i s depen-dence o f the i n t e r ac t ion on t he s t a te o f t he who le shou ld ex t end t o i nc lude

the en t i re un iverse (See Bell , ~5) w ho a l so u ses the w ave func t ion of theuniverse bu t in ano ther way) . However, fo r su i tab le l a rge-sca le sys tems , onecan see there i s in genera l a c lass ica l l imi t , in which the wave func t ionapp rox ima te ly f ac to r i ze s so t ha t t he fo r ce s be tween pa r t i c l e s i n each sys t emm ay be re fe r red on ly to the s ta te o f tha t sys tem a lone . (~2) But any sucht rea tm ent in te rms of la rge-sca le sys tems i s ev iden t ly a g ross s impl i f ica t iono f a mu ch m ore com plex and sub t le p roces s . Fo r exam ple , seve ra l quan tumsys t ems may sepa ra t e i n such a way tha t t he i r wave func t i ons f ac to r i ze , sotha t t hey behave i ndependen t ly. Ye t , l a t e r, such sys t ems may combine aga inwi th each o the r and w i th o the r sy s tems t o fo rm new quan tu m s t at e s o f as ing le l a rger sys tem. When th i s happens , the sys tems tha t were independentnow cease t o be so , and a new quan tum po t en t i a l a r i s e s , i n wh ich t hein t e r act i ons be tween pa r t ic l e s w i th in any one o f t he i n it i al ly i ndependen tsys t ems a r e now found to be d ependen t on t he s t a te o f t he who le l a rge rsys t em in wh ich t hey t ake pa r t .

From the above , i t i s c lea r tha t in the long run on ly the en t i re un iversecan b e r ega rded a s s e l f de t e rmina t e wh i l e any pa r t m ay be i ndependen t i ngenera l on ly for som e l imi ted per iod o f t ime . Th is i s no t on ly in the ra thersuper f ic ia l c lass ica l sense tha t the pos i t ions and momenta of i t s var iouscons t i t uen t pa r t s o f a g iven sys t em may be a l t e r ed i n i n t e r ac t i on w i th o the rsys t ems . Ra the r, i t i s i n t he more fundamen ta l s ense t ha t t he ve ry mode o f


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i n t e r ac t i on be tween cons t i t uen t pa r t s depends on t he who le , i n a way t ha tcan no t be sp ec i f ied w i thou t fi r s t spec i fy ing the s ta te o f the whole . The re fo rethe s imp le no t i on o f m echan i ca l de t e rmin i sm no l onge r app l ie s , bu t r a t he r,

some th ing m uch m ore sub t le i s invo lved ( fo r f u r t he r deve lopm en t s a longthese l ines see Bohm and Hiley(~2)) .

Of cou r se , i t mus t be po in t ed ou t t ha t t h i s t heo ry i s t hus f a r i nhe r en t l ynon re l a t i v i s t i c , because t he quan tum po t en t i a l imp l i e s an i n s t an t aneouscon nec t ion be tw een d i s tan t pa r t i c les . D e B rogl ie , l ike E ins te in , the re fore d idno t f e e l t ha t t he ex t ens ion t o t he many -body sy s t em a s p roposed he r e ,p rov ided a genera l ly sa t i s fac to ry t rea tment . We sha l l d i scuss th i s i s sue inmore de ta i l in the nex t sec t ion , bu t b road ly speak ing , the re a re two poss ib leapproaches tha t may be adopted . In the f i r s t , one accep ts re la t iv i ty ( in i t s

f i e ld theore t i ca l fo rm) as be ing a genera l ly cor rec t s t a r t ing po in t , and t r i esf r o m t h e r e t o e x p l a i n t h e m a n y - b o d y q u a n t u m w a v e e q u a t i o n b y m e a n s o ffu r t he r pa r t i cu l a r a s sum p t ions w i th in t h is f r amew ork . Th i s i s, o f cou r se , t hea t t it ude t ha t wou ld be f avo red by E in s te in and de Brog li e . I n t he s econdapp roa ch , h oweve r, one r ega rds t he ba s ic concep t s o f bo th r e l a t i v i ty andquan tum t heo ry a s ab s t r ac t i ons f rom, and l im i t i ng ca se s o f , a more gene ra lconcep tua l f r amework . I n pa r t i cu l a r, we have p roposed fo r t he f r amework anew n o t ion o f imp l ica te o rder. We sha ll no t , howev er, go fu r ther here in tothis n ot io n, w hich is t rea ted in deta i l e lsewh ere . (16)

3 . F U R T H E R D E V E L O P M E N T S F R O M D E B R O G L I E ' S I D E A S

One o f the f i r st deve lopm en t s a f t e r t he work de sc r i bed i n Sec t i on 2 wasto ex p la in the p rob ab i l i ty d i s t r ibu t ion , P = ~ ,*~ , (which had thus fa r jus tbeen demons t r a t ed t o be cons i s t en t w i th t he conse rva t i on equa t i on ) . Th i s wasdone i n t e rms o f a hy d ro dy nam ica l m ode l by Bo hm and Vig ie r. ~6) A f lu idwas a s sumed , a l ong t he l ine s o r i g ina l l y p rop osed by M ade lung , (~7) wh osemean dens i t y was P = ~*~ , , and w hose mean cu r r en t was j = ~*g tVS/m.Inaddi t ion , i t was supposed tha t a par t i c le was be ing ca r r ied a long the f lowl ines o f the f lu id . But the flu id was fu r ther assum ed to be a deeper " sub-q u a n t u m - m e c h a n i c a l " m e d i u m , w h o s e i n n e r s t r u c t u r e w a s i n r a n d o mm o v e m e n t , w h i c h w a s c o m m u n i c a t e d t o t h e p a r t i c l e t o p r o d u c e B r o w n i a nmo t ion . I t was t hen sho wn tha t f o r a w ide va r i e t y o f a s sum p t ions conce rn ingth is r and om mo t ion , an a rb i t r a ry i n it ia l d i s tr i bu t i on o f p robab i l i t y,Po(x)w ast ran s for m ed ev en tua l ly in to P = ~*~ , , and tha t th i s was the re fo re in e ffec t anequ i l i b r i um d i s t r i bu t i on . Unde r o rd ina ry c i r cums t ances a l l ma t t e r w i l l h aveth is d i s t ri bu t i on ( i nclud ing , f o r examp le , bo th t he measu r ing appa ra tu s andwhat i s observed) . So genera l ly speak ing , the ana lys i s g iven in the p rev ious


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sec t ion wi ll s t il l ho ld . H ow ever, fo r a suff ic ien t ly fas t d i s tu rbanc e , a d i ffe ren td i s t r i bu t i on may p r eva i l f o r a sho r t t e rm , and new r e su l t s may be ob t a ined(for more deta i ls see Bohm and Vigier~6>) .

These no t i ons we re fu r t he r deve loped i n a somewha t f o rma l way byNe lson ~8) and by de l a Pen a . (9) Ho w ever, de B rog l ie used th i s app roa ch as aphy s ica l bas i s fo r h i s th ink ing , b y pro pos in g tha t , m ore genera l ly, th i s f lu idi s a space- f i l l ing background ana logous to e ther, bu t wi th new qua l i t i e s ( i tm ay be r ega rded , f o r examp le , a s t he phys i ca l ba s is o f t he " ze ro po in t " f luc -t ua t i ons o f quan tum mechan i ca l f i e l d t heo r i e s ) . He de r i ved a r e l a t i v i s t i ce x te n s io n o f th e q u a n t u m p o t e n ti a l o f t h e K l e i n - G o r d o n e q u a t io n f r o m t h ea s sumed t he rma l p rope r t i e s o f t h is m ed ium in a new way i n wh ich t hemechan i ca l a c t i on v a r i ab l e was i n t r i n s ica l ly conn ec t ed t o the en t ropy o f t he

backg round f l u id .Cu fa ro Pe t ron i , Droz -Vincen t , an d Vig i e r (18) wen t f u r t he r a l ong t he se

l i ne s t o cons ide r t he r e l a t i v i s t i c many -body sy s t em. They p roposed t ha t t hequan tum po t en t i a l is a r e f le c t i on o f a s t ochas t i c p roce s s , ba sed on t he " ze ropo in t " f l uc tua t i ons . Fo r t he spec i a l c a se o f a two -bod y sy s t em, t hey a s sum edtha t t h i s p o t en t i a l p rov ide s a con nec t i on t ha t i s i n s t an t aneous i n t he cen t r e o ft he m ass f r ame o f t he two bod i e s . W hi l e t h is v io l a t e s t he sp i ri t o f loca l i t ywh ich pe rvad es t he wo rk o f E in s te in and de B rog li e , t hey g iven an a rgum en timp ly ing t ha t a cons i s t en t a cc oun t o f c ausa l i ty c an neve r the le s s be m a in -

t a ined . W e f ee l how eve r t ha t t he i r a rgum en t i s no t a s c l e a r a s i t m igh t be ,wh i l e i n t he manybody sy s t em, i t appea r s t o be i nde f inab l e a t wha t t imes t hepar t i c les wi l l be connec ted , espec ia l ly cons ider ing tha t in genera l they a l l ac tt oge the r i n sepa rab ly. They a r e s t i l l howeve r work ing a long t he se l i ne s , andhope ev en tua l l y t o c l e a r up ques t i ons o f th i s k ind .

We a l so have been work ing on t h i s sub j ec t , and have p roposed a mode lo f a h igh ly non l i nea r f i e l d w i th s t r ong vacuum f l uc tua t i ons ( s ee Bohm andHi ley (19 )) , wh i ch i n m any ways r e semb le s t he i dea t ha t de Brog l i e ha ssugges ted , espec ia l ly in tha t we take par t i c les as s ingu la r i t ie s o f the f i eld ,r a t he r t han a sa p r i o r i given en t it i e s. H ow eve r, the m ain d i ffe rence is tha t wefocus ou r a t t en t i on on t he pos s ib i l i ty o f s t ab l e l im i t cyc l e s f o r such non l i nea requa t i ons . We g ive a rgum en t s showing qua l i t a t i ve ly t ha t a com p le t e ly l oca lse t o f equa t ion s o f th is k ind m ay g ive r ise , in a su i tab le l imi t , to the resu l t so f q u a n t u m m e c h a n i c s . T h i s m e a n s t h a t , c o n t r a r y to w h a t i s c o m m o n l yas sumed , Be l l ' s i nequa l i t y may be v io l a t ed , w i thou t imp ly ing t he need fo rany non loca l f e a tu r e s o f t he fun dam en ta l f ie l d equa t i ons .

In add i t ion , one o f us (Bo hm (16'z)) has deve lo ped a m ode l o f ther e l a t i v i s t i c " sub -quan tum" backg round i n wh ich de Brog l i e ' s no t i on t ha teach pa r t i c l e ha s an i n t e rna l " c lo ck" p l ays a key pa r t . Wh a t i s f u r t he rassum ed i s a sense o f l eve ls , in which ea ch osc i l l a to r ( tha t i s equ iva len t to ac lock ) i s a co l lec t ive coord ina te o f a se t o f lower l evel osc i l l a to rs to w hich


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de Broglie P i lo t W ave The ory and the Fur ther D evelo pm ent 1013

the sam e c lo ck re la t ions ap p ly (i.e ., these in tu rn a re co l lec t ive coo rd ina tes o fa s t il l deep er leve l ). The con s i s tenc y of the de B rog l ie re la t ion

~ ~ pl dq, = nh

f o r a ll l eve ls was t hen dem ons t r a t ed . T h i s m ode l is ab le t o r ep rod uce ma nyfu r the r f e a tu re s o f the quan tum theo ry. I n pa r ti cu l a r, by suppos ing t ha t e achl eve l ha s a z e ro po in t m o t ion t ha t i s r and om excep t fo r s a ti s fy ing t he c lockcond i t i on , one i s ab l e t o deduce t he He i s enbe rg unce r t a in ty r e l a t i onsh ip sd i r ec t l y (so t ha t t he appea rance o f P l anck ' s cons t an t i n t hese r e l a ti onsre f lec t s i t s p resence in the de Brog l ie condi t ion) . Th is sugges t ion in p r inc ip leopens t he way fo r dea l i ng r e l a t i v i s t i c a l l y w i th t he " sub -quan tum" med ium,

exp l a in ing a ll o f i ts quan tum p rope r t i e s na tu r a l l y t h rough de Brog l i e ' s c l ockcond i t i on , and connec t i ng t h i s cond i t i on w i th t he ba s i c quan tum s t a t i s t i c a ld i s t r ibu t ions . A n essen t ia l s t ep tha t rem ains to be don e is to der ive theq u a n t u m p o t e n t ia l f r o m t h e s e a s s u m p t i o n s o r s o m e e x te n s io n o f t h em . W ehope t ha t some work may be done on t h i s i n t he fu tu r e .

A l though we a r e g iv ing cons ide r ab l e a t t en t i on t o t he ques t i onsd iscussed in th i s sec t ion , our m ain w ork i s s ti ll a s ind ica ted in S ec t ion 2 ,a long the l ines o f deve lo pm ent o f the imp l ica te o rd er. (16) In pa r t i cu la r, wea r e c o n s id e r in g t h e m e a n in g o f at t em p t i n g t o m a k e a q u a n t u m t h e o r y o f t h e

genera l i zed s t ruc tu re o f space- t ime . W hee le r (21) has in fac t a l re ady beenlook ing a t t h is p rob l em f rom a t opo log i ca l po in t o f v i ew, i n wh ich he s t a rt sw i t h h is " p r e - g e o m e t r y, " a n d h o p e s t o a r ri v e a t o r d i n a r y g e o m e t r y as al im i t i ng ca se. I n ou r app ro ach , we a l so have a k ind o f no t i on o f "p r e -geo m e t ry, " bu t we s t a r t w i th con cep t s t ha t a r e ba s i ca l l y a - l oca l, and de r i veloca l i ty a s an app rox ima t ion . Th i s w il l p rov ide a deepe r unde r s t and ing o ft hose r e l a t ionsh ip s wh ich we have t hus f a r exp l a ined t h rough t he quan tumpo ten t i a l and t he de Brog l i e c l ock cond i t i on .

4 . C O N C L U S I O N

Firs t o f a l l , we would l ike to ca l l a t t en t ion to the fac t tha t the ea r lywork o f de B rog li e , as r epo r t ed h e r e , p l aye d a key pa r t i n ma k ing pos s ib let h e d e v e l o p m e n t o f th e m a t h e m a t i c a l f o r m o f t h e q u a n t u m t h e o r y i ts el f.Unfo r tuna t e ly, h i s phys i ca l i n tu i t i on and imag ina t i ve i n s igh t s we re no tgene ra l l y t aken up , and t he r e fo r e d id no t hav e a w idesp read e f f ec t on w ha twas subsequen t l y done w i th t he t heo ry. Neve r the l e s s , t he r e i s an app rec i ab l enum ber o f peop l e wh o a r e d i s s at i sf i ed w i th such an a pp ro ach t o phys i c s , inwh ich ph ys i ca l i n tu i t ion ha s c om e t o p l ay a r e la t i ve ly m ino r r o l e ( pe rhapsm a i n l y a s a c o n v e n i e n t a id t o w o r k i n g w i t h m a t h e m a t i c a l f o r m a l i sm s a n d


14/16

1 0 1 4 B o h m a n d H i l e y

compu ta t i ons ) . And indeed some phys i c i s t s a r e s t i l l work ing w i th ex t ens ionso r dev e lopmen t s o f de Brog l ie idea s , o r w i th ye t o the r i deas , wh ich a r e a imeda t a be t t e r unde r s t and ing o f t he und e r ly ing phys i ca l r ea l i ty t han is tr e a t ed by

the ma them a t i ca l f o rm a l i sm a lone . W ha t ha s been done by de Brog l i e ha sbeen muc h app rec i a t ed by t he se peop le a s an i n sp i r a t ion , an d ve ry o f t en a s asource of ideas .

A P P E N D I X 3

B e c a u s e s o m e e r r o n e o u s a c c o u n t s o f h o w I c a m e t o d e v e l o p th et ra jec tor ies - in te rpre ta t io n of the qu ant um the ory a re in c i rcu la t io n , (22) I

t hough t i t adv i s ab l e t o s t a t e i n some de t a i l how I c ame to do t h i s work .T o beg in wi th , I w ro te a bo ok C23) f rom Bo hr ' s p o in t o f v iew, main ly in

o rde r t o t r y t o unde r s t and t he quan tum theo ry. Bu t a f t e r I had wr i t t en t hebook , I f e l t t ha t I s ti ll d idn ' t r e a l l y unde r s t and t he qu an tum theo ry, and so Ibegan t o l oo k fo r new ap p roaches . M eanwhi l e , I had s en t cop i e s o f t he bookto B ohr, Pau l i , E in s t ein , and o the r s c i en ti s ts . Bohr d id no t r e spond , bu t P au l is en t an en thus i a s t i c r ep ly, s ay ing he l i ked t he book ve ry much . E in s t e in a l sogo t i n t ouch w i th me , s ay ing t ha t t hough the book exp l a ined t he quan tumtheo ry a bou t a s we l l a s wou ld eve r be pos s ib l e , he st il l was no t conv inced ,

bu t wan ted t o d i s cus s t he sub j ec t w i th me .W e had s eve ral d i s cus s ions , t he net r e su lt o f wh ich w as t ha t I was

cons ide rab ly s t r eng thened i n my f ee l i ng t ha t t he r e was some th ingf u n d a m e n t a l t h a t w a s m i s si n g i n q u a n t u m t h e o r y. T h i s m a y p e r h a p s h a v emade me work w i th g r ea t e r ene rgy, bu t Y. Ne ' eman ' s s t a t emen t t ha t I was" s h a k e n " b y m y c o n v e r s a t i o n w i th E i n st e in a n d " h a d n o t r e c o v e r e d t o th i sda y" is en t i r e ly f a ls e . I n any ca se , wh a t a c tua l l y hap pene d w as t ha t I sooncame upon the t r a j ec to r i e s - i n t e rp re t a t i on , and p repa red a p r ep r in t , cop i e s o fwh ich were s en t t o m any phys i c i s t s i nc lud ing de Brog li e , Pau l i , and E ins t e in .

I l e a rn t sho r t l y t he r ea f te r f rom de B rog l ie t ha t he ha d deve loped th i s i deam uch ear l ie r, and so , in l a te r vers ions o f the pape r, I adk now ledg ed th i s fac t .Pau l i was ve ry nega t i ve i n h is r ep ly, s ay ing a l so t ha t de B rog l ie haddeve loped t he s ame m ode l m any yea r s ea rl i er, and t ha t i t had been show n byh im to be wro ng a t t he So lv ay Confe rence . (3)

As a r e su l t o f Pau l i 's l e tt e r, I deve loped a t heo ry o f t he ma ny -bo dyprob l em answer ing h i s ob j ec t i ons , wh ich was i nco rpo ra t ed i n a s econd pape r(Bohm(5) ) . I had severa l fu r ther d i scuss ions wi th Eins te in , bu t he was no t a ta l l en thus i a s t ic ab ou t t he i dea , p ro bab ly ma in ly beca use o f t he f ea tu r e o fnon loca l i t y o f t he qua n tum po t en t i a l, wh ich con f l i c t ed w i th h i s ba s i c no t i on

3 On the B ackground of the Pape rs on T rajectories-Interpretationby D. J. Bohm.


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de Brogile Pilot Wave Theory andthe Further Development 1015

t h a t c o n n e c t i o n s h a d t o b e u n i v e r s a l l y l o c a l i n t h e f u n d a m e n t a l l a w s o fp h y s i c s .

W h i l e I c a n u n d e r s t a n d E i n s t e i n ' s o b j e c t i o n s f u l l y, I f e e l t h a t i t m a y

h a v e b e e n a ta c t i c a l e r r o r o n h i s p a r t t o d is m i s s s u c h i d e a s b e c a u s e t h e yc o n f l i c te d w i t h h i s o w n n o t i o n s a s t o th e n a t u r e o f r e a l it y. F o r t h o u g hp e r h a p s u n s a t i s f a c t o r y i n m a n y r e s p e c ts , th e y m a d e p o s s i b le , a s e x p l a i n e d i nt h e p r e s e n t p a p e r , c e r t a i n i m p o r t a n t i n s ig h t s i n t o t h e m e a n i n g o f t h e q u a n t u mt h e o r y. I n a d d i t i o n , t h e y h e l p e d k e e p a l i v e a n i n t e r e s t i n f i n d i n g a d e e p e ru n d e r s t a n d i n g o f t h e q u a n t u m t h e o r y. I f ee l t h a t a c o r r e c t a p p r o a c h m i g h th a v e b e e n t o e n c o u r a g e s u c h w o r k a s a p u r e l y p r o v i s i o n a l a p p r o a c h , b u tr e c o g n i z i n g t h a t i t w a s n o t l i ke l y in i ts e l f t o b e a f u n d a m e n t a l t h e o r y,w i t h o u t f u r t h e r r a d i c a l l y n e w i d ea s . T h e r e s u lt o f n o t d o i n g t h is s o r t o f t h i n g

w a s t h a t , f o r t h e m o s t p a r t , f u n d a m e n t a l p h y s i c s w a s r e d u c e d t o its p r e s e n ts t a te o f r e l y i n g a l m o s t e x c l u s i v e l y o n f o r m u l a e a n d r e c i p e s c o n s t i t u t in ga l g o r i t h m s f o r th e p r e d i c t i o n o f e x p e r i m e n t a l r e s u lt s , w i t h o n l y t h e v a g u e s tn o t i o n s o f w h a t t h e se a l g o r it h m s m i g h t m e a n p h y s i c a l ly.

R E F E R E N C E S

1. N. Bohr,Atomic Phys ics and Human Knowledge(Science Editions, New York, 1961).2. L. de Broglie,Une interpretation causale e t non lindaire de la mdcha nique ondulatoire: la

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Documents

The de Broglie Pilot Wave Theory and the Further Development of New Insights Arising Out of It D. J. Bohm 1 and B. J. Hiley ~