Pergamon

(']laos, Solitott.~ & Frac#al~ Vol. 4, No. 3, p p 403 4{19. 1994 Copyright © 1994 Elsevier Science Ltd

Printed m Great Britain. All rights reserved (It)60 0779,'9457.1)1) + .i~l

0960-0779(94)E0078-4

Young Double-slit Experiment, Heisenberg Uncertainty Principle and Cantorian Space-Time

M. S. EL NASCHIE*

Abstract We give an ad hoc explicit derivation of the Heisenberg uncertainty relation based on a Peano Hilbert planar cur*e. Subsequently we show how some of the logical difficulties in the interpretation of the double-slit experiment with quantum particles may be circumferenced using the proposed Cantorian model of space-t ime geometry.

1. INTRODUCTION

The use of a Peano-like Cantorian geometry as a model for quantum space- t ime was considered in several previous publications [1-3, 6, 7] and other articles by Nottale and Le M6haut6 in the present volume. There, it was argued that while space- t ime is curved in the large and considerably flat at the scale of classical physics, it is probably wrong to assume that this flatness must hold for the extremely small scale [6]. On the contrary, quantum space- t ime may well be similar to a four-dimensional analogue of a Menger sponge [5] and this is already at the de Broglie length scale [7]. This point was made plausible recently by deriving the Heisenberg uncertainty principle using the Menge r - Urysohn dimensional system in conjunction with the concept of Hausdorff velocity and assuming the validity of the de Broglie relation [7]. In what follows we give a limited but explicit derivation of the Heisenberg uncertainty principle based on a Peano-Hi lbe r t curve. The analysis is essentially like the earlier more general but also slightly more abstract derivation [7]. Subsequently, a discussion of the double-slit experiment is given in the light of an earlier discussion by Feynman et al. [8] as well as the present picture of a Cantorian Peano-like space- t ime. Finally, an analogy between the emergence of a fractal pattern through random construction and the well-known quantum interference is drawn.

2. DERIVATION OF HAUSDORFF DIMENSION OF A MULTI-DIMENSIONAL CANTOR SPACE

Propos i t ion . If an m-dimensional Cantor space represents a set of n- independent events arid if the Hausdorff dimension of each of the tl-dimensions of the same set in an additive

/ ( 0 ) representation is a , then dl, '') is the Hausdorff dimension of the m-dimensional and space

1 I " I

, C > : m : t U~,5 I' !

where

0 < d{<! 'i ~< 1.

*Department of AppLied Mathematics and Theoretical Physics, University of Cambridge. UK.

41)3

41~4 \1 ~ I i N\,< ~m

/'r<~(U'. Sincu the c \cn is arc indct)cndcni , lhu probabi l i t ) ot f inding ,me l~Oini if,. lh .... m-dimcnsiop, al ( a n t o r <,pace is found lr t)nl the mul t ip l ica t ion theorem as t i le pruducl ~1 lh . prohal~il i tx o1 f inding a point in one e l the dimcn.sions I ~ ..... : 1 i~l aild tile_, prol~al~ilii~ i t f inding the poin! ~ i t l i i n ~li1\ t) l l ' ,_' t / I ' the (TalltOr st'Is forming the (L in tor space ...~;/'~: {;.,in,.~ tilt_" Hausdor f f d imension U ~n, of a onc-d imcnsiomt l Cantor set il l c!cfinine. ~i pSctldo-~coillcl-. rical p robah i l i t \ ctut~iiCnl ,au can ,aritc thai

U '~' i

(+tmscqucntl 5. al~d using the mul t ip l ica t ion l l l corcm, thu i~robal~ilit} el I indln u the: p~,lnl i t dn\ ~)1" lhm /~#-dimcnsions of the ,q~i-'c t,,

, # H ,

l~>tll on the t / lhcr hand. f in an #l-dimcnsion~il addi l ive represt_'ntalitm ,~1 the ',;.lille' ( Hl lh, i :q~ace. the ~,tilllC probabi l i t~ is e i \cn ~tlso b\ (l~t' mu l t ip l i ca l ion lh0, ircrn as

I l l i

l-Xltiating the rigl~i-hand -,idc of both uxprcssions. \~c linci ilk<i!

( ( l , : ¢1 ,

P l l ~ i t 111 t.'~.ill~,

z

~ h i c h p r o x e s the propositi,,m It shou ld hc n o t e d tha i tn cou ld t~lke the , ,ainu of aip,: real n u m b u r , bui ltv- lbc tti.';div' ',¢t

x~c can set m - ~¢ if and on ly if r e - , s - 4 . W c max use this fact t~* a f f i rm file pt: ius,hiJit)

~fl rising fOUl-di lncl ls i tmaJ sets for m u d e l l i n g spacL, finlc. [ ; '£1rlhCl ' I l l l ) l 'k , t,~<C h~t ' t2 sh,t)','~ll {t

p r e v i o u s w o r k [2 .3] thai lhc lri.tidic set

Hh} I n ( ¢ ::-

I n 3

m a y be t h o u g h t of as a kind t)| w e i g h l e d m e a n of a B o r e l - l i k c sol ~lich :is lhc ( ' , lnt~ri~m

,,pace t ime p r o p o s e d he re T h u s c ~ a h i a l i n e t h e s imp le fo rmu!n

for UI!" -: In 2: ln 3. x~c l ind thai U'. *~ - t~ - 4 and also that U'[" -- 5<, ' -- 25 1 * ;here N*[" ::, !.he

i n f o r m a t i o n e n t r o p y t>i: the set. It migh t b<: of in t e res t to i l tqc that t <1~": W " i,, :~

m i n i m u m at n - S.

I 3. I)ERI%ATION ~}I ~ THE 1_TN(TERTAINTY PRINCIPLE FOR A I EANO-HILBi- ;RI

( ' ( )NSTR!;CTI( )N

R c l c r r i n g to Fig. 1. il is clea~ lilaI a " C a n t o r i o n ' " m u x i n g on ibis Pc,:n,.~ ! l i l b c l l ,vl i which c o n s t i t u t e s now a g e o d e s i c for the (.';.tllt()riOll. \viii revcr<,e the ,d~n ,~1 {he h,.>riz~mla~

c o m p o n e n t of its xe loc i t v t h r ee t imes . / \SSuln ing any sign c o n x c n t i o n . ~, : set: Iron1 t:ig. '.

Young double-slit experiment 405

®/--;

®

,--/ " @ ~ AX

+ /

zLX

~ X

Fig. 1.

that six of the nine s t ra ight d is tances will have, say, a posi t ive hor izonta l c o m p o n e n t , and the o the r three negat ive ones . F r o m the condi t ion that the p robab i l i ty of a posi t ive d i rec t ion (P~) or a negat ive d i rec t ion (P2) must satisfy the no rma l i ty equa t ion

~.~t~ = P~ + P~ = 1, l

and since Pt = ~ one finds that P1 = ~ and P, = 5- It is an equal ly d e m e n t a r v ma t t e r to d e t e r m i n e the d i spers ion in the veloci ty using the s t anda rd dev ia t ion

where Xl = V . x - - - - V , PI

CI 2 ~ E ( X i - t ? l l ) 2 p l l

IVI l = E A ' P I i

2 I = ~- P2 ~ ~, and we find that

D

and

o ~ v b + v d) ~ " . . . . . ~ V - .

C o n s e q u e n t l y the d i spe rs ion is

A v = o = 2-v2v. 3

Next we have to dis t inguish be tween the Euc l idean veloci ty V, and our Can to r i an space

4(1(~ Y,1, ",. I-~ N \~< iHt

x c l o c i t v l . It] c tm i l og ) tt~ the H; . iusc lo r i f l e n g t h , l hc { vc loc i t~ ln~i~, he' l c l i ncc t th lu<,c to i ! i v c l o c i t ) so [h~lI we l l laV ~vi-itc

),

w h e r e t t is the i - cso lu l i on ; lnd 2 p lays the : ,mnc ro le o f ;~ ~ z l ~ c l c n g t h h i i l hc Pca ln t~ - t t i l l ) c i~ Ct ) l l s t r t l c t io l / .

S ince Ot l l P o a n o - l t i l J , ) c r [ ct l r \ t_ o f [:i 7. 1 il; i ', U' " 2, # ] - \ t Llil¢J \ t - / .~ t l l l t : ! ln t i~ 1 h;.il

! ',, )).

~',;oling [h:.il l h0 l l lO i l lC l l l t l l l l i', P :- !1/1 ~ v,'hc'rc: in i~, ;i<,Tilnlc(I Io I~c' ~1 fc~,~)itdi~tll inctc'l)c'nctvi/ i no l~ - rch i l i v i<d ic ill;.l~,',. Ollc' l incV, t r l ) l l l \ 1 lh:,ll

\ t ) : i l l /~ I, - - \ - (~1, , !11t - ] ~ 2| ~

NC\ I xAC I_lkC the t ic B l~ lg l i c rt_'Jalion P : tl >. \ ; h o l e // is l i l t t)Jcll~Ck c't~ll~,titnl ut ~llil\'~ i i , nt lc l lo~w I~ct,, tecn l i l t ' role > o1 lt~c Pc{ i l lO H i i h c r / x<x.n~<olt_.ll~th ;l l lcl ttl<_' \\;i ',c_'ic'll~lJl ~!t < t l t i~ i l l l t l l~ l p ro- t i t les ( 'on< ;cqucn i l~ , . ~vc in<ix, x ~ i ! c l l l u l

\ I~ .] ,, 7 /

Mullipl_vin~ no',~ l~oth ,,ich_'~ v,hh ,\~- A b \',c lind ',hm :xp\,t l~cc~mc~ i<'~oluii~m ;~ncl

v, ~i,, clcngih indcpendcnl:

t

J II~il iI I t_';.ill <~

4, ~'()I!NG I)OUIUE-~I , IT I;.XPERIMF~N'I

Jtl tJIuir CiLl+Sic_';Ai Uxl+o',it~m ol i.ltl;illltl111 111cCh~illlC', LIIItI pLtth intcur~d'- , l :c ' \ t inIL l l +, +'I +i/ !:+~;

db, uus~<t_+d the l og i ca l d i f f i cu l t i c> , u n u o u n t c r c d it+ the l,+',t>-slit cx j+cr imc ' ! i t ;mcl tilt+, p ' , i i~ tdoxk.~i COllch_r~ion thLtt " v<hun l+t+tJl bo lus ~tt+c ~+pc'llutl, it i', i1ot i f f l u lhLlt t i le l+~trt~clc' gt>us tlltt+u+_,h t i l iC hob,_" o r tJIC o t h e r " , J'hc'_~ ~t+ o11 to ~.~t x, LI litth_' ]kllC! li lLtt ' t o CXtl iCtlU. (+ul+~Cl'¢t.~ .. 1I-O111 th.,+ logic~d d i t l i cu l t i c '~ i t+d toduccd I+', th i~ uXmnl+~lu ,+:c m i g h t tr,, \ a i i ou~ , , l l t i l ]cc'- , . \Vc n l i ~h t ",~i\ f o r c : , ,anip lc th: t t perhaps, ti~c c l u c t r o n tl-~t+vt.+J~ Jn c t~mplux t l~l juctoi~. !:~+Jl1~ t J l rou~ i l h~>iv i t hen back to ho lu 2 and i in~ i l l~ t h n + u g h I in ,,¢m+u ct+ inpl ic~f lcd m ~ i n n c i '

No,A rc te r rmg to the Pc'~tno col! of Find. ". il i',, c~, i t iet l l lJl;tt i l l "~tlCJl LI (_'.tlllOt'lLl{l

Pc~mo-likc space t ime , c los ing the lo\ver hole rnm,! immcd ia t c lx cl~cmgc the pc~ih lhrt~ueh the u p p e r hole. Fhus bo th holc~, c~rc ¢ o m p l c l c l v i n / c r c o n n c c l c d wi th in this (Ymh~imll p ic ture . To put it in a d ia lec t ic f o r m u l a t i o n . ~\c tmi\ ~cl,,' Ibm Pc~itao-likc ,,pcicc l ime l/)l~.ill hc the na tu ra l mo d e l l~r F e \ n m ~ m artif ice. Ih~u, 'c \cr . ",~c Lifo ' , l rcss ing hcrc th~li Ihc. c o m p l e x i t x oI the qtu+tt ion is in t roducccl to the d x n a m i c s h\ the ~ct+nic'lr\ of lhc g¢,+clc'~,ic ,. of the spz~cc- t imc itself.

This get)111cll-.~ IS higldy i ) tm-I incar ~lncl compIc:,, I<, ~,uch cx icn l lh~i/ ~,i/ ;~ l iner ;lmi llll<~ <,cilic the re is no hope :it all t+t ccdcuhit ing an v th ing except pr~+l+~thill,,ticallx. Nt)lc ,ll,-,~, ill;t!

Young double-slit experiment 407

I " -

Screen

Fig. 2.

D e t e c t o r

the mere existence of the apparatus changes the geometry of micro space- t ime instantly. Following Fig. 2 for instance, a hypothetical observer who can see in straight lines only could not explain at all why closing hole No. 2 will immediately prevent him from observing the " 'quantum particle" hitting the detector behind the screen. Thus, within this Cantorian geometry, space- t ime , apparatus and observer form one wholeness similar to what Bohr described in general philosophical terms but was not able to formu!ate mathematically. On the other hand. Bohm [9, 12] was able to give an analytical t reatment for essentially the same idea but lacked a simple intuitive geometrical model. We feel that Cantorian geometry as a model for micro space- t ime could be of definite help in advancing towards this goal [9, 12].

5. INTERFERENCE AND FRACTAL PATTERN ANALOGY

There is also all interesting but limited analogy between Cantorian geometry and the two-slit experiment which in spite of being indirect may still have a deeper connection with the two-slit experiment than that which is revealed at first sight.

The analogy is best explained through direct reference to a fractal construction procedure. The object of this demonstration is the Sierpinski gasket which plays an important role in deriving the bijection formula [31 . The essence of the procedure is as follows, We start by locating three corners of a triangle and plot a point anywhere on one side. Then we choose at random one corner and plot a point in the middle of the imaginary line connecting the first point and the corner. We repeat this procedure using another corner point chosen again at random and so on [10, 11].

After a sufficiently large number of points we will see how the hazy picture of a Sierpinski gasket starts taking sharp features as shown in fig. 3, which is originally due to a construction found by Hutchinson in 1981 and developed by Barnsley [10, 11]. The amazing thing is, however, and here lies the clue to the connection we are proposing, that if the corners are chosen in a definite regular sequential order instead of at random, then we will end up with just a few points and not with anything resembling the attractive and orderly Sierpinski pattern. The rest of the would-be points disappear probably into some fixed points. This simple experiment for which nothing more is needed than time, pencil and a sheet of paper was described by E. Lorenz in his recent book [10] as "'a remarkable thing". O. Peitgen et a l . commented in the same context [11] " ' . . . we are inclined at first not to

+ll+ M . ~. [!i N \+< Illi

( ] )

. + .

o+ 2

+ .

( + > )

:+ ,¢N "+'

+r ~ ,

r . ¢ + ,+'~ ", • +

• +

~ , , . + : . ,7 + .

$ ) ',

<' r +'~t ,+ ~++7

* ' , , . +J+. + .

;+~. +., +,+ m+~;,++

d 7 • ~ ,+~, i , . . < •

+ " .,7+-," ~ z : . ~ \ .... ':- ~<.7' ~.~ +~ +', i ; P. ~.+ .. .~+ :+

1 1 _ + ~. l i l t ' C'II~C>IL+.+'.t.+:ICC ~+! l t : ~ t l i ~ l i t+ult<_.in t h P t + t i # h I ~ ! t + t + a l + i h > t i c ct)It<~tl+l lc. ' t l~!n l t i l ]~+~,~, l i iL '<lie' !+ l t+cc- tJ i l l ,+1 I f i< J

l h i . I l l< t \ i+c.' \iu~c_ci ~l', ,i+a~+'. H! ~,rdu~l~. c l i~Li i~c h +, ; m + i l l { t i c + l f ; t n < ~ l t + l l l / ~ t i i ~ + l ,+! -+t,, i<c i ! ~ll+uii ~<>mc <++ +I, c'~'+c'liti~il ;l~,t c'c'+~ ~,f l i l t ' illlt ' l '/c+tt'l]ct +t! t!c + I l l l l l l i .~til cltl:illltit++ l+;tlt{,'lc + ",p< lil'.lt+lll< ' l~,\ h t

~tll~.til,)~,tlt.lS i ' , u h a \ i ~ u r ~d '-,c'mlu 1 7 > u r l l t m l l i - s h i l t mi l l ;< . . N u \ u r l h c l u s . , il ,-. ;t l i U l u . ,u ; - l~ r t< .~ . l t ; ; i !

H t + V , c \ CY. t i r e : ~i l t l++t t+t>l l +~, r u a l l v ~tll+l+it+~,l ~+it1~.i]t~<t>t)i+is. ] l i t ] I c +.+~.is, c t)+ t++tkin~_, +i++ +Vi,+:~t '+t i lC+i lcHU

il~ ~i t'+~t+-<,lit c x l + c r i t l l C i + t ~ u tit+ i l t i i k t l~+\t l ~ , l >tit+u ' t h r~>uTh ~ h l c i + ll~+lu lh<' Ht,t iulc x;~i,><

~<~+ckllt CillCt '+x<C ~i l t h a \ u t h e %vml l - kn t : l \ \ l l i i - l t t :Flcrci+c:c, p++itlCl-li, l 'h i '~ +~ tt~t+. Utllll+lt<..!-lt~il+ ,~

f i i ~ d i n 7 t h u Siurt+inski t~,tltlt?rl~, W l l e l l I c t l i l l ~ thiugh [+,,+,luuud +It l a l l t i t i t i + , t + , tUt / t l t t t i '~! . $[ +,to' l~+lt

ct<)~.tF, it c l t i a i l t l . l i l ! !++lt+tic'lu iI+ t>rc lu i l~+ k l l t )x. t x~hlc: l i hulu ,I t~ul+t ti-i i+~+uglt, lhmi~ i h ,

i t l t u l - f+ iC , llmt: pat tmi- l° l <_ti+cil+t+cai,+ ; i l l t l ~,c' h~+\c, t+~irt i¢!c , I-+ci+~t~it+t+.+i i i l l - , ~t)il<k>,i')tUlct<. It+ t i le '

C~.i'+c' \~+'ht_'t+l '+t,C cht',t/.~m thin c.'til+l}t++l+ ++i l i l t l r i + i l l T I c + i l l ;i t l t . ' tcYi l l i l t i+~t lc ,+ , :c luc l / / i ; t l t , .< i \ : l l l t i i i+ t '>iutl+in,~ki p~_iiteri~ t t i '+al+l)o~lt<,.

In t ~ t h o l + \~t i l-Cls. t ) t l i ' i ~ l l t ) l + i l l c ' c ' I ) l t l i u ~ i \ +,%hici] ~i t+~ i r t iu lu -u+{ i~u h:i '++ l . ~ l k c I ) c ' t ) i i c s , , t + t ) l l t t k l ,

I'+.tllctt)lllnt'~+~, +.trlcl l+)rt ibal~i l i t+~ + + |he i c< ,uh i,. u \ t u l i i t h ~)! l l i l t ) t+ i~ ia t i t ) n ii~ l l l c l~+i ~l! ~+i ciil :)! ,_it_.i Ix

I o o k i n . u ]~)~tttcl+rl+ I ' h u si t t i++i l i ( ) i l i-, t t l c t , rc '~ ,+r~c i i i t ~ . - k l+c)~ l+rcci>cl+~ +~h ich ~cl~ i.t iu i ~u r t !u f ,

l i t l k l;ikt_+l] w l i i u h <.;t)ll-t.'~,l-~l)lltl. t , ! d u l i n i t u t+rc tur . Ii1 t h i , u~i~u lit> p ; . i l l C l t i 7llCc~.ctucl i l } l t i l - i l )~ t i l~ ) i i I', I'U', UalC'Ct.

li~ t h u ua+c + o f t l i u $ ic r f+ i t l<>k i ut i l~+t t+t iu{ l t ) i~ t l+u l+~.lll.Cl-i1 i +++ uh_~url~ t i i c i-e,+ti l t ~ll t_+ll{i<~tl,

~l\'nl.il+lim~. 5o t h a t t h e \ . i~, t ia l o i + d u r l+lla\ t+lc in tc , rt+rmtc+ct ~i>+ c l u u c l + t i ~ o t>r l+ i i ract<)xim~i l i1 +4. 1_!+ i

( ' o n s i d e r i i + 1 7 t h e l.ltlc++i '+ i i+ l ]p l ic i i ) c>l o u r l i+ac+lal u o n s i r t i c i i ~ i i + 71 i,, +it,~i c l i f i i c u l t t~i sum ttic:

l l l l l l l t t+os,~il+ilitiu+ t+l+c,l+eCl l o F llOV+ i i ~ t e r p r c t a t i c > t + + eft the: t ~ t i - s ! i t C X i + I u I i I I l C I I I ~v i ih i i~ ~ i~ l l+~ . i l l l c '~Olk t+f it ( ' a l l l t l r i { i l l tllit~l+O ,>pciuu l i r a 0 .

l h i + p a p c u + i+ ~.ill 0x t c i l ~ , i ~ )n o f ~_111 u n t + u l + l i + h o < i t l i L i l lU~<c i ip t I I $ i l ; ) t i r i117 :l '++~,it t , , {lcit.+in7,c_+~

( ) . R o s ~ i o r c i ¢c i c l c l l t + i l l \ ~av+ t h u i l l a i l t i~>ur i l+ t ; i l / tJ +trt>n<+~l+~ r o c t > m i + i c u i d e c l t i l~t t i t n l~ot i l~ t I+<

i+ut+l i ,~t~ed. P i n a l l v ~ I ~t>~ t-~cu-,+u~ittc<i to; i n c l u c l u it in tl~i~+ v o l u r i + c dC, , l+ i tu t+~ iu<_,lil+~ ~hut i+ ,<. <~till ii+ a "+oi+\ , t+rc'l in+ii-lttr+~ I tu-n+.

Young double-slit experiment 409

REFERENCES

1. M. S. El Naschie. Complex dynamics in a four-dimensional Peano-Hilber t space, I1 Nuoro Cimento 107, 583-594 (1992),

2. M. S. El Naschie, Mathematical model for chaos and ergodic criticality in four dimensions, Math. Co,zputer Modell. 15, 77-8(/(199l).

3. M. S. El Naschie, On dimensions of Cantor set related systems, Chaos, Solitons & ~)'actals 3,675 685 (1993). 4. W. Heisenberg, Die Physikalshen Prinzipien der Quatztum Theorie. Hinzel. Stuttgart (1958). 5. B. B. Mandelbrot, l'he Fractal Geometry of Nature. W. H. Freeman, New York (1983). 6. M. S. E1 Naschie, Quantum mechanics and the possibility of a Cantorian space-t ime. Chaos, Solitons &

Fractals 1, 485-487 (1991). 7, M. S. E1 Naschie. On Heisenberg uncertainty principle and Cantorian space-t ime, Chaos, Solitons & ~)'aetals

2,437-439 (1992). 8. R. Feynman and A. Hibbs, Quantunz Mechwzics" and Path hztegrals. McGraw-Hill. New York (1965). 9. B. J. Hiley and D. Peat (eds), Quantum Implications. Routledge, London (1987).

10. E. N. Lorenz, The Essence of Chaos. UCL Press, London (1993) (see, in particular, p. 174). 11. O, Peitgem H. Jiirgens and D. Saupe, Chaos and Fractals. Springer, Berlin (1992). 12. D. Bohm and B. J. Hiley, The Undecided Uni~,erse, Routledge, London (1993). 13. M. S. El Naschie, On certain "empty" Cantor sets and their dimensions, Chaos, Solitons ~ b)'aetals 4,

293-296 (1994). 14. M. S. E1 Naschie, ls quantum space a random Cantor set with a Golden Mean dimension at the core? Chaos,

Solitons" & Fractal~ 4, 177-179 (1994). i5. M. S. El Naschie and S. A1 Athel, Young two-slit experiment, Heisenberg uncertainty principle and

Peano-likc space-t ime. (Unpublished manuscript 1992).