Quantum cellular automaton in 1-D

Pergamon

(Tluo*. Soliton~ & Fractal; Vol. 4, N o 3, pp. 439 460, 1994 Copyright © 19~4 Elsevier Scicl~cc Lid

Printed m Great Britain. All rights reserved I)96[)-0779,,'9457.[10 + .(Ill

0960-0779(93)E0023-I

Quantum Cellular Automaton in 1-D

D. W. B E L O U S E K

Program in History and Philosophy of Science, University of Notre Dame. Notre Dame, IN 46556. USA

and

E. HENIGIN, D. HOTT and J. P. KENNY

Department of Physics and Astronomy, Bradley University, Peoria, 1L 61625, USA

Abstract Simple algebraic cellular automata and a diffusion confinement are used to model quantum behavior of a relativistic particle in a I-D box. Comparisons and contrasts are drawn between such a quantum cellular automaton (QCA) and both orthodox and unorthodox interpret- ations of quantum mechanics. The persistent tension between deterministic pictures and probabilistic descriptions of quantum reality are clarified for the case of a 1-D quantum particle viewed as a cellular automaton surging for survival in space and time.

1. INTRODUCTION

W h e n v iewed his tor ical ly over the ent i re cen tury since its initial fo rmula t ion it has to be a d m i t t e d that quan tum mechanics still has c loudy and confus ing concep tua l founda t ions . It still hovers be tween a de te rmin i s t i c and wel l -def ined Schrod inge r view and a probabi l i s t i c or s tochast ic descr ip t ion exempl i f i ed by the B o r n - H e i s e n b e r g pic ture [1]. All systems in na tu re do obey quan tum mechanics and the cont inuing e xpe r ime n t a l ver i f ica t ions of its subt le traits to r e m a r k a b l e degrees of prec is ion have fai led to clarify any of the compe t ing mode l s which a t t emp t to show how the smal le r t idbi ts of rea l i ty behave in space and t ime.

With the recen t d e v e l o p m e n t of t rac tab le chaot ic dynamics and its poss ible re la ted and newly def ined discipl ine of " 'quantum chaos" , some novel possibi l i t ies p resen t themse lves t oday [2]. A kerne l of t ruth emerg ing from this new and fert i le s tudy seems to be that , while classical dynamica l systems tend to be chaot ic , the co r r e spond ing qua n tum systcms are much less st). It is th rough adher ing to qu a n tum mechanics that na ture is s table and cohe ren t in t ime and space.

A n o t h e r fea ture that can be g leaned f rom these s tudies is that n e a r b chaot ic systems of na ture seem to succumb to a single d imens iona l f r e e dom in the i r desc r ip t ion if an obse rve r or a c o m p u t e r can wait long enough. O n e - d i m e n s i o n a l ( I - D ) solu t ions to p rob l ems seem naive ini t ial ly, but na tura l b o u n d e d systems in complex mani fes ta t ions seem to d iscard higher d imens iona l i ty even tua l ly .

This p a p e r deve lops a hypo the t i ca l one -d imens iona l ent i ty ca l led a q u a n t u m cel lu lar a u t o m a t o n ( Q C A ) as a p laus ib le mode for r e in t e rp re t ing the t rad i t iona l " 'quantum par t i c l e" in s t anda rd fo rmula t ions of quan tum mechanics . The Q C A was in t roduced in an ear l ie r p a p e r [3] where hints e m e r g e d f rom its rich and va r i ega ted behav io r that some

43~

44il l) ~.\. B l l ~ n ' , t l , . , v , d

q t l a n t u n l e f f e c t s c o u l d s t l c c u n l b to Stlch a d e s c r i p t i o n , ( 'clJul~lr ;Ju|onlal~l o b c \ ]~CLll

de te rmin i s t i c spa t io t empora l rules bnt exhibi t pa t t e rns e l cohe rence and dccohcrcnc< ,ahich are s imilar to quantunl mechanica l sxstcms.

,Section 2 of thi'~ pape r def ines the Q ( ' A and its conf inement within ~l I-D he>x. Sexeral cxamples of O C A spat io t empora l behav io r arc examined in Secti~m 3 ~ i th genera l tea turcs such as rec tu rencu , classif icat ion types, r e la t ionsh ip to l ractals , arid dominan l t rends no ted as re la ted to poss ible quan tu ln in te rp re ta t ions .

%cotton 4 exp lo res path integral fo rmal i sn> and probabi l i s t i c i n t e rp re t a t ions and de \ t i e r> , thu uncer ta in ty re la t ions h o s p i t a H e to bo th O C A behav io r tu ld quant t in l mechaniv~. En t ropy . in fo rmat ion and contex tua l cons ide ra t ions are discussed in Sect ion 5. where it i,, d l o ~ n that no inft+rmation is ava i lab le for a classical par t ic le in :l I+ox x~hilc the confinL'd O ( ' A exhibi ts a intr insic qt lantuin po ten t i a l i ty in being d i sposed to hc at cer ta in posi t i tms iu the 1-D box ra ther tll;.tn tethers. It is here that a ~avc - l iku picture scctns to c lnergc llt~ln the Q C A Ioca l i za t ions .

Fhc Q ( ' A p roduces some pecul ia r non- local cf lects which can hc x ic~cct as e i ther bum,_, p roscopic in t ime ~r panspa t ia l and these are discussed i l l Sect ion ~. Atteml~ts 1{, ident i lx t.hc O ( ' A with e l e m e n t a r y par t ic les arc dcxchq~cd in Sect ion 7 Sccti~m ,~ then c~msidcrx St)lllC al{crn{llC a n d e~.cn t l l lOr l l lodox \Jew,,, tff qUallltn11 n l e c h t l n i c s b a s e d tln nlul t iple worh l , ;lnd h idden \ a r i a b l c s \~hich a p p c a r under :l ne~ perspec t ive when the O ( ~ \ is cons ide red

Sect ion 9 p roposes cl ncw Q C A picture as an :.licl to unde r s t and ing the CopenhagcI~ i .qtcrpretat ion and such fea tures as m e a s u r e m e n t , co l laps ing wa~e funct ions, non- loca l i t \ and intr insic pa r t i c l e /wa ' , c fea tu res are d e v e l o p e d in this light. The p a p e r conc ludes ii~ Sect ion 1(1 with a case being made for the Q C A process as being respons ib le for some ~i the subt le quan tum effects which ha~c pers is ted in concep tua l qt lantt lnl mechanics . Nc~ areas nf inves t iga t ion and some p r o p o s e d expe r imen t s are ' ,uggestcd in this c tmcludinu sect ion based tm the f ledgling not ions d e v e l o p e d in this paper .

2. THE QUANTUM CELIJILAR AITT(}rMAT(}N {QCA} DEFINED

Cel lu la r a u t o m a t a tire s imple n la thcmat i ca l mode l s of dynamica l sys tems consis t ing ~1 ~l lat t ice of ident ica l , d iscre te sites each tak ing a finite set of in teger wllucs and evolving r e e l d iscrete t ime s teps accord ing to a de te rmin i s t i c rule [4]. As ou t l ined in the prev ious papul [31 we choose a pa r t i cu la r ce l lu lar a u t o m a t o n , he r ea f t e r cal led the (..)CA for conven ience . which is conf ined in a 1-D box of length L = Nx<,. and obe y ing the rule

A ( r , , ~ + 1) = A { r - 1 , , ~ ) -r- A ( r + 1. s ) m o d 2 i i i

~ h e r c A ( r . s) is the value of cell r tit lhnc step s. N and r are posi t ive in tegers with N g rea te r than or equal to 2 and s is u non-nega t ive integer . We now apply a b o u n d a r \ condi t ion to confine the Q C A within the box such that

,4{1. , f l) -- ,-t(2, ~t a n d .q(,X'. s + I) - A{, \ ' - 1..~). (2

O u r initial seed ing cond i t ion , which is a rb i t r a ry for the presen t , will he

. 4 { i n t [ ( N + 11,,'2]. t~', - 1. !~

where int [ ] is the in teger funct ion with all o t h e r cells ass igned a value of It. Fhe Q C A i te ra t ion rule (1) is essent ia l ly the s implest form which descr ibes a p robab i l i ty

a m p l i t u d e diffusion [5]. A ( r . ~ + 1)l is i n t e rp re t ed as a t ransi t ion amp l i t ude and hence A { r . ~ ~ 1)12 is the Q C A ' s nans i t i on p robab i l i t y from cells ( r - i..v) and (t + 1.~t t~

( r . ~ l) . An impor t an t fea ture of ( l ) is th:4l it mode l s a local diffusion, thai is. lhc

Quantum cellular automaton 441

transition amplitude at ( r , s + 1) is determined only by transition amplitudes within the backward light cone of that cell (the Q C A moves at the speed of light c). The dynamics of the Q C A diffusion are inherently local. The chosen boundary condition (2) represents an infinite potential wall at each box extremity which serves as a perfectly absorbing probability reservoir. The boundaries dissipate the Q C A diffusion and so act as a decoherence mechanism for the local phase correlation of the Q C A state vector [6]. The initial condition (3) represents an initial seeding or "localization" of the QCA. Alterations in this condition to generate different seeding configurations and subsequent developments will be discussed later.

The dimensions of the space- t ime cells are determined from the quantum diffusion equation

(Ax) 2 = ( h / m ) A t (4)

where h i m is the diffusion constant (h is Planck's constant and m is the mass of the diffusing particle). Since the Q C A is a relativistic particle we apply the condition implied by the Dirac equation such the Xo = Cro, and for convenience we borrow on previous work [7] to set xo = h/me, the Compton wavelength of the part icle/automaton, and this is the spatial extent of each cell. The temporal duration of each cell is then

ro = h / m c 2. (5)

The width of a given box is then Nxo and the time intervals surge forward in units of r,,. A-values of 1 in each box can be viewed as virtual probabilities since they indicate in

what cells the Q C A may be located. This would not represent the relative likelihood of a given cell being occupied since no normalization is at tempted. The possible values of 1 and 0 are of course consistent with fermion localizations. The above development is just the diffusing Pascal Triangle in mod 2 and the diffusion is constrained at the boundary by a chosen condition (2) which must be considered ad hoc for the present.

3. QCA EXAMPLES

Figure 1 shows a typical Q C A for the case of N = 10. We see that it returns to the original cell with the same A-values after 62 steps. We define this recurrence time as k = As - 62. From an examination of Fig. 1 we find that it tries out every possibility or arrangement of ls and (Is as it develops its recurrence cycle. In Fig. 1 values of 0 in cells are neglected with only the 1-values listed.

From considering possible spa t io - tempora l paths over a single recurrence a remarkable fact emerges. There is one and only one continuous swath in space and time f rom s = ! to 62. Some shorter space- t ime swaths seem promising for several time iterations (see dashed lines of Fig. 1), but they collapse before a single recurrence period is reached. The spa t io - tempora l development is hereafter described as a swath rather that a path in space and time due to the natural girth Xo and temporal surge to. Paths usually designate a series of successive points on a trajectory and so are not an accurate desciption of QCA progression. Selected groups of these shorter time swaths are shown with dashed lines while the unique space- t ime swath is represented with a solid line. It is this unique space- t ime feature which markedly strengthens the case for such a cellular automaton as that defined by (1) and (2) emulating quantum space- t ime behavior for a Dirac particle. In performing a path or swath integral, as prescribed by Feynman [8], we see that all swaths give null probabilities, save the unique swath marked in Fig. 1 with the solid line. Thus

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Fhc ,\' - 1() case is one' ()I ihosu in ~ h i c h al l t~ossible ct )n lh i i la t i~ ins , / i Is and lib ~.il-U l l lc ' t i | ) t i t ]~V t l lU ( J C ' , ' \ I~cf lCl i 'e h Ic '¢LII 'S. x \ n \ i n i t i a l ~c'c'clin 7 w i l l t -)r()dt ict_, i h c i d e n t i c a l l C c ' l i l M t + l ~


time and this independence of initial seeding feature is found to be valid for even N-valued box widths with few exceptions. It is noticeable from looking at the unique swath shown in Fig. 1 that certain symmetries can be considered vis-a-vis parity and time reversal or time phasing. At s = 3 l - -ha l f the recursion t ime-- there is a parity reversal. Every cell in the box is occupied, some more than others, and a hint of possible wave-like features can be noted strictly from cell occurrence frequency notions.

Several other even N-valued box widths do not exhibit the space-t ime features evident in the N = 10 case and for sizable values of N a rich variety of behaviors can be produced. In all even N-valued cases the allowable space-t ime swath for a recurrence cycle with a single seed is unique. All other possible swaths quench after some time. And the QCA's unique swath does recur at the initially seeded cell with a recurrence time, or k-value, independent of the initial seeding site.

For odd N-values the initial seeding choice will produce a considerable variety of possible behaviors and a selected box width of N = 11 is chosen to illustrate this. Figure 2 shows how a QCA develops in the case where N - 11. Here we see a bifurcated or diplodic recurrrence with k = 4 and the initial central seeding specified by (3). In the bifurcated recurrence two space-t ime swaths are continuous as shown in Fig. 2(a) and the QCA, although periodic in time with a unique k-value, never returns to the initial siting. However, as we change the initial seeding we see a variegated spatio-temporal behavior as shown in Fig. 2(b)-(d) . A total of 8 of the 11 possible initial seedings produce unique spat io- temporal swaths while just 3 of the 11 produce a variegated bifurcated recurrence. As can also be noted from looking at Fig. 2 only the initial seeding with r (0) = 6 produces recursion with k = 4. All other 10 possible initial seedings double this recursion time to k = 8 whether they are bifurcated or not.

The N = 11 case represents typical spat io-temporal behavior for odd N-valued box widths. Since the four cases represented by Fig. 2(a)-(d) exhaust all the possibilities for the N = 11 case with a single initial seeding, it is worthwhile to note the following:

(1) Eight of the eleven possible initial seedings produce unique space-t ime swaths, all of which have k = 8.

(2) Three of the eleven possible initial seedings produce diplodic space-t ime swaths and these occur for initial seedings in cells adjacent to the central r = 6 cell.

(3) Of the three bifurcated space-t ime swaths only the centrally seeded r = 6 cell produces the rapidly recurring k = 4 case. The two others have the more character- istic k = 8 recurrence.

(4) A diverse parity and time phasing behaviour can be noted in the four distinct cases

shown in Fig. 2. (5) In Fig. 2(a) and (c) the Q C A never enters the central three space cells, r - 5, 6,

or 7. (6) Only in (c) does the QCA ever reside at r = 1 and 11.

Several other features could be noted in Fig. 2 but for the present we consider only the ones itemized above, since they will be used in developing some quantum/automaton correspondences in the subsequent sections of this paper.

Figures 1 and 2 are illustrative examples of typical QCA behavior for both even and odd N-valued 1-D box dimensions. From these two we can further generalize spatio- temporal behavior. It is the symmetry considerations which cause the initial seedings in the N = 11 case to produce the highly variegated space-t ime symmetries, cell occurrence frequencies and recursion times. A slight change in a boundary dimension, for example changing from N = 11 to 10, could produce a marked change in behavior of a particle/ automaton in a typical 1-D box. One could envisage a case where the N = 10 box is

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a) r ( 0 ) = 6 b} r (0 )=3 or 9 k = 4 k = 8

= I ( )

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c) r (0 )=5 or 7 d) r (O)=l or 2,4,8,10,11 k = ~ k = 8

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" ! t l t C ~ l t C d O11,:.:.

-:it..urc ~ ~how,, recurrence tdnle~ tor ,,eeciing~ ~.pecitied in (3) for box wichhs ranging lp:~lll i !o "q) The mi l ia l ,,eeding has no effect on recurrence tmlw, for ';he even i~ox ,aldth,,

~q~i ~n ~hc ca',e of the odd box sizes recurrence t imes o f k and 2k are admissable x~ith the ~ " , . :,-+,aiue ;dw, avs being val id for ccntrai seeding. From a cursory analysis of the rusuit~, !~ ,~,, in Fitzs ~ 3 certain ~encralizalu.m', can he made on the behav io r t+f the O ( A .

: r ' -~ ~!: all. nil <.)CA'~ ;tre c~lher I~he, s ! <)r ('his,, 2 ce l lu lar au tomata usin£ the Wn l f ram


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e.,

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7.501

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.301,~ 2

m

m

1--quenching Q C A

0- -e rgod ie Q C A

A--luminal QCA

14 26

Box Width--N

I

38 50

Fig. 3. Graph of recursion times, k-values, as a function of box width N, for the QCA. Solid boxes show quenching QCAs. Open circles indicate ergodic N-values while triangles indicate luminal QCAs with low recursion

values.

classification scheme [4]. The Class 1 cases all go to zero and are quenched. Here zero can be viewed as a "grea t a t t rac tor" . So for N-values of 3, 7, 15, 31, 63 . . . . (2" - 1) (where n is a positive i n t ege r - - a series which incidentally includes all Mersenne primes when n is a prime number) the Q C A is quenched as soon as the A-value first encounters the edge of the box. Changing the initial seedings does not r emove the quenching but extends the quenching period until both virtual probabilit ies encounte r the box edge.

All the Q C A s which are not Class 1 fall into the Class 2 category. Here they all recur with characterist ic p e r i o d s - - k - v a l u e s - - s h o w n in Fig. 3. However , as a glance at Fig. 3 shows, some of the N-va lued box widths experience sizable recurrence times whereas some others recur ra ther quickly. For the sake of analysis and interest we have fur ther bracketed these Class 2 au tomata into sub-orders. These are:

Class 2--ergodic: For N-values of 2, 6, 10, 12, 18, 22, 28, 46 . . . the Q C A tries out all possible a r rangements of ls and 0s as it calculates its spa t i o - t empora l trajectory. For the sake of discussion this order will be described as ergodic since all possibilities are tried out. Actual ly, the recurrence time can be predicted by the formula k = 2 (2 '~/2 - 1 ) = ~ for these specific orders a l though at present no simple predictive formula indicates at which N-values the Q C A will be ergodic (92 is the number of possible configurat ions or number of accessible states for the Q C A ) .

Class 2--bifurcated or diplodic: Bifurcated recurrences occur for all the odd integer values centered initially. As N-values increase it is found that only the center initial seeding produces bifurcated swaths. Some of these move at luminal velocities and recur extremely rapidly even though they never re turn to the initial seeded cell. Figure 2(a) is an example of such a luminal bifurcated recurrence. Examples of these can be seen in Fig. 2 where (a) and (c) show such bifurcated recurrences.

Class 2 - - l u m i n a l : By luminal we mean traveling at a net macro-speed equivalent to the speed of light and it is these Q C A which recur most rapidly with the lowest possible k-values. In Fig. 2 we can see readily that (a) and (d) are luminal by this classification

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Fig. 5. Plol el anuther selected section of the QCA swath for the N-S01i box width. This is a "near-chaotic'" section and few fractal-like pattcrns appear as in Fig. 4. Fhe unique haplodic swath is marked with the dark line

and again about 875 time surges are displayed in the plot.

cells. There is no birth rule in the case of the Q C A . The Game of Life is actually a 2-D game wherc the picture and popula t ion chanoes with discrete t ime intervals. The Q C A is a sarvival game with one spatial d imens ion and the adjacent squares coming from the previous genera t ion and the subsequent one so that three genera t ions are presented in each hox. Ul t inmtelv , as shown in Figs l and 2, only one swath in space and time survives for tile even N-va lued boxes and the bulk of the odd N-va lued boxes over an entire r,2currence cycle.

What is interest ing about the Q C A when viewed as a survival game similar to Lit'e is that it admits two genera t ion survivors. The survival fate of the 1 is to be displayed two r<, intervals ahead in time. Migrat ion or leaving the site pat terns can also be discerned two genera t ions ahead.

The Q C A has a near soli ton-like behavior as no ted already [10] in filter cellular au tomata . Actual ly , as the diffusing Q C A hits the bounda ry a reflected wave, which is equivalent to the probabi l i ty absorpt ion , passes through subsequent incoming waves very much as soli tons pass through each other. Much previous work on pari ty au tomata [ l l ] seems to indicate similar pa t terns to that of the Q C A in the cases considered here.

x 4 s i ) \:~ l~ , i / ,> i <,i k ,'~ a l

4. C ( ) M P A T I B I I . I T Y ~%ITH ( ) R T H O I ) O X Q M

In this sec t ion we cons ider three aspects (~t o r t h o d o x Q M where we ma'. appl~, !I,. s t ~ n l d g l r d fo rmal i sm and concepts to the O ( ' A . First . the path integral t 'ormallsm c.:ln , , . d i r ec th app l i ed to Q ( ' A s p a c e - t h n c swaths. F e y n m a n and Hibbs discuss the i -[) !)i~:i, equa t ion and fo rmula te path in tegrals over space t ime paths s imilar to those el Ihw O( \ IS I. The t rans i t ion p robab i l i t y P for a space- - t ime path from initial point .v(i) to final p~+ln~

( l } is

f ' l . v ( f ! . . v ( i ) I = Bl_v(S), v(i}l: ~xhurc t.~l_x(f)..v(i)] is the t rans i t ion amp l i t ude for that same path . For a spacu t ime {,:11~ c o m p o s e d of ,S" d iscre te s teps of finite t e m p o r a l length the t rans i t ion amp l i t ude tg ~,- : par t ic le m o v m g from .v(i) to v ( ! ) is given I~x

/ ~ l . v ( l ) . . , ( i l l - [lIB(., + 1. ,)J

x~i~cic H( ~ -+- l, s ) is the t rans i t ion ampli tucte of the (x + 1 }st s tep and the ,,cries !>r,.>duv c J i ' , l ~ l k c n f r o l l l .',' () [c, <S' - 1.

The t rans i t ion a m p l i t u d e for each s tep is the suln of the con t r ibu t ions from ct~c~: p~,ss~i,i, p;.llh in ih~lt s tep. Thus

tT (v + l . ~) E A ( , ' , ' )

> h e r e A ! n ) is the single s tep t rans i t ion amp l i t ude for the tzth path t rom v t~, , - :m i lhc ,,urn is t aken over all such poss ible paths . To any O C A pos i t ion (.v, , ,-~ i} then_ ~;!<. I x ~ and onh ' two poss ible pa ths , one f rom ( . t - - 1 . . ~ ) and one from (.t ~ I. , } 9,,. max in tc rp re t .4(.v.~} as the total s tep t rans i t ion amp l i t ude from (.v i . , i} .~n,I {.t +- 1. s - 1). Hence

f ~ ( ~ - ~ - 1. ~i - [ A ( . v - i . ~) . A ( . . ~- 1. v) i = A ( ~ . ~ ~ - i l . "

r ind the prohal~il i t~, P but,,vcen in i t ia l and f ina l sl;.itu,, is

1-herc lorc , the t rans i t i t )n a n l p l i l u d e l r t )nl t (.'l It> . t l / I ~ i l l I~c i l t i n -L t : r t ) i t ~iilcl ctnl,, II ti l l ~,i the t tq l l lS in the series p r o d n c l aru l l t ) l l -zero. Th is is t rue t m h fo r <i ' ,~'aih u f tuinp~>rall:+ '.,ClCevtssive Is: such ci swath will hcl\c ci iransit icm p robab i l i t y o i un i t \ , l ) c~e lop ing ()(W\ space t ime swaths by connec t ing ls is ct)n~,istuilt with the path intcgrnl formuln[it ln ' , f i

OM. ,*\ccording to tht.' Born v ie~ o i q t iantunl nleeh~inics the sq t ia lcd mt)dlllus el the ,,l~l,.

~ector is physical ly i n t e rp re i ah l e as a prohahilit .x dens i t \ .

t' - ~i,::(x. ~)~¢,(.t. ~! - tl,: , i .

~ h c t e t / ,*( . t ,+) is the c o m p l e x con juga te e l ! l , ( t . +). For ihe p~trticlc in it i - l ) i+t+x ,+!

h :ng lh ,/

I / ' ( . t ) -- ( 2 L l i : s J i l ( s l * 7 . t L } + ', +

~ h c l c #~ is the energy level quanitnl~ nu ,nber V'~c have neglcclecl lhe t empora l part el (h~ , ,olulion in (13) for conven ience sinc<, ii is iusi gin osci l la t ing contril~uti~ln. Thcl-eh)ic

q,:::(t I~i,{~ ~ ( 2 l . ) s m : l # ~ t 1. ) ' i :


We can divide the box up into intervals and approximate the probability of finding the particle in each interval by calculating

P d x = [~p*(x)~(x)] dx (15)

where dx is the width of the interval and the value of x used is the midpoint of the interval. Hence, if the box has a length L = 10x,,, then we can divide it into 10 intervals of unit length and calculate the probability of finding the particle in each interval by

1 3 successively substituting x = ~xo, 5xo, etc., into equation (15). We find that the probability density tends to give a distribution corresponding to the odd energy quantum numbers tr = 1, 3, 5, etc., but fails to give any correspondence to even energy quantum numbers for reasons not clearly understood at present.

We now consider the uncertainty relations

dx dp >~ t7/2 (16)

and

dE dt > t7,/2. (17)

derivable from a QCA space-t ime swath. Consider a dynamical These relations are measurement of the QCAs" momentum by "'localizing" it in a particular cell at some time step. At that step there is an uncertainty as to the location of the QCA, dx = xo, and there is an uncertainty as to the QCA momentum since within that time step it is not known if it is moving to the left or to the right (this can only be determined by looking at subsequent time steps). The uncertainty in momentum can be determined from de Broglie's momentum-wavelength relation

p = h /~ . ( i s )

The uncertainty relation (16) is derivable bv requiring that the minimum uncertainty in momentum would occur when the box width xo is equivalent to the smallest half wavelength allowable. The uncertainty in time during a dynamical measurement of the QCA's energy will be d t = to. The uncertainty in energy (17) is derivable from the relativistic energy equation

E = [ ( p c ) 2 q- (t71(.:2)2112 (19)

and substituting p = h/)~, dp >~ h/2xo, and xo = h/mc. Therefore, the uncertainty relations are natural consequences of the QCA space-t ime

swath.

5. ENTROPY, INFORMATION AND QCA CONTEXTUAL POTENTIALITY

A basic feature of a classical thermodynamic system is that all possible states are equally accessible and thus the system is equally likely to be found in each state. Each state contributes equally to the total system entropy

S = k In ~ (20)

where k is Boltzmann's constant and £2 the number of accessible states. The occupation frequency distributions F(x) for N = 10 is displayed in Fig. 6 and it shows the typical features of quantum wave-like behavior when the unique swath is considered. Certain cells are more likely than others to be occupied by the QCA when considered over a single recurrence cycle. Each cell does not contribute equally to the total entropy. Figure 6 also

C..,

e" C~

C~

r- •

1

I !

t

!()i

,5

( ' l a s s i c u l P r o l ) a b i l i t y

0 - 0 - . . . . ~ - 0 - - 0 . . . . . . . 4 t . - - - O - - - - - - ~ . . . . O , - - - - - 0

/ .~" ~ v ¸

0 "

( ~ u a n t u t n P o t e n t i a l i | y - - X i x ~

( e l l P o s i | i o n N = I ( )

- [ ~ l l i l ~ i r ~ L ¢ , ! ~ 1 1 ) ) ~ . . L , t [ } ~ ~ , ',:, ; 1 . ~ ; ; i t i [ [ ~ \',~. I ] ) : : , , , . L , [ i i T L I ) ~

) ..,,-

25 -Z

a7

di',l"lcl~>, lhL: Ci:.l,,',l~.;cll tlt_'Ullp,Allt, lll li't. 'ulLl,.:nL:x,, c ' t~tll l l i l l t2, ;Ill l h c ~ i ' , I~ ',;;lu'Jl , 't. '![ i~ \ c ' i ~ ~,11]~i

~,.:c,'.t:vum_c , ' , t i c : t ~ [b.~. \ !il ~:b..," f :~,.'i~, cel l ]-. C¢ltl~llx c~q,: ,bh ' ,,I i~um? ,wct l? ic~l ' ,x{i.:~

~'(, ) 1] ~i dL? I'C(.l { l \ k ' ] I ! ] t : ' 111 i I~ ' C'\ CiC' ,~" \ \ t Silittz !,"~ L X I ) L ' ( ' i C L I , ~ ic] i ,1111 ii~,[IX( . . . . ! ! ! : '

It Ix m~liccd'~]c i!'t F i g ~, Ill:.tt lilt ',,.: ~cct lp~l t l (m lFcClUCllc ~ di' ,IFIJ)l.lll(~llS tl/clic~llt. ' th;,! !i/~ ( ) ( ~ A bchc~\c ' , el', it it " ' Ice!. , + ~ l i'. guilt_loci bk ~! guc~si-F~ott:tltictl ( l c c , d l tlutt t]lc !~;~Flich" - h-cc xxith t iw cl:~s'.ic:ll i+otct+ti:d t : - ( ) m~,idc t h e b<+\) ~hct+ ~'c~icitl:ttil~g tt., -.i+;t~', +its+ -,,.vltths cmd h c l l c c il~ c t c t c~miml~g ~hct t '.l:ttc". il ~xill t,ccut-~ > Ih i ' . ,ittc+,,l-p,~tct)th+i \ C . +. ~,clch ~'h:~H t i i " , l l i l~ [ i l i~ ) l l ]'~CH~.'- ill cel l ' , ) 'CCtIII~'IIL'C:' , " , " ,h ich C()IFL",I)I~II t . i 1<1 LIULI",I-[')(~I~,'ilII:IJ "~', +I

Q u a n t u m cellular a u t o m a t o n 451

and vice versa and can be wr i t ten conven ien t ly as

X(x) = 1/F(x). (21)

A graph of F(x) for N = 10 is also inc luded in Fig. 7. The above equa t ion is a choice which would def ine such a po ten t i a l , its main fea tu re be ing tha t it would have m a x i m a where F(x) would have min ima and vice versa. A high p robab i l i t y of occur rence would

1 1 5

1 1 1

1 1 10 ~. 1 1

1 1 1

"" C 1 ] 1 ~ %/:1 1

1 1

~ 1 1 1 20

1 1 1 1 1 1

1 25 1 1

1 1 1 $ 1 1 1 30

1 1 1 1

1 1 1 1 1 i 35 11%>__

1 1 1 40

1 1 1 1 11[ 1

1 1

1 45 1

1 1 1

1 1 1 50 1 1 1

1 .

i i 55

1 1 1

1 60 1

1 1 62

1 I 1

L 1 1 70

L = l l x o ~-,

©

©

~z II

s = O

Fig. 7. Plot of Q C A surging in t ime for a box which is r a n d o m l y var ied b e t w e e n N = 10 and 1 [. The unique swath devia tes f rom the normal N = lfl case at s" = 19 but the ex t ended b o u n d a r y is first e n c o u n t e r e d by a 1 on the right

at .s = 14. T he reverse light cone is shown and the Q C A swath stays away f rom the right bounda ry .

4c~ 1 ) ~,~.' f:ii I,>1 <,1 I.. c'+' ~U'

i11dicatu ;! t+t~lcnt iul \~el l ~llltt tt~e i11(+1-c I i k c l \ the o c c u r r e n c e the d e e p e r the q u a s i - p o t e t l i l a i

~ c l l . \~"c II1~_t\ \ ~. r i l l : 21) cls

I, ii~ .V(.t ) . . . . k In / : ( .e l t - e 2

\<here k is 13oltztnunn'<+ cons tan t . Nov , X+ It+ t ' ( . t ) !+eprcscnts t i le t t t t i l l cnt tOl+ ), d i 'qr i t+utet+ ~+\~.'i- t t lc b o x ~_lntl thti~; rc'\ecll,, the t_ ' l l t r t t l3\ ' c t m t r i h u t i t m h\ ' each ,+,t:.llc ~tl+cl hence

/ , t i l . \ ( t l ; 7 ( . t ) - / ( . e l ] ' .

\+,here ] ( t i i', the in fo r t l l a t i t+ l ; <lr nu : cn t r t t l + ) l ' h i ' , t_tua~,i-l+otunti;_il . \ ( . t } di~,clo<,c ', the in lorn l ; . t t i<) ! l c o n t e n t ~>l tht-' (<)( .+\+<- c t l l l i i i l t , c+

diffu~,i+.+n. ,<\', t ha t d i f i u s i o r l i', dcpcndt-u+t t l l ; , t in the OC_'iV'> ph) 'dcc l l c . ' l l \ i l tHl lYlCl l t +w id th <+1 t+~t'~x a n d I+ti!4ndai+v c tmd i t i on> ; ) . \ \u ,,hMI t i l l thi~; ClUa,',i-t)Olel~ti~ll the ( ) ( . , \ ct+nlu×! i ict l t+otcnt ial i t_\+ F(.t-) rc\c+.tl <, the possit+ilitie~ ,>i the ( ) (_ ' i \ in ;i prt)t+al++ilistic ,,,,u\ \ i~l t i le , ,p~. icc- t in lc <,v,,~.tth; .\ '( . t i cli~,closc,, the po t cn t i ; ! l f o r the Q C ' \ tt+ ;iccu>,~, lhcl~,c pt>>,sil~ilitic:- ~tlltl hcr lcc charaeter izc~, tilt+, i i+ f t+rmat i t )n c r e a t e d i+\ the i n t e r a c t i o n o t I l l c O ( . \ cliltt_l,+i<+n ~ i t h t h e e l l \ i t ' t ) l l l l l t t l l l . O n l \ t.lt_lCllltLlll/ ~lllCl O ( ' k \ ~V'+,tc1115, c o n f i n e d in the ill; i i lnc_'l cli~>cti-,xcd c_Xllltain i n f o r l l l a l i O l l . T h e cl~lssicul t~t'CtillellCC" l r0qt lC' l lC)CC)l l [a i ! l>, lus~, iillc)rnl+.lti~Ul< t \ l luru; t , . l hc O ( ' A tllliCltlt-' l~ath t lc~C/l l !et lcu l tut_l t ie l lc ) C~llt~iil+l ", in i~ortn~it iot ! s in l i h ! r tt~ their ~+4iilcci l i o m tL?,in 7 tilt_" ' , \ave lU l lC t i o l l a i l d the c l t i t i n l t i i l l c~ i l cu l a t i ona l pr t )cudt t ru ' , .

6. N ' r R A N ( ; I { :liND NI)N-I.O~_+.-%I. QtYA E F F E t " I . S

%inct-' t h e B e l l i n c q u u l i t 5 v,;l~+ l+i-++l+~+,+c<.l i i2[ ~md +t~+ u x p u r i m t - ' n t ~ i l \ut+iti,.:utit+n +~+ t h e

, . !xpet+i l i lel / ts t l l Asj-~ect j l3J nu t he~ t l e t i ca l uxpl;4t1~.ltittllS can c l M m ~m\ qtl;illtU.I11 ci>t!-us['~tl i ld t+'11C~.t t l l I l c s s it ~Id111its it+ tilt_" p1"C~+t_'11Ut. ' t t J 110!l-II+c~.i l i111-']tlellCe +++, ill t+t)ie+.;tlllg lh~.' ~+lit,.Tt+111~. + . i

quan tuYn ',+,,tel+]',. l h c ( ~ ) ( ' + \ d r i e s [1FOdl.lk'L' ~tt l l lC el ' feet <, \~hu.'h u~ul hC c[ii+li~lctutiZ,+_'c[ .~'- mt l l - Ioc ; . l l , t;+tlJ+,'cI' lt l l l l i l l~t]. I->rttscitpic ,~i l++ln'+,l+t~lti'<tl, d e p c n d i n 4 on ;el ,+[+',ur\cr .+- I'tCH+",p~++cli ~':

F i 4 u r u T i l lu 'q ra t+ ,+ a n t+ l l -h t ca l t e l - , U l + c r l u n l i n a l eft~.tct. F h u riuh+, i + o u n d u r \ in Fi<_' r .~+.

Cll~.tl1~t.'d ~+i~i ~i b i l l a r \ tallcltu11 Kll+1111hc'1 ~Ctlt+'!-~ttor i'!,.H~£1 I f )× , tit If>,. l h N , q l o u i d c,+r!+cH~,>n~i

!o pl1.vsical ~It i_l~.I t iOlls "+.illC<._' i+t+tllit_l~.11i,..:\ ~\~,ul,.+ \ e l \ i11 r e a l >,v<+,t t_+nlt ,+ m ~ ~ ~ + t I + ~ + U ~ ~+ ~? t ~ t ~ + ~ ~ ~ ~

cl+111pLlri!1~ [: ,g. ? w i t h F ig . !. tht-' t i l rq t t l l lC t i l t ' i++,+tillcl',It-.v cll~_111~t_'tl '.<+ UH'cCt tilt+' C)( "..\ <+~+;t', +I

, =; 14 ~ind t h e l i t s t lit11!+' t i l e uniclUU .+~;ith d t ' \ i ~ i t c d ir+.tnl tilt: pu t -c \ ell ~,l'<ttu +~.;t- i+

+ = i t} l l t t+\<+' \u t . ;it + : : l u ! h e u n i H u c -,~+~Li~'h <,\~i? in t h e r = d c~,j] "~.JliJt: !i+ic_ ' ++i l~+l ! l l ; t t l~+ ,

',',hi<.'l'i lee!+<_luted thi~< clc'~,i;.ition ~.~,~i <~ +!! tJlc' ]'+ cu l l . [ h t - ' l ! lhTi! t i la t i lql e l flit+' t~tt)ttllti~ti.

~:ttcct r e a c h e d the t.Hliqtic n',,.;tth ,+t lpc'rJutt l l t laJi\ its can I++ '-+cot! b~. <ha\~,i!i~ ihu +c~t+tt-<,u ++ui++ . t m u in t-i<._,. ~ Y!tc: i11t t+rni l l t i tm \~J-lic_'h ~Jl;~iti~ud t h e unicl t . lc <++\~, i th i+c-+ t+tlt~.idc tilt+' i-~. - ,r < i x:

li<_,ht cc!l~t-' ~lnd th is is ~,ho~\!+ tilst+ in ! t+, -

P+,,+,,u< pie uH+:ut?, ~il~<o <,uutn it+ hc :, +qll)t>t'l+t\+ ,d i hc (_J(' , \ ;is il ',tie-urn,• lt+f ',ti,, +\ +~1 i l l ,+i~<it_c. ; l i ld t i m e . FiTul+c ;4 >,ht)\~<,, +rich it1! Cl'I 'CCI. H u r t . <_ind i l l y, ul','.,,.:CttleiH kii~,t;u'.,~,ltin ',- iut>~,t+,>!>i~ DleCi l lS 'kt_ 'c ' i l l< 4 ahccid m t i n i c " +i', dclm<_'d !~x Riutdilk i141 \~ ; i l l u \urc i t~ t i!! l l l l l t ic-ncin,4 ()(<-\~ tl[liClLiU s\~<ath ++, the \ - 1(I. i+t+x ',tcik unlai-<~t_'d., l'tir. '<t ,dn~lc. t i the' ~ i l tc i t : iJ tit + + lf~ ,..i,+ the t r ; l c in 7 t ) l the un ic tuu ,,+~it l l ~<\u ',uc' ci tt+t~il legit+it',< ' c h a n ~ c w h e n ~,.+_. c_x+111[+~.li+c! [--+<~ "- ~.~tt+! t. i ,4 l+ Fhu UlliClUU ~,v,'ath SU_ltts iron+, lt+ic i+ight i l l F ig . ,,4 "+klltl~'i117"" ~)r ~c'i l~',int D th<~t ~,i+lllt. ' ]= +;i!Yic i i l l e r \ ; l l , , ahc;. id it ~ i l l tn~+xu thrt+ti<,t+ the cxlundt_'cl t ~ounda r \ it -,c.t_'t+/iii~l\ ~ntv.\,, , , ,unc ~t) i n l o r \ a l <, ahe;. id tha t i l il t;tko!, ~t- +i,,ual \ + + I I I ro t ! to in i t i~ l l i v it w i l l hc qttut+lchud il

-; " t )

i t tia~, been >+uggc'stud Iw t t i c t d i j k ii4! t ha t . i l l clU+illtu111 i l lec i l~ !n ica i i n t c r4c t i tm , , c t lns t ' , te i i t ~ i t h ,,peci;tt r e h l t i x i t \ ' arc !+l+ti~,eopic ln t i !hi,, ~+,eci11~.. !+t)l-ilo <ltll h+Clc. I.~11.: < i ) (< \ .


4 L= 1 0 x o - - ~

1 % 1 s=O 1

1 1 1 1 i 1

1 1

1 10 1

1 1

1 111111 1 1 ~ 1,5

1 1 1 1 < 1 20

1 1 1 1 1 1 1 25

1 i

1 1 ~ 1 l i 30

1 ~ 1 1 1 11 45

Fig. 8. A Q C A d e v e l o p m e n t where a normal N = ill box is extended to N - 11 at s = I(~. The original N = 10 swath would start on the left (see Fig. l) . The swath acts proscopical ly in that it "'knows" it will have to go to the

right as shown to survive in space and t ime.

relativistic particle and its behavior in this case is consistent with it being a quantum relativistic process. The Q C A is able to use global information in calculating its survival swath into the future.

Actual physical systems could hardly be expected to present an unchanging box for its quantum occupants whether it be the Q C A or a physical particle obeying mechanics. Although the situations used to produce Figs 7 and 8 might appear contrived mathematic- ally it is not unreasonable when physical systems are considered. The box boundaries would have to be produced by Q C A s which behave in a manner similar to the Q C A in the box and a fluctuation of one or more cell sizes could be a normal boundary condition. It is noticeable in Fig. 7 that the fluctuating boundary on the right side caused the Q C A to calculate its unique swath more to the left side and avoid the fragile confinement parameters. The Q C A in this and some other cases used as examples tends to survive in a unique swath removed from happenstance.

Additional Q C A swaths can be created in selected sets of box dimensions. Multiple swaths, subsequently called odes for convenience, can be created by changing the initial seedings. Figure 9 shows a quadruplode swath for N = 11. At n = 8. with proper initial seeding, it is possible to have a triplode, three distinct swaths similar to the four in Fig. 9.

This means that it is possible for a Q C A proceeding in a single swath to multiply into several additional swaths if changing boundary conditions or the introduction of extra

p*--- L = l l x o --i

i

~migu<: ~x,.t i ih <-, ' , ' ,hk 'h , no <,h~'~xx n v, h h dc l l k Ih lc ' -

(){',&<~ hike' , plclcu in the" pt tg+ct col in <it the l+r,+pcr t i t / i t \ hii+th i u i c . + l+t+ii-~ +<~

phrascolo<g', l r o i l l ('t'H1W~l\'k (;dl~'h:' f ' i ]~i]{' IcJl. i~ .~upplaiHod b\ ti h~il~pCn~t~ii~cc cll~ltl~c i i !~titlnclaric_ TM ~il the I i gh t thl/c'. ; \ ~inTIc '~\~,~llh is cM l cd ~i hl l l~loclc. ;i clollt~l¢ "~tt~ilh ~i diph~; i< ;i t r i p le , ~ : ~ i t h ~i l i i i qoc tc i l l \ : ,~, ;l q u ~ l c h u l ~ l o d o ~ll k - i ] :l<, ~,mCll i!i t:i 7. ., c j t i in lut~loclc ~,\~,'cilh ~.1! \ ' 1-4. c tc Ftfi~ II]c_'LIn ~. lh~.ll h ) l i.II1\ in lm~ci <i box of cJilllCll~,ioil <, \ 3<+ - I Ci l l l l l t l { [~c crTocl ic. Fhu . \ ~(}(I (~( ' / \ ,~ ~,ht)WI1 ii/ [ i7~,-4 ;IIIcl D il lc' nt',i, c i~ t l t l ! ,

~iilcl cl~ I1~tl 11\ ~ t l l ~lll p~ l~ ib l c c~Hlli~ti l~ilh>n'~ <!l [', and (I,~ {11 c'~tlc'iil~iliFi~ il!-~ :miguL. '.~11~ ~ilicc' :,l cou ld I~c. t inc lc i the r i gh l ~ccclin<.z c~mcliti~m~. :l 21~7-~c1< -.

7. I~]I,I~]Mi~]NTAR't' P tRTI ( ' i , I~ ] ( ' ( )RRESP( )NI )F ;N( ' I , .

&l ~.i c o n c e p t u a l lc~,c] t i le O ( ' A c a n bL' L:olllt '~arcd \v i th ~ d l r t l d i n ~ u r ~ , l i tHiOl l (11 l i t

clc lncni~lr '~ p~u-iiclc I151. ttc: c h a r a c i c i i z c d clcction<~ not a <, pcl-q~tin 7 i ndh i du~ l i cnt i t ic , - I~!i l~ i lhor ci <, '!o11~ s t l i i lTk t l l s u c c o ~ h u h occup ied ~l~ilc"~'" w h i c h p rod t i cc l hc imt~rc~i~>n <>! ,,~ i d c n t i f h i h l c h~dhic t t ia l ~ im i la r lo the o luoc i s t~l c~¢r \c l~ i \ exper ience ' . \ 1 . ~ . p<>~qtqc, c'ic'c~,r~l '~l~itc"~ t;~ln on l y bc well definod OllCc ,7iw_,n the ph)qcal Cl l~, i ro i l i l lc 'n l t/1 ;_in Cxpor i i l l c ' l l l

In ~i l~'ttor It) L o r c n t z . $ c h r o d i n g c r I1~1 quc~, t ionod Who lhc r o r not ; i l l ¢ l oc l r on ~i~ f i o ld - f r co ~to~.icm ~ o u i d lt_'l~ihl its a l~i l i tv i~> c ' x i~ i ~l~ ',i l~mri l l~incnl c 'n t i l \ . \Vh~ll thi~ li-~lll,,l~ilt:. h~h~ is tha t thE" t~lcctron rcq t i i r cs a physical confincn~cnl ~hl l:iolcls or po lcn l i~ i l s in orctcr i ~ {I l t l t~m able It) occups w o l i - d o f i n u d st~ito', cind hc i l co cx i~ l tl ~, ~tll en t i t y L ikc~vi~c. lhc: possit~lc s~iatOS ~icce~sible to the (.)C,-\. Foprc'4ClltCd bv the ,~i~il~.' ~cct~+r conligl_llCitioli~> ttnd oh~uaclo l - izod by the (..) (_ " r \ "~ c o n i c x t u a l po ton t i~ l l i t b . ::lro c lc lcrmi i '1od b \ ii~ diftu<,i~m Ct)lllint~lT~t.'llt zind ~llc wel t cJol:incd cml\ ~vhun cl I~t)Lllld~.ll-V o f st)lilt_" ht l r l ih pt-t_'~,clll, ,,\is{). l t tc ( ) ( ' , . \ ~,pac'o Iin10 ~wath i<4 simply :l lompor~ i l soqLiOilco o f s t icccss i \cJy OCCUpiOc] nc ighl~>i in .~ cel ls.

As or ig inal l_v s l l o ~ n t~\ Schroc l i ngc r . l hc l) i r~tc u q u a t i o n for / i /u re la t i v i s t i c clcct l~>n implies that s u p e r i m p o s e d i_11)Oll tt~¢ eJccl ro i l "h l r~in~l~i lorv m o t i o l l i ~, Cll] i l rOgt l l~ l r circtl l~iF m o t i o n o f i ns tan tancou , , \ c l o c i t x c (Zitwrbewe[.:.ngi w h i c h is i i l t c r p r o l c d zis ~i~in~ i-i~c: !<, the e lec t ron ' s spin i nagne t i c l l lOl l l t? l l t 1171. T h e Q ( ' A si01_lCC-liltlC' stvaths displa_vcd in PiT~ I ] . 4 ~illCJ .~ c l ca rh ' i l l us t ra te th is t0hcllOll/tH1a, F h e Q ( ' A n l i c r o v e h ) c i t v b\ ' design is :_il~c,,~- l hc speed o l l igh t ~tntl th is CCilrcsl3ond~, io the c i gcnva lues o l the ~c loc i t v in LIII~ ~Hlc, t l i i - cc l ion in I ) i ic lc '~ m a t r i x l :or l l l l_ l la l ion o1: ClrLtzinttlm 111t, challic,., I I,~ I

F h c b o h a v i o r c l isp laycd b \ t hu O ( V \ ~p~lcc -t inlC ,~w~ttll,~ i', cons i~ tcn l ~ i t h lhc c.lt izit i(~!l q o c h a s t i c il+c'~.lllllc'llt <tl cJt_'tllClll~_tr\ pa t t i c l c~ in l lv.>dcl l l \it_'\>,'~ !i<>l, I | c r c the tcl~l t i~i ' . i :

O u a n t u m cel lu lar a u t o m a t o n 455

electron is considered as governed by a Poisson stochastic process whereby it acts like a two-component neutrino randomly reversing direction by 180 ° and flipping handedness with the rate of such events being interpreted as inertial mass. It has also been suggested that such treatment of all massive particles at the fundamental level may resolve some of the space- t ime difficulties of quantum theory [201 .

The QCA space- t ime swath, in Fig. l for example, clearly shows such reversals of direction and flips in handedness. We can assert that the stochastic process underlying the QCA is a direct consequence of the confinement of its diffusion and arises out of the interaction of that diffusion with the physical environment which naturally precipitatcs the observed randomfirregular behavior.

Particle content, the types of allowable physical particles, is another feature which can be ga~tuged for the QCA based on its spatio-temporal behavior. For example, a luminal QCA would have to correspond to a massless particle since all massive particles must have macroscopic velocities less than the speed of light, c. A particle with a sizable inertial mass would linger longer in a certain region of the 1-D box when compared with a lighter particle which would bounce back and forth from one side to the other at near-luminal net speeds. However , if we equate certain degrees of freedom with spatial dimensions there are constraints relating spin, mass and dimensional freedoms, abbreviated as stud here, which survive in modern formulations of particle theories based on sound physics which are digested in Table 1. These constraints are based on the following relations:

D - 2s, when m = 0 and D = 2s + 1, when m 4- II.

Specific experimental candidates are forwarded in three of the four categories of Table 1 and they range from the massless laminal neutrinos to massive pseudoscalar mesons. These latter particles having spins of 0 do not obey the Dirac equation but succumb to the K le in -Gordon equation [21]. The QCA is derived by analogy with the Dirac equation which admits only fermions of spin ~ in its purest form. The fact that at least half the cells have to be empty at any given time could be viewed as available cells for antimatter states.

Looking over the QCA developments it is evident that massive particles would have to be confined within boxes where L would have to be equal or greater than 6x,,. All smaller boxes would either quench or have luminal QCA and massless particles. The case where N - 6 is the first ergodic and non-luminal Q C A and it has a quenching barrier at N = 7.

The presence of the pseudoscalar mesons in Table 1 is worthwhile when it is considered that all pseudoscalars such a pions, kaons, etc., have microvelocities near the speed of light embedded as they are in the innards of nuclear matter. Pions, being the lightest of these particles, would have primordial importance if the QCA is to represent elementary particles. However , several e lementary particle descriptions and analyses already recognize the primitive nature of pions [22].

What emerges from identifying QCAs with elementary particles is the conviction that

Table 1. Par t ic le con ten t of I -D Q ( ' A

Q C A proper ty ,stuD cons t ra in t Expe r imen t a l cand ida tes

Lumipa~ m - 0, ~ - 1/2, D - 1 Un ique Neu t r inos Bi furca ted None

Sub- luminal m ,~ 0, ,~ = 0, D = i Un ique Pions. kaons , etc.

Pseudosca la r mesons Any pa r t i c l e / an t ipa r t i c l e pa i r Bifurcated

+15<, I ~ \t+ i+,i +, ,t '-,{ {,, ,~' ,+,'

thcl'~_' Lirc \ c r \ lt.",~ i l l l l i l l S i c pat-t ic lc pr t )p t . ' l - l iCx a n d LtJi m c a s u t a l q c p~_iralll¢l.t_'lS ,,uct~ ~l~, tli~.i~,,>.

<q+)in. ~.ii lgUlLir i l l o n l c n l t i i l l , e t c . art_' m ( / n t e x l t i a l ; l h a i i~. d c p c n d (m t h e gh~ba! ,~tir i+t/t i i tci i i l~'~.

l -here m igh t hc no ?,uch t h i n u as l i t t le t ictbits o i I l l ; ittt_' l + t i t t.'llC_'l~+ el l l l lO lnmn l t . l i i l Lll <ili htl! pl+OCeS<,c>, v,h}ch s t i r \ i \ c },~ , < ' i u c t c d s p a c c x ai l , . l l } m c s Lind \~J+ltl~,t. ' >,\'+';ttJ+t~, c ; in I~t.' c~H!<,llt!c'~{ i -

h a \ i n 7 p i~>pc r t i c - Thci+c ..~lc nt, i n t c rna { dt-'ercc~, <~i frc¢c!t) l l l li! ( )(- '+. \ ,part ic le. . . b t i l t.~i.+i\ ~ i i t \ i v i n 7 ,+~i t l~, i i l ,;p+.tt;c v.nd t in !c

S+ ! NI)RI 'H()I ) ( )X QM A N I ) ' I H E O('X ('¢)MP%REI)

{{C}'+,t-'KII+,cT~ cll',i>,{oilcci the xl;li~+" tH ~t ~jl.lLllt. lt.{lll >",+',|Clll {)l}ti! t¢~ {,qCLI~LIICIIIL' II t ~ LI -~'~ )

-\F}'+,I()[ICLII] /)~YlWIIlI(I Lind L,tHu-,ld(_q'ct.i 111,..'Ll%tlFClllt-'lll it", ;! {lilll',il},)li (~[ {ilL' ".\~l<qll [l',Hli ,

pos',iblt- ' tc, ~tp+ a c t u a l statt-" [ 2 .~ { L %}milarl ' , . tht-' ( ) ( , \ 'qult-' \c'ci,+, conf i~ul ; , t i t+t>, ruprc,,t'+++ {x+tcntialiti;_'>+ ~hich arc :ictt,ulizcd ~hcn the ht+tllldCd ,]itftt?,it,li ,_,+llap~,c~ i,~ :+ ~,inek +++

h}fut-catcd pos~,ibi}it> {-+gtlrc ] <+,llt+v,,. tha t ch.irmg tilt_! C' , t+ i t l t i tHl ,~! tilt+' ( ) ( ' . ' \ hi-,t,+r', ! ; ' . , }il t , . r t . '~, t t i l~ tc;itur,_':,

dOt. el(l[-< ({1 b C ~ } l l l l i n E ~,~,+til ;ill ' , { '+;lrt ictl lLtl ] ;! t.llll(_illi. ",'a.;it]1 t+l c'.Hiih:'t't;ii-~Jt_" ] ' , c_';tll i'~c . . . . , <, c~:+

/,,,.~<k;;~lrU in t imu it+ the in i t i a l O C ' . \ {>,+,,ition ; ind (2+ hCgil l l l+IDZ "+",itJl ;lilt+ Ui';+.+I i Xt". t+l . t i

",'-+,:+lh ~, ¢Ltn i,¢+' tr;i,.'(_'t_i /}n-li '~lr+i Ill t H ' n c . I ,+r t-'x;Hl+ip],+_'. I t-q.t l l l l i t l t2 ' I(~ {;lU i . + 'I t .~il l l t l l lL + 11[ , +<

,+Ill', ' .)lit ' >,",\LIT]I llllt"+ I+C ttLtCCc{ J'~;.I,+.'L t , , ', ++ f l t ' , t t ~+ '..YLlt_'il ', ill t i t ; ~on l iuu i+<! t}~m {{ ,+ \ : c , , c+

' , '+C~i l l l l i l l~ LI{ ', : (! l i \ t - ' di>,I l i lc: l ,~k ; i t l l x c';l l l i•c l lLICc't] I t + , [:; "+c\<'r ; i } i<>l~.\~ilc~ l l l l l t ' '.\ '+;it+i-

c+\i>l+ t)\Ci ",C",CILI! ti l l]t_' i l l t C i \ t l l b

[ ] \ t .+rc t t+x " ' i l l / i l l ' + ~ t + i l t l <-'" c+, + "ic'{:.l l i~c_' - . l i l l e ' " i i ! lc_ ' i { - , lc t ; i t i t+I t c+! cll_!;lntti l l+ rllc'c2h~.llllL'> i . ] ~

t+l+t_'~,Clll +-, COll lpLt l ; l t ; l t_" CtH1CCIII +', It+ lhLil+ t i t t i lt. ' Q ) ( " \ ht_'h; i \ i t+F [~ '~c tc t ! p<l '+tt 'J; i tc". ;i tilll'.c:l+'~ ti

",ILII<' \ t_ 'ch)T \vh ich C~t~!\c"- cxH l l i i l L IO t l ' , h Lic',_'t)l~lill~ t~+ i l l o %chi~,di l ;gCl + ~ALi\c.! c.'citiLlllt>l, Lli l , l

llt-'t+cF Ct)iiLIp'-,t_"- int<, ;i p a r t i c u i a i SILIiU \ dc tc rm i l l i , , i t c cxo lu t i+m <+t "+rc.l;lti\c qatc+.. '+ t+,--ut- !c' l l t l t_' l+hlg i l l C i i + t l r h l g i t l ' , l l t i l l l L ' l l t ' - ; i i /ct c x l c r n ; t J td+',t_'l+\c'l+X il lc'7,'~mtll i, i i It+ the i '(~til ltJLtli(+l "- + i

qt l~ i l l lU i i l t hc t+ r~ Thu>- ni l lx+~,>dhh_ ' 'q~ilt- '<, du, ,cr}hcd t+~. I h c t l t l i v c i , q i ! x iL ih: ',~'c't<lt ;ii+c cclu~til,, ! c a l . aJth~+ttgt~ o n l \ (tnt_. ix uccc>,,+il+lc" I++ iliC;i~,ttl-t_'lllt_.lll o! + h t ln l : . l l l c ,+ l> .c i~u>nc, ,+ .

F h c ( ) ( . ' \ l+i~tt>!\ i,~ F lu I h>llo\~', l hc "+IlILIII\ ;+~i}<.l'.'" clc~cripi i<m+ <~ith (+nu c',c+ct+!,,,++ \ ! \'<il'h:,tis+ i i l l lc. ' sic'ill'+. ~ ,'(,. +_.~" ..,~'+ . . . . 17 ++] t i c . ~iJl c'clC~.lSt}ll~< ( . ) ( ' , \ ~,]'+{IC'L' ! i l l l t

' , ~ a t h ' , co lh ip . ' , c t,+ ~l s ing lu ~,\vLI th I h c ( ) ( L X '+;l~.ilL' t+cc+ttlr t_'\t/l'~t-'., d i< ,c rc to l~ xc' l '-, l l l l l¢+tl l i ' ,

~>~+c'; ~,I l lHt l i l l l c i i l t t : i+ \ : t i c b t l t ;It \L l i ' l t ! t i ' . t i n / c x ChLilI~C:-. d l ~ c O l l t i i l t l O t l ' > i \ +)r " coJl;IpSC<, .` ~!lt,+

t i l l iq t ic l+~ o b s c r \ ; I b l c x l ; i t c

I h c pe r i od i c O('A kl'<ltc \ o c ' h > r i c ' c h l c l i o i l ' - oh+ n,+l r c , , u l l l r o i p l i l t i I - i lc rL ic l i ( l i+ ~+i l h c ~1~ '~,

. , l ; i l c \ t - ' chH ' - %\iti~ ~i i l lC ;p , i . l l i l l ~ il!~ti-tlll++c+nl l i l c ' rC t | t lu ' t i t l l l <, HFt_' i l l t i t ;p t . ' l l t l t - ' l l t t+l ,._+Xic'lil;+

, t l+ '< ' i \cr ' . T h e ()C . ':\", v'(HllLit l t_' l l lCnl alit+tt~ i t , t t l t tu ' , }o l ; It+ i n t c r i c r c \~ilt,+ H<.cll ' , ~

cjct-'+.ijlt_'tt-'llt'C I l l t i le ' ] t /Ci l ] [ l t l i l~L' C t l l l c ' i i l l l i H ! (+I l i l t Q)( \ ,,t.;llC t<',+l( l ! k \ h i c i ! I!, ( F i t l l ' q l l l i l c

t l l t+t+uTh It- i t c n t l t t - ` ( ) ( . \ d i i l , i s i < > n 1; l , t l~ i , , , e l l - a c t i o n , ~ h i c h p l C C i p i l L i t c ' , t h e -.tl +:.,cilt c •

i , + ' t h t c l i ( l l l , . [ h t llIC.'C.'IILtI/i~IIt ~++ "~U.IIL \,.'l_'{,+l < + , l l a l + x c i- i n l l c i c l l ! ~A itt+{:'., f l + , t+)e ,

i ',~+unclt-'d cll{lt,+',lt+;t \c . ' t t l { lJ i /d+t l ( ) l ] t i t {he J { ) ( + \ , . , \ x ; i | { l ix lli)+ , i t : , , t l l { t+{ . ' , l i t ,~i<../+i7

i l}{c!{ ic_'{I t , I t (1~ IllL'{I'-+UICqltCI!! !+tit l ; t l h O f i', ;l II:.itUl:_ii ,.'tHI~,C'tItIcllCc' +,1 l h c ( ) { & - . Jl l{t ' !+;It '{If):

\ t i l h t he i r I l+ys ct + C l l \ i l+ t l l l i t l c i l l l hc I'~t+tllld~iric'. lT, ohn~ ,. causal il+tCrl+!-clati~m in tel+Ill <, i l l hicldc + \ar i~ ih lc> iiict+ti,<>i;+tc<., r l , /~- l , lc~l l i tx l ! l t t i

l i l t + t( l i l l i+. lJlSil+ (It (+)~v'| \ i ; t ' t l t l c l n l t l l l l {+,c/icnli ' ,t[ +< ~ki+ici'.. ;it-'l', <lil c l t l i l t t t i l ! + !+;trtici,..+, ~r t i c l d i l i , ) n tt+ l i l t . <i;l,,,<,ic~,} {+,+t+tt.'llliLil [_]5 1 l h u I l lLi thc' i ! lLi t iCLi! h ; l i l ! c>l t i iC ( . I t iL i l l ' t i l l ) p(>Ic_'illi~ii dc ! i \ ' . t b l c { ' t t ) l l l ti~c" ",lLilc' \ c c h + r \~hich i ', i i l l t - ' Ip lCt t - 'c ! ; ix i+c+'!li-<:~c'l~tlll~ , ; i l l ,+hit- 'ct i \cl~ ic~il ii,, i,,] ,~]l(,'.c f iu ld t - 'qu ; i th+n i <, tJ lc v, c i l r ( x l i n g c ~ \\Lt~c ct_}tl:.! l i(Hi. [ lc , h l l l k -,I;ttt_. \ t - 'uhH ~\ ,,~_'c_'ll el', .i ,.~,

lgr(,.<.,lit- L pi},~t ~ l x c ' ~t i~ ich " ' i u c l - ~ l}it-' ph~ ' , i c ; i ! c T l \ l t ~ H l l l l c ' l l t , i !t ic_ h tc~. l l i zcd !~a;+ttclc +~i,+

i i ; l i l ' + l l l i l : - n ( m - l o c n l (~ , i . i pmr lun l inc t l l i l+ l f lucf t<.c, ,< l i i c i x u l i c l c , i u c t,, chan,.'.c,+ ,t~ ~i~.


environment. Bohm defines a quantum potential which provides a quantum force on the particle and this potential is how the particle "feels" its environment. X(x), the quantum potentially defined in Section 5, is virtually identical to Bohm's quantum potential.

In a similar way the Q C A diffusion, although dynamically local, responds non-locally to changes in the boundary by altering the QCA swath for its survival. The local dynamics of the QCA is correlated to the global structure of the Q C A space- t ime history. The QCA diffusion is not an external field effect but rather represents the Q C A spreading itself out to encounter its boundaries and reacting to changes therein so as to maintain its survival surge in its space- t ime swath. Global information is processed by the QCA in calculating its local future.

9. QCA: A MOVE TOWARDS COPENHAGEN?

The influences of Bell's inequality [12] and Aspect 's experiments [13] have undoubtedly revived the conceptual debates in quantum mechanics over the past two decades. Working physicists keep oi1 using the calculational aspects of quantum mechanics while ignoring non-local effects, superluminal signal possibilities, quantum encoding of ultimate information, and a series of other features that seem to be emerging out of reworking foundational quantum concepts. Like any good orthodoxy, the Copenhagen interpretation of quantum mechanics, as it is often called, becomes a secure repository for these novel effects and their explanations by widening its comprehensive sway. Since it encompasses nearly all effects, logics, novel explanations, and suggested experiments it remains an extensive " interpreta t ion" without producing any models or favoring any calculational methodo- logies.

The following are some features where the Q C A adds some insights if it is to prove to be a viable conceptual model:

Measurement. In the Q C A view measurement is a " 'recounting" of the space- t ime swaths, tF(.r, t) proceeds forward in time and ~F*(x, t) proceeds backward in time admitting one unique or haplodic swath as in the case of N = 10, for example. Recounting the swaths can only be performed going backwards with ¢F*(x, t). In the Schrodinger picture t/,*(x, t) = g,(x, - t ) . Several possible swaths proceed forward: only one proceeds backward. A measurement results on the parameter y when the parameter is bracketed in a unique swath via the ~F*yqJ which " 'recounts" the swath. All measurement processes start in fact at a later time and proceed backwards towards the time at which a discernment is made on a given parameter . The QCA is proceeding forward in time via q~ and undergoing an ontological counting process which defines its very survival and existence. The measurer via tF* proceeds backwards in time epistemologically " 'recounting" the swaths touched by the QCA in its survival process. There is no need of a distinct projection postulate needed to define measurement as proposed by early quantum theorists like von Neuman [26].

Wave~particle duality. The QCA is neither a wave nor a particle. It is a calculational process first and foremost and has no intrinsic properties. All parameters that can be measured or assigned to such a process such as mass, magnetic moments , etc., are contextual. Perhaps spin is its intrinsic property, giving it its fermionic traits, moving locally at the speed of light and being of mod 2.

Interference effi'cts. The QCA interferes with itself and does not interact with other entities via external potentials. The case of the double slit provides an illustrative example. From preliminary work addressing such problems it is evident that most of the time the Q C A process passes through both slits. The incoming Q C A is a fractal, not being confined. but on entering the slits it moves through two distinct I-D transverse confinements for

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i{ Ctl!II~, lliI+IFll]tl!hVi al(,!-i;!il\ ! \ \!~(!X~.ll liL'{t


To be a successful model in miming quantum behavior the Q C A picture will have to be tractable in 2-D. It is impossible to imagine the angular momentum of a particle, whether related to the Q C A or not, being fit into a 1-D description. So the Q C A will move on in future work to higher dimensions and the boundary rules may become more complex. Ultimately, 3-D pictures will have to emerge with some consistency. In addition, some experiments can possibly be performed as those suggested by Prosser [27] which will determine if an electron or a photon interferes with itself or another electron or photon. The Q C A interferes with itself only. Many wave pictures of electrons and photons explain both destructive and constructive interference diffraction patterns by having the particles interfere with each other.

The promise of the Q C A picture arises from the feature that it presents a material universe composed of parts each of which are calculating, continually counting, where they are to go in space if they are to survive in time. We who are physical and mathematical scientists are "'recounting" their space- t ime histories in interpreting our experiments and must recognize (Latin: re-cognare, to know again) the swath of the Q C A in its surge to survive in space and time. Human minds in the persons of scientists do the same thing, count the new possibilities in surging towards the future. Our computers (corn putare: to think with) are engaged in a similar process and somehow a surging in existence seems evident and understandable in these and many more disciplines which ultimately are quantum processes.

Acktsowh'd~,,,'met~tv One of us (DWB) wouM like to thank PIof. J. ( 'ushing of N,.~tre l)amc Uifiversity for informed discm, sions. Thanks are also duc to B. Williams of thc Bradley Physics Department for assistance.

R E F E R E N C E S

i M.. lanlmcr. 77u, ('om'eptua/ l)eve/opmettt 0/" QsmHtu.t Mec/mniev. ell. 5 and ~, MeGraw-Ltill. Ne~ Yolk

2. M ('. (;utzwillcr. ('haotic Ula~'vical am/Quantum Sv~tem~. Springer. New York (IU~.JI) L 5;. D, W. Bclousek. N. Clcmcnts, L. Goodman and J. Kenn~. Quantum autonl,ltion in a l-l) box.

('hao.~. ,S'o/ittm~ & k)'actal.s 2. 6{~I-673 (1992). -:. S. Wolfram. Universality and complexity in cellular automata. Phv,sie~ 10D, 1 (1984), 5. G. Bavm, I.eeturev on Quantunl Meehatsie,~. oh. 3. Benjamin. Ne~,,, York (106c)). t,. R. Omne~,, ('onsisterlt interpretation of quantum mechanics. Rev. Modern Phys. 64, 33 ~) I 1~,121. 7. [3. (}a\cau, Y. Jacobson, M. Kac and k. S. Schtllmall, Relativistic extension o! the analogy between quaiHuln

mechanics and Brownian motion, Phys. Rev. Lett. 53. 419 (1984) ,";. R. P. Fcvnman and A. R. Hibbs, Quantu#tz .4h'e/umic~ arid Path hm'gral~, pp 2~ 3tL McGiaw-Hill, New York

( 1 t ~ 5 ). cp. M. (iardncr. Mathematical games. Seienti[TcAmerican 223. 120 123 (ItJ70).

I l l . . I .K . Park. S. Steiglitz and W. P. Thurston. Soliton-like behavior in automata, l'hvsiea 19D.423 432(i~S{~). I1. ('. It. Goldbcrg. Parity filter automala. Cot,tph,x Svs'tem~' 2, c q_141 (IUSS). 12 .l.S. Bell..Speakah/e amt Unspeakable its Quantu.z Mechattie~. oh. [. Cambridge Uni~crsil,, Picss, New York

! lt,~S7): L~riginal article appeared in Phvsica 1. 195 200 ([964). 12. A. Aspect. P (;rainger and ('. Roger. Experimental tests oI realistic lcoal theories xia Bcll's theorem,

l'hv~. Rev. l,cl; 47, 461}-4[~3 (1981). l d. ('. W. Rictdijk. 0#,, Wave,~. P,u'ticles and Hiddeti t'ariahle.s, oh. 6. Van (3~rctnn. Asscn {I9711. 15 E. %chrodinger. What is an elementary particle? Endeavor 6, 35 11950). l(:. E. Schrodingcr, Letter to Lorentz, in l,etwrv ott Wave :'rice/tattles, edited h', K Prihram and M. J. Klein,

p. 5tL Philosophicai Library. New York (i967). 17. V. F. Weisskopf, On the self-energy and the electromagnetic field of the electron, Phv.s. Rev. 56, 72 (ItJ3t,ll. IS. P. ix,. M. Dirac, Principles ~ Quantum Mechanics (4th Edn), pp. 261-2¢~3. ( 'larcndon Press. Oxford (lCJ5S) It#. M. J. Ablowitz, J. M. Keiser and L. A. Takhtajan, Class of multistable fiTnC revclsible cellular automata with

rich particle content, Phv.s. Rev. 44A, 690tt (10911. 211. I. Biahaicki-Bilula, Quantum Concepts its .glmee a#td Time, editcd by R. Pcnrose and ('. J. lsham,

pp. 226 235. Clarendon Press. Oxford (1968). 21. S. Schwcbcr, An httroduction to Quantum bTeht lTteorv. oh. 3. Harper. New York (19~1). 22. M. MacGregor, Can 35 pionic mass intervals among related resonances bc accidental?. II ;~r;;~t'~ (';;;~e,,;t, fl

58A, 15')- 192 (198(I).

23. M . . l ammc l , 17u' Phi lo~* l~tn ,!l ()m~pmot~ Mcchanic~. pp. 5(~5 5()h. X,~,ilc~. Nc~ "~<wk {1~741 24. t t [ i ' ,crctt 111, "'Rclati',c SmI¢'" tormulations of quantum mechanics. Roy. Al~d<t~ i'hv.s. 29. 454 41,2 t i~ '~ i 25 D. B~hnl, A suggested h~tcrr~rctation of lhc quantum fllcorv in terms o! "'Iliddcn \"ariablc,,'" l'hv~. R<'. 8.~

[h(~ 174 ( l U 5 2 } 2 h J . A l b c r t s t m , '~'~11 N c u m a n n "~, h i d d e n p a ] a m c t c l p r o o l , . Im, , '* . . / [ 'hvL 29, 4 7 4 4~4 I I~)t~lj 2 7. I).. D. Pn~sscr. ( ) u a n t u m "thctw,. tl]ltl Ihc ] l a l u r c (',[" intcM'crctlcu. I n t . . 1 / / u ' o t c t P/tl s, lS . 1Sl !~).:. I tu7¢,~ ~S. M. A l i ' , a h . ( ) u a n t l m l f lu id lh~.:or',. ;111(_1 I o u - d i m c n s i o r l a l g c n m c t i } . I'r(,k' l h , ' , , ' c z P I I ~ .'iUl~p/ 11)2. ! !

l'gt~(t L 2 u [.. N. ( OOllCl+ |'},.'~und c lcc t r t ) [ i p a i r s in ~t d c . ~ c n c t a l c c l cc t~ , , n ga:, . l'll~s t~c+ 1t)4. I ],"-;tJ (It~S('+l ~t1. 1 ) \~: ~,ci[llll~i. | h e [ ' )hysical s i g n i f i c a n c e o f IhL. x a c u u m ' , t a lc ~)I a kttlalltklIlt I iuhl . m lYlc I'¢IH~,~,,;,':~ ,

lac~z~nz, cdilcd b\ S. 'gamlcrs and H R. Brown. ch (, ('la~cndun Pro,,',. ()xh~d (lU~l)

Quantum cellular automaton in 1-D

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