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Chaos, Solitons & Fractals Vol. 4. N-o. 3. pp. 439-'160. 199'1

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Quantum Cellular Automaton in l-D

D. W. BELOUSEK

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, IL 61625, USA

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

1. INTRODUCTION

When viewed historically over the entire century since its initial formulation it has to be

admitted that quantum mechanics still has cloudy and confusing conceptual foundations. Itstill hovers between a deterministic and well-defined Schrodinger view and a probabilisticor stochastic description exemplified by the Born-Heisenberg picture [1]. All systems innature do obey quantum mechanics and the continuing experimental verifications of itssubtle traits to remarkable degrees cf precision have failed to clarify any of the competingmodels which attempt to show how the smaller tidbits of reality behave in space and time.

With the recent development of tractable chaotic dynamics and its possible related and

newly defined discipline of "quantum chaos", some novel possibilities present themselvestoday [2]. A kernel of truth emerging from this new and fertile study seems to be that,while classical dynamical systems tend to be chaotic, the corresponding quantum systems

are much less so. It is through adhering to quantum mechanics that nature is stable andcoherent in time and space.

Another feature that can be gleaned from these studies is that nearly chaotic systems ofnature seem to succumb to a single dimensional freedom in their description if an observeror a computer can wait long enough. One-dimensional (1-D) solutions to problems seem

naive initially, but natural bounded systems in complex manifestations seem to discardhigher dimensionality eventually.

This paper develops a hypothetical one-dimensional entity called a quantum cellularautomaton (aCe; as a plausible mode for reinterpreting the traditional "quantumparticle" in standard formulations of quantum mechanics. The QCA was introduced in an

earlier paper [3] where hints emerged from its rich and variegated behavior that some

,+39

440 D' W' Ber-ousrx et al'

quantum effects could succumb to such a description. Cellular automata obey local

deterministic spatio-temporal rules but exhibit patterns of coherence and decoherence

which are similar to quantum mechanical systems'

Section 2 of this paper defines the QCA and its confinement within a 1-D box' Several

examples of eCA sf,atio-temporal behavior are examined in Section 3 with general

features such as ,"",rrr"n"", ciassification types, relationship to fractals, and dominant

trends noted as related to possible quantum interpretations'

Section 4 explores path integral formalisms and probabilistic interpretations and develops

the uncertainty retaiions hoJpitable to both QCR Oefravior and quantum mechanics'

Entropy, information and coniextual considerations are discussed in Section 5' where it is

shown that no information is available for a classical particle in a box while the confined

QCA exhibits a intrinsic quantum potentiality in being disposed to be at certain positions in

the 1-D box rather than tthers. Ii is here tirat a wave-like picture seems to emerge from

the QCA localizations.The QCA prod.rces some peculiar non-local effects which can be viewed as either being

proscopic in time o, punrputiul and these- are discussed in Section 6' Attempts to identify

the QCA with elem;tary particles are developed in Section 7' Section 8 then considers

some alternate and even unorthodox views of quantum mechalics based on multiple worlds

and hidden variables which appear under a new pelspective when the QCA is considered'

Section 9 proposes u n"* QCA picture ut itt ila to understanding the Copenhagen

interpretation and such features as measurement, collapsing wave functions' non-locality'

and intrinsic particle/wave features are developed in this light' The paper concludes in

Section 10 with u.ur" being made for the QCA process as being responsible,for.some of

the subtle quantum effects l"nl.n have persisted ln conceptual quantum mechanics' New

areas of investigation and some proposed experiments are suggested in this concluding

section based on the fledgling notions developed in this paper'

2. THf' QUANTUM CELLULAR AUTOMATON (QCA) DEFINED

cellular automata are simple mathematical models of dynamical systems consisting of a

lattice of identical, discrete s^ites each taking a finite set of integer values and evolving over

discrete time steps according to a deterministic rule [a]' As outlined in the previous paper

[3] we choose a particular ."llulut automaton, hereafter called the QCA for convenience'

*hi"h i, confinedin a 1-D box of length L : Nxo, and obeying the rule

A(r,s +1): A(,- 1,s)+ A(r+ 1,s)mod2 0)

where A(r, s) is the value of cell r at time step s, N and r are positive integers with N

greater than or equal to 2 and s is a non-negitive integer. We now apply a boundary

Iondition to confine the QCA within the box such that

A(1, s + 1) : A(2, s) and A(N, s + 1) : ,4(N - 1, s)' (2)

our initial seeding condition, which is arbitrary for the present, will be

A{int[(N + r)12],0] : 1, (3)

where int [ ] is the integer function with all other cells assigned a value of 0.

The eCA iteration iule (1) is essentially the simplest form which describes a probability

u-p-fi,ua" diffusion t5]. lA(;, s + 1)l is interpreted- as a transition amplitude and hence

f if", r illi-ir ,fr"'OCA't transition probability from cells (r - I,s) and (r * 1' s) to

(r,s * 1). An important feature of (fj is that it models a local diffusion, that is, the

Quantum cellular automaton

transition amplitude at (r, s + 1) is determined only by transition amplitudes within the

backward light cone of that cell (the QCA moves at the speed of light c). The dynamics ofthe QCA 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 QCA diffusion and so act as a-decoherence

mechanism for the local phase correlation of the QCA state vector [6]' The

initial condition (3) represents an initial seeding or "localization" of the QCA. Alterationsin this condition to generate different seeding configurations and subsequent developments

will be discussed later.The dimensions of the space-time cells are determined from the quantum diffusion

equation

(Lx)z : (hlm)A,t (4)

where hf m is the diffusion constant (h is Planck's constant and m is the mass of the

diffusing particle). Since the QCA is a relativistic particle we apply the condition implied

by the birac equation such the xo :er6; and for convenience we borrow on previous work

[i] to set r, :i/-", the Compton wavelength of the particle/automaton, and this is the

ipatial extent of each cell. The temporal duration of each cell is then

ro : hlmcz (s)

The width of a given box is then Nxo and the time intetvals 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 QCA may be located. This would not represent the relative likelihood of a

given cell being occupied since no normalization is attempted. 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 QCA 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 orarrangement of 1s and 0s 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 spatio-temporal paths over a single recurrence a remarkable

fact emerges. There is one and only one continuous swath in space and time from s:I to

62. Some shorter space-time swaths seem promising for several time iterations (see dashed

lines of Fig. 1), but they collapse before a single recurrence period is reached. The

spatio-temporal development is hereafter described as a swath rather that a path in space

and time due to the natural girth xo and temporal surge ro. Paths usually designate a series

of successive points on a trajectory and so are not an accurate desciption of QCAprogression. Selected groups of these shorter time swaths are shown with dashed lines

while the unique space-time swath is represented with a solid line. It is this unique

space-time feature which markedly strengthens the case for such a cellular automaton as

that defined bV (1) and (2) emulating quantum space-time behavior for a Dirac particle. Inperforming 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

441

D. W. Brrousr' et al .

t tl:rI 1 \1--

-.zr:./! i

s=o

5

10

15

20

25

30

40

45

50

55

60

65

70

75

t (r.-

r ,<1""1 1/1'1.1

Fig. 1. Display of space-time development of a quantum cellular automaton (QCA) with N = 10' It recurs after

k=62, A,s:62 iterations *i,t-itr" "riqr"

.pu."-ti-" .wath shown with soiid line. Dashed lines show virtual

swaths which .offufr.^uft.i u t#it.a n"-Uer of time surges. There is a parity reversal at s :31'

there is no need to renormalize the virtual probabilities or A(r, s)-values since only one

value of 1 at any given time interval lies on tire actual space-time trajectory of the QCA' a

fact which can be determined after several time iterations.

The N : L0 case is one of those in which all possible combinations of 1s and 0s are tried

o,r, ty the QCA before it recurs. Any initial seeding will produce the identical recursion

Quantum cellular automaton

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-d-vis parity and time reversal or time

phasing. At s:31-half the recursion time-there is a parity reversal. Every cell in the

tox 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-time 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-time swath for a recurrence cycle with a

single seed is unique. All other possible swaths quench after some time. And the QCA's

uniqrr" swath does recur at the initially seeded cell with a recurrence time, or ft-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:11is chosen to illustrate this. Figure 2

i6o** 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-time swaths are continuous as shown in Fig. 2(a) and the

eCA, although periodic in time with a unique k-value, never returns to the initial siting.

However, ur *" ihunge 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

spatio-temporal swaths while just 3 of the 11 produce a variegated bifurcated recurrence.

A, .un 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 spatio-temporal behavior for odd N-valued box

widths. Since the four cises represented by Fig. 2(a)-(d) exhaust all the possibilities for the

N : lL case with a single initial seeding, it is worthwhile to note the following:

(1) Eight of the eleven possible initial seedings produce unique space-time swaths, all ofwhich have k:8.

(2) Three of the eleven possible initial seedings produce diplodic space-time swaths and

these occur for initial seedings in cells adjacent to the central r:6 cell.

(3) Of the three bifurcated space-time 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 QCA 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 cleveloping some quantum/automaton

correspondences in the subsequent sections of this paper.

Figures I 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

,lg : f f case to produce the highly variegated space-time symmetries, cell occurrence

frequencies and recursion times. A slight change in a boundary dimension, for example

changing from N : 11 to 10, could pt-odu." a marked chaltge in behavior of a particle/

automaton in a typical 1-D box. One could envisage a case where the N : 10 box is

443

D. W. Ber-ousp.x et al.

x__JotI,ltr I

1lt1

1t11

1 --1-

./.7/'\

:

1111

1I

l*--t-=I1,,\

111

I111

.r'l-/!

./1

/\r tI

1I11

11

11((

1

L=llxo- 11

s= 10

s=20

s=25

a) r(0)=5k=4

L= llxo

c) r(O)=$ s1 7

k=8

b) r(0)=3 e19k=8

s= IO

s=2O

I s=25

d) r(O)= 1 or 2,4,8,7O,11

k=8Fig.2. Display of space-time development of quantum cellular automaton (QCA) with N: 11 for direrse initialseedings. In (a) the initial seed is r(0) = 6 and a bifurcatecl or diplodic swath is produced rvith k:4.In (b)-(d) other initial seedings produced a recursion time k : 8 with only (c) being diplodic. Unique swathsoccur in 8 of the 11 seeded cases each of which has k = 8. Solid lines indicate unique/bifircated suarhs and dashed

lines show selected virtual paths which collapse after a limited number of time surges.

extended to N : 11 at some time and a unique space-time swath could proceed as abifurcated one.

Figure 3 shows recurrence times for seedings specified in (3) lbr box widths ranging fromN :2 to 50. The initial seeding has no effect on recurrence times for the even Uox wlOtnsand in the case of the odd box sizes recurrence times of ft and 2k are admissable with thelow k-value always being valid for central seeding. From a cursory analysis of the resultsshown in Figs 1-3 certain generalizations can be made on the behavior of the ecA.

First of all, all QCAs are either Class 1 or Class 2 cellular automata using the Wolfram

l*- r,= rr*o

Quantum cellular automaton

I-quenching QCA

O -ergodic QCA

A-luminal QCA

Box Width-N

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

ralues.

classification scheme [a]. The Class l cases all go to zero and are quenched. Here zero canbe viewed as a "great attractor". So for N-values of 3, 7, 15, 31, 63, . . . (2" - I) (where nis a positive integer-a series which incidentally includes all Mersenne primes when n is aprime number) the QCA is quenched as soon as the A-value first encounters the edge ofthe box. Changing the initial seedings does not remove the quenching but extends thequenching period until both virtual probabilities encounter the box edge.

All the QCAs which are not Class 1 fall into the Class 2 category. Here they all recurwith characteristic periods-k-values-shown in Fig.3. However, as a glance at Fig.3shows, some of the N-valued box widths experience sizable recurrence times whereas someothers recur rather quickly. For the sake of analysis and interest we have further bracketedthese Class 2 automata into sub-orders. These are:

Class 2-ergodic: For N-values of 2, 6, 10, 72, 18,22,28, 46 . . . the QCA tries out allpossible arrangements of 1s and 0s as it calculates its spatio-temporal trajectory. For thesake of discussion this order will be described as ergodic since all possibilities are tried out.Actually, the recurrence time can be predicted by the formula k:2 (2N12 - 1): Q forthese specific orders although at present no simple predictive formula indicates at whichN-values the QCA will be ergodic (Q is the number of possible configurations or numberof accessible states for the QCA).

Class Z-bifurcated or diplodic: Bifurcated recurrences occur for all the odd integervalues centered initially. As N-values increase it is found that only the center initialseeding produces bifurcated swaths. Some of these move at luminal velocities and recurextremely rapidly even though they never return 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- luminal: By luminal we mean traveling at a net macro-speed equivalent to the

speed of light and it is these QCA which recur most rapidly with the lowest possiblek-values. In Fig. 2 we can see readily that (a) and (d) are luminal by this classification

445

,jtb0o

o

Fq)I

6)!

br

a

T//\_/111 Stt\/t_ /!/x\fr I,l \/\r/\i \/sbAl/

446 D. W. Brr-ousl.s et al .

scheme with (a) being bifurcated luminal and (d) with a unique luminal swath. Forbifurcated luminal QCAs the recursion time k < N f2, while for the case of the uniqueluminal paths it is always such that k < N.

Larger values of N produce fractal-like displays for the QCA as shown in Figs 4 and 5where N : 800. These two figures are not fractals since the diffusion is limited both on thelarge scale, x:L:800xo, and on the small scale, x: lxo. llowever, it could be suggestedthat as N-+ oo the resulting pattern would be a fractal with dimension 1.59. Figure 4 shows

a relatively well-observed sequence similar to that seen with clear Sierpinski triangles,while Fig. 5 displays another set from the same QCA which appears highly disordered. Theunique space-time swath is marked with the dark solid line in each case.

In an attempt to define a near fractal dimension for patterns such as those of Figs 4 and

5 it is noticable that the QCA could be viewed as a restricted subset of Conway's Game ofLde l9).If a square of width 3xo and height 3ro is centered on any 1 on any of the QCAplots, Conway's "genetic rules" of survival and death are obeyed. Survival occurs if a given

1-site has 2 or 3 other ls in adjacent cells. Death occurs with only a single 1 in adjacent

Fig. 4. Plot of a selected section of the QCA swath for the N : 800 box width. The near fractal-like patterns areshorvn with Sierpinski triangles evident. This is an ordered segment of the N = 800 QCA. About 875 time surges

are shown here with the un'ique swath marked with the dark line.

Quantum cellular automalon 44'l

Fig.5. Plot of another selected section of the QCA srvath for the N : 800 box width. This is a "near-chaotic"section and few fractal-like patterns appear as in Fig. 4. The 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 QCA. The Game of Life is actually a 2-Dgame where the picture and population changes with discrete time intervals. The QCA is asurvival game with one spatial dimension and the adjacent squares coming from theprevious generation and the subsequent one so that three generations are presented in each

box. Ultimately. as shown in Figs I and 2, only one swath in space and time survives forthe even N-valued boxes and the bulk of the odd N-valued boxes over an entirerecurrence cycle.

What is interesting about the QCA when viewed as a survival game similar to Life is

that it admits two generation survivors. The survival fate of the 1 is to be displayed two ro

intervals ahead in time. Migration or leaving the site patterns can also be discerned twogenerations ahead,

The QCA has a near soliton-like behavior as noted already [10] in filter cellularautomata. Actually, as the diffusing QCA hits the boundary a reflected wave, which isequivalent to the probability absorption. passes through subsequent incoming waves verymuch as solitons pass through each other. Much previous work on parity automata [11]seems to indicate similar pattems to that of the QCA in the cases considered here.

448 D' W' Berouser et al'

4. COMPATIBILITY WITH ORTHODOX QM

In this section we consider three aspects of orthodox QM where we may apply the

standard formalism and concepts to th; QCA. First, the path integral formalism can be

Oi.".,fV applied to QCA spac;-time swaths' Feynman and Hibbs discuss the l-D Dirac

"qrrution u.ra tor.nutut" puttt integrals over space-time paths. similar to those of the OCA

[S]. The transition proUatitity P {or a rpu."-ii-" path from initial point x(l) to final point

x(/) is

Pl*(f). x(t)l : la['ff)' r(t)]l' (6)

where Blx(f), x(l)] is the transition amplitude for that same path. For a space-time path

"o-por"d of'S atlr"te steps of finite temporal length the transition amplitude B for a

particle moving from x(l) to x(/) is grven by

B[*(f),x(i)] : n[B(s + 1' s)] Q)

where B(s * 1, s) is the transition amplitude of the (s + 1)st step and the series product fIis taken from s:0 to S - 1.

The transition amplitude for each step is the sum of the contributions from each possible

path in that step. Thus

B(s + 1, s) : zA(n) (8)

where A(n) is the single step transition amplitude for the .nth path from s to s-* 1 and

the sum is taken over"all such possible paths. To any QCA position (-r. s * 1) there are

twoandonlytwopossiblepaths,onefrom(*-!'s)andonefrom('t*1's)'We,,'uy int"rp."i a1r,r; as the total step transiiion amplitude from (-r- - 1's - 1) and

(x*1,r-1).Hence

(10)

(1 1)

B(s+1,s):lA(*- 1's)+ A(x+ 1's)l :A('i-'s+1)

So

Bl*(f),x(i)l : rI[A(x, s + 1)]

and the probability P between initial and final states is

Plr(f),x(i)l : ln[, (x' s + 1)]12'

Therefore, the transition amplitude from x(i) to x(/) will be non-zero if and only if all of

the terms in the series product are non-zero. This is true only for a srvath of temporally

successive 1st such a swath will have a transition probability of unitl'. Developing QCA

space-time swaths by connecting Ls is consistent with the path integral formulations of

QM.According to the Born view of quantum mechanics the squared modulus of the state

vector is physically interpretable as a probability density:

(e)

p : 4f (x, t)V@, t) : 4|

where V*(x,t) is the complex conjugate of tp(x, r)' For the particle in a

length L

V@) : Ql rlttz sin(nnx f L)

where n is the energy level quantum number. We have neglected the temporal part of the

solution in (13) for convenience since it is just an oscillating contribution. Therefore

(12)

l-D box of

(13)

,t*@)ltt@) : (21 L) sinz (nnx f L). (14)

Quantum cellular automaton 449

We can divide the box up into intervals and approximate the probability of finding the

particle in each interval by calculating

Pdx : fp*(x)tp(x)ldx (1s)

where dr is the width of the interval and the value of x used is the midpoint of the

interval. llence, if the box has a length L:10x,, then we can divide it into 10 intervals ofunit length and calculate the probability of finding the particle in each interval by

successively substituting *:i*o,]xo, etc., into equation (15). We find that the probabilitydensity tends to give a distribution corresponding to the odd energy quantum numbers

n:1,3, 5, etc., but fails to give any correspondence to even energy quantum numbers forreasons not clearly understood at present.

We now consider the uncertainty relations

dx dp > hf2

and

dE dt > hlz. (r7)

These relations are derivable from a QCA space-time swath. Consider a dynamicalmeasurement of the QCAs' momentum by "localizing" it in a particular cell at some timestep. At that step there is an uncertainty as to the location of the QCA, dx : x,, and there

is an uncertainty as to the QCA momentum since within that time step it is not known if itis 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 momen-tum-wavelength relation

p : hl|. (18)

The uncertainty relation (16) is derivable by requiring that the minimum uncertainty in

momentum would occur when the box width x, is equivalent to the smallest halfwavelength allowable. The uncertainty in time during a dynamical measurement of the

QCA's energy will be dt : ro. The uncertainty in energy (17) is derivable from the

relativistic energy equation

(16)

A:l(pc), * (mc2121tlz

and substituting p : hl)', dp > hf2x,, and xo : hlmc.Therefore, the uncertainty relations are natural consequences of the QCA

swath.

(1e)

space-time

5. ENTROPY, INFORMATION AND QCA CONTEXTUAL POTENTIALITY

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

S: klnQ (20)

where k is Boltzmann's constant and Q the number of accessible states. The occupationfrequency distributions F(x) for N : 10 is displayed in Fig. 6 and it shows the typicalfeatures of quantum wave-like behavior when the unique swath is considered. Certain cellsare more likely than others to be occupied by the QCA when considered over a singlerecurrence cycle. Each cell does not contribute equally to the total entropy. Figure 6 also

D. W. Brrous:.r et al

ao

910tuoc)trc)Itrj

or\ o

X

XII

d

q)

A

e)d

@

Quantum PotentialitY--X(x)

\-//E---*- *t"'t-/=.

___.s__g/

10

Cell Position N= 1O

Fig.6. Plot showing QCA cell occurlence probabilities during a single recurrence cvcle for N : 10 box width' The

classical probability is ttre number of ls in each box ror u i..i.i*.8cycle (k:62)Jhe quantum probability F(x)

is the number of times ""d;;li;;; u ,rniq"" swath i; it over a ,."urr"n." cycle. The quantum potentiality X(r)is defined as the reciprocal of F(x) as shown on the right'

displays the classical occupation frequency, counting all the 1s in each cell ovel a single

recurrence cycle, for the N = 10 case. Each cell is equally capable of being occupied when

considered over the entire cycle, as would be expected, and contributes equally

to (20).It is noticable in Fig. 6 that these occupation frequency distributions indicate that the

eCA behaves as if it l'feels" or is guided by a quasi-potential (recall that the particle is

free with the classical potential y : 0 inside the box) when calculating its space-time

swaths and hence in deiermining what states it will occupy. This quasi-potential X(x) is

such that distribution peaks in "Lll

o""rr.t"nces which correspond to quasi-potential wells

Classical ProbabilitY

Quantum ProbabilitY--F(x)

Ouantum cellular automaton

and vice versa and can be wdtten conveniently as

X(x) : 1,lF(x). (2t)

A graph of F(r) for N : L0 is also included in Fig. 7. The above equation is a choicewhich would define such a potential, its main feature being that it would have maximawhere F(x) would have minima and vice versa. A high probability of occurrence would

p-oID

o

o

oB

FO

zil

Ho@It0

s

.----__-->1

1111

11

11

11

1 I1

111

11

11

11

1

s=O

5

10

15

20

25

30

35

40

45

50

55

60

62

11

11

11

1111

II11

11

1

L= roxo*lI/r, II I trl

11I

1

1

L=llxo---*l

Fig. 7. Plot of QCA surging in time for a box which is randomly varied between N : 10 and 11. The unique swathdeviates from the normal N : 10 case at s: 19 but the extended boundary is first encountered by a 1 on the right

at s = 14. The reverse light cone is shown and the QCA swath stays away from the right boundary.

452 D. W. BEr-ousEK et al.

indicate a potential well and the more likely the occurrence the deeper the quasi-potentialwell.

We may write (21) as

kln X(x) : - ktn F(x) (22)

where ft is Boltzmann's constant. Now kln F(x) represents the total entropy distributedover the box and thus reveals the entropy contribution by each state and hence

ktnX(x): -s(") : I(x) (23)

where 1(x) is the information or negentropy.This quasi-potential X(;r) discloses the information content of the QCA's confined

diffusion. As that diffusion is dependent upon the QCA's physical environment (width ofbox and boundary conditions), we shall call this quasi-potential the QCA contextualpotentiality. F(r) reveals the possibilities of the QCA in a probabilistic way via the

space-time swath; X(r) discloses the potential for the QCA to access those possibilitiesand hence characterizes the information created by the interaction of the QCA diffusionwith the environment. Only quantum and QCA systems confined in the manner discussed

contain information. The classical occurrence frequency contains less information, whereasthe QCA unique path occurrence frequency contains information similar to that gained

from using the wave function and the quantum calculational procedures.

6. STRANGE AND NON.LOCAL QCA EFFECTS

Since the Bell inequality was proposed [12] and its experimental verification in theexperiments of Aspect [13] no theoretical explanations can claim any quantum correspond-ence unless it admits to the presence of non-local influences in effecting the outcome ofquantum systems. The QCA does produce some effects which can be characterized as

non-local, superluminal, proscopic or panspatial, depending on an observer's perspective.Figure 7 illustrates a non-local or superluminal effect. The right boundary in Fig. 7 was

changed via a binary random number generator from 10x, to 11xo. This should correspondto physical situations since boundaries would vary in real systems. As can be seen fromcomparing Fig. 7 with Fig. 1, the first time the boundary changed to effect the QCA was ats : 14 and the first time the unique swath deviated from the pure N : 10 state was ats : 19. However, at ,r : 19 the unique swath was in the r :4 cell while the informationwhich produced this deviation was in the r : 11 cell. The information of the boundaryeffect reached the unique swath superluminally as can be seen by drawing the reverse lightcone in Fig. 7. The information which changed the unique swath lies outside the reverselight cone and this is shown also in Fig. 7.

Proscopic effects also seem to be a property of the QCA as it surges for suvival in space

and time. Figure 8 shows such an effect. Here, and in subsequent discussions, proscopicmeans "seeing ahead in time" as defined by Rietdijk [14]. As an exercise in influencingQCA unique swaths the N : 10 box was enlarged for a single time interval at s : 16. Inthe tracing of the unique swath we see a total parity change when we compare Fig. 8 withFig. 1. The unique swath starts from the right in Fig. 8 "knowing" or sensing that some 15

time intervals ahead it will move through the extended boundary. It seemingly knows some29 intervals ahead that if it takes its usual N : 10 route initially it will be quenched at.-to

It has been suggested by Rietdijk [14] that all quantum mechanical interactions consistentwith special relativity are proscopic and this seems borne out here. The QCA is a

l1l1

11

1111

1111

1

I

I

I

I

Quantum cellular automaton

s=o

5

10

20

25

30

35

40

45

Fig'8'AQCAdevelopmentwhereanormalN=l0boxisextendedtoN:11ats:16.TheoriginalN:10swath would start on the left (see Fig. 1). The swath acts.proscopically in that it "knows" lt wifLfrave-iolo to the

right as shown to survive in space and time.

relativistic particle and its behavior in this case is consistent with it being a quantumrelativistic process' The QCA is able to use global information in calculatin! its;urvivalswath into the future.

Actual physical systems could hardly be expected to present an unchanging box for itsquantum occupants whether it be the eCA or a physical particle obeying mechanics.Although the situations used to produce Figs 7 and 8 might aipear contrived mathematic-ally it is not unreasonable when physical systems ur" .onrid"red. The box boundarieswould have to be produced by ecas which behave in a manner similar to the ecA in thebox and a fluctuation of one or more cell sizes could be a normal boundary condition. It isnoticeable in Fig. 7 that the fluctuating boundary on the right side caused the ecA tocalculate its unique swath more to the left side and av6id the fragile confinementparameters. The QCA in this and some other cases used as examples t"nJ, to survive in aunique swath removed from happenstance.

Additional QCA swaths can be created in selected sets of box dimensions. Multipleswaths, subsequently called odes for convenience, can be created by changing the iniiialseedings. Figure 9 shows a quadruplode swath for N : 11. At n:'g, with pioper initialseeding' it is possible to have a triplode, three distinct swaths similar to the four in flg. e.

This means that it is possible for a QCA proceeding in a single swath to multipiy intoseveral additional swaths if changing boundary conditions or the introduction of"""t.u

454 D. W. Brr-ousp.x et al

s=o

5

10

15

Fig.9. A quadruplodic set of QCA swaths shown for the N: 11 box width. This multiplodic cannot result from asingle initial seed, the cases for which are shown in Fig. 2. Cells with r = 3. 6 and 9 are never occupild by the

unique swaths, which are shown with dark lines.

QCAs takes place in the proper cells at the proper time. A birth rule, to borrou,phraseology from Conway's Game of Life [9], is supplanted by a happenstance change inboundaries at the right time. A single swath is called a haplode. a double swath a dipiode.a triple swath a triplode at N : 8, a quadruplode at N : 11 as seen in Fig. 9, aquintuplode swath at N : 14, etc. This means that for any integer a box of diminsionsN :3a - 1 cannot be ergodic. The N : 800 QCAs shown in Figs 4 and 5 are not ergodicand do not try out all possible configurations of 1s and 0s in calculating its unique swathsince it could be, under the right seeding conditions, a26j-ode.

7. ELEMENTARY PARTICLE CORRESPONDENCE

At a conceptual level the QCA can be compared with Schrodinger's notion of anelementary particle [15]. He characterized electrons not as persisting individual entities butrather as "long strings of successively occupied states" which produce the impression of anidentifiable individual similar to the objects of everyday experience. Also, poisible electronstates can only be well defined once given the physical environment of an experiment.

In a letter to Lorentz, Schrodinger [16] questioned whether or not an electron infield-free space would retain its ability to exist as a permanent entity. What this translatesinto is that the electron requires a physical confinement via fields or potentials in order forit to be able to occupy well-defined states and hence exist as an entity. Likewise, thepossible states accessible to the QCA, represented by the state vector configurations andcharacterized by the QCA's contextual potentiality. are determined by its diffusionconfinement and are well defined only when a boundary of some sort is present. Also, theQCA space-time swath is simply a temporal sequence of successively occupied neighboringcells.

As originally shown by Schrodinger, the Dirac equation for the relativistic electronimplies that superimposed upon the electron's translatory motion is an irregular circularmotion of instantaneous velocity c (Zitterbewegung) which is interpreted ur giuitrg rise tothe electron's spin magnetic moment [17]. The QCA space-time swiths displayed ii figs 1,2, 4 and 5 clearly illustrate this phenomena. The eCA microvelocity by iesign is alwaysthe speed of light and this corresponds to the eigenvalues of the veiocity

-in u.ry on"

direction in Dirac's matrix formulation of quantum mechanics [1g].The behavior displayed by the QCA space-time swaths is consistent with the quantum

stochastic treatment of elementary particles in modern views [19]. Here the reiativistic

[ltl

Quantum cellular automaton

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 withthe rate of such events being interpreted as inertial mass. It has also been suggested thatsuch treatment of all massive particles at the fundamental level may resolve some of thespace-time difficulties of quantum theory [20].

The QCA space-time swath, in Fig. 1 for example, clearly shows such reversals ofdirection and flips in handedness. We can assert that the stochastic process underlying theQCA is a direct consequence of the confinement of its diffusion and arises out of theinteraction of that diffusion with the physical environment which naturally precipitates theobserved randomf rregular behavior.

Particle content, the types of allorn'able physical particles, is another feature which can begauged for the QCA based on its spatio-temporal behavior. For example, a luminal QCAwould have to correspond to a massless particle since all massive particles must havemacroscopic velocities less than the speed of light. c. A particle with a sizable inertial masswould linger longer in a certain region of the 1-D box when compared with a lighterparticle which would bounce back and forth from one side to the other at near-luminal netspeeds. However, if we equate certain degrees of freedom with spatial dimensions there areconstraints relating spin, mass and dimensional freedoms, abbreviated as smD here, whichsurvive in modern formulations of particle theories based on sound physics which aredigested in Table 1. These constraints are based on the following relations:

D :2s, when m :0 and D :2s * 1, when m * 0.

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. Theselatter particles having spins of 0 do not obey the Dirac equation but succumb to theKlein-Gordon equation [21]. The QCA is derived by analogy with the Dirac equationwhich admits only fermions of spin I in itr purest form. The fact that at least half the cellshave 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 tobe confined within boxes where Z would have to be equal or greater than 6xo. All smallerboxes would either quench or have luminal QCA and massless particles. The case whereN : 6 is the first ergodic and non-luminal QCA and it has a quenching barrier aI N :7.

The presence of the pseudoscalar mesons in Table 1 is worthwhile when it is consideredthat all pseudoscalars such a pions, kaons, etc., have microvelocities near the speed of lightembedded as they are in the innards of nuclear matter. Pions, being the lightest of theseparticles, would have primordial importance if the QCA is to represent elementaryparticles. However, several elementary particle descriptions and analyses already recognizethe primitive nature of pions [22].

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

Table 1. Particle content of l-D QCA

4.55

QCA property snrD constraint Experimental candidates

LuminalUniqueBifurcated

Sub-luminalUnique

Bifurcated

m=0, s:112, D:l

m*0, s=0, D:lNeutrinosNone

Pions, kaons, etc.Pseudoscalar mesonsAny particle/antiparticle pair

456 D. W. Ber-orrsEx et al .

there are very few intrinsic particle properties and all measurable parameters such as mass,

spin, angular momentum, etc., are contextual; that is, depend on the global surroundings.

There might be no such thing as little tidbits of matter or energy or momentum at all butprocesses which survive in selected spaces and times and whose swaths can be construed as^having

properties. There are no internal degrees of freedom in QCA/particles, but only

surviving swaths in space and time.

8. TNORTHODOX QM AND THE QCA COMPARED

Heisenberg envisioned the state of a quantum system prior to measurement as a set ofAristotlean potentia and considered measurement as a transition of the system from a

possible to an actual state [23]. Similarly, the QCA state vector configurations representpotentialities which are acluahzed when the bounded diffusion collapses to a single orbifurcated possibility.

Figure 1 shows that during the evolution of the QCA history two interesting featuresdevelop: (1) beginning with any particular 1 a unique swath of connectable 1s can be tracedbackward in time to the initial QCA position, and (2) beginning with any given 1 severalswaths can be traced forward in time. For example, returning to Fig. 1, beginning at s: L5

only one swath may be traced back to s :0 from each 1 in the configuration. However.beginning at s:0 five distinct swaths can be traced to s:15. Several forward time swaths

exist over several time intervals.Everett's "many worlds" or "relative state" interpretation of quantum mechanics [24]

presents comparable concepts to that of the QCA behavior. Everett postulates a universalstate vector which evolves continuously according to the Schrodinger wave equation and

never collapses into a particular state, A determinisitc evolution of "relative states" occurs,rendering measuring instruments and external observers inessential to the foundations ofquantum theory. Thus all possible states described by the universal state vector are equallyreal, although only one is accessible to measurement or human consciousness.

The QCA history in Fig. 1 follows the "many worlds" description, with one exception.At various time steps, s:16, 23, 37, 47, 54, 62, etc., all coexisting QCA space-timeswaths collapse to a single swath. The QCA state vector evolves discretely yet smoothlyover short time intervals but at various times changes discontinuously or "collapses" into a

uniquely observable state.The periodic QCA state vector reductions do not result from the interaction of the QCA

state vectors with a measuring instrument. The reductions are independent of externalobservers. The QCA's confinement allows its diffusion to interfere with itself via a

decoherence of the local phase correlation of the QCA state vector which is transmittedthrough the entire QCA diffusion. It is this self-action which precipitates the subsequentreductions. The mechanism of "state vector collapse" is inherent within the QCA'sbounded diffusion. Actualization of the QCA's swath is not a result of any specialinteraction of measurement but rather is a natural consequence of the QCAs interactingwith their physical environment-the boundaries.

Bohm's causal interpretation in terms of hidden variables incorporates non-locality intothe formalism of QM via a "quantum potential" which acts on quantum particles inaddition to the classical potential [25]. The mathematical form of the quantum potential is

derivable from the state vector which is interpreted as representing an objectively real fieldwhose field equation is the Schrodinger wave equation. Bohm's state vector is seen as a de

Broglie pilot wave which "feels" the physical environment of the localized particle andtransmits non-local (superluminal) influences to the particle due to changes in the

Quantum cellular automaton 45'7

environment. Bohm defines a quantum potential which provides a quantum force on the

particle and this potential is how the parlicle "feels" its environment' X(x), the quantum

iotentially definecl in Section 5, is virtually identical to Bohm's quanturF potential'

In a similar \\'av the QCA diffusion, although dynamically local' responds non-locally to

changes in the bounaary uy altering the QCA swath for its survival. The local dynamics of

the QCA is correlated io ifr" globil structure of the QCA space-time history' The OCA

diffusion is not an external fieid effect but rather represents the QCA spreading itself out

to encounter its boundaries and reacting to changes therein so as to maintain its survival

surse in its space-time 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 on using the calculational aspects of quantum mechanics while ignoring

non-tocat effects, superliminal signal possibilities, quantum encoding of ultimate informa-

tion, and a series of other featurei thai seem to be emerging out of reworking foundational

quantum concepts. Like any good orthodoxy, the Copenhagen-interpretation of quantum

mechanics, as ii is often catlJd. becomes a secure repository for these novel effects and

their explanations by widening its comprehensive sway. Since it encompasses nearly all

effects, iogics, nouei explanuiiont. and suggested experiments it remains an extensive,,interpreta"tion" rvithout producing any models or favoring any calculational methodo-

logies.the following are some features where the QCA adds some insights if it is to prove to be

a viable conceptual model:

Measuremenl. In the QCA view measurement is a "recounting" of the space-time

swaths. tl,G,t) proceeds forward in time and 4,*Q,t) proceeds backward in time

admitting one unique or haplodic swath as in the case of N : 10, for example. Recounting

the swaihs .an only be performed going backwards with rp*(x, r)' In the Schrodinger

picture V*(x, t): iQ, -i;. S"u"tul possible swaths proceed forwardl only one proceeds

backward. A measurement results on the parameter y when the parameter is bracketed in

a unique swath via the rp*yrir which "recounts" the swath. AII 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 rp and undergoing

an ontologiJal counting process which defines its very survival and existence. The measurer

r.i3 ,-.. pr-oceeds backwards in time epistemologically "recounting" the swaths touched by

th; eCA in its survival process. There is no need of a distinct projection postulate needed

to c.:::: measurement ui ptopot"d by early quantum theorists like von Neuman [26]'I\-;.. :,irricle tluality. The-QCA is neither a wave nor a particle. It is a calculational

proc-s> :::st and foremost and has no intrinsic properties. All parameters that can be

meas-::: J: assigned to such a process such as maSS' magnetic moments, etc', are

coni;:r:-=- p:rhaps spin is its intrinsic property, giving it its fermionic traits, moving locally

at ti:= :::.: ,.: lisht and being of mod2.Ir:.-'.-=':-. ,+rrrr. 1.ne QCe interferes with itself and does not interact with other

enti::-: -.-. :.-:::lal potentials. The case of the double slit provides an illustrative example.

Fror- ::=-.::-::.r u.ork addressing such problems it is evident that most of the time the

eC-{ :- -:=.: I -sScS through both slits. The incoming QCA is a fractal, not being confined.

but c- :-,:::-:::le slits it -ou". through two distinct 1-D transverse confinements for

458 D. W. Blloustx et al '

some time interval. while in the slits it loses its fractal nature. After passing through the

slits it never regains its fractal natule. Each slit could be a N: 10 bor' for example' Upon

passing througli the slits the unique QCA swaths now diverge into ser-eral s*.aths. none of

which ever cross the central plane of the slits. Swaths from the right slit never cross to the

screen on the left or vice versa. This neglected feature is true for photons [27] and is a

feature that should be experimentally verifiable with present instrumentation for either

photons or electrons .,.. -.^_+Space-time quantizatiors. Theoretical physics as a whole has_been reluctant to quantlze

space and time even though they complement momentum and energY in the uncertainty

.itution, (16) and (17). Momentum. angular momentum, and energ)' are quantized as a

matter of course, often in an ad /roc fashion. Each space-time cell can be r.ieu'ed as an

action quant as defined by Rietdijk [14]. This quantization can be vie$.ed akin to

renormalization mechanisms in quantum iield theories where a dimensional scaling is

inserted, whether a momentum or a distance, to make the interactions considered

renormalizable. In the quantum field theories renormalization is performed-a-posteriori

when the energy or momentum is sizable in the interactions' Here in the QCA formu-

lations the quantization is performed a priori. There is considerable theoretical compulsion

that no 0-D entities .un "tirt in the physical world and all quantum field theories rvhich

employ ..virtual,' point-like exchange puiti.t"t to carry fields must consider ertending their

dimensionality [28].Globatflocitioiriarrotions. The QCA processes global information via X('r) to calculate

it next step locally. Ail of its dynamics aie local within the light core' Yet' as indicated in

Section 6, X(x) behaves as if it were an extrinsic potential carrying an interaction energy'

Actually it is generated by the QCA local dynamics and arises^out of its striving or surging

forward in time to survive. Information is supplied so that the QCA can sun'ive'

Novel space-time questions. Could Coopii pairs, the bastions of superconductirit.v the

orists [29i be diplodic QCAs passing througtr a crystal lattice or between t$o confining

planes? Could antimatler states, whici emerge out of the seminal Dirac equations [18] as

did the QCA, be the vacant space-tlme cellsl Is the vacuum state' the paradium on rvhich

modern tractable field theories [30] depend, just the 1s that do not lie on measurable

space-time swaths?

These and many other questions cannot be answered here but there are such rich

behavior patterns. phasings, Iecurlences and multiplodic swaths in the QCA rvhich

resonate with other Ais.lpti-r"r such as chaotic dynamics, fractals, calculational games' and

soliton-like behavior which show promise' The QCA insights do not clarit\ quantum

mechanics concepts but at least in several of the aspects mentioned above thev change the

viewing of the problems, whether visible from Copenhagen or not'

10. CONCLUSIONS

euantum mechanics is a universal set of phenomena which at present partialil'' succumbs

to many pictures and many interpretations. The QCA picture, albeit of a fledgling nature

u, prerented here, could become one of the possible future representations of quantum

systems when developed more completely.If there is merit to the QCA picture then the smallest tidbits of the universe are

calculating and exploring new possibilities in their surging to survive. There are no external

interactions of the typ-"t t,rgg"sted by field theories, but each QCA obeys its local

dynamics. The eCA iJp,rr" process, pure drive or surge. Its dynamic process is local, but

it gains information globally as shown here.

Acknowledgernertls-One of us (D\\'Binformed discussions. Thanks are also d';'

Quantum cellular automaton

iike to thank Prof. J. Cushing of Notre Dame University for\\'iliiams of the Bradley Physics Department for assistance'

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459

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tractable in2-D.It is impossible to imagine the angular momentum of a particle' whether

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future work to higher dimensiois and the boundary rules may become more complex'

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