D. Finkelstein- Space-Time Structure in High Energy Interactions

8/3/2019 D. Finkelstein- Space-Time Structure in High Energy Interactions

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SPACE-TIME STRUCTURE IN HIGH ENERGY INTERACTIONS

Belfer Graduate School of ScienceYeshiva University, New York, New YorkThe preceding two talks were motivated by the courageous

faith that our present ideas of the continuum and the gravita-tional field extend into the range of elementary particlesizes and far below. Equally interesting to high-energyphysicists is the possibilitv that these ideas of saace andtime are already at the very edge of their domain and arewrong for shorter distances and times, and that high-energyexperiments are a probe through which departure from theclassical continuum can be discovered. The present talk isdevoted to this alternative. The difficulty is that all ourpresent theoretical work is based on a microscopic continuumand one is faced by the rather formidable problem of re-doingall physics in a continuum-free manner. Yet I think thosewho cope with the conceptual problems of quantum field theoryfor enough years eventually get sick of the ambiguities anddivergencies that seem to derive from the continuum and aredriven to seek some way out of this intellectual impasse.I would like to describe a program of this kind that I havebeen led into after vainly trying to extract some information,;about the small from extremely non-linear field theories.

The starting point here also is Riemann, who explicitlyposes the question of whether the world is a continuous ora discrete manifold in his famous inaugural lecture. Hepoints out certain philosophical advantages of the discretemanifold, in fact argues for it more strongly than for thecontinuous. and set devotes his life to the continuous. m Y


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I th in k pr ima r i ly on grounds of s im pl i c i ty . Why i s thecontinuous s impler than the d is cr e t e mani fold? For onething because i t has more symmetries av ai la bl e t o i t . Acontinuum can have r ot at io na l symmetry, th e world has ro ta t io n-a l symmetry, a checkerboard cannot have r ot at io na l symmetry.Today, however, th er e ar e more o ptio ns open t o us than t he rewere t o Riemann. Faced wi th th e app are nt dilemma of th edi sc re te and the continuum, our exp erien ce wi th quantum theor yleads us t o plunge boldly between thes e two wi th the synth es iscal led quantum. By a quantum I mean an object whose proposi-t i o n a l c a l c u l u s i s is om or ph ic t o t h e l a t t i c e o f s ub sp ac es of

quantum concepts ar e inj ec te d a t the t op . We make up a com-of spa ce and p l e t e c l a s s i c a l p i c tu r e o f t he world , a geomet r i ca l anddynamical s t r uc tu re which cou ld i n p r in c ip l e desc r ibe a r e a lworld, and th en we tak e th i s beautiful theory and amend i t byq u a n t iz a t i o n . I s i t not poss ib l e t h a t i n f a c t quantum concep tse n t t a l k i s b el on g i n t h e f ou n da ti o n? I n p a r t i c u l a r , t h a t i n s t e a d o ft h a t a l l our tak ing geometry and qua ntiz ing i t , so t o speak, we should takep i c o ntin uu m quanta and geom etrize them? Should we no t t r y t o make aspace -t ime theory i n which from th e s t a r t t he e lementa ryt h i n k t h o s e ob je ct s which make up t h e spac e-time a r e de sc rib ed by quantume l d t h e or y laws and th e space- time i t s e l f i s assembled out of the se bythe quantum-logical procedure s t h a t we have mastered alread ynd a re i n t h e quantum many-body problem?

I was fo rc e fu l l y i n t roduced to t h i s whole i d e a by FeynmanI have some eig ht years ago. He di dn ' t b el ie ve i n continuum then.

(He s t i l l doesn ' t b e l i eve i n t he con tinuum. S ee h i s " Charac t erof Ph ys i ca l ~ a w s " ) ~ He sugge sted th a t a re asona ble modelho ex pl ic i t ly f o r t h e w or ld i s a computer, a gian t d ig i t a l compute r. Thething s we c a l l events are process es of computat ion, and thetu re . He fundamental f ie ld s repres ent s to red informat ion. The continuumt e theory of th e world i s t o t a l l y absurd from t h i s po in t o f v iew.It impar t s t o each poi nt of space- time an in f i n i t e memoryc a p ac i ty , i n t h a t a n i n f i n i t e number of b i t s a r e r e q u ir e dt o d e f i n e a f un da me nt al f i e l d l i k e t h e e l e c t r i c v e c t o r po te n-t i a l . It t akes an i n f in i t e channe l capac i ty t o communicatethese numbers from one po in t of space t o anoth er. An i n f i n i t enumber of comp utati ons must be done by t h e computing elementa t each po in t t o work ou t t he f i e l d equa t ions and pass on t he

, o ut pu t t o t h e f u t u r e .


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i s not The f i r s t th i ng i s t o make our computer ou t of quantumth e world i s e lements. Computers a r e o r d i na r i ly made ou t o f b inaryom the id ea e lements, b i ts , ze ros and ones . Le t us s imply suppose th a tb ac k t o th e in s t e a d of d e a l in g wi th th e s e by th e to o l s o f o rd in a ry(I am indebted Boolean alg ebr a we use th e alg eb ra of subspaces of Hi lb er t

e working ' i n space. In more fa m i l i a r t e rms , i f a b i t c a n t a k e on th ev a lu e s 0 and 1 i t can a l so ta ke on coheren t superposi-

mpu te r which t i o n s o f th e s e s t a t e s . In b r i e f , th e n a tu r a l b u i ld in g b lo cke s of Quantum f o r a quantum computer i s t h e s p i n -a t h eo r y , a s a n a t u r a l

i s t o b u i l d i n g b lo ck f o r a c l a s s i c a l co mp ut er i s t h e s e t w i t h twoe t h e b a s i c e l em e nt s .

, in s te ad of Table I i n d i c a t e s i n f a i r d e t a i l how t h e p r oc e ss e s ofi n d oi ng . p r o p o s i t i o n a l c a l c u l u s w hi ch f i g u r e i n t h e s y n t h e s i s ofo p re s e n t a computers i n modern au tomata theory a re t o be t ra ns l a t edly in d i ca te th e f rom Boolean a lgebr a in to Hi lb er t space theory . Perhaps the

t h a t t o my only element of novel ty involved i s th e need t o go beyondI w i l l the lowes t-order p ro pos i t i ona l ca lcu lus . We make up computersth e end of out of th in gs l i k e words, sequences of b i t s , and we must look

wandered i n t o forward t o making a pa th ou t of sequences of quantum elementsa n d u n l imi ta b le f o r example, s o we n ee d th e p re s c r ip t io n s f o r making such

assemblages ou t o f in d iv idu a ls .P u t t i n g i t d i f fe r e n t ly , we ne ed t o go d ee p er in to th e

p ro p o s i t io n a l c a l c u lu s a n d d e a l w i t h s y stems h a v in g in t e r n a lg p r i m a r i l y s t r u c t u r e .mass f igure . This i s d e a l t w i t h i n a hi g h e r o r d e r o r p r e d i c a t e c a l c u -

l u s i n o rd inary log ic , bu t i n quantum mechanics we encounte rn which t h e th e same problems eve ry time we do many-body th eo ry . We know

ion of t ime, how, given a theor y of two ob je ct s , t h at i s two Hilber t spaces ,t o make a theory of t he pair, which i s o ne of th e b a s ic s t e p s

of l i g h t . i n the syn the s is o f computers. Another o f th e importan to p e ra t io n s i s going from an obj ect t o a new one cal le d ana r b i t r a r y s e t of s uc h o b j e c t s . Th i s i s so me times c a l l e d th e

e m pt s ta r p ro c e ss i n a uto ma ta s y n th e s i s . The corre spond ing pro-t h a t much c ed ur e i n s e t t he or y i s going from a Set S t o 2 . I n au e and a case where an ob jec t i s descr ibed by a Hilb e r t s p a c e S t h e

immediate obs ta cl e is : what do we mean by two t o th e powera t of a ~ i l b e r t pace? I n f a c t , t h er e i s a b e a u t i fu l c o r re s p o n -

and th e dynami- dence between th e laws of s e t the ory and th e laws of thef a i r l y e as y e x t e r i o r a l g e b r a o v e r a H i l b e r t s p a c e . T hi s i s t h e a lg eb ra

e y st ep s . t h at comes i n whenever we do th e theor y of many fermions.


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when we t h a t we si ng le d ou t a t t he log ica l l ev e l . What a re thes t a t i s t i c s , c or re sp on di ng t h i n g s a t the geometr ic leve l?

A l l th e concepts of space and time can be expressed al son o t f o r i n a t h eo r y of a p a r t i a l l y o r de r ed s e t . ( I t ' s odd tha t theer en t same to o l should work twi ce. ) Space-t ime can be regarded

I a s a c a us a l meas ur e sp ac e: t h a t i s t o s a y, a l l t h e m e t r ici s concepts can be expressed i n terms of two thin gs : the measure

or i s n ' t , o f a space-t ime volume, and the re la t i on of cau sa l i ty . Forn the Fermi- example, th e dis tan ce between two eve nts can be defi ned i n

terms of the measure of the cau sal i nt er va l between the twos of auto- events. The topology of space-t ime can be defined i n termsproper ty of a system of neighborhoods consi st i ng of c aus al in te rv al s.as s ub - I n f a c t , I would s ay t h a t r e l a t i v i t y g a i n s by t h i s t r a n s l a -

r a l o b j e c t s t i o n i n t h a t t h e t h e o r i e s of i t s e n t i r e s t r u c t u r e , from t h ed i f f i c u l ty topo log ica l l e ve l up t o the l e ve l o f c ausa l i ty and geodesic s,

me con cep ts can a l l be expressed i n terms of thes e two t h ing s i n ana logyntum tr an sl a- wit h th e way we develop th e correspo nding th eo ri es of th eth e noti on one-dimensional t ime axi s; whereas i n th e ordi nary developmente usua l the topology and the metr ic st ru ct ur e ar e somehow divorced

I found i n th a t we do not use the th in gs corresponding t o spheres i necedence gene ra l r e l a t i v i ty t o de f ine t h i s topology. It would giveI n c l a s s i - r a t h e r odd r e s u l t s i f we s a i d t h e d i s t a n t s t a r s a r e i n t op o-

one ra t he r log ica l con tac t w i th u s ju s t because we rece ive nu l l r aysa from them.

an s i t i v i ty ) So: Space -t ime i s a ca us al measure spacs5. Where i nof Hi lb er t the theo ry of quantum automata do we f i n d th e correspo nding

immediate concepts? Right on the surf ace, wait ing f o r us t o grasp them.ce. We know Fi r s t , the ru l e fo r t r an sl a t in g measure i s derived from

the the lower l ev e l . Measure i s ba s i ca l ly a l og ica l concept i nn of neg- the theory of. automata: you ju st count proc esse s. So I w i l l

consider t ha t our cau sal measure space i s t o have a measureeach le ve l der ived by count ing . In the quantum theory count ing i s done

number by the t r ac e opera t ion . Sta tements of loc a t i on i n space- timei n t h e hope i n a t h e or y of t h i s k i n d a r e r e p r e s e nt e d i n H i l b e r t s pa ce ,

l e l se w i l lwhole mechanics ar e repr esen ted i n Hi lb er t space i n quantum mechanics.

y ampl i tudes , The only th ing th a t r emains i s t o spec i fy a c a u s a l s t r u c t u r e .e r t space , Where i n au tomata do we f i nd th e caus a l re la t i on but i n

la t i on s the f ac t tha t some computa t ions must take p lac e before o the rspace compu ta tions, i n the re l a t i on o f log ica l dependence. I n f ac t ,


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i n von Neumannls be au ti fu l comparison of th e computer andth e brain , one comes very cl os e to d isc uss ing th e geometryof th e bra i n or of the automaton i n terms of ju s t th e onenotion of what I w i l l c a l l lo gi ca l precedence. The measure-ments t h a t von Neumann makes on comnuters and br ai ns he

Arithm etic depth i s t he maximum number of lo gi ca l l y dependentpro ces ses . Arithm etic br ea dt h i s th e maximum number oflog i ca l l y independent , concur ren t p rocesses i n t he cha in .

compute rs a r e s t i l l e s s en t i a l l y one d imensional , hav ing pe r -h a s a l o g i c a l b r e a d t h o f - 1 0 . F or t h e b r a i n i t ' s abouti;10 i n both di rec t io ns . You might say th at man i s a two-dimensional cr ea tur e. I f we wish t o model space-t ime we w i l lhave t o t h in k even more i n t e rms o f such h igh ly pa ra l l e l o rasynchronous computat ional models. Each event i n space-t imei s somehow a ca lc ula t io n going on independent ly of thoset h a t o cc ur i n s p ac e li k e s u r f a c es r e l a t i v e t o i t , and i f youpush th e d ur ati on of th e fundam ental s te p down below 10 -14cmf o r s a f e t y , t h e n t h e a r i t h m e t i c d e p th of t h e u n iv e r se i s a t

number of course, expr essing t he four-dimensional n atu re of thecomputat ional proce ss t h at we must seek t o model . The poi nti s , g iven these two bas ic no t ions of cause and measure fo rautomata a complete log ic al th eory of th e geometry of automatacan be worked out.

Let me ju s t mention two examples, one motiv ated e n t ir e l yby th e i de a o f a lo gi ca l model wi thout any consid erat ion ofr e l a t i v i s t i c i n va r ia n c e, and t h e n a m o d i f ic a t i o n of t h i s t omake i t e x a c t l y L o re n tz i n v a r i a n t .

Le t ' s cons ide r t he s imples t k ind of s e r i a l computat ion,which I c a l l the binary code. Suppose we s t a r t f rom a s ing l ek ind o f b ina ry d ig i t r epresen t ing two a l t e rn a t i ve s which youcan thi nk of as 0 , a move forward i n t ime and a s te p t o ther i g h t o r 1, a move forward i n time and a s t ep t o t he l e f t i na kind of checkerboard diagram. We bu i ld a pat h as a sequenceof such t h ings , so l e t me c a l l t h i s bas i c t h in g ou t of whichwe w i l l assemble space-t ime a l in k. Let us pass t o th e quantum


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th e geometryus t th e one. The measure-

number ofn t h e c h a in .

ements ofo r space. Today

h a vi ng p e r -i t ' s about

i s a two-e-time we w i l l

h l y a r a l l e l o rn space -t ime

ly of t hosei t , and i f you

i s a to f t h e

l na tu re o f t he. The po in t

n of t h i s t o

from a s i n g l ea s t e p t o t h e

e l e f t i na sequence

n g out of whichs t o t he quantum

theory by desc r ib ing t h e l i nk 1 by a two-dimensional Hi lb er tspace. The no tio n of a p a t h IT i s t h e n a w e l l- d e fi n edHi lber t - space concept , a quantum sequence of such l i n k s :n = seq 1. For the end-point of th e path , i t suf f ices s implyt o i gnore o rde r , t o s ay t h a t two pa ths cor r espond t o t he samepoi nt i n space- time, have the same end-point, i f they di f fe ronly by a permutat ion of t h e i r l in ks . The Hi l ber t spacedescr ibing the poi nt of space- t ime p i s then got ten f romthe Hi lbe r t space desc r ib ing t he pa th n i n t h i s p ri mi ti vespace-t ime by a symmetrizat ion procedure t ha t l ea ds from theusua l d i r ec t p roduc t t o t he f am i l i a r Bose-E ins t ein quan t i za-t i o n . A poi nt of space- t ime of t h i s model i s a Bose-Eins te inensemble o r s e r i e s of two- s ta t e ob j ec t s , l i nk s desc r ibed bya two-dimensional Hi lb er t space: p = s e r .

I ' v e g iven t he l owes t l e ve l o f t h i s mode l. The nex tt h i n g i s t he causa l s t ru c tu re : wha t does i t mean t o say onesuch ensemble 1 comes bef ore anoth er p i n th e assembly2proce ss? The simpl est procedure i s t o s ay t h a t p 1 P 2 i fthe number of l i nk s of each kind i n p i s smal l e r , so t h a t

1you can ge t to p , nt u i t i ve ly speak ing, by j us t adding2more l in ks and not sub t rac t ing . This can be expressed i n

terms of quantum symbolic log ic qui te t r iv ia l l y . Consider ingtwo po in ts p 1, P 2 and a kind o f l i nk 6 , f o r e a ch p o i n tt h e r e i s a number opera tor n6 (1 ) , n6 (2 ) . Then the pr i mi t i vemodel f o r t h e c a u s a l r e l a t i o n i s

I f we go over t o t he c l a s s i ca l l i m i t of t h i s quantumtheory, which I do very chi ldishly jus t by dropping thecommutators, we o bt ai n a c l as s i ca l caus a l measure space S,t h e r e f o r e a c la s si c al geometry which we can then lo ok a t a s ag eo me tr ic al o b j e ct i n i t s e l f . What i s i t s s t r u c t u r e ? I t ss t r u c t u r e i s t h e f u t u r e n u l l c on e N of s p e c i a l r e l a t i v i t ywi th exact ly the fami l iar Minkowsky measure and causal order ingtogeth er wi th what has t o be counted as an i n t e r n a l c o o r d i n a t e ,a s i n g l e a n g l e: S=NX S 1 .

From the fu tu re n u l l cone N+ i t i s of cour se a t r i v i a -l i t y t o a ss em ble a l l of space-time by very sim ple formal pro-cedures. I n pa r t i cu l a r , two words i n t he b ina ry code p ro -


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of the null cone. The theory is not exactly covariant be-cause the commutation relations of two harmonic oscillators,which is what we are dealing with here are not invariantunder SL even though the causal ordering C is invariant

2under SL2 . The non-covariance# f the commutation relationsdisappears in the classical limit. That's why we end ur, with

We then are faced with a decision between two models;

1; one covariant only in the classical limit, which suggests6 that if we look at fine enough regions in space-time it is' i not inconsistent to imagine that departures from special:I! relativistic covariance show up; the other, exactly covariantat all distances, obtained by replacing the commutation relations for two harmonic oscillators by the commutation re-lations of the Majorana representation. If you like you cansay that what we have here is a new interpretation for theMajorana representation and algebra. The idempotents in itcan be regarded as statements of location in a space whichcould be regarded as a quantized null cone in which thecovariant relation p Cp within the Majorana algebra plays

1 2the role of the causal order. So I call this exactly invari-ant theory the Majorana space-time. Two of these modeis suffice to make up a four-dimensional space-time, but when youdouble up on the number of external coordinates, you alsomultiply the number of internal coordinates. It turns outthat the full structure of the algebra of the automaton

1 generating what I would call two words in the binary codeis that of the solid future cone C+ multiplied not into S1I, but into U(2,C)=S1 x SU . These are internal coordinates

2in the simple sense that changing the values of these coordi-iI; nates does not produce any causal separation.Let's ascend to the dynamical level, where the action is. 11I'I What is the form in which the laws of nature should be ex- // pressed in such a space-time? Guided by our success in deal-

I;! ing with partial orderings of the two lower levels, it is 44 suggested that we try and express the laws of mechanics also

I /I r by the theory of a partially ordered set. (1t is encouraging It,,I that in the theory of thermodynamics, which really belongs1 I, to the same level as dynamics, such a formulation in terms of /I partial ordering is actually more unified and beautiful than


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n t b e- t h a t , s ay , i n t er m s of an en t ropy p r in c ip l e , o r t he morec l a s s i c a l o n es . A l l of th e thermodynamics h as been e q r e s s e da n t a s a t h eo r y of a p a r t i a l l y o r d e re d s e t i n w hi ch t h e o b j e c t sa r e t h e s t a t e s a nd t h e p a r t i a l o r d e r in g i s t h e r e l a t i o n o fla t i on s the ex i s t ence o f a na tur a l p rocess go ing from one s t a t e t oanoth er. The entropy appears a s a v a lu a ti o n o f t h i s p a r t i a l

order ing i n the same way t ha t t ime app ears as a valua t ion ofde ls ; t he impl i ca t i ve o rde r ing of t he p rev ious two l ev e l s )i t i s Feynman has exhibited a model of quantum theor y i n a dis cr et espace, which i s a b ig s t ep t owards a quantum space. I t i s

i n f a c t a two-dimensional model of th e Dirac equa tion. I fr e - you think of a two-dimensional space-t ime i n the form of acheckerboard, i f you suppose th a t only th e black squares can

be occupied as i n the game of check ers and th a t a man can s te pt h e f or wa rd t o t h e r i g h t o r s t e p f o rw a rd t o t h e l e f t , t h en a g a i n

s i n it a p a t h i s obviously a binary sequence, and the elementarylink s a bina ry d i g i t . Feynman po in t ed ou t t h a t t h e Di racequa tion on t h i s simple model cou ld be de rived from th e lawt h a t t h e t r a n s i t i o n a mp li tu de f o r a p a t h i s (im)R where Ri s the number of re f l ec t i on s of mot ion a long th e path7. Thed i f f e r e nce form of t h i s law i s t he s t a t ement t h a t t he ampl i-tude y f o r t h e p a t t e r n

code t e r ns wi th a ce r t a i n coheren t phase :S1

Thi s i s i n f a c t t h e D ir ac e q ua ti on i n t h i s s pa ce , a l -i s . though i t i s n o t e n t i r e l y f a m i l i a r l o o ki n g . I n t h e c a s e

be ex- where we go over t o wave-functions which change s l i g h t l yi n dea l - ove r a s i n g l e square , so t h a t t h e i r changes can be accura t e ly

i t i s r epresen t ed i n t e rms o f de r iva t i ves , t he f i r s t two ampl it udes

l on gs i n t e r p r e t a t i o n as t h e p r o b a b i l i t y a mp li tu de o f j i t t e r ( z i t t e r )t e rms of pe r chronon , t he p roba b i l i t y ampli t ude fo r r e f l ec t i o n o f

f u l t h a n m ot io n i n one u n i t of time on t h i s l a t t i c e . The argument i n


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the ampl i tude y i s not mere ly a po in t i n space , bu ta point and a di r ec t i on , which i s s u f f i c i e n t t o d ef i nea p a t h i n t h e c l a s s i c a l mo tio n.

The space of pa ths of t h i s model i s the co nf igu rat ionspace of the l in ea r I s i ng model and the qu ant i ty R i nFeynman's t ransi t ion ampli tude i s t he pa i r Hami l ton i an fo rthe l in ea r I s in g model . The gen era l iza t ion f rom two di sc re tedimensions t o a covariant model i s s imply t o r ep l ac e t heIs in g model by th e l i ne ar Heisenberg model, th e I s in g Hamil-ton ian op erator by th e Heisenberg pa i r Hami l tonian.

But I must st o p now.


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t TABLE I. CONCEPTS OF QUANTUM LOGICProposi t ional Sys tem

Notat ion Representa t ioni n System, Object a,b , ..., A,B ...P ropos i t i ondi sc re te p ,&, . . . subspacei n c l u s i o nI d e n t i t y p r o p o s i t i o n I , I a t he Hi lbe r t spaceH a m i l -Nul l p ropos i t i on t he ze ro vec to rO r , ad junc t ionAnd, conjunction i n t e r s e c t i o n

N orthocomplementor thogonaldimension

P,q, . . opera to rF unct ion of a quan t i t y f ( p ) , p2 , . . . c f . f u n c t i o n a lc a l c u l u sP ropos i t i ona l func t ionof a quan t i ty P(P) PX , . " pred i ca t e"

s i n g l e t , p u re s t a t eCalculus of Pro pos i t io nal Systems

Sum, disjunction d i r e c t sumd i r e c t p r od u ct

Theory of Binary Rela t ion ssubspace of Iax Ibexchange subspace

T r a n s i t i v e a ~ bl ~ R C C aRcFun cti on, Mapping F: (Pc Q) c F ( P ) c l i n e a r

~ F ( Q ), I F ( P ) I I I P I t r ans format ions e t a, 2" (F.D) ex t e r i o r a lgebra

S e r i e s of a ' s s e r a (B.E.) symmetricSequence of a ' s seq a (M.B.) ten so rs on I&


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Concept

QuantifiersNumericalUniversalExistential


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C

337

TABLE: 11. CONCEPTS OF GEOMETRY OF LOGICAL NETS

Concept Notation Representation

Point (Event) P Computational step

Measure (of point set) \ P I CardinalityCausal precedence PCP Logical precedenceTime-like path 7~ Maximal well-ordered set

Space-like surf ce Z Maximal non-ordered set


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REFERENCES1. J.M. Jauch, Foundations of Quantum Mechanics, Addison-

~esley, 967. ,2. R.P. Feynman, The Character of Physical Laws, M.I.T.

Press, 1967.3. See J. Hartmanis Lectures on Automata Theory, Tata Insti-

the Brain, Yale UP, 1958.4. For further references see D. Finkelstein, The Physics

of Logic, IC/68/35, International Center for TheoreticalPhysics, Trieste, 1968.

5. For related studies of quantized geometry see especiallyH. Snyder, Quantized Space-Time, Phys. Rev. B, 38 (1947);C.F. v. Weizsacker, E. Scheibe and G. Sussman,Komplementaritat und Logik 111, Zeits, f. Naturforschung13a, 705 (1958); and C.F. Weizsacker, Quantum Theory and-eyond (preprint)For further references see D. Finkelstein, The Space-TimeCode IC/68/19, International Center for TheoreticalJPhysics, Trieste, 1968.

6. R. Giles, Mathematical Foundations of Thermodynamics,Pergamon Press (1964)

7. See R. P. Feynman and A. R. Hibbs, Quantum Mechanics andPath Integrals, McGraw-Hill, 1965.


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DISCUSSIONM.I.T.

JAUCH The exposition of Professor Finkelstein offers suchTata Insti- dazzling possibilities that if I make a simple question or

two it will not do justice to all the richness of what hehas presented us; but I would like to make one statement and

Physics one question.Theoretical First, the motivation of all this was given in terms of

Riemannls question about the replacement of the continuum bydiscreteness. However, when you start out you throw continui-

9,38 (1947); ty out the front door and it comes in the back door again,, namely through Hilbert space. Hilbert space, of course, isconstructed with coefficients from the continuum, and so

m heory and continuity comes in; at the lower level to be sure, butstill it is there, somehow you will not be able to get rid

Space-Time of it, and you have to live with it.The second is a question, simply a technical question.

I did not quite understand: did you get the three-plus-onedimensionality of space, or is that an input that you haveFINKELSTEIN It was put in by hand when I said the binarycode. If you take the singulary code, you get a one-dimensional space-time consisting of nothing but a time axis,plus a single internal degree of freedom having nothing todo with causality, a kind of Newtonian world. If you wanted,say, a nine-dimensional world yould nly have to use aternary code with three basic characters.

As for the continuity, remember it is not discretenesswhich is the goal, but finiteness. I don't ever want tohave to do an integral again as long as I live. I want todo nothing but finite sums, and if we work with finite dimen-sional Hilbert spaces, we find nothing but finite sums tobe computed, even though they possess the full continuoussymmetry group of, in this case, the Lorentz group. In-cidentally, I should mention that in this kind of a model,and in fact all the ones I've exhibited, time is the numberof chronons. This is also the dimensionality of the Hilbert


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At the FINKELSTEIN Exactly. Back in the quantum theory there mustbe some remnants of this invariance and I have not fullyexplored it yet. There is always available something like

in el~g tro n) ,

the ruture tlme-translat~on. ~ n l s S the seml-group, ratherthan group, of future-time-like translations by discreteamounts, multiples of course of the fundamental constant r .The spectra that one gets for the coordinates in the quantum

what one

we know

ppens in

theory before going to the classical limits varies slightlyfrom model to model, but typically, in spite of the exactLorentz covariance, one might have that t is an integermultiple of the fundamental constant, whereas x,y and zpossess purely continuous spectra.WIGNER I am a good deal confused by a number of thingsthat you said, particularly about the invariance of the theory.You don't postulate Poincare' invariance, but you do postulateLorentz invariance and time-displacement invariance, ordid I mishear that?FINKELSTEIN In fact, I've described several theories sothat it's understandable that the hypotheses could get garbled.In my initial work I postulated no invariance at all. I simply

I looked for quantum models of binary computation procedures,I and was rather shocked to discover that the simplest non-

I

trivial model possessed Lorentz invariance in the classicallimit. Then I noticed that one could restore full Lorentzinvariance in the quantum theory by a slight change in thecommutation relations and exhibit a whole class of othermodels. These models still lack time-transitional invarianceas unitary transformations, as one might expect for a theory

sum. Rinht which contains only the future light cone. A time-translation-re- f does exist, which is not unitary but an isometric linear trans-e evi- formation in the quantum theory.WIGNER I see, but you don't have time-translation invariance.

FINKELSTEIN Time-translation is not represented by a unitaryZI I transformation. Time-translation by discrete quantities is+mm e 9 1 represented by an operation something like the excitation

WIGNER The question which is not terribly clear is simplythis: if you have time-translation invariance and Lorentzinvariance, by the combination of the two you have alsoinvariance with respect to every other translation. Now,


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Coral Gables Conference onI1 FUNDAMENTAL INTERACTIONS

AT HIGH ENERGY

CENTER FOR THEORETICAL STUDIESJANUARY 22-24, 1969 UNIVERSITY OF MIAMI

Timm Gtldehus, G eoflrey Kaiser,and Arnold Perlmutter

EDITORS

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D. Finkelstein- Space-Time Structure in High Energy Interactions

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