Time Topology for Some Classical and Quantum Non-Relativistic Systems

Stanisław Olszewski

Time Topology for Some Classical

and Quantum Non-Relativistic

Systems

Studia Philosophiae Christianae 28/1, 119-135

S tu d ia P h ilosop h iae C hristianae A TK

28(1992)1

ST A N IS Ł A W OLSZEW SKI

TIME TOPOLOGY FOR SOME CLASSICAL AND QUANTUM NON-RELATIVISTIC SYSTEMS

1. Introduction. 2. The observer; h is birth, life and d eath. 3. T w o k in d s of the scale of tim e. 4. S im p le e x a m p les of an e x te r n a l observer. 5. P ertu rb ation of a q u an tu m -m ech a n ica l sy stem and top ology of tim e. S. P ertu rb ation en erg y obtained from a circular scale of tim e. 7. E li­ m in a tio n p rin cip le for eq u al tim es. 8. S u rvey.

1. IN T RO D U C TIO N

A ccording to E. K a n t tim e an d space a re tw o ’’fo rm s” of perception. The background of th eir m eaning is m ain ly in ­ tu itio n al \ In physics tim e is — in principle — a coordinate: in n o n -relativ istic physics, th is coordinate m ay v a ry in d ep en ­ d en tly of th e spatial coordinates, w h ereas in a relativ istic ca­ se the tim e coordinate is coupled w ith its sp atial co u n terp arts. This situ atio n holds eq u ally fo r classic a n d q u a n tu m theories. In a sta tio n a ry state of a system described by th e n o n -re la ­ tivistic q u a n tu m m echanics, tim e e n te rs only th e phase fac to r of the w ave fu n ction w hich has no influence on th e calculated av erag ed q u a n titie s corresponding to th e observed data. In effect, in a sta tio n a ry sta te no change in tim e of th e observed q u a n titie s m ay occur.

In tu itio n ally, in classical a n d q u a n tu m m echanics (as w ell as in special relativ istic theory) we assum e th e tim e coordina­ te m ay v a ry from a m inus in fin ity called ’’the p a st” to a p lus in fin ity called ’’th e f u tu r e ”. A tim e scale of this k in d w e call linear; see Fig. 1. The purpose of th is p ap er is to indicate th a t a tim e scale w hich is d iffe re n t th an lin e a r m ay be — a t least m some cases — of a b e tte r use. In our considerations w e disreg ard fu lly th e relativ istic theo ry : w e n eglect the coupling betw een tim e and o th er (spatial) coordinates into one space- -tim e m etrics as w ell as the pro b lem of tim e in te rv a ls and

1 W. S. F ow ler, T h e D e v e l o p m e n t of S cie n tif ic M ethod, P ergam on P ress, O xford 1962.

th e ir m easurem ents. The tim e coordinate is assum ed (sim ilarly to L e ib n iz 2) solely to be a p a ra m e te r w hich allow s us to disting u ish b etw een ’’e a rlie r” an d „ la te r”.

The p roblem of tim e topology has its rich bibliography \ In this p a p e r we try to ap p roach th is p roblem on the basis of an analysis of sim ple physical system s, both classical an d q u antu m . For th e classical case we lim it ourselves to periodic system s. A ccording to T h irrin g 4 an alm ost periodic tim e evo­ lu tio n is a general p ro p e rty of sm all system s, w hereas fo r larg e system s this evolution exhib its a chaotic behav iour con­ nected to a m ixing o f th e observables. F or the q u a n tu m case w e exam ine th e tim e evolution of a system su b m itted to the action of a sm all e x te rn a l field called p e rtu rb a tio n an d deno­ ted b y V; w e consider only a system having n o n -d egenerate q uan tu m -m ech an ical states, so th e system energies correspond­ ing to d iffe re n t states are d ifferen t. B eyond the p e rtu rb a tio n V, w e assum e th e system is a fu lly isolated object.

We fo rm u late th e problem of tim e topology for a q u a n tu m - -m echanical : system as follows: In a v e ry d ista n t past, th e system w as in a u n p e rtu rb e d sta tio n a ry sta te (P) in w hich :— in th e absence of V — it could rem ain in fin itely. A t some m om en t th e system w as p e rtu rb e d by V, an d in a v e ry d ista n t fu tu re -it w ill be found in a n o th e r sta tio n a ry sta te (F); w e

2 G: W. L eibniz, Confe ss io Philosophi. Ein, Dialog, K losterinarm ,

F ra n k fu rt a/M., 1967.

3 H, Reichenba.ch, T h e P h il o s o p h y o f S p a c e an d T im e , D over P u b li­ cation s, N e w Y ork 1958; J. J. C, Sm art (ed.), P r o b l e m s of S p i c e an d

T im e , M acM illan, L ondon 1964; T. G old (ed.), T h e N a t u r e o f T im e,

C orn ell U n iv ersity P ress, Ithaca 1967; H. R eich en b ach , T h e D irectio n

of'-'Time, U n iv ersity of C alifornia P ress, B erk eley 1971; A.. Griinfoaum, P hilo so p h ica l P r o b l e m s o f S p a c e an d T im e , 2nd ed., A. R ied el,. D o r­

d recht 1973; J. R.. Lucas, A T r e a ti s e on T i m e an d S pace, M ethuen, L on ­ don 1973; L. Sklar, Space, T i m e an d S p a c e -t im e , U n iv e r sity of C a li­ fornia' P ress, B erk eley 1974; I. H in ck fu ss, T h e E x iste n c e of S p a ce 'an\d

T im e , C larendon P r e s s /O x f o r d 1975; G. N erlich, T h e S h a p e of .Space,

U n iv e r sity P ress, C am bridge 1976; W. H. N ew to n -S m ith , T h e S t r u c t u ­

re of T im e , R ou tled ge an d K eg a n P au l, L ondon 1980; G. J. W hitrow , T h e N a tu r a l P h il o s o p h y of T im e , C larendon P ress, O xford 1980;

D. H. M ellor, R eal T im e , U n iv e r sity P ess, C am bridge 1981; R. S w in ­ burne, S pace a n d T im e, 2nd ed., M acM illan, L ondon 1981; E. Jaques, . T h e F o r m of Time,. R u ssa k -H ein em a n n , N ew Y ork 1982; M. F ried m an n ,

F o u n d a tio n s of S p a c e - T im e T heorie s, P rin ceton U n iv ersity Press, P r in ­

ceton 1983; R. L e P o id ev in , R e la t io n i s m an d T e m p o r a l T o p o lo g y : P h y ­

sics, or M eta p h ysics ? , „The P h ilo so p h ica l Q uarterly” 40 (1990), 419—432.

' 4 W. T h ir r in g ,’ L e h r b u c h d e r M a t h e m a t is c h e n P h y s ik . B a n d ' _4:

assum e P φ F. The question w e p u t h e re is w h a t is th e kind of tim e p a th w hich is follow ed by th e system on its w ay fro m P to F. B efore we approach th e problem of a tim e p a th fo r a q u an tu m -m ech an ical case w e exam ine — in Section 2— 4.

— the tim e topology for classical periodic system s. 2. THE OBSERVER; H IS B IR TH , LIFE A N D D EA TH

As a firs t step, let us point out th a t a tim e toplogy can be a subjective notion depending both on th e p ro p erties of the ex am ined system as w ell as th e p erception ab ility of the ex am in er (an observer). We define th e o bserver as a being who can do (and register) m easurem ents. F o r exam ple, we m ay im agine an o bserver w h o can reg iste r solely th e coordinate- X of th e position of a body in a C artesian coordinate system . A nother observer can reg ister, say, th e x an d y coordinates of th is system ; th e th ird observer can m easure all th re e kinds of coordinates: x, y an d z. B eyond these coordinates w e m ay h av e observers w ho can m easure — for exam ple — one, o r m ore, velocity com ponents of the body; o th er observers can reg iste r o th er body prop erties, for exam ple its m ass. Finally , w e m ay im agine an o bserver whose percep tio n an d m easuring a b ility is infin ite. We call Him an E x te rn a l O bserver.

A ny observer who is able to accept only a fin ite set of observations of a system is an o bserver connected — in some w ay — w ith th a t sytem . He d iffers fro m a n o th e r observer w hose p erception ability, i.e. th e (finite) set of m easurem nts, w hich can be done by him on the system and reg istered, is differen t. It is easy to note th a t th e perception a b ility of an observer depends on th e system properties, in p a rtic u la r th e system com plexity. F o r exam ple, fo r a system w hich is a sta rig h t line identical w ith the x axis, th e re is no problem of how to m easu re th e space coordinates o th er th a n x; sim i­ la rly in o rd er to m easu re a position in the system w hich is a p lan e th ere is no p roblem of how to m easure m ore th a n two kinds of in d ep en d en t coordinates, say x an d y. L e t us assum e th a t beyond observables d iffe re n t th an tim e an y observer can also m easure tim e. This m eans th a t he is able to distinguish betw een an e a rlie r and a la te r event; an ev en t for an o bser­ v e r m eans — as a ru le — a m easu rem en t. The tim e m easu­ re m e n ts enable him to establish th e sequence of events. T he s e t of all m easu rem en ts in in sta n ts t ; enclosed w ith in an in te rv a l of tim e done in sequence by an observ er w e call th e life in te rv a l of th is observer. The e arliest of th e observations

done by th e o bserver (corresponding to th e sm allest tj) w e call his b irth ; the la te st observation (corresponding to th e larg est t;) w e call his death. We assum e th e n u m b er of o bser­ vation s in th e set of m easu rem en ts rep re sen tin g th e life of an observer is finite, although no lim it can be im posed a p rio ri on it.

The in sta n ts of tim e t; m easu red by an o bserver durin g his life can be rep re sen te d on the tim e scale given in Fig. 1. These in sta n ts fill·th e in te rv al BD; see Fig. la.

3. TW O K IN D S OF THE SC ALE OF TIM E

L et a sim ple (linear) h arm onic oscillator oscillate along th e X axis. W ith th e oscillator an o bserver is connected w ho can m easu re th e values of the x coordinate; a n y m ea su rem e n t of X is accom panied b y a corresponding m easu rem en t of t. T he oscillator moves, say, firs t in th e direction of positive x, n e x t it goes to negativ e x. We assum e th e o bserver s ta rts his m ea­ su re m e n ts im m ed iately a fte r th e oscillator m oves across th e point x = 0. This m eans th e life of th e o bserver s ta rts a t c e rta in position

χΒ = ε δ;0 (1)

for w h ich t = t B. N ex t x increases u n til th e positive value close to th e am p litud e of th e oscillator is atta in e d ; th e n x begins to decrease, becom ing in some in sta n t, a negative num b er. A t the sam e tim e th e sequence of th e m easured in sta n ts tj is an increasing set of num bers. A fter th e sm allest x is a tta in e d the v alu e of x increases. The larg e st of the increasing b u t negativ e x is some

XD = - s ' ^ o. (2) The n e x t m easu red valu e of x is

Xd+ 1 = s £ 0 ( 3 )

so

XD+1 == X

(4)

because w e assum e th a t th e ry th m of the oscillator as w ell as th a t of th e m easu rem en ts of χ rem ain s unchan ged in th e course of th e oscillations. This situ atio n rep e a ts in fin ite ly in accordance w ith th e d efinition of th e oscillator as a p erfectly perio dic system . An im p o rta n t fe a tu re of a p e rfe c t periodic system is th a t we have no physical p a ra m e te r w hich allow s

us to distinguish b etw een one oscillaltion period a n d an o th er. If we assum e th a t th e re exists a one-to-one correspondence betw een th e m easurem en ts of x an d those of t, the full perio­ dicity of the sytem im plies th e sam e periodic p ro p e rty fo r th e tim e variable. In effect, th e tim e scale for the oscillator should be n o t lin ear b u t form s a closed line leading to th e eq u ality

Id-;] — tB (5)

sim ilar to (4). This p ro p e rty is rep re sen te d in Fig. lb. Topolo­ gically the scale is eq u ivalen t to a circle; see Fig. 2. The life

betw een the b irth (tB) and th e death (tD) of the observer con­ n e c te d w ith the oscillator will re p e a t in fin ite ly a n d n o th in g in the system allow s for him to d etect the repetitio ns. These rep e titio n s can be discovered only by a n o th e r observer whose ability to do m easu rem en ts is larg er th a t th a t of the o bserver connected w ith the oscillator. Me call such an observer an e x te r­ nal observer (not capitalized). In o rd er to discover th e re p e ti­ tions of th e life of the observ er connected w ith th e oscillator, th e e x te rn al observer should have th e possib ility of counting th e oscillations. A n exam ple of such an e x te rn al o bserver is given in th e n e x t section.

4. SIM PLE E X A M PLE S OF A N E X T E R N A L OBSERVER L et us consider e a rth Z w hich circles abou t sun S along an ellipse. A n observer is connected w ith th e system . W e call him an in te rn a l o bserver (i.o.) an d assum e th e observer can m easu re only the distance r betw een S an d Z a n d can re g iste r th e in sta n ts of tim e t corresponding to d iffe re n t r. L et us assum e th e m easurin g ab ility (perception) of th e observer begins to w ork a t the sm allest possible r — r min. The firs t m easu rem en t r B gives th e sm allest valu e of the increasing r, w hereas th e last m easu rem en t, r D, gives th e sm allest valu e of th e decreasing r. The set of m easu rem en ts rep e a ts in fin ite ly w ith o u t th e possibility of detection of th is fac t by th e obser­ ver. Because of th is periodicity, th e p ercep tio n of th e obser­ v e r goes to zero and begins to w ork again ap p ro x im ate ly in th e sam e in sta n t of tim e, viz. w h en th e distance r D яь r B is attain ed . The tim e scale is v e ry m uch sim ilar to th a t obtained in Sec. 3 for th e harm onic oscilator: it begings a t t B, th e tim e w hen th e distance r B w as m easured, a n d ends a t t D, th e tim e w h en th e distance r D w as attain ed . The to ta lity of th e m easu­ rem e n ts rep re sen ts th e life of i.o. a n d — since th e system is

fu lly periodic — a c irc u la r Scale of tim e should be applied. The i.o. cannot d etect th a t t D æ t B.

Now let us introd u ce an e x te rn al observer (e.o.) w ith resp ect to th e in te rn a l one; the e.o. can m easu re m ore kinds of p a­ ra m e te rs th a n i.o. We assum e also th a t e.o. can rea d th e m easu rem en ts of i.o. b u t n o t vice versa. For exam ple, besides th e distance r = |S Z | th e e.o. can m easu re th e angle of th e direction SZ in resp ect to certain co n stan t direction SA w hich lin k s sun S w ith some s ta r A; see Fig. 3. If th e re is no precession of th e ellipse ab o u t th e ax is p e rp e n d icu la r to the ellipse plan e and going th ro u g h th e point S, th e angles

θ = <£ASZ (8)

m easu red in each course of Z abou t S w ill re p e a t exactly. L e t us assum e th e percep tio n of e.o. is sw itched on at th e angle

Hm = (<ïASZ)m (7) w hich is th e angle b etw een SA and SZ in th e case of

|sz|

C ' ·θ* = * . + « " (9)

an d th e last m easu rem en t gives certain

aD. = am+ 2K -e"',

(10)

because th e n e x t m ea su rem e n t of ϋ· is # B providing w e assum e th e e a rth m ovem ent a n d th e ry th m of the m easu rem en ts a re fu lly periodic. So th e life of e.o., though ric h e r th a n th e life of i.o. by th e set of m easu rem en ts of â b e ttw e e n $ B an d i?D, has th e tim e scale identical to th a t of i.o.

Now le t us assum e th e ellipse p erform s a slow precession in its plane. The life of i.o. does not change, b u t e.o. m ay now see a n d label d iffe re n t lives of i.o. This is so because th e a n g le -&m, th ere fo re th e w hole set of th e observed angles b etw een # B an d ê D, is now d iffe re n t for an y course of Z about S. T h e distinction b etw een two d iffe re n t courses allow s e.o. to discover th a t th e n e x t course of Z abo ut S begins im m edia­ tely a fte r th e end of th e fo rm er course. This m eans a p ro p e rty ■of th e system (precession) accom panied by th e corresponding

increase of th e ab ility to do m easu rem en ts (angles i9), m ake th e tim e scale of e.o. d iffe re n t th an th e tim e scale of i.o. We m ay assum e, for exam ple, th a t precession of th e ellipse covers a fu ll angle 2π ex actly durin g v courses of Z ab o u t S. Then e.o. w ill discover th a t his life is v tim es longer th a n the life of i.o. B ut q u alitativ ely , the tim e scale of e.o. rem ain s th e sam e as th a t of i.o., w hich m eans th a t th e scale of e.o. is also of a c irc u la r shaipe. T his is so because a fte r th e precession covered a fu ll angle 2π th e set of th e m easu rem en ts obtained b y e.o. rep eats ex actly 5. The e.o. cannot discover th is re p e ti­ tio n by him self, b u t th is can be done b y a n o th e r observer, le t us call him e.e.o., an d h aving a la rg e r ab ility fo r doing m easu rem en ts th a n e.o.; sim ultaneously, th e change of a new physical p a ra m e te r is necessary. F or exam ple, the system Z, S and A is a p a r t of a galaxy. We assum e th e plane ZSA on w hich Z, S an d A a re placed, ro ta te s ab ou t th e axis SA. We assum e e.e.o. is connected w ith the ro ta tin g system and — beyond r an d & ■— he can m easu re th e ro ta tio n angle φ of the p lan e ZSA ab o u t SA. Then e.e.o. w ill discover th a t d iffe re n t fu ll precessions of th e ellipse end a t d iffe re n t positions of the p lane ZSA. So e.e.o. has his ow n tim e scale, larg e r th a n th a t of e.o. The e.e.o. m ay count d iffe re n t fu ll precessions and discover th a t th e end of one life of e.o. is im m ed iately follow ed b y his — e.o. — n e x t life.

We sum m arize th is section by concluding th a t the tim e scale of an observ er connected w ith a m echanical sytem m ay depend both on his m easu ring ab ility and th e physical p ro p erties of th e system . If th e notion of th e observer is dropped out, we m ay speak ab o u t a convenient tim e scale for a system . A set of m easu rem en ts can be p erfo rm ed on a sy stem and th e tim e scale for an y system can be dep en d en t on this set.

5. PER T U R B A T IO N OF A Q U A N T U M -M EC H A N IC A L SY STEM A N D TO POLO GY OF TIM E

In n o n -relativ istic m echanics, both classic a n d q u antum , a s tra ig h t-lin e a r tim e scale is u su ally assum ed; Fig. 1. In preceding sections w e considered some m echanical system s an d gave arg u m e n ts for a d iffe re n t th a n linear, n am ely c ir­ cular, scale of tim e; Fig. 2. In fact, th e topology of the tim e scale is, to a larg e ex ten t, a m a tte r of convenience. A w

ell-5 L. D. L andau, E. M. L ifszits, Mechanics, Vol. 1 of C ourse o f T h e o r e ­

-kn o w n problem of sim ilar n a tu re concerned spatial coordina­ tes a n d was given in astronom y: th e p lan ets m ovem ent can be described eq ually in th e Ptolem aic, or geocentric, system as w ell as in th e Coperniean, or heliocentric, system , b u t the second sy stem is m ore co n ven ien t th an the first. The purpose of the rem a in d e r of th is p a p e r is to exam ine th e topology of the tim e scale fo r a n on -relativ istic q u antu m -m ech anical system . To th is p u rp o se w e choose th e problem of a p e rtu rb a tio n of a n o n -d eg en erate sta tio n a ry sta te of a q u an tu m -m ech anical system .

If th e system is in its sta tio n a ry state, no m ea su rem e n t do­ ne for it can provide a distinction b etw een a la te r an d an ea rlie r in sta n t of tim e, see Sec. 1. In fact, for such a system th e notion of tim e looses its sense. In re a lity th e sta tio n a ry sta te s on only a few q u an tu m -m ech an ical system s a re exactly know n. To such system s belong, fo r exam ple, a p article in th e field of a co n stan t poten tial, th e harm onic oscillator, and th e sta te s of th e no n -relativ istic hydro gen atom (athough in description of th e last tw o cases some special functions are necessary). We often seek for states w hich differ only slig h tly fro m th e e x a ctly know n states, for exam ple sta te s of th e h y d ro ­ gen ato m in th e w eak electric or m agnetic e x te rn al field. The sta te s w h ich a re w ell-k n o w n a re called u n p e rtu rb ed , w hereas sta te s obtained from the action of a su p p le m e n ta ry field a re called p e rtu rb e d states. S chrödinger gave a m ath em atical p ro ­ ce d u re w hich leads fro m u n p e rtu rb e d to p e rtu rb e d s t a t e s s. This is a com plicated ite ra tiv e process w hich re p re se n ts the p e rtu rb e d energies an d p e rtu rb e d w ave functio ns as a com bi­ n atio n of series based on th e u n p e rtu rb e d q u antities. Ite ra tio n m eans th a t contrib u tion s of th e n e x t step (higher order) can be ex pressed successively by those obtained in a fo rm e r step (low er order). This m akes th e w hole calculation e x tre m e ly com plicated an d in p ractice lim ited to only a few steps, on condition th a t w e assum e th a t re su lts calculated in these few steps ad e q u ate ly ap p ro x im ate th e ex act solutions.

A m ore system atic ap proach to th e p e rtu rb a tio n m ethod can be done on th e basis of field th eo ry \ A lthough w e look fo r a p e rtu rb e d sta tio n a ry state, th e tim e p a ra m e te r is in tro ­

8 L. I. S ch iff, Quantum, M echan ics, 3rd ed., M cG raw H ill, N e w Y ork 1968.

7 S. R aim es, M a n y - e le c t r o n T h e o r y , N orth -H ollan d P ubl. Comp,, A m sterd am 1972.

duced into th e calculation. This iparam eter is tak en along the lin ear scale of tim e. L et us consider th e sta te of a system w hich has th e low est energy, so it is called the gro un d state. It is assum ed th a t a t tim e t being v e ry far in the past ( t = — oo) th e sta te w as u n p e rtu rb e d w h ereas now ( t= 0 ) it is p ertu rb ed . T here is an o p erato r (the tim e-developm ent operator) w hich leads from the situ atio n a t t = — oo to th a t a t t = 0 . The in te rac tio n o p erators a re th e p e rtu rb a tio n ope­ ra to rs rep re sen te d in the so-called in te rac tio n p ictu re and ordered w ith th e help of the chronological operator. The sense of this last o p erato r is th a t it a rra n g e s th e sequence of the in teractio n op erato rs from th e e a rlie r to th e late r tim es. So, for exam ple, the p ro d u ct H A (ti) H B (ta) of tw o in te r­ action operators does not change on condition ti > U, b u t changes upon th e action of th e chronological op erato r into Hb (ta) H A (ti) if ta > ti. The in te rv als of th e in te g ratio n p e r­ fo rm ed over th e tim e v ariab les ex te n d from — oo to zero; therefo re, th ey correspond to a half of th e scale given in Fig. 1. The tim e-d ev elo p m en t op erato r obtained in th e above w ay can be av eraged over th e u n p e rtu rb e d w ave fun ction of the gro u n d state. T he im ag in ary p a rt of th e tim e derivative of the lo g arithm of th is average re p re se n ts exactly th e cor­ rection to th e energy of the u n p e rtu rb e d state, so w h en added to this energy, th e co rrection gives th e energy of th e p e r­ tu rb e d state. Such a tre a tm e n t, alth o u g h elegant, seem s to m ake th e p e rtu rb a tio n calculation still m ore laborious th an the S chrödinger ite rativ e procedure. Especially, the tre a tm e n t does no t give a clear in sig h t into how d iffe re n t com ponents term s (su m s),. en terin g th e S chrödinger p e rtu rb a tio n series, can be obtained. This situ atio n changes, how ever, w hen a cir­ c u la r scale of tim e — sim ilar to th a t given in Fig. 2 — is tak en into account instead of the lin e a r scale. This k in d of p e rtu rb a tio n th eo ry has been p rese n ted in some detail else­ w h e re 8. In the n e x t two sections w e outline those of its fe a tu re s w hich seem to be im p o rta n t from th e poin t of view

of the topology of tim e.

8 S. O lszew ski, T i m e Scale and, i t s A p p li c a ti o n in P e r tu r b a ti o n

6. PER T U R B A T IO N ENERGY O BT A IN E D FROM A C IRC ULA R SCALE OF TIM E

The tim e evolution of a q u an tu m -m ech anical system w hich goes on according to th e circu lar scale of tim e can be r e ­ p rese n ted by cycles of collisions of this system done w ith some p e rtu rb a tio n p o ten tial V.

We assum e th a t a system w hich is o riginally in a sta tio n ­ a ry s ta te n is tra n sfe re d u n d e r th e action of th e firs t collision in a cycle w ith th e p o ten tial V into some o th er s ta te p. A fter a su fficiently long tim e, sta te p can be considered as sta tio n ­ a ry sim ilarly to n. It is convenient to assum e th a t in sta te n th e system is ch aracterized by its own tim e v ariable t„, in sta te p th e system has its ow n tim e v ariab le tD, etc. Con­ sequ en tly , th e collision w ith V changes t n into tp. A n o th er collision tra n sfe rs th e system from a sta tio n a ry sta te p to a sta tio n a ry sta te q; th en th e tim e v ariab le t„ is changed into t„. F u r th e r collisions w ith V can tra n s fe r th e system to s ta te r, sta te s, etc.; accordingly, th e tim e p a ra m e te r w ill assum e th e tim e v ariab les tr, ts, etc. The last collision in a cycle is n ecessarily th a t w hich tra n sfe rs th e system back into sta te n. The n e x t collision w ith V begins a n ew cycle. The n u m b er of cycles is u n lim ited and tend s to in fin ity . A n exam ple of a cycle composed of fou r collisions (th ree in te r ­ m ed iate collisions) is rep re sen te d in Fig. 4.

A ny collision gives its co n trib u tio n to th e energy change of th e system . This change is p ro p ortio nal to th e in teg ral of the p ro d u ct of (a) th e com plex co njugate of th e w av e fu n c ­ tion of th e system before collision, (b) the p e rtu rb a tio n po­ ten tial, (c) th e w ave fu n ctio n a fte r collision. The in te g ratio n is perfo rm ed over the position-dependent, or spatial, as well as th e tim e-d ep en d en t coordinates. A ny collision defines th e space and tim e variab les over w hich th e in te g ratio n has to be done. In general, th e re s u lt of the succesive in tegrations, corresponding to succesive collisions, tends to zero. The ex ­ ception is th e situ atio n w hen a cycle ends w ith some col­ lision w hich tran sfo rm s th e system back to its in itial sta te n. We say th en th a t a cycle is closed and th en w e obtain a non - -zero co n trib u tio n to energy know n from th e S chrödinger p e rtu rb a tio n theory.

The in teg rals over space and tim e can be separated. The re s u lt of th e in te g ratio n over tim e for a (closed) cycle of collisions w e call a k in etic p a rt, w hereas a sim ilar in teg ral done over space v ariab les in a cycle is called a static p a rt.

In o rd er to get d iffe re n t contrib utio n s to th e S chröd ing er p e rtu rb a tio n energy, th e system m u st be su b m itte d to d if­ fe re n t (closed) cycles of collisions w hich begin an d end a t sta te n. The n u m b er of collisions in a cycle corresponds w ith th e o rd er of. th e S chrödinger p e rtu rb a tio n te rm . If th e tim e evolution in the collision cycle is re p re se n te d by a non- -b ran ch ed p ath , th e n w e obtain only one S chrödinger term for each p e rtu rb a tio n order. This is no t a satisfacto ry state of affairs, since for any p e rtu rb a tio n o rd er larg e r th a n 2 th e n u m b er of th e S chrödinger term s (sums) is larg e r th a n u n ity; for a high p e rtu rb a tio n o rder th is excess in the n u m b er of term s becomes v ery high. In o rd er to g et a correct n u m b er of the S chrödinger p e rtu rb a tio n term s, b ran ch ed p a th s of the tim e evolution of a system during its collision cycle should be as­ sum ed. G raphically th e n o n-b ran ch ed tim e p a th for a cycle can be rep re sen te d by a single loop (circle), called a m ain loop, w h ereas the b ran ch ed tim e p ath s are composed of se­ v e ra l loops, called side loops, w hich sprin g out of th e tim e loop having th e begin n ing -en d sta te n. The b ran ch ed tim e p ath s can be obtained w ith th e aid of th e elim ination p rin ­ ciple discussed in Sec. 7.

L et us note th a t in o rd er to get a c o rre c t rep re sen ta tio n of S ch rö d in g er’s re su lt for p e rtu rb a tio n energy, all collision cycles have to be d ifferent, e ith e r in th e ir shape or in the indices w hich label the sta te s m et in th e collisions. A r e ­ petitio n of a cycle g iv es.n o co n trib u tion to energy.

7. ELIM IN ATIO N PR IN C IPLE FOR E Q U A L TIM ES

In q u a n tu m m echanics of m an y -electro n system s w e have a principle given by P auli w hich m akes referen ce to sy m ­ m e try p ro p erties of th e w ave fun ctio n of the system . If the w ave fun ctio n of a m an y -electro n system is ap p ro x im ated by a com bination of p ro d ucts of th e one-electron w ave fu n c ­ tions th e P au li principle becom es an exclusion principle w hich states th a t an y o n e-electron w ave function cannot be occupied by m ore th an one electron. The exclusion principle is of a fu n d am en tal im portance for describing physical and chem ical p ro p erties of m atte r. H istorically, the ex p lan atio n of th e periodic system of elem ents, as w ell as th e electric an d th erm al p ro p ertie s of m etals, re p re se n te d its firs t suc­ cess. The assu m p tio n th a t electrons in a system behave like iden tical p articles w as also a t the basis of th e principle.

A lthough p hysically it has a com pletely d iffe re n t sense,

th e fo rm u latio n of th e elim ination p rin ciple in this p a p e r h a s some sim ilarities to th a t of th e P a u li principle. T he elim ina­ tion prin cip le concerns th e tim e in sta n ts of collisions w ith V a n d sta te s th a t th e en erg y co n trib u tio n given by any cycle in w hich the system has two or m ore sim u ltan eo us collisions w ith V should be su b tra c te d from th e p e rtu rb a tio n energy. A t th e sam e tim e, w e p o stu late th a t th e static p a rt of tw o cycles h av ing th e sam e p a tte rn of collisions w ith V a re identical in d ep en d en tly of th a t w h e th e r a cycle is re p re s e n t­ ed by a m ain loop, or a side loop of tim e.

In applying th e elim ination principle, as w ell as the po­ stu la te of id e n tity for the static p arts, the beginning-end sta te n is considered in a d iffe re n t w ay th an the in te rm e d ia te sta te s in a cycle. F irst, th e ow n tim e of sta te n can nev er be equal to th e ow n tim e of an in te rm e d ia te state; second, the beg inn ing -end sta te for an y static p a rt in a cycle re ­ p rese n ted by a side loop should be p u t equal to sta te n. T his eq u ality m akes th e begin n ing -end sta te for a static p a rt th e sam e fo r all cycles.

The elim ination principle n eed s to lau n ch a com binatorial analysis for a n y collision cycle. This analysis has, as its p u r­ pose, to select all cases for w hich th e in te rm e d ia te collision tim es can be equal. Two or m ore equal collision tim es divide th e original (main) loop of tim e into tw o or m ore loops. For. exam ple, for a cycle w hich has th re e in te rm e d ia te collisions (rep resen ted by a loop on Fig. 4) w e can have th e follow ing cases of equal tim es: th e collision in sta n t 1 is sim ultaneous w ith collision in sta n t 2; collision in 1 is sim ultaneou s w ith th a t in 3; collision 2 is sim u ltan eou s w ith 3; fin ally w e can h ave sim u ltaneo usly all th re e in te rm e d ia te collisions (in stan ts 1, 2, 3). The colon on Fig. 5— 5c re p re se n ts a sym bol of equal tim es. A n im p o rta n t p o in t is th a t an y tw o tim es, w h en p u t equal on a given p ath , should give loops w hich m ay touch; b u t no crossing of th e tim e p a th is allow ed. F or exam ple, a p a th is n o t allow ed on w hich sim u ltan eou s tim e a re select­ ed in th e w ay re p re se n te d in Fig. 6. The allow ed k in d of selection of the tim e p a th s im plies th a t the sequence of col­ lision tim es w ith V should be preserv ed . If w e p u t collision tim e 1 equal to tim e 3, th e n tim e 2 (w hich is in te rm e d ia te b etw een 1 an d 3) cann o t be equal to tim e 4 w h ich is la te r th a n 3.

The elim ination p rin cip le can lead to tim e p a th s composed of m an y loops. F o r exam ple, fo r a cycle of fo u r in te rm e d ia te

collisions, w h ere a collision tim e 1 is p u t equal to collision tim e 4, we obtain a p a th rep re sen te d by tw o loops. H ow ever, the elim ination principle req u ires also to tak e into account th e case of tim e 2 equal to tim e 3 on th e p a th given on the r ig h t side of Fig. 7. This leads to th e p a th rep re sen te d in Fig. 7a an d a sep arate Schrödinger term corresponding to it is attain ed ,

A consequent application of the elim ination principle, to­ g e th e r w ith th e p o stu la te of id en tity of th e static p arts, gives a full set of term s of the Schrödinger p e rtu rb a tio n series for energy. The sign w ith w hich the term s e n te r th e series is also given by th e elim ination principle. Two collision tim es assu m ed equal on a given n on -b ran ch ed p a th m eans th a t the re su lte d term has to be su b tra c te d from the te rm re p re s e n t­ ing th e cycle having a n o n-b ran ch ed p ath . F or exam ple, th re e collision tim es assum ed equal together, can be considered as com ing from a p a th of tw o equal collision tim es, so th e re su lt fo r th re e equal collision tim es should be su b tra c te d from, th a t obtained for tw o equal collision tim es, etc.

8. SU R V EY

In this p ap er w e ex am ined a problem of th e topology of th e tim e scale. As an a lte rn a tiv e to an open, or linear, scale th e re a re p re se n te d arg u m e n ts for a closed, or circular, scale of tim e.

As th e firs t step, w e p o in ted out th a t the tim e scale can be a sub jective notion d ep en d en t on th e physical p ro p erties of a given system an d th e percep tio n a b ility of an o bserver connected w ith th a t system . In a fu lly periodic classical sy ­ stem , th e re is no p a ra m e te r w hich allow s for an observer to distinguish b etw een one cycle of ev en ts (observations) and a n o th e r cycle. In this case th e c ircu lar scale tim e is n a tu ra lly fitte d to th e periodic p ro p ertie s of th e system . The len g th of th e scale m ay d epend on: (i) the n u m b er a n d k in d of p a ra ­ m ete rs ch aracterizin g th e system , (ii) th e p ercep tio n ab ility of a n o bserver. In th e case of a langer scale boith (i) and (ii) a re la rg e r th a n in the case of a sh o rte r scale. O nly an obser­ v e r lin a periodic system h aving a la rg e r scale c a n detect th e periodicity of a sy stem having a sh o rte r scale.

As a second step, in o rd er to have an idea ab o u t th e topo­ logy of th e tim e scale for m icrophysical system s, th is topo­ logy w as ex am ined for a n o n -d eg en erate q u antu m -m echanical system p e rtu rb e d by a sm all poten tial. It is p o in ted o u t th a t

the S ch rö d in g er p e rtu rb a tio n en erg y of this system can be obtained in a r a th e r sim ple w ay on condition th a t a circu lar scale of tim e is assum ed. B eginning w ith some u n p e rtu rb e d q u a n tu m state, th e system is su b m itted to a cycle of collis­ ions w ith th e p e rtu rb a tio n p o ten tial; a fte r th e last collision in a cycle th e system re tu r n to its beginnig sta te an d a new cycle of collisions begins. A n elim ination p rinciple is tak en into account in o rd er to su b tra c t th e co n tribu tion s given by th e cycles in w hich two or m ore sim ultaneous collisions w ith th e p o ten tial occur. W ith the aid of this principle th e re exists a s tric t correspondence b etw een th e colllision cycles a n d th e term s of the S chrödinger p e rtu rb a tio n series. On th e o th er hand , th e application of a lin e a r scale of tim e to a no n-dege­ n e ra te ground sta te can repro d u ce S ch rö d in g er’s th e o ry of p e rtu rb a tio n energy of a sy stem in a m uch m ore com plicated w ay.

I dedicate this pap er to th e m em ory of m y fa th e r, P aw el Olszewski, who stim u la te d m y in te re st in th e prob lem of tim e. I am g rate fu l to Tadeusz K w iatkow ski for his assistance in checking the com ponents of th e p e rtu rb a tio n series an d to Joseph A. D ziver for his collaboration in p rep a rin g the E nglish version of th e m an u script.

p --- » . --- F

(past)

(future)

Fig. 1. L in ear scale of tim e.

P

—---9---в

---

_D

--- :

---/

/

_\

_\

_*

---В

D

Fig. la , lb . L in ear scale of tim e for a cla ssica l p eriod ic system . В — th e first ob served p oin t; D — th e la st o b served p oin t. For a s tr ic tly periodic sy stem th e o b serv a tio n of D ta k es p la ce im m ed ia ­ t e ly b efo re th e ob servation B.

F ig. 2. C ircular sca le of tim e.

Fig. 3. T he m otion of Z a lo n g an e llip se about poin t S w h ich is in one o f th e e llip s e foci. T he d ashed orbit rep resen ts p recessio n o f the e llip s e a b ou t th e a x is goin g across S and b ein g p erp en d icu lar to th e F ig u re p lane.

n

2

F ig. 4. A n ex a m p le o f a c y c le o f co llisio n s h a v in g th ree in term ed ia te c o llis io n s w ith th e p ertu rb ation p o te n tia l V.

O

2

1 = 3

n

Ô

2

1

2:3

F ig. 5— 5c. C ycles o f th ree in term ed ia te co llisio n s w ith p o ten tia l V in case w h en sim u lta n eo u s co llisio n s occur.

n

F ig. 6. A n ex a m p le o f a forb id d en set of sim u lta n eo u s co llisio n s for a cycle of fou r in term ed ia te collision s.

n

1 U

Rg .7

Fig. 7a

F ig . 7—7a. E x a m p les o f a llo w ed sets o f .sim ultaneous co llisio n s for a c y c le o f fou r in term ed ia te collision s.