L E T T ] ~ AL ~trOVO OlM]~NTO VOL. 30, >r. 1 3 Genna io 1981

The Uncertainty Relations, the Wave-Particle Dualism and Timeless Paths in Spacetime.

S. ANTOCI

Isti tuto di _Fisica Generale << A. Volta >> - Pavia, I tal ia Gruppo Nazionale di Struttura della Materia del Consiglio Nazionale delle Rieerehe - I tal ia

( r i cevn to il 10 S e t t e m b r e 1980; m a n o s c r i t t o r ev i s iona to r i c evu to il 26 N o v e m b r e 1980)

A l t h o u g h t h e m a t h e m a t i c a l f o r m a l i s m of genera l r e l a t i v i t y is wel l su i t ed to deal w i t h a m e t r i c t e n s o r gik of a r b i t r a r y s igna tu re , i t is genera l ly t h o u g h t t h a t such an a r b i t r a r i n e s s m u s t be l i m i t e d if one w a n t s to descr ibe t h e rea l wor ld (1.z). I t is in f ac t r e q u i r e d t h a t a t e ach p o i n t of t h e f ou r - d i m ens i ona l c o n t i n u u m t h e m e t r i c t enso r m a y b e r e d u c e d to t h e fo rm

(1) gik--

( 00 - - 1 0

0 - - 1

0 0 - -

b y a su i t ab l e co -o rd ina te t r a n s f o r m a t i o n ; a local i n e r t i a l re ference f r a m e can t h u s b e def ined eve rywhere . L e t us t e n t a t i v e l y explore w h i c h new fea tu re s a p p e a r if t h i s c o n s t a n t is r emoved , a n d if we a l low t h a t , for all t h e va lues of t h e f i rs t co -ord ina te , t h e m e t r i c t e n s o r m a y be r e d u c e d to t h e fo rm

(2) gi/: :

( oo) - - 1 0

0 - - 1

0 0 - -

in a f ini te reg ion of t h r e e - d i m e n s i o n a l space, whi le i t is r educ ib le to t h e fo rm of eq. (1), e lsewhere . F o r t h e sake of s impl ic i ty , we i n t r o d u c e a Car t e s i an re fe rence f r a m e et, x, y, z

in t h e f o u r - d i m e n s i o n a l c o n t i n u u m a n d we a s sume t h a t , for all t h e va lues of t h e ct co-ord ina te , gi~ is g iven b y eq. (2) w i t h i n a f ini te region a t r e s t in t h r e e - d i m e n s i o n a l space, whi le i t t a k e s t h e va lues of eq. (1) e lsewhere. A t w o - d i m e n s i o n a l p i c t u r e of t h i s s i t u a t i o n in t h e (x, e t ) -plane is g iven in fig. 1; ou t s ide t h e s h a d e d a rea t h e p rope r t i e s

(1) L. LANDAU and E. LIFCHITZ: Th~orie des champs (Moscow, 1970), Io. 302. (~) 2k. S. EDDINGTON: The Mathematical Theory o] Relativity (Cambridge, 1954), p. 25.

20

THE UNCERTAIIN-TY RELATIONS, THE WAVE-PARTICLE DUALIS~I ETC. 2 ]

of o rd inary space- t ime can be experienced, while wi th in the shaded p a t h t ime does not exist , and real lengths are measured ins tead of t ime intervals . Imag ine now to measure t he dis tance be tween two points for which the spat ia l co-ordinates are x, y, z and x ~ Ax, y + Ay, z + Az, respect ively . If the segment jo ining the two points lies in ~he region where g,:1~ has the values of eq. (1), the dis tance l is

(3) l = [(Ax)2 § (Ay)2 q - (Az)2]~

ct lJ~J

Fig. 1. - The t ime le s s p~ th of a t ime le s s r eg ion a t res t .

as usual. I f t he segment lies ins tead in the region where giT: has the values of eq. (2), the dis tance

(4) l '= [(Act) 2 + (Ax) 2 + (Ay) ~ + (Az)2]~

can range be tween l and infinity, according to the values tha t ct assumes at the end points of the segment . Since we cannot exer t on the ct co-ordinate in the t imeless region the same control t h a t is possible outside, the resul t of any measure of length, pe r fo rmed inside tha t region, shall be unpredictable . As a consequence, when we measure t he spat ia l extension of the t imeless region, a dual behav iour appears : if the measure is pe r fo rmed f rom the outside, a finite, well-defined va lue is found, while, if the measu remen t is pe r fo rmed inside, an unpredic table , possibly infinite resul t is obta ined.

Le t us per form a Loren tz t rans format ion to a reference sys tem mov ing wi th respect to the rest f rame of the t imeless region at a cons tan t speed - - v. In the new reference system, where the t imeless region appears to m o v e wi th speed v, the metr ic tensor keeps the values of eq. (1) in the outside region; in the inner region it has no longer the values of eq. (2), bu t i ts s ignature is of course unchanged. Therefore, when the size of the t imeless region is measured f rom the outside, a well-defined resul t is found, while wi th in the t imeless p a t h any length up to inf ini ty can stil l be measured.

The dual behav iour descr ibed above is s imilar to t ha t occurr ing to the mater ia l par t ic les deal t wi th by q u a n t u m mechanics and we are t e m p t e d to see wha t happens if we assume (s) t ha t t he size of t he t imeless region is kh/mc, where h/mc is the Compton

(~) P . A. M. DIRAC: The PrincipZcs o] Quantum Mechanics (Oxford , 1976), p. 263. See a l so : L. L. FOLDu a n d S. A. XcVOUTHUYSEN: Phys. Rev., 78, 29 (1950); M. B~TNGE: NUOVO Cimento, 1, 977 (1955).

22 s. ANTOCI

reduced wave-length of a particle with rest mass m, and k is a real number close to unity. Let us further assume that some distinctive feature, indicating the presence of the particle, can be found anywhere within the timeless region. Suppose now that the timeless part of space is moving at a constant speed v <<c along some direction within a Cartesian reference frame, and that we can find the particle anywhere within a segment of the x-axis of length A/~; this behaviour is possible, provided that the end points of the segment lie within the timeless path, represented in fig. 2 by its projec-

ct' / H r / K r /

x

i / / I H / K 1

F i g . 2. - P r o j e c t i o n o n t h e (x, c t ) - p l a n e of t h e p a t h of a t i m e l e s s r e g i o n m o v i n g w i t h c o n s t a n t s p e e d v.

t ion HH' , K K ' on the (x, ct)-plane. The end points can be placed anywhere within the path, provided that their distance is h 4 ; for example, they can be represented in fig. 2 by the pairs of points A, A' or B, B'. Let us consider these points from the outside point of view. When a particle is found at events which correspond to the points A and A', we at tr ibute to the particle an average speed along x

A F (5) vA = ~ c.

CF

If the particle is found at events which correspond to the points B and B', the average speed along x at tr ibuted to the particle is, instead,

B F ( 6 ) v ~ : ~ c .

CF

All the other possible pairs of events yield for the x-component of the speed a value between v~ and v, . We can therefore assert that when the particle can be localized within a segment of the x-axis of length A4 its speed along x is known with the uncer- ta in ty Av~ = v~--vA. Since we consider speeds much smaller than c, we have

(7) L4 ~- 2. C F .

TIIE U N C E R T A I N T Y RELATIO1NS, THE WAVE-PARTICLE DUALISM ETC, 2 3

Therefore

B F - - A F 2kh (8) A v e - ~ c - ~ - - .

CF m Al~

But Ap~ = m Av~ is the uncertainty of the momentum which is at t r ibuted to the par- ticle by Newtonian mechanics. Equat ion (8) then gives

(9) Ap~ A/~ = 2/oh.

The same relation can be obtained for the y and z axes. Suppose now that the timeless region is moving at a constant speed v = v~ along the x-axis alone. The segment of the axis where the particle can be localized moves together with the three-dimensionM timeless region with the speed v. From fig. 2 we find

(10) v = v~ _~ - - vA-I- vB

Therefore the t ime of transit of the interval Al~ at a given point of the x-axis is

Al~ 2kh ( l l ) At . . . . .

i ' x ~ V x A V x

But AE='mv~Av~ is the uncertainty of the kinetic energy which is attr ibuted to the particle by Newtonian mechanics. Equation (11) then gives

(12) A E A t ~ 2 k h .

We have so far shown that, if a timeless region of space exists, with a size of the order of h/me, and if it contains somewhere the material particle, then uncertainty relations can be found, but a wavelike behaviour is not yet apparent. The hypothesis advanced before, that the material particle can be found anywhere within the timeless region may be further specified. Let us assume, from analogy with the results of ref. (3), that the material particle describes in the timeless region a periodic trajectory around an arbitrarily oriented axis, and that one turn is completed when the ct co-ordinate is varied by the anmunt h/mc. Since the particle has a proper energy associated with it, a nonzero energy density is present within the timeless region. But when the metric tensor has the form of eq. (2), the energy density is neither positive definite nor negative definite, as can be seen by considering, for example, the electromagnetic stress-energy tensor. A negative-energy region can therefore coexist with a positive-energy region within the timeless part of space. We can think that both these regions describe the periodic t rajectory performed by the particle, and that the very presence of the particle is related to the mutual position of these regions. Consider now a reference frame ct', x', y', z' in which the timeless region is set at rest at the origin. To visualize the situation using only the three axes ct', x', y' assume that the axis of the periodic tra- jectory is parallel to the z'-axis. The timeless paths associated with the two energy regions then appear as helices wound along the ct'-axis with pitch h/mc. Assume that they are wound in the same sense, so that the mutual position of the two regions is the same for any value of the ct co-ordinate. When we at tempt to localize the particle at a given point of, say, the x'-axis, we may find it there, even if the timeless region

2~ S. ANTOCI

is very far apart, since we can experience as lengths along x' the intervals within the timeless path. If the presence of the particle depends on the mutual position of the two energy regions, it is reasonable to expect that the probability to find the particle between x' and x'q- dx' does not depend on the value of x', since the mutuM position in space of the two energy regions does not depend on ct ' . To proceed further, imagine that a timeless region containing the particle is created at rest at a given time. As t ime elapses, the helical paths grow along the cV-axis and the segment of the x'-axis, within which the particle can be localized, lengthens with speed c. Let us observe the same process from a new reference frame ct , x , y , z , moving with respect to the primed one at a constant speed - - v along the x'-axis. Suppose that Ivl <<c as before, in the primed reference system the pitch of both helices was h / m c ; in the moving system the pitch of the helix associated with the positive-energy region, and measured along the x-axis results to be

h (13) p+ ~ - - - (c @ v)

~/:C2

while the pitch of the helix associated with the negative-energy region, measured along x, is

h _~ - - ( c - - ~ ) , (14) p_ ~c 2

since a negative-energy region moving in the new reference system with speed v behaves as a positive-energy region with speed --v. If the two energy regions have a given mutuM position at a given value of x, they shall resume the same mutuM posi- tion at values of x increased or lowered by the amount

p + p _ n h c 2 ~ v ~ n h (15) Ax. = n - - - ~_ ~ - - ,

p + - - p _ m c 2 2v 2 m y

where ~ takes the values n = 0, • 1, • 2, . . . . As a Consequence, the probabili ty to find the particle between x and x • dx shall be a per iodic function of x, and its period P is half the wave-length ~ of the probability wave associated with the particle according to wave mechanics

h (16) ~ = 2 . P _ ~ - - .

The arguments used above can readily be extended to deal with a timeless region moving with a constant speed along an arbitrary direction.

We have suggested, with the use of heuristic arguments, that the uncertainty rela- tions and the wave-particle dualism may find their origin in the existence of timeless regions of space. A rigorous statement of this idea is obviously required, and can eventually be provided by the general-relativity theory.