L]~TT~RE AL NUOVO CI~N~O voL 41, ~. 11 1O :Novembre 1984

Maximal Acceleration as a Consequence of Heisenberg's Uncertainty Relations.

]~. R . CAIAI~I]~LLO

Dipartimento di Eisica Teorica e sue Metodologie per le Scienze Applicate dell' Universit~ Salerno, Italia

(ricev~to il 30 Luglio 1984)

PACS. 03.65. - Quantum theory; quantum mechanics.

Summary. - The existence of a maximal acceleration for physical particles, proposed in 1981 by the author, is shown to be a consequence of Heisenberg's uncer ta inty relations.

Previous work (1) on the development of a quantum geometry in relativistic phase space led, quite unexpectedly, to the notion of a maximal (proper) acceleration. This appears because in our theory <( particles ~> are extended objects, never to be identified with mathematical points in ordinary space (and it is then immediately intui t ive: a~ = v2/r<c2/r, which is essentially our formula Am~x~c2/~, where ~ is the ((linear dimension ~ of the particle). The subject has attracted some at tent ion (2), because infinities and collapses would then be forbidden, and in several ways the assumption of a maximal acceleration contains some of the conceptual elements of quantum mechanics.

We propose here to point out thut this maximal acceleration appears to be a straight- forward consequence of Heisenberg's uncertainty relation

h (1) A E . A t > -

2 '

LANDAU was the first, in his theory of fluctuations, to use the consequence of (1)

(2) AE. A/(t) • h d] ~t ' Z

(1) E. JR. CAIANIELLO: Lelt . N u o v o Gimento, 32, 65 (1981); E. t~. CAIANIELLO, S. DE FILIPPO. G. MAI~MO and G. VILASI: Left . Nuovo Cimento, 34, 112 (1982). (~) P. CALDIROLA: Lett . N u o v o Cimento, 32, 264 (1981); ]=I. E. B•ANDT: Left . Nuovo Cimenlo, 38, 522 (1983); G. SOA~PETTA: Nuovo Cimenlo, 41, 51 (1984).

370

M A X I M A L A C C : E L : E R A T I O N A S A C O I ~ S : E Q U : E : N C E : E T C . 371

where /(t) is different iable. Consider a pa r t i c le nea r ly a t res t (accelerat ion can thus be la rges t (1)) and t ake ](t) = v(t). U n d e r the assumpt ions V<~c and

(3)

(2) yields i m m e d i a t e l y

A E ~ E , Av<~e

h Ial < A E . Av < m c ~.c ,

o r

m c 3

(4) A . . . . = 2 - - h

A t t i t u d e s v a r y about the impor t ance to ge b iven to smal l factors such as the 2 in (4). Suppose i t should be t aken as meaningfu l . Our p rev ious works gave for A ~ (neglect ing re la t iv i s t i c factors which cause A ~ 0 when v -+ c) the equ iva len t ex- pressions

(5)

al l der iv ing f rom

(6)

A r e a x - -

#~ c a c h # c 2

m h m ~ 2 mA

h = ~#c (or h ! ) .

A t t e m p t s at numero logy requ i red the ad hoc a s s u m p t i o n t h a t # = m, mass part icle , or re la ted somehow to it . ;~ is t hen a Compton (< d i ame te r )), dependen t of course ou the t y p e of force ac t ing on the par t ic le . The l i m i t a t i o n (4) coincides w i t h (5) if we take /~ = 2 m (no more a n a s s u m p t i o n ) ; i t tel ls t h a t

2 c 2 c 2

A r e a x - - ). r '

which was our i n tu i t i ve expec t a t i on for v << c. (Limits on h igher der iva t ives , chronons, etc. , would fol low as eas i ly ; we do no t pursue here th is point , because i t migh t lead, w i thou t a deeper scrut iny, to unwar r an t ed conclusions.)

Our work would have been much s imple r i f we had been able to s t a r t the o ther w a y a round; we confess t h a t we were no t t h e n able to (( guess ,> a (~ m a x i m a l accelera t ion ,>, and found i t ha rd to swal low when first met .

A few words of c o m m e n t m a y be in order. Not ions such as a (( f u n d a m e n t a l l eng th )) or (( t i m e >) (chronon) have been and are be ing proposed as i n tu i t i ve means to counte r the chal lenge p resen ted b y Heisenberg ' s re lat ions. ~ o object io~ is i n t ended here agains t such views, especia l ly because they, when pursued w i t h sound physical or ma thema t i ca l ins ight , m a y p rove of g rea t in teres t , whe the r as facts or tools (a,a). W e wish only to r e m a r k t h a t the first r evo lu t ion came about w i t h E ins t e in ' s m a x i m a l ve loci ty: what p roved i n s t r u m e n t a l to i t was no t the K a n t i a n a p r i o r i << space >> or <( t ime ~>, bu t the

(a) P. CALDI~OL~: S u p p l . ~Vuovo Cimento, 3, 297 (1956) and following papers. (a) L. BI:EDENHAItI'~: Some remarks on us ing octonions in quan tum mechanics, in A m a t f i Meet ing (May 1983).

3 7 2 E. R. CAIAlqI:ELLO

<< derived ~> concept of velocity. That was ent irely against long-established mental patterns, and it is to be doubted whether, still now, may find i t obvious to class (( velocity >> as an a pr ior i concept (~ more fundamental than space or time.

Such it has of course proved to be, bu t the decay t ime of mental pat terns lasts generations. With Heisenberg's uncer ta inty h the situation has not changed; human mind has resisted the second revolution of physics t rying to maintain, again, as fun- damental the same age old (( common sense ~) concepts as primitive.

The simple thought contained in this note expresses instead the at tempt to regard h (as well as c) as <~ primit ive >), and all else <( derived )>. In support of it, we may remark that this a t t i tude brings to a conceptual unification of quantum and information geo- metry (5,6).

The most interest ing remark as regards (~ length ~> is, in our opinion, that presented by F~nR~TTI in a recent note (v). His argument, as simple as our, proves that the measurement of any length ~ cannot be more precise than a (~ quantum noise ~> lower l imit At that he computes: it is firm and unobjectionable, since no ((fundanlental length )> is mentioned. Just because of its simplicity, Ferret t i ' s proof adds strength to the question, whether it makes sense to use theories in which lengths smaller than his l imit appear (and cause trouble!). Coming to numbers, we remark that, when gravitat ion is implied, his l imit coincides exactly with that obtained from our maximal- acceleration hypothesis, i.e. the Planck length (~).

I t seems to us that the present, naive approach bypasses this type of argumentations. I t also appears that in model which should use this notion much has to be re-thought: if for nothing else, because acceleration is gravitation, and accelerated frames belong already there.

(5) E. 1%. CAIANIEI,LO: Left . N u o v o Cimenlo, 38, 539 (1983). (~) :E. 1:~. C~_IANIELLO : Enlropy , in /ormat ion and quan lum geometry, to a p p e a r in Proceeding o] S a n t a F e In ternat ional Con]erence on Non l inear Phenomena ( J u l y 1984). (7) B. ~EI%I%ETTI: Lett . Nuovo Cimenfo, 40, 169 (1984).