L]~TT]~R]~ AL NUOVO CIM]~NTO VOL. 32, N. 3 19 Sot tombro 1981

Is There a Maximal Acceleration?

]7,. R . CAIANIELLO (*)

Hochschule der Bundeswehr , ~achbereich Elektrotechnik - Mi inchen

(r icevuto il 27 Lugl io 1981)

In previous occasions (l-a) I h a v e proposed as a game, or (~ tool for t hough t ,), a mode l which geometr izes QM by endowing phase space (extended wi th t and E dimen- sions) w i th a met r ic : ~ p lays in i t the same role as c in Minkowski space. I t was shown to y ie ld t he Bohr -Sommer fe ld quan t iza t ion bo th for in tegers and half- integers, t he Kle in-Gordon equa t ion as defining the null-cone, a Dirac equa t ion which s tays inva- r i an t only under CPT, (~ free par t ic les )) which have ex tended s t ructures and pos i t ive mass spectra de termined , w i thou t fu r the r assumptions , by a pseudoharmonie poten- t ial . The loose not ion of (x, p)-space as a fibre a t t ached to t is rep laced wi th the much more s t r ingent met r ica l r equ i r emen t ; as a consequence, the (( in tu i t ive ~) not ion of par- t icle as a ((pointlike object ~) is comple te ly lost ; a par t ic le can only be ~ex tended ~) and capable of r ich s t ructures , as indica ted in (2). I propose to p lay here this game a l i t t le fur ther , in a stil l more simplified version. I t was shown in the w, orks ment ioned above tha t QM can be der ived by using a connect ion which can be t aken ra ther f reely as (( met r ic ~ or (( gauge )): we take i t here as gauge, and then forget about it. We are thus lef t wi th only an 8-dimensional flat metr ic , which we take , in appropr ia te units , as twice Minkowski , once in each subspace; t he connect ion be tween the (t, x) and (E, p) space, f rom which QN[ arises, is forgot ten about . W e are thus at a (( p r equan tum, quasi-classical )) level , which we propose here to explore, bo th to supply a more expl ic i t and in tu i t ive mo t iva t i on to our model , and because a l ready at this level one meets t he possibi l i ty of a r e fo rmula t ion of mechnics in which physical objects arc l imi ted no t only by a m a x i m a l ve loci ty , bu t also by a m a x i m a l accelerat ion, in a sense t h a t wil l be specified. The discussion is qu i te e l emen ta ry and self -contained; no reference to t he ci ted papers is necessary, except ing of course the a l ready ment ioned met r iza t ion pos tu la te , on which the whole g a m e rests. I t m a y be re levan t to remark , at th is point , t ha t t he concept of a (~ m a x i m a l accelerat ion )) proposed here migh t wel l be an object of in te res t per se, regardless of the con tex t in which i t happens now to arise: i t works

(*) Universitk di Salerno. (l) E. R. CAIAN1ELLO: Left. Nuovo Gimento, 25, 225 (1979); 27, 89 (1980); Nuovo Cimento B, 59, 350 (1980). (B) ]~. R. GAIANIELLO and G. VILASI: Lett. Nuovo Cimento, 30, 469 (1981). (a) E. R. CAIANIELLOI • game with geometry and quantum mechanics, invited talk at the V I J . J . N . R . International Con]erence, Alushta, May 1981.

65

~ "E,. R. CAIANI~LLO

better than ordinary cut-offs, is not objectionable on relativistic grounds and prevents , particles ,~ from totally collapsing into one another.

To begin with, it is clear tha t any at tempt at describing physics through a metric geometry requires first of all a decision about what (~ length ~ should mean. Any meas- urable quantities (provided they are given the same dimension) can be put together so as to form a (~ metric structure ,); thus one could define in Newtonian physics

(I) S ~ = t 2 + ~x 2 ~ t 2 ~- ax e (~ arbitrary real) .

The problems are 1) relevance, 2) compatibility with the physical model this S ~ pur- ports to describe. I shall formulate explicitly this notion, which is plain common sense to every physicist, by posing a Postulate o] metric measurability: ~ length must be measurable with arbitrarily high precision ~) (i.e. l imited only by available experimental know-how); to allow for local changes of frame, (~ any partial sum of elements of length must be likewise measurable,). Let us go back to the Newtonian example (1). I emphasize that we are here not concerned with descriptions of measuring apparatuses, bu t only with the fact that iV independent measurements, say of the quant i ty x, give (for iV arbitrarily large) a distribution

I [ (2) pa,(x) -~ p(x; ~, az) = V ~ a x e x p 2a~ J

around the arithmetical mean ~ of the measured values x, so that

(3) <$2> = ~ + ~ + al + ~a~.

Both dispersions a~ and a~ in a Newtonian world can decrease independently: a metric (1) is irrelevant because x and t are not mutually dependent, though not contradictory with this postulate.

Better experimental evidence has shown that ax and as cannot be independent:

(4) ~ < v. fit

The measurabili ty postulate clearly binds now ~ and ~ in the length (1): the choice = - 1/c a, which gives

(5) <s~> = ~-- ~ 1 C2 X2

needs no comment. Uncer ta in ty is defined by informational entropy. Suppose that the minimum avail.

able dispersion in measuring x is a and that a device is used having a' > a. The information loss caused by the use of the less precise device is, from (2),

+ } o (6) AI = ~,(x) leg p~,(x) dx -t- a(x) log/~(x) dx = log - -

ff

(for continuous distributions only differences are free from ambiguity), so tha t

(7) AI > O.

IS THERE A MAXIMAL ACC]~L]~RATION~ ~ 7

Remark, next, t ha t to say tha t x can be anywhere within an interval 2L may be rendered equivalent to saying tha t the quant i ty x is measured with an appara tus having dispersion a'= _5. Then (6) tells, when maximal precision is a, t ha t log (L/a) is the best information one can have on the location of x in Z. One might crudely express this by saying tha t (~ L is divided into segments each of length a 7), and con- coct thereby (~ lat t ice models ~) of a continuum.

Wi th our phase space the s i tuat ion is ent irely analogous. Suppose we want to measure x and p. We can always give i t a metric

(8) 1

S ~ = t 2 _ _ x ~ q - C 2 ~

According to our discussion, in classical theory this is feasible bu t i r relevant because %, a~, etc. are not mutual ly related. In quantum theory, though,

(9) %a~ >~h, e tc . ,

so tha t (8) is not measurable according to our postulate, unless %, a~ are sui tably re la ted: (9) acts like (4) before. We may take

2

where % is some number of which only the sign e is relevant, since I%l can always be scaled away.

By calling a~ = 2, a~ = / z c and setting

(lO) h = ~t/~e,

(8) is thus reduced to the relevant and unique form

1 h ~ e [ 1 2 ] ( 1 1 ) S ~ = t 2 - - x 2 q - e2 ~ ~ E - p ~ .

2 and pc were regarded by BORN (4) as fundamental (~ length 7) and (~ momentum ,); for us the only meaningful s ta tement is t ha t expressed by relations (10) and (9). Uncer- t a in ty of posit ion in volume I2 of phase space x, p is given, as with (7), by

(12) AI = log_--,

a famil iar result, h --~ 0 disconnects configuration and momentum parts , as i t should in the classical l imit .

(') M. BOR~: Rev. Mod. Phys. , 21, 463 (1949).

E . R. CAIANIELLO

Note now that, by adding to the Minkowskian metrization of space-time the require- ment tha t for any particle

( 1 3 ) c 2 d s 2 ---- c ~ d t 2 - d x 2 > 0 ,

one has the fundamental result of relativity, tha t no physical velocity can exceed the maximal value c.

Wha t happens when the same requirement is imposed upon the metric element of our space (no other is allowed because when ~ ~ 0 one wants to retrieve classical mechanics)? I t reads now

(14) ~* r l 3

c' ds ' = c ' d t ' - - d x ~ + e ;~c, [ ~ d E ' - - d p ' ] > 0 �9

The answer to this question depends crucially upon the value of ~. I t was shown in (2) tha t the choice 8 = + 1 leads to positive mass spectra through a Born-like equation; i t would be tr ivial to verify tha t the alternate choice ~ = - 1 does not lead to any such result (3); we know thus already, from our past game, that the two situations are profoundly different. We have to compute, from (14), the value of

(15) + - > o .

Whatever, if any, the correct formal t reatment of this problem (we shall re turn to this point at the end), i t will make sense only if, under ordinary circumstances, i t reduces to the standard formalisms; furthermore, since v = c cannot be exceeded, it is to be expected that a maximal value of the acceleration should appear when v << c.

Call , v rest frame ~ of the particle the one obtained by an ordinary Lorentz trans. formation leading from the observer frame to that in which the velocity v of the par- ticle vanishes at that instant . We may compute thus (15), to begin with, with the s tandard formulae of special relativity

(16)

whence

fields

(17)

, , , ) - 1 > o ,

e = - - 1: no maximal acceleration,

~ = + I

a = %/a~ + a*, A- ada<A~ = (c~--v2) t#~<A, ,=o, moh

#%3 c~ pc 2 ( 1 8 ) A~ ,=o = A = - - =

mo h mo~ mo~t

is the maximal acceleration (s).

~) (18) states that the orbital speed of an extended structure with radius ~ )~ cannot exceed G.

IS TH]~RE A MAXIMAL ACC~L~RATION~ 6 9

The same m a x i m a l accelerat ion A in the v res t f rame (relat ive to which the conccp~ wil l be f rom now on defined) obta ins of course also b y comput ing wi th NewConian me- chanics. A~ is m a x i m u m when v = 0, as expec ted ; for v = e, A = 0. B y choosing

= + 1 we obta in thus a typ ica l ins tance of wha t m a y be cal led an ~, a posteriori i n tu i t i ve ,) concept : (~ ve loc i ty )~ in our 8-space is made of two 4-vectors, o rd inary ve loc i ty v and accelera t ion a : w h y should only its first half , v, h a v e an upper bound?

Before g iv ing a fu r the r look to the mat te r , i t is conven ien t to check at t he crudes t level , nonre la t iv i s t ic mechanics, whe the r the game is af ter all wor th playing. W e mus t first note tha t , since our par t ic les are in no w a y (~ m a t h e m a t i c a l po in ts ,), t he s t a t emen t t h a t t h e y cannot get (( closer t h a n some dis tance ~ allows a degree of i nde te rminacy : different forces m a y lead to different numer ica l va lues for such a (, d is tance ~ for the same par t ic les (which we m a y conceive, if we have to, as (( clouds ~) which m a y more or less in terpenet ra te) .

I shal l consider th ree typ ica l s i tua t ions :

a) Two identical particles attracting one another gravitationally.

In no case t he force can become larger t h a n p e r m i t t e d by t h e assumed m a x i m a l accelera t ion (18).

T a k e # = m o (rest mass); in t he c.m. sys tem one finds

m0 (19) - - a = G ,

2 r a

so tha t , since f rom (18) A = c2/~, r cannot in (19) become smal ler t h a n al lowed by

m~ A m~176 G m~ 2 2 4 ~2 '

i .e.

(20) ~ = 2Gm~ C2 '

which is of course the Schwarzschi ld radius : complete collapse is ]orbidden.

b) Two attracting electric charges o/ same magnitude e and mass mo.

In the c.m. sys tem

(21) m ~ c2 - - e2

2 ~. ~ '

whence

e 2 (22) 4 . - - 2 -

m 0 c 2

which is t he classical (~ e lectron ~) d iamete r (s).

(6) ~te c a n n o t of cou r se s a t i s f y (18); b y s e t t i n g )le = r162 = ~(~]mec), (18) y i e ld s ~ = e']~c. I d e n t i f y i n g

t h e n th i s w i t h (20), one f inds for t he pa r t i c l e * r a d i u s * ~/2 = % / ~ , t h e P l a n e k l e n g t h .

70 ~. R. CAIANI~,LLO

c) The Bohr orbits.

Consider the high-school Bohr orbits. They have diameters and binding energies

2 ~ 2 e 2 ,mo e 4 (23) ~.,~ ~ n 2 , E~ -= - - -

~no e 2 4 . = - - 2 ~ 2 n ~;

E . measures the energy defect with respect to the free electron. Set now ~---- X."/~e and compute the maximum acceleration which we assume is

reached, as before, when r ---- ~ in

i.e. from (18)

we find. t ha t

e 2

q n o a ~ ~-~,

~ n c2 e2

n0 n o ~ ~t~

e 2 (24) tt~ e 2 . . . . E .

expresses the kinetic binding through the relat ivist ic mass formula, which does not belong to Bohr ' s theory.

These est imates indicate that , in this game and at this level, the main effect of QM is tha t of forbidding the to ta l collapse of a t t rac t ing particles; ~longitudinal,~ and

t ransversal ~ masses and forces should then create orbit ing situations, the quantiza- t ion of which requires of course the full use of QM geometry (neglected here since no a t tent ion is pa id to the gauge connection).

Passage from {15) to (16) was made w i ~ special re la t iv i ty : this amounts to associating to the part icle, as proper t ime, t ha t determined by only the (t, x)-subspaee metric. A deeper discussion would involve issues such as arise in connection with the Dirae- ]~liezer equation for the classical electron, and several nontr ivial points would have to be settled. A complete recourse to QM is probably preferable; i ts results are expected to be consistent with the est imates given here.