Volume 210, number 1,2 PHYSICS LETTERS B 18 August 1988

W O R L D F U N C T I O N D Y N A M I C S IN G E N E R A L I Z E D S P A C E T I M E S

Enrique ALVAREZ Departamento de Fisica Te6rica, Universidad Autdnoma, E-28049 Madrid, Spain

Received 16 February 1988

A generalization of general relativity is proposed in which the dynamical variable is the two-point distance in a metric space which is not necessarily homeomorphic to ff~", even locally. In the present framework it is possible to compute such things as, for example, the local physical field associated with the Hausdorff dimension, so that the question of the actual value of the dimen- sionality of spacetime can be meaningfully addressed.

Shortly after Einstein p romoted the geometry o f spacet ime to the rank of a dynamica l var iable of a physical theory, Kaluza and Klein contempla ted the possibili ty of f ive-dimensional worlds, introducing in this way for the first t ime the problem of explaining in one sense or another the actual value of the d imen- sionali ty of spacetime.

With the recent revival of interest in mul t id imen- sional theories (cf. ref. [ 1 ] ), the possibi l i ty of reach- ing a dynamica l explanat ion of the effective number of d imensions at low energies has appeared. The present work is an a t t empt to bui ld up a theory in which the number of d imensions is related to a true dynamical variable. In doing so we shall try to be as conservat ive as possible, and to keep closest to the s tandard picture of spacet ime as a different iable manifold.

The concept of an "even t" is usually associated to the coordinates employed in describing it in a local patch of the manifold. We shall abstract from this no- tion, and consider the spacet ime as the set of all events, which we assume equipotent with the cont inuum.

The only structure a pr ior i o f space t ime will be that of a metr ic space; this means that given two points, x and y, there is a real function d(x, y) such that it is posi t ive semidefini te , d(x, y) >1 O; it is different from zero if the two points are different: d(x, y) = O--*x=y; it is a symmetr ic function o f its arguments d(x, y) = d(y, x) and, besides, it satisfies the t r iangular ine- quality, d(x, z) <~d(x, y) + d(y, z).

Every r iemannian manifold has a distance defined on it; it suffices to consider the geodesic distance be- tween two points. (At least locally. There can be global ambiguit ies; for example, in a compact space, there are always two ways o f going through a geodesic from x to y. ) But there are many metr ic spaces which are not manifolds, because they are not homeomorph ic to ~", even locally. A simple example is any space obeying the "u l t ramet r ic inequal i ty" (cf. ref. [2] ) which is a stronger version of the t r iangular one, namely d(x, z) ~<max{d(x, y), d(y, z)}. This is the case when a hierarchical structure is present, so that the d is t r ibut ion of points in the corresponding space is a dendrogram, and also for the p-adic distance in the field o f the rationals. Another physical example is p rov ided by the structure of the order pa ramete r in Parisi 's solution of the mean field theory of spin glasses. In those cases, it is never possible to define a homeomorph i sm to ~"; perhaps the simplest way to see this is to realize that in the case in which the space considered is linear, then ultrametrici ty implies d(2x, O) <~ d(x, 0), an inequal i ty of the non-archimedean type. It is worth stressing that, weird as it is, there are physical indicat ions that the metr ic structure at short distances should be very different from the one asso- ciated with classical physics (cf. ref. [3] ). After all, the uncer ta inty principle forbids any actual measure of distances smaller than the Planck length and it is not at all evident what should happen i f we a t tempt to "measure" a macroscopic length with a sub- planckian "rule".

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Volume 210, number 1,2 PHYSICS LETTERS B 18 August 1988

Given any metric space without any further struc- ture, there are two main concepts of dimension which can be defined (cf. ref. [ 4 ] for a clear discussion from a physical point of view): the Hausdorff dimension dimH (A) (related to the "capacity" of the set A; that is, to the minimum number of open balls needed to cover A), and the topological dimension dimL(A) (which is an essentially combinatorial concept ).

It will be also assumed that global quantities can be defined in our spacetime; this means that we have further structure, namely, a measure # defined in it. It is natural to assume that for every point, except for a set of #-measure zero, we can define a "local" di- mension, n(x), that is, that the corresponding limit makes sense:

n(x) =l im dimH (B (x, r) ) . ( 1 ) ,~o

This, in turn, means that for sufficiently small R

#(B(x , R) ) ~ m(x )R '"-'~ . (2)

The natural path for a test particle to follow will be the one extremizing the action, which generalizes the geodesic principle of ordinary riemannian geometry:

I

S(x,y; C):=J dz(s) d(x, z(s)) , (3) 0

where the curve z=z(s) joins the two given points x and y: z (0) =x ; z( 1 ) =y.

The dynamics of the spacetime itself will be de- duced from Einstein's program of relating geometry to gravitation. In order to do so, let us first introduce a natural hierarchy of fields defined as coincidence limits of the square of the metric distance:

A(x) = l im d2(x, y) = 0 , . v ~ ) ,

B(x) =l im d [d2(x, y) ] , A'~V (Jy

C(x)=l!m . .,dy--5 [ d 2 ( x , y ) ] ,

D(x ) =}ira d3 . dy---53 [d~-(x, y) ],

E(x) =l im ~ [d2(x, y) ] . (4) \" * i, U]2 ' ) ?

The simplest geometric lagrangian contains terms of the form

S ' = f d,u[c,C(B L ,B -~ )+c2D(B-J ,B -~ ,B - I )

+¢3E(C -I, C - ~ ) + c 4 E ( B - I , B - ~ , B - ~ , B -~ )

+...] (5)

(where we interpret the nth derivative as a linear op- erator from X" into ~).

In a riemannian manifold the geodesic distance squared is Synge's world function (cf. ref. [5] ). The derivatives defined above get now indices on them (because we can characterize a given event by its co- ordinates in a local patch). It is not difficult to show that under those conditions

A(x)=B,(,~c)=Dz,,/,(x)=O, C~,,(x)=g,,~(x),

E,,,,,,~(x) = - ~ (R,,,p~ + R,,~,,~ ) . ( 6 )

When the space is locally homeomorphic to ~", the natural measure is just the invariant pullback of the ordinary Lebesgue measure in ~". Our action then re- duces to

f ---~X 2 ) ~ g d 4 x R , (7) S =

namely, the ordinary Hilbert lagrangian. In the general case, however, we have a spacetime

defined as a set of events, with a topological structure given by the distance, and an integration defined by a certain measure. The distance - or better still, its second derivative, C(x) - and the measure [that is, the dimension field n ( x ) ] are just the dynamical fields of the system.

The variation with respect to C(x) gives the dy- namics of the distance field (that is, the generaliza- tion of Einstein's equations), which are complicated ones, and not very illuminating; whereas the varia- tion with respect to the measure d# should give (a priori) the dynamics of the dimension. Unfortu- nately, it is not difficult to check that with an action of the form postulated by us, the only way to get sta- tionarity is when the integrand itself is zero, so that it does not actually impose any constraint on the dimension.

An intuitive picture of the structures involved in our idea can be obtained by considering the trivial example, which interpolates between dimensions 1

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Volume 210, number 1.2 PHYSICS LETTERS B 18 August 1988

and 2, of the union of the unit interval in the real line with the unit disc:

K:= [0, 1]wB(2, R = 1) . (8)

It is easy to check that d imHK= 2, but the dimension field n(x) is defined in all points, except in x = 1:

0 < x < l , n ( x ) = l ,

l < x < 3 , n ( x ) = 2 . (9)

The measure # in this case would be the Lebesgue measure in ~ for 0 < x < 1 and the Lebesgue measure in ~2 for x > 1.

To summarize, we have presented a generalization of Einstein's theory of general relativity in which the (local) dimension of spacetime is a dynamical vari- able. We think that this set of ideas should be most likely relevant in the construction of quantum grav- ity models superseding the category of differential manifolds.

I am grateful to Dr. A. Cordoba, Dr. J. Diaz-Alonso, Dr. A. Gonzalez-Arroyo, Dr. J. Madore, Dr. M.A.R. Osorio and Dr. I. Peral for stimulating discussions and to Dr. B. Lopez-Melero for his patient mathematical tutoring.

References

[1 ] T. Appelquist, A. Chodos and P.G.O. Freund, Modern Kaluza-Klein theories ( Benjamin, New York, 1987 ).

[2] R. Rammal, G. Toulouse and M. Virasoro, Rev. Mod. Phys. 158 (1986) 765.

[ 3 ] I.V. Volovich, Number theory as the ultimate physical the- ory, preprint CERN-TH-4781-87; J.P. Serre, A Course in arithmetic (Springer, Berlin, 1979); Z.I. Borevich and I.R. Shafarevich, Number theory (Aca- demic Press, New York, 1966 ).

[ 4] J.P. Eckman and D. Ruelle, Rev. Mod. Phys. 57 ( 1985 ) 617. [5 ] J.L. Synge, Relativity: the general theory (North-Holland,

Amsterdam, 1960).

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