QUANTUM MECHANICS AND QUANTUM COSMOLOGY

							Bulgarian Journal of Physics 27 No 3_ (200O) 4~49



             QUANTUM MECHANICS AND QUANTUM COSMOLOGY
                                      AT THE PLANCK SCALE
                               G. S. Djordjevi6, B. Dragovichl), Lj. Ne~i6

          Department of Physics, University of Nis, P.O.Box 91 18000 Nls Yugoslavla

                   l) Institute of Physics, P.O.Box 57, IIOOI Belgrade, Yugoslavia
                                      e-mail: dragovicCphy.bg.ac.yu


 Abstract. We present a short review of adelic quanturn mechanics and adelic quantum cosmol-
 ogy, which may be also regarded as an introduction to adelic string theory. As an illustration,
 a,delic cosmologlcal model with a homogeneous scalar field is considered.



 l. Introduction

       According to classical mechanics, space and time are continuous, and distances can be in
 principle measured by any accuracy. However, in quantum mechanics there is the uncertainty
 relation Z\xAk Z h/2 that imposes a restriction on simultaneous measuring of position x and
 momentum k. Morever, quantum mechanics combined with general relativity yields [1] Aa; >
 lpl = (Gh/c3)1/2 _ 10-33cm; i.e. the Planck length is the minimum one which can be measured.
 Thus, nothing can be said about the structure of spac,>time beyond the Planck scale. In fact,
 this result is derived using concepts of archimedean geometry and real numbers. For this reason
 it seems to be quite natural at the Planck scale to take into account also non-archimedean
 geometry based on ~adic numbers. The corresponding mathematical instrument to do that is
 adellc analysis. A ~adic number [2] x e Qp (P = a prlme number) can be presented as

                                           oo
                                   x = pry ~xiiPO, l, ..,P - 1.
                                                        xi =                                         (1)

                                           i=0

 An adele [3] a is an infinite sequence


                                             ( a2 ' '
                                           a = aoo "',ap"" '        )                                (2)


 where aoo e R E Qoo is a real number and ap e Qp, with restriction laplp ~ I for all but a finite
 number of p. The set of all adeles A is a ring under componentwise addition and multiplication.
 An additive character on A is

                        X
              X(xy) = H ~ (x~y~) exp( 2lrix_y_)rHlexp(2lri{xpyp}p), x,ye A                           (3)



 where {ap}p denotes the fractional part in expansion of ap. The characteristic function ~,
 defined by ~(u) = I if u ~ I and ~(u) = O,u > 1, plays a role of a vacuum state in 3~adic
 quantum mechanics. Since 1987, there has been a significant investigation in construction of
 physical models with ~adic numbers and adeles (for a review, see [2]).

 2. Adelic quantum mechanics

       In our 1~adic quantum mechanics, canonical variables are l~adic numbers and wave func-
 tions are complex-valued. Since wave functions and their variables belong to dlfllerent valued
 fields,'the usual quantizatlon does not work. However, p-adic quantum mechanics [4] can be
                                                                                                          47

formulated using the Weyl quantization procedure. Adelic quantum mechanics [5], which unifies
standard and all ~adic quantum mechanics, is defined as a triple (L2(A), W(z), U(t)), where z
is an adelic point of a classical phase space, and t is an adelic time. L2(A) is the Hilbert space
on A, W(z) is a unitary representation of the Heisenberg-Weyl group on L2(A), and U(t) Is
a unitary representation of the evolution operator on L2(A). An adelic eigenfunction has the
form

                                 ~(x) = I~oo(xco) (4)
                                            n l~p(xp) ~ll ~(1 xp lp), x e A

                                       pes pcs
where S is a finite set of primes p, and W(oo) e L2(R), ~r(p) e L2(Qp). The evolutlon operator
U(t) and its integral kernel are defined by


                                U t" W(X u) ry fQ.t'~1; X" t~)W(')(X~, t~)dX'
                                 () =    K~(X'~"
                                                                                               ~,
                                                                                                          (5)




            1; = the probability amplitude tlvl;xt,t~), also go from space-time point
wherereprcsents oo,2,3t5,.•.. for a particle to called thea quantum-mechanlcal
propagator,
                        The kernel K;v(xfvf'

(x~, t~) to aspace-time point (x:,1, t:,/). It is useful to introduce ~adic generalization of Feynmah's
path integral as
                                          f X - S(xu,t/1;xl,t/) h
                                 'Cp(xu, tn;p x', t/) =Dx(t), ) (6)
                                                              (
                                                                    1



                                                                                       ,,
where h is the Planck constant and an action S(xn, tn; x/, t/) = J;: p(x, ~,t)dt. In paper [6] it
was shown that for one-dimensional 8ystems with a classical action S(x", tn; :~', t/) quadratic in
x/ and x'/ the kernel ;Cv has the form


             Kv(xu tu xl t') Av_ ~ ::7~ aXuaXla2S Illl2 v ~S(x// tn x' t )
                               ~ I a2S ) Il I ~ eaylax/ 1 X                         ( ,;'                 (7)



For a particu]ar physical system, ~adic eigenfunctions are subject of the spectral problem

                                           Up(t)~(p") (:1;) = Xp(at)~(p") (x) ,                           (8)


where a is ~a,dic analogue of energy. Adelic generalization of (8) is straightforw~Lrd. p-Adic
vacuum states are such futlctions Wo e L2(Qp), for which


                                                       f
                                      Up(t)~O(x) = Kt(x, y)~O(y)dy = ~o(x).                                (9)



 In accordance with (4), the simplest adelic vacuum state has the form


                                         ~o(x, t) = ~(*) (x~, t~) ~(Ixplp),
                                                                           n                              (10)



 where l~{oo)(xco' too) is a ground state for the real counterpart. As an illustration of the above
 formalism we considered a harmonic oscillator with time-dependent frequency [7] and a (rela-
 tivistic) free partlcle [8].


 3. Adelic quantum cosmology

      Adelic quantum cosmology [9] is an application of adelic quantum mechanics to the universe
 as a whole. It unifies ordinary and radic quantum cosmology. Here, path integral formalism oc-
 curs to be quite appropriate too] to take integration over both archimedean and non-archimedean
48

geometries on the equai footing. In this approach we introuduce adelic complex-vaiued cosmcF
logical amplitude by a functional integral [9]


                                             f D(9/!u)v~)(~))vXv(~Sv[9,sv'~']) '
                           (h;'j' ipn, ~)nlh{j' ip/, ~:/>v = (1 l)
In practice, it is not possible to deal with full superspace (the space of all 3-metrics and mat-
ter field configurations). Instead, one exploits minisuperspace (a finite number of coordinatcs
(htj' ip)). After this simp]jfication, v factors of adelic minisuperspace propagator are given by
the relation
                                        au lq
                                       <q a' )v= dN'Ct'(qau Nlqa/ O) (12)
where

                                               ,,f-
                                      'Cv(qa" Nlqa/ O) = 1)qaXv( St'[qa]) (13)
is an ordinary quantum-mechanical propagator with fixed minisuperspace coordinates qa (qau _
qa(tn), qa/ =: qa(t')) and lapse function N.


4. Adelic miniusperspace model with a homogeneous scalar field

     This model over the field of real numbers was considered in [1l]. Metric element is
           (
 ds2 = U2 -N2(t)~T + a2(t)d~~ , and )the corrcsponding action for the gravitational and
 homogeneous scalar field c is


                                    1 d ' 2 N2 a2V(c) + I . (14)
                              S1'[a,c]2~tN+ a4~2
                                            aa
                                             N2

 The change of variables in the form x = a2 cosh(2c) and y = a2 sinh(2c), and taking into
 account only class of models with the scalar field potential V(c) = acosh(2ip) + psinh(2c)t
 where a and p are arbitrary rational parameterst leads to the classical action

                                               a2 -
                                   n rt 24p N3 + ~(2 ~ a(x +xu)
                                                         l
                             Sv(x , y , Nlxl,y',O) =



                                    -p(y/ + yn))N + ( (15)
                                               8N
                                                          - x" - x')2 + (y - y')2
                                                                                n


 and to the corresponding propagator

                                                           1
                                 u n 14NlvXv(~Sv(xn, yn, Nlx!, y/, O)). (16)
                           'C1'(x ,y ,Njx/,yl,O) =


 Investigating existence of the vacuum states we obtain

                                       { ~(lxlp)~(lylp), INlp ~ 1,Ialp ~ 1,Iplp ~ 1,p~ 2,
                    ipp(x,y,N) = ~(lxl2)~(Iyl2), IN12 ~ ~,lal2 ~ 2, jpjp ~ 2. (17)

 We also found states in the form ~(pvlxlp)~(plllylp):


                              ;
                           { ~(2:jx[ )~(2'1lyl ) IN12 ~ 2- 11N12 ~ 2-2,,,Ial2 ~ 20~v,lpl2 < 23,s
     ipp(x,y,N) = ~(p lxl )~(p'!lylp),lNlp ~p~ ,jNlp ~ p~2/i jal ~ p3v,lplp ~ p3/t, (18)
                                                         2v



 where v, p = 1, 2, 3, .
                                                                                                          49

5. Concludmg remarks
     Adelic quantum mechanics and adelic quantum cosmology are establlshed. They are ex-
emplified by minisuperspace cosmological model with homogeneous scalar field. Some other
minisuperspace models can be found in [10]. The obtained adelic eigenfunctions have the form
(4). In particular, one has an eigenstate of the form

                                 ~(x, y) = ip_(x*, y*) ~1 1 ~(Ixplp)~(Iyplp),                            (19)



where ip~o(x=,y*) is the wave function of the universe in the standard                     quantum cosmology.
To interprete W(x, y) consider I~(x, y)]~ in rational points x, y, that is

                                IW(x' y)1~ = Iip*(x' y)1~ 11 ~(lxlp)~(1*Ip)'                             (20)
                                                                  p

Due to the properties of ~ function, we have

                                  l~r(x,y)1~- { , 2      x,yeZ,
                                           _ Iip~(:c,y)1=,                                               (21)
                                                     O x, y e Q\Z'
According to the usual interpretation of a wave function, from (21) follows discreteness of
quantities xl y at the natural (Planck) scale h = c = G = l. This kind of discreteness is a 2~adic
effect and depends on adelic quantum state. There are enougll indications [12] that the above
formalism of adelic quantum mechanics can be also extended to superstring theory.

References

 [ 1] L. J. Garay, Int. J. Mod. Phys., AIO, 145 (1995).
 [ 2] V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-adic Analysis and Mathematical
        Physics, World Scientific, Singapore 1994.
 [ 3] I. M. Gel'fand, I. M. Graev. I. I. Piatetskii-Shapiro: Representation Theory and Automor-
       phic Functions, Saup.ders, London, 1966.
 [ 4] V. S. Vladimirov and I. V. Volovich, Commun. Math. Phys. 123, 659 (1989).
 [ 5] B. Dragovich, Teor. Mat. Fiz. 101, 349 (1994); Int. J. Mod. Phys. A 10, 2349 (1995).
 [ 6] G. S. Djordjevi6 and B. Dragovlch, Med. Phys. Lett. A12, 1455 (1997).
 [ 7] G. S. Djordjevic and B. Dragovich p-Adic and Adelic Harmonic Oscillator with Time-
      Dependent Frequency, To be published in Theor. and Math. Phys., quant-ph/O005027.
 [ 8] G. S. Djordjevi6, B. Dragovich and Lj. Nes*i6, Mod. Phys. Lett. A14, 317 (1999).
 [ 9] B. Dragovich and Lj. Ne~i6, Grav. Cosm. 5, 222 (1999).
[lO] B. Dragovich and Lj. Ne~i6, On Adelic Generalization of C0$mological Minisuperspace
      Models. Presented at BPU-4, Veliko Turnovo (Bulgaria) (2000).
[11] L. J. Garay, J. J. Halliwel and G. A. Mena Marug~n, Phys. Rev. D 43, 2572 (1991).
[12] B. Dragovich, On Adelic Strings, heFhth/O005200.