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- A more formal statement of the axiom would bethat if 0 < | x | < | y | then there is some positiveinteger n such that | nx | > | y | .The properties of space and time:- continuous, homogeneous, infinitely divisiblequantities.- Below this scale, distances and durations cannotscaled up in order to produce macroscopicdistances and durations.- In other words, we must abandon thearchimedean axiom at very small distances.- How can one construct a physical theorycorresponding to a non-archimedean geometry?NONARCHIMEDEAN NORM- There is a stronger inequality for an absolutevalue |.| than triangle inequality which is knownas the ultrametric inequality or strong triangleinequality:| x + y | ≤ max{| x |, | y |}- Any norm |.| satisfying this is callednonarchimedean or ultrametric.3


(4+D)-DIMENSIONAL COSMOLOGICAL MODELS OVERTHE FIELD OF REAL NUMBERS4+D – dimensional Kaluza-Klein cosmology with RW typemetric having two scale factors a and R corresponding to aD-dimensional internal space and four dimensional universei a ia{ t,r , ρ }. r - space coordinates, ρ − internal spacecoordinates.ds2~= −N2dt2+ R2( t)dr dri2( )2 221+kr ( 1+k'ρ ) 24i+ a( t)dρdρaaThis model is describing an accelerating universe withdynamical compactification of extra dimensions.• The form of the energy-momentum tensor isTAB= diag( −ρ,p,p,p,pDpD,..., p, D),If we want the matter is to be confined to the fourdimensionaluniverse, we set all p = 0.We assume the energy-momentum tensor of spacetime tobe an exotic χ fluid with the equation of statepD⎛ m ⎞= ⎜ −1⎟⎝ 3 ⎠χ ρ χDimensionally extended Einstein-Hilbert action (without ahigher-dimensional cosmological term) is~ 3 DS = ∫ − g R dt d R d ρ + Sm= κ ∫ dtL5


1 DL = ~ Ra R&2N~ D− kNRa +1 12 6For closed universe ( k = 1D(D −1)+ ~ R12N~ 3 DN R a2ρ χ3aD−2), continuity equation& ρ R + 3 ( p + ) R&χ χ ρχ= 0energy densitym⎛ R0⎞ρ χ ( R ) = ρχ( R0 ) ⎜ ⎟ .⎝ R ⎠If we define Λ ≡ ρ χ (R)1 D 2 D(D −1)L = ~ Ra R&+ ~ R2N12N1 ~ D 1 ~ 3 D− NRa + NΛRa2 63aD−2a&2a&2D+ ~ R2ND+ ~ R2N2aD−12aRa & &D−1Ra & &Growth of the scaling factor R, leads to the decrease of thecosmological constantmΛ R ) = Λ(R )( R / R).( 0 0(the universe evolves form its small to large size, the largeinitial value of Λ decays to small values)20 0 =2~ 3D )If we take m = 2, Λ(R ) R 3, N(t)= R ( t)a ( t N ,L =R&N R12Classical solutions2D(D −1)a&+12Na22+D2NRa & &RaαtR(t)= Ae , a(t)=Beβ tFor( 0) = a(0)lαtβtP R(t)= lPe, a(t)= lPeR =6


For D = 1Ht−HtR( t)= lPe, a(t)= lPe, H = R&/ R − Huble parameteraccelerating (de Sitter) universe and a contracting internalspace with exactly same rates.for D > 1R ( t)2Ht= [ −1±1−2/3(1−1/ )], ( ) ± =DHtlDPea t lPe−1R(t)±= lPeDβt[ −1±1−2/3(1−1/ D)]2 βt, a(t)= lPe,Quantum solutionsWheeler-DeWitt equationHψ ( R,a)= 0X = ln R , Y = ln a⎡2 26⎤⎢(D ∂ ∂ ∂ ∂−1)+ − 6 ( , ) = 02 2 ⎥ X Y⎣ ∂XD ∂Y∂X∂Yψ⎦3 Dx = X + Y ,D + 3 D + 3yX − Y= D + 3⎧ ∂⎨−3⎩ ∂x22+2D + 2 ∂D ∂y2⎫⎬ψ( x,y)= 0⎭with four possible solutions±Dψ ( x,y)=±Dψ ( x,y)=AB±±ee±±γ / 3x±γD/( D+2) yγ / 3xmγD/( D+2) y,,7


P-ADIC NUMBERS AND ADELES- Concept (notion) of distance.- We measure distances (between real and rational numbers)using the absolute value on the real numbers, (natural metric onthe field of the real numbers R).- How did the real numbers enter analysis?- The field of real numbers R is the result of completing the fieldof rationals Q with the respect to the usual absolute value |.|.- The field Q is Causchi incomplete with respect to the usualabsolute value |.|{1, 1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414213, …}(A sequence {x n } is called a Caushy sequence with the respect to the norm|.| if it satisfies the following property: Given any a>0, there exists some Nsuch that m,n > N implies | x m - x n | < a. Basically, a sequence is Caushy if itsterms became “arbitrarily close” with respect to the norm |.|)- A map |.| from the rationals to the non-negative reals is called anorm (or valuations) if it satisfies the three following conditions:(1) | x | = 0 if and only if x = 0(2) ∀ x, y we have | xy | = | x || y | (norm of product is product ofnorms)(3) ∀ x, y we have | x + y | ≤|x | + | y | (the triangle inequality)- The usual absolute value |.| clearly satisfies these properties,but what other kinds of norms can exist?- There is trivial norm |x|=1 for all rationals x except 0, with |0|=0.- Besides the usual completion of field of rationals (which leadsto the R) there is a non-obvious completion of rationals . Theseare the p-adic fields Q p where p is some fixed prime number(Kurt Hensel in 1897).8


- Each p-adic field Q p is defined by completing Q with respect tothe “new” norm |.| p which is defined as follows:k m- 0 ≠ x ∈Q, x = p where m and n are nonzero integers,nneither divisible by the prime p, and k is an integerp−k| x | = p .- If we further define |0| p =0, then that |.| p satisfies the necessaryconditions above to be a norm on Q.- The set of equivalence classes of Caushy (with respect to |.| p )sequences has a natural field structure. It is a completion of Qwhich we call the field of p-adic numbers- There is a stronger inequality for an absolute value |.| thantriangle inequality which is known as the ultrametric inequalityor strong triangle inequality:| x + y | ≤ max{| x |, | y |}- Any norm |.| satisfying this is called nonarchimedean orultrametric.- How can one construct a physical theory corresponding to anon-archimedean geometry?- p-adic fields are nonarchimedean, it is natural to considerformulations of physics in terms of Q p rather than R- “Which p do we use?”- The adeles constitute a locally compact topological ring A Q ,individually taking the form( a ∞5; a2,a3,a , L)In all but a finite number of cases a p is a p-adic integer.- There are primary two kinds of analyses on Q p : Q p Q p(class.) and Q p C (quant.).9


PHYSICS ON ADELIC SPACESReasons to use p-adic numbers and adeles:♦ the field of rational numbers Q, which contains all observationaland experimental numerical data, is a dense subfield not only inR but also in the fields of p-adic numbers Q p♦ there is an analysis within and over Q p like that one related to R♦ general mathematical methods and fundamental physical lawsshould be invariant [I.V. Volovich, Number theory as theultimate physical theory, CERN preprint, CERN-TH.4781/87(July 1987)] under an interchange of the number fields R andQ phG♦ there is a quantum gravity uncertainty ( ∆ x ≥ l30=c), whenmeasures distances around the Planck length l 0, which restrictspriority of Archimedean geometry based on the real numbersand gives rise to employment of non-Archimedean geometryrelated to p-adic numbers♦ it seems to be quite reasonable to extend standard Feynman'spath integral method to non-Archimedean spaces,♦ adelic quantum mechanics [B. Dragovich, Adelic Model ofHarmonic Oscillator, Theor,. Math. Phys, 101 (1994) 1404-1412] is consistent with all the above assertions.Adelic quantum mechanics( L 2( A),W ( z),U ( t))♦ L ( ) - adelic Hilbert space,2A♦ W (z)- Weyl quantization of complex-valued functions onadelic classical phase space,♦ U (t)- unitary representation of an adelic evolution operatoron L ( )2A12


Dynamics of p-adic quantum model?( p)U p ( t)ψ ( x)= ∫ Kt( x,y)ψ ( y)dyQ p( p)p-adic quantum mechanics is given by a triple( L 2( Q p), W p( z p), U p( t p))Adelic evolution operator U (t)is defined byU ( t)ψ ( x)=∫K( x,y)ψ ( y)dy =∏A t Qvv=∞,2,3,...,p,...The eigenvalue problem for U (t)reads∫( p)U ( t)ψ ( x)= χ(E t)ψ ( x)αααKvt( xv, yv) ψ( v)( yv) dyThe main problem in our approach is computation of p-adicpropagator K p ( x",t";x',t') in Feynman's path integral method, i.e.Kp( x",t";x',t')=∫( x",t")( x',t')χ ( −Exact general expression for propagatorp1 t"∫ L(q&, q,t)h)t'DqvKp( x",y",t";x',y',t') = λ ( −p21 ∂ S2h∂x'∂x") ×2 1/ 21 ∂ Sh ∂x'∂x"pχp( − S ( x",t";x', t'))where S ( x",y",z",t";x',y',z't') is the action for classical trajectory.Illustration of p-adic and adelic quantum mechanical models,• a free particle and harmonic oscillator [VVZ, Dragovich]• a particle in a constant field, [Djordjevic, Dragovich]• a free relativistic particle[Djordjevic, Dragovich, Nesic], aharmonic oscillator with time-dependent frequency[Djordjevic, Dragovich]13


Resume of AQM: AQM takes in account ordinary as well as p-adiceffects and may me regarded as a starting point for construction ofmore complete quantum cosmology and string/M theory. In the lowenergy limit AQM effectively becomes the ordinary one.An adelic wave function has formψ∏∏= ψ ∞( ∞ ) ⋅ ψ ⋅p∈Mp ( x p )p∉MADELIC QUANTUM COSMOLOGYx Ω(|x | )- The main task of AQC is to describe the very early stage in theevolution of the Universe.- At this stage, the Universe was in a quantum state, whichshould be described by a wave function (complex valued anddepends on some real parameters).- But, QC is related to Planck scale phenomena - it is natural toreconsider its foundations.- We maintain here the standard point of view that the wavefunction takes complex values, but we treat its arguments in amore complete way!- We regard space-time coordinates, gravitational and matterfields to be adelic, i.e. they have real as well as p-adicproperties simultaneously.- There is no Schroedinger and Wheeler-De Witt equation forcosmological models.- Feynman’s path integral method was exploited andminisuperspace cosmological models are investigated[Djordj.Drag.Nes. Vol.].- Adelic minisuperspace quantum cosmology is an application ofadelic quantum mechanics to the cosmological models.- Path integral approach to standard quantum cosmology, thestarting point is Feynman’s path integral methodp14


'''''''''< hij, φ , Σ | hij, φ , Σ > ∞ = ∫ D(gµν ) ∞D(φ)∞ χ∞(−S∞[gµν, φ])The standard 3+1 decomposition2 µ νµν2ds = g dx dx = −(N − N N ) dt + 2N dx dt + h dx dxp-adic complex valued cosmological amplitude'''''''''i< hij, φ , Σ | hij, φ , Σ > p = ∫ D(gµν ) pD(φ)p χ p ( −Sp[gµν, φ])(4+D)-DIMENSIONAL MODEL OVER THE FIELD OF P-ADICNUMBERSi2iiijijLagrangian of the modelL=1 22X&2ND(D −1)+ Y&12N+D2NX&Y&SolutionsClassical actionX +Y ( t)= C t + C= C1tC 2 , 3 4S ( X",Y",N;X ', Y ',0)1= ( X"−X')2N2D(D −1)+ ( Y"−Y')12N2+D2N( X"−X')( Y"−Y')Kernel of p-adic operator of evolutionKp( X",Y",N;X ', Y ',0)1/ 2⎛ D(D + 2) ⎞ D(D + 2)= λp⎜( S ( X",Y",N;X ', Y ',0))2⎟χ2 p −⎝ 48N⎠ 12Npchange3 Dx = X + Y ,D + 3 D + 3yX − Y= D + 315


KpS ( x",y",N;x',y',0)1 D(D + 5) 2 1= (1 + )( x"−x')− D(D + 3)( y"−y')2N62N⎛ 6 + D(D + 5) ⎞ ⎛ D(D + 3) ⎞( x",y",N;x',y',0)= λp⎜⎟λp⎜−⎟⎝ 12N⎠ ⎝ 2N⎠D(D + 3) ⎛ D(D + 5) ⎞× 12⎜ + ⎟2N⎝ 6 ⎠1/ 2pχ ( −S).p-adic ground state wave function in (x,y) minisuperspaceConditionspψ p( x,y)= Ω(|x | p)Ω(|y | p)| N | ≤ |1+D(D + 5) / 6 |pp2| N | ≤|D(D + 3) | , p ≠pp2.p-adic ground state wave function in (X,Y) minisuperspace⎛ D D ⎞ ⎛ X − Y ⎞ψ p(X , Y ) = Ω⎜|(1 − ) X + Y | p ⎟Ω⎜|| p ⎟.⎝ D + 3 D + 3 ⎠ ⎝ D − 3 ⎠16


CONCLUDING REMARKSFor this model we construct corresponding p-adic model and showexistence of p-adic ground state wave functions.In addition, we explore quantum evolution of the model and possibilityfor its adelic generalization.It is necessary for the investigation of space-time discreteness at veryshort distances.It is shown that it is possibile to construct adequate adelic (4+D)-Kaluza-Klein model.It is shown that there exist adelic ground states of the form±D ∞ ( x∞,y∞)∏ p∈Mp(xp,y p)∏ p∉(| xp| p)Ωψ ( x , y)= ψ , ψM Ω (| y | )pp- Adopting the usual probability interpretation of the wavefunction, we have2∞±D,∞∞∞2∞∏| ψ ( x , y)| = | ψ ( x , y ) | | ψ ( x , y ) | Ω(|x | ) Ω(|y | )2( Ω (| u | p )) = Ω(|u | p )p∈M- At the rational points x and y and for M = ∅ppp2∞∏p∉Mpppp⎪⎧± 22 | ψ= ∞⎨ ∞ x yx y D ( , ) || ψ ( , ) | ,∞⎪⎩ 0x,y ∈ Zx,y ∈Q\ Z- Discretization of minisuperspace coordinates- P-Adic and Adelic Quantum cosmology give possibility toconsider number of extra dimensions and their structure.17

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