ON SOME ACCELERATING COSMOLOGICAL KALUZA-KLEIN MODELS

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1 ON SOME ACCELERATING COSMOLOGICAL KALUZA-KLEIN MODELS GORAN S. DJORDJEVIĆ and LJUBIŠA NEŠIĆ Deartment of Physics, University of Nis, Serbia and Montenegro nesiclj, Received February, 005 We consider (4 + D)-dimensional Kaluza-Klein cosmological model with two scaling factors, as real, -adic and adelic quantum mechanical one. One of the scaling factors corresonds to the D-dimensional internal sace, and second one to the 4-dimensional universe. In standard quantum cosmology, i.e., over the field of real numbers R, it leads to dynamical comactification of additional dimensions and to the accelerating evolution of 4-dimensional universe. We construct corresonding -adic quantum model and exlore existence of its -adic ground state. In addition, we exlore evolution of this model and a ossibility for its adelic generalization. It is necessary for the further investigation of sace-time discreteness at very short distances, i.e., in a very early universe.. INTRODUCTION On the basis of the results of quantum gravity [], theoretical uncertainty of measuring distances is greater or equal to the Planck distance. It leads to the conclusion that on very short distances Archimedean axiom is not valid, i.e. the sace can osses ultrametric features. Because geometry is always connected with a concrete number field, in the case of the nonarchimedean geometry it is used to be the field of -adic numbers Q. In the high energy hysics, these numbers have been used almost twenty years. The motivation comes from string theory []. Generally seaking, -adic aroach should be useful in describing a very early hase of the universe and rocesses around Planck scale. A significant number of aers, motivated by [], has been ublished u to now (for a review see [3, 4]). In this aer we are esecially interested in alication of -adic numbers and analysis in quantum cosmology [5]. -Adic quantum mechanics [6] (QM) has been successfully alied in minisuersace quantum cosmology [7]. We have treated many cosmological models, mainly constructed in four sace-time dimensions [5]. Based on that one can ask: is it ossible to extend this aroach to the multidimensional quantum cosmological models? In this article we briefly formulate such a model. Rom. Journ. Phys., Vol. 50, Nos. 3 4, P , Bucharest, 005

2 90 Goran S. Djordjević, Ljubiša Nešić After a brief mathematical introduction in Section, a short review of -adic and adelic quantum mechanics and cosmology is given in Section 3. Section 4 is devoted to the classical (4+D)-dimensional cosmological models filed with exotic fluid [8]. The corresonding -adic model is considered in Section 5. This aer is ended by short conclusion, including adelic generalization, and suggestion for advanced research.. -ADIC NUMBERS AND ADELES Perhas the most easier way to understand -adic numbers is if one starts with the notion of norm. It is well known that any norm must satisfy three conditions: nonnegativity, homogeneity and triangle inequality. The comletion of the field of rational numbers Q with resect to the absolute value, or standard norm., gives the field of real numbers R Q. Besides this norm there are another ones which satisfy the first two conditions and the third one in a stronger way x+ y max( x, y ), () so called strong triangle inequality. The most imortant of them is -adic norm. [3]. The feature (), also called ultrametricity, is one of the most imortant characteristics of the -adic norm ( denotes a rime number). The number fields obtained by comletion of Q with resect to this norm are called -adic number fields Q. It is known that any nonzero rational number x can be exressed as x =± γa/ b, where a γ is the rational number, and a and b are the natural numbers which are not divisible with the rime number and have no common divisor. Then, -adic norm of x is, by definition, x = γ. Because Q is local comact commutative grou the Haar measure can be introduced, which enables integration. In articular, the Gauss integral will be emloyed x x x / β χ ( α +β )d =λ ( α ) α χ, α 0, Q 4α () where χ = ex π ix { }, is an additive character ({x} is the fractional art of x), and λ( α ) is an arithmetic comlex-valued function [3]. It should be noted that there is the real counterart of λ ( ) x = ( isign a), x Q. (3) λ

3 3 Accelerating cosmological Kaluza-Klein models 9 Commonly the main roerties of λ υ, (υ denotes or any ) are: (0) ( υ υ a ) υ ( ) (4) λ =, λ α =λ α, ( ) ( ) ( ) ( υ υ υ υ ) υ ( ) (5) λ α λ β =λ α+β λ α +β, λ α =. Very simle but rather imortant function in -adic analysis and -adic QM is, x, Ω ( x ) = (6) 0, x >, which is the characteristic function of Z. Note that Z = { x Q : x } is the ring of -adic integers. Simultaneous treatment of real and -adic numbers can be realized by concet of adeles. An adele a A is an infinite sequence a= ( a, a,, a, ), where a R and a Q, with the restriction that a Z for all but a finite set S of rimes. The set of all adeles A can be written in the form A = U A ( S), A ( S) = R Q Z. (7) S Also, A is a toological sace. Algebraically, it is a ring with resect to the comonentwise addition and multilication. There is the natural generalization of analysis on R and Q to analysis on A [9]. S S 3. -ADIC AND ADELIC QUANTUM MECHANICS AND COSMOLOGY -Adic quantum mechanics is quantum mechanics over the field Q, and -adic quantum cosmology is the alication of -adic QM on the universe as a whole [5] ADIC AND ADELIC QUANTUM MECHANICS -Adic QM has been develoed in two different ways: in the first one wave function is comlex valued function of -adic variable [3], and in the second one, -adic wave function deends on -adic variable [0]. If we want simultaneous (adelic) treating of standard quantum mechanics and all -adic s, including usual robabilistic interretation develoed in standard QM, then the first formulation is the referable one. Because -adic fields Q are totally disconnected [3], dynamics of -adic quantum models is described by a unitary evolution oerator Ut ( ) without using

4 9 Goran S. Djordjević, Ljubiša Nešić 4 the Hamilton oerator and infinitesimal dislacement. Therefore, Ut ( ) is formulated in terms of its kernel Kt ( x, y) U() t ψ ( x) = K t( x, y) ψ ( y)dy. (8) Q In the above equation K t is the kernel of the -adic evolution oerator which is defined by the functional integral (in units G = = c= ) K t t( x, y) = χ L( q, q)dt d q( t), 0 (9) with roerties analogous to the roerties existing in standard QM. In that way, -adic QM is given by a trile [3] ( L( Q), W( z), U( t)). Keeing in mind that standard QM can be also resented as the corresonding trile, ordinary and -adic QM can be unified in the form of adelic quantum mechanics [] t ( L ( A ), W( z), U( t)), (0) where L ( A ) is the Hilbert sace on A, Wz () is the unitary reresentation of the Heisenberg-Weyl grou on L ( A ) and Ut ( ) is the unitary reresentation of the evolution oerator on L ( A). The adelic evolution oerator Ut ( ) is defined as ( ) ( ) t υ υ υ υ () Ut () ψ ( x) = K t( xy, ) ψ ( y)d y= K υ ( x, y) ψ υ ( y)dy. A The eigenvalue roblem for Ut ( ) reads v Q v Ut () ψ ( x) =χ( Et) ψ ( x), () αβ α αβ where ψ αβ are adelic eigenfunctions, Eα = ( E, E,..., E,...) denotes adelic energy, indices α and β denote energy levels and their degeneration, resectively. Any adelic eigenfunction has the form A (3) Ψ ( x) =Ψ ( x ) Ψ ( x ) Ω ( x ), x, S S S where Ψ L( R) and Ψ L ( Q ) are ordinary and -adic eigenfunctions, resectively. We would like to arahrase Manin s oinion [] that our universe cannot be described fully by real numbers nor -adic s only, than by adeles. In other words, our universe is inherently adelic.

5 5 Accelerating cosmological Kaluza-Klein models ADIC AND ADELIC QUANTUM COSMOLOGY Adelic quantum cosmology can be understood as an alication of adelic quantum theory to the universe as a whole [5]. In the ath integral aroach to standard quantum cosmology, the starting oint is Feynman s ath integral method, i.e. the amlitude to go from one state with intrinsic metric h ij and matter configuration φ on an initial hyersurface Σ to another state with metric h ij and matter configuration φ on a final hyersurface Σ is given by a functional integral hij,φ,σ hij,φ,σ = ( gµν ) ( ) ( S [ gµν ]) D Φ χ,φ, D (4) over all (four)-geometries g µν and matter configurations Φ, which interolate between the initial and final configurations [3]. In above exression Sg [ µν,φ] is an Einstein-Hilbert action for the gravitational and matter fields. To erform -adic and adelic generalization we first make -adic counterart of the action using form-invariance under the inter-change of the real to the -adic number fields. Then we generalize (4) and introduce -adic comlex-valued cosmological amlitude hij,φ,σ hij,φ,σ = ( gµν ) ( ) ( S[ g ]) D Φ χ D µν,φ, (5) where gµν ( x) and Φ ( x) are the corresonding -adic counterarts of metric and matter fields continually connecting their values on Σ and Σ. The standard minisuersace ground state wave function in the Hartle-Hawking (no-boundary) roosal [4], will be attained if one erforms a functional integration in the Euclidean version of Ψ [ hij ] = D ( gµν ) ( Φ) χ ( S [ gµν,φ ]), D (6) over all comact four-geometries g µν which induce h ij at the comact threemanifold. This three-manifold is the only boundary of all the four-manifolds. If we generalize the Hartle-Hawking roosal to the -adic minisuersace, then an adelic Hartle-Hawking wave function is an infinite roduct Ψ [ h ] = ( g ) ( Φ) χ ( S [ g,φ]), ij D D υ υ υ υ µν υ µν (7) where the ath integration must be erformed over both, archimedean and nonarchimedean, geometries. If evaluation of the corresonding functional integrals gives as a result Ψ [ h ij ] in the form (3), we will say that such cosmological model is adelic one.

6 94 Goran S. Djordjević, Ljubiša Nešić 6 A more successful -adic generalization of the minisuersace cosmological models can be erformed in the framework of -adic and adelic quantum mechanics without using the Hartle-Hawking roosal. In such cases, we [5] examine the conditions under which some eigenstates of the evolution oerator exist QUANTUM MINISUPERSPACE MODELS IN -ADIC AND ADELIC QUANTUM MECHANICS In the study of the homogeneous universes, the metric deends only on the time arameter. It enables formulation of the models with a finite dimensional configuration sace, so called minisuersace [3]. Their variables are the three-metric comonents, and corresonding material fields, if any. The quantization of these models can be erformed adoting of the rules of QM. One of the main reasons for using minisuersace models in describing the universe is their simlicity. While the full (suersace) quantum cosmological models are usually unsolvable, the minisuersace ones can be handled with available mathematical tools. In -adic miniusuersace quantum cosmology, we investigate necessary conditions for the existence of the -adic ground state in the form of Ω-function. Some other tyical eigenfunctions like δ-function can be also of interest. The existence of these functions enables construction of -adic models and their adelization. Usually, this leads to the some restrictions on the arameters for the many exactly soluble minisuersace cosmological models [5]. The necessary condition for the existence of an adelic quantum model is the existence of -adic ground state Ω ( ) defined by (6), i.e., q q α K ( qα, N; qα, 0)d qα =Ω ( qα ), (8) α (q α are minisuersace coordinates, α =,,, n and N is the lase function). Analogously, if a system is in the state Ω ( ν q ), where Ω ( ν q ) = if qα and Ω ( qα ) = 0 if qα >, then its kernel must satisfy equation K ( q N q 0)d q ( ν q ) ν α, ; α, α =Ω α. (9) ν q α ν If -adic ground state is of the form of the δ-function, where δ-function is defined as, if a= b, δ( a b) = 0, if a b, then the corresonding kernel of the model has to satisfy equation α ν α

7 7 Accelerating cosmological Kaluza-Klein models 95 ( q N q 0)d q ( ν K q ) ν α, ;, α =δ α. (0) q = α Equations (8) and (9) are usual -adic vacuum ingredients of the adelic eigenvalue roblem (), where χ ( Et ) = in the vacuum state { Et } = 0. The above Ω and δ functions do not make a comlete set of -adic eigenfunctions, but they are very simle and illustrative. Since these functions have finite suorts, the ranges of integration in (8) (0) are also finite. The las function N is under the kernel K ( qα, N; qα, 0) and N is restricted to some values on which eigenfunctions do not deend exlicitly. 4. (4+D)-DIMENSIONAL COSMOLOGICAL MODELS OVER THE FIELD OF REAL NUMBERS The old idea that four dimensional universe, in which we exist, is just our observation of hysical multidimensional sace-time attracts much attention nowadays. In such models, the comactification of extra dimensions lay the key role and in the some of them leads to the eriod of accelerated exansion of the universe [5, 8, 6]. This aroach is suorted and encouraged with the recent results of the astronomical observations. We briefly recaitulate some facts of the real multidimensional cosmological models, necessary for -adic and adelic generalization. The metric of such Kaluza-Klein model with D-dimensional internal sace can be resented in the form [8, 7] d ad a dridri ρ ρ = + + d s N ()d t r R () t a () t () ( k ) kr + ρ + 4 where is Nt ( ) is a lase function, Rt ( ) and at ( ) are the scaling factors of 4-dimensional universe and internal sace, resectively; r r i r i (i =,, 3), ρ ρaρ a ( a=, D), and k, k = 0,±. The form of the energy-momentum tensor is T = diag( ρ,,,,,,..., ), () AB D D D where indices A and B run over both sacetime coordinates and the internal sace dimensions. If we want the matter is to be confined to the four-dimensional universe, we set all D = 0. Now, we examine the case for which the ressure along all the extra dimensions vanishes D = 0 (in braneworld scenarios the matter to be confined to the four-dimensional universe), so that all comonents of T AB are set to zero

8 96 Goran S. Djordjević, Ljubiša Nešić 8 excet the sacetime comonents [8]. We assume the energy-momentum tensor of sacetime to be an exotic fluid χ with the equation of state χ m ( ) = ρ 3 χ, (3) ( χ and ρ χ are the ressure and the energy density of the fluid, arameter m has value between 0 and ). 4.. CLASSICAL MODEL Dimensionally extended Einstein-Hilbert action (without a higherdimensional cosmological term) is d 3 D m d (4) S = gr td Rd ρ+ S =κ tl where κ is an irrelevant constant, R is the scalar curvature of the metric, so we can write off the lagrangian of the model (for flat internal sace) as L = D DD ( ) 3 D D (5) = Ra R + R a a + R ad Ra knra D + N ρ R3aD χ. N N N 6 For closed universe (k = ), substitution of the equation of state in the continuity equation ρ χ R+ 3( χ +ρ χ) R = 0 (6) leads to the energy density in form R0 ρ χ( R) =ρ χ( R0 ), R (7) where R 0 is the value of scaling factor in an arbitrary reference time. If we define cosmological constant as Λ=ρ ( R), the lagrangian becomes χ L = D DD ( ) = Ra R + R3aD a + D RaD Ra NRa D + N Λ R3aD. N N N 6 m (8) Growth of the scaling factor R, according to (7) leads to the decrease of the cosmological constant by the relation m R0 Λ ( R) =Λ ( R0 ). R (9)

9 9 Accelerating cosmological Kaluza-Klein models 97 This decaying Λ term may also exlain the smallness of the resent value of the cosmological constant since, as the universe evolves form its small to large size, the large initial value of Λ decays to small values. If we take that m =, initial condition for cosmological constant and scaling factor Λ ( R 0) R0 = 3, for las function Nt () = R3() tad () tnt (), the lagrangian (8) becomes R DD ( ) L = + a D Ra +. (30) N R N a N Ra It should be noted that there are no arameters k and Λ in the lagrangian. That means that although they are not zero in this model, it is equivalent to a flat universe with zero cosmological term. In another words, there is not difference between four dimensional universe which looks flat and not filled with fluid and closed universe filled with exotic fluid. The solutions of corresonding equations of motion are Rt () = Ce αt, (3) at () = Ce βt, (3) where the constants C, C, α and β deend of initial conditions. It is a reasonable assumtion that the size of all satial dimensions is the same at t = 0. It may be assumed that this size would be the Planck size, i.e. R(0) = a(0) = lp. Above solutions, can be written in terms of Huble arameter H = R / R [8] Rt () = l e Ht P, (33) at () = l e Ht. (34) Deending of the dimensionality of the internal sace we have, for D = and P Rt () = l e Ht P, (35) ( ) Ht D ± 3 D a () t l P e ± =, (36) ( ) Dβt ± 3 D R () t l P e ± =, (37) at () = l eβ t, (38) P for D >. The solution corresonding to D = redicts an accelerating (de Sitter) universe and a contracting internal sace, with exactly the same rates. In the case D > analysis is comlicated, but results are similar.

10 98 Goran S. Djordjević, Ljubiša Nešić QUANTUM MODEL Quantum solutions are obtained from the Wheeler-DeWitt equation HΨ ( R, a) = 0, (39) where H is the Hamiltonian and Ψ is the wave function of the universe. For this model above equation is written in the new variables ( X = ln R and Y = ln a) ( D ) Ψ ( X, Y) = 0. X D Y X Y (40) By the new change x = X 3 + Y D, y = X Y, Wheeler-DeWitt equation D+ 3 D+ 3 D + 3 takes a simle form 3 D + + Ψ ( x, y) = 0. x D y Equation (4) has the four ossible solutions [8] (4) γ γd ± x y ( ) e 3 D D x y A ± ± ± Ψ, = +, (4) γ γd ± x y ( ) e 3 D D x y B ± ± Ψ, = +, (43) where A ± and B ± are the normalization constants. It is ossible to imose the boundary conditions to get a Ψ D( Ra, ) = 0. For further details see [8]. 5. (4+D)-DIMENSIONAL MODEL OVER THE FIELD OF -ADIC NUMBERS Consideration of (4+D)-dimensional Kaluza-Klein model over the field Q will be started from the lagrangian in the form (30). All quantities in this lagrangian will be treated as -adic ones. Taking again relacement X = ln R and Y = ln a, it becomes DD ( ) L = D XY X + N N Y +. (44) N The corresonding -adic equations of motion are X + D Y = 0, X + D Y = 0. (45) 3 It is not difficult to see that above system can be rewritten in the following form X = 0, Y = 0. (46)

11 Accelerating cosmological Kaluza-Klein models 99 If we omit from consideration seudoconstant solutions and concentrate on the analytical ones we get Xt () = Ct + C, Yt () = Ct 3 + C4. The calculation of the -adic classical action gives S( X, Y, N; X, Y, 0) = DD ( ) = ( X X ) + ( Y Y ) + D ( X X )( Y Y ). N N N (47) Because this action is quadratic with resect to both variables X and Y, we can write down the kernel of -adic oerator of evolution [8] K( X, Y, N; X, Y, 0) = DD ( + ) DD ( + ) =λ ( ( 0)) 48 χ S X, Y, N; X, Y, N N (48) Let use again the change x = X 3 + Y D, y = X Y to searate the D+ 3 D+ 3 D + 3 variables and make the further analyses of this model rather simle. In these variables the classical action and the kernel of evolution oerator writes S( x, y, N; x, y, 0) = DD ( + 5) = ( x x ) D( D 3)( y y ) N + 6 +, N 6 + DD ( + 5) DD ( + 3) K( x, y, N; x, y, 0) =λ 6 λ N / DD ( + 3) DD ( + 5) ( ( 0)) + χ S x, y, N; x, y,. N 6 (49) (50) Now, we can examine when -adic wave function has the form corresonding to the simlest ground state [9] Ψ ( x, y) =Ω ( x ) Ω ( y ). (5) Putting the kernel of the oerator of evolution (50) in equation (8) we get that the required state exists if both conditions DD ( + 5) N +, N D( D+ 3),, (5) 6 are fulfilled. To answer the question: can this conditions be useful in determination of dimensionality of the internal sace? needs further careful analysis. Going back to the old variables, -adic ground state wave function for our model is

12 300 Goran S. Djordjević, Ljubiša Nešić ( ) Ψ ( x, y) =Ω D X D Y X Y + D 3 D 3 Ω. D We can also write down the solutions in the variables R and a. (53) 6. CONCLUSION In this aer, we demonstrate how a -adic version of the quantum (4+D)-Kaluza-Klein model with exotic fluid can be constructed. It is an exactly soluble model. From equations (4), (43) and (5), i.e. (53), it is ossible to construct an adelic model too, i.e. model which unifies standard and all -adic models []. The investigation of its ossible hysical imlication and discreteness of sace-time deserves much more attention and room. Let us note that adelic states for the (4+D)-dimensional Kaluza-Klein cosmological model (for any D which satisfies (5)) exist in the form (54) ± S D, S S ±, Ψ ( xy, ) =Ψ ( x, y ) Ψ ( x, y ) Ω ( x ) Ω ( y ), where Ψ D ( x, y ) are the corresonding real counterarts of the wave functions of the universe. In the ground state Ψ ( x, y) are roortional to Ω ( ν x ) Ω ( µ y ) or to δ( ν x ) δ( µ y ). Adoting the usual robability interretation of the wave function (54), we have (55) S x y ± D x y x y x y, S S Ψ (, ) = Ψ (, ) Ψ (, ) Ω ( ) Ω ( ), because ( Ω ( x )) =Ω ( x ). As a consequence of Ω-function roerties, at the rational oints x, y and in the (secial) vacuum state ( S =, i.e. all Ψ ( x, y ) =Ω ( x ) Ω ( y ), we find Ψ ± D ( x y) x, y Z ( x y),,,, Ψ, = 0, x, y Q\ Z. This result leads to some discretization of minisuersace coordinates x, y. Namely, robabilty to observe the universe corresonding to our minisuersace model is nonzero only in the integer oints of x and y. Keeing in mind that Ω-function is invariant with resect to the Fourier transform, this conclusion is also valid for the momentum sace. Note that this kind of discreteness deends on adelic quantum state of the universe. When some -adic states are different from Ω ( x ) Ω ( y )( S ), then the above adelic discreteness becomes less transarent. (56)

13 3 Accelerating cosmological Kaluza-Klein models 30 Performing the integration in (55) over all the -adic saces, and having in mind that eigenfunctions should be normed to unity, one recovers the standard effective model over real sace. However, if the region of integration is over only some arts of -adic saces then the adelic aroach manifestly exhibits -adic quantum effects. Since the Planck length is here the natural one, the adelic minisuersace models refer to the Planck scale. Further investigations will be focused on determination of conditions for existence of the ground states in the form Ω ( ν x ) Ω ( µ y ) and -adic delta function. We should emhasize that investigation of dimensionality D of internal sace from the conditions (5) and seudoconstant solutions of the equation (46) deserves attention. It could additionally contribute to better understanding of the model, esecially from its -adic sector. Acknowledgments. The authors would like to thank the organizers for the kind invitation to give talk at the Conference. Lj. Nesic is thankful to D. Grecu and M. Visinescu for their kind hositality during the Conference. The research of both authors was suorted by the Serbian Ministry of Science and Technology, Project No 643. REFERENCES. L. J. Garay, Int. J. Mod. Phys., A0 (995) 45.. I. V. Volovich, Theor. Math. Phys., 7 (987) V. S. Vladimirov, I. V. Volovich and E. I Zelenov, -Adic Analysis and Mathematical Physics, World Scientific, Singaore, L. Brekke and P. G. O Freund, -Adic numbers in hysics, Phys. Re., 33 (993).. 5. G. S. Djordjevic, B. Dragovich, Lj. Nesic, I.V.Volovich, Int. J. Mod. Phys., A 7 (00). 43, gr-qc/ G. S. Djordjevic, B. Dragovich and Lj. Nesic, Modern Physics Letters, A 4 (999) B. Dragovich and Lj. Nesic, Grav. Cosm., 5 (999). 8. F. Darabi, Classical and Quantum Gravity, 0 (003) I. M. Gel fand, M. I. Graev and I. I. Piatetskii-Shairo, Reresentation Theory and Automorhic Functions, Saunders, London, A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Acad. Publ., Dordrecht, Yu. I. Manin, in Conformal invariance and string theory, Acad. Press, (989) 93.. B. Dragovich, Theor. Mat. Phys., 0 (994) D. L. Wiltshire, An Introduction to Quantum Cosmology, gr-qc/ J. Hartle and S. Hawking, Phys. Rev., D8 (983) P. K. Townsend, M. N. R. Wohlfarth, Accelerating cosmologies from comactification, he-th/ S. Jalalzadeh, F. Ahmadi, H. R. Seangi, Multi-dimensional classical and quantum cosmology, he-th/ J. Wudka, Phys. Rev., D35 (987) G. S. Djordjevic, B. Dragovich, Lj. Nesic, Int. Conference: FILOMAT 00, Nis, 6 9 August 00. FILOMAT 5 (00) 33, G. S. Djordjevic, B. Dragovich, Mod. Phys. Lett., A (997) G. S. Dordjevic and Lj. D. Nesic, in Proc. of the XI Congress of Physicists of Serbia and Montenegro, Petrovac, SCG, 3 5. June 004, 6/9.