Quantum Universe on Extremely Small Space-time Scales

The semiclassical approach to the quantum geometrodynamical model is used for the description of the properties of the Universe on extremely small space-time scales. Under this approach, the matter in the Universe has two components of the quantum nature which behave as antigravitating fluids. The f...

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Datum:	2010
Hauptverfasser:	Kuzmichev, V.E., Kuzmichev, V.V.
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Veröffentlicht:	Відділення фізики і астрономії НАН України 2010
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Schlagworte:	Астрофізика і космологія
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spelling	irk-123456789-562072014-02-14T03:12:01Z Quantum Universe on Extremely Small Space-time Scales Kuzmichev, V.E. Kuzmichev, V.V. Астрофізика і космологія The semiclassical approach to the quantum geometrodynamical model is used for the description of the properties of the Universe on extremely small space-time scales. Under this approach, the matter in the Universe has two components of the quantum nature which behave as antigravitating fluids. The first component does not vanish in the limit ħ → 0 and can be associated with dark energy. The second component is described by an extremely rigid equation of state and goes to zero after the transition to large space-time scales. On small space-time scales, this quantum correction turns out to be significant. It determines the geometry of the Universe near the initial cosmological singularity point. This geometry is conformal to a unit four-sphere embedded in a five-dimensional Euclidean flat space. During the consequent expansion of the Universe, when reaching the post-Planck era, the geometry of the Universe changes into that conformal to a unit four-hyperboloid in a five-dimensional Lorentz-signatured flat space. This agrees with the hypothesis about the possible change of geometry after the origin of the expanding Universe from the region near the initial singularity point. The origin of the Universe can be interpreted as a quantum transition of the system from a region in the phase space forbidden for the classical motion, but where a trajectory in imaginary time exists, into a region, where the equations of motion have the solution which describes the evolution of the Universe in real time. Near the boundary between two regions, from the side of real time, the Universe undergoes almost an exponential expansion which passes smoothly into the expansion under the action of radiation dominating over matter which is described by the standard cosmological model. Квазiкласичний пiдхiд до квантово-геометродинамiчної моделi застосовано для опису властивостей всесвiту на екстремально малих просторово-часових масштабах. У цьому пiдходi матерiя у всесвiтi має двi компоненти квантової природи, якi поводять себе як антигравiтуючi рiдини. Перша компонента не набуває нульового значення в границi ħ → 0 та може бути асоцiйована з темною енергiєю. Друга компонента описується екстремально жорстким рiвнянням стану i прямує до нуля пiсля переходу до великих просторово-часових масштабiв. На малих просторовочасових масштабах ця квантова поправка вiдiграє значну роль. Вона визначає геометрiю всесвiту бiля точки початкової космологiчної сингулярностi. Ця геометрiя є конформною до одиничної 4-сфери, зануреної у 5-вимiрний евклiдовий плоский простiр. Пiд час наступного розширення всесвiту, пiсля досягнення пост-планкiвської ери, геометрiя всесвiту перетворюється на геометрiю, конформну до одиничного 4-гiперболоїда у 5- вимiрному плоскому просторi з лоренцiвською сигнатурою. Це узгоджується з гiпотезою про можливу змiну геометрiї пiсля виникнення всесвiту, що розширюється з областi поблизу точки початкової сингулярностi. Виникнення всесвiту може бути iнтерпретовано як квантовий перехiд системи з областi у фазовому просторi, забороненої для класичного руху, але де iснує траєкторiя в уявному часi, в область, де рiвняння руху мають розв’язок, що описує еволюцiю всесвiту у реальному часi. Поблизу межi мiж двома областями, з боку реального часу, всесвiт зазнає майже експоненцiального розширення, яке гладко переходить у розширення пiд дiєю випромiнювання, що домiнує над матерiєю, у вiдповiдностi iз стандартною космологiчною моделлю. 2010 Article Quantum Universe on Extremely Small Space-time Scales / V.E. Kuzmichev, V.V. Kuzmichev // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 626-635. — Бібліогр.: 20 назв. — англ. 2071-0194 PACS 98.80.Qc, 04.60.-m, 04.60.Kz http://dspace.nbuv.gov.ua/handle/123456789/56207 en Український фізичний журнал Відділення фізики і астрономії НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Астрофізика і космологія Астрофізика і космологія
spellingShingle	Астрофізика і космологія Астрофізика і космологія Kuzmichev, V.E. Kuzmichev, V.V. Quantum Universe on Extremely Small Space-time Scales Український фізичний журнал
description	The semiclassical approach to the quantum geometrodynamical model is used for the description of the properties of the Universe on extremely small space-time scales. Under this approach, the matter in the Universe has two components of the quantum nature which behave as antigravitating fluids. The first component does not vanish in the limit ħ → 0 and can be associated with dark energy. The second component is described by an extremely rigid equation of state and goes to zero after the transition to large space-time scales. On small space-time scales, this quantum correction turns out to be significant. It determines the geometry of the Universe near the initial cosmological singularity point. This geometry is conformal to a unit four-sphere embedded in a five-dimensional Euclidean flat space. During the consequent expansion of the Universe, when reaching the post-Planck era, the geometry of the Universe changes into that conformal to a unit four-hyperboloid in a five-dimensional Lorentz-signatured flat space. This agrees with the hypothesis about the possible change of geometry after the origin of the expanding Universe from the region near the initial singularity point. The origin of the Universe can be interpreted as a quantum transition of the system from a region in the phase space forbidden for the classical motion, but where a trajectory in imaginary time exists, into a region, where the equations of motion have the solution which describes the evolution of the Universe in real time. Near the boundary between two regions, from the side of real time, the Universe undergoes almost an exponential expansion which passes smoothly into the expansion under the action of radiation dominating over matter which is described by the standard cosmological model.
format	Article
author	Kuzmichev, V.E. Kuzmichev, V.V.
author_facet	Kuzmichev, V.E. Kuzmichev, V.V.
author_sort	Kuzmichev, V.E.
title	Quantum Universe on Extremely Small Space-time Scales
title_short	Quantum Universe on Extremely Small Space-time Scales
title_full	Quantum Universe on Extremely Small Space-time Scales
title_fullStr	Quantum Universe on Extremely Small Space-time Scales
title_full_unstemmed	Quantum Universe on Extremely Small Space-time Scales
title_sort	quantum universe on extremely small space-time scales
publisher	Відділення фізики і астрономії НАН України
publishDate	2010
topic_facet	Астрофізика і космологія
url	http://dspace.nbuv.gov.ua/handle/123456789/56207
citation_txt	Quantum Universe on Extremely Small Space-time Scales / V.E. Kuzmichev, V.V. Kuzmichev // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 626-635. — Бібліогр.: 20 назв. — англ.
series	Український фізичний журнал
work_keys_str_mv	AT kuzmichevve quantumuniverseonextremelysmallspacetimescales AT kuzmichevvv quantumuniverseonextremelysmallspacetimescales
first_indexed	2025-07-05T07:26:16Z
last_indexed	2025-07-05T07:26:16Z
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fulltext	V.E. KUZMICHEV, V.V. KUZMICHEV QUANTUM UNIVERSE ON EXTREMELY SMALL SPACE-TIME SCALES V.E. KUZMICHEV, V.V. KUZMICHEV Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine (14b, Metrolohichna Str., Kyiv 03680, Ukraine) PACS 98.80.Qc, 04.60.-m, 04.60.Kz c©2010 The semiclassical approach to the quantum geometrodynamical model is used for the description of the properties of the Uni- verse on extremely small space-time scales. Under this approach, the matter in the Universe has two components of the quantum nature which behave as antigravitating fluids. The first compo- nent does not vanish in the limit ~ → 0 and can be associated with dark energy. The second component is described by an ex- tremely rigid equation of state and goes to zero after the transi- tion to large space-time scales. On small space-time scales, this quantum correction turns out to be significant. It determines the geometry of the Universe near the initial cosmological singularity point. This geometry is conformal to a unit four-sphere embed- ded in a five-dimensional Euclidean flat space. During the conse- quent expansion of the Universe, when reaching the post-Planck era, the geometry of the Universe changes into that conformal to a unit four-hyperboloid in a five-dimensional Lorentz-signatured flat space. This agrees with the hypothesis about the possible change of geometry after the origin of the expanding Universe from the region near the initial singularity point. The origin of the Universe can be interpreted as a quantum transition of the system from a region in the phase space forbidden for the classical motion, but where a trajectory in imaginary time exists, into a region, where the equations of motion have the solution which describes the evo- lution of the Universe in real time. Near the boundary between two regions, from the side of real time, the Universe undergoes almost an exponential expansion which passes smoothly into the expansion under the action of radiation dominating over matter which is described by the standard cosmological model. 1. Introduction It is accepted that the present-day Universe as a whole can be considered as a cosmological system described by the standard model based on general relativity [1, 2]. According to the Standard Big-Bang Model [2, 3], the early Universe was very hot and dense. In order to de- scribe that era, one must take into account that the Uni- verse has passed in the course of its evolution through a stage with quantum degrees of freedom of the grav- itational and matter fields before turning into the cos- mological system, whose properties are described well by general relativity. This means that a consistent de- scription of the Universe as a nonstationary cosmological system should be based on quantum general relativity in the form admitting the passage to general relativity in semiclassical limit ~→ 0 [4, 5]. A consistent quantum theory of gravity, in principle, can be constructed on the basis of the Hamiltonian for- malism with the application of the canonical quantiza- tion method. The first problem on this way is to choose generalized variables. Following, e.g., Refs. [6–8], it is convenient to choose metric tensor components and mat- ter fields as such variables. But the functional equations obtained in this approach prove to be insufficiently suit- able for specific problems of quantum theory of gravity and cosmology. These equations do not contain a time variable in an explicit form. This, in turn, gives rise to the problem of interpretation of the state vector of the Universe (see, e.g., the discussion in Ref. [3] and ref- erences therein). A cause of the failure can be easily understood with the help of Dirac’s constraint system theory [9]. It is found that the structure of constraints in general relativity is such that the variables which cor- respond to true dynamical degrees of freedom cannot be singled out from canonical variables of geometrodynam- ics. This difficulty is caused by the absence of a prede- termined way to identify space-time events in generally covariant theory [10]. One of the possible versions of a theory with a well- defined time variable is proposed in Refs. [11, 12] in the case of the homogeneous, isotropic, and closed Uni- verse. The Universe is supposed to be filled with a homogeneous scalar field which stands for the primor- dial matter1 and (macroscopic) relativistic matter asso- ciated with the material reference frame. As calculations have demonstrated [11, 12], the equations of the quan- tum model may be reduced to the form, in which the matter energy density in the Universe has a component which is a condensate of massive quanta of a scalar field. 1 Since we deal with the quantum theory, we should describe the matter content of the Universe by some fundamental La- grangians of fields. 626 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 QUANTUM UNIVERSE ON EXTREMELY SMALL SPACE-TIME SCALES Under the semiclassical description, this component be- haves itself as an antigravitating fluid. Such a property has the quantum nature, and it is connected with the fact that the states with all possible masses of a con- densate contribute to the state vector of the quantum Universe. If one discards the corresponding quantum corrections, the quantum fluid degenerates into a dust, i.e. the matter component of the energy density which is commonly believed to make a dominant contribution to the mass-energy of the ordinary matter in the present Universe in the standard cosmological model. Let us note that the presence of a condensate in the Universe, as well as the availability of a dust representing an ex- treme state of a condensate, is not presupposed in the initial Lagrangian of the theory. An antigravitating con- densate arises out of the quantum description of fields. If one supposes that the properties of our Universe are described in an adequate manner by such a quantum theory, an antigravitating condensate being found out can be associated with dark energy [12]. Assuming that particles of a condensate can decay to baryons, leptons (or to their antiparticles), and particles of dark matter, one can describe the percentage of baryons, dark matter, and dark energy observed in the Universe [13]. In the semiclassical limit, the negative pressure fluid arises as a remnant of the early quantum era. This anti- gravitating component of the energy density does not vanish in the limit ~ → 0. In addition to this compo- nent, the stress-energy tensor contains the term vanish- ing after the transition to general relativity, i.e. to large space-time scales. However, on small space-time scales, quantum corrections ∼ ~ turn out to be significant. As is shown in this paper, the effects caused by these cor- rections determine the equation of state of matter and the geometry near the initial cosmological singularity point. They define a boundary condition that should be imposed on the state vector at the origin, so that a nucleation of the Universe from the initial cosmological singularity point becomes possible. In this paper, we use the modified Planck system of units: lP = √ 2G~/(3πc3) is taken as a unit of length, ρP = 3c4/(8πGl2P ) is a unit of energy density, and so on. All relations are written for dimensionless quantities. 2. Equations of Motion 2.1. Classical model Let us consider the homogeneous, isotropic, and closed Universe which is described by the Robertson–Walker metric ds2 = dτ2 − a2dΩ2 3, (1) where τ is the proper time, a is the cosmic scale factor, and dΩ2 3 is a line element on a unit three-sphere. It is convenient to pass to a new time variable η, dτ = aNdη, (2) where N is the lapse function that specifies the time reference scale. We assume that the Universe is originally filled with a homogeneous scalar field φ and a perfect fluid as a macroscopic medium which defines the so-called material reference frame [10, 11, 14]. In the model of the Universe under consideration, the action has the form S = ∫ dη { πa da dη + πφ dφ dη + πΘ dΘ dη + πλ̃ dλ̃ dη −H } , (3) where πa, πφ, πΘ, πλ̃ are the momenta canonically con- jugate with the variables a, φ, Θ, λ̃, H = N 2 { −π2 a − a2 + a4[ρφ + ρ] } + +λ1 { πΘ − 1 2 a3ρ0s } + λ2 { πλ̃ + 1 2 a3ρ0 } , (4) is the Hamiltonian, ρφ = 2 a6 π2 φ + V (φ) (5) is the energy density of a scalar field with the potential V (φ), and ρ = ρ(ρ0, s) is the energy density of a perfect fluid which is a function of the density of the rest mass ρ0 and the specific entropy s. The quantity Θ is the thermasy (potential for the temperature, T = Θ, νU ν), λ̃ is the potential for the specific free energy f taken with an inverse sign, f = − λ̃, νUν , and Uν is the four- velocity. The momenta πρ0 and πs conjugate with the variables ρ0 and s vanish identically, πρ0 = 0, πs = 0. (6) Hamiltonian (4) of such a system has the form of a lin- ear combination of constraints and weakly vanishes (in Dirac’s sense [9]), H ≈ 0, (7) ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 627 V.E. KUZMICHEV, V.V. KUZMICHEV where the sign ≈ means that the Poisson brackets must all be worked out before the use of the constraint equa- tions. The quantities N , λ1, and λ2 are Lagrange multi- pliers. The variation of action (3) with respect to them leads to three constraint equations −π2 a − a2 + a4[ρφ + ρ] ≈ 0, πΘ − 1 2 a3ρ0s ≈ 0, πλ̃ + 1 2 a3ρ0 ≈ 0. (8) It follows from the conservation of these constraints in time that the conservation laws hold, E0 ≡ a3ρ0 = const, s = const, (9) where the first relation describes the conservation law of a macroscopic quantity which characterizes the number of particles of a perfect fluid, and the second equation represents the conservation of the specific entropy. With regard for these conservation laws and Eqs. (6), one can discard degrees of freedom corresponding to the vari- ables ρ0 and s and convert the second-class constraints into first-class constraints [11] in accordance with Dirac’s proposal. The equation of motion for the classical dynamical variable O = O(a, φ, πa, πφ, . . . ) has the form dO dη ≈ {O, H}, (10) where H is Hamiltonian (4), and {., .} are Poisson bra- ckets. 2.2. Quantum model In quantum theory, the first-class constraint equations (8) become constraints on the state vector Ψ. Passing from the classical variables to corresponding operators and using the conservation laws (9), we obtain three equations{ − ∂2 a + a2 − a4[ρφ + ρ] } Ψ = 0, { − i∂Θ − 1 2 E0s } Ψ = 0, { − i∂λ̃ + 1 2 E0 } Ψ = 0. (11) It is convenient to pass from the generalized variables Θ and λ̃ to the non-coordinate co-frame h dτ = s dΘ − dλ̃, h dy = s dΘ + dλ̃, (12) where h = ρ+p ρ0 is the specific enthalpy which plays the role of inertial mass, p is the pressure, τ is proper time in every point of space, and y is a supplementary vari- able. The corresponding derivatives commute between themselves, [∂τ , ∂y] = 0. It follows from the first equation in system (11) that it is convenient to choose a perfect fluid in the form of relativistic matter. Introducing the quantity E ≡ a4ρ = const, (13) we come to the equations which describe the quantum Universe [11]{ − i ∂τc − 1 2 E0 } Ψ = 0, ∂yΨ = 0, (14) { − ∂2 a + a2 − 2aĤφ − E } Ψ = 0, (15) where τc is the time variable connected with the proper time τ by the differential relation dτc = h dτ , Ĥφ = 1 2 a3 [ − 2 a6 ∂2 φ + V (φ) ] (16) is the operator of mass-energy of a scalar field in a co- moving volume 1 2 a 3. It follows from Eqs. (14) that Ψ does not depend on the variable y. The first equation of system (14) has a particular solution in the form Ψ = e i 2 Eτ̄ \|ψ〉, (17) where τ̄ = E0 E τc is the rescaled time variable. The state vector \|ψ〉 is defined in the space of two variables a and φ and is determined by the equation( − ∂2 a + a2 − 2aĤφ − E ) \|ψ〉 = 0. (18) This equation describes the state of the Universe with a definite value of the parameter E. The vector \|ψ〉 represents the dynamical state of the Universe at some instant of time η0 which is connected with time τ̄ by the relation τ̄ = 4 3 ∫ η0 Ndη. Considering the vector \|ψ〉 as an immovable vector of the Heisenberg representation, one can describe the motion of the quantum Universe by the equation 〈ψ\| 1 N d dη Ô\|ψ〉 = 1 N d dη 〈ψ\|Ô\|ψ〉 = 1 i 〈ψ\|[Ô, 1 N Ĥ]\|ψ〉, (19) 628 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 QUANTUM UNIVERSE ON EXTREMELY SMALL SPACE-TIME SCALES where [., .] is a commutator, and Ĥ is determined by expression (4), in which all dynamical variables are sub- stituted with operators. The observable Ô corresponds to the classical dynamical variable O. For Ô = a, we obtain 〈ψ\| − i∂a\|ψ〉 = −〈ψ\|ada dτ \|ψ〉. (20) In the classical theory, the corresponding momentum has the form πa = ∂aS = −ada dτ ≡ −aȧ, (21) where S is the action. For Ô = −i∂a, we find 〈ψ\| − i 1 N d dη ∂a\|ψ〉 = 〈ψ\|a− 2 a3 ∂2 φ − 2a3V (φ)\|ψ〉. (22) 3. Scalar Field Model Equation (18) can be integrated with respect to φ, if one determines the form of the potential V (φ). As in Ref. [12], we consider the solution of Eq. (18) in the era when the field φ oscillates with a small amplitude near the minimum of its potential at the point φ = σ. Then V (φ) can be approximated by the expression V (φ) = ρσ + m2 σ 2 (φ− σ)2, (23) where ρσ = V (σ), m2 σ = [d2V (φ)/dφ2]σ > 0. If φ = σ is the point of absolute minimum, then ρσ = 0, and the state σ corresponds to the true vacuum of a primordial scalar field, while the state with ρσ 6= 0 matches with the false vacuum [15]. Introducing a new variable x = ( mσa 3 2 )1/2 (φ− σ), (24) which describes a deviation of the field φ from its equi- librium state, and defining the harmonic oscillator func- tions 〈x\|uk〉 as solutions of the equation( −∂2 x + x2 ) \|uk〉 = (2k + 1)\|uk〉, (25) where k = 0, 1, 2, ... is the number of a state of the oscillator, we find Ĥφ\|uk〉 = ( Mk + 1 2 a3ρσ ) \|uk〉, (26) where the quantity Mk = mσ ( k + 1 2 ) (27) can be interpreted as an amount of matter-energy (or mass) in the Universe related to a scalar field. This energy is represented in the form of a sum of the ex- citation quanta of spatially coherent oscillations of the field φ about the equilibrium state σ, k is the number of these excitation quanta. The mentioned oscillations correspond to a condensate of zero-momentum φ quanta with the mass mσ. We look for a solution of Eq. (18) in the form of a superposition of the states with different masses Mk, \|ψ〉 = ∑ k \|fk〉\|uk〉. (28) Using the orthonormality of \|uk〉, we obtain the equation for the vector \|fk〉:( −∂2 a + Uk − E ) \|fk〉 = 0, (29) where Uk = a2 − 2aMk − a4ρσ (30) is the effective potential. In the case ρσ = 0, this equa- tion is exactly integrable [11]. The corresponding eigen- value is equal to E ≡ En,k = 2n+ 1−M2 k , (31) where n = 0, 1, 2, ... is the number of a state of the quan- tum Universe with the mass Mk in the potential well (30). The vectors \|fk〉 and \|fk′〉 at k 6= k′ are, generally speaking, nonorthogonal between themselves. So that, the transition probability w(n, k → n′, k′) = \|〈fk′ \|fk〉\|2 is nonzero. For example, the probability of the transition of the Universe from the ground (vacuum) state n = 0 to any other state obeys the Poisson distribution w(0, k → n′, k′) = 〈n′〉n′ n′! e−〈n ′〉, (32) where 〈n′〉 = 1 2 (Mk′ −Mk)2 is the mean value of the quantum number n′. Substituting Eq. (28) into Eq. (22), we obtain 〈fk\|− i N d dη ∂a\|fk〉 = 〈fk\|a−2a3ρσ−4Mk\|fk〉+Δk, (33) where Δk = −3mσ〈fk\| ∑ k′ 〈uk\|∂2 x\|uk′〉\|fk′〉. (34) ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 629 V.E. KUZMICHEV, V.V. KUZMICHEV Here, k′ takes the values k and k±2. This term describes the component of the pressure of the condensate caused by the motion of φ quanta in the phase space with the momentum −i∂x. Using Eq. (25), we find Δk = 3Mk〈fk\|fk〉− −3 2 √( Mk + 3 2 mσ )( Mk + 1 2 mσ ) 〈fk\|fk+2〉 −3 2 √( Mk − 3 2 mσ )( Mk − 1 2 mσ ) 〈fk\|fk−2〉. (35) In the case k � 1, the masses Mk±2 'Mk � 1 2 mσ and, according to Eq. (29), the vectors \|fk±2〉 ' \|fk〉. Then Δk = 0 at k � 1. (36) This means that the contributions to the sum with re- spect to k′ in Eq. (34) from the different k-states of the Universe are mutually cancelled. As a result, the pressure of a condensate is determined only by its quan- tum properties (see Eq. (44) below). We note that if one discards the contributions from the transition am- plitudes 〈fk\|fk±2〉, a condensate turns into an aggregate of separate macroscopic bodies with zero pressure (dust) [12]. The existence of this limit argues in favor of the reliability of this quantum model. 4. Semiclassical Approach 4.1. Einstein-type equations In order to give the physical meaning to the different quantities emergent in this theory, we reduce Eqs. (29) and (33) to the form of the Einstein equations. To this end, we choose the vector \|fk〉 in the form 〈a\|fk〉 = const√ ∂aS(a) eiS(a), (37) where S is the unknown function of a (we omit the index k here and below). Substituting Eq. (37) into (29) and (33) and taking (36) into account, we obtain the equa- tions which can be represented in the “standard” form for the perfect fluid source 1 a4 (∂aS)2 − ρm − ρu + 1 a2 = 0, (38) 1 a2 d dτ (∂aS) + 1 2 (ρm − 3pm) + ρ̃u − 1 a2 = 0, (39) where ρu = 1 a4 { 3 4 ( ∂2 aS ∂aS )2 − 1 2 ∂3 aS ∂aS } (40) and ρ̃u = i 2a2 d dτ ( ∂2 aS ∂aS ) (41) are the quantum corrections to the stress-energy tensor, ρu ∼ ~2, and ρ̃u ∼ ~ (in ordinary units [12]), ρm = ρk + ρσ + ρ, pm = pk + pσ + p (42) can be interpreted as the (effective) energy density and the isotropic pressure having the form of the sum of the components, ρk = 2Mk a3 , ρσ ≡ V (σ) ≡ Λ 3 , ρ = E a4 , (43) Λ is the cosmological constant. The equations pk = −ρk, pσ = −ρσ, p = 1 3 ρ, (44) stand for the equations of state. The equations of state for the vacuum component ρσ = const and the relativis- tic matter ρ are dictated by the formulation of the prob- lem. The vacuum-type equation of state of a condensate with the density ρk, which does not remain constant throughout the evolution of the Universe but decreases according to a power law with increase of a, follows from the condition of consistency of Eqs. (38) and (39). From (42)–(44), we can conclude that a condensate behaves itself as an antigravitating medium. Its anti- gravitating effect has a purely quantum nature. Its ap- pearance is determined by the fact that the Universe’s state vector (28) is a superposition of quantum states with all possible values of the quantum number k. In the classical limit (~ = 0), the terms ρu and ρ̃u can be discarded, and Eqs. (38) and (39) reduce to the Einstein equations which predict an accelerating expan- sion of the Universe in the era with ρk > 2 3 ρ, even if Λ = 0. Since ρ ∼ a−4 decreases with a more rapidly than ρk ∼ a−3 (or even ∼ a−2 [12]), the era of acceler- ating expansion should begin with increasing a, even if the state with ρk < 2 3 ρ and Λ ∼ 0 existed in the past, when the expansion was decelerating. The condensate of a quantized primordial scalar field can be identified with the dark energy [12, 13]. In this connection, it should be emphasized that the energy density dynamics of the condensate ρk has to be 630 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 QUANTUM UNIVERSE ON EXTREMELY SMALL SPACE-TIME SCALES considered in accordance with quantum theory. A dust component, as such, is a limiting case of the quantum description, when the transition amplitudes 〈fk\|fk±2〉 in Eq. (35) are not taken into account. According to quan- tum theory, the quantum fluctuations of density which depend on time and the scale factor in an explicit form are imposed on a classical background. Equation (35) implies that it is convenient to choose the energy den- sity of a dust as such a background. Then the energy density of the condensate can be written in the form ρk = ρdust k + δρk, where δρk describes quantum fluc- tuations. At such a semiclassical description, the con- densate is a source of ordinary matter and dark matter [12] which together form ρdust k . Quantum fluctuations δρk contribute to the vacuum component, so that the latter appears to be non-zero, even if ρσ = 0. Thus, tak- ing quantum fluctuations into account guarantees the satisfaction of the energy conservation law for the total stress-energy tensor. Let us calculate the corrections ρu and ρ̃u. These terms are essential in the very early Universe at a < 1 2. This quantum theory predicts the quantum origin (nucleation) of the Universe from the region a ∼ 0 [11]. This means that the state vector in this region is con- stant, 〈a ∼ 0\|fk〉 = const. For such a state, S = i 2 ln a+ const (45) and the quantum corrections (40) and (41) are equal to ρu = − 1 4a6 , ρ̃u = iȧ 2a4 = − i 2a5 ∂aS = 1 4a6 , (46) where we used representation (21) for the calculation of ρ̃u. It can be done in the semiclassical approach under consideration3. With regard for Eqs. (46), Eqs. (38) and (39) can be reduced to the form of the standard Einstein equations for the homogeneous, isotropic, and closed Universe:( ȧ a )2 = ρtot − 1 a2 , ä a = − 1 2 [ρtot + 3ptot] , (47) where the quantities ρtot = ρm + ρu, ptot = pm + pu (48) 2 For the present-day Universe, we have a ∼ 1061 in accepted dimensionless units. 3 Let us note that the presence of a minus sign in ρu (46) is not extraordinary. According to quantum field theory, for instance, vacuum fluctuations make a negative contribution to the field energy per unit area (the Casimir effect). describe the total energy density and the pressure of the matter in the Universe which take its quantum nature into account in the semiclassical approximation. The quantum correction ρu may be identified with the ultra- stiff matter with the equation of state pu = ρu, (49) where pu is the pressure. This ‘matter’ has quantum origin. Let us estimate the ratio of the energy density \|ρu\| to ρm. Passing to the ordinary units, we have R ≡ [( 2G~ 3πc3 )2 1 4a6 ]/[ 8πG 3c4 ρm ] . (50) where ρm and a are measured, respectively, in GeV/cm3 and cm. For our Universe today, ρm ∼ 10−5 GeV/cm3, a ∼ 1028 cm, and Rtoday ∼ 10−244, (51) i.e. the quantum correction may be neglected to an ac- curacy of ∼ O(10−244). In the Planck era, ρm ∼ 10117 GeV/cm3, a ∼ 10−33 cm, and the relation RPlanck ∼ 1 (52) shows that the densities ρm and ρu are of the same order of magnitude. 4.2. Quantum effects on sub-Planck scales On sub-Planck scales, a < 1, the contributions from the condensate, cosmological constant, and curvature may be neglected. As a result, the equations of the model take the form 1 2 ȧ2 + U(a) = 0, ä = −dU da , (53) where U(a) ≡ 1 2 [ 1 4a4 − E a2 ] . (54) These equations are similar to ones of Newtonian me- chanics. Using this analogy, they can be considered as equations which describe the motion of a ‘particle’ with unit mass and zero total energy under the action of the force −dUda , U(a) is the potential energy, and a(τ) is a generalized variable. A point ac = 1 2 √ E , where U(ac) = 0, divides the region of motion of a ‘particle’ into the subregion a < ac, where the classical motion of ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 631 V.E. KUZMICHEV, V.V. KUZMICHEV (a) E = 1 (b) E = 3 Fig. 1. Scale factor ã vs imaginary time t for the cases E = 1, which corresponds to the ground state n = 0 (a), and E = 3 for the state with n = 1 (b) a ‘particle’ is forbidden, and the subregion a > ac, where the classical trajectory of a ‘particle’ moving in real time τ exists. In the subregion a < ac, there exists the classi- cal trajectory of a ‘particle’ moving in imaginary time t = −iτ + const in the potential −U(a). Denoting the corresponding solution as ã, we find ã = ac sin z, t = a3 c 2 [2z − sin 2z]. (55) At small z, i.e. in the region ã ∼ 0, we have ã = ( 3 2 t )1/3 . (56) Comparing Eq. (56) with the standard model solution (see, e.g., [3, 16]), we conclude that it agrees with the fact that the ‘matter’ near the point ã = 0 is described by the equation of state of the ultrastiff matter (49). In Fig. 1, the scale factor ã is shown as a function of imaginary time t. It demonstrates that the scale factor oscillates in imaginary time near its zero value. The amplitude and the frequency of these oscillations depend on the parameter E. In the subregion a ≤ ac, where the contributions from the condensate and the cosmological constant can be neglected, the eigenvalue E of Eq. (29) is quantized according to expression (31) with Mk = 0. The case E = 1 corresponds to the Universe which is in the ground state n = 0 (see Fig. 1,a). The value E = 3 refers to the quantum state n = 1 (see Fig. 1,b). The amplitude of oscillations which determines the size of the region forbidden for the classical motion de- creases as n−1/2, while the frequency increases with n. The case of very high values of n describes the classical motion of the system. In the limit n → ∞, the region forbidden for the classical motion shrinks to the point producing the initial singularity, in which the Universe is characterized by the infinitely large frequency of oscil- lations of the space curvature. In the subregion a > ac, the solution of Eqs. (53) can be written as a = ac cosh ζ, τ = a3 c 2 [2ζ + sinh 2ζ]. (57) At ζ � 1, this implies that the scale factor at τ � 2a3 c increases almost exponentially a = ac [ 1 + ( 1 2a2 c )3 τ2 + . . . ] ≈ ac exp { τ2 8a6 c } . (58) The almost exponential expansion of the early Uni- verse in that era is related to the action of quantum effects which, according to Eqs. (46) and (49), cause the negative pressure, pu < 0, i.e. produce an antigravitat- ing effect on the cosmological system under considera- tion. At ζ � 1, solution (57) takes the form a = ( τ ac )1/2 . (59) It describes the radiation-dominated era and corre- sponds to time τ � 2a3 c . In Fig. 2, the scale factor a is shown as a function of real time τ . With increase of τ , it increases at first by law (58) and then in accordance with Eq. (59). The initial value a(τ = 0) depends on the quantum number n. Figure 2,a demonstrates the case where n = 0, while Fig. 2,b shows the case n = 1. In the limit n → ∞, the initial singularity a(τ = 0) = 0 will be the reference point of the scale factor a as a function of τ as in general relativity. 632 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 QUANTUM UNIVERSE ON EXTREMELY SMALL SPACE-TIME SCALES Solutions (55) and (57) are related to each other through the analytic continuation into the region of com- plex values of the time variable, t = −iτ + π 2 a3 c , z = π 2 − iζ. (60) The scale factors ã (55) and a (57) are connected through the condition a(τ) = ã (π 2 a3 c − iτ ) , (61) which describes the analytic continuation of the time variable τ into the region of complex values of Euclidean time t. This analytic continuation can be interpreted as a quantum tunneling of the Lorentzian space-time from the Euclidean one. 4.3. Transition amplitude The model determined by Eqs. (53) allows us to describe the origin (nucleation) of the Universe as the transition from the state in the subregion a < ac to the state in the subregion a > ac. The corresponding transition am- plitude can be written as follows [15]: T ∼ e−St , (62) where St is the action on a trajectory in imaginary time t, St = 2 ∞∫ −∞ dtU(ã). (63) Let us proceed to the integration with respect to the time variable z. According to Eq. (55), the scale fac- tor ã is a periodic function of z. At first, we consider the oscillations of a ‘particle’ on the finite time inter- val [−z0, z0] with the boundary conditions ã(z0) = ±ac and ã(−z0) = ∓ac. Supposing that z0 = π 2 ν, where ν = 1, 3, 5, . . . numbers the quantity of half-waves of the function ã(z), centered at the points z = ±πq, q = 0, 1, 2, . . . , which cover the interval [−z0, z0]. Then the action St takes the form St = 2 π 2 ν∫ −π2 ν dz dt dz U(ã(z)). (64) Using the explicit form of solution (55), we find St = − √ Eπν, (65) (a) n = 0 (b) n = 1 Fig. 2. Scale factor a vs real time τ for the values n = 0 (a) and n = 1 (b) and amplitude (62) becomes T ∼ e √ Eπν , (66) i.e. a ‘particle’ which is the equivalent of the Universe leaves the subregion forbidden for the classical motion with an exponential probability density. It is pushed out of the forbidden subregion into the subregion of very small values of a in real time τ by the antigravitating forces stipulated by the negative pressure which cause quantum processes at a ∼ 0 (see Eqs. (46) and (49)). This phenomenon can be interpreted as the origin of the Universe from the region a < ac. It is possible only if the probability density that the Universe is in the state with a ∼ 0 is nonzero. In the limit ν →∞, the transition amplitude T → e∞. This result can be interpreted so that the origin of the Universe occurs with probability 1 during the infinite imaginary time interval. ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 633 V.E. KUZMICHEV, V.V. KUZMICHEV 4.4. Geometry Let us consider how the geometry of the Universe changes as a result of its transition from the region a < ac into a > ac. In the model under consideration, the metric has the form (1). According to solutions (55) and (57), metric (1) takes the form ds2E = −a2 c sin2 z { 4a4 c sin2 z dz2 + dΩ2 3 } at a < ac (67) and ds2L = a2 c cosh2 ζ { 4a4 c cosh2 ζ dζ2 − dΩ2 3 } at a > ac, (68) where the interval with the Euclidean signature is de- noted by the index E, and the one with the Lorentzian signature is marked by L. Introducing the new time variables ξ and ς according to dξ = 2a2 c sin zdz, dς = 2a2 c cosh ζdζ, (69) metrics (67) and (68) can be reduced to the conformally flat form ds2E = −a2 c [ 1− ( ξ 2a2 c )2 ]{ dξ2 + dΩ2 3 } , (70) ds2L = a2 c [ 1 + ( ς 2a2 c )2 ]{ dς2 − dΩ2 3 } . (71) Both metrics are related to each other through the an- alytic continuation into the region of complex values of the time variable ς = iξ. The conformal factor in metric (70) varies from zero value at ξ = −2a2 c to the maximum value a2 c at ξ = 0 and then vanishes again at ξ = 2a2 c . The conformal factor in metric (71) ranges from its min- imum value a2 c at ς = 0 to infinity increasing with ς. Metric (70) is conformal to the metric of a unit four- sphere in a five-dimensional Euclidean flat space. With increasing a, the Universe transits from the region a < ac into the region a > ac, where the geometry is conformal to a unit hyperboloid embedded in a five-dimensional Lorentz-signatured flat space. Such a picture of changes in the space-time geometry during the transition of the Universe from the region near the initial singularity into the region of real physical scales agrees with the hypoth- esis [17, 18] widely discussed in the literature (see, e.g., the reviews [19, 20]) for the de Sitter space about a pos- sible change in the four-space geometry after the spon- taneous nucleation of the expanding Universe from the initial singularity point. 5. Conclusions In this paper, we study the properties of the quan- tum Universe on extremely small space-time scales in the semiclassical approach to the well-defined quantum model. We show that quantum gravity effects ∼ ~ ex- hibit themselves near the initial cosmological singularity point in the form of an additional matter source with the negative pressure and the equation of state as for the ul- trastiff matter. The analytical solution of the equations of the theory of gravity, in which matter is represented by radiation and the additional matter source of the quan- tum nature, is found. It is shown that the geometry of the Universe is described at the stage of the evolution of the Universe, when quantum corrections ∼ ~ domi- nate over radiation, by the metric which is conformal to a metric of a unit four-sphere in a five-dimensional Eu- clidean flat space. In the radiation-dominated era, the metric is found to be conformal to a unit hyperboloid embedded in a five-dimensional Lorentz-signatured flat space. One solution can be continued analytically into another one. The origin of the Universe can be interpreted as a quantum transition of the system from the region in the phase space forbidden for the classical motion, but where a trajectory in imaginary time exists, into the region where the equations of motion have the solution which describes the evolution of the Universe in real time. Near the boundary between two regions, from the side of real time, the Universe undergoes almost an exponential ex- pansion which passes smoothly into the expansion under the action of radiation dominating over matter which is described by the standard cosmological model. As a result of such a quantum transition, the geometry of the Universe changes. This agrees with the hypothesis about a possible change of the geometry after the ori- gin of an expanding Universe from the region near the initial singularity point. In this paper, this phenomenon is demonstrated in the case of the early Universe filled with radiation and the ultrastiff matter which effectively takes into account quantum effects on extremely small space-time scales. This work was supported partially by the Program of Fundamental Research “The Fundamental Proper- ties of Physical Systems under Extreme Conditions” of the Division of Physics and Astronomy of the National Academy of Sciences of Ukraine. 1. R.M. Wald, General Relativity (Chicago University Press, Chicago and London, 1984). 634 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 QUANTUM UNIVERSE ON EXTREMELY SMALL SPACE-TIME SCALES 2. K.A. Olive and T.A. Peacock, Phys. Lett. B 667, 217 (2008). 3. E.W. Kolb and M.S. Turner, The Early Universe (Addison-Wesley, Redwood City, 1990). 4. C.J. Isham, Proc. GR14 Conference (Florence, 1995); gr-qc/9510063. 5. C. Kiefer, in Lecture Notes in Physics 541: To- wards Quantum Gravity, edited by J. Kowalski-Glikman (Springer, Heidelberg, 2000); gr-qc/9906100. 6. J.L. Anderson, in Gravitation and Relativity, edited by Hong-Yee Chin and W.F. Hoffman (Benjamin, New York, 1964). 7. J.A. Wheeler, in Battelle Rencontres, edited by C. De- Witt snd J.A. Wheeler (Benjamin, New York, 1968). 8. B.S. DeWitt, Phys. Rev. 160, 1113 (1967). 9. P.A.M. Dirac, Lectures on Quantum Mechanics (Belfer Graduate School of Science, Yeshiva University, New York, 1964). 10. K.V. Kuchař and C.G. Torre, Phys. Rev. D 43, 419 (1991). 11. V.E. Kuzmichev and V.V. Kuzmichev, Acta Phys. Pol. B 39, 979 (2008); gr-qc/0712.0464. 12. V.E. Kuzmichev and V.V. Kuzmichev, Acta Phys. Pol. B 39, 2003 (2008); gr-qc/0712.0465. 13. V.E. Kuzmichev and V.V. Kuzmichev, in Trends in Dark Matter Research, edited by J.V. Blain (Nova Science, Hauppage, 2005); astro-ph/0405455. 14. J.D. Brown and D. Marolf, Phys. Rev. D 53, 1835 (1996); gr-qc/9509026. 15. S. Coleman, Phys. Rev. D 15, 2929 (1977). 16. I. Dymnikova and M. Fil’chenkov, Phys. Lett. B 545, 214 (2002). 17. J.B. Hartle and S.W. Hawking, Phys. Rev. D 28, 2960 (1983). 18. S. Hawking and R. Penrose, The Nature of Space and Time (Princeton University Press, Princeton, 2000). 19. B.L. Al’tshuler and A.O. Barvinsky, Usp. Fiz. Nauk 166, 459 (1996). 20. C. Kiefer and B. Sandhöfer, in Beyond the Big Bang, edited by R. Vaas (Springer, Berlin, 2008); gr-qc/0804.0672. Received 01.10.09 КВАНТОВИЙ ВСЕСВIТ НА ЕКСТРЕМАЛЬНО МАЛИХ ПРОСТОРОВО-ЧАСОВИХ МАСШТАБАХ В.Є. Кузьмичов, В.В. Кузьмичов Р е з ю м е Квазiкласичний пiдхiд до квантово-геометродинамiчної моделi застосовано для опису властивостей всесвiту на екстремально малих просторово-часових масштабах. У цьому пiдходi матерiя у всесвiтi має двi компоненти квантової природи, якi поводять себе як антигравiтуючi рiдини. Перша компонента не набуває нульового значення в границi ~→ 0 та може бути асоцiйована з темною енергiєю. Друга компонента описується екстремально жорстким рiвнянням стану i прямує до нуля пiсля переходу до великих просторово-часових масштабiв. На малих просторово- часових масштабах ця квантова поправка вiдiграє значну роль. Вона визначає геометрiю всесвiту бiля точки початкової космо- логiчної сингулярностi. Ця геометрiя є конформною до одини- чної 4-сфери, зануреної у 5-вимiрний евклiдовий плоский про- стiр. Пiд час наступного розширення всесвiту, пiсля досягне- ння пост-планкiвської ери, геометрiя всесвiту перетворюється на геометрiю, конформну до одиничного 4-гiперболоїда у 5- вимiрному плоскому просторi з лоренцiвською сигнатурою. Це узгоджується з гiпотезою про можливу змiну геометрiї пiсля виникнення всесвiту, що розширюється з областi поблизу то- чки початкової сингулярностi. Виникнення всесвiту може бути iнтерпретовано як квантовий перехiд системи з областi у фа- зовому просторi, забороненої для класичного руху, але де iснує траєкторiя в уявному часi, в область, де рiвняння руху мають розв’язок, що описує еволюцiю всесвiту у реальному часi. По- близу межi мiж двома областями, з боку реального часу, все- свiт зазнає майже експоненцiального розширення, яке гладко переходить у розширення пiд дiєю випромiнювання, що домi- нує над матерiєю, у вiдповiдностi iз стандартною космологi- чною моделлю. ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 635

Quantum Universe on Extremely Small Space-time Scales

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