Macroscopic Effect of Quantum Gravity: Graviton, Ghost and Instanton Condensation on Horizon Scale of Universe

Journal of Modern Physics, 2013, 4, 48-101 http://dx.doi.org/10.4236/jmp.2013.48A007 Published Online August 2013 (http://www.scirp.org/journal/jmp)

Leonid Marochnik1, Daniel Usikov1, Grigory Vereshkov2,3 1Physics Department, University of Maryland, College Park, USA 2Research Institute of Physics, Southern Federal University, Rostov-on-Don, Russia, 3Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia Email: [email protected], [email protected], [email protected]

Received June 11, 2013; revised July 12, 2013; accepted August 3, 2013

Copyright © 2013 Leonid Marochnik et al. This is an open access article distributed under the Creative Commons Attribution Li- cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT We discuss a special class of quantum gravity phenomena that occur on the scale of the Universe as a whole at any stage of its evolution, including the contemporary Universe. These phenomena are a direct consequence of the zero rest mass of gravitons, conformal non-invariance of the graviton field, and one-loop finiteness of quantum gravity, i.e. it is a direct consequence of first principles only. The effects are due to graviton-ghost condensates arising from the interfere- ence of quantum coherent states. Each of coherent states is a state of gravitons and ghosts of a wavelength of the order of the horizon scale and of different occupation numbers. The state vector of the Universe is a coherent superposition of vectors of different occupation numbers. One-loop approximation of quantum gravity is believed to be applicable to the contemporary Universe because of its remoteness from the Planck epoch. To substantiate the reliability of macroscopic quantum effects, the formalism of one-loop quantum gravity is discussed in detail. The theory is constructed as follows: Faddeev-Popov path integral in Hamilton gauge  factorization of classical and quantum variables, allowing the existence of a self-consistent system of equations for gravitons, ghosts and macroscopic geometry  transition to the one-loop approximation, taking into account that contributions of ghost fields to observables cannot be eliminated in any way. The ghost sector corresponding to the Hamilton gauge automatically ensures of one-loop finiteness of the theory off the mass shell. The Bogolyubov-Born-Green-Kirckwood-Yvon (BBGKY) chain for the spectral function of gravitons renormalized by ghosts is used to build a self-consistent theory of gravitons in the isotropic Universe. It is the first use of this technique in quantum gravity calculations. We found three exact solutions of the equations, consisting of BBGKY chain and macroscopic Einstein’s equations. It was found that these solutions describe virtual graviton and ghost condensates as well as condensates of instanton fluctuations. All exact solutions, originally found by the BBGKY formalism, are reproduced at the level of exact solutions for field operators and state vectors. It was found that exact solutions correspond to various condensates with different graviton-ghost compositions. Each exact solution corresponds to a certain phase state of graviton-ghost substratum. We establish conditions under which a continuous quantum-gravity phase transitions occur between different phases of the graviton-ghost condensate.

Keywords: Quantum Gravity

1. Introduction roscopic quantum effect under discussion in this paper is condensation of gravitons and ghosts in the self-consis- Macroscopic quantum effects are quantum phenomena that occur on a macroscopic scale. To date, there are two tent field of the expanding Universe. A description of known macroscopic quantum effects: superfluidity at the this effect by an adequate mathematical formalism is the scale of liquid helium vessel and superconductivity at the problem at the present time. scale of superconducting circuits of electrical current. We show that condensation of gravitons and ghosts is These effects have been thoroughly studied experimen- a consequence of quantum interference of states forming tally and theoretically understood. A key role in these the coherent superposition. In this superposition, quan- effects is played by coherent quantum condensates of tum fields have a certain wavelength, and with different micro-objects with the De Broglie wavelength of the amplitudes of probability they are in states corresponding order of macroscopic size of the system. The third mac- to different occupation numbers of gravitons and ghosts.

Intrinsic properties of the theory automatically lead to a identified and quantified in one-loop approximation. Fourth, characteristic wavelength of gravitons and ghosts in the as was been shown by t’Hooft and Veltman [3], the one- condensate. This wavelength is always of the order of a loop quantum gravity with ghost sector and without distance to the horizon of events1. fields of matter is finite. For the property of one-loop In this fact, a common feature of macroscopic quan- finiteness, proven in [3] on the graviton mass shell, we tum effects is manifested: such effects are always formed add the following key assertion. All one-loop calcul- by quantum micro-objects, whose wavelengths are of the ations in quantum gravity must be done in such a way order of macroscopic values. With this in mind, we can that the feature of one-loop finiteness (lack of diver- say that macroscopic quantum gravity effects exist across gences in terms of observables) must automatically be the Universe as a whole. The existence of the effects of implemented not only on the graviton mass shell but also this type was first discussed in [1]. outside it. Quantum theory of gravity is a non-renormalized the- Let us emphasize the following important fact. Be- ory [2] and for this reason it is impossible to calculate cause of conformal non-invariance and zero rest mass of effects with an arbitrary accuracy in any order of the the- gravitons, no conditions exist in the Universe to place ory of perturbations. The program combining gravity gravitons on the mass shell precisely. Therefore, in the with other physical interactions within the framework of absence of one-loop finiteness, divergences arise in ob- supergravity or superstrings theory assumes the ultimate servables. To eliminate them, the Lagrangian of Ein- formulation of the theory containing no divergences. To- stein’s theory must be modified, by amending the defi- day we do not have such a theory; nevertheless, we can nition of gravitons. In other words, in the absence of one- hope to obtain physically meaningful results. Here are loop finiteness, gravitons generate divergences, contrary the reasons for this assumption. to their own definition. Such a situation does not make First, in all discussed options for the future theory, any sense, so the one-loop finiteness off the mass shell is Einstein’s theory of gravity is contained as a low energy a prerequisite for internal consistency of the theory. limit. Second, from all physical fields, which will appear These four conditions provide for the reliability of in a future theory (according to present understanding), theory predictions. Indeed, the existence of quantum only the quantum component of gravitational field (gra- component of the gravitational field leaves no doubt. viton field) has a unique combination of zero rest mass Zero rest mass of this component means no threshold for and conformal non-invariance properties. Third, phy- quantum processes of graviton vacuum polarization and sically meaningful effects of quantum gravity can be graviton creation by external or self-consistent macro- 1Everywhere in this paper we discuss quantum states of gravitons and scopic gravitational field. The combination of zero rest ghosts that are self-consistent with the evolution of macroscopic ge- mass and conformal non-invariance of graviton field ometry of the Universe. In the mathematical formalism of the theory, leads to the fact that these processes are occurring even the ghosts play a role of a second physical subsystem, the average contributions of which to the macroscopic Einstein equations appear on an in the isotropic Universe at any stage of its evolution, equal basis with the average contribution of gravitons. At first glance, it including the contemporary Universe. Vacuum polariza- may seem that the status of the ghosts as the second subsystem is in a tion and particle creation belong to effects predicted by contradiction with the well-known fact that the Faddeev-Popov ghosts are not physical particles. However the paradox, is in the fact that we the theory already in one-loop approximation. In this have no contradiction with the standard concepts of quantum theory of approximation, calculations of quantum gravitational pro- gauge fields but rather full agreement with these. The Faddeev-Popov cesses involving gravitons are not accompanied by the ghosts are indeed not physical particles in a quantum-field sense, that is emergence of divergences. Thus, the one-loop finiteness they are not particles that are in the asymptotic states whose energy and momentum are connected by a definite relation. Such ghosts are no- of quantum gravity allows uniquely describe mathemati- where to be found on the pages of our work. We discuss only virtual cally graviton contributions to the macroscopic observ- gravitons and virtual ghosts that exist in the area of interaction. As to ables. Other one-loop effects in the isotropic Universe virtual ghosts, they cannot be eliminated in principle due to lack of ghost-free gauges in quantum gravity. In the strict mathematical sense, are suppressed either because of conformal invariance of the non-stationary Universe as a whole is a region of interaction, and, non-gravitational quantum fields, or (in the modern Uni- formally speaking, there are no real gravitons and ghosts in it. Ap- verse) by non-zero rest mass particles, forming effective proximate representations of real particles, of course, can be introduced for shortwave quantum modes. In our work, quantum states of short- thresholds for quantum gravitational processes in the ma- wave ghosts are not introduced and consequently are not discussed. croscopic self-consistent field. Furthermore, macroscopic quantum effects, which are discussed in our Effects of vacuum polarization and particle creation in work, are formed by the most virtual modes of all virtual modes. These modes are selected by the equality H 1 , where  is the wave- the sector of matter fields of J  0,1 2,1 spin were well length, H is the Hubble function. The same equality also character- studied in the 1970’s by many authors (see [4] and re- izes the intensity of interaction of the virtual modes with the classical ferences therein). The theory of classic gravitational gravitational field, i.e. it reflects the essentially non-perturbative nature waves in the isotropic Universe was formulated by of the effects. An approximate transition to real, weakly interacting particles, situated on the mass shell is impossible for these modes, in Lifshitz in 1946 [5]. Grishchuk [6] considered a number principle (see also the footnote 2 on p. 4). of cosmological applications of this theory that are result

Copyright © 2013 SciRes. JMP 50 L. MAROCHNIK ET AL. of conformal non-invariance of gravitational waves. mathematically rigorous procedure for the separation of Isaacson [7,8] has formulated the task of self-consistent classical and quantum variables is discussed in Sections description of gravitational waves and background ge- 2.3 and 2.4; 3) The derivation of differential identities, ometry. The model of Universe consisting of short gra- providing the consistency of classical and quantum equa- vitational waves was described for the first time in [9,10]. tions performed jointly in any order of the theory of per- The energy-momentum tensor of classic gravitational turbations is given Section 2.5. Rigorously derived equa- waves of super long wavelengths was constructed in [11, tions of gauged one-loop quantum gravity are presented 12]. The canonic quantization of gravitational field was in Section 2.6. done in [13-15]. The local speed of creation of short- The status of properties of ghost sector generated by wave gravitons was calculated in [16]. In all papers listed gauge is crucial to properly assess the structure of the above, the ghost sector of graviton theory was not taken theory and its physical content. Let us immediately em- into account. One-loop quantum gravity in the form of phasize that the standard presentation on the ghost status the theory of gravitons defined on the background space- in the theory of S-matrix can not be exported to the the- time was described by De Witt [17]. Calculating methods ory of gravitons in the macroscopic spacetime with self- of this theory were discussed by Hawking [18]. consistent geometry. Two internal mathematical pro- The exact equations of self-consistent theory of gra- perties of the quantum theory of gravity make such ex- vitons in the Heisenberg representation with the ghost port fundamentally impossible. First, there are no gauges sector automatically providing a one-loop finiteness off that completely eliminate the diffeomorphism group de- the mass shell are obtained in our work [19]. In [19], it is generacy in the theory of gravity. This means that among shown that the Heisenberg representation of quantum the objects of the quantum theory of fields inevitably gravity (as well as the Heisenberg representation of arise ghosts interacting with macroscopic gravity. Sec- quantum Yang-Mills theory [20]) exists only in the ondly, gravitons and ghosts cannot be in principle situ- Hamilton gauge. The ghost sector corresponding to this ated precisely on the mass shell because of their confor- gauge represented by the complex scalar field with minimal non-invariance and zero rest mass. This is because mal coupling to gravity. there are no asymptotic states, in which interaction of One-loop finiteness provides the simplicity and ele- quantum fields with macroscopic gravity could be ne- gance of a mathematical theory that allows, in turn, dis- glected. Restructuring of vacuum graviton and ghost covering a number of new approximate and exact solu- modes with a wavelength of the order of the distance to tions of its equations. This paper is focused on three ex- the horizon of events takes place at all stages of cosmo- act solutions corresponding to three different quantum logical evolution, including the contemporary Universe. states of graviton-ghost subsystem in the space of the Ghost trivial vacuum, understood as the quantum state non-stationary isotropic Universe with self-consistent geo- with zero occupation numbers for all modes, simply is metry. The first of these solutions describes a coherent absent from physically realizable states. Therefore, direct condensate of virtual gravitons and ghosts; the second participation of ghosts in the formation of macroscopic solution describes a coherent condensate of instanton observables is inevitable2. fluctuations. The third solution describes the self-polar- Section 3 is devoted to general discussion of equations ized condensate in the De Sitter space. This solution al- of the theory of gravitons in the isotropic Universe. It lows interpretation in terms of virtual particles as well as focuses on three issues: 1) Canonical quantization of in terms of instanton fluctuations. gravitons and ghosts (Sections 3.1 and 3.2); 2) Cons- The principal nature of macroscopic quantum gravity truction of the state vector of a general form as a product effects, the need for strict proof of their inevitability and of normalized superpositions (Section 3.3); 3) The proof reliability impose stringent requirements for constructing of the one-loop finiteness of macroscopic observables a mathematical algorithm of the theory. Sections 2 and 3 (Section 3.4). The main conclusion is that the quantum are devoted to the derivation of the equations of the the- ghost fields are inevitable and unavoidable components ory with a discussion of all the mathematical details. In of the quantum gravitational field. As noted above, Section 2, we start with exact quantum theory of gravity, one-loop finiteness is seen by us as a universal property presented in terms of path integral of Faddeev-Popov [21] of quantum gravity, which extends off the mass shell. and De Witt [22,23]. Key ideas of this Section are the The requirement of compensation of divergences in following. 1) The necessity to gauge the full metric (be- terms of macroscopic observables, resulting from one- fore its separation into the background and fluctuations) loop finiteness, uniquely captures the dynamic properties and the inevitability of appearance of a ghost sector in of quantum ghost fields in the isotropic Universe. The the exact path integral and operator Einstein’s equations existence of Quantum Gravity in the Heisenberg re- (Sections 2.1 and 2.2); 2) The principal necessity to use presentation in the Hamilton gauge is a nontrivial pro- normal coordinates (exponential parameterization) in a perty of the theory. Exactly this property automatically

Copyright © 2013 SciRes. JMP L. MAROCHNIK ET AL. 51 provides a one-loop finiteness of the theory off mass ness of procedures for BBGKY structure is entirely a shell. question of the existence of moments of the spectral Sections 4.1 and 4.2 contain approximate solutions to function as mathematical objects. A positive answer to obtain quantum ensembles of short and long gravitational this question is guaranteed by one-loop finiteness (Sec- waves. In Section 4.3 it is shown that approximate solution 3.4). The set of moments of the spectral function tions obtained can be used to construct scenarios for the contains information on the dynamics of operators as evolution of the early Universe. In one such scenario, the well as on the properties of the quantum state over which Universe is filled with ultra-relativistic gas of short-wave the averaging is done. The set of solutions of BBGKY gravitons and with a condensate of super-long wave- chain contains all possible self-consistent solutions of lengths, which is dominated by ghosts. The evolution of operator equation, averaged over all possible quantum this Universe is oscillating in nature. ensembles. At the heart of cosmological applications of one-loop A nontrivial fact is that in the one—loop quantum quantum gravity is the Bogolyubov-Born-Green-Kirck- gravity BBGKY chain can formally be introduced at an wood-Yvon (BBGKY) chain (or hierarchy) for the spec- axiomatic level. Theory of gravitons provided by BBGKY tral function of gravitons, renormalized by ghosts. We chain, conceptually and mathematically corresponds to present the first use of this technique in quantum gravity the axiomatic quantum field theory in the Wightman for- calculations. Each equation of the BBGKY chain con- mulation (see Chapter 8 in the monograph [25]). Here, as nects the expressions for neighboring moments of the in Wightman, the full information on the quantum field is spectral function. In Section 5.1, the BBGKY chain is contained in an infinite sequence of averaged correlation derived by identical mathematical procedures from gra- functions. Definitions of these functions clearly relate to viton and ghost operator equations. Among these proce- the symmetry properties of manifold on one this field is dures is averaging of bilinear forms of field operators defined. Once the BBGKY chain is set up, the existence over the state vector of the general form, whose mathe- of finite solutions for the observables is provided by in- matical structure is given in Section 3.3. The need to herent mathematical properties of equations of the chain. work with state vectors of the general form is dictated by This means that the phenomenology of BBGKY chain is the instability of the trivial graviton-ghost vacuum (see more general than field operators, state vectors and gra- [24], Section 3.6). Evaluation of mathematical correct- viton-ghost compensation of divergences that were used 2Once again, we emphasize that the equal participation of virtual gra- in its derivation. vitons and ghosts in the formation of macroscopic observables in the Exact solutions of the equations, consisting of BBGKY non-stationary Universe does not contradict the generally accepted chain and macroscopic Einstein’s equations are obtained concepts of the quantum theory of gauge fields. On the contrary it follows directly from the mathematical structure of this theory. In order to in Sections 5.2 and 5.3. Two solutions given in 5.2, des- clear up this issue once and for all, recall some details of the theory of cribe heterogeneous graviton-ghost condensates, con- S-matrix. In constructing this theory, all space-time is divided into sisting of three subsystems. Two of these are condensates regions of asymptotic states and the region of effective interaction. Note that this decomposition is carried out by means of, generally of spatially homogeneous modes with the equations of speaking, an artificial procedure of turning on and off the interaction state p   3 and p   . The third subsystem is a adiabatically. (For obvious reasons, the problem of self-consistent condensate of quasi-resonant modes with a constant con- description of gravitons and ghosts in the non-stationary Universe with formal wavelength corresponding to the variable physical λH = 1 by means of an analogue of such procedure cannot be considered a priori.) Then, after splitting the space-time into two regions, it is wavelength of the order of the distance to the horizon of assumed that the asymptotic states are ghost-free. In the most elegant events. The equations of state of condensates of quasi- way, this selection rule is implemented in the BRST formalism, which resonant modes differ from p   3 by logarithmic shows that the BRST invariant states turn out to be gauge-invariant automatically. The virtual ghosts, however, remain in the area of inter- terms, through which the first solution is p   3 , action, and this points to the fact that virtual gravitons and ghosts are while the second is p   3 . Furthermore, the solu- parts of the Feynman diagrams on an equal footing. In the self-consis- tions differ by the sign of the energy density of conden- tent theory of gravitons in the non-stationary Universe, virtual ghosts of equal weight as the gravitons, appear at the same place where they sates of spatially homogenous modes. The third solution appear in the theory of S-matrix, i.e. at the same place as they were describes a homogeneous condensate of quasi-resonant introduced by Feynman, i.e. in the region of interaction. Of course, the modes with a constant physical wavelength. The equ- fact that in the real non-stationary Universe, both the observer and virtual particles with λH = 1 are in the area of interaction, is highly ation of state of this condensate is p  and its self- nontrivial. It is quite possible that this property of the real world is consistent geometry is the De Sitter space. The three manifested in the effect of dark energy. An active and irremovable solutions are interpreted as three different phase states of participation of virtual ghosts in the formation of macroscopic properties of the real Universe poses the question of their physical nature. graviton-ghost system. The problem of quantum-gravity Today, we can only say with certainty that the mathematical inevitabil- phase transitions is discussed in Section 5.4. ity of ghosts provides the one-loop finiteness off the mass shell, i.e. the Solutions obtained in Section 5 in terms of moments of mathematical consistency of one-loop quantum gravity without fields of matter. Some hypothetical ideas about the nature of the ghosts are the spectral function, are reproduced in Sections 6 and 7 briefly discussed in the final Section 8. at the level of dynamics of operators and state vectors. A

Copyright © 2013 SciRes. JMP 52 L. MAROCHNIK ET AL. microscopic theory provides details to clarify the struc- ations of the theory (except the gauge condition) are ture of graviton-ghost condensates and clearly demon- represented in 4D form which is general covariant with strates the effects of quantum interference of coherent respect to the transformation of the macroscopic metric. states. In Section 6.1, it is shown that the condensate of The mathematically consistent system of 4D quantum quasi-resonant modes with the equation of state and classic equations with no restrictions with respect to p   3 consists of virtual gravitons and ghosts. In graviton wavelengths is obtained by a regular method for Section 6.2 a similar interpretation is proposed for the the first time. The case of a gauged path integral with condensate in the De Sitter space, but it became ne- ghost sector is seen as a source object of the theory. cessary to extend the mathematical definition of the mo- Important elements of the method are exponential para- ments of the spectral function. meterization of the operator of the density of the con- New properties of the theory, whose existence was not travariant metric; factorization of path integral measure; anticipated in advance, are studied in Section 7. In consequent integration over quantum and classic com- Section 7.1 we find that the self-consistent theory of ponents of the gravitational field. Mutual compliance of gravitons and ghosts is invariant with respect to the Wick quantum and classic equations, expressed in terms of turn. In this section, we also construct the formalism of fulfilling of the conservation of averaged EMT at the quantum theory in the imaginary time and discuss the operator equations of motion is provided by the virtue of physical interpretation of this theory. The subjects of the the theory construction method. study are correlated fluctuations arising in the process of tunnelling between degenerate states of graviton-ghost 2.1. Path Integral and Faddeev-Popov Ghosts systems, divided by classically impenetrable barriers. Formally, the exact scheme of quantum gravity is based The level of these fluctuations is evaluated by instanton solutions (as in Quantum Chromodynamics). In Section on the amplitude of transition, represented by path in- 7.2, it is shown that the condensate of quasi-resonant tegral [21,22]: modes with the equation of state p   3 is of purely i 4 ˆ i out in expLLxgrav d det M k instanton nature. In Section 7.3, the instanton condensate  theory is formulated for the De Sitter space.  Potential use of the results obtained to construct sce- ˆ ˆîk i ˆ AggBk d xi narios of cosmological evolution was briefly discussed in (1) Sections 4-7 to obtain approximate and exact solutions. i 4 expLLLxgrav ghost d Future issues of the theory of the theory are briefly dis-  cussed in the Conclusion (Section 8).  A system of units is used, in which the speed of light is ˆ ˆ ˆîk i  AggBk dd , c 1 , Planck constant is  197.327 MeV fm ; Ein- xi stein’s gravity constant is where 42 1  8G  81.324  10 MeV  fm . 1 LL   ggRgˆ ˆîk ˆ    grav  ik 2. Basic Equations 2 is the density of gravitational Lagrangian, with cos- According to De Witt [17], one of formulations of one- mological constant included; Lghost is the density of loop quantum gravity (with no fields of matter) is re- ghost Lagrangian, explicit form of which is defined by duced to the zero rest mass quantum field theory with i i localization of det Mˆ ; Aˆ is gauge operator, Bx  spin J  2 , defined for the background spacetime with k k ˆ i classic metric. The graviton dynamics is defined by the is the given field; M k is an operator of equation for in- interaction between quantum field and classic gravity, finitesimal parameters of transformations for the residual ii and the background space geometry, in turn, is formed by degeneracy    x ; the energy-momentum tensor (EMT) of gravitons.  52 ik  In the current Section we describe how to get the self- ddˆˆg gˆ  (2) xik consistent system of equations, consisting of quantum operator equations for gravitons and ghosts and classic is the gauge invariant measure of path integration over

C-number Einstein equations for macroscopic metrics gravitational variables; d is the measure of integra- ˆ i with averaged EMT of gravitons and ghosts on the right tion over ghost variables. Operator M k is of standard hand side. The theory is formulated without any con- definition: strains on the graviton wavelength that allows the use of MAˆ ik ˆ  ggˆ ˆ ik 0, (3) the theory for the description of quantum gravity effects kk  at the long wavelength region of the specter. The equ- where

ˆˆik ˆ ˆ ik l The construction of the ghost sector, i.e. finding of the ggl  gg (4) Lagrangian density Lghost , is reduced to two operations. il k kl i ˆ i ggˆˆ ll  ggˆ ˆ  First, det M k is represented in the form, factorized over independent degrees of fr eedom for ghosts, and then the is variation of metrics under the action of infinitesimal localization o f the obtained expression is conducted. transformations of the group of diffeomorphisms. Ac- Substitution of (6) and (4) to (3) gives the following cording to (1), the allowed gauges are constrained by the system of equations condition of existence of the inverse operator 1  0 Mˆ i .     0, k  t (7) The Equ ation (1) explicitly manifests the fact that the   0   source path integral is defined as a mathematical object ggˆˆ  0. only after the gauge has been imposed. In the theory of t gravity, there are no gauges completely eliminating the According to (7), with respect to variables  0 ,   degeneracy with respect to the transformations (4). ˆ i the operator-matrix M k reads Therefore, the sector of nontrivial ghost fields, interacting with gravity, is necessarily present in the path inte-    gral. This aspect of the quantum gravity is important for ˆ i t M k   (8) understanding of its mathematical structure, which is    ggˆˆ  fixed before any approximations are introduced. By that t reason, in this Section we discuss the equations of the (Note matrix-operator is obtained in the form (8) without theory, by explicitly defining the concrete gauge. the substitution of transformation parameters if Leut- The mathematical procedure of transition from path willer measure ddˆˆˆˆ gg 00  is used. The measure dis- integral (1) to the equations of Quantum Gravity in the L cussion see, e.g. [28].) Functional determinant of matrix- Heisenberg representation (with the canonical quantiza- operator det Mˆ i is represented in the form of the deter- tion of gravitons and ghosts) is described by us in detail k minant of matrix Mˆ i , every element of which is a func- [19]. The first step of this procedure is to represent the k tional determin ant of differential operator. As it is integral (1) as a path integral over the canonical variables. follows from (8), Such an integral was proposed by Faddeev [26] on the basis of the general theory of Hamilton systems with  ˆ iˆ ˆ ik detMk det iggkdet  det . (9) explicitly unsolvable constraints [27]. The second step is tt to introduce the normal coordinates of the gravitational field using the exponential parameterization of metric. One can see that the first multiplier in (9) is 4-in- The Hamilton gauge of the normal coordinates specifies variant determinant of the operator of the zero rest mass the Faddeev path integral in such a way that the ghost Klein-Gordon-Fock equation, and two other multipliers sector (corresponding to it) allows to introduce canonical do not depend on gravitational variables. variables of ghost fields and to represent the ghost La- Localization of determinant (9) by representing it in a grangian in the Hamilton form. In the third step of this form of path integral over the ghost fields is a trivial procedure, the transition from the gauged path integral to operation. As it follows from (9), the class of synchro- the canonical Hamilton formalism in the Heisenberg nous gauges contains three dynamically independent representation is made (using the standard definition of ghost fields ,,, two of each ,  do not interact the operator of evolution). The results of the [19] are ri- with gravity. For the obvious reason, the trivial ghosts gorous basis of the simplified procedure for obtaining ,  are excluded from the theory. The Lagrangian gauged equations of quantum gravity with ghosts, which density of nontrivial ghosts coincides exactly with La- is described below. grangian density of complex Klein-Gordon-Fock fields Hamilton gauge is that of synchronous type: (taking into account the Grassman character of fields  , ): 00 0 ggˆ ˆ  ,0.  ggˆ ˆ  (5) 1 Lg ˆ gˆ ik  . (10) For that gauge ghost 4 i k

Aˆ 1,0,0,0 , Bi  ,0,0,0 , (6) The normalization multiplier 14 in (10) is cho- k  sen for the convenience. The integral measure over ghost where  x is the metric determinant of the basic fields has a simple form: 3D space of co nstant curvature (for the plane cos- ddd.    mological model   1 ). x

The calculations above comply with both general re- trivializatio n of measure (12) takes place for the expo- quirements to the construction of ghost sector. First, path nential parameterization that reads integration should be carried out only over the dynami- k ggˆˆik gg il exp ˆ cally independent ghost fields. Second, in the ghost sec-  l tor, it is necessary to extract and then to take into account (13) il kˆ k1 ˆˆ m k only the nontrivial ghost fields, i.e. those interacting with gg ll lm , 2 gravity.  where g ik is the defined metric of an auxiliary basic 2.2. Einstein Operator Equations space. In that class of our interest, the metric is defined by the inte rval Let us take into account the fact that the calculation of 22   gauged path integral should be mathematically equiva- ds dt dx dx , lent to the solution of dynamical operator equations in where  is the metric of 3D space with a constant the Heisenberg representation. It is also clear that opera-  curvature. (For the flat Universe  is the Euclid tor equations of quantum theory should have a definite  relationship with Einstein equations. In the classic theory, metric.) it is possible to use any form of representation of Ein- The exponential parameterization is singled out among all other parameterizations by the property that ˆ k are stein equations, e.g. i the normal coordinates of gravitational fields [30]. In that i n il km 1 ik lm ik ˆ ˆˆ respect, the gauge conditions (5) are identical to 0  0 . gˆ ggˆˆ Rlm gg ˆˆ Rlm g ˆ 0,  a 2 The fact that the “gauged” coordinates are the normal coordinates, leads to a simple and elegant ghost secto r n 1 km ˆˆk lm k k gˆ gRˆˆim i gR lm i  0, b (11) (10). The status of ˆ , as normal coordinates, is of 2 i principal value for the mathematical correctness while n ˆ 1 lmˆ ik separating the classic and quantum variables (see Section gˆˆRggRgik ikˆˆ lm  0, c 2 2.4). Besides, in the framework of perturbation theory the where, for example, n  0,1 2,1 . Transition from one to normal coordinates allow to organize a calculation proce- another is reduced to the multiplication by metric tensor dure, which is based on a simple classification of non- and its determinant, which are t rivial operations in case linearity of quantum gravity field. It is important that this when the metric is a C-number function. If the metric is procedure is mathematically non-contradictive at every an operator, then the analogous operations will, at least, order of perturbation theory over amplitude of quantum change renormalization procedures of quantum non-po- fields (see Section 2.5 and 2.6). lynomial theory. Thus, the question about the form of Operator Einstein equations that are mathematically notation for Einstein’s operator equations has first-hand equivalent to the path integral of a trivial measure are relation to the calculation procedure. Now we show that derived by the variation of gauged action by variables ˆ k . The principal point is that the gauged action neces- in the quantum theory one should use operator equations i (11b) with n  12, supplemented by the energy-mo- sarily includes the ghost sector because there are no mentum pseudo-tensor of ghosts. gauges that are able to completely eliminate the degen- In the path inte gral formalism, the renormalization eracy. According to (10), in the Hamilton gauge we get procedures are defined by the dependence of Lagrangian S of interactions and the measure of integration of the field (14) 4  11ik ˆ  operator in terms of which the polynomial expansion of  d.xg ˆ gRˆˆik i k g  22  non-polynomial theory is defined [29]. The introduction  of such an operator, i.e. the parameterization of the In accordance with definition (13), the variation is metric, is, generally speaking, not simple. Nevertheless, done by the rule it is possible to find a special parameterization for which ggˆ ˆ ik  ggˆ ˆ il ˆ k . the algorithms of renormalization procedures are defined l only by Lagrangian of interactions. Obviously, in such a Thus, from (14) it follows parameterization the measure of integration should be ˆkkˆ ˆˆlˆ k trivial. It reads: iigg Rli   g  1 (15) dd,ˆ ˆ k (12) kl ˆˆghost k ml ghost  i  ggˆ ˆˆTil i ggˆ Tml 0. xik 2 ˆ k where i is a dynamic variable. The metric is ex- After subtraction of semi-contraction from (15) we ob- pressed via this variable. It is shown in [29] that the tain a mathematically equivalent equation

1 pressions (17) c oincide with the respected properties of ˆˆkk kl ˆ ii2 il equations and EMT of co mplex scalar fields. This fact is of great importance when concrete calculations are done kl ˆˆ1 k ml ggˆ ˆ Ril i  ggˆ ˆ Rml (16) (see Section 3). 2 ˆˆkl ˆghost ˆ k  gg Til  gi 0. 2.3. Factorization of the Path Integral In (15), (16) there is an object Transition from the formally exact scheme (18)-(21)) to the semi-quantum theory of gravity can be done after Tˆghost ik some additional hypotheses are included in the theory. (17) 1 lm The physical content of these hypotheses consists of the  ik   ki  ggˆ iklmˆ    , 4 assertion of existence of classical spacetime with metric g , connectivity i and curvature R . The first which has the status of the energy-momentum pseudo- ik kl ik tensor of ghosts. hypothesis is formulated at the level of operators. ˆ ik In accordance with the general properties of Einstein’s Assume that operator of metric g is a functional of C-number function g ik and the quantum operator ˆ k . theory, six spatial components of Equations (15) are con- i sidered as quantum equations of motion: The second hypothesis is related to the state vector. Each state vector that is involved in t he scalar product l ˆ ggˆˆ Rl  gˆ  | inout  , is represented in a factorized form (18)  , where  are the vectors of quantum lmˆ ghost 1 lˆ ghost   ggˆˆ Tlm ggˆˆ T l. states of gravitons;  are the vectors of quasi-classic 2 states of macroscopic metric. In the framework of these (Everywhere in this work the Greek metric indexes stand hypotheses the transitio na l amplitude is reduced to the for ,1,2,3  .) In the classic theory, equations of product of amplitudes: constraints ˆ0  0 and ˆ  0 are the first integrals of 0 0 out in   . (22) equations of mo tion (18). Therefore, in the quantum out in out in theory formulated in the Heisenberg representation four Thus, the physical assumption about existence of primary constraints from (16), have the status of the classic spacetime formally (mathematically) means that initial conditions for the Heisenberg state vector. They the path integral must be calculated first by exact inte- read: gration over quantum variables, and then by approximate integration over the classic metric.  0lmˆ 1 lˆ  ggˆ ˆ R0lm ggˆ ˆ R l Mathematical definition of classic and quantum vari-  2 ables with subsequent integrations are possible only after  ˆˆ0l ˆ ghost ˆ the trivialization and factorization of integral measure are  gg T0l  g  0, (19)  done. As already noted, trivial measure (12) takes place in exponential parameterization (13). The existence of ˆˆllˆ  ˆˆ ˆ ghost gg R00ll gg T  0. in   vector allows the introduction of classic C- number variables as follows If conditions (19) are valid from the start, then the internal properties of the theory must provide their vali- kkˆ ,exp. gg ikil   gg k ii l dity at any subsequent moment of time. Four secondary relations, defined by the gauge non containing the higher Quantum graviton operators are defined as the diffe- ˆ kkkˆ order derivatives, also have the same status: rence  iii . Factorized amplitude (22) is calculated via the factorized measure ˆ ik i AggBk ˆˆ 0. (20)  ddˆ g d  ,

The system of equations of quantum gravity is closed 52 (23) dgdgd  ik ,dˆ k . by the ghosts’ equations of motion, obtained by the va- g   i xikxik riation of action (14) over ghost variables: Factorization of the measure allows the subsequent in- ggˆˆik  0, tegration, first by d , d , then by approximate ik (21)   ik integration over d . In the operator formalism, such ggˆˆ  0. g ik consecutive integrations correspond to the solution of Ghost fiel ds  and  are not defined by Grassman self-consistent system of cl assic and quantum equations. ghost scalars, therefore Tik is not a tensor. Nevertheless, Classical equations ar e obtained by averaging of operator all mathematical properties of Equation (21) and ex- Equation (16). They read:

ˆk product of measure of in tegration over independent irre- i 0. (24) ducible representations. At the next order, factorization is Subtraction of (24) from (16) gives the quantum dy- done over coordinates, because the classical background namic equations and the induced quantum fluctuations have essentially ˆkˆk 0. (25) different spacetime dynamics. Note, to factorize the mea- ii sure by symmetry criterion we do not need to go to the Synchronous gauge (20) is converted to the gauge of short-wave approximation. classical metric and to conditions imposed on the state Background metric of isotropic cosmological models vector: and classical spherically symmetric non-stationary gra- gg00 ,0,=0. gg0 ˆ i  (26) vitational field meet the constrains described above. 0 These two cases are covering all important applications Quantum Equation (21) of ghosts’ dynamics are added of semi-quantum theory of gravity which are quantum to Equations (24)-(26). effects of vacuum polarization and creation of gravitons Theory of gravitons in the macroscopic spacetime with in the non-stationary Universe and in the neighborhood self-consistent geometry is without doubt an approximate of black holes. theory. Formally, the approximation is in the fact that the single mathematical object gˆˆg ik is replaced by two 2.4. Variational Principle for Classic and objects—classical metric and quantum field, having es- Quantum Equations sentially different physical interpretations. That “coer- Geometrical variables can be identically transformed to cion” of the theory can lead to a co ntroversy, i.e. to the the form of functionals of classical and quantum vari- system of equations having no solutions, if an inaccurate ables. At the first step of transformation there is no need mathematics of the adopted hypotheses is used. The to fix the parameterization. Let us introduce the nota- scheme described above does not have such a contro- tions: versy. The most important element of the scheme is the exponential parameterization (13), which separates the ik ˆˆik 11 gˆ ggXgˆ ,,ˆik  Yik classical and quantum variables, as can be seen from (23). ggˆ (28) After the background and quantum fluctuations are in- YXˆ ˆ lk   k . troduced, this parameterization looks as follows: il i

ik il k According to (27), formalism of the theory allows de- ggˆ ˆ gg exp ˆ k l finition of quantum field î as symmetric tensor in (27) l il k physical space, g ˆˆ ˆ. Objects, introduced in gg expˆ , kl k ik ki l (28), have the same status. With any parameterization the Note that the auxiliary basic space vanishes from the following relationships take place: theory, and instead the macroscopic (physical) spacetime lim XˆîkgYg ik , lim . mmik ik with self-consistent geometry takes its place. ˆll0ˆ 0 If the geometry of macroscopic spacetime satisfies We should also remember that the mixed components symmetry constrains, the factorization of the measure (23) kk of tensors Xˆˆ,Y do not contain the background metric becomes not a formal procedure but strictly mathematical ii as functional parameters. For any parameterization, these in its nature. These restrictions must ensure the existence k tensors are only functionals of quantum fields ˆ of an algorithm solving the equations of constraints in the i which are also defined in mixed indexes. For the ex- framework of the perturbation theory (over the amplitude ponential paramet erization: of quantum fields). The theory of gravity is non-poly- 1 nomial, so after the separation of single field into classi- Xˆ kkkˆˆˆ  lk , cal and quantum components, the use of the perturbation iii2 il theory in the quantum sector becomes unavoidable. The 1 Yˆ kkkˆ ˆ lkˆ , (29) classical sector remains non-perturbative. In the general iii2 il case, when quantum field is defined in an arbitrary Rie- ˆ ggˆ dgˆ e, mann space, the equations of constraints is not explicitly ˆ i solvable. The problem can be solved in the framework of where dX det k . One can seen from (29), that the perturbation theory if background gik and the free determinant of the full metric contains only the trace of k (linear) tensor field î belong to different irreducible the quantum field. representations of the symmetry group of the background Regardless of parameterization, the connectivity and l i spacetime. In that case at the level of linear field we curvature of the macroscopic space ik , Rklm are ex- obtain (23), because the full measure is represented as a tracted from full connectivity and curvature as additive

Copyright © 2013 SciRes. JMP L. MAROCHNIK ET AL. 57 terms: namically independent variables due to the doubling of the number of equations. The variation should be done ˆ llˆ li,RRˆ  i ˆ i. ik ik ik klm klm klm separately over each type of variables. The formalism of Quantum contribution to the curvature tensor, the path integration suggests a rigid criterion of dynamic independence: the full measure of integration, by defi- ˆ iiiˆˆˆ ininˆ ˆˆ klm km;; l  kl m nl km nm kl , nition, must be factorized with respect to the dynamically is expressed via the quantum contribution to the full con- independent variables. Obviously, only the exponential nectivity: parameterization (27), leading to the factorized measure (23), meets the criterion. 1 ˆ lmYXˆˆlm  YX ˆˆlm  YYX ˆˆˆl X ˆjn The variation of the action over the classic variables is ik 2 im;; k km i ij kn ;m  (30) done together with the operation of averaging over the 1 ljnljn mljn quantum ensemble. In the result, equations for metric of YXjn i;; k k XYXX i ik ;m . 4 the macroscopic spacetime are obtained: The density of Ricci tensor in mixed indexes reads  S in kl ˆ  g ggˆ ˆ Ril (33) 1   2 gg  Gˆ kkl Gˆ 0, ˆ kl nk i i l gX Ril 2  ˆˆkk where Ggii   . Variation of the action over 11 background variables, defined as kkˆ  , YXXˆ ˆ ml ˆ nk XXˆˆ mk nl X ˆ lk k YXXˆˆˆ ml jn ii in ;;;m m i i nj ;m yields the equations: 22;l  S 11 1 ˆ k ˆˆ ˆ ˆ ˆkl ˆ jm ˆ ns ˆ ˆkm ˆ nl   i 20. gG i  (34) YYjn sm YY jm sn X X;;i X l Y ml X;; n X i .   422  k Equations (33) and (34) are mathematically identical. We (31) should also mention that if the variations over the back- Symbol “;” in (30), (31) and in what follows stands for ground metric are done with the fixed mixed compo- the covariant derivatives in background space. The den- nents of the quantum field, these equations are valid for sity of gauged gravitational Lagrangian is represented in any parameterization. a form which is characteristic for the theory of quantum Exponential parameterization (27) has a unique pro- fields in the classical background spacetime: k perty: the variations over classic i (before averaging) and quantum ˆ k (without averaging) variables lead to 4 ˆ 1 ˆ ik i Sxgd,grav  d X,, i k the same equations. That fact is a direct consequence of  4 the relations, showing that variations  k and ˆ k 1 l l ˆ ik are multiplied by the same operator m ultiplier: grav XR ik 2 ik il k k ggˆ ˆ ggˆ ˆ li,const,ˆ  11kl jm sn il km XYˆ ˆ Yˆ YYXXˆ ˆˆˆ2 YXX ˆˆˆ ik il k k jnsm jmsn;; k l ik;; m l ggˆˆ ggˆ ˆ ˆ , co nst. 82  li (32) By a simple operation of subtraction, the identity allows the extraction of pure background terms from the equa- When the expressi on for grav was obtained from contraction of tensor ( 31), the full covariant divergence tion of quantum field. The equations of graviton the ory in the background space have been excluded. Formulas in the macroscopic space with self-consistent geometry (31), (32) apply for at any parameterization. are written as follows: Let us discuss the variation meth od. In the exact quan- ˆ kkkˆ 1 ˆ l  EGGiii  l 0, (35) tum theory of gravity with the trivial measure (12), the 2 variation of the action over variables ˆ k leads to the 11 i LGˆkk ˆˆˆˆ kl GGG k kl 0. (36) Einstein equations in mixed indexes (15) and (16). In the ii2 il i2 il exact theory, the exponential parameterization is con- With the exponential parameterization, the formalism venient, but, generally speaking, is not necessary. A prin- of the theory can be expressed in an elegant form. Let us cipally different situation takes place in the approximate go to the rules of differentiation of exponential matrix self-consistent theory of gravitons in the macroscopic functions spacetime. In that theory the number of variables doubles, YXˆ ˆ mkˆˆ k ,.XXˆ ikˆ im k (37) and with this, the classical and quantum components of im;; l i l ;l m ; l gravitational fields have to have the status of the dy- Taking into account (37), we get the quantum con-

Copyright © 2013 SciRes. JMP 58 L. MAROCHNIK ET AL. tribution to the full connectivity (30) as follows Let us introduce the following notations: 1 1 ˆ llllˆ ˆ ˆˆmnˆ ˆˆik ik ikˆ ikˆ ilˆ k ik ik;;  ki YX kn  im ; XXg1 l , 2 (38) 2 (41) 1 1 llˆ ˆˆYXˆ ˆ lm  . XXgˆ ikˆ ik ikˆ ikˆˆ il k . 4 ik;; ki ik; m 2 2 l Formulas (32) could be rewritten as follows: With use of (41), let us extract from (40) the terms not l containing the quantum field, and the terms linear over Xdˆ l expˆ ,ˆ eˆ 2 , k k the quantum field:

ˆ 1 42  ˆ lk; ˆ kk1 k k Sxgdegrav  X k; l , ER R   4 ii2 i i (39) 1 1  ˆ lk kl;;; kl lk k lm ; grav XR k l ˆˆîl;;; li i l i ˆ ml ; 2 2 (42) 11 11 ˆ lmkn;; k kmn; ˆˆkl kml k ˆ ˆ k X knmlˆˆ;;  ˆˆ l2.  ˆˆ nml; liRR i l m  i Ti, 82  22 ˆ kkˆˆ k As is seen from (39), for the exponential parameteriza- TTi i grav T i  ghost , tion, the non-polynomial structures of quantum theory of where gravity have been completely reduced to the factorized k 3 Tˆ exponents . igrav The explicit form of the tensor, in the terms of which 11 ˆ kl ˆ nˆˆ m ˆˆˆn m the self-consistent system of equations could be written X mi;; nl  ;; i l2  lm ; ni ; 42 is as follows 11 krlnXˆ ˆˆm ˆˆ2 ˆˆn m ˆkkˆ 11 klklklmkˆ ˆˆ ˆ 2 imrnlrllmnr;; ;; ; ; EGiiilGXR lii  XR mli  e 82 22 1 (43) 1  ˆ lmˆ kˆˆ kˆ km l  ˆ lm k kˆ km l XX11im;; mi  im ; XXîm;;ˆˆ mi im; 2  2  1 kmnl lmn 1 kmnl lmn XXˆˆˆ ˆ XXˆˆˆ ˆ in11;;mm n in;;mmn 2 ;l 2 ;l (40) kl 1 k lmkˆ 2 1 11kl n m n m XRˆˆ XR   e1 ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ 22li i  ml i  X m;;i nl ;;i l2  lm ; ni ; 2 2 42 is the EMT of gravitons; 11 krlnXˆ ˆˆ m ˆˆ2ˆˆnm imr;;nlrl;; lm;; nr 1 82TXˆˆkk l    k X ˆml  (44) i ghost  ;;li ;; il i;; ml 1 4 XXˆ kl   kˆ ml   . ;;li ;; il i ; ml ; is the EMT of ghosts. In the averaging of (42), it was 4 k taken into account that  î 0 by definition of 3We are using the standard definitions. Matrix functions are defined by the quantum field. Averaged equations for the classic their expansion into power series as any operator functions: fields (35) take form of the standard Einstein equations ˆ ˆ ˆ n . UV  cVn containing averaged EMT of gravitons, renormalized by n The derivative of n-th degrees of matrix by the same matrix is ghosts: defined as ˆ k Ei Vˆ n  nVˆ n1. (45) Vˆ kkk1 ˆ k  RRiii T i 0. The derivative by numerical (non matrix) parameter z is 2 VVˆ n ˆ nVˆ n1 . Quantum dynamic equations for gravitons (36) could zz be rewritten as follows: If matrix function UVˆ ˆ and its derivative WUVˆ ˆˆn  are ele-   ˆkklkllkklm1 ;;; ; Liilliilimlˆ ;;;ˆˆ ˆ ; mentary functions, then 2 UVˆ ˆ Wˆ . 11kklkml zz i ˆ ˆlRii ˆlRm (46) Formulas (37)-(39) are the consequence of these definitions. It worth 22 to mention, that in matrix analysis in all intermediate formulas one  TTˆ kkˆ 0. should be careful with the index ordering. ii

As is seen in the Equations (46), in the theory of gra-  Tˆ k 0. (54) igrav ;k vitons all nonlinear effects are in the difference between ˆ k ˆk the EMT operator and its average value. System of Equ- Take notice, that tensors Ei and Li in (51), (53) ations (45), (46) is closed by the quantum dynamic equ- are multiplied by the linear forms of graviton field ations for ghosts, which could be also written in 4D operators only. Such a structure of identities is only valid covariant form: for the exponential parameterization. This fact is of key value for the computations in the framework of pertur- ˆîk ik XX;k0, ;k0 (47) ;;i  i bation theory. The order n of the perturbation theory is defined by the highest degree of the field operator in the Equations (47) provide the realization of the conserva- quantum dynamic equations for gravitons (46). The EMT tive nature of the ghosts’ EMT: of gravitons which is consistent with the quantum equ- Tˆ k 0. (48) ighost ;k ation of order n contains averaged products of field operators of the order n 1 (e.g., the quadratic EMT is 2.5. Differential Identities consistent with the linear operator equation). We see that by defining the order of the perturbation theory, we have In the exact theory, which is dealing with the full metric, identity (53), in which all terms are of the same maximal there is an identity: order of the quantum field amplit ude: kn 1  kl 1 k ml k ˆ ˆ ˆ ˆ Tigrav DgRkliilmi ˆˆ gR   ;k  2 (55) (49) 11lkˆkn ˆ mn 1  ˆki; LL l l m . kl k ml  22 ggˆˆ;;li ;; il  i ;; ml   0  4  Such a structure of the identity automatically provides ˆ where Dk is the covariant derivative in the space with the conservation condition (54) at any order of perturba- 4 metric gîk . This identity is satisfied by Bianchi identity tion theory . and by the ghost equations of motion. In terms of covariant derivative in the background space, identity (49) 2.6. One-Loop Approximation could be rewritten as follows: In the framework of one-loop approximation, quantum ˆ kk1 ˆ ˆˆ kˆlˆ lˆk fields interact only with the classic gravitational field. EdEik; ln i klEE i ik l 2 ;k (50) Accordingly, Equation (46) are being converted into li- ˆkˆ lˆk near operator equations: Eik; ikE l 0. 1 For the exponential parameterization, taking into ac- ˆkkllkklkmlkl;;ˆ ;ˆ ;ˆ LRiililliilmlˆ ;;;ˆ  ; i count (38), the expression (50) can be transformed to the 2 (56) 11 following form klˆ R m  k ˆ 0. 22iml i 11 ˆ klkklˆ ˆ ˆ EEEik;;ki l l l 0. (51) Of course, these equations are separated into the equ- 22 ations of constraints (initial conditions): Identity transformation EELˆ kkk ˆˆ and the iii ˆ0ˆˆ 0 subsequent averaging of (51) yields: LLL00 0, 0,  0, (57) 11 and the equations of motion: ELˆklˆ ˆkkLˆm 0. (52) iki;k ; llm 22 ˆˆ1 l LL  l  0. (58) Here we have used explicitly the fact that 2 k ˆk î 0 , Li 0 , by definition. Next, The equations for ghosts (47) are also transformed into Expression (45) is substituted into (52). Taking into 4In the framework of the perturbation theory, any parameterization, account the Bianchi identity and the conservation of the except the exponential one, creates mathematically contradictory mod- ˆ kn1 ghost EMT, we obtain: els, in which the perturbative EMT of gravitons Tigrav is not 11 conserved. In our opinion, a discussion of artificial methods of solu- ˆˆklˆ kkˆmtions of this problem, appeared, for example, if linear parameterization TLigrav ki;  l lL m . (53)  ;k 22  ggik ikˆ ik is used, makes no sense. The algorithm we have sug- As is seen from (53), quantum equations of motion (46) gested here is well defined because it is based on the exact procedure of separation between the classical and quantum variables in terms of provide the conservation of the averaged EMT of gra- normal coordinates. We believe there is no other mathematically non- vitons: contradictive scheme.

Copyright © 2013 SciRes. JMP 60 L. MAROCHNIK ET AL. the linear operator equations: An important technical detail is that in the perturbation theory the graviton field should be consistent with an ;;ii0, 0. (59) ;;ii additional identity. In one-loop approximation that iden- In the one-loop approximation, the state vector is re- tity is obtained from the covariant differentiation of Equ- presented as a product of normalized state vectors of gra- ation (56): vitons and ghosts: ˆ kkl QRil    lki ˆ ; 0. (67)   . (60) ggh The appearance of conditions (67) reflects the fact that Equations for macroscopic metric (45) take the form: we are dealing with an approximate theory. As it was already mentioned in Section 2.3, the partition of the kk1 k metric into classic and quantum components, and, re- RRii i 2 (61) spectively, the factorization of the path integral, can be  ˆˆkk only done under the condition that additional constrains g TTi grav g gh i ghost gh . are applied to the geometry of background space. These The averaged EMTs of gravitons and ghosts in Equ- constrains are manifested through the structure of the ation (61) are the quadratic forms of the quantum fields. Ricci tensor of the background space which should pro- ˆ ik ik ˆ ikˆ ik ˆ ikˆˆ il k Assuming that X  g , X 1  , X 2 l 2 vide the identity (67) for the solutions of dynamic equa- in (43), (44), we obtain: tions for gravitons. In the Heisenberg form of quantum theory the additional identity can be written as conditions Tˆ k igrav on the state vector: 11  lm;;k k lm;k kml;ˆ ˆ kklˆ ˆˆmi;; l ˆ iˆ ˆ im;ˆ l ˆl  mi; QRil   lk ;i 0. (68) 42  Status of all constrains for the state vectors are the same 11klmn;; n lmn ; imnlˆˆ;;; ˆˆ n2  ˆˆ nml (62) and are as follows. If (57), (65), (68) exist at the initial 22 moment of time, the internal properties of the theory 1 ;m ˆ lkmklmlmkˆ ;;ˆ ˆ ˆˆ k ˆˆ nl should provide their existence at any following instance 2 mi  mi  mi;  i  mn 2 ;l of time. While one is conducting a concrete one-loop calcula- kml knml 1 k 2  2ˆ ˆ RRˆ ˆ  ˆ , tion, there is a problem of gauge invariance of the total ml i i l n m 2 i  EMT of gravitons and ghosts. As was mentioned by De Witt [17], after the separation of the metric into back- ˆ kk1 ;;;kkl Ti ghost ;;i i il;. (63)  4 ground and graviton components, the transformations of the diffeomorphism group (4) can be represented as trans- Quantum Equations (56), (59) provide the conservation formations of the internal gauge symmetry of graviton of tensors (62), (63) in the background space: field. In the framework of one-loop approximation, these Tˆ k 0, transformations are as follows: ggigrav ; k (64) ˆ kklkk   ; . (69) Tˆ k 0. iilii;; gh i ghost gh ; k The problem of gauge non-invariance is twofold. First, The ghost sector of the theory (56)-(64) corresponds to the EMT of gravitons (62) is not invariant with respect to the gauge (26). Note, however, that all equations of the transformations in (69). Second, the ghost sector (the theory, except gauges, are formally general covariant in ghost EMT), inevitably presented in the theory, depends the background space. That provides a way of expanding on the gauge. Concerning the first problem, it is known the class of gauges for classic fields. Obviously, we can that the operation removing gauge non-invariant terms move from the initial 4-coordinates, corresponding to the from the EMT of gravitons belongs to the operation of classic sector of gauges (26), to any other coordinates, averaging over a quantum ensemble. In the general case conserving quantum gauge condition of arbitrary background geometry and arbitrary graviton i wavelengths we encounter a number of problems (when ˆ0 0. (65) conducting this operation), which should be discussed It is not difficult to see, that in the classic sector any separately. gauges of synchronous type are allowed: In the particular case of the theory of gravitons in a gN2 ,0.t g (66) homogeneous and isotropic Universe, the averaging pro- 00 0 blem has a consistent mathematical solution. It was where tN )( is an arbitrary function of time. shown in Section 3.1 that removing the gauge non-

Copyright © 2013 SciRes. JMP L. MAROCHNIK ET AL. 61 invariant contributions from the EMT of gravitons from field. Therefore, the satisfaction of these equalities is the quantum ensemble has been set gauge-invariantly. provided by the isotropy of graviton spectrum in the k- To address the second aspect of the problem, we should space and by the equivalence of different polarizations. ˆ  ˆ k take into account that the theory of gravitons in the 3-tensor modes   t and their EMT ttTit , macroscopic space with the self-consistent geometry oper- respectively, are gauge invariant objects. Gauge non-in- ates with macroscopic observables. Therefore, in this ˆˆkk variant modes  iv, is are eliminate d by conditions theory one-loop finiteness, as the general property of that, imposed on the state vector, read one-loop quantum gravity, should have a specific em- ˆ kk ˆ 0. (73) bodiment: by their mathematical definition, macroscopic iv vsis observables must be the finite values. This requirement Note that the conditions (73) automatically follow from on the theory in the Heisenberg representation is realized Equation (56) and conditions (63). As a result of this, a in Hamilton gauge (26) only. gauge non-invariant EMT of 3-scalar and 3-vector modes 3. Self-Consistent Theory of Gravitons in the is eliminated from the macroscopic Einstein equations, Isotropic Universe and we get  TTˆ kk0, ˆ  0. (74) 3.1. Elimination of 3-Vector and 3-Scalar Modes vviv ss is by Conditions Imposed on the State Vector The important fact is that in the isotropic Universe, the We consider the quantum theory of gravitons in the space- separation of gauge invariant EMT of 3-tensor gravitons time with the following background metric is accomplished withou t the use of short-wave approxi - 2 ik mation. In connection with this, note the following fact. ds gik dx dx (70) In the theory, which for mally operates with waves of N222222 t dt a  t dx  dy dz . arbitrary lengths, the problem of existence of a quantum ensemble of waves with wavelengths greater than the In this space the graviton field is expanded over the irre- distance from horizon is open [13]. In cosmology, the ducible representations of the group of three-dimensional existence of such an ensemble is provided by the follow- rotations, i.e. over 3-tensor ˆ  , 3-vector  t ing experimental fact. In the real Universe (whose pro- ˆˆk ˆ iv 0 v,  v and 3-scalar perties are controlled by observational data beginning from the instant of recombination), the characteristic ˆˆk  0 ,,ˆ ˆ modes. Equations (56) are split is00  s  s  s scale of casually-connected regions is much greater into three independent systems of equations, so that each (many orders of magnitude) than the formal horizon of of such systems represents each mode separately. The events. The standard explanation of this fact is based on state vector of gravitons is of multiplicative form that the hypothesis of early inflation. Taking into account reads these circumstances, we do not impose any additional restrictions on the quantum ensemble. . gtvs The procedure described above is based on the exi- The averaged EMT (60) is presented by an additive form stence of independent irreducible representations of gra- that reads: viton modes only. But in this procedure, gauge-non- ˆ k invariant modes are eliminated by using of a gauge, i.e. ggTi grav (71) they are eliminated by using of gauge-non-invariant pro- ˆ kkˆ ˆ k cedures. The gauge-invariant procedure of getting the ttvvsTTit  iv   Tis s. same results is presented in [24], Section 3.1. The averaged EMT contains no products of modes that belong to different irreducible representations. This is 3.2. Canonical Quantization of 3-Tensor ˆ k because the equality gi  g 0 is divided into Gravitons and Ghosts three following three independent equalities The parameters of gauge transformations do not contain ˆ kk0,ˆ 0, ssis vviv terms of expansion over transverse 3-tensor plane waves. (72) ˆ k Therefore, Fourier images of tensor fluctuations are gauge- tt it 0. invariant by definition. We have Equalities (72) are conditions that provide the consis- 00 tency of properties of quantum ensemble of gravitons ˆ0kk  0, ˆˆ   0k0, with the properties of homogeneity and isotropy of the  ˆˆkk Q k, (75) background. In the homogeneous and isotropic space, the  same equalities hold for Fourier images of the graviton kQkk0, Q 0,

Copyright © 2013 SciRes. JMP 62 L. MAROCHNIK ET AL. where  is the index of transverse polarizations. The obtained by variation of gauged action over classic and operator equation for 3-tensor gravitons is quantum variables. To canonically quantize 3-tensor gra-

ikx vitons and ghosts, one needs to make sure that the vari-  t tQ,exk  k  t, k ational procedure takes place for Equations (76)-(79) (76) directly. To do so, in the action (39) we keep only back- k 2   k30H kk2 , ground terms and terms that are quadratic over 3-tensor a fluctuations and ghosts. Then, we exclude the full deri- where H  aa is Hubble function and dots mean derivative from the background sector and make the transi- vatives with respect to the physical time t . tion to Fourier images in the quantum sector. As a result The special property of the gauge used is the following. of these operations, we obtain the following The differential equation for ghosts is obtained from the 3aa2 equation for gravitons by exchange of graviton operator 3 Std, aNLLgrav ghost with the ghost operator. It reads N 2 1 a3 ikx k 2 ˆ ˆ tt,e,x  3H  0. (77) LLgrav ghost ˆkkˆ Nak kk (81)  kk  kk2 8 N k a k  3 Macroscopic Einstein Equation (61) read 1 a   2 kk Nak kk . 2 4 k N 3,H   g (78) In (81), the background metric is taken to be in the form  2 23H Hp g , (79) of (70), and the N function is taken to be a variation variable (the choice of this function, e.g. N  1 , to be where made after variation of action). Here and further on, the 1 k 2 ˆˆ ˆ ˆ normalized volume is supposed to be unity, so gg kk2 kk  g 3 8 k a Vx d1 . The terms which are linear over the gra- 2 viton field are eliminated from (81) because of zero trace 1   k  gh kk2 kk gh , of 3-tensor fluctuations. Variations of action over N and 4 a k (80) a are done with the following averaging. These pro- 1 k 2 ˆˆ ˆ ˆ cedures lead to Equations (78), (79) and Expression (80). pg  ggkk 2  kk  8 k 3a Variation of action over quantum variables leads to the 2 quantum equations of motion (76) and (77). 1   k  gh kk 2 kk gh In accordance with the standard procedure of canoni- 4 3a k cal quantization of gravitons, one introduces generalized are the energy density and pressure of gravitons that are momenta renormalized by ghosts. Formulas (80) were obtained La3 after elimination of 3-scalar and 3-vector modes from ˆ ˆ  . (82) kk  4  Equations (62) and (63). We also took into account the ˆk following definitions Then, commutation relations between operators that are ˆ 0 defined at the same instant of time read T0  g ,  a3 ˆˆ  ˆ ,,ˆˆˆ    TTpg . kk kk  3 4  (83) i . Also we have the following rules of averaging of bilinear kk forms that are the consequence of homogeneity and iso- Formulas (82) and (83) are presented for the N  1 case. tropy of the background Note also that the derivative in (82) should be calculated taking into account the   condition. ˆ ˆˆˆ kk  gggkk kkgkk , The ghost quantization contains three specific issues.

First, there is the following technical detail that must be   . ghkk gh ghkk gh kk taken into account for the definition of generalized mo- The self-consistent system of Equations (76)-(79) is a menta of ghost fields. The argument in respect to which particular case of general equations of one-loop quantum the differentiation is conducted needs to be considered as gravity (56), (59), (61)-(63). In turn, these general equ- a left co-multiplier of quadratic form. Executing the ations are the result of the transition to the one-loop appropriate requirement and taking into account Grass- approximation from exact Equations (43)-(47) that were man’s character of ghost fields, we obtain

33 La L a Suppose gk , hk are linear independent solutions to  ,.  (84) kkk k(88), so that their superposition with arbitrary co- k 4 4 k efficients gives the general solution to (88). With no loss Second, the quantization of Grassman’s fields is car- of generality, one can suppose that these solutions are ried out by setting the following anti-commutation renormalized in some convenient way in each concrete lations case. From the theory of ordinary differential equations it is known that g , h functions are connected to each a3 k k  ,, i , other by the following relation kk kkkk 4  (85) C a3   k  ghkk hg k k 3 , (89) kk,,  kki kk. a  4  where C is a normalization constant. The comparison Third is the bosonization of ghost fields, which is carried k of (88) with (76) and (77) shows that solutions of out after quantization of (85). The possibility of the operator equations are presented by the same functions. bosonization procedure is provided by Grassman algebra, For operators of graviton field we have which contains Grassman units defined by relations ˆ ˆ uu uu 1 . Therefore, conjunctive Grassman fields ˆkkkkkA gBh , (90) can be always presented in the following form ˆ ˆ where Akk , B  are operator constants of integration.  kkuu,,k k (86) Directly fro m th es e operator constants, one needs to build the operator which gives rise to the full set of basis where k is Fourier image of complex scalar field which is described by the usual algebra. The substitution vectors. of (86) in (85) leads to the following standard Bose com- It is important to keep in mind that commutation property of operator constants Aˆ , Bˆ and physical mutation relations kk  interpretation of basis state vectors are determined by 3 a  the choice of line ar independent solutions of Equation  ,,i 4 kk kk (88). The simplest basis is that of occupation numbers. (87) 3 The choice of linear independent solutions as self-con- a   ,.i kk kk jugated complex functions corresponds to this basis. 4 * In accordance with (89), if gkkkkfh, f the nor- The Hermit conjugation transforms one of them to the malization constant is pure imaginary. Let’s take Cik  , other. so we obtain i 3.3. State Vector of the General Form ff** ff . (91) kk k k a3 To complete the self-consistent theory of gravitons in the isotropic Universe, one needs to present the algorithm of To build the graviton operator over this basis, one need introduction of the graviton-ghost ensemble into the the- to carry out th e multiplicative renormalization of op- ory. Propert ies of this ensemble are defined by Heisen- erator constants taking into account that field is real. This berg’s state vector which is expanded over the basis that yield has a physical interpretation. Any possible basis is the AcBc4,ˆˆ 4 . system of eigenvectors of an appropriate time independ- kkkk   ent Hermit operator. The existence of such operators can As result of these operations, we get the graviton op- be proved in a general form. Let us consider the follow- erator and its derivative that read ing operator equation which is an analog of operator ˆ 4, cfˆ cˆ f* equations of gravitons and ghosts kkk kk  (92) 2 ˆ * k  kkk4. cfkk c f yH30y y . (88) kka2 k Standard commutation relations for operators of gra- Coefficients of Equation (88) are continuous and diffe- viton creation and annihilation are obtained by the sub - rentiated functions of time along all cosmological scales stitution of (92) into (83) and taking into account (91). except for the singularity. Thus, with the exception of the They read singular point, the general solution of Equation (88) ˆˆ definitely exists. Below we will show that the existence cckk,,   kk    (93) of a state vector follows only from the existence of gen-  ccˆˆ,0,,0.   ccˆˆ  eral solution of Equation (88) (see also [13]). kk  kk 

In accordance with (93), the operator of occupation meterizes Heisenberg’s state vector actually determines  numbers nccˆˆkkk  ˆ exists that gives rise to basis vec- the initial condition of quantum system of gravitons and tors nk of Fock’s space. Non-negative integer num- ghosts. bers nk  0,1,2, are eigenvalues of this opera tor. Formulas (94) and (97) can be also used in case when In accordance with (80), the observables are additive real functions are chosen as linear independent solutions over modes with given k . Therefore, the state vector of Equation (88). The justification for this is due to the is of multiplicative structure that reads fact that real linear independent solutions can be obtained from complex self-conjugated ones by the following li-   , g  k near transformation k where  is state vector of k -subsystem of 1 **i k gkkkkkkff,. h  ff (98) gravitons of momentum pk  and polarization  . 22

In turn, in a general case, k is an arbitrary super- After transition to the basis of real functions in (92) position of vectors that corresponds to different occu- and (95), we get pation numbers but t he same k values . Suppose that ˆ ˆ ˆ  is the amplitude of probability of find ing the k -  kkk4, Qgkk Ph nk (99) subsystem of graviton s in the state with t he occupation ˆ kk4, qgˆ kk phˆ k number nk . If so, then the state vector of the general form is the product of normalized superpositions where 2 1  n ,1. (94) ˆ ˆ ˆ gnkkk  n QQkk cc kk    , k nnkk 2 After the bosonization in the ghost sector is done, one ˆˆi PPkk ccˆkk  , gets equations of motion and commutation relations that 2 (100) are similar to those for graviton. The same set of linear 1  * qabˆˆkkk , independent solutions fkk, f that was introduced for 2 operators of grav iton field is used for op erators of ghost i  fields. What is necessary to take into acc ount here is pˆkk aˆbk. originally complex character of ghost fiel ds, which leads 2  to kk  . As a result, operat ors of ghost and anti- Relations (100) allow to work with real linear inde- ghosts creation and annihilation appear in the theory. pendent solutions and to use simultaneously state vec- They read tors (94) and (97) for the representation of occupation numbers. Note that in the framework of the basis of real  4, afˆ bˆ f* kkkkk functions, operator constants are operators of generalized (95 ) * coordinates and momenta: kkk4. afkk b f ˆ ˆ PQk,,,.k ikk  pqˆˆk k   ikk (101) The substitution of (95) into (87) leads to standard com-   mutation relations To complete this Section, let us discuss two problems that are relevant to intrinsic mathematical properties of aaˆkk,,,ˆ  kkaaˆ kkˆ   aaˆ k,0,ˆk   the theory. First of all, let us mention that “bosonization” bbˆˆ,,,,0, bbˆˆ bb ˆˆ(96) of ghost fields is a necessary element of the theory kkkk  kk  k k   because only this procedure provides the existence of ˆˆˆ ˆ abˆˆkk,,0,,,0 ab kk abˆkk   abˆ kk state vector in the ghost sector. Mathematically, it is   because the structure of the classic differential equation Applying the reasoning which is similar to that described (88) and properties of its solution (91) are inconsistent above, we conclude that in the ghost sector, the state with the Fermi-Dirac quantization. In terms of original vector of the general form is also given by product of ghost fields we have normalized superpositions. It reads * kkk4,  fkk f  gh n nnkkn , (102) kk  * kknn  kk kkk4. fkk f 22 (97) 1. nnkk Substitution (102) into (85) and taking into account (91) nnkk leads to anti-commutation relations for operator con- The set of am plitudes  ,  ,  , which para- stants that read nk nk nk

 ,, ,. Effects of vacuum polarization and particle creation are  k k kk k k kk  negligible for the subsystem of shortwave gravitons. In The  ,  relation can formally be consi- this sector, quanta of gravitational field can be con- kk kk dered as anti-commutation relation for operators giving sidered, with a good accuracy, as real gravitons that are rise the Fermi space of ghost states. There is no such a situated at their mass shell. The conservation of the num- possibility for kk, operators because their anti- ber of such real gravitons takes also place with a good commutation is negative. If one considers these operators accuracy. In the shortwave limit, choosing linear in- as complete mathematical objects that are not subject to dependent solutions of the (104) form, occupation num- any transformations, then it is impossible to build an bers nk are interpreted as numbers of real gravitons operator over them that gives rise to some space of states, with energy  k  ka, momentum pk  a and and this is because of non-standard anti-commutation polarization  . The possibility of such an interpretation relation. The problem is solved by the fact of the exis- is the principle and the only argument in favor of choice tence of Grassman units which are necessary elements of of this basis. For the subsystem of shortwave gravitons, Grassman algebra. At the operator constants level, the initial conditions are permissible not in the form of pro- bosonization is reduced to the following transformation ducts of superpositions but in the form of products of state vectors with determined occupation numbers. In ua,, ua  ub,. ub kkkkkkkk accordance with the usual understanding of the status of This leads to operators with (96) commutation pro- shortwave ghosts, their state can be chosen in the va- perties. cuum form. The gas of shortwave gravitons is described The choice of basis is the most significant prob lem in in more detail in Section 4.1. the interpretation of theory. In the theory of quantum In the kaa2   vicinity, there is no criterion al- fields of non-stationary Universe, the choice of linear lowing a choice of preferable basis. It is impossible to * independent basis fkk, f is ambiguous, in principle. introduce the definition of real gravitons in this region This differentiates it from the theory of quantum fields in because there is no mass shell here. This is the reason the Minkowski space. In the latter, the separation of field why we will use the term “virtual graviton of determined into negative and positive frequency components is Lo- momentum” in discussions of excitations of long wave- rentz-invariant procedure [31]. A natural physical postu- lengths. Under the term “virtual graviton” we mean a late in accordance to which the definition of particle graviton whose momentum is defined but whose energy (quantum of field) in the Minkowski space must be re- is undefi ned. Each set of linear independent solutions lativistically invariant leads mathematically to corresponds to the distrib ution of energy for the d eter- 12 it mined moment um. This distribution can be set up, for f  2e k . In the non-stationary Universe with kk example by the expansion of basis function in the Fourier the metric (70), the similar postulate can be introduced integral. Thus, the choice of basis is, at the same time, only for conformally invariant fields and at the level of the definition of virtual graviton. One needs to mention auxiliary Minkowski space. At the same time, the gra- that different sets of probability amplitudes  corre- nk viton field is conformally non-invariant. This can be seen spond to different definitions of the virtual graviton for from the following. Using the conformal transformation the same initial physical state. Note also that limitations  ykk ya and transition to the conformal time that are defined by asymptotes (104) do not fix basis ddt  a, one can see that Equation (88) is transformed functions completely. to the equation for the oscillator with variable frequency that reads 3.4. One- Loop Fin iteness  2 a  The full system of equations of the theory consists of op- ykkk  y 0. (103) a erator equations for gravitons and ghosts (76), (77), Effects of vacuum polarization and graviton creation in macroscopic Einstein equations (78), (79) and Formula the self-consistent classic gravitational field correspond (80) for the energy density and pressure of gravitons. The to parametric excitation of the oscillator (103). averaging of (80) is carried out over state vectors of The approximate separation of field on negative and general form (94) and (97). The one-loop finiteness is positive frequency components is possible only in the satisfied automatically in this theory. The finiteness is short wavelength limit. Regardless of the background dy- provided by the structure of ghost sector, and it is a result namic, linearly independent solutions of Equation (103) of the following two facts. First, in the space with metric exist, and they have the following asymptotes (70) the ghost Equation (77) coincides with graviton Equation (76). Second, the number of internal degrees of  11ik*2ik a freedom of the complex ghost field coincides with that of ffkkke, e, . (104) 22kka 3-tensor gravitons. We will show this by direct cal-

Copyright © 2013 SciRes. JMP 66 L. MAROCHNIK ET AL. culations. Heisenberg state vector. It reads Let us introduce the graviton spectral function which k 2n 4  Wffkffk0 8d. *22*n (110) is renormalized by ghosts. It reads ngrav  222nnkk kk k aa 0  Wkk g ˆˆk g 2.gh kk gh (105)  The integral (110) describes the contribution of zero oscillations whose spectrum is deformed by macroscopic Zero and first moments of this function are the most gravitational field. Asy mptotic (104) shows that this important objects of the theory. They are integral is diverges. In such a situation, the usual way is to use regularization and renormalization procedures. As ˆˆ W0 g kk g 2,gh kk gh k  a result of these operations, quantum corrections to Ein- stein equations appear. These corrections are the con- k 2  ˆˆ formal anomalies and terms that came from Lagrangian W1 2 g kk g 2.gh kk gh k a  22  RRln  g  where g is a scale parameter that (106) comes from renormalization (see Section 10.1). The theory that we present here does not use such operations. The energy density and pressure of gravitons that are There is a contribution of ghost zero oscillations in the expressed via moments (106) can be obtained by trans- moments of spectral function. Its sign is opposite to formations identical to (80) with use of equations of mo- (110). It reads tion (76) and (77). They read k 2n 11 1 1 W 2    DW,, p  D W n ghost  2n gh kk gh gg16 411 16 12 (107) k a 2n   k * DW003.HW 8 2n ffkk a In addition, the following relation between moments is k (111) k 2n derived from equations of motion 8 aaˆˆ bbˆ ˆ f* f  2n  gh kk kk gh k k  k a DHDW6411160.HW  (108)  abˆˆˆ f*2  bˆ a  f2 . This relation ensures that the graviton energy-momen- gh kkgh k gh kk gh k  tum tensor is conservative: The observables (107) are expressed via sums

WWn grav n ghost . In those sums, the exact graviton- ggg30Hp    ghost compensation takes place in the contribution from As it was shown above, field operators can always be zero oscillations. chosen from the basis of complex self-conjugated func- The final expressions for the moments of spectral tions that are the same both for gravitons and ghosts. One function are obtained by using the explicit form of state needs to also mention that the interpretation of short vectors (94) and (97). They read wave gravitons as real gravitons determines the asymp- 2n k 2 WN8, fU**2fUf 2 (112) totic of basis functions (see (104)). After the com- nkkk 2n kkk mutation of operators of creation and annihilation are k a done, graviton contributions to the moments of the spec- where tral function Wnn ,0,1 can be presented in the follow- 222 Nnnn (113) ing form kkkknnnkkk     nnnkkk 111 k 2n ˆˆ Wngrav2n ggkk and k a   2n **1  k * Un  1 kknnkk1   8 ffkk  2n 2  nk 0 k a  2n (109)  k  * * 42 ccˆ ˆ ff  n  1 2n  g kkgkk  nnkk1  k  k a  nk 0 (114) ccˆˆ  f*2   ggkk k * n 1 .  nnkk1 k 2 ccˆ ˆ  f nk 0  ggkk k  * n 1 In the right-hand-side of (109), the first term is the  nnkk1 k  functional which is independent of the structure of nk 0

Copyright © 2013 SciRes. JMP L. MAROCHNIK ET AL. 67 are spectral parameters. They are defined by initial con-   Pnexp i , ditions for the chosen normalized basis of linear in- nnkkkg  dependent solutions of Equations (88). (For sake of bre-   Pnexp i , vity, in (114) and below we use the following notation nnkkkgh  kk,  .) Note that the relation (112) does not (117)   Pnexp i , contain divergences. Divergences in the relation (112) nnkkkgh  may appear only because of non-physical initial condi- nk nk tions. The spectrum of real gravitons that slowly de- Pnkkexpn . n ! creased for k  is an example of such a non-phy- k sical initial conditions. The substitution of (116), (117) to (113), (114) leads to The spectral function (105) depends of three arbitrary NN2. n  n  n (118) constants as it is averaged over the state vector of general k k kg kgh kgh form. It reads ** UUk  k 2 **2 2 gghii WNfkk8 kkkUfUfkk. (115)  kk nnnkg kkee kgh kgh , (119) ggh   In (115), the basis of normalized linear independent kk1, 1, solutions contains information on the dynamics of ope- where rators of graviton-ghost field; integration constants * NUU,, contain information on the initial ensemble g i 1  kkk eexpk Pn i i of this field. Due to the background’s homogeneity and knn kg kk1, 2  nk isotropy the moduli of the amplitudes and average occu-  pation numbers do not depend on the directions of wave Pnexp i i ,  kg nnkk 1,  vectors and polarizations: nk (120)  2  gh ik  nn  , kneePnkgh xpi in kg  nk k   kk1 n nk 1 k  2  nn n k , (116) Pnexp i i kgh  k  kgh nnkk1 n 1  k nk   2 ggh   n  n. Limit equalities kk  1 are satisfied if the kgh  nk k nk 1 phase difference between states of neighboring occupation numbers does not depend on values of occupation Phase of amplitudes, in principle, may depend on the numbers. It is also easy to see that (118) and (119) apply, directions and polarizations. One must bear in mind that with somewhat different definitions, to any ensemble in the pure quantum ensembles, for which the averaging g ik gh ik with  k e and  k e parameters. over the state vector is defined, phases of amplitudes are We already mentioned above that different basis func- determined. If the phases are random, then the additional tions that correspond to different definitions of the virtual averaging should be conducted over them, which corre- graviton can be used for the same initial physical state. sponds to the density matrix formalism for mixed ensem- Limitations due to the prescriptions on the asymptotic bles. The question of phases of amplitudes is clearly expression (104) allow to fix only asymptotic expansions linked to the question of the origin of quantum ensembles. of basis functions for k . These expansions can be In particular, it is natural to assume that the ensemble of used, however, only for description of shortwave modes long-wavelength gravitons arises in the process of re- (Section 4.1). Meanwhile, all non-trivial quantum gravity structuring graviton vacuum. This process is due to con- phenomena take place in spectral region where cha- formal non-invariance of the graviton field and can be racteristic wavelengths are of the order of the horizon described as particle creation. In this case, there is a scale. The choice of basis functions to describe these correlation between the phases of states with the same waves is not unique, and the set of amplitudes of pro- occupation numbers, but mutually opposite momenta and bability  depends significantly on this set. At the nk polarizations: the sum of these phases is zero. level of Equations (118), (119), the ambiguity in the de- If the typical occupation numbers in the ensemble are finition of the virtual graviton reveals itself in the am- g ik large, then squares of moduli of probability amplitudes biguity of values of parameters nkg and  k e . are likely to be described by Poisson distributions. For Similar ambiguity exists in the ghost sector. Two con- this ensemble we get clusions follow from that. First, it is necessary to work

Copyright © 2013 SciRes. JMP 68 L. MAROCHNIK ET AL. with the state vector of general form, at least during the the following boundary condition5 first stage of the study of the system that contains kk,,,   k , ,, , 0. (122) excitations of long wavelengths. Concretization of the amplitudes  is possible only after using of addi- There are no arbitrary constants in this expansion if the nk tional physical considerations that are different for each (122) condition is satisfied. concrete case. Second, a theory would be extremely de- The following linear ordinary differential equation of the third order with respect to 1  functional follows sirable which is invariant with respect to the choice of k from the Equation (103) for y  af, af * functions linear independent solutions of Equation (88), and, corre- kkk spondingly, is invariant with respect to the choice of 11  1 1  20,22 amplitudes of prob ability  defining the structure of kk  nk 2 kk  k (123) Heisenberg’s st ate vector, respectively. In Section 5, we 22k . will show that such a formulation of the theory exists in k the form of equation for the spectral function of gravitons The solution of Equation (123) satisfying to the asymp- renormalized by ghosts. The mathematically equivalent totic condition (122) reads  formulation of theory exists in the form of infinite 11 s  11.Jˆ s (124) BBGKY chain or hierarchy where joint description of k kk s0 gravitons an d ghost i s carrie d out in terms of moments of Powers of Jˆ operator from (124) are defined as fol- spectral function Wn,0,1,2,, N . k n lows 4. Approximate Solutions 1d  Jˆ  , k 4   3 4.1. Gas of Short Wave Gravitons 0 k k 2 Let us consider the gas of grav itons of wavelength that is ˆ 0 15  JJkk11,  1  46  , (125) much shorter than the distance to the cosmological ho- 84kk rizon. We exclude the long waves from the model. Also, JJ2111,1J  JJJss  1. the calculation of observables is done approximately, so kkk kkk that non-adiabatic evolution of quantum ensemble is not The integral is calculated explicitly for arbitrary s , so s taken into account. In the framework of these appro- that Jk 1 is a local functional of  and its derivatives. ximations, it is possible to save the pure vacuum status of It follows from (125) that a small parameter of asymptotic ghosts because their role is just to provide the one-loop expansion is of the order of 1 k 2 . The (124) solution finiteness of macroscopic quantities. Long wave exci- is approximate because non-local effects are not included tations we will consider in Section 4.2. to the local asymptotical series. Calculation of these ef- The calculation of observables fo r the gas of sh ort fects is beyond of limits of this method. wave gravitons can be done by general formulas (107), The asymptotic expansion (124), (125) defines the

(112)-(114) after the definition of basis functions and the 1 k functional, and hence, it defines basis functions state vector. For the short wave approximation, the full (121). The substitution of (121) to (112) produces asymp- asymptotic expansion of basis functions exists that sa- totic expansions of moments of spectral function that tisfies the normalization condition (91) and asymptotes read (104). Of course, to use the method of asymptotic expan- 4 W  sions, basis functions must be taken in the following n a22 n form 2n k    g cckkgghaakk bb kk gh 11iikk* k k   ffkke, e, aa22kk 1 cc ab e2ik  (121)  g kk g gh kk gh 2    d, kk  0 1 2ik g cckk g gh bakk gh e. 2  where   (126) a kk,,,  ,  a 5Note that the n  0 asymptotic exists for cosmological solu-

n is a real functional of scale factor and its derivatives. In tions of usual interest. For instance, 0  0 as 0   for the the short wave approximation, this functio nal is ex- inflation solution. For 0   it takes place for the FRW solution for panded into the lo cal asymptotic series, which satisfies to the Universe filled with ordinary matter.

State vector s from (126) can be concretized from the read general considerations. It was mentioned in Section 3.3 n  kn H 2 k , that such terms as vacuum, zero oscillations and quantum g 42k aakk2 k wave excitations are well defined for the n  0 (130)   n condition. Under the same condition, state vectors that  2 k pkg 42nk 2. HH are built on basis vectors of the Fock sp ace are easily 36aakkk interpreted. First of all, this statement is relevant to Relations (129) and (130) are valid if 1 gravitons. Eige nvalues nk and eigenvectors nk of 22 2 2   ak   H, H , i.e. the square of ratio of ncˆˆkk  cˆk operator describe real gravitons in asymptotic states. In the short wave approximation, the con- graviton wavelength to horizon distance is much less cept of r e al gravitons is valid for all other stages of the than unity. In case of large occupation numbers, these Universe evolution. Thus, in this particular case, the state results are of the quasi-classical character and can be vector of the general form can be reduced to the product obtained by the classical theory of gravitational waves of vectors corresponding to states with definite graviton [9]. numbers nk  0,1,2, possessing definite momentum As can be seen from (130), the high-frequency gra- and polarization. It reads viton gas differs from the ideal gas with the equation of state p   3 by only so-called post-hydrodynamic cor-  n . (127) g  k rections. In accordance with the approximation used, k these corrections are of the order of  22H  1 in com- In asymptotic states, short wave ghosts are only used to parison with main terms. Thus, the following simple for- compensate non-physical vacuum divergences. In ac- mula can be used cordance with such an interpretation of the ghost status, we suppose that ghosts and anti-ghosts sit in vacuum Cg1  gg3,pCkn4 g1   k . (131) states that read a k

gh  00.kk (128) kk 4.2. Quantized Gravitons and Ghosts of Super-Long Wavelengths Averaging over the quantum state that is defined by (127) and (128) vectors, we get In the framework of this theory, it is possible to describe the ensemble of super-long gravitational waves cc n, ggkk k kaa2    by an approximate analytical method.  ghaakk gh ghbbkk gh 0, Such an ensemble corresponds to the Universe whose observable part is in the chaotic bunch of gravitational  ggcckk waves of wavelengths greater than the horizon distance.  cc ab  The chaotic nature of the bunch is provided by non-zero g kk gghghkk wave vectors of these waves, so that observable pro-

ghbakk  gh 0, perties of the Universe are formed by superposition of waves of different polarizations and orientations in the 4 k 2n space. Such a wave system can produce an the isotropic Wnn  22 n  k . a k k spectrum and isotropic polarization ensemble consistent To calculate macroscopic observables in this approxi- with the homogeneity and isotropy of the macroscopic mation, it is sufficient to keep only the first terms of space. expansion of moments of spectral function that contain Such an ensemble of super-long waves can be formed no higher than second derivative of scale factor. In this only if the size of causally-bounded region is much approximation, moments of spectral function read greater than the horizon distance, which is possible in the framework of the hypothesis of early inflation (or other 4 n k scenarios (see, e.g. [32]). However, the problem of kine- W0  2  , a k k matical stability of an ensemble exists even in the frame- 8  2 nk work of the hypothesis of early inflation. The case is due DHH2    , (129) a k k to the fact t h a t the ensemble of long waves is destroyed during the post-inflation epoch if the Universe is ex- 42 2 nk Wk1 42nHHk 2, panded with a deceler ation. When long waves come out aa k kkof horizon, they are transformed to t he s hort w aves. Taking into account (129), we get energy density and Below we show that the k inematical self-stabilization of pressure of high-frequency graviton gas from (107) that an ensemble is possible in the framework of self-con-

Copyright © 2013 SciRes. JMP 70 L. MAROCHNIK ET AL. sistent theory of long waves. The iteration procedure over  k 2 parameter for Long waves under discussion correspond to virtual Equation (133) is constructed accordingly to the follow- gravitons. To describe them approximately, one needs to ing rules use asymptotic expansions of basis functions over the 3 0 2 small param eter  k . As well as in the case of short d k 0 1 2 3 0,kkPRk Qk , waves, th e basis can be chosen in the representation of d a20 k (136) self-conj ugated functions that are parameterized by the 3 n d  d k 4,1.ka22 a2 n1 n  universal real functional k a . This preserves the 3 k definition (121) and Equation (123). However, due to of d d our interest in the asymptotical expansion 1 k a over In particular, we get  k 2 parameter, it is necessary to rewrite Equation (123) d2 1 in the following form k 2.kPa24  (137) d 2 k   The virtual graviton is defined by integration constants   221122 PQ,, R of the main term of asymptotic expansion. aa4.ka  (132) kkk a2  k k Because the k functional of (121) is real (and therefore  0  the k functional of (136) is also real), we obtain Let us introduce the geometric-dynamic time following inequality 2 dda and the following functional 2 40,0,0.PQkk R k P k Q k  (138)  1 n The dependence of constants PQkkk,, R and phase  k 2 kk 2k a n0 on k fo r k  0 is defined by the finiteness condition for kW2 and dd22W  , and taking into ac- Note that the  time coordinate corresponds to the ori- k k ginal gauge gg00 1. Equation (132) and the spec- count the inequality (138) we obtain tral function (115) now read 21 Pkkk ,, Rk 3 02 d k 22d 2 Qkk,kk. 3 4,ka ak (133)   d d The main terms of asymptotical expansions of moments * ik i k WNUUkk8ee,k kk (135), energy density and pressure of long wave gra-  0  vitons can be obtained from (136) for k and (137) for d (134) 1  , k . They read k   k k 16CC 16 8 C DWgg23  ,, g 2 where  is a numerical parameter. Its value is unim- aa26 1 a2 k (139) portant because the constant’s contribution to phases of CCgg23 CC gg23 basis functions is absorbed by phases of contributors that  gg,.p  aa26 3 aa26 form vectors of the general form (94) and (97). Ob- servables (107) are expressed via moments of spectral where function. The latter read 2* CkPNUUgk2   kkk, 2 k 1 d Wk (140) D  * 62 CQNUU . a k d gk3  kkk k 81 dd22 kk* iikkFor the first time, approximate solutions for the energy 62  NUUkkk 2ee a k  ddk density and pressure in the (139) form were obtained for d  classical long gravitational waves in [11,12]. In the theo- i k * ik i k UUkkee , ry of classical gravitational waves [11,12], the constants k d  of integration Cg 2 and Cg3 must be positive. The cru- 1 2 cial formal difference between classical and quantum WkW1  2  k a k long gravitational waves is in the fact that the last ones allow an arbitrary sign of Cg 2 and Cg3 (negative as 8 2*iikk 2 kNk kkUUee.k well as positive). The physics of this crucial difference a k will be discussed below (Section 4.3). (135) In a particular case of  -type graviton spectrum,

which is localized at the region of very small conformal It follows from (142) that CCgg230, 0 if the gra- wave numbers, (139) can be considered as exact solu- viton contribution dominates over ghosts in the quantum tions. One needs to to go over from summation to inte- condensate. We will name such a condensate “quasi- gration classical”. Its energy density is positive, and it can be 11 formed by usual super-long gravitational waves. If the ... d32kkk ... d ....  32 ghost contribution dominates over gravitons in the quan- k 2 2 0 tum conden sate, then CCgg230, 0 . Such a con- After that, these solutions can be obtained by the fol- densate of negative energy density has no classical ana- lowing limits logy. k Summarizing the results of Sections 4.1 and 4.2, we kP2 1 const, k QQ const,k see that in co smological applications of one-loop quan- kka 1 tum gravity we deal with the multi-component system 2 * 2 consisting of short wave graviton gas g1 and two sub- NUU  k , (141) kkk k 2 00 systems of graviton-ghost condensate g2,g 3 . Taking into account (131) and (139), we get the following equ-  00const k , 0. ation for the scale factor In (141 ) k1 and a1 are the constants of di mension of a2 CCC conformal wave number and scale factor, respectively. 3.gg123 g (143) They provide the correct dimension to parameter aaaa4426 2 limkk0 kP. In the first scenario, the long wavelength condensate is of negative energy, which means that the contribution of 4.3. Scenarios of Macroscopic Evolution ghost dominates over gravitons. The evolution of such a In accordance with (139), the system of long wave Universe is of oscillating type. The solution reads gravitons behaves as a medium consisting of two sub- CCC2  44 C 2 Cg1 ggg123 g 2 systems whose equations of state are p11 3 and a  sin , p22  . But, the internal structure of this substratum 22CCgg22 3 cannot be determined by measurements that are con- (144) 2 ducted under the horizon of events. The substratum effect Cg1 CCgg12 4 Cg3 a2   (139) on evolution of the Universe, is seen by an ob- 1,2 22CCgg22 server as an energy density and pressure of the “empty” (non-structured) spacetime, i.e. vacuum. The question is: There is no classic analogy to the solution (144). It can does the graviton vacuum have a quasi-classic nature, or be used for scenarios of evolution of the early quantum has its quantum gravity origin been revealed in some Universe. In the region of minimal values of the scale cases? factor aamin 1 , the g3 condensate bounces the Uni- Let us review the situation. First, the superposition of verse back from a singularity. The transition from the quantum states in state vectors of the general form (94) expansion to the contraction epoch at the region of maxi- and (97) could be essentially non-classical. Second, the mal scale factor aamax 2 is provided by g2 con- clearly non-classical ghost sector is inevitably presented densate. Because of correlation of signs of Cg 2  0 and in the theory. Its properties are determined by the con- Cg3  0 , the non-singular Universe oscillates. Recent dition of one-loop finiteness of macroscopic quantities scenarios of oscillating Universes based on condensates (Section 3.4). The ghost sector is directly relevant to the of hypothetical ghost fields are under discussion in the (139) solution. Let us consider (118) and (119), assuming current literature as an alternative to the idea of inflation for the sake of simplicity nnkgh kgh. Parameters (see, e.g. [32])). Actually, we have shown that the same- of solution (139) are expressed via parameters of gra- type scenario is constructed with the standard building viton-ghost ensemble as follows blocks of quantum gravity the well-known Faddeev- Popov’s ghosts located far from the mass shell. Thus, a CkPn21c 2  g os gk2  kg  kk very attractive idea is that one and the same mechanism k  of graviton-ghost condensate formations in the frame- gh  nkgh 1ckkos work of one-loop quantum gravity based on the “stan-  C  2 dard” Einstein equations (without hypothetical fields and g3 generalizations of Einstein’s general relativity) could be Qn1 ggcos n 1h cos  responsible for cyclic evolution of the early Universe  kkg kkkkkgh k (instead of inflation). (142) The second type of scenario applies if gravitons

Copyright © 2013 SciRes. JMP 72 L. MAROCHNIK ET AL. dominate over ghosts in the condensate of positive The theory that allows quantitatively describe quasi- energy. The solution reads resonant quantum gravitational effects is constructed in the following way. For the spectral fun ction of gravitons 42 2 22CCaCaCg 2213ggggg C2a  C1  and ghosts Wk , as defined in (105), a differential equ-  (145) ation is derived. For this, the first Equation (76) is mul- 4 Cg 2 tiplied by the ˆ  (and then by the ˆ  ), conjugated 2eCC C xp. k k gg23 g 1 3 Equation (72) is multiplied by the ˆ (and then by the   k ˆk ); and the equations obtained are aver aged and added. The g3 condensate forms the regime of evolution in Similar action is carried out with equations for ghosts, the vicinity of singularity; meanwhile the asymptote of after which the equations for ghosts are subtracted from cosmological solution for   is formed by g2 the equations for gravitons. These operations yield: condensate. Short wave gravitons g1 dominate during 2   2k the intermediate epoch. The ratio of graviton wavelength WFkk 23HWWk2 k 0, (147) to horizon distance is constant during the following asymp- a 2 totical regime k Fkkk6,HF  2 W (148)  a Cg 2 atexp , where 3   Wkk g ˆˆk g 2,gh kk gh This means that the long wave cond ensate g2 forms   ˆ   the self-consistent regime of evolution that provides its Fkk g ˆ kg 2.gh  kk gh kinematic stability.  Further, Equation (147) is differentiated. Expressions for  5. BBGKY Hierarchy (Chain) and Exact Fkk, F via Wk are substituted into the results of dif- Solutions of One-Loop Quantum Gravity ferentiation. For the spectral function the third-order Equations equation is produced 5.1. Constructing the Chain   2  WHWkk93HHW6 k Approximate methods used in Sections 4.1 and 4.2 pro- 2 (149) 4k  WHkk20W. vide an opportunity to describe only limit cases which a2 are ultra shortwave gravitons and ghosts against the background of almost stable Fock vacuum and super- It is now necessary to draw attention to the fact that long wave modes, forming nearly stable graviton-ghost Wtk   is Fourier image of the two-point function, taken condensate. Now we are examining self-consistent theory at tt  : of gravitons and ghosts with the wavelengths of the order Wtt ,; xx  of distance to the horizon: ki  ˆiktt,,2,,,xxˆ    tt xx  (150) 2 k 2  H ,.H (146) 1 3 iky 2 Wt d,;e. yWtty a k V  When describing modes (146), one should keep in mind An infinite set of Fourier images is mathematically two factors. First, in the area of the spectrum (146), there equivalent to the infinite set of moments of the spectral are no reasonable approximations, which could be used function to solve Equations (76) and (77), if the law of cos- W mological expansion at, H t is not known in n advance. Second, the (146) modes are quasi-resonant. k 2n  ˆ  ˆ 2, Quantum gravity processes of vacuum polarization, 2n g kk g gh kk gh k a  spontaneous graviton creation by self-consistent field and n 0,1,2, , . graviton-ghost condensation are the most intensive in this region of spectrum. From (146) it is also obvious that the (151) threshold for quantum gravitational processes involving Therefore, from the equation for Fourier images (149), zero rest mass gravitons and ghosts is absent. These we can move to a n infinite system of equ ations for the processes at the scale of horizon occur at any stage of moments. For this, Equation (149) is multiplied by evolution of the Universe, including, in the modern Uni- ka2n followed by summation over wave numbers. verse. The result is a Bogoliubov-Born-Green-Kirkwood-Yvon

(BBGKY) chain. Each equation of this chain connects 1) In one-loop quantum gravity, the BBGKY chain can the neighboring moments: be formally introduced at an axiomatic level; DHDW64160,HW  (152) 2) The internal properties of Equations (152)-(155) 11 provide the existence of finite solutions to this system; B1, 2 3) In finite solutions, there are solutions which do not 2 meet the “classic” condition of positiveness of moments WHWHH15   3 22  3 W  (153) 111 (see Sections 5.2 and 5.3). 3   It follows from these facts that there should be an 240HHHHWW 18 124 24HW2 0, opportunity and the need to implement a renormalization Bnn,1 procedure to the theory. This procedure should be able to   redefine the moments of the spectral function to finite WnHWnn32 3 values, but that leaves them sign-undefined. As it can be 34nnHn22 126 2 1HW n seen from the theory which is presented in Sections 2 and (154) 3, in the one-loop quantum gravity such a procedure is 2n22n23 9nH 9 6 nHHHW 2   n contained within the theory under condition that the 482WnHW   0, ghost sector automatically provides the one-loop fini- nn11 teness. n 2, , . We found three exact self-consistent solutions of the Equations (152)-(154) have to be solved jointly with the system of equations consisting of the BBGKY chain (152)- following macroscopic Einstein equations (154) and macroscopic given below in Sections 5.2 and 5.3. The existence of exact solutions can be obtained  11 HDW  1, through direct substitution into the original system of 16 6 (155) equations. The microscopic nature of these solutions, i.e. 2 11 3.HDW1  dynamics of operators and structure of state vector is 16 4 described in Sections 6 and 7. Note that an infinite chain of Equations (152)-(154) contains information not only on the space-time dy- 5.2. Graviton-Ghost Condensates of namics of field operators, but also about the quantum Constant Conformal Wavelength ensemble, over which the averaging is done. The mul- In Section 4.2 the exact solution was found for the titude of solutions of the equations of the chain includes graviton-ghost condensate, consisting of spatially uni- all possible self-consistent solutions of the operator equ- form modes (see (139)-(141)). This solution satisfies to ations, averaged over all possible quantum ensembles. the first two BBGKY Equations (152), (153) for an Theory of gravitons presented by BBGKY chain, con- arbitrary law of evolution H t and under condition ceptually and mathematically corresponds to the axio- that W  0 for n  2 . (Recall that in this solution D matic quantum field theory in the Wightman formulation n and W1 must be understood as the result of limit transi- (see Chapter 8 in monograph [25]). Here, as in Wight- 2 tion k  0 ; and equality to zero of higher moments man, full information on the quantum field is contained follows from the spatial uniformity of modes.) Now we in an infinite sequence of averaged correlation functions, describe the exact self-consistent solutions for the system, definitions of which simply relate to the symmetry pro- in which in addition to spatially uniform modes, quasi- perties of manifold, on which this field determines. resonant modes with a wavelength equal to the distance In BBGKY chain (152), (153) and (154), unified gra- to the horizon of events are taken into account. In terms viton-ghost objects appear which are moments of the of moments of the spectral function, the structure of spectral function, renormalized by ghosts. The ghosts are solutions under discussion is not explicitly labeled so that the chain is can be built formally in the model not containing ghost fields. Mathe- DDg 234, Dg   Dg  matical incorrectness of such a model is obvious only WWg24 Wg , WWgn   4,2,  with a microscopic point of view because in the quantum 11 1 nn 16CC16 8C theory all the moments of spectral function diverge the Dg3,2,2,gg32DgW g g2 stronger, the more the moment number is. The system of aaa6221 Equations (152)-(154) does not “know”, however, that 48CC24 gg41 aa004n without the involvement of ghosts (or something other Dg4l n,4lWn  g  n, aa214e a2n a renormalization procedure) it applies to the mathema- n 1, , . tically non-existent quantities. The three following mathematical facts are of principal importance. (156)

Here CCCgg32,,gn4 , a 0 are numerical parameters. other words, in the space of conformal wave numbers the Restrictions on their values follow from the condition of spectrum of quasi-resonant wave modes is localized in the existence of the exact self-consistent solution. the vicinity of the fixed value kk 0 .

The solution is found by using of the consistency of Depending on the sign of Cg 4 , we get two exact so- functions (156) with the relations arising from the macro- lutions to the macroscopic observables of graviton-ghost scopic Einstein’s equations (we are discussing model media in the form of functionals of scale factor. with 0 ): 1) Oscillating Universe. 14 Suppose that C  0 . In accordance with (160), in CCC e a g 4 H 2 ggg324ln 0 , this case all C  0 . The positive sign of all moments 33aa62a2 a gn4 Wgn  40  suggests tha t grav ito ns dominate over CCC e34a H gg324   gln 0 , (157) ghosts in the ensemble of quasi-resonant modes. We also aaa6223 a see that the parameter of spatially uniform mode g2 is 54 3CCC negative, i.e. Cg 2  0 . As was shown in Section 4.3,  ggg324e a0 HH2l622n.signs of parameters of g2 and g3 modes are the aaa3 a same, so Cg3  0 . From this it follows that ghosts are In (157) as well as further, we use notation dominant in case of spatially uniform modes. The energy density and pressure of graviton-ghost substratum read CCg 41  g 4 . Functions D and W1 from (156) trans- form the equation (152) to an identity. The substitution Cg3 3C a of W and W into (153), taking into account (157),    g 4 ln0 , 1 2 g aa62a leads to the following expression (162) Cg3 C a 48 p   g 4 ln 0 . BH1, 2  g 62 a4 aaea

a 4 The parameter C is not explicitly showed up in (162) 4ln20.CC220  CCCC g 2 g42 gg4244ggg 42 a 3 because it is expressed via Cg 4 in accordance with (160). There is an oscillating solution to the Einstein (158 ) 2 equation 3H   g if solutions for the turning points The infi nite chain (154), in contrast to the Equation (153), aaam  min, max exist, i.e. contains moments of spectral functions of quasi-resonant 3Ca4 44 modes. Nevertheless, it does result, only including (158) g 40 aa00  bbe,  ln  . (163) as a particular case 4 Cg3 aamm  Bnn,1 In the vicinity of turning points energy density is formed 48  a by contributions of ghosts and gravitons, which are HCCC4ln0 22n  gn41 g 4 gn 4  a  a comparable in their absolute values, but have opposite signs. Far from turning points, graviton quasi-resonant 4  CC CC20 C ,modes dominate. Simplifying the situation, we can say gg24gn44 gn gn4 1  3  that in the oscillating Universe spatially uniform modes n 2, , . have essentially quantum nature, and quasi-resonant modes (159) allow semi-classical interpretation. The following relations between parameters follow from In the absence of a spatially homogeneous subsystem (158) and (159) g3 , the infinite sequence of oscillations degenerates into one semi-oscillation. Indeed, with Cg3  0 the scale 3 CCn ,.CC (160) factor, as a function of cosmological time, reads g 4(ng ) 4 g2 4 g4 C  2 Thus, moments of the spectral function of quasi-resonant aa  expg 4 ,C 0. (164)  0g 4 modes satisfy to the following recurrent relation 4 n In accordance with (164), the Universe originates from a CCgg44 Wgnn114422 Wg Wg 4. (161) singularity, reaches the state of maximal scale factor aa aamax 0 and then collapses again to si ngularity. Comparison of (161) with (151) shows that in the exact 2) Birth in Singularity and Acce lerating Expansion. solution under discussion all quasi-resonant modes Accordingly to (161), moments of the spectral function of quasi-resonant modes form an alternating se- have the sa m e wavelength  aCg 40 ak. In quence if Cg 4  0 . It reads

n finite width aa21 . In the sub-region of small values of n 24 Cg 4 a Wg41    ln,1,,.n  (165) the scale factor, the Universe is born in a singularity, n a2n a 0 reaches the state with a maximum value of aa 1 , and It is clear that the result (165) can not be obtained for the then returns to the singularity. In the limit C3g  0 the quasi-classical ensemble of gravitational waves. The possibility of such an evolution disappears because of microscopic nature of this solution is discussed in Sec- a1  0 . In sub-region of the large scale factor, the tion 6. It is appropriate here to emphasize one more time evolution of the Universe starts at the infinite past from that the theory, which is formulated in the most common the state of zero curvature. At the stage of compression, way in the BBGKY form, captures the existence of such the Universe reaches the state with a minimum value of a solution. aa 2 , and then turns into an accelerated mode of It is not difficult to notice that the solution which we expansion. With Cg3  0 , this branch of cosmological are now discussing is in a sense, an alternative to the solution is described by the following function of cosmological time previous solution. With Cg 4  0 , parameters of spatially homogeneous modes are positive Cgg230,C 0 . 2 Cg 4  Thus, spatially uniform modes admit semi-classical inter- aa  exp , C 0. (170) 044 g pretation, but quasi-resonant modes have essentially quan-  tum nature. The e nergy density and pressure of graviton-ghost substratum are Note that degenerate solutions (164) and (170) differ only in the sign under of exponent. Cg3 3 Cg 4 a  g 62ln , aaa0 5.3. Self-Polarized Graviton-Ghost Condensate (166) in De Sitter Space C Cg 4 ea p g3 ln . g 62 It is easy to find that the system of Equations (152)-(155) aaa0 has a simple stationary solution H  const , Specific properties of solutions to Einstein’s equations D  const , W  const . This solution describes the high- 2 n 3H   g depend on initial conditions and relations ly symmetrical graviton-ghost substratum that fills the between the parameters of graviton-ghost substratum. De Sitter space. It reads First of all, let us mention a scenario that corresponds to 11 a singular origin with the strong excitation of spatially HW2  ,e, aaHt 3610 3 uniform modes (171) 4 1 3 Cag 40  ggpW  1. CHg3 0, 0,  e. (167) 12 4Cg3 This solution exists both for the 0 case and for In the case (167), the Universe is born in the singularity   0 . The first moment of the spectral function satisfies and fairly quickly reaches the area of large scale factor the inequality W1 12  is the only independent values, where it expands with the acceleration: parameter of the solution. The remaining moments are expressed through by recurrence relations: 12 ta Cg 4 aC tln12 , , g 4 2 8 ta0 2a DW 1, 14 (168) 3 C (172) 3g nn23n3 aa 0 ,. 2 3 C WHn1  Wn,1n. g 4 22n 

Branch of the same solution, with H  0 describes the From (171) and (172) it clearly follows that the so- collapsing Universe with a singular end-state. lution has essentially vacuum and quantum nature. The Two other scenarios correspond to the weak excitation first can be seen from the equation of state pg   g . of graviton spatial ly uniform modes The second can be seen from the fact that the signs of the 4 moments WW 0 alternate. Another sign of the 3 Cag 40 nn1 Cg3 0, e. (169) quantum nature of the effect is contained in the pro- 4C g3 perties of graviton spectrum. The first of recurrence rela- In the case of (169), the region of legitimate values of the tions allows estimating of wavelengths of graviton s and scale factor is divided into two sub-regions ghosts that play a dominant part in the formation of

0 aa1 and aa2  separated by a barrier of observables

3N a W1 13 g 4   const. (173)  ggpH  2 , (176) k WH2 10 8 As can be seen from (173), during the exponential ex- The macroscopic Einstein’s equation is transformed into pansion of the Universe typical values of k rapidly the equation for the inflation exponent shift to the region of exponentially large conformal wave 3N numbers. The physical wavelength and macroscopic 24g 3.HH 2  (177) observables are unchanged in time. Such a situation 8 occurs if the following two conditions apply. Because Ng is a macroscopic parameter, the solution 1) In the k-space of conformal wave numbers spectra under discussion can be directly relevant to the asymp- of graviton vacuum fluctuations are flat; totic future of the Universe. In this case, the number of 2) In the integration over the flat spectrum, divergent gravitons and ghosts under the horizon of events and  - components of integrals excluded for reason to be dis- term in the Equation (177) should be considered as para- cussed in Section 6.2. Observables are formed by finite meters, whose values were formed during the earlier residuals of these integrals. stages of cosmological evolution. According to Zel’dovich In Section 6.2, we will show that these conditions are [33],  -term is the total energy density of equilibrium actually satisfied on the exact solution of operator equ- vacuum subsystems of non-gravitational origin. The pro- ations of motion, with special choice of Heisenberg’s blem of the  -term formation is so complex that little state vector of graviton-ghost vacuum. Microscopic cal- has change d since the excellent review of Weinberg [34]. culation also allows expressing the first mo ment o f spec- We are limited only to showing the order of magnitude tral function through the curvature of De Sitter space of  310 47 3 GeV4 allowed by observational data. 9N g 4 Some possibilities of co-existence of graviton conden- WH1  2 , (174) 2 sate and  -term will be discussed for  0,N g  0 . where Ng is a functional of parameters of state vector, (For other poss ibilities see Section 6.2.) The curvature of which is of the order of the number of virtual gravitons the De Sitter space for the asymptotical state of the Uni- and ghosts that are situated under the horizon of events. verse is calculated by means of the solution to the Equa- Their wavelengths are of the order of the distance to the tion (177). It reads horizon. It must be stressed that the number of gravitons 22 2 41 1  and ghosts Ng is a macroscopic value. H , The order of magnitude of N is determined by gra-   N NN226 (178) g g gg viton and ghost numbers in the condensate. Let us em- RH12 2 . phasize that numbers of gravitons and ghosts and hence, 

Ng parameters are macroscopic qualities. Further The energy density of vacuum in this state contains con- down in this sectio n it is assumed that the gravitons tributions of subsystems formed by all physical inte- dominate in the condensate and that the parameter ractions including the gravitational one N  0 . g 3N Note that the resu lt (174) can be easily predicted from    g H 4 . (179) the general considerations, including considerations of vac 82  dimension. Indeed, the general formula (112) shows that 2 The relative input of graviton-ghost condensate into the moment W is of dimension Wl ( l is of 1 1   asymptotic energy density of the vacuum depends on dimension of length). It also contains the square of the parameters of the Universe. If the following inequality Planck length as a coefficient. Because W1 is a func - tional of the metric, desired dimension can be obtained  2N g  1, (180) only using metric’s derivatives. It follows from this that 2 4 6 WC1  H where C dimensionless constant that contains parameters of vacuum condensate. Given (174), applies because of a small number of gravitons and ghosts, the solution in its final form is as follows: then the quantum-gravitational term is small and one must use the following solution 12N g 4 DH 2 , 2  2 1   n1 HN  1.2 g (181) 1 3 24 Wnn 2n 21!21nn2 (175) 2 If the inequality (180) is satisfied because of a small 2N g Hn22n ,1.  -term then the asymptotic state is mostly formed by  2 the graviton-ghost condensate

2 2 8 3) De Sitter Universe, H   . (182) 2 N g 3 III HtIII 2 8 aa0 e, HIII , It can be seen from (178 ) for 0 , the number of Ng gravitons an d gh osts that can appear in the Universe is a 2224 limited by maximum value 3,2  g (187) 2 a N g 6 122  2 Ngmax  2 10 . (183)    a 48 63  ggp  . aN In this limiting case (183), the equipartition of the va- g cuum energy takes place between graviton-ghost and If arbitrary shifts in time axis are excluded, then (185) non-gravitational vacuum subsystems and (186) are 3-parameter solutions aC043,,gg C . 2  42  1 2   Meanwhile (187) does contain one free parameter Ng . H  ,. g vac (184) Ngmax 32Also one can see that three exact solutions correspond to the spaces of different symmetries. The solution (187) 5.4. The Problem of Quantum-Gravity Phase describes 4-space of constant curvature, with the highest Transitions possible symmetry. Solution (186) (in the vers ion of appropriate unlimited expansion) describes 4-space, the Three exact solutions of the equations of quantum gra- geometry of which tends asymptotically to the geometry vity (with no matter fields and in the absence of  -term) of the Milln space. Finally, the solution (185) (in the are, in our view, impressive illustrations of physical conversion corresponding to oscillations) describes 3-geo- tent of the theory. (Of course, we can not exclude the metry, which is translation-invariant along the axis of existence of other exact solutions). The sets of basic time. Different symmetries of different solutions are the formulas (that characterize each of solutions) have the rationale for the introduction of phases of graviton-ghost form: vacuum. It is supposed to be continuous phase transitions 1) Oscillating Universe, between phases with different symmetries. a da Representations of phase transitions are, of course, tHa,0,2   Ha I min only heuristic nature. In the one-loop quantum gravity, amin I multi-particle correlations in the system of gravitons and III  CCC e14aI ghosts are not taken into account. For this reason, in this Ha2 gg324gln 0 , I 33aaa622 a theory it is impossible to define the order parameter that plays the role of the master parameter when choosing a III 3  I CCCggg4240, 0, Cg30, (185) phase state. Phase transitions that were discussed above, 4 were actually initiated by disparity between the choice of I 2 I I a Cg3 3C a the asymptotic state and set of the initial conditions. Of  g 4 0 3l26 g  2n,course, such operations are meaningful only within the aaaa suggestion that the effect of non-equilibrium phase transi- 4 CI I a g3 3Cg 4 tion will be contained in future theory. 63  gg p   . a aa62 Staying on the heuristic level, we can use the exact 2) Birth in Singularity and Accelerating Expansion, solutions (185), (186), (187) to demonstrate in principle the possibility of the existence of equilibrium phase a da t   , transitions. Let us consider the exact solutions as the 0 HaII  various branches of a general solution. A rough phase transition model is the passage from one branch to CCCII II II 14 II 2 ggg324e a0 another while maintaining continuity of scale factor and HaII ln , 33aa622a a its first and second derivatives. As can be seen from

II II 3 II II (185), (186), (187), these conditions provide the equ- CC0,  CC 0, 0, (186) gg424 gg43 alities of volumes, energies and pressures of graviton- II ghost systems on both sides of the transition point. It is 2 II aa C 3 Cg 4 3 g3 ln,easy to see that these conditions correspond to the phase 26g 2II aaaa0 transitions of the second kind. The microscopic theory makes it possible to see that at the point of transition the II 3 CII a 4Cg3 g 4 internal structure of graviton-ghost substratum is changed 63  gg p   . a aa62 (see Sections 6 and 7).

Consider consistently simplified models of all of the II III  gc()a 2  g phase transitions. Graviton-ghost vacuum is of the lowest II II 2 symmetry in phase (185). This phase is invariant under C 3 Cg 4 a 24 g3 lnc2  , condition that the shift on the time axis is of the 62II aacc22a Ng oscillation period only. Phase (186) is of higher sym- 0 (190) II ()3()apaII III 3 p  III metry. It is Milln space asymptotically in time at t . gc22 gc g g II Phase (187) (graviton-ghost vacuum in the De Sitter II 3 C 2 2Cg3 g 4 24 space) has the highest symmetry.    aa622 N Suppose that the symmetry of the graviton-ghost cc22 g vacuum increases in the process of the universe evolution, In this case, we have the following formulas for the that is, the phase transitions occur in the sequenc e transition point and the relationship between the para- IIIIII . According to (185), (186), the point of meters of the phases: transition from the initial state of the oscillating universe 4 e2 aII II I to the state of unlimited expansion II is determined 0 4Cg3 ln  , 4 4 II by the following relations ac2 aC cg24 (191) I II II gcaa11 g c 14 2 Cg 4 e ac2 12 I II ln  . 2 II IIaa00II 1 II  a N CCln ln  CC , c2 a0 g gg44aa3a4  gg 33 cc11c1 (188) According to (191), a continuous phase transition II II II gcapa111133 gc g apa c g c II III is possible if the parameters of Phase II satisfy the inequality II II  4 II  CC  C  C, 4 gg444  g 3 g 3 II  II 3a e aC04g c1   1. (192) II where ac1 is the value of the scale factor at the fitting 4Cg3 point, common to the two phases. From (188) one can If the phase transition took place, then in Phase III the get the formula for the fitting point and relationship be- number of gravitons under the horizon of events is tween the parameters of different phases: unambiguously defined by parameters of Phase II . The I  I  II  II  I  II  CCC C CC phase transition looks like a “freezing” of the distance to IIggg444I g 4  gg 44 aa    00   the horizon and of the value of the physical wavelength 14 of quasi-resonant modes. III (189) 4 CCgg33 Finally, we note that a continuous phase transition 14 ac e. I III from the oscillating Universe to De Sitter space III 1 3 CC   gg44 is possible if the following conditions are met 44 2 II  II  As we know, in Phase I gravitons dominate in quasi- e4eaC0304gg aC ln ,  1, resonant modes, and ghosts dominate in spatially uni- 4 4 I I ac3 aC 4 C form modes. Following the transition, in Phase II qua- cg34 g3 (193) I si-resonant modes are dominated by ghosts, but spatially C e14a 122 g 4 lnc3  . uniform modes are dominated by gravitons. Formulas 2 I ac3 a N g (189) provide constraints on the range of allowed values 0 of the transition point ac1 and the parameters II II II 5.5. Gravitons in the Presence of Matter. CC,,  a of the Phase II for given values of the gg340 Nonlinear Representation of the BBGKY III  parameters CCagg340,, of Phase I . According to Chain (189), whatever the parameters of Phase I are there is The full system of equations of self-consistent theory of a some set of parameters of a Phase II . Thus, a gravitons in the isotropic Universe consists of the continuous phase transition III from the state of BBGKY chain (152)-(154) and macroscopic Einstein oscillating Universe to the state of the Universe in a equations. In Equations (152)-(154), the Hubble function phase of the unlimited expansion and the asymptotic H and its derivatives H, H are coefficients multiplied acceleration is inevitable. by the moments of the spectral function. In such a form Further, let us consider the phase transition II III . the chain conserves its form even if besides of gra- The conditions of sewing together of solutions (186) and vitons, other physical fields are also sources of the (187) read macroscopic gravitational field. We are interesting in the

Copyright © 2013 SciRes. JMP L. MAROCHNIK ET AL. 79 evolution of the flat isotropic Universe at a stage when In the general case, the system of equations (195) and the contributions of gravitons and non-relativistic parti- (197) (to which the definition aa  H is added) should cles, baryons and neutralinos, are quantitatively signi- be solved numerically with initial conditions determined ficant. (The latter are presumably carriers of the mass of by the scale factor, moments of the spectral function and Dark Matter.) We assume also that non-gravitational their derivatives physical interactions created the equilibrium vacuum sub- aDWWW0;  0;  0, 0, 0, systems with full energy (a n effective  -term) of the nnn (198) order of  31047 3Ge V 4 . The macroscopic Eins- n 1,,. tein equations containing all so urces mentioned above The initial condition for the Hubble function should be read calculated via the equation of the constraint (194) 0 1 RR0 tot 111M 2 HDW000    . (199) (194) 48 121 3 3a3 0 2 11 M  HDW1 3 , 48 12 3 a Any solution of Equations (195) and (197), which cor- 13 responds to initial conditions (198), (199), satisfies the RR0   p 0 tot tot identity which is local in time 44 (195) 11M HD  W  . 2 111M 1 3 Ht Dt Wt  . (200) 16 6 2a 48 121 3 3at3  Equation (195) should be differentiated with respect to time, and then D from (152) should be substituted into the result of differentiation. These operations produce 6. Exact Solutions: Dynamics of Operators one more equation and Structure of State Vectors  33M 1 In this section, we get the exact solutions for field opera- H HDW113 W. (196) 812a 2 tors and expressions for the state vectors that correspond to exact analytical solutions of BBGKY chain (185) and The BBGKY chain (152)-(154) takes into account the (187). Microscopic studies of exact solutions allow interaction of gravitons with the self-consistent classical greater detail to identify their physical content. Solutions gravitational field which is represented by the Hubble (185) and (187) are formed as a result of certain spec- function and its derivatives. According to Einstein Equ- trally dependent correlations between graviton and ghost ations (194)-(196), a self-consistent gravitational field is contributions to the observables. These are full gravi- created by gravitons and other components of cosmo- ton-ghost compensation of contributions of zero oscilla- logical medium, i.e. by the matter and non-gravitational tions (one-loop finiteness); full compensation of contri- vacuum subsystems. Therefore, the self-consistent gra- butions in all parts of the spectrum, except the region of vitational field is a way of describing of significantly quasi-resonant (QR) and spatially homogeneous (SH) non-linear properties of the system that are the result of modes; incomplete compensation of contributions of QR gravitational interaction of elements of the system. After and SH modes with non-zero occupation numbers; cor- excluding higher derivatives of the metric from the relations between excitation levels and graviton-ghost BBGKY chain (153) and (154), the true non-linear character of the theory emerges. Substitution of (194)- contents of QR and SH modes, and, finally, some corre- (196) into (153) and (154) gives the non-linear repre- lations of phases in quantum superpositions of graviton sentation of BBGKY chain: and ghost state vectors. The physical nature of solution (186) turned out to be  DHDW64160,11 HW  unexpected and nontrivial. In Section 7, it will be shown    1 2 2 that mathematically this solution describes instanton Wn32nHWnnDnW 3n  4  6 3 1 1 16 condensate, which physically corresponds to the system M  of correlated fluctuations arising during tunneling of 81nn2289  2263nn  W 3  n 2a  graviton-ghost medium between states with fixed differ-  2 ence of graviton and ghost numbers. We explain also that nn1   2  WH11 D 233 n n W self-polarized graviton-ghost condensate in the De Sitter 32  2 space also allows instanton interpretation. M  8189nn22  4299 nn  W 3  n a  6.1. Condensate of Constant Conformal  Wavelength 482WnHWnnn11  0,1,,. 2 (197) Let us consider the solution (185) for CC34gg0, k0:

222production of solutions (186.I) at the microscopic level, 2 ka00 k0 Haln , a0 exp. (201) this fact is crucial. We will show that the choice of a state a2 a 4  vector, satisfying the condition of coherence leads to the The graviton wave equation with the (201) background fact that only this solution takes part in the formation of reads the observables. The second solution, containing a func-

22 tion F a , is a mathematical structure that does not ˆˆkk  ˆ0. (202) kk0 k correspond to the exact solution to the BBGKY chain. The equation for the ghosts looks similar. Fundamental Averaging of bilinear forms of operators (203) and solutions of Equation (202) are degenerate hypergeo- (204) over the state vector of the general form leads to metric functions. It is unnecessary to consider those the following spectral function 2 solutions for all possible values of the param eter k . ˆˆ Wkk g ˆ ˆk gg2 h kk  gh First of all, it is obvious that the macr oscopic ob-  (205) servables can be formed only by simplest hy pergeometric 16 a 2 2 ABFaCFa2 ln 0 . functions. Values k that are k  0 (spatially uni- 2 kk k  22 ka00 a form modes) and kk 0 (quasi-resonant modes) stand out. For all other modes there is a precise graviton-ghost The constants appearing in (205) are expressed through compensation. The reason why it is a mathematically averaged quadratic forms of operators of generalized co- possible follows from the general formulas (115), (118), ordinates and momentums: 6 (119) . ˆ  AQQqqkk g k g 2,gh ˆ kˆk gh Let us start with quasi-resonant modes. Exact solu-  tions of the Equation (201) and similar equation for  BPPpˆ 2,pˆ ˆ  ghosts for kk22 read kk g kg gh kkgh 0  (206) ˆ ˆ ˆ k CQPPQkk ggk k k     4k 22 Pˆ 22 0 QkPˆ ˆ edkk0022 k e ˆˆ ˆ kk0  2.gh qpkk  pq kk gh ak000 Following the transition to the ladder operators in for- 16 a QPFaˆ  ˆ ln12 0 , mula (100) and calculations, carried out similar to (112)- 2 kk ka00 a (120), we get (203) An21cosg k kg  kk  4k022kk22pˆ 22 gh ˆ ˆˆ00k 21cos,n  kkkqkp0 ed   e kghkk ak000 g Bn21cos (207) 16 a k kg kk qpFaˆˆ  ln12 0 , 2 kk gh ka00 a 21cos,n  kghkk (204) C  0. ˆ ˆ k where QPkk ,  and qpˆkk, ˆ are operators whose properties are defined in (99), (100), (96); For sake of simplicity, in (207) average numbers of ghosts and anti-ghosts are assumed to be the same: a da a2 2 0 nn . Faa0  . kgh kgh 312aa00 212 a0 aaln 2 ln Let us go back to the expression (205). Obviously, th e aa spectral function (205) creates moments (161) only if

Note that one of fundamental solutions to Equation BCkk  0 . The condition Ck  0 is satisfied automatically as a consequence of isotropy of macroscopic (202) is the Hermite polynomial H1  , which corre- 22 state, i.e. because of in depen dence of average occupation sponds to positive eigenvalue kk0  1 . In the re- numbers of the direction of vector k . Bk  0 imposes 6Formally, all modes except with k 2  0 and kk22 , look like “fro- 0 the conditions on amplitudes and phases of quantum zen” degrees of freedom, which are excluded from consideration by the superpositions of state vectors with different occupation model postulate. By virtue of the principle of uncertainty, postulates of this type are outside the formalism of quantum field theory. We want to numbers. It is necessary to draw attention to the fun- emphasize that in the finite one-loop quantum gravity there is no need damental fact: the solution under discussion does not to “freeze” degrees of freedom not participating in the formation of exist, if phases of superpositions are random. Indeed, particular exact solutions. Instead of mathematically incorrect operation averaging the expression (207) over phas es, we see that of “freezing”, the formalism of the theory offers mathematically consistent operations of graviton-ghost compensations. condition Bk  0 is satisfied only if nkg  n kgh.

The last equality automatically leads to Ak  0 , i.e. compressed to the singularity. In this scenario, valu e a0 which eliminates the nontrivial solution. can be defined as the size of the Universe, accessible for Thus, the condition of the existence of the solution observation in the end stage of expansion. As can be seen under discussion is the coherence of the quantum state. It from (212), if a0 is a macroscopic value, the difference is easy to notice (see (120)), that equality Bk  0 , as a in numbers gravitons and ghosts Ng  1 is also a condition of coherence, is satisfied for zero phase diffe- macroscopic value. rence of states with the neighboring occupation numbers Contributions of SH modes to the expressions for the of gravitons and ghosts: moments are shown in (156), and the relation between the parameters C and C is shown in (160). As a ggcosh  cos 1 2g 4g kkkk (208) part of the microscopic approach, the construction of ggh   kk1, cosk cosk  1. exact solutions for these modes is performed by the method of transaction to the limit, described at the end of Taking into account (208), we get th e following final Section 4.2. The parameter of spatially homogeneous expression (209) for the spectral fu nction of quasi-re- condensate is introduced similarly to (210): sonant gravitons and ghosts 64 a nn1 gh cos 1 g cos  WW n  n ln0 . (209) 00gh  00 g 00 k k 2 kg kgh ka a 2 (213) 00 2 Nk,0. In calculating moments, summation over wave numbers is k 2 gh 00 replaced by integration. Account is taken of that the The index “ gh “ in N  0 indicates the dominance of spectrum as the delta-form with respect to the modulus of gh ghosts over the gravitons in the spatially homogeneous k  k . Also a new parameter N is introduced where g condensate. The moments are: Ng is the difference of numbers of gravitons and ghosts in the unit volume of Vxd13 in the 3-space, which  16kN1 gh Wg2, is conformally similar to the 3-space of expanding Uni- 1 aa22 verse. Index “ g “ in designation of N parameter in- 1 (214) g kN dicates the dominance of gravitons in quasi-resonant 32 1 gh Dg2. 22 modes. In accordance with this definition, the following aa1 replacement is performed Definitions of parameters k1 and a1 are given in (141). 22 The energy density and pressure of the system of QR and nnNkk , (210) kg kgh k 2 g 0 SH modes are given by (211) and (214): Results of calculating of moments are equated to the  8 kN00gga0 2 kN kN1gh relevant expressions of (156) and (161), which were  g ln  aa22 a a2 a 2 a 2 obtained by exact solution of the BBGKY chain: 001 8kN a 1   0 g ln 0 , Wg4d Wk22n k 22 nk 22n  aa0 a 2 a 0 (215) 21n 8kN a  kN kN 64Nk 2n 00g 0 2 ggh1 g 0 ak0024 a0 p  ln   ln ln , g 33aa22 ea a2 a 2 a 2 aa22nnaaa2 001 0 (211) 8kN a 1 a  0 g ln0 . Dg42WW  22  2 00 3aa ea a a 0 128Nk aka48 2 In Formula (215), the terms in brackets are eliminated by  g 0 ln 0 0ln0 . aa22ee 14 a a 2 14 a the condition (160), which is rewritten in terms of mi- 0 croscopic parameters In accordance with (211), there is a relation between kN01g kNgh parameters ka00, and Ng that appear in the micro- 22 . (216) scopic solution aa01 2 3ka00 Ng  . (212) The solution (215), (216) describes a quantum coherent 8 condensate of quasi-resonant modes with graviton do- Recall that in the solution under discussion, the Universe minance, parameters of which are consistent with that of was born in singularity, expands to a state with a ma- spatially homogeneous condensate with the ghost do- ximum scale fa ctor aamax 0 , and then is again minance.

6.2. Condensate of Constant Physical * UUk  Wavelength kcr 1   The De Sitter solution for plane isotropic Universe reads  g cckk g gh abkk gh ; 2 

Ht 1 UUk  kann (222) aa0e , H const. (217) H 1  g cckkg gh bakk gh For the background (217), the gravitons and ghost equa- 2  tions and their solutions read *  Ukcr . 1 ˆˆˆ  k 2 0, Here U is the spectral parameter of quantum kkk kwave (218) waves, which become real gravitons if k  1; Ukcr , 12 U are the spectral parameters of quantum fluc- ˆ cfx c f*  x, kann kkkak tuations that emerge in the processes of graviton (and ghost) creation from the vacuum and graviton (and ghost) 1 ˆˆ  k 2 ˆ 0, annihilation to the vacuum. kkk Obviously, at the first stage of calculations we assume (219) that the averaging in (221), (222) is conducted over the ˆ 12  * kkaf x bk f x , state vectors of the general form (94), (97). This allows ak us to go to formulas (113), (114) or (118)-(120). Then it where is necessary to take into account that the moments Wn must not depend on time, and that they also should be i ix f xx1e,. k free of divergences. When analyzing the conditions for x  these demands, the specific form of the expression (220) Ladder operators in (218), (219), have the standard plays an important part. The measure of integration and property of (93), (96), which allow their use of in the dependence of field operators on the wave number constructing build basic vectors for the Fock space from and time can be represented in the terms of the variable which the general state vectors are constructed. x  k . A separate (additional) dependence on the wave The self-consistent dynam ics of g ravitons and ghosts number can be connected with the structure of spectral in the De Sitter space are not trivial in the sense that the parameters. After substitution of the variable kx  averaged bilinear forms of operators (218), (219) which in the equation (221), it is seen that the first term in (220) are explicitly and essentially depending on time, must is time-independent only if Ukwave is independent on lead to time-independent macroscopic observables. It the wave number. This means that the graviton and ghost must be emphasized, that the existence of such, at first spectra must be flat. However, with the flat spectrum glance unlikely solution, is guaranteed by the existence there is danger of divergences: if of the solution for the BBGKY chain. The key to the U  constk   0 , then the first integral in (220) kwave 2 solution lies in the structure of the state vectors of gra- does not exist, because fx 1 with x . vitons and ghosts. The divergences can be avoided only with exact com- Substitution of operator functions (218), (219) into the pensation of contributions from gravitons and ghosts to general expression for the moments (151) yields: the spectral parameter Ukwave . Let us point out, that in 2 that case we are not talking about zero oscillations but WH 22n n 2 about the contributions from the states with non-zero  occupation numbers. The compensation condition lead- 2 2 d,xx21n U f x U f* x (220) ing to U  0 is:   kkwave cr  kwave 0 2 . (223) Ufx nnnkkk  kann   The result (223) has a simple physical interpretation. The where quantum waves of gravitons and ghosts with the equation

NUk  kwave of state which differs from p  can not be carriers  of energy in the De Sitter space with the self-consistent  g cckk gghgh aa kk  (221) geometry. The total energy of quantized waves is equal  to zero due to exactly the same number of gravitons and bb  ; gh kk gh ghosts in all regions of the spectrum:

nnnn. (224) 12Ng 4 kk12kk DH , 2 With equal polarizations of gravitons and the equality of 1 n1 numbers of ghosts and anti-ghosts, it follows from (224) Wn 21!21nn 2 n 22n (228) that nnkkgg h. Exact equality of the average number of gravitons and ghost is a characteristic feature 2N g  H 22n , of the De Sitter space with the self-consistent geometry. 2 Let us mention that for the solution discussed in the n  1, previous Section 6.1, that equality is absent in principle. It means that different solutions have different micro- where scopic structures of the graviton-ghost condensate. Nnggcos gh cos , Based on the reasoning analogous to the one described (229) ,.    nnkk1, nnkk1 above, spectrum parameters Ukcr , Uk()ann also must not depend on the wave vector k . However, the corre- Zero moment W0 , which has an infrared logarithmic sponding integrals in the second and third terms of (220) singularity, is not contained in the expressions for the are not divergent. The absence of divergences is due to macroscopic observables, and for that reason, is not the fact that with x  the integration is taken over calculated. In the equation for W , the functions are 2ix 0 the fast oscillating functions  e . To calculate these differentiated in the integrand and the derivatives are integrals, they should be additionally defined as follows: combined in accordance with the definition    DW003 HW. At the last step the integrals that are n 21!n  lim dxx21n e 2ix  1 , calculated, already posses no singularities.  0  2211n 0 (225) Averaging of the parameter (229) over the phases  n1 2!n yields N  0 . Therefore the solution under discussion 2limdixx2n e 2ix 1 . g  0  2n does not exist if the superposition of the phases are 0 2 random. The coherence of the quantum ensemble, i.e. the At every instant of time, the procedure of redefinitions of correlation of phases in the quantum superposition of the in teg r als (225) selects the contributions from virtual basic vectors, corresponding to the different occupation gravitons and ghosts with a characteristic wavelength numbers, points to the fact that the medium is in the (173) and eliminate the contributions of all other gravi- graviton-ghost condensate state. The gravitons are do- ton-ghost modes. This redefining procedure provides the minant in the condensate if N g  0 , and the ghosts are existence of recursive relations (172) in the exact solu- dominant if N g  0 . tion of the BBGKY chain. The duality of the condensate and the indeterminate Thus, in (220) we have a flat spectrum of gravitons sign of the  -term create different evolutional scenarios. * Of course, all these scenarios are present in the ex- and ghosts, Ukwave  0 , UUkcr kann  Uconst  k . The expression for the spectral parameter takes the form: pression (178), which is obtained as a solution of the macroscopic Einstein equation (177). In addition to the 2  scenarios described in the Section 5.3, we will show the Un* 1  nn1 possibility of strong renormalization of energy of non- n gravitational vacuum subsystems by the energy of the **graviton-ghost condensate7. nn11nn11,  nn  (226) nn We have in mind a situation, in which the modulus of  -term exceeds the density of vacuum energy in the nnn  n, asymptotic state of the Universe by many orders of mag- where n is a normali zed statistical distribution. The nitude: average value of the number of gravitons and ghosts,  having the wavelength in the vicinity of characteristic   1, (230)  3H 2 values (173), are calculated by the formula vac 

 where  is a huge macroscopic number. From (178) it nngghn nn . (227) follows that the effect of strong renormalization takes n0 place if Using the Poisson distribution in (226), (227), the values 7 of integrals (225) and the formulas (119), (120), we get Mechanisms that are able to drive the cosmological constant to zero have been discussed for decades (see [34,38] for a review). Any par- the moments ticular scenarios were considered in [37,39-42].

6 2 condensation of gravitons and ghosts into the state with a  0, N  , g 2 certain wavelength of the order of the horizon scale— Ng    (231) plays a significant role in the formation of the asymptotic 6  values of energy density of cosmological vacuum. The    2. vac  2 N current theory explains how the strong renormalization of g the vacuum energy occurs, but, unfortunately, it does not Let us mention that the strong renormalization of the explain why this happens and why the quantitative cha- positive  -term is provided by a condensate in which racteristics of the phenomenon are those that are ob- the ghosts are dominant, and for the negative  -term— served in the modern Universe. Of the general con- by a condensate for which the gravito ns are dominant. siderations one can suggest that the coherent graviton- For clarity and for the evaluations let us introduce the ghost condensate occurs in the quantum-gravitational 12 19 phase transition (see Section 5.4), and the answers to Plank scale M Pl 81.2210    GeV, the 14 3 questions should b e sought in the light of the circum- scale of  -term M   , and the scale of the stances. density of Dark Energy in the asymptotical state of the 14 7. Gravitons and Ghosts as Instantons Universe, M  3  . We discuss the case when DE vac 7.1. Self-Consistent Theory of Gravitons in M  M . DE  Imaginary Time If non-gravitational contributions to  -term are self- ompensating, then a realistic estimate of the M  -scale 7.1.1. Invariance of Equations of the Theory with can be based on the Zeldovich remark [33]. According to Respect to Wick Rotation of Time Axis [33], non-gravitational  -term is formed by gravita- As has been repeatedly poi nted out, the complete system tional exchange interaction of quantum fluctuations on of equations of the theory consists of the BBGKY chain the energy scale of hadrons. In terms of contemporary (152)-(154) and macroscopic Einstein’s Equation (155). understanding of hadron’s vacuum, the focus should be On the basis of common mathematical considerations, it on non-perturbative fluctuations of quark and gluon can be expected that solutions to these equations covers fields, forming a quark-gluon condensate (see [35,36]). every possible self-consistent states of quantum sub- In this case,  -term is expressed only through the mini- system of grav itons and ghosts and the classical sub- mum and maximum scales of particle physics which are system of macroscopic geometry as well. In examining the QCD scale M QCD  215 MeV and Planck scale the model that operates with the pure gravity (no matter 19 M Pl 1.22 10 GeV : fields and  -term), one can identify the following unique property of the theory. Equations of the theory (152)- M 6 34QCD  424 (155) are invariant with respect to the Wick time axis  M   2 10 GeV . (232) M Pl rotation, conducted jointly with the multiplicative transformation of moments of the spectral function: In terms of these scales, it is turns out that a large number of   MM4410 5, which is defined in (230), can ti ,H i,  DE 12 be obtained by the huge number of N , for the same n (235) g DW,1  . number of orders of magnitude greater than the ratio nn 2 MMPl  . Indeed, choosing  , we find Rules of transformation of time de rivatives are obtained the value of Ng , which determines the ratio of vacuum from (235) energy density to the true cosmological constant in the HHiDi,,,  asymptotic state: nn Winnnn 1,W  1, (236) 2 n M 2 Ng    Winn1.   2 . (233) M Pl 3 In (236) and further on we use the notation   dd . The vacuum energy density of asymptotical state is cal- The statement about the invariance of the theory can culated as follows proved by direct calculations. As a matter of fact, transformations of quantities that appear in (152)-(155) by the  32 2 3 use of the rules (235) and (236) lead to the BBGKY vac   MMPl  . (234) 2 N g chain with imaginary time  Thus, the macroscopic effect of quantum gravity—the  64160,11   (237)

  functions that correspond to solutions of Equation (241). nn32n 3  We have been left with the infinite series only. These 34nn22 126 2 n 1 n series and integrals over spectrum of products of these (238) series can not be made consistent with the simple ma- 22nn223 9n 9 6(2) n   n thematical structure of the exact solutions (186.II). For  this reason the solution (186.II), as the functional of scale 482nn11nn  0,1,,,  factor is not relevant to solving operator equations in real and to macroscopic Einstein’s equations with imaginary time. time 11 7.1.2. Imaginary Time Formalism    , 16 6 1 As is known, the imaginary time formalism is used in (239) 11 non-relativistic Quantum Mechanics (QM) (examples see, 3.2  e.g., in book [43]), in the instanton theory of Quantum 16 4 1 Chromodynamics (QCD) [44-49] and in the axiomatic It is easy to see that for 0 Equations (152)-(155) quantum field theory (AQFT) (See Chapter 9 in the mo- identically coincide with (237)-(239) after some trivial nograph [25]). The instanton physics in Quantum Cos- renaming. mology was discussed in [51,52]. The invariance of the theory with respect to the Wick In QM and QCD the imaginary time formalism is a rotation of the time axis leads to the nontrivial con- tool for the study of tunnelling, uniting classic inde- sequence. Having only self-consistent solution of the pendent states that are degenera te in energy, in a single BBGKY chain and macroscopic Einstein’s equations, we quantum state. In AQFT, the Schwinger functions are can not say whether this solution is in real or imaginary defined in the four-dimensional Euclidian space—Euclid time. Nevertheless, having a concrete solution of BBGKY analogues of Wightman functions defined over the Min- chain, we can view the status of time during further study. kowski space. It is believed that using properties of To do so, it is necessary to explore the opportunity to Euclid-Schwinger functions after their analytical con- obtain the same solution at the level of operator functions tinuation to the Minkowski space, one can reconstruct the and state vectors. If this opportunity exists, the appro- properties of Wightman functions, and thereby restore priate self-consistent solution of BBGKY chain and ma- the physical meaning of the appropriate model of quan- croscopic Einstein’s equations is recognized as existing tum field theory. in real time. In the previous Section 6, we showed that All prerequisites for the use of the formalism of ima- two exact solutions (186.I) and (186.III) really exist at ginary time in the QM and QCD on the one hand, and in the level of operators and vectors, and thus have a AQFT, on the other hand, are united in the self-con- physical interpretation of standard notions of quantum sistent theory of gravitons. Immediately, however, the theory. specifics of the graviton theory under discussion should The problem is: What a physical reality reflects the be noted. Macroscopic space-time in self-consistent existence of solutions to Equations (152)-(155) (or that theory of gravitons, unlike the space-time in the QM, the same thing, (237)-(239)), not reproducible in real QCD and AQFT, is a classical dynamic subsystem, time at the level of operators and vectors? The existence which actually evolved in real time. If in QCD and of the problem is explicitly demonstrated by the ex- AQFT Wick’s turn is used to examine the significant ample of exact solutions (186.II). Assume that this properties of quantum system expressed in the pro- solution for CCk0, 220,Ck3 4 exists 34gg0g20 babilities of quantum processes, then in relation to the in real time: deterministic evolution of classical macroscopic sub- kk2a 2 2system this turn makes no sense. Therefore, after solving Ha2 0ln ,a exp0 . (240) 2 0  equations of the theory in imaginary time, we are obliged a a0 4 to apply (to the solution obtained) the operation of analy- The wave equation for gravitons with the (240) back- tic continuation of the space for the positive signature to ground reads the space of negative signature. It is clear from the outset ˆˆkk22  ˆ0. (241) that the operation is not reduced to the opposite Wick kk0 k turn, but is an independent postulate of the theory. The equation for the ghosts looks similar. Equation (241) Before discussing the physical content of the theory, differs from (202) just in the sign of coefficient before let us define its formal mathematical scheme. The theory the first derivative. However, this difference is crucial: if is formulated in the space with metric kk22 0 it is impossible to allocate the finite Hermit 0 ds222222 d  a  dx  dy  dz . (242) H1  polynomial from degenerate hypergeometric

Note that in our theory, that is suppose to do with time by conversion of ti  . Quantum equations of cosmological applications (as opposed to QCD and motion for field operators in the imaginary time read AQFT), one of the coordinates is singled out simply dd2ˆˆk 2 because the scale factor depends on it. This means that in kkˆ 22 30,  k (246) the classical sector of the theory time  , despite the fact d d a that it is imaginary, is singled out in comparison with the dd2ˆ ˆ k 2 3-spatial coordinates. In the quantum sector the  coor- kk 30, ˆ (247) 22dd  a k dinate also has a special status. Operators of graviton and ghost fields with nontrivial commutation properties where   aa . are defined over the space (242). Symmetry properties of Equations (246), (247) differ from (76), (77) by only space (242) allow us to define the Fourier images of the replacement of kk22 . At the level of analytic operators by coordinates x,,yz, and to formulate the properties of solutions of the equations this difference, of canonical commutation relations in terms of derivatives course, is crucial. However, formal transformations, not of operators with respect to the imaginary time  : dependent on the properties of analytic solutions to Equations (76), (77) and (246), (247), look quite similar. 3 ˆ  a d k Therefore, all operations to construct the equation for the ,ˆk ikk . (243) 4d   spectral function in imaginary time (analogue to Equa- tion (149)) and the subsequent construction of BBGKY 3 ˆ a dk ˆ chain coincide with that described in Section 5.1 with the ,, i 22 22 4d  kkk replacement of kk . Replacing kk changes  (244) the definition of moments only parametrically: instead of 3 ˆ a dk ˆ (151) we get ,.kkki 4d  n  k 2   Note that (243), (244) are introduced by the newly n 2 k a independent postulate of the theory, and not derived from  standard commutation relations (83), (87) by conversion ˆ ˆ 2,ˆˆ   gkk g gh kk gh (248) of ti  . (Such a conversion would lead to the dis-  appearance of the imaginary unit from the right hand n 0,1,2, , , sides of the commutation relations.) Thus, the imaginary dd2 time formalism can not be regarded simply as another 00 2 3. way to describe the graviton and ghost fields, i.e. as a d d mathematically equivalent way for real time description. Further actions lead obviously to the BBGKY chain In this formalism the new specific class of quantum (237), (238) and to the macroscopic Einstein equations phenomena is studied. (239). The system of self-consistent equations is produced by To solve Equations (246) and (247), we will be using variations of action, as defined in 4-space with a positive only the real linear-independent basis signature: ˆ 4, Qgˆ Phˆ 2 kk kkk 1dddd aaaNaaa22 2  S d3  22  ˆ  ˆˆ   kk4,qgkkphk (249)  NNd N dd d  1 3   1 a ddˆ ˆ ghkk hg k k. kk 2  a3 Nak ˆˆkk (245) 8ddk N  As will be seen below, one of the basic solutions satisfies 1 a3 ddˆˆ  the known definition of instanton: an instanton is a kkNak 2ˆ ˆ . kk solution to the classical equation, which is localized in 4ddk N   the imaginary time and corresponds to the finite action in Note that the full derivative with respect to the imaginary the 4-space with a positive signature. We will call the time is not excluded from Lagrangian. In (245) the operator functions (249) the quantum instanton fields of integrand contains the density of invariant gˆ Rˆ . The gravitons and ghosts. Operator constants of integration ˆ ˆ Lagrange multiplier N after the completion of the QPkk ,  and qpkk, satisfy commutation relations (101). variation procedure is assumed to be equal to unity. The Ladder operators are imposed by Equation (100) and system of equations corresponding to the action (245) then used in the procedure for constructing the state can also be obtained from the system of equations in real vectors over the basis of occupation numbers. State

Copyright © 2013 SciRes. JMP L. MAROCHNIK ET AL. 87 vectors of the general form in graviton and ghost sectors Expressions for the moments are simplified and read are already familiar structure (94) and (97). Only the 2 n interpretation of occupation numbers is changed: now it n k 2 nk41Ak 2 g . (255) is number of instantons nk , nk , nk of graviton, ghost k a and anti-ghost types, respectively. Direct calculation of the moments of the spectral func- Exact solutions, with the stable instanton configur- tion leads to the expression: ations, are described in the following Sections 7.2 and 7.3. In principle, for a limited imaginary time interval, 2 n n k there might be unstable configurations, but in the present 41 A gBh22 , (250) nk2 kkkwork such configurations are not discussed. (The ex- k a ample of the unstable instanton configuration see in where [53]). ˆ  Note that moments (255) can be obtained within the AQQqqkg kk  gg2 hˆˆkk gh  classical theory, limited, as generally accepted, to the g 21n cos (251) solutions localized in imaginary time. In doing so, Ak kg kk acts as a constant of integration of classical equation. 21n gh cos, The above approach is the quantum theory of in- kgh kk stantons in imaginary time. Here are present all the ele- ˆ  BPPppkg kk  gg2 hˆ kkˆ  gh ments of quantum theory: operator nature of instanton  field; quantization on the canonical commutation rela- 21cosn g tions; basic vectors in the representation of instanton kg kk (252) occupation numbers; state vectors of physical states in gh 21cos.nkghkk the form of superposition of basic vectors. With the quantum approach, a significant feature of instantons is The term containing products of basis functions gkkh is displayed, which clearly is not visible in the classical eliminated from (250) by the condition of homogeneity theory. It is the nature of instanton stable configurations of 3-space. In (251) and (252) average values of numbers as coherent quantum condensates. of instantons of ghost and anti-ghost types are assumed Construction of the formalism of the theory is com- to be equal: nnkgh kgh. One needs to pay atten- pleted by developing a procedure to transfer the results of n the study of instantons to real time. It is clear that this tion to the multiplier 1 in (250): the alternating procedure is required to match the theory with the experi- sign of moment s is a common symptom of instanton mental data, i.e. to explain the past and predict the futu re nature of the spectral function. of the Universe. As already noted, the procedure of Instanton equations of motion (246), (247) are of the transition to real time is not an inverse Wick rotation. hyperbolic type. This fact determines the form of asymp- This is particularly evident in the quantum theory: in totics of basis function for k   1 where   d a  (243), (244) the reverse Wick turn leads to the com- is conformal imaginary time. One of basis functions is mutation relations for non-Hermitian operators, which localized in the imaginary time and the other is in- can not be used to describe the graviton field. creasing without limit with the increasing of modulus of The procedure for the transition to real time has the the imaginary time status of an independent theory postulates. We will for- kk ee mulate this postulate as follows. ghkkk,,1.  (253) ak22ak 1) Results of solutions of quantum equations of motion In this situation, it is necessary to differentiate between (246), (247), together with the macroscopic Einstein’s stable and unstable instanton configurations. We call a Equation (239) after calculating of the moments (that is, configuration stable, if moments of the spectral function after averaging over the instanton state vector) should be are formed by localized basis functions only. Without represented in the functional form limiting generality, we assigned hk to the class of in-  a,,, , nn a,,,  . (256) creasing functions. It is easy to see that the condition of stability Bk  0 that eliminates contributions of hk 2) It is postulated that functional dependence of the from (250) is reduced to the condition of quantum co- moments of the spectral function on functions describing herence of instanton condensate: the macroscopic geometry must be identical in the real ggcosh  cos 1 and imaginary time. Thus, at the level of the moments of kkkk (254) the spectral function, the transition to the real time is ggh   kk1, cos kcos  k  1. reduced to a change of notation

aDaH,,,  ,H ,, , tunnelling processes form a macroscopic quantum state. (257) The Josephson effect is a characteristic example: fluc-   na,,, WaHHn ,,, . tuations of the electromagnetic field arise when a super- conductive condensate is tunnelling across the classically DaHH,,,  WaHH,,,  3) Moments  and 1  ob- impenetrable non-conducting barrier. Here, the tun- tained by operations (257), are substituted to right hand nelling can be formally described as a process develop- side of macroscopic Einstein equations that are consi- ing in imaginary time, but the fluctuations arise and exist dered now as equations in real time. Formally this means in the real space-time. Experimental data show that that the transition to the real time in the left hand side of regardless of the description, the tunnelling process Equation (239) is reduced to changing of the following forms a physical subsystem in the real space-time, with notations perfectly real energy-momentum.  H ,.22H (258) In Quantum Chromodynamics (QCD) physically similar phenomena are studied by sim ilar methods [49]. Thus, the acceptance of postulates (256)-(258) is equi- The vacuum degeneration is an internal property of QCD: valent to the suggestion that in real time the self-con- different classical vacuums of gluon field are not topo- sistent evolution of classic geometry and quantum ins- logically equivalent. In the framework of the classic dy- tanton system is described by the following equations namics any transitions between different vacuums are impossible. In that sense the topological non-equivalence 11  H DaHH,,, W1  aHH ,,, , plays role of the classical impenetrable barrier. There is 16 6 (259) an heuristic hypothesis in quantum theory—that the 2 11 3H DaHH,,, W1  aHH ,,, , probability of tunnelling transition between different 16 4 vacuums can be calculated as = ew S , where S is the under the condition that the form of functionals in right action of the classical instanton. The instanton is defined hand sides of (259) is established by microscopic cal- as a solution of gluon-dynamic equations localized in the culations in imaginary time. It is obvious also that in the Euclidian space-time connecting configurations with framework of these postulates any solution of equations different topologies. As in the case of Josephson Effect, consisting of BBGKY chain and macroscopic Einstein it is assumed that the tunnelling processes between equations (obtained without use of microscopic theory) topologically non-equivalent vacuums are accompanied can be considered as the solution in real time. by generation of non-perturbative fluctuations of gluon and quark fields in real space-time. Let us notice that in 7.1.3. Physics of Imaginary Time QCD the instanton solutions, analytically continued into Mathematical and physical motivation to look for the real space-time, are used to evaluate the amplitude of formalism of imaginary time comes from the fact that fluctuations. The fluctuations in real space-time are con- there are degenerate states separated by the classical sidered as a quark-gluon condensate (QGC). The exi- impenetrable barrier. In non-relativistic quantum mecha- stence of QGC with different topological structure in nics the barriers are considered, that have been formed “off-adrons” and “in-adrons” vacuums, is confirmed by by classical force fields and for that reason they have the comparison of theoretical predictions with experimental obvious interpretation. It is well known, that the cal- data. One of remarkable facts is that the carrier of appro- culation of quantum tunnelling across the classical im- ximately the half of nucleon mass is in fact the energy of penetrable barrier can be carried out in the following the reconstructed QGC. order: 1) the solution of classical equation of motion Now let us go back to the self-consistence theory of inside the barrier area is obtained with imaginary time; 2) gravitons. In that theory, due to its one-loop finiteness, from the solution obtained for the tunnelling particle, one all observables are formed by the difference between gra- calculates the action S for the imaginary time; 3) the viton and ghost contributions. That fact is obvious both tunnelling probability, coinciding with the result of the from the general expressions for the observables (see solution for Schrodinger equation in the quasi-classical (118), (119)), and from the exact and approximate so- approximation, is equal w  eS . Obviously, the se- lutions (described in the previous sections) as well. The quence described bears a formal character and cannot be same final differences of contributions may correspond interpreted operationally. Nevertheless, a strong argu- to the totally different graviton and ghost contributions ment toward the use of the formalism of imaginary time themselves. All quantum states are degenerated with in the quantum mechanics is the agreement between the respect to mutually consistent transformations of gra- calculations and experimental data for the tunnelling vitons and ghosts occupation numbers, but providing micro-particles. unchanged values of observable quantities. Thus the mul- A new class of phenomena arises in the cases when titude of state vectors of the general form, averaging

Copyright © 2013 SciRes. JMP L. MAROCHNIK ET AL. 89 over which leads to the same values of spectral function, We will work with the imaginary conformal time is a direct consequence of the i nternal mathematical    d a . The cosmological solution is: structure of the self-consistent theory of gravitons, sa- 1 tisfying the one-loop finiteness condition. aa e,    0. (261) 0  In that situation, it is very natural to introduce a hypothesis about the tunnelling of the graviton-ghost At the level of the BBGKY chain, due to the fact that the system between quantum states corresponding to the theory is invariant with respect to the Wick rotation, the same values of macroscopic observables. By the analogy calculations performed to get the solutions coincide with with the effects described above, one may suggest that 1) the those described in Section 5.3. At the microscopic the tunnelling processes unite degenerate quantum states level we use the exact solutions (246), (247) with the into a single quantum state; 2) tunnelling is accompanied background (261): by creation of specific quantum fluctuations of graviton 12 and ghost fields in real space-time. With regard to the ˆ Qgx Ph x, kkkak  mathematical method used to describe these phenomena, (262) today we may use only those methods that have been 12 ˆ qgx ph  x , tested in adjacent brunches of quantum theory. It is easy kkkak to see that this program has been realized in Sections where xk   0 , 7.1.1 and 7.1.2. We solve the equations of the theory for imaginary time, but the amplitude of the arising fluc- 11x x gx1e,hx   1e. tuations we evaluate by the analytical continuation (256)- xx  (258), analoguos to the ones used in QCD. The specific of our theory lie in the fact that at the final step of The expressions for the moments of the spectral function calculations we use the classical Einstein equation (259) are reduced to the form: 0 describing the evolution of the macroscopic space in real n  22nn 22 2 2 nkk1d2 xxAgBh . (263) time. The possibility of using these equations is deter-   mined by the action (245), which, when calculated by  means of the instanton solutions and averaged over the Equations for Akk, B are given in (251), (252). From state vector of instantons, is identically equal zero. As a (263) it is obvious that the self-consistent values matter of fact, after using instanton Equations (246) and n  const can be obtained only for a flat spectrum of (247) and averaging, the action (245) is reduced to the form: instantons. However, with the flat specter and Bk  0 , the second term in (263) creates a meaningless infinity. 1132 Therefore B  0 , and that, in turn, leads to the con- Sad3     . (260) k  16 dition (254), i.e. to quantum coherence of the instanton condensate. The quantitative characteristics of the con- The integrand in (260) is equal zero in the Einstein densate are formed by instantons only, localized in equations with imaginary time (239). The fact that imaginary time. wSexp 1 means that the macroscopic  It is easy to calculate of the converging integrals in evolution of the Universe is determined. That feature (263): allows the use of Equation (259), after the moments are 0 2 analytically continued into the real time. 22nx 1 2 d1xx  e  x 7.2. Instanton Condensate in the De Sitter Space  (264) 1 21!!21nnn 2. Among exact solutions of the one-loop quantum gravity, 221n a special status is given to De Sitter space if the space After analytical continuation into the real space-time, curvature of this space is self-consistent with the quan- following the rules (256)-(258), we obtain the final tum state of gravitons and ghosts. In Section 6.2, it was result: shown that in the self-consistent solution, gravitons and ghosts can be interpreted as quantum wave fields in real 12N DH inst 4 , space-time. Nevertheless, it should be mentioned, that 2 the alternating sign of the moments (228) points to a n1 1 possibility of instanton interpretation of that solution.  Wnn  2n 21!21nn 2 (265) Methods described in Sections 7.1.1 and 7.1.2, when 2 2N applied to De Sitter space, show that such interpretation inst Hn22n ,1, is really possible. 2

Copyright © 2013 SciRes. JMP 90 L. MAROCHNIK ET AL. where vacuum8. Nnn. (266) inst gh g 7.3. Instanton Condensate of Constant The comparison of the two models of graviton-ghost Conformal Wavelength condensate in the De Sitter space reveals some interest- The exact solution (186.II) has a pure instanton nature. ing features. In both cases we deal with the effect of Now we will obtain that solution with the value C3g  0 . quantum coherence. Expressions (265) differ from (228) One can rewrite the formulas (240), (241) for the ima- only in the formal substitution NNg  inst . However the ginary time: conditions leading to the quantum coherence are different 222 2 kka  in these models. According to (229), in the condensate of  00ln ,aa exp . (267) 2 a 0 4 virtual gravitons and ghosts, the average value of gra- a 0  viton and ghost occupational numbers are the same, and dd2ˆ ˆ kk kk22ˆ 0, the non-zero effect appears due to the fact that the phase d 2 0 d k correlation in the quantum superposition in the graviton’s (268) dd2ˆ ˆ and ghost’s sectors are formed differently. As it follows kkkk22ˆ 0. from (254), (266), in the instanton condensate the phases d 2 0 d k in the graviton and ghost sectors correlate similarly, but As we already know, the spatially homogeneous modes the non-zero effect appears due to the difference of participate in the formation of the solution for the average occupation numbers for graviton’s and ghost’s Equation (186.II). As follows from (268), when k 2  0 , instantons. The absence of the macroscopic structure of the description of the spatially homogeneous modes in the condensates does not allow the detection of the diffe- imaginary time does not differ from their description in rences by macroscopic measurements. In both cases the real time. The contribution from modes g2 is present 22 graviton-ghost vacuum possess equal energy-momentum in (267), with the relations CkCkgg40 0, 20  3 4 characteristics. taken into account. These relations are necessary to The question about the actual nature of the De Sitter provide the existence of the self-consistent solution. In space is lies in the formal mathematical domain. In these what follows we are considering the quasi-resonant modes circumstances one should pay attention to the following only. 22 facts. While describing the condensate of virtua l gra- For kk 0 , the signs of the last terms in the Equ- vitons and ghosts, we were forced to introduce an addi- ation (268) provide the existence of instanton solutions tional definition of the mathematically non-existent inte- we are looking for: grals (225), i.e. to introduce into the theory some ope- 4k rations that were not present from the beginning. It is the ˆ  0 k a additional operations that have provided a very specific 0  property of the solution—the alternating signs in the  22 Pˆ 22  Qkˆ Pˆ edkk0022 k e (269) sequence of the mom ents of the spectral function. By  kk0    0 k0  contrast, the theory of the instanton condensate has a   completely different formal mathematics. The theory is 16 12 a QPFaˆ ˆ ln , 2 kk motivated by the concrete property of the graviton-ghos t ka00 a0 system which is degeneration of quantum states, and the 4k construction of th e theory is constructed by the intro- ˆ 0 k  duction of mathematically non-contradictory postulates. a0 The moments of the spectral function’s with alternating   22 pˆ 22  signs is an internal property of the graviton-ghost in- qkpˆˆedkk0022 k e (270)  kk0   stanton theory. When we considered the instanton con-  0 k0  densate in the De Sitter space, no additional mathe- 16 a matical redefinitions were necessary (compare the for- qpFaˆˆln12 , ka2 kk a mulas (225) and (264)). We have the impression that the 00 0 instanton version of the De Sitter space is more mathe- 8A cosmological scenario based on this solution was proposed in [50]. matically comprehensive. Therefore, one may suggest In this scenario, birth of the flat inflationary Universe can be thought of that the key role in the formation of the De Sitter space as a quantum tunneling from “nothing”. As the Universe ages and is (the asymptotic state of the Universe) belongs to the emptied, the same mechanism of tunneling that gave rise to the empty Universe at the beginning, gives now birth to dark energy. The empty- instanton condensate, appearing in the tunnelling pro- ing Universe should possibly complete its evolution by tunneling back cesses between degenerated states of the graviton-ghost to “nothing”.

a 2 sity and pressure for the system of quasi-resonant and 2 da a0 Fa a0  . spatially homogeneous instantons: a 312aa 212 0 aaln 2 ln gh aa 8kN a 00   0 inst ln g aa22 a Calculations which follow contain the same mathema- 0 0 tical operations we have already described several times 2 kNgh kN g 01inst inst in the previous sections. After we remove contributors to aa22 a 2 the spectral function which contains F a , we obtain 01 the condition for the coherence of the condensate. Some 8kNgh a  0 inst ln , details of the calculations is r elated to the alternating 22 aa0 a0 signs of the moments, i.e. with the multiplier 1 n , (273)  gh characteristic for the instanton theory. Particularly, in the 8kN0 inst ea pg  22 ln expression for 1  g4 , there is a general si gn “minus” . 3aa0 a0 But, according to the Einstein equations in imaginary gh g 2 kN kN time   g40  . The positive sign of the first moment 01inst inst 1 222 is provided by the dominant contribution of ghost in- 3aa01 a stantons over the contribution of graviton instantons. 8kNgh ea With that taken into account, we obtain the final equa-  0 inst ln . 3aa22 a tions for the moments of quasi-resonant modes, ob- 0 0 tained after the analytic continuation into the real In Formula (273), the terms in brackets are eliminated by space-time: the condition (160), which is rewritten in terms of ma- gh 21n croscopic parameters n1 64Nkinst 0 a Wgn 41 22n ln  gh g aa0 a0 kN01inst kNinst 22 . (274) 2n aa n1 24k a 01 1ln,0 a2n a Solutions (273), (274) describe a quantum coherent con- 0 (271) gh densate of quasi-resonant instantons with the ghost do- 128Nk e14a Dg4l inst 0 n minance. The parameters of the condensate are in ac- 22 a aa0 0 cordance with parameters of a spatially homogeneous 48k 2 e14a condensate with graviton dominance.  0 ln . a2 a 0 8. Discussion Here the following definition has been used: From the formal mathematical point of view, the above 22 nn Nkkgh   , theory is identical to transformations of equations, deter- kgh kg k 2 inst 0 mined by the original gauged path integral (1), leading to 3ka2 exact solutions for the model of self-consistent theory of N gh  00. inst 8 gravitons in the isotropic Universe. To assess the validity of the theory, it is useful to discuss again but briefly the The graviton instantons are dominant for the spatially three issues of the theory that are missing in the original homogeneous modes: path integral. 16kNg 1) The hypothesis of the existence of classic spacetime Wg2, 1 inst 1 aa22 with deterministic, but self-consistent geometry is intro- 1 (272) duced into the theory. It is not necessary to discuss in g 32kN1 inst detail this hypothesis because it simply reflects the Dg2. 22 aa1 obvious experimental fact (region of Planck curvature and energy density is not a subject of study in the theory The parameter of the spatially homogeneous condensate under discussion). Note, however, that the introduction is defined as follows: of this hypothesis into the formalism of the theory leads gghto a rigorous mathematical consequence: the strict defi- nn00gg1 00 cos h1 0cos  0 nition of the operation of separation of classical and 22 Nkqqg  ,0. quantum variables uniquely captures the exponential pa- k 2 inst 00 rameterization of the metric.

2) The transfer to the one-loop approximation is con- shell into the structure of the theory. ducted in the self-consistent classical and quantum sys- a) Future theory that will unify quantum gravity with tem of equations. Formally, this approximation is of a the theory of other physical interactions may not belong technical nature because the equations of the theory are to renormalizability theories. If such a theory exists, it simplified only in order to obtain specific approximate may only be a finite theory. One-loop finiteness of quan- solutions. After classical and quantum variables are iden- tum gravity with no fields of matter that is fixed on the tified, the procedure of transition to the one-loop appro- mass shell [3] can be seen as the prototype of properties ximation is of a standard and known character [17]. In of the future theory. reality, of course, the situation in the theory is much b) Because of their conformal non-invariance and zero more complex and paradoxical. On the one hand, the rest mass, gravitons and ghosts fundamentally can not be quantum theory of gravity is a non-renormalized theory located exactly on the mass shell in the real Universe. (see e.g. [2]). Specific quantitative studies of effects off Therefore, the problem of one-loop finiteness off the one-loop approximation are simply impossible. On the mass shell is contained in the internal structure of the other hand, the quantum theory of gravity without fields theory. of matter is finite in the one-loop approximation [3]. The c) In formal schemes, which do not meet the one-loop latter means that the results obtained in the framework of finiteness, divergences arise in terms of macroscopic one-loop quantum gravity pose limits to its applicability physical quantities. To eliminate these divergences, one that is mathematically clear and physically significant. needs to modify the Lagrangian of the gravity theory, The existence of a range of validity for the one-loop entering quadratic invariants. This, in turn, leads to quantum gravity without fields of matter is a conse- abandonment of the original definition of the graviton quence of two facts. First, there are supergravity theories field that generates these divergences. The logical in- with fields of matter which are finite beyond the limits of consistency of such a formal scheme is obvious. (The one-loop approximation. Second, the quantum graviton mathematical proof of this claim is contained at Section field is the only physical field with a unique combination 10.2.) of such properties as conformal non-invariance and zero d) In the self-consistent theory of gravitons, one-loop rest mass. For this field only there is no threshold for the finiteness off the mass shell can be achieved only through vacuum polarization and particle creation in the isotropic mutual compensation of divergent graviton and ghost Universe. Therefore, in the stages of evolution of the contributions in macroscopic quantities. The existence of Universe, where gauges, automatically providing such a compensation, is H 22, Hm  (m is mass of any of the elementary par- an intrinsic property of the theory. ticles), quantum gravitational effects can occur only in From our perspective, the properties of the theory the subsystem of gravitons. It is also clear that in any identified in points a), b), c) and d), clearly dictate the future theory that unifies gravity with other physical need to use only the formulation of self-consistent theory interactions, equations of theory of gravitons in one-loop of gravitons, in which the condition of one-loop fini- approximation will not be different from those we dis- teness off the mass shell (the condition of internal con- cuss in this work. Therefore the self-consistent theory of sistency of the theory) is performed automatically. We gravitons has the right to lay claim be a reliable descrip- also want to emphasize that, as it seems to us, the scheme tion of the most significant quantum gravity phenomena of the theory given below has no alternative both logi- in the isotropic Universe. cally and mathematically. 3) The need to use the Hamilton gauge, which pro- Gauged path integral  choosing the Hamilton gauge, vides a transition from the path integral to the Heisen- which provides one-loop finiteness of the theory off mass berg representation [19], and then to the self-consistent shell of gravitons and ghosts  factorization of classic theory of gravitons in the macroscopic spacetime. It is and quantum variables, which ensures the existence of a important that in the Hamilton gauge the dynamic pro- self-consistent system of equations  transition to the perties of the ghost fields automatically provide one-loop one-loop approx im ation, taking into account the fun- finiteness of the theory off mass shell of gravitons and damental impossibility of removing the contributions of ghosts. The condition of one-loop finiteness off the mass ghost fields to observables—appears to us logically and shell largely determines the mathematical and physical mathematically as the only choice. content of the theory. Given that the main results of this As part of the theories preserving macroscopic space- work are exact solutions and exact transformations, the time being clearly one of its components, we see two evaluation of he proposed approach is reduced to a dis- topics for further discussions. The first of these is the cussion of this point of the theory. Let us enumerate once replication of the results of this work by mathematically more logical and mathematical reasons, forcing us to equivalent formalisms of one-loop quantum gravity. Here include the condition of one-loop finiteness off the mass we can note that, for example, in the formalism of the

Copyright © 2013 SciRes. JMP L. MAROCHNIK ET AL. 93 extended phase space with BRST symmetry, our results finiteness, leads to the discovery of a new class of phy- are reproduced, even though the mathematical formalism sical phenomena which are macroscopic effects of quan- is more cumbersome. The second topic is the reproduc- tum gravity. Like the other two macroscopic quantum tion of our results in more general theories than the one- phenomena of superconductivity and superfluidity, ma- loop quantum gravity without fields of matter. Here is croscopic effects of quantum gravity occur on the ma- meant a step beyond the limits of one-loop approxima- croscopic scale of the system as a whole, in this case, on tion as well as a description of quantum processes in- the horizon scale of the Universe. Interpretation of these volving gravitons, while taking into account the exi- effects is made in terms of gravitons-ghost condensates stence of other quantum fields of spin J  32. In the arising from the interference of quantum coherent states. framework of discussion on this topic, we can make only Each of coherent states is a state of gravitons (or ghosts) one assertion: in the one-loop N  1 supergravity con- with a certain wavelength of the order of the distance to taining graviton field and one gravitino field, the results the horizon and a certain occupation number. The vector of our work are fully retained. This is achieved by two of the physical state is a coherent superposition of vec- internal properties of N  1 supergravity: 1) The sector tors with different occupation numbers. of gravitons and graviton ghosts in this theory is exactly A key part in the formalism of self-consistent theory of the same as in the one-loop quantum gravity without gravitons is played by the BBGKY chain for the spectral fields of matter; 2) The physical degrees of freedom of function of gravitons, renormalized by ghosts. It is im- gravitino with chiral h 32 in the isotropic Universe portant that equations of the chain may be introduced at are dynamically separated from the non-physical degrees an axiomatic level without specifying explicitly field of freedom and are conformally invariant; 3) The gauge operators and state vectors. It is only necessary to assume of gravitino field can be chosen in such a way that the the preservation of the structure of the chain equations in gravitino ghosts automatically provide one-loop fini- the process of elimination of divergences of the moments teness of N  1 supergravity. As for multi-loop calcul- of the spectral function. Three exact solutions of one- ations in the N  1 supergravity and more advanced loop quantum gravity are found in the frame work of theoretical models, we have not explored the issue. BBGKY formalism. The invariance of the theory with Of course, a rather serious problem of the physical respect to the Wick rotation is also shown. This means nature of ghosts remains. The present work makes use in that the solutions of the chain equations, in principle, practice only of formal properties of quantum gravity of cover two types of condensates: condensates of virtual Faddeev-Popov-De Witt, which point to the impossibility gravitons and ghosts and co ndensates of instanton fluc- in principle of removing contributions of ghosts to ob- tuations. servable quantities off the mass shell. A deeper analysis All exact solutions, originally found in the BBGKY undoubtedly will address the foundations of quantum formalism, are reproduced at the level of exact solutions theory. In particular, one should point out the fact that for field operators and state v ectors. It was found that the formalism of the path integral of Faddeev-Popov-De exact solutions correspond to various condensates with Witt is mathematically equivalent to the assumption that different graviton-ghost microstructure. Each exact solu- observable quantities can be expressed through derivation we found is compared to a phase state of gravi- tives of operator-valued functions defined on the clas- ton-ghost medium; quantum-gravity phase transitions are sical spacetime of a given topology. On the other hand, introduced. finiteness of physical quantities is ensured in the axio- We suspect th a t the manifold of exact solutions of matic quantum field theory by invoking limited field one-loop quantum gravity is not exhausted by three solu- operators smoothed over certain small areas of spacetime. tions described in this paper. Search for new exact solu- Extrapolation of this idea to quantum theory of gravity tions and development of algorithms for that search, re- immediately brings up the question on the role of space- spectively, is a promising research topic within the proposed theory. Of great interest will also be approximate time foam [18] (fluctuations of topology on the micro- solutions, particularly those that describe non-equili- scopic level) in the formation of smoothed operators, and brium and unstable graviton-ghost and instanton con- consequently, observable quantities. To make this pro- figurations. blem more concrete, a question can be posed on collec- tive processes in a system of topological fluctuations that 9. Conclusions form the foam. It is not excluded that the non-removable Faddeev-Popov ghosts in ensuring the one-loop finite- 1) The equations of quantum gravity in the Heisenberg ness of quantum gravity are at the same time a pheno- representation and the equations of semi-quantum/semi- menological description of processes of this kind. classical self-consistent theory of gravitons in the mac- Study of equations of self-consistent theory of gra- roscopic Riemann space, respectively, can exist only in vitons, automatically satisfying the condition of one-loop the exponential parameterization and Hamilton gauge of

Copyright © 2013 SciRes. JMP 94 L. MAROCHNIK ET AL. the density of the contravariant metric; Astrophysics and Space Science, Vol. 34, 1975, pp. 249- 2) Equations of semi-quantum/semi-classical theory 263. doi:10.1007/BF00644797 necessarily contain the ghost sector in the form of a [10] L. S. Marochnik, N. V. Pelikhov and G. M. Vereshkov, complex scalar field providing one-loop finiteness to the Astrophysics and Space Science, Vol. 34, 1975, pp. 281- theory; 294. doi:10.1007/BF00644799 3) In case of isotropic Universe, in one-loop approxi- [11] L. R. W. Abramo, R. H. Brandenberger and V. F. Muk- mation the theory can be presented as a set of equations hanov, Physical Review D, Vol. 56, 1997, pp. 3248-3257. doi:10.1103/PhysRevD.56.3248 including Einsteins equations for the macroscopic metric with the energy-momentum tensor for gravitons and [12] L. R. W. 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Appendix To avoid mathematical contradictions that could arise at 10. Renormalizations and Anomalies the limit d  3 , Einstein equations in D-dimensional spacetime should be written down in exactly the form in In Sections 10.1 and 10.2, we discuss a self-consistent which they were obtained from the variational principle: theory of gravitons in isotropic Universe with the ghost sector not taken into account. As has been repeatedly 110 gddRR0  d stated, we believe that such a model is not mathema-  d 2 tically sound. Gauges, completely removing the dege- 1 dd1 ad 32 a neracy, are absent in the theory of gravity. Thus, in the 2 self-consistent theory of gravitons the ghost sector is d 1 inevitable present. Now, however, let us assume for the akd 12ˆˆˆ ˆ , 8  gkk kk  g moment that the self-consistent theory of gravitons with- d k (276) out ghosts is worth at least as a model of mathematical d 1  gR physics. The purpose of this Section is to get the pro- 2 dd perties of this model and to show that it is mathema- d tically and physically internally inconsistent. 1 dd232 dd12a a   d3 a a 2 d 10.1. Gravitons with No Ghosts. Vacuum d 1  akd 12ˆˆˆˆ   , Einstein Equations with Quantum  gkk kk g 8 k Logarithmic Corrections d a It clear from the outset that in the non-ghost model the 2 ˆˆˆkkk dk10. (277) calculation of observables will be accompanied by the a emergence of divergences. It is therefore necessary to Here  d is the Einstein gravitational constant in D - formulate the theory in such a way that the regularization D2 dimensional spacetime (Dimension  d   l ). The and renormalization operations are to be contained in its left hand sides of Equation (276) satisfy the Bianchi mathematical structure from the very beginning. We talk identity: here about changes in the mathematical formulation of 1 the theory. The relevant operations should be introduced dd1  ad 32 a into the theory with care: first, in the amended theory, 2  d (278) coexistence of classical and quantum equations should be 1 a ensured automatically; second, the enhanced theory  dd12 add232 a  d 3 a a 0. 2 a  should not contain objects initially missing from the d theory of gravity. In the right hand side of Equation (276), the identity (278) The dimensional regularization satisfies both above- generates condition of the graviton EMT conservation mentioned conditions. Important, however, is the follow- that satisfies if the equations of motion (277) are taken ing fact: the use of dimensional regularization suggests into account. Regarding the origin of the system of Equa- that the self-consistent theory of gravitons in the iso- tions (276) and (277), we should make the following tropic Universe is originally formulated in a spacetime of comment. In this case it is inappropriate to invoke the dimension D = 1+ d, where 1 is the dimension of time; reference to the path integral and factorization of its d 32 is the dimension of space. The special status measures because the path integral inevitably leads to the of the time is due to the two factors: 1) all the events in theory of ghosts interacting with the macroscopic gravity. the Universe, regardless of its actual dimension, are or- We can only mention a heuristic recipe: one should refer dered along the one-dimensional temporal axis; 2) the to the density of Einstein equations with mixed indices, canonical quantization of the graviton field in terms of define the exponential parameterization of the metric, the commutation relations for generalized coordinates and expand the equations into a series of metric fluc- and generalized momenta also presuppose the existence tuations with an accuracy of the second-order terms. De- of the one-dimensional time. As for the space dimension, viations from this recipe (for example, linear parame- the limit transition to the true dimension d = 3 is imple- ˆ terization gˆik gh ik ik ) lead to a system of inconsistent mented after the regularization and renormalization. classical and quantum equations. To remove this sort of Thus, we are working in a space with a metric inconsistency, one is forced to use artificial transactions 22 2   outside the formalism of the theory (see, for example, ds a d dx dx ,, d [11]). da a 2 (275) d 1 While working with the system of Equations (276) and gaRdd,2 22d3. aaa (277), we face with two mathematical problems. The first

Copyright © 2013 SciRes. JMP L. MAROCHNIK ET AL. 97 problem is that in the framework of that system of parameter of the asymptotic expansion is the effective equations, except in very special cases, it is impossible to frequency formulate the dynamics of operators on a given back- 22k , ground that is to get the solution of the Equation (277) as k an accurate operator function of time. This is due to the dd11aa  2 (279)  aR2 23d . fact that formulae of (276) in reality are not yet specific d 2 44daa equations. They are only a layout of Einstein equations with radiation corrections. These equations can only be In this method, the integrals over the wave numbers can be obtained after regularization and renormalizations of the defined in terms of the principal value. Contributions of ultraviolet divergences. In addition, the functional form the poles at k   can not be mathematically of equations depends on which quantum gravitational verified if only because there are such contributions from effects are to be taken into account outside the sector of each term of the infinite asymptotic series. The inability vacuum (i.e. zero) fluctuations of the graviton field. The to describe infrared effects is the principal disadvantage only possible way to study the system of Equations (276) of a theory with divergences, which uses only asymptotic and (277) is 1) to obtain the solution of operator equation expansions with respect to the wavelength. Meanwhile, (277) in a form of a functional of the scale factor without as general considerations and the results of this work specifying the dependence on a  with a clear em- show, in the physics of conformal non-invariant massless phasis on zero fluctuations in this functional; 2) to field the most interesting and innovative effects occur in substitute the obtained functional in (276) under certain the infrared spectrum. The method of describing these assumptions about the state vector; 3) to regularize and effects, based on the exact BBGKY chain, can not be renormalize and finally 4) to solve the macroscopic Ein- used in the theory with divergences, because a method stein equations, obtained after these operations. Imple- regularizing the infinite chain of moments of the spectral mentation of the program, an essential element of which function does not exist. is the allocation of zero fluctuations generating ultra- The above problems automatically reduces the interest violet divergences, is possible only when using the me- toward the theory with divergences. However, given that thod of asymptotic expansions of solutions of operator all previous works in this area have been implemented in equation in the square of wavelength of the graviton the framework of regularization and renormalization, let modes. Thus, the problem of the lack of macroscopic us conduct our analysis to the end. In calculations, it is Einstein equations in the original formulation of this enough to consider the equation for the convolution. After theory with divergences limits the methods of this theory identity transformations, using the equation of motion to the short-wave approach. Note that this fact was clear- (277), we get ly indicated by DeWitt [17]. 1 dd232 The second problem is related to the infrared insta- dd12 a a  d3 a a   2   bility of the theory, with the object of the theory being a d (280) conformal non-invariant massless quantum field. The d 1 d 1  Wak , problem is due to the fact that not every representation of 16 d k the asymptotic series can be substituted into energy- where momentum tensor to perform the summation over the  wave numbers. For example, if in the explicit form, a Wkgkkg ˆˆ term in the asymptotic series contains a large parameter k 2n in the denominator, then starting from n  2 in the is the spectral function of gravitons. The calculation of integration over the wave numbers the infrared diver- the spectral function by the method of asymptotic ex- gences will appear. Such an asymptotic series can not be pansion with respect to the square of wavelength was used even for the renormalization of ultraviolet diver- described in Section 4.1. Now we need to repeat this gences, because when it is used in the space of the calculation excluding the ghosts, but with input from physical dimension d  3 , the logarithmic divergences zero fluctuations in the spacetime of dimension D = d + 1. arise simultaneously at the ultraviolet and the infrared The relevant calculations do not require additional com- limits. In the method of dimensional regularization the ments. A spe ct ral functi on is represented as: problem is reduced to the fact that it is impossible to WWvac W  exc , (281) choose an interim dimension d in a way such that the kk k  vac integral exists at both limits. where Wk is the vacuum component of the spectral exc Formally, the technical problem described above is function and Wk is the spectral function of excita- partly solved by reformatting the asymptotic series. In tions. After passage to the limit d  3 , the contribution exc particular, the following method will be used, in which of Wk to the EMT of short gravitons is exactly the

Copyright © 2013 SciRes. JMP 98 L. MAROCHNIK ET AL. same as (129), (130). In the future, we discuss only the equations is written in D-covariant form: contribution from vacuum components of the spectral 1 function. In the calculations, we must keep in mind that RRkk  id 2 i d in the d -dimensional space the number of internal de- d 1 grees of freedom of transverse gravitons is 2  d dd12 d 13  d  wdg 122 d  . The solution for the vacuum d 1  22d  2 spectral function is expressed in terms of the functional 2 d 2 (124): ;;kl  dd11 d1 kkk 1 vac  RR22 R RR2 W ddii  id d d  k  d 1   ;;il  

4 d  dd12  0. d 1 (282) a 4 k (285)   dd12 Equation (285) are obtained by the variation of action 4 d  s ˆ s  11.Jk d 1  D a 4k s0 Sgx d vac  d ˆ s The powers of operator Jk 1 are d efined by formulas d 1  d 1  (286) 2  dd212 (125), in which k has the form (279). After sub-  13 d 2  RRd d 1 d . stitution of (282) into (280), the zero-term in the asymp-  22 22d   d 2 d 2 totic expansion creates an integral, calculated by the rules    of dimensional regularization: It is obvious from (285), (286) that the method of di- 1 mensional regularization retains overall covariance of the   theory. Of course, quantum corrections, appearing in k k (285), satisfy the condition of conservation. 12dd21 kk d  (283) Renormalization and removal of regularization (limit d  d 2  2 12 d  3 ) are held at the level of action. A parameter with 2 0 k   the dimension of length, which will eventually acquire 32d the status of renormalization scale, is contained within   d 12. d 1 d 12 the theory. This parameter, referred to as L , is appears 21d g in the D -dimensional constant of gravity: The  -function in (283) diverges for d  3 . Therefore, Ld 3. (287) calculation of the integral (283) and transformation of dg expressions with  -functions are carried out with those The technique of removal the regularization assumes values of d which provide the existence of the integral conservation of dimensionality for those objects in which and  -functions. At the final stage, the result of these the limit operation is performed. There are two such calculations is analytically continued to the vicinity objects: the measure of integration d and the density d  3 . All other terms of the asymptotic expansion (282) of the Lagrangian  . As can be seen from (286), (287), generate finite integrals and do not require a di- the first (Einstein) term of the action is written down as mensional regularization. For reasons of heuristic ra- 1 ther than mathematical nature, it is considered that SR11 d, (1) , vac  d these terms are negligible compared to the contribution 2 (288) of the principal term of the asymptotic expansion (see 4DD dgLdx  g , below the effective Lagrangian (293)). Convolution of d D-dimensional Einstein’s Equation (280), containing where D -dimensional objects  and d have the the main term of the vacuum EMT of gravitons, has the same dimensions as the corresponding 4-dimensional form: objects. In this sector of the theory the limit transition is trivial: RR , dgdx  4 . In the sector of 1 dd232 d dd12 a a  d 3 a a  quantum corrections to the Einstein theory, we introduce 2 d (284) the same measure and obtain the density of the Lagran- dd123  d  d 12 gian: ad 1 . d 3 d 12 d 1 2 a d 1 2   d 3  2 d 1 2 Ldg 21 d 3 d R 2 (289) Other Einstein equations can be obtained using the Bi- d 1 d 22d  2 anchi identities. A complete system of Einstein vacuum 2 d 2

It is necessary to emphasize that the operations of re- 2882 22 exp normalizations and removal of regularization have to be gg f 2 mathematically well-defined and generally-covariant. The condition of mathematical certainty assumes that the is the renorm-invariant scale. There is a heuristic argu- renormalization is conducted before the lifting of regu- ment allowing to use the obtained expression: quantum larization. At the same time, the general-covariance of corrections in the Lagrangian (ref (12.19)) dominate over the procedure is automatically fulfilled if the counter- all other neglected terms of the asymptotic series over 2 terms imposed in the Lagrangi an are the D -dimensional The logarithmic parameter lng R  1 . invariants. Note also that if the mathematical value is The renormalized Einstein vacuum equations with finite at d  3 , then the abov e formulated conditions do quantum corrections obtained from the Lagrangian (293) not prevent the expansion of this quantity in a Taylor are as follows: series over the parame ter 32 d . In particular, we kk1 can write: RRii  2 d 1 d 3  22;;kl d 3222 2  LR R LR   ggk ggd dd  2 RRlni ln (294) (290) 288 RR 3  d  2  ;;il 2 g Rd 1ln, 2  11g  2 Rd kk22k  RRii Rlni R  0. 48R  where g  1 Lg ; ellipsis designate the terms which do not contribute to the final result. The substitution (290) in Note that exactly the same equations are obtained from (289) provides: D-dimensional Equation (285), provided that the operations are performed in the same sequence: first a d 1 2 renormalization with the introduction of D-covariant 2 dd21 3 d  2 d 1 Rd counter-terms is conducted, and then a limit transition to 22d  2 2 d 2 the physical dimension is performed. (291) d 1 2 2 dd21 5  d g 2  10.2. Intrinsic Contradiction of Theory with No d 1 Rd ln . 22d  2 R Ghosts: Impossibility of One-Loop 2 d 2 d  Re normalization According to (291), the source Lagrangian of the theory We are still discussing a formal model—self-consistent requires a D-invariant counter-term, which removes the theory of gravitons with no ghosts. In the previous sec- contribution proportional to the diverging  -function it was shown that the renormalization of divergences, tion: that inevitably arise in this model, requires the imposition d 1 of an additional term quadratic in the curvature in the 2 2 dd21 3  d  2 Lagrangian. It is now necessary to draw attention to two 0 d 1 Rd 22d  2 mathematical facts: 1) the need for a modification of 2 d 2 (292) Einstein theory is caused by quantum effects contained in   R2 . the Lifshitz operator Equation (277); 2) the original La- 4 f 2 d grangian and operator equations of the modified theory have the form: In (292), there is a new finite constant of the theory of gravity 1 f 2 . The removal of the regularization in the 1   RRˆˆ 24 gˆd,x (295) renormalized Lagrangian is conducted by the regular 2 2 4 f transition:  11 122   gˆ RRˆˆkk ren lim0  ii d 3  2 2 1 g 22  kkl k1 k2  RR 2  2 Rln (293) DDˆˆ R ˆ DDR ˆˆˆ RR ˆˆ R ˆ (296) 2 41f 152 R 2 iilii f 4  2 1  2 g  0, RR  2 ln , 2 1152 R ˆ where Di is a covariant derivative in a space with the where operator metric gˆik . It is quite obvious that these facts

Copyright © 2013 SciRes. JMP 100 L. MAROCHNIK ET AL. contradict each other: the quantum effects in the Lifshitz um Equation (296), after their polynom ial expansion in equation lead to a theoretical model that contradicts the powers of curvature, look as follows (fi nite logarithmic Lifshitz equation. Let us demonstrate that the contradic- corrections are omitted): tion is a direct consequence of the non-renormalizability kk1 of the model (295) off the graviton mass shell. RR  ii2 Equation (296), after their linearization describe quantized waves of two types—tensor and scalar. It makes  11;;kklk k2 22R;iiR;lRR i iR sense to discuss the problem of the scalar modes only in 288 f0 4  the event that at least preliminary criteria for consistency 2  of modified the ory will be obtained. Therefore, first of all,  22 we should reveal properties of the tensor modes. Here is 48 f0 an expression for the Lagrangian of a system consisting ;;lk k;; lm k ; l11 ; k k; l RRRRRRRR;;li i;; lm i;; l i i; l 0. of self-consistent cosmological field and tensor gravi- 24 tons: (301)  22 2 3  39aaNaa  Here   1  is a divergent  -function obtained StNad    2  Na22 fN24aNaa2 by dimensional regularization; 1 f is a seed constant   0 of a theory with quadratic invariant. The complete quan- 11 6 aNaa   2 tum Lagrangian corresponding to Equation (301) has the 22 2 (297) 8  Nf aNaa form: 1 ddˆ  ˆ k 2   11 kkˆˆ .  RRˆ   ˆ 2 22kk   22 k Naddtt   2 1152 4 f0   (302) The equation for gravitons is produced either by the   2;4RRˆˆl gˆd. x linearization of the Equation (296), or from (297) by the 22 ;l  192 f0  variation procedure: Renormalization of the second term in (302) is per-  a ˆˆˆ  2 formed by selecting the seed constant: 122 Rkkkk f a (298) 11   . Rˆ 0. f 22288 f 2 f 2 k 0 However, a divergent coefficient forms before the third Please note that the last term in (298) makes it im- term. To overcome this divergence, it is necessary to possible to retain the Lifshitz equation. After the trans- introduce a new seeding “fundamental” constant of the formation 2 modified theory of gravity 1 h0 with a renormalization 12 12 rule: ˆk aR1  fˆk 111 Equation (298) has a form     22 222. hf0 48 288 h R ˆˆ 22 kkaP k 0. (299) Further actions are obvious and pointless: Lifshitz 6 equation is the subject of the next modification; quantum In (299), the deviation from the Lifshitz equation is corrections generate another new divergence; to renor- manifested in the effective frequency of gravitons—the malize the new divergence a new theory of gravity is latter contains an additional function of curvature’s deri- introduced, etc. The only conclusion to be drawn from vatives this procedure is that based on the criteria of quantum field theory, the one-loop self-consistent theory of gra- 11;l PRln 1  f2  vitons in the isotropic Universe, and not possessing the 24;l (300) property of one-loop finiteness outside of mass shell, ;l ln 1Rf22 ln 1Rf . does not exists as a mathematical model. In such a theory  ;l it is impossible to quantitatively analyze any physical When calculating quantum corrections to the macro- effect. The theory of gravitons without ghosts is non- scopic equations, the modification of the effective fre- renormalizable even in the one-loop approximation. It is quency leads to additional divergences. Averaged vacu- also important to stress that the correct alternative to a

Copyright © 2013 SciRes. JMP L. MAROCHNIK ET AL. 101 non-renormalizable theory is only a finite theory with the quantum theory of gravity should be obvious from the graviton-ghost compensation of divergences. latest development trends in the theories of supergravity In the future, from our perspective, the method of and superstrings. On the other hand, however, in all regularization and renormalization in general will be works we know on cosmological applications of one- excluded from the arsenal of quantum theory of gravity, loop quantum gravity theoretical models are used, which, including one from the theory of one-loop quantum ef- according to the criteria of quantum field theory, do not fects involving matter fields. Correct alternatives to exist. We cannot comment on the specific results ob- existing methods of analysis of these effects to be found tained in these models by the reasons clear from the con- in extended supergravities, finite at least in one-loop tent of this section. Once again we should emphasize that approximation. the self-consistent theory of gravitons, if it exists as a The situation prevailing in the scientific literature is a theoretical model, must be finite outside the mass shell of paradoxical one. On the one hand, inadequate nature of gravitons. Effects arising in the finite theory are des- the regularization and renormalization methods in the cribed in the main text of this work.