Graviton, Ghost and Instanton Condensation
On Horizon Scale of Universe. Dark Energy as
À Macroscopic Effect of Quantum Gravity

Leonid Marochnik Physics Department, University of Maryland, College Park, MD 20742, USA    Daniel Usikov 36477 Buckeye St., Newark, CA 94560, USA    Grigory Vereshkov Research Institute of Physics, Southern Federal University, Rostov-on-Don 344090, Russia

We show that cosmological acceleration (Dark Energy effect) falls into to a special class of quantum phenomena that occur on a macroscopic scale. Dark Energy is the third macroscopic quantum effect, the first after the discovery of superfluidity and superconductivity. Dark Energy is a phenomenon of quantum gravity on the scale of the Universe as a whole at any stage of its evolution, including the contemporary Universe. The effect is 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. Therefore, no hypothetical fields or ”new physics” are needed to explain the Dark Energy effect. This macroscopic effect of one–loop quantum gravity takes place in the empty isotropic non–stationary Universe as well as in such a Universe filled by a non–relativistic matter or/and radiation. The effect is due to graviton–ghost condensates arising from the interference 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 – De Witt gauged path integral 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 choice of ghost sector, satisfying the condition 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. Quantum–gravity phase transitions are introduced. In the formalism of the BBGKY chain, the generalized self–consistent theory of gravitons is presented, taking into account the contribution of non–relativistic matter in the formation of a common self–consistent gravitational field. In the framework of this theory, it is shown that the era of non–relativistic matter dominance must be replaced by an era of dominance of graviton–ghost condensate. Pre–asymptotic state of Dark Energy is a condensate of virtual gravitons and ghosts with a constant conformal wavelength. The asymptotic state predicted by the theory is a self–polarized graviton–ghost condensate of constant physical wavelength in the De Sitter space. The Dark Energy phenomenon of such a nature is presented in the form of GCDM model that interpolates the exact solutions of equations of one–loop quantum gravity. The proposed theory is compared with existing observational data on Dark Energy extracted from the Hubble diagram for supernovae SNIa. We show that GCDM model has advantages over CDM model if criteria for the statistical probability are in use. Result of processing of observational data suggests that the graviton–ghost condensate is an adequate variable component of Dark Energy. We show that its role was significant during the era of large–scale structure formation in the Universe.

Dark energy 95.36.+x; Quantum gravity 04.60.-m

I  Introduction

Macroscopic quantum effects are quantum phenomena that occur on a macroscopic scale. To date, there are two known macroscopic quantum effects: superfluidity at the scale of liquid helium vessel and superconductivity at the scale of superconducting circuits of electrical current. These effects have been thoroughly studied experimentally and theoretically understood. A key role in these effects is played by coherent quantum condensates of micro-objects with the De Broglie wavelength of the order of macroscopic size of the system. The third macroscopic quantum effect under discussion in this paper is condensation of gravitons and ghosts in the self–consistent field of the expanding Universe. Hypotheses on the possibility of graviton condensate formation in the Universe proposed by Hu 01 () and Antoniadis–Mazur–Mottola 02 () in a general form. A description of these effects by an adequate mathematical formalism is the problem at the present time.

We show that condensation of gravitons and ghosts is a consequence of quantum interference of states forming the coherent superposition. In this superposition, quantum fields have a certain wavelength, and with different amplitudes of probability they are in states corresponding to different occupation numbers of gravitons and ghosts. Intrinsic properties of the theory automatically lead to a characteristic wavelength of gravitons and ghosts in the condensate. This wavelength is always of the order of a distance to the horizon of events111Everywhere in this paper we discuss quantum states of gravitons and ghosts that are self–consistent with the evolution of macroscopic geometry of the Universe. In the mathematical formalism of the theory, the ghosts play a role of a second physical subsystem, the average contributions of which to the macroscopic Einstein equations appear on an equal basis with the average contribution of gravitons. At first glance, it may seem that the status of the ghosts as the second subsystem is in a contradiction with the well–known fact that the Faddeev–Popov ghosts are not physical particles. However the paradox, is in the fact that we have no contradiction with the standard concepts of quantum theory of gauge fields but rather full agreement with these. The Faddeev–Popov ghosts are indeed not physical particles in a quantum–field sense, that is, they are not particles that are in the asymptotic states whose energy and momentum are connected by a definite relation. Such ghosts are nowhere to be found on the pages of our work. We discuss only virtual gravitons and virtual ghosts that exist in the area of interaction. As to virtual ghosts, they cannot be eliminated in principle due to lack of ghost–free gauges in quantum gravity. In the strict mathematical sense, the non–stationary Universe as a whole is a region of interaction, and, formally speaking, there are no real gravitons and ghosts in it. Approximate representations of real particles, of course, can be introduced for shortwave quantum modes. In our work, quantum states of shortwave ghosts are not introduced and consequently are not discussed. Furthermore, macroscopic quantum effects, which are discussed in our work, are formed by the most virtual modes of all virtual modes. These modes are selected by the equality , where is the wavelength, is the Hubble function. The same equality also characterizes the intensity of interaction of the virtual modes with the classical gravitational field, i.e. it reflects the essentially non–perturbative nature of the effects. An approximate transition to real, weakly interacting particles, situated on the mass shell is impossible for these modes, in principle (see also the footnote on p. 6)..

In this fact, a common feature of macroscopic quantum effects is manifested: such effects are always formed by quantum micro–objects, whose wavelengths are of the order of macroscopic values. With this in mind, we can say that macroscopic quantum gravity effects exist across the Universe as a whole. The results of this work suggest that the existence of the graviton–ghost condensate is directly responsible for the Dark Energy effect, i.e. for the observational data of the acceleration of the expansion of the Universe 03 (); 04 (). Most significantly, a graviton–ghost condensate formation is direct consequence of the first principles of the theory of gravity and quantum field theory, so that no hypothetical fields are needed to explain the Dark Energy effect.

Quantum theory of gravity is a non–renormalized theory and for this reason it is impossible to calculate effects with an arbitrary accuracy in any order of the theory of perturbations. The program combining gravity with other physical interactions within the framework of supergravity or superstrings theory assumes the ultimate formulation of the theory containing no divergences. Today we do not have such a theory; nevertheless, we can hope to obtain physically meaningful results. Here are the reasons for this assumption.

First, in all discussed options for the future theory, Einstein’s theory of gravity is contained as a low energy limit. Second, from all physical fields, which will appear in a future theory (according to present understanding), only the quantum component of gravitational field (graviton field) has a unique combination of zero rest mass and conformal non–invariance properties. Third, physically meaningful effects of quantum gravity can be identified and quantified in one–loop approximation. Fourth, as was been shown by t’Hooft and Veltman 3 (), the one–loop quantum gravity with ghost sector and without fields of matter is finite. For the property of one–loop finiteness, proven in 3 () on the graviton mass shell, we add the following key assertion. All one–loop calculations in quantum gravity must be done in such a way that the feature of one–loop finiteness (lack of divergences in terms of observables) must automatically be implemented not only on the graviton mass shell but also outside it.

Let us emphasize the following important fact. Because of conformal non–invariance and zero rest mass of gravitons, no conditions exist in the Universe to place gravitons on the mass shell precisely. Therefore, in the absence of one–loop finiteness, divergences arise in observables. To eliminate them, the Lagrangian of Einstein’s theory must be modified, by amending the definition of gravitons. In other words, in the absence of one–loop finiteness, gravitons generate divergences, contrary to their own definition. Such a situation does not make any sense, so the one–loop finiteness off the mass shell is a prerequisite for internal consistency of the theory.

These four conditions provide for the reliability of theory predictions. Indeed, the existence of quantum component of the gravitational field leaves no doubt. Zero rest mass of this component means no threshold for quantum processes of graviton vacuum polarization and graviton creation by external or self–consistent macroscopic gravitational field. The combination of zero rest mass and conformal non–invariance of graviton field leads to the fact that these processes are occurring even in the isotropic Universe at any stage of its evolution, including the contemporary Universe. Vacuum polarization and particle creation belong to effects predicted by the theory already in one–loop approximation. In this approximation, calculations of quantum gravitational processes involving gravitons are not accompanied by the emergence of divergences. Thus, the one–loop finiteness of quantum gravity allows uniquely describe mathematically graviton contributions to the macroscopic observables. Other one–loop effects in the isotropic Universe are suppressed either because of conformal invariance of non–gravitational quantum fields, or (in the modern Universe) by non–zero rest mass particles, forming effective thresholds for quantum gravitational processes in the macroscopic self–consistent field.

Effects of vacuum polarization and particle creation in the sector of matter fields of spin were well studied in the 1970’s by many authors (see 52 () and references therein). The theory of classic gravitational waves in the isotropic Universe was formulated by Lifshitz in 1946 13 (). Grishchuk 40 () considered a number of cosmological applications of this theory that are result of conformal non–invariance of gravitational waves. Isaacson 41 () has formulated the task of self–consistent description of gravitational waves and background geometry. The model of Universe consisting of short gravitational waves was described for the first time in 20 (). The energy–momentum tensor of classic gravitational waves of super long wavelengths was constructed in 42 (); 43 (). The canonic quantization of gravitational field was done in 11 (); 44 (); HuParker1977 (). The local speed of creation of shortwave gravitons was calculated in 45 (). In all papers listed above, the ghost sector of graviton theory was not taken into account. One–loop quantum gravity in the form of the theory of gravitons defined on the background spacetime was described by De Witt 7 (). Calculating methods of this theory were discussed by Hawking 37 (). For the first time, the approach based on the fact that the ghost sector of graviton theory is determined by the condition of one–loop finiteness off the mass shell is presented in the present paper.

One–loop finiteness provides the simplicity and elegance of a mathematical theory that allows, in turn, discovering a number of new approximate and exact solutions of its equations. This paper is focused on three exact solutions corresponding to three different quantum states of graviton–ghost subsystem in the space of the non–stationary isotropic Universe with self–consistent geometry. The first of these solutions describes a coherent condensate of virtual gravitons and ghosts; the second solution describes a coherent condensate of instanton fluctuations. The third solution describes the self–polarized condensate in the De Sitter space. This solution allows interpretation in terms of virtual particles as well as in terms of instanton fluctuations. All three solutions are directly related to the physical nature of Dark Energy.

The principal nature of macroscopic quantum gravity effects, the need for strict proof of their inevitability and reliability impose stringent requirements for constructing a mathematical algorithm of the theory. In our view, existing versions of the theory of gravitons in macroscopic spacetime with self–consistent geometry do not meet these requirements. In this connection, note the following fact. Because of conformal non–invariance, the trace of the graviton energy–momentum tensor is not zero simply by definition of graviton field. Naturally, the information on macroscopic quantum effects (that are the subject of study in this paper) is contained in this trace. After the first publication of preliminary results of our research MUV () we feel that a number of problems under discussion needs a much more detail description. Due to some superficial similarities, effects of graviton and ghost condensation in the De Sitter space are perceived sometime as another method of description of conformal anomalies, which are calculated by a traditional method of regularization and renormalization. (Calculations of such anomalies see in, e.g., 48 (); 49 (); 50 (); 51 (); TW (); MV ().) We would like to emphasize that such analogies have neither physical nor mathematical basis. Conformal anomalies describe the effect of reconstruction of the spectrum of zero oscillations of quantum fields. Their contributions to the energy–momentum tensor are independent of state vector of the quantum field. They are parameterized by numerical coefficients of the order of unity that are factors of quadratic form of the curvature. Conformal anomalies that are microscopic quantum effects are able to contribute to the macroscopic evolution of the Universe only if its parameters are close to Planck ones. In contrast, the graviton and ghost condensation is a macroscopic quantum effect in the contemporary Universe. It is quantitatively described by macroscopic parameters that govern the structure of a state vector. These parameters are averaged numbers of quanta in coherent superpositions. Note also that in all papers known to us, calculations of anomalies were conducted in the framework of models with no one–loop finiteness off the graviton and ghost mass shell. It is shown in Appendices XII.1, XII.2 that such models are internally inconsistent. The rigorous theory of one–loop quantum gravity presented here shows that the effect of conformal anomalies on gravitons must be zero (see Appendix XII.3). In the empty Universe where the only gravitons exist, the De Sitter space can be formed by only a graviton–ghost condensate.

Sections II and III are devoted to the derivation of the equations of the theory with a discussion of all the mathematical details. In Section II, we start with exact quantum theory of gravity, presented in terms of path integral of Faddeev–Popov 5 () and De Witt 4 (). Key ideas of this Section are the following. (i) The necessity to gauge the full metric (before its separation into the background and fluctuations) and the inevitability of appearance of a ghost sector in the exact path integral and operator Einstein’s equations (Sections II.1 and II.2); (ii) The principal necessity to use normal coordinates (exponential parameterization) in a mathematically rigorous procedure for the separation of classical and quantum variables is discussed in Sections II.3 and II.4; (iii) The derivation of differential identities, providing the consistency of classical and quantum equations performed jointly in any order of the theory of perturbations is given Section II.5. Rigorously derived equations of gauged one–loop quantum gravity are presented in Section II.6.

The status of properties of ghost sector generated by gauge is crucial to properly assess the structure of the theory and its physical content. Let us immediately emphasize that the standard presentation on the ghost status in the theory of –matrix can not be exported to the theory of gravitons in the macroscopic spacetime with self–consistent geometry. Two internal mathematical properties of the quantum theory of gravity make such export fundamentally impossible. First, there are no gauges that completely eliminate the diffeomorphism group degeneracy in the theory of gravity. This means that among the objects of the quantum theory of fields inevitably arise ghosts interacting with macroscopic gravity. Secondly, gravitons and ghosts cannot be in principle situated precisely on the mass shell because of their conformal non–invariance and zero rest mass. This is because there are no asymptotic states, in which interaction of quantum fields with macroscopic gravity could be neglected. Restructuring of vacuum graviton and ghost modes with a wavelength of the order of the distance to the horizon of events takes place at all stages of cosmological evolution, including the contemporary Universe. Ghost trivial vacuum, understood as the quantum state with zero occupation numbers for all modes, simply is absent from physically realizable states. Therefore, direct participation of ghosts in the formation of macroscopic observables is inevitable222Once again, we emphasize that the equal participation of virtual gravitons and ghosts in the formation of macroscopic observables in the non–stationary Universe does not contradict the generally accepted concepts of the quantum theory of gauge fields. On the contrary it follows directly from the mathematical structure of this theory. In order to clear up this issue once and for all, recall some details of the theory of matrix. In constructing this theory, all space–time is divided into regions of asymptotic states and the region of effective interaction. Note that this decomposition is carried out by means of, generally speaking, an artificial procedure of turning on and off the interaction adiabatically. (For obvious reasons, the problem of self–consistent description of gravitons and ghosts in the non-stationary Universe with by means of an analogue of such procedure cannot be considered a priori.) Then, after splitting the space–time into two regions, it is assumed that the asymptotic states are ghost–free. In the most elegant way, this selection rule is implemented in the BRST formalism, which shows that the BRST invariant states turn out to be gauge–invariant automatically. The virtual ghosts, however, remain in the area of interaction, and this points to the fact that virtual gravitons and ghosts are parts of the Feynman diagrams on an equal footing. In the self–consistent theory of gravitons in the non–stationary Universe, virtual ghosts of equal weight as the gravitons, appear at the same place where they appear in the theory of matrix, i.e. at the same place as they were introduced by Feynman, i.e. in the region of interaction. Of course, the fact that in the real non–stationary Universe, both the observer and virtual particles with are in the area of interaction, is highly nontrivial. It is quite possible that this property of the real world is manifested in the effect of dark energy. An active and irremovable participation of virtual ghosts in the formation of macroscopic properties of the real Universe poses the question of their physical nature. Today, we can only say with certainty that the mathematical inevitability of ghosts provides the one–loop finiteness off the mass shell, i.e. the mathematical consistency of one-loop quantum gravity without fields of matter. Some hypothetical ideas about the nature of the ghosts are briefly discussed in the final Section X..

Section III is devoted to general discussion of equations of the theory of gravitons in the isotropic Universe. It focuses on three issues: (i) The gauge invariant procedure for eliminating gauge non–invariant modes by conditions imposed on the state vector (Section III.1); (ii) Construction of the state vector of a general form as a product of normalized superpositions (Section III.3); (iii) The allocation of class of legitimate gauges that are invariant with respect to transformations of the symmetry group of the background spacetime while providing the one–loop finiteness of macroscopic observables (Sections III.4 and III.5). The main conclusion is that the quantum ghost fields are inevitable and unavoidable components of the quantum gravitational field. As noted above, one–loop finiteness is seen by us as a universal property of quantum gravity, which extends off the mass shell. The requirement of compensation of divergences in terms of macroscopic observables, resulting from one–loop finiteness, uniquely captures the dynamic properties of quantum ghost fields in the isotropic Universe.

The treatment set out in Sections II and III, in essence, is a sequence of transformations of equations defined by original gauged path integral. Basically, these transformations are of mathematical identity nature. There are only three elements of the theory, missing in the original integral:

(i) The hypothesis of the existence of classical spacetime with the deterministic but self–consistent geometry;

(ii) A transition to the one–loop approximation in the self–consistent system of classical and quantum equations;

(iii) A class of gauges that automatically provides the one–loop finiteness of self–consistent theory of gravitons in the isotropic Universe.

Conceptually discussing of the first two additional elements was not necessary. Their introduction to the formalism is to the factorization of measure of path integral and expansion of equations of the theory in a series of powers of the graviton field. The existence of appropriate correct mathematical procedures is not in doubt. Agreement on choosing of a gauge is also mathematically consistent. Moreover, a gauge is necessary for the strict definition of path integral as a mathematical object.

The existence of class of gauges, automatically providing one–loop finiteness off the mass shell is itself a nontrivial property of the theory. The assertion that only such gauges can be used in the self–consistent theory of gravitons in the isotropic Universe is actually a condition for the internal consistency of the theory. In Appendix XII.2, we present the formal proof that an alternative formulation of the theory (with no one–loop finiteness) does not exist.

From the requirement of the one–loop finiteness it follows that the quantum component of the gravitational field is of a heterogeneous graviton–ghost structure. In further Sections of work, it appears that this new element of the theory prejudge its physical content.

Sections IV.1 and IV.2 contain approximate solutions to obtain quantum ensembles of short and long gravitational waves. In Section IV.3 it is shown that approximate solutions obtained can be used to construct scenarios for the evolution of the early Universe. In one such scenario, the Universe is filled with ultra–relativistic gas of short–wave gravitons and with a condensate of super–long wavelengths, which is dominated by ghosts. The evolution of this Universe is oscillating in nature.

At the heart of cosmological applications of one–loop quantum gravity is the Bogolyubov–Born–Green–Kirckwood–Yvon (BBGKY) chain (or hierarchy) for the spectral function of gravitons, renormalized by ghosts. We present the first use of this technique in quantum gravity calculations. Each equation of the BBGKY chain connects the expressions for neighboring moments of the spectral function. In Section V.1 the BBGKY chain is derived by identical mathematical procedures from graviton and ghost operator equations. Among these procedures is averaging of bilinear forms of field operators over the state vector of the general form, whose mathematical structure is given in Section III.3. The need to work with state vectors of the general form is dictated by the instability of the trivial graviton–ghost vacuum (Section III.6). Evaluation of mathematical correctness of procedures for BBGKY structure is entirely a question of the existence of moments of the spectral function as mathematical objects. A positive answer to this question is guaranteed by one–loop finiteness (Section III.4). The set of moments of the spectral function contains information on the dynamics of operators as well as on the properties of the quantum state over which the averaging is done. The set of solutions of BBGKY chain contains all possible self–consistent solutions of operator equation, averaged over all possible quantum ensembles.

A nontrivial fact is that in the one–loop quantum gravity BBGKY chain can formally be introduced at an axiomatic level. Theory of gravitons provided by BBGKY chain, conceptually and mathematically corresponds to the axiomatic quantum field theory in the Wightman formulation (see Chapter 8 in the monograph 6 ()). Here, as in Wightman, the full information on the quantum field is contained in an infinite sequence of averaged correlation functions. Definitions of these functions clearly relate to the symmetry properties of manifold on one this field is defined. Once the BBGKY chain is set up, the existence of finite solutions for the observables is provided by inherent mathematical properties of equations of the chain. This means that the phenomenology of BBGKY chain is more general than field operators, state vectors and graviton–ghost compensation of divergences that were used in its derivation.

Exact solutions of the equations, consisting of BBGKY chain and macroscopic Einstein’s equations are obtained in Sections V.2 and V.3. Two solutions given in V.2, describe heterogeneous graviton–ghost condensates, consisting of three subsystems. Two of these are condensates of spatially homogeneous modes with the equations of state and . The third subsystem is a condensate of quasi–resonant modes with a constant conformal wavelength corresponding to the variable physical wavelength of the order of the distance to the horizon of events. The equations of state of condensates of quasi–resonant modes differ from by logarithmic terms, through which the first solution is , while the second is . Furthermore, the solutions differ by the sign of the energy density of condensates of spatially homogenous modes. The third solution describes a homogeneous condensate of quasi–resonant modes with a constant physical wavelength. The equation of state of this condensate is and its self–consistent geometry is the De Sitter space. The three solutions are interpreted as three different phase states of graviton–ghost system. The problem of quantum–gravity phase transitions is discussed in SectionV.4.

Solutions obtained in Section V in terms of moments of the spectral function, are reproduced in Sections VI and VII at the level of dynamics of operators and state vectors. A microscopic theory provides details to clarify the structure of graviton–ghost condensates and clearly demonstrates the effects of quantum interference of coherent states. In Section VI.1 it is shown that the condensate of quasi–resonant modes with the equation of state consists of virtual gravitons and ghosts. In Section VI.2 a similar interpretation is proposed for the condensate in the De Sitter space, but it became necessary to extend the mathematical definition of the moments of the spectral function.

New properties of the theory, whose existence was not anticipated in advance, are studied in Section VII. In Section VII.1 we find that the self–consistent theory of gravitons and ghosts is invariant with respect to the Wick turn. In this Section we also construct the formalism of quantum theory in the imaginary time and discuss the physical interpretation of this theory. The subjects of the study are correlated fluctuations arising in the process of tunnelling between degenerate states of graviton–ghost systems, divided by classically impenetrable barriers. The level of these fluctuations is evaluated by instanton solutions (as in Quantum Chromodynamics). In Section VII.2 it is shown that the condensate of quasi–resonant modes with the equation of state is of purely instanton nature. In Section VII.3 the instanton condensate theory is formulated for the De Sitter space.

Potential use of the results obtained to construct scenarios of cosmological evolution was briefly discussed in Sections IVVII to obtain approximate and exact solutions. A main application of the theory of macroscopic effects of quantum gravity is to explain the physics of Dark Energy. As a carrier of Dark Energy, quantum gravity offers graviton–ghost condensate of quasi–resonant modes. In Section VIII the theory of gravitons and ghosts is formulated in a self–consistent field, in the formation of which heavy particles (baryons and particles of Dark Matter (neutralino as a example)) are involved. In Section VIII.1 it is shown that the cosmological solution describing the evolution of the scale factor, non–relativistic matter and graviton–ghost subsystem, automatically satisfies the chain integral identities contained in the equations of this theory. From these identities it follows that the stage of the Universe evolution, which is dominated by non–relativistic matter, must inevitably be replaced by a stage at which substantive contribution to the energy density of the cosmological substratum will produce graviton–ghost condensate of quasi–resonant modes of a constant conformal wavelength. This stage of the Universe evolution we consider as the pre–asymptotic one. Conversion of pre–asymptotic condensate to a condensate of constant physical wavelength occurs in the process of quantum–gravity phase transition.

In Section VIII.2 the exact solutions of equations of the one–loop quantum gravity are used to construct model of Dark Energy, intended for interpretation of experimental data. The model is based on simple interpolation formulae, describing the energy density and pressure of graviton–ghost condensates at various stages of cosmological evolution. For the cosmological model based on the physical considerations presented here, we suggest the abbreviation ”GCDM model.” Here ”CDM” means Cold Dark Matter; ”G” means a graviton–ghost condensate in the role of Dark Energy; symbol indicates an asymptotic state of the Universe in which the energy density of cosmological substratum goes to a constant value. The following cosmological Einstein equations correspond to the GCDM model




are energy density and pressure of graviton–ghost medium; is the density of non–relativistic matter; is Hubble function; is acceleration of the Universe expansion. The symbol indicates the asymptotic value of total energy density of graviton–ghost condensate and equilibrium vacuum of non–gravitational physical fields. Note that , even if the non–gravitational vacuum energy (”standard” Einstein’s cosmological constant) vanishes by some compensation mechanisms of the type of supersymmetry.

As can be seen from (I.2), the theory of graviton–ghost condensates in an extremely simplified version predicted by a three–parameter model of Dark Energy. A quantitative comparison of theory with the observational data is conducted in Section IX. Section IX.1 devoted to a brief review of the physics of Dark Energy problems. Synopsis of observational data on supernovae SNIa is given in Section IX.2. Our focus is on the choice between CDM and GCDM models based on general principles of fundamental physics without involving hypothetical elements. In Section IX.3 a comparative analysis of the results of processing the Hubble diagram for SNIa by formulas for both models is conducted. Here we show that the GCDM model not only explains quantitatively the effect of Dark Energy, but also reproduces a number of specific details of observational data.

The results and problems of the theory are briefly discussed in the Conclusion (Section X). Appendix XI is devoted to the cosmological constant problem within the framework of its interpretation as the energy density of equilibrium vacuum of non–gravitational physical fields. An internal inconsistency of the theory which lacks one–loop finiteness off the mass shell is proved in Appendices XII.1,XII.2. In XII.3, it is shown by the method of dimensional transmutation that the one–loop self–consistent theory of gravitons and ghosts is not only finite but is also free of anomalies.

A system of units is used, in which the speed of light is , Planck constant is MeVfm; Einstein’s gravity constant is MeVfm.

Ii Basic Equations

According to De Witt 7 (), one of formulations of one–loop quantum gravity (with no fields of matter) is reduced to the zero rest mass quantum field theory with spin , defined for the background spacetime with classic metric. The graviton dynamics is defined by the interaction between quantum field and classic gravity, and the background space geometry, in turn, is formed by the energy–momentum tensor (EMT) of gravitons.

In the current Section we describe how to get the self–consistent system of equations, consisting of quantum operator equations for gravitons and ghosts and classic –number Einstein equations for macroscopic metrics with averaged EMT of gravitons and ghosts on the right hand side. The theory is formulated without any constrains on the graviton wavelength that allows the use of the theory for the description of quantum gravity effects at the long wavelength region of the specter. The equations of the theory (except the gauge condition) are represented in 4D form which is general covariant with respect to the transformation of the macroscopic metric.

The mathematically consistent system of 4D quantum and classic equations with no restrictions with respect to graviton wavelengths is obtained by a regular method for the first time. The case of a gauged path integral with ghost sector is seen as a source object of the theory. Important elements of the method are exponential parameterization of the operator of the density of the contravariant metric; factorization of path integral measure; consequent integration over quantum and classic components of the gravitational field. Mutual compliance of quantum and classic equations, expressed in terms of fulfilling of the conservation of averaged EMT at the operator equations of motion is provided by the virtue of the theory construction method.

ii.1 Path Integral and Faddeev–Popov Ghosts

Formally, the exact scheme of quantum gravity is based on the amplitude of transition, represented by path integral 4 (); 5 ():



is the density of gravitational Lagrangian, with cosmological constant included; is the density of ghost Lagrangian, explicit form of which is defined by localization of ; is gauge operator, is the given field; is an operator of equation for infinitesimal parameters of transformations for the residual degeneracy ;


is the gauge invariant measure of path integration over gravitational variables; is the measure of integration over ghost variables. Operator is of standard definition:




is variation of metrics under the action of infinitesimal transformations of the group of diffeomorphisms. According to (II.1), the allowed gauges are constrained by the condition of existence of the inverse operator .

The equation (II.1) explicitly manifests the fact that the source path integral is defined as a mathematical object only after the gauge has been imposed. In the theory of gravity, there are no gauges completely eliminating the degeneracy with respect to the transformations (II.4). Therefore, the sector of nontrivial ghost fields, interacting with gravity, is necessarily present in the path integral. This aspect of the quantum gravity is important for understanding of its mathematical structure, which is fixed before any approximations are introduced. By that reason, in this Section we discuss the equations of the theory, by explicitly defining the concrete gauge.

In cosmological applications of the quantum gravity it is convenient to use synchronous gauges of type:


For that gauge


where is the metric determinant of the basic 3D space of constant curvature (for the plane cosmological model ). More general approach to the choice of gauge used for cosmological problems is discussed in Section III.5.

The construction of the ghost sector, i.e. finding of the Lagrangian density , is reduced to two operations. First, is represented in the form, factorized over independent degrees of freedom for ghosts, and then the localization of the obtained expression is conducted. Substitution of (II.6) and (II.4) to (II.3) gives the following system of equations


According to (II.7), with respect to variables , the operator–matrix reads


(Note matrix–operator is obtained in the form (II.8) without the substitution of transformation parameters if Leutwiller measure is used. The measure discussion see, e.g. 8 ().) Functional determinant of matrix–operator is represented in the form of the determinant of matrix , every element of which is a functional determinant of differential operator. As it is follows from (II.8),


One can see that the first multiplier in (II.9) is 4–invariant determinant of the operator of the zero rest mass Klein–Gordon–Fock equation, and two other multipliers do not depend on gravitational variables.

Localization of determinant (II.9) by representing it in a form of path integral over the ghost fields is a trivial operation. As it follows from (II.9), the class of synchronous gauges contains three dynamically independent ghost fields , two of each — do not interact with gravity. For the obvious reason, the trivial ghosts are excluded from the theory. The Lagrangian density of nontrivial ghosts coincides exactly with Lagrangian density of complex Klein–Gordon–Fock fields (taking into account the Grassman character of fields ):


The normalization multiplier in (II.10) is chosen for the convenience. The integral measure over ghost fields has a simple form:

The calculations above comply with both general requirements to the construction of ghost sector. First, path integration should be carried out only over the dynamically independent ghost fields. Second, in the ghost sector, it is necessary to extract and then to take into account only the nontrivial ghost fields, i.e. those interacting with gravity.

The extraction of dynamically independent nontrivial ghost fields can be done not only by factorization of functional determinant (as it made in (II.9)), but by means of researching the equations for the ghosts as well. It is well known 9 (), that from the definition of it follows that the ghost equations coincide with equations for parameters of infinitesimal transformations of residual degeneracy. Therefore, according to (II.3), we can immediately get , where are the Grassman fields. For the gauge (II.5) with we get


From (II.12) it follows that those transversal components of vector yield the equations , i.e. these degrees of freedom correspond to two trivial ghosts non–interacting with gravity. Note, that in (II.11) only the longitudinal component is present. Equation (II.12) has a status of equation connecting the longitudinal component and function . It means that the longitudinal field is not dynamically independent. Equation for dynamically independent degrees of freedom is obtained as follows. First, the operator is applied to the equation (II.12), next the result is substituted into time–differentiated equation (II.11). After the substitution, one gets Klein–Gordon–Fock equation,


where . Reconstruction of the ghost Lagrangian (II.10) from dynamical equation (II.13) does not require an additional explanation.

Thus, the prove of wave properties of ghosts in the class of synchronous gauges is done by two methods with clear correlations between objects and operations used in these methods.

We will return to the discussion of gauges and ghosts in the Section III.5.

ii.2 Einstein Operator Equations

Let us take into account the fact that the calculation of gauged path integral should be mathematically equivalent to the solution of dynamical operator equations in the Heisenberg representation. It is also clear that operator equations of quantum theory should have a definite relationship with Einstein equations. In the classic theory, it is possible to use any form of representation of Einstein equations, e.g.


where, for example, . Transition from one to another is reduced to the multiplication by metric tensor and its determinant, which are trivial operations in case when the metric is a –number function. If the metric is an operator, then the analogous operations will, at least, change renormalization procedures of quantum non–polynomial theory. Thus, the question about the form of notation for Einstein’s operator equations has first–hand relation to the calculation procedure. Now we show that in the quantum theory one should use operator equations (II.14) with , supplemented by the energy–momentum pseudo–tensor of ghosts.

In the path integral formalism, the renormalization procedures are defined by the dependence of Lagrangian of interactions and the measure of integration of the field operator in terms of which the polynomial expansion of non–polynomial theory is defined 9 (). The introduction of such an operator, i.e. the parameterization of the metric, is, generally speaking, not simple. Nevertheless, it is possible to find a special parameterization for which the algorithms of renormalization procedures are defined only by Lagrangian of interactions. Obviously, in such a parameterization the measure of integration should be trivial. It reads:


where is a dynamic variable. The metric is expressed via this variable. It is shown in 9 () that the trivialization of measure (II.15) takes place for the exponential parameterization that reads


where is the defined metric of an auxiliary basic space. In that class of our interest, the metric is defined by the interval

where is the metric of 3D space with a constant curvature. (For the flat Universe is the Euclid metric.)

The exponential parameterization is singled out among all other parameterizations by the property that are the normal coordinates of gravitational fields 10 (). In that respect, the gauge conditions (II.5) are identical to . The fact that the ”gauged” coordinates are the normal coordinates, leads to a simple and elegant ghost sector (II.10). The status of , as normal coordinates, is of principal value for the mathematical correctness while separating the classic and quantum variables (see Section II.4). Besides, in the framework of perturbation theory the normal coordinates allow to organize a calculation procedure, which is based on a simple classification of nonlinearity of quantum gravity field. It is important that this procedure is mathematically non–contradictive at every order of perturbation theory over amplitude of quantum fields (see Section II.5, II.6).

Operator Einstein equations that are mathematically equivalent to the path integral of a trivial measure are derived by the variation of gauged action by variables . The principal point is that the gauged action necessarily includes the ghost sector because there are no gauges that are able to completely eliminate the degeneracy. According to (II.10), in the class of synchronous gauges we get


In accordance with definition (II.16), the variation is done by the rule

Thus, from (II.17) it follows


After subtraction of semi–contraction from (II.18) we obtain a mathematically equivalent equation


In (II.18), (II.19) there is an object


which has the status of the energy–momentum pseudo–tensor of ghosts.

In accordance with the general properties of Einstein’s theory, six spatial components of equations (II.18) are considered as quantum equations of motion:


(Everywhere in this work the Greek metric indexes stand for .) In the classic theory, equations of constraints and are the first integrals of equations of motion (II.21). Therefore, in the quantum theory formulated in the Heisenberg representation four primary constraints from (II.19), have the status of the initial conditions for the Heisenberg state vector. They read:


If conditions (II.22) are valid from the start, then the internal properties of the theory must provide their validity at any subsequent moment of time. Four secondary relations, defined by the gauge non containing the higher order derivatives, also have the same status:


The system of equations of quantum gravity is closed by the ghosts’ equations of motion, obtained by the variation of action (II.17) over ghost variables:


Ghost fields and are not defined by Grassman scalars, therefore is not a tensor. Nevertheless, all mathematical properties of equations (II.24) and expressions (II.20) coincide with the respected properties of equations and EMT of complex scalar fields. This fact is of great importance when concrete calculations are done (see Section III).

ii.3 Factorization of the Path Integral

Transition from the formally exact scheme (II.21) — (II.24)) to the semi–quantum theory of gravity can be done after some additional hypotheses are included in the theory. The physical content of these hypotheses consists of the assertion of existence of classical spacetime with metric , connectivity and curvature . The first hypothesis is formulated at the level of operators. Assume that operator of metric is a functional of –number function and the quantum operator . The second hypothesis is related to the state vector. Each state vector that is involved in the scalar product , is represented in a factorized form , where are the vectors of quantum states of gravitons; are the vectors of quasi–classic states of macroscopic metric. In the framework of these hypotheses the transitional amplitude is reduced to the product of amplitudes:


Thus, the physical assumption about existence of classic spacetime formally (mathematically) means that the path integral must be calculated first by exact integration over quantum variables, and then by approximate integration over the classic metric.

Mathematical definition of classic and quantum variables with subsequent integrations are possible only after the trivialization and factorization of integral measure are done. As already noted, trivial measure (II.15) takes place in exponential parameterization (II.16). The existence of vector allows the introduction of classic –number variables as follows

Quantum graviton operators are defined as the difference . Factorized amplitude (II.25) is calculated via the factorized measure


Factorization of the measure allows the subsequent integration, first by , , then by approximate integration over . In the operator formalism, such consecutive integrations correspond to the solution of self–consistent system of classic and quantum equations. Classical equations are obtained by averaging of operator equations (II.19). They read:


Subtraction of (II.27) from (II.19) gives the quantum dynamic equations


Synchronous gauge (II.23) is converted to the gauge of classical metric and to conditions imposed on the state vector:


Quantum equations (II.24) of ghosts’ dynamics are added to equations (II.27) — (II.29).

Theory of gravitons in the macroscopic spacetime with self–consistent geometry is without doubt an approximate theory. Formally, the approximation is in the fact that the single mathematical object is replaced by two objects — classical metric and quantum field, having essentially different physical interpretations. That ”coercion” of the theory can lead to a controversy, i.e. to the system of equations having no solutions, if an inaccurate mathematics of the adopted hypotheses is used. The scheme described above does not have such a controversy. The most important element of the scheme is the exponential parameterization (II.16), which separates the classical and quantum variables, as can be seen from (II.26). After the background and quantum fluctuations are introduced, this parameterization looks as follows:


Note that the auxiliary basic space vanishes from the theory, and instead the macroscopic (physical) spacetime with self–consistent geometry takes its place.

If the geometry of macroscopic spacetime satisfies symmetry constrains, the factorization of the measure (II.26) becomes not a formal procedure but strictly mathematical in its nature. These restrictions must ensure the existence of an algorithm solving the equations of constraints in the framework of the perturbation theory (over the amplitude of quantum fields). The theory of gravity is non–polynomial, so after the separation of single field into classical and quantum components, the use of the perturbation theory in the quantum sector becomes unavoidable. The classical sector remains non–perturbative. In the general case, when quantum field is defined in an arbitrary Riemann space, the equations of constraints is not explicitly solvable. The problem can be solved in the framework of perturbation theory if background and the free (linear) tensor field belong to different irreducible representations of the symmetry group of the background spacetime. In that case at the level of linear field we obtain (II.26), because the full measure is represented as a product of measure of integration over independent irreducible representations. At the next order, factorization is done over coordinates, because the classical background and the induced quantum fluctuations have essentially different spacetime dynamics. Note, to factorize the measure by symmetry criterion we do not need to go to the short–wave approximation.

Background metric of isotropic cosmological models and classical spherically symmetric non–stationary gravitational field meet the constrains described above. These two cases are covering all important applications of semi–quantum theory of gravity which are quantum effects of vacuum polarization and creation of gravitons in the non–stationary Universe and in the neighborhood of black holes.

ii.4 Variational Principle for Classic and Quantum Equations

Geometrical variables can be identically transformed to the form of functionals of classical and quantum variables. At the first step of transformation there is no need to fix the parameterization. Let us introduce the notations:


According to (II.30), formalism of the theory allows definition of quantum field as symmetric tensor in physical space, . Objects, introduced in (II.31), have the same status. With any parameterization the following relationships take place:

We should also remember that the mixed components of tensors do not contain the background metric as functional parameters. For any parameterization, these tensors are only functionals of quantum fields which are also defined in mixed indexes. For the exponential parameterization:


where . One can seen from (II.32), that the determinant of the full metric contains only the trace of the quantum field.

Regardless of parameterization, the connectivity and curvature of the macroscopic space , are extracted from full connectivity and curvature as additive terms:

Quantum contribution to the curvature tensor,

is expressed via the quantum contribution to the full connectivity:


The density of Ricci tensor in mixed indexes reads


Symbol ”;” in (II.33), (II.34) and in what follows stands for the covariant derivatives in background space. The density of gauged gravitational Lagrangian is represented in a form which is characteristic for the theory of quantum fields in the classical background spacetime:


When the expression for was obtained from contraction of tensor (II.34), the full covariant divergence in the background space have been excluded. Formulas (II.34), (II.35) apply for at any parameterization.

Let us discuss the variation method. In the exact quantum theory of gravity with the trivial measure (II.15), the variation of the action over variables leads to the Einstein equations in mixed indexes (II.18) and (II.19). In the exact theory, the exponential parameterization is convenient, but, generally speaking, is not necessary. A principally different situation takes place in the approximate self–consistent theory of gravitons in the macroscopic spacetime. In that theory the number of variables doubles, and with this, the classical and quantum components of gravitational fields have to have the status of the dynamically independent variables due to the doubling of the number of equations. The variation should be done separately over each type of variables. The formalism of the path integration suggests a rigid criterion of dynamic independence: the full measure of integration, by definition, must be factorized with respect to the dynamically independent variables. Obviously, only the exponential parameterization (II.30), leading to the factorized measure (II.26), meets the criterion.

The variation of the action over the classic variables is done together with the operation of averaging over the quantum ensemble. In the result, equations for metric of the macroscopic spacetime are obtained:


where . Variation of the action over background variables, defined as , yields the equations:


Equations (II.36) and (II.37) are mathematically identical. We should also mention that if the variations over the background metric are done with the fixed mixed components of the quantum field, these equations are valid for any parameterization.

Exponential parameterization (II.30) has a unique property: the variations over classic (before averaging) and quantum (without averaging) variables lead to the same equations. That fact is a direct consequence of the relations, showing that variations and are multiplied by the same operator multiplier:

By a simple operation of subtraction, the identity allows the extraction of pure background terms from the equation of quantum field. The equations of graviton theory in the macroscopic space with self–consistent geometry are written as follows:


With the exponential parameterization, the formalism of the theory can be expressed in an elegant form. Let us go to the rules of differentiation of exponential matrix functions


Taking into account (II.40), we get the quantum contribution to the full connectivity (II.33) as follows


Formulas (II.35) could be rewritten as follows:


As is seen from (II.42), for the exponential parameterization, the non–polynomial structures of quantum theory of gravity have been completely reduced to the factorized exponents333We are using the standard definitions. Matrix functions are defined by their expansion into power series as any operator functions: . The derivative of –th degrees of matrix by the same matrix is defined as The derivative by numerical (non matrix) parameter is