How to unify the general theory of relativity with quantum physics ?

Appendix B
UNITARY FIELD THEORY AND QUANTUM GRAVITY
B.1. The process of unification in physics
B.2. Einstein's visions of geometric unitary field theory
B.3. Wheeler's geometrodynamics. Gravity and topology.
B.4. Quantum geometrodynamics
B.5. Gravitational field quantization
B.6. Unification of fundamental interactions. Supergravity. Superstrings.
B.7. General principles and perspectives of unitary field theory

B.5. Gravitational field quantization

In the previous paragraph on quantum geometrodynamics, we have already partially "bitten" the issue of quantum gravity, but from a very specific and unusual point of view. Here we will mention in a few words the general aspects and basic approaches to the quantization of gravity (for further details see eg [279], [280], [177], [258]) .
  The vast majority of phenomena in nature are caused by either gravity or quantum phenomena, but not both at the same time. Quantum phenomena are crucial in the microworld of molecules, atoms and elementary particles, but gravity is negligible. In astronomical scales, on the other hand, planets, stars, and galaxies are firmly dominated by gravity, while quantum fluctuations are unobservable. Therefore, the connection between gravity and quantum physics is not important for almost all natural sciences. We do not know how gravity works at the level of atomic and subatomic particles. We are not yet able to combine general theory of relativity and quantum mechanics into "one formula" ...
  In short: Gravity is so weak that its local effect on small objects such as quantum systems is immeasurably small. Conversely, quantum effects are immeasurably small for objects large enough to exhibit more significant gravity.
  At the very beginning (see §5.4 - 5.5), however, the universe could be so "small" that quantum fluctuations could "shake" the entire universe. And the concentration of matter and gravity so great that gravity could play an important role in the formation of quantum fields and their further behavior. The quantum concept of gravity can thus become key to understanding the earliest stages of the evolution of the universe. In general, it would be highly desirable to combine the general theory of relativity and quantum physics into one theory that would describe the universe at all levels.
  Quantum gravity can manifest itself significantly in basically three (astro) physical situations :
-> The initial "singularity" in the formation of the universe by the big bang (§5.5 "Microphysics and cosmology. Inflationary universe.") .
-> "Singularity" inside a black hole (§4.2 "Final stages of stellar evolution. Gravitational collapse. Formation of a black hole.") .
-> Spacetime quantum foam in microscales of quantum geometrodynamics (§B.4 "Quantum geometrodynamics") .
  Although the gravitational field in Einstein's general theory of relativity differs considerably in its geometric character from the fields with which other parts of physics work, it also has some significant properties in common with them. Since all these other fields are governed by the laws of quantum physics, it is necessary to investigate the consequences of the application of quantum laws to gravity. The quantum approach to gravity basically has two stages : 

a) Semi-quantum approach - semiclassical gravitation ,
which applies quantum laws only to a non-gravitational field, serving either as a source of a gravitational field or responding to a given gravitational field.
  Here includes quantum field theory in curved space that studies the behavior of the quantized non-gravitational fields at a given (fixed and not quantized) gravitational field, i.e. against the background of curved spacetime. This method has celebrated great success in the Hawking effect of the evaporation of black holes (see §4.7 "Quantum radiation and thermodynamics of black holes"). According to Einstein's equations R_ik - ¹/₂ g_ik R = 8p T_ik , the quantum behavior of the mass exciting the gravitational field causes that even the excited gravitational field will carry significant traces of this quantum behavior. Here, however, these are exclusively "induced" quantum properties, have nothing to do with the nature of the gravitational field itself.
  Such an approach can not be considered entirely consistent, because the question arises of quantizing those "degrees of freedom" of the gravitational field, that are not directly related to the excitation matter. In fact, there are independent excitations (gravitational waves) transmitting positive energy in the gravitational field. It is unlikely that the gravitational field has any property that avoids the quantization of quantities, which in all other physics are subject to universal quantum laws (discussed below in the passage "Quantum or non-quantum nature of gravity?").
  There is also a mathematical reason for quantizing gravity: in Einstein's GTR equations, one side characterizes the gravitational field, the other side the mass. If matter is quantized, the energy-momentum tensor T_ik would be expressed in terms of quantum-mechanical operators. And classical gravity is described by ordinary numbers (the values of the components of the metric tensor and the curvature tensor). There is no known way to consistently relate the non-quantized gravitational field on the left side of the equation to the source of quantized matter on the other side (like if we wanted to figure out how many pears it takes to make a watermelon...).
  If we did not require gravity to be an intrinsically quantum theory, we would use the average-expected values of energy, momentum, mass density in the energy-momentum tensor. There would then be the usual numbers on both sides of the equation, the equation is consistent and solvable. At the same time, this semiclassical (or semiquantum) theory of gravity is capable of very precisely describing all gravitational phenomena observed in nature, in the laboratory, in space - now and in the foreseeable future. Apart from hypothetical singularities. From the side of quantum physics, the semiquantum approach is criticized for its inconsistent approach and incompatibility with interpretations that consider the superposition of states and the collapse of the wave function as a real physical process. However, we do not share this interpretation (it is discussed in the passage "Interpretation of quantum physics" and "Disinterpretation").
  In §3.4-3.9, §4.9 and §5.3, we saw that some (otherwise realistic) solutions of GTR equations have singularities indicating that the general theory of relativity ceases to function in their vicinity. GTR alone is no longer enough to describe the enormous mass densities that occur here, but a new theory involving the laws of the microworld is needed. It is hoped that it is the quantum theory of gravity that could shed light on these difficulties of standard GTR, especially the elimination of spacetime singularities. Quantum theory of gravity, as a theory of the microstructure of space and time, could perhaps also help to overcome the basic problems of contemporary quantum field theory with "ultraviolet" divergences *). We see that the quantum theory of gravity promises the possibility of better penetration into the structure of elementary particles and their interactions, the possibility of revealing new important laws of nature.
*) According to some research [65], for example, elementary black micro-holes could play an important role in the physics of elementary particles: Contributions of intermediate states of arbitrarily large energy are included in the calculation of the particle's own energy, which usually leads to divergences. Inclusion of gravitational interaction respective virtual particles and the possibility of virtual black micro-holes in intermedial statecould perhaps potentially lead to the elimination of these divergences..?..

The basic principles of contemporary physics therefore require a more consistent approach :

b) Quantization of the gravitational field itself ,
in which gravitational energy would be transmitted in quantums, as well as all other forms of energy.
  We will first mention very briefly some basic methods of quantization in general and then the gravitational field :

Canonical quantization
The most common and straightforward way to move from classical theory to quantum theory is the method of canonical quantization. Applying this method to the mechanics of mass particles creates quantum mechanics. In such quantization (sometimes called "primary"), field-forming probability waves are assigned to the mass particles.
  When using the canonical method on the physical field (or continuum mechanics), particles appear as quantum excitations of these systems - whether they are quasiparticles (phonons) existing only against the background of the mechanical continuum, or especially real particles (such as photons, fermions etc.) existing against the background of the "vacuum" of the respective fields. This method is called secondary quantization [164], [177].
  The first step in canonical quantization is to convert the equations of the investigated system into Hamilton's (canonical) form. Normal level approach to the theory of dynamic variables q_A (generalized coordinates, A = 1,2,3, ...) and a Lagrangian L(q, q^.) can be converted to canonical form, it consists in introducing s canonical momenta p_A = ¶L/¶q ^._A, expressions of (generalized) velocities q^._A using the momentums p_A and in the construction of the Hamiltonian function H(p,q) = _ASq_Ap_A - L(q,p) expressing the total energy of the system using the quantities q and p. The equations of motion of the system in the variables paq then have a simple and symmetrical form q^._A =¶H/¶p^._A , p_A = - ¶H/¶q^._A - Hamilton's canonical equations.
  For the functions of generalized coordinates and momentums f(p, q, t), g(p, q, t) the so-called Poisson brackets {f,g} = _AS ((¶f/¶q_A)(¶g/¶p_A)- (¶f/¶p_A)(¶g/¶q_A)) are introduced, satisfying simple algebraic relations: {f,g} =-{g,f}, {f1 +f2,g} = {f1,g} + {f2,g}, {f,{g,h}} + {g,{h,f}} + {h,{f,g}] =0, etc. Eg. Poisson brackets formed from coordinates and momentum have the form {p_A, q_B} = d_AB ·
  During the actual transition to quantum theory, the state of the system is described by a certain (generally complex) function y(q) of generalized coordinates - a wave function, and the quantities appearing in classical theory are replaced by the appropriate operators. The wave function gives the probability distributions of the different system configurations: |y|².dq is the probability that the value of the generalized system coordinates will fall into the dq element of the configuration space during the measurement. The mean value of the observable quantity f, characterizing the state of the systems , is then equal to <f> = <y|F^{^}|y> ş ňy(F^{^}y) dq, where F^{^} is the Hermitian operator assigned to f .
  The behavior of the quantized physical system is then determined by a simple one the wave equation ih ¶y/¶ t = H^{^}y, where H^{^} is the Hamiltonian (operator assigned to the Hamiltonian function H). Operators assigned to classical physical quantities satisfy (except for equations of motion) certain commutation relations in which Poisson brackets appear: F^{^}G^{^} - G^{^}F^{^} ş [F,G] = ih{f,g}; these commutation relations are an expression of the quantum (Heisenberg) uncertainty principle.
  When quantizing a physical field, the field equations are usually converted to a wave equation, so that the field (in some finite region of space) can be expressed as a superposition of plane waves; thus the field is described by a discrete series of variables - amplitudes and frequencies of waves. Based on these amplitudes, canonical field variables are defined - generalized coordinates q_A and momentum p_A, which are used to express Hamiltonian as the sum of independent members of the shape of a one-dimensional harmonic oscillator corresponding to individual waves with appropriate wave vectors and polarizations. During a quantum transition, canonical variables (generalized coordinates q_A and momentum p_A) become operators with commutation relation [^{^}P_A ,^{^}Q_A] = - i h. The use of these commutation relations to determine the Hamiltonian eigenvalues leads to discrete energy levels of the field E_n = (n + 1/2) h.w.
  The application of this canonical method of quantization to the electromagnetic field is well known (quantum electrodynamics) and leads to the idea of a free electromagnetic field as a set of particles - photons, each of which has energy h.w and momentum h.w/c. An analogous quantization procedure for a weak gravitational field within a linearized theory, performed in 1930-36 [221], [34], [6], leads to the existence of gravitons as quantum of the gravitational field. Gravitons are particles with zero rest mass and spin 2 (compare with the mention of the symmetry of gravitational waves in §2.7), which are transversely polarized (only the maximum spin values +2 and -2 are realized). For the general case of nonlinear tensor theory, quantization was developed by Dirac [70], [6].
  Several variants of canonical quantization of the gravitational field have been developed. These modifications differ both in the way of introducing time (as time is taken either directly x⁰ coordinates or time with respect to certain non-rotating "normal" reference systems), as well as by choice and relations between generalized coordinates and momentum (due to the singularity of Lagrangians, there are certain coupling equations between p and q to reduce the number of independent canonical variables).

Feynman quantization of path integrals
Quantum physics has achieved impressive success in explaining the structure and function of atoms, atomic nuclei, and elementary particles. However, it is by no means easy to understand the internal causes of the quantum behavior of microsystems based on our experience with the classical behavior of the macroworld. Feynman's formulation of quantum theory [178] is characterized by a very close relationship to classical physics *) expressed by the principle of least action. In classical physics (mechanics, electrodynamics, GTR) between a given initial x₁ and final x₂ state of the investigated system, only such a always take movement, for which the integral action S = _x1ň^x2L dt is extreme. On the other hand, in quantum physics, as is well known, such processes also takes place that do not comply with this principle and are impossible according to classical physics - for example, the tunneling phenomenon.
*) The transition from classical to quantum physics is so elegant and straightforward that J.A.Wheeler [277] used this approach to persuade A. Einstein, but to no avail, to revise his opposition to the stochastic principles of quantum mechanics.
  In Feynman's approach, all trajectories leading from the initial state x₁ to the final state x₂ are considered equally and simultaneously, regardless of whether they are permissible or not according to classical physics.  As if the particle were moving along each imaginary trajectory at the same time as it traveled between the two states - it is the set of all virtual trajectories ("all histories"). If the integral _x1ň^x2L dt is calculated for each trajectory, the probability of transition of the system from the initial state x₁ to the final state x₂ will be given by the square of the quantity

^,

obtained as sum and taken through all trajectories - the sum through all possible "histories". It is evident that the largest contribution to this sum is made by those trajectories that have a phase coefficient (i/h) ň Ldt almost the same (exponents add up), while for trajectories with large differences in the exponents (i/h) ň Ldt in the sum cancel each other out. The most probable trajectory (corresponding to close values of ň Ldt) will therefore be a classical trajectory with extreme behavior of the integral of the action. Trajectory here means "path" in the space of the given configurations systems; if it is a complex system described by a large number of parameters, it will be a trajectory in multidimensional space. Feynman showed that this formulation is equivalent to the usual Schrödinger and Heisenberg concept of quantum mechanics. As with the classical principle of the smallest action, the extreme of the integral of ň Ldt is not immediately sought in practice, but Lagrange's equations of motion are derived, even when using the Feynman method, the total sum over all trajectories is not directly calculated. Feynman's procedure is rather used as a means for deriving and elaborating quantum theories, as well as their physical interpretation.
Path quantization of universes in "superspace" 
Misner [277] and deWitt [179] tried to use Feynman's concepts to quantize the "most classical" object we can imagine: the universe as a whole. They introduced the so-called superspace - infinite-dimensional space, whose "points" represent all possible geometries of space (states of the universe). The line - the trajectory - in this superspace then represents a certain variant of the evolution of the universe. It is clear that the practical use of superspace is possible only under very simplistic conditions. Misner therefore proposed to study the evolution of a closed homogeneous universe (generalized Kasner's models - §5.4) , for the description of whose state three parameters are sufficient; infinitely the dimensional superspace is reduced here to a three-dimensional "mini-superspace". The superspace of Fridman's homogeneous isotropic universes is even one-dimensional - all spatial sections are characterized by the value of the parameter a(x°). In the framework of superspace is possible also to mathematically formulate Wheeler quantum geometrodynamics (mentioned above §B4 " Quantum geometrodynamics ") [275].
  Quantum approaches in cosmology, leading to the hypothesis of more universes, are discussed in §5.5, section "Chaotic inflation", passage "The emergence of multiple universes". 

Difficulties and perspectives of quantum gravity
In the case of a weak gravitational field, it is not necessary to take into account the geometric character of gravity (curvature of spacetime) and it can be investigated like any other field against Euclidean (or other masses curved) spacetime. From this perspective, gravity can be considered in the context of the standard field theory of spin 2. In the general case, however, a number of mathematical and physical problems of quantum theory of gravity stem from the geometric nature of the gravitational field and its nonlinearity.
  Due to the universality of the gravitational interaction, expressed by the principle of equivalence in GTR, the intensity of the gravitational field Gⁱ _kl or (canonical) momentum does not form tensors. The respective commutation relations are then not covariant and therefore have no direct physical meaning. In §2.8 "Specific features of gravitational energy", we showed that the density and flow of energy and momentum t^ik gravitational field has no tensor character, energy of the gravitational field is not localizable, of it from here leads to difficulties in interpreting the gravitational field quanta - gravitons. The fact that gravity does not have a clearly localized energy density in the GTR is a major obstacle to the usual method of quantization. This is because energy is a quantity that generally plays a key role in the quantization process.
  Due to the covariance of GTR, a larger number of variables is used than corresponds to the number of dynamic degrees of freedom. It is therefore difficult to orientate oneself in the geometric and physical meaning of superfluous variables and to correctly use arbitrariness in the choice of coordinates. The nonlinearity of the general theory of relativity prevents the superposition of individual partial solutions, which is a significantly different situation than in electrodynamics. In addition, nonlinearity leads to very complex relationships between the components of the metric tensor (taken as the generalized coordinates of the system) and the canonically associated velocities and momentums.
  From the universality of gravity follows another remarkable and unusual property, which was already highlighted in §2.5 "Einstein's equation of the gravitational field" in deriving Einstein's equations of the gravitational field and in the following text, it has been discussed many times in the analysis of gravitational phenomena: it is self- gravity of the gravitational field. That the gravitational field is able to generate another gravitational field by its energy. This causes great difficulties in the quantization process, as both real and virtual gravitons can emit more virtual gravitons, then more, etc., which leads to expressions diverging to infinity *). In other words, there are problems with the renormalizability of the quantum theory of gravity.
*) Quantum fluctuations of a vacuum can be imagined as a pair of particles that are born together at a certain moment, move separately for a short time, but then meet again and annihilate each other. The closed loops in the Feynman diagrams correspond to these virtual particles. According to the general theory of relativity, the energy of an infinite number of these virtual pairs would curve space to an infinitesimal dimension, contrary to reality. Unlike the strong, weak, and electromagnetic interactions in GTR gravity, this infinity cannot be eliminated by renormalization because the appropriate parameters are not available. A possible solution was found in the so-called supergravity, mentioned below in §B6 "Unification of fundamental interactions. Supergravity. Superstrings.".

It can be said that at present the main problem is not the construction of a quantum theory of gravity. On the contrary, there are a number of variants of the quantum general theory of relativity, but they show certain ambiguities (eg in the choice of canonical variables, renormalization, physical interpretation, etc.). Further stages of the development of quantum GTR should result in a situation where gravitational and non-gravitational quantum theories will be special cases of general quantum theory. Until 20 years ago, it seemed that the starting point of this unified theory should be the quantum general theory of relativity. Later successes of quantum field theory in elementary particle physics, however, set a new trend: the construction of unitary theories of fundamental interactions, including gravitational (supergravity), on the soil of quantum field theory. Recently, the possibilities of geometric formulation of supergravity unitary theories in multidimensional spaces have been explored (eg generalized Kaluza-Klein theories, superstring theories - see the following §B.6 "Unification of fundamental interactions. Supergravities. Superstrings.").

(no)Possibilities of experimental verification of quantum gravity
Physics, as a natural science, necessarily requires that every theory be verified by experiment or observation. And here perhaps there is even more difficulty than in the actual formulation of the quantum theory of gravity. The smallest spatial and mass scales on which it is possible to verify general relativistic gravitational phenomena in today's experimental technique are of the order of about 10⁵ km and 10²² kg. At smaller spatial scales (eg kilometers) and for smaller masses (of the order of kilograms) we cannot experimentally distinguish between Newton's theory of gravitation and the general theory of relativity *). And there can be no talk of experimental analysis of gravity in subatomic microscales in the foreseeable future! So far, we have had difficulty detecting gravitational waves from black hole fusion; but there is no hope of experimental demonstration of their quanta, gravitons. Quantum theories of gravitation are likely to remain at the level of experimentally unproven hypotheses for a long time to come. The only way to confront these theories with reality for the time being will probably be to observe its consequences for phenomena in the very beginnings of the evolution of the universe (similar to that of unitary theories, such as superstrings) - cf. §5.5 and 5.8.
*) Other interactions, on the other hand, are experimentally verified on significantly smaller scales, while not on a larger scale. For weak and strong interactions, of course, this is on a scale comparable to the atomic nucleus. Although the electromagnetic interaction has an unlimited range, it is experimentally directly verified, except for the microworld, only at scales of the order of kilometers. Celestial bodies are practically electrically neutral. The only verification of electrodynamics on a large scale is the propagation of electromagnetic waves in space.
  The question of what role gravity plays in the structure and interactions of subnuclear elementary particles will probably remain open for a long time. No role, marginal or determining role in particle structure? (discussed in §B.6, passage "Superunification and Supergravity").

Quantum or non-quantum nature of gravity ?
In current physics, gravity is generally assumed to be quantum at a fundamental level, just like other interactions. In principle, however, it is possible to explore the alternative possibility that gravity is intrinsically non-quantum ("classical"), but the interpretation of some of the dynamical laws of quantum field theory would be suitably modified so as to maintain agreement with experiments and observations while at the same time achieving theoretical correspondences between GTR and quantum physics.
 Quantum superposition of particles and their gravity
The basis of the understanding of quantum mechanics is the so-called double-slit experiment, where individual particles are directed to two very narrow and closely spaced slits in an otherwise impermeable shielding (described in §1.1, passage "Corpuscular-wave dualism"). The particles behave as if they pass through both slits at the same time and on the other side of the partition they would interfere with themselves and fall into the places described by the wave function. However, if we measure which slit the particle passes through, there is no interference and the particles fall into places opposite the slits. This works for photons, electrons, and heavier and more complex particles.
  From the point of view of the relationship between gravity and quantum physics, it may be interesting to ask the question: "What happens to the gravitational field of a mass particle when it passes through a double slit?". If a particle passes straight through, without interference, its gravitational field will match where it is currently on its trajectory in front of and behind the slit. A more complicated situation occurs when a particle passes seemingly through both slits, with interference on the other side. Does the particle's gravitational field also split as if it were distributed according to a stochastic wave distribution and then converge at the point of positive interference? This would be the case if gravity were quantum in nature. However, if the particle's gravity followed a classical trajectory, it would indicate that gravity is not quantum in nature...?..
  We can imagine other implications of a thought experiment with a particle passing through a double slit that quantumly interferes behind the slits. If the gravitational field generated by this particle is classical, we can in principle measure it with high enough accuracy to determine the position of the particle without disturbing it. This measurement should make it possible to reveal which slit the particle passed through. Would that prevent the particles from superposing? - why? Or would we violate the quantum uncertainty principle by knowing the particle's position and momentum with too much precision? The relations between the structure of spacetime - gravity - and quantum physics could also show the hidden determinism lying somewhere deep in quantum physics..?..
  No physical experiments of this kind will be feasible in the foreseeable future, so far these are only hypothetical thought experiments.

The mystery of the quantum energy of the vacuum <-> Cosmological constant
According to quantum field theory (§1.2, passage "Quantum field theory"), even an ideal vacuum is not a completely empty space, but quantum fluctuations take place in it - it is filled with virtual quanta, particles and antiparticles, which are constantly created and instantly disappear. The vast majority of interactions of virtual quanta that arise in this way cancel each other out on average, and the vacuum basically appears as empty space. Quantum physics requires that the interactions of all possible quanta and particles be included, giving a sum total of infinite values, which are removed by renormalization.
  From the point of view of the general theory of relativity, these quantum fluctuations create energy, which due to universal action is added to the cosmological constant, considered the immanent energy of the vacuum. Using the astronomically determined cosmological constant, the energy density of the vacuum is determined to be approximately 10^-9 Joule/m³ (~5 GeV/m³). If we wanted to explain the cosmological constant as vacuum energy coming from virtual quantum fluctuations, quantum physics would provide an incomparably larger absurd value of ~10⁶⁰ -10¹¹³ J/m³ by including a huge amount of quantum fluctuations! This huge mismatch of up to 120 orders of magnitude is sometimes metaphorically called a "vacuum catastrophe"...
  A completely "meticulous" quantization of the gravitational field, including the inclusion of virtual particles of quantum fluctuations that either diverge at Planck microscales or give a completely unrealistically high vacuum energy value relative to the cosmological constant, may not be justified. It is likely that only real particles created by interactions can contribute to gravity (including dark energy), not all the virtual ones that could be drawn in Feynman diagrams. After all, it should be kept in mind that we don't have (and probably never will have) any experimental data about the real shape of gravitational action in quantum microscales, it's all just far-reaching extrapolations from large scales..?..
  From a realistic point of view, we can consider the cosmological constant as another independent astrophysical constant that no theory can explain.
  From another point of view, the vacuum quantum energy problem is discussed in §B.4, the passage "Giant quantum microfluctuations".

Some new alternative theories related to quantum gravity :

Discrete structure of spacetime
A common denominator of a number of problems with singularities and infinities in nonquantum and quantum field theory are limit transitions of the type r ® 0, x ® 0, t ® 0, etc., in which field values (or even space-time metrics) often diverge. These limit transitions are made possible by the continuous nature of spacetime, in which we can investigate the behavior of physical fields in principle up to infinitesimal scales of a geometric point.
  But what if, however, the continuity of space-time is the same illusion, as it was until the 19th century the continuity of matter? As modern physics has come to know about the discrete quantum structure of matter, the hypothesis is offered that space-time is also quantized - consists of a huge but numerous number of very small, indivisible elementary "cells", of a kind of "space-time dust". If these hypothetical "quantum geometries" are small enough, e.g. of the order of Planck's length 10^-33 cm, spacetime appears to be completely continuous, as no physical processes investigated so far can distinguish finer distances than about 10^-15 cm.
Note: We have already encountered the quantum structure of spacetime in the previous §B.4 in the field of quantum geometrodynamics. However, it was an "induced" quantum structure created by using the laws of quantum physics on gravity as a curved spacetime. However, this is the primary one, an axiomatically postulated discrete structure of spacetime itself (so to speak "from God").
  In such a quantum geometry with a discrete spacetime structure, there are no limit transitions to zero spatial distances and time intervals, so infinite divergent values of fields and non-physical singularities of spacetime should not arise.
This idea could be important for quantum field theory in general, as the energy of quantums is inversely proportional to the wavelength of the respective "wave ball". If the wavelengths cannot be less than a certain lower limit, because a shorter length simply does not exist, then the energy of the quanta is limited from above - quanta with infinite energy, which cause such difficulties in quantum field theories, are precluded.
 Loop quantum gravity
On the development of the above-mentioned ideas about the nature of space for discrete connection with quantum field theory is based so called loop quantum gravity theory [...]. Since the rotational momentum - spin plays an important role in quantum physics, not Cartesian coordinate networks were used to quantify spacetime, but it is modeled using the so-called spin network . ....
  The founder of loop quantum gravity is Abhay Ashketar, a physicist of Indian descent who works at the University of Pennsylvania. The theory originated and developed from the 1980s and 1990s and T.Jacobson, L.Smolin, C.Rovelli, J.Baez, Ch.Isham, M.Boyowald and other pioneers of loop quantum gravity work on it together with Ashtekar.
  A. Ashketar first expressed the 3-dimensional space metric using new variables using the SU(2) (or SO(3)) formalism of the calibration field symmetries. His collaborators then showed that the Hilbert space of the quantized calibration field SU(2) can be generated by so-called spin networks, based on the twistor theory *) of R.Penrose [...]. These spin networks are taken as basic fundamental elements in loop theory  creating a structure of space that thus becomes discrete. It can be imagined as a kind of "skeleton" or "kit" of one-dimensional fibers (graphs). The spin network does not exist in some space into which it would be nested, but the space itself creates (or creates what we perceive as space on a larger scale). At the submicroscopic level, space is no longer homogeneous, but has a fine-grained structure - it consists of countless interconnected "rings" or "loops" of Planck's dimensions .
*) We did not introduce twistors in our book, the author is not convinced of their importance... Otherwise, however, we used and discussed a number of other ideas and concepts of the excellent relativistic physicist R.Penrose.
  The development of the spin network over time creates a kind of "spin foam", which can be put into the context of the above-mentioned quantum approach Feynman path integrals.
................
.....? ...... here comes a more specific outline of the methodology of loop gravity ....? .....
.........
The loop quantum theory of gravity is considered to be a certain alternative approach, to some extent competitive with the theory of superstrings (briefly outlined at the end of the following §B.6 "Unification of fundamental interactions. Supergravity. Superstrings."). Ashketar's loop quantum gravity does not aim to unify the four basic interactions - electromagnetic, strong, weak and gravitational, but "only" to unite quantum theory with gravity as space-time physics. Therefore, we have included a brief mention of it in this §B.5 on quantum gravity, while superstrings in the following §B.6 on unitary field theories ...
  The real contribution of the loop theory of gravity is the concept of a discrete structure of spacetime. However, the current specific formulation of "spin foam", with its excessive mathematical complexity and "artificiality", probably disqualifies it as a credible theory of the real physical nature of gravity and other fields (in the spirit of the principle of simplicity of fundamental knowledge)..?..

Entropic nature hypothesis of gravity
In §4.7 "Quantum radiation and the thermodynamics of black holes" were from the point of view of thermodynamics and information statistics were analyzed the connection between entropy and gravity of a black hole, which in the 70s and analyzed J.D.Bekenstein and S.Hawking is added concepts of quantum evaporation of black holes. This created the "thermodynamics of black holes". All this was within the curved spacetime of the general theory of relativity, against which the processes of quantum field theory were analyzed. Entropy S (defined as the ratio of the heat dQ supplied to the system to the absolute temperature T: dS = dQ/T) here it was considered to be an important quantity, which, however, arose secondarily from specific interactions of quantums and particles with each other and with the gravitational field - as a close analogy.
  From the completely opposite side to this issue came up the Dutch physicist Erik Verlinde [E.P.Verlinde: On the Origin of Gravity and the Laws of Newton; arXiv: 1001.0785 (2010)] , which was based on the well-known fact that with each macroscopic process in a continuous environment, entropy changes from a thermodynamic point of view. E.g. when we stretch the spring or rubber band, it tends to return to its original length - we feel an elastic force (macroscopically described by Hooke's law), whose physical origin is in the electromagnetic interaction between the atoms of matter. At the same time, from a thermodynamic point of view, it is accompanied by a change in entropy: a stretched spring or rubber band has a lower entropy than in the equilibrium contracted state, into which it tries to get in accordance with the 2nd Act of Thermodynamics. The energy balance during stretching by Dx can be expressed here by the entropy S using the relation:
            F. Dx = T. DS,
where F is the elastic force, T is the thermodynamic temperature. The product T.DS is in thermodynamics the increase of thermal energy DQ due to the work F.Dx performed when the spring is stretched by Dx.
  When this analogy is logically reversed, the elastic force can be considered as a consequence of the growth of entropy - a kind of "entropic force". It was this logical inversion that E.Verlinde used in 2010 for his peculiar hypothesis about the origin of gravity. Verlinde declared entropy to be the primary very fundamental quantity whose growth causes the "entropic force that is the essence of gravity. So there is no gravity ! *) - we observe it only as an apparent force, a macroscopic manifestation of entropy growth with random statistical behavior of microscopic particle-quantum field systems. The entropic force causes the motion of bodies, which leads to mechanical work dA = F.dx. This is manifested by heat leading to a change in entropy: T.dS = F.dx (ie F = T. grad S) .
*) This concept is heuristically somewhat similar to LeSage's hypothesis of the mechanical origin of gravity, discussed in §1.3 "Mechanical LeSage hypothesis of gravity", according to which gravity as a separate force also does not exist, is caused by impacts of small moving particles ...
  The second basic starting point of Verlinde theory is a holographic principle (see §4.7, passage "Loss of information in a black hole. Holographic principle."): information - entropy - is located on the border of the investigated system - holographic projection screen. This made it possible to "derive" Newton's law of gravity of inverted squares in a simple spherical model with a material body in the middle :
A brief outline: Around the central body mass M to define a spherical surface of radius R. This area in Verlinde theory serves as a holographic projection screen, which is located all the information on the bodyM. To the surface of this sphere can then in this model formally assign thermodynamic temperature  T = Mc²/(2.N.k_B), where k_B is Boltzmann's constant, and N is the number of particles - the body of information bits M. The density of information bits on the projection screen is inversely proportional to the size of the screen, i.e. R^-2, so T ~ M/R². We insert a test particle of mass m into the investigated sphere. When we move it in the radial direction by a distance Dx, the entropy of the sphere changes by DS = m.Dx. Substituting these relations into the above-mentioned Verlind relation F Dx = T.DS for the entropic force we get the ratio F ~ mM/R², which corresponds to the standard Newton's law of gravity inverted squares of distance. Newton's law of gravitation "emerged" from these thermodynamic considerations without assuming the existence of a gravitational force at the microscopic level ...
  In 2016, E.Verlinde further incorporated quantum fluctuations of vacuum into his entropic hypothesis as another source of entropic force. This created another contribution to the "entropic force - gravity", which generally does not decrease with the square of the distance. Verlinde, using his appropriate modeling, tries to alternatively explain the problem of dark matter ("The future evolution of the universe. Hidden matter") - this would be caused by quantum fluctuations inside the holographic projection surface, and dark energy ("Accelerated expansion of the universe? Dark energy?") in space - caused by quantum fluctuations outside this surface and in the whole space [Verlinde E .: Emergent Gravity and the Dark Universe ; arXiv: 1611.02269v2 (2016)] ..?..
  Opponents' frequent objections focus on the spherical symmetry of Verlinde's models, which are not concise for the actual distribution of matter in space; but this could be addressed in further generalizations. However, the fundamental objection against to Verlind's theory is physico-gnoseological: a change in entropy is not the cause gravitational or other forces in physical systems, but only one of the consequences of specific physical interactions taking place in the system. And besides, the second basic postulate, the holographic principle, is only an unconfirmed hypothesis (theoretically based only on the horizons of black holes, see §4.7). So it's actually the opposite: the changes in entropy are caused by, among other things, the gravitational force...
  It will probably be different with that gravity ..!..

B.4. Quantum geometrodynamics		B.6. Unification of fundamental interactions. Supergravity. Superstrings.

Vojtech Ullmann