The Uniquely Consistent and Finite Local Quantum

The Uniquely Consistent and Finite Local Quantum Field Theory of Gravity YVAN LEBLANC Email: [email protected] Affiliation: eFieldTheory.COM PACS: 4.60.+n, 11.17.+y, 97.60.Lf URL: eFieldTheory.COM/abs/?eFTC-180901 c 2018 Yvan Leblanc. All rights reserved Copyright September 27, 2018 Abstract Overviewing various approaches attempting at resolving the Quantum Gravity (QG) problem over the last several decades, I present here a lucid and cold analysis about the current state of affairs regarding the unification of gravity with quantum theory. I easily distinguish the garbage from the good stuff and finally isolate the uniquely workable theory already fully capable of unambiguous and finite predictions at all orders of perturbation theory. Such a theory of gravity is not a String Theory (ST) but the local Yilmaz Quantum Field Theory (QFT) of gravity, Quantum Gravidynamics (QGD). New interpretations and finite evaluations of loop diagrams are presented which can be applied to all QFTs, whether renormalizable or not. Surprising consequences emerge such as the non-existence of anomalies in local QFTs.

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1

Introduction

In his 1992 article entitled Canonical Quantum Gravity and the Problem of Time, Isham [1] discusses the problem of time and its inclusion into Quantum Gravity. He writes: Let me emphasise once more that most of the problems of time in quantum gravity are not associated with the existence of ultraviolet divergences in the weak-field perturbative quantisation; in particular, many interpretative difficulties arise already in infinity-free, minisuperspace models. Therefore, I feel it is correct to say that the problems encountered in unravelling the concept of time in quantum gravity are grounded in a fundamental inconsistency between the basic conceptual frameworks of quantum theory and general relativity. In his abstract, he similarly declares: This problem originates in the fundamental conflict between the way the concept of ‘time’ is used in quantum theory, and the role it plays in a diffeomorphism-invariant theory like general relativity. Isham [1] actually introduces three schemes for resolving this problem: (I) schemes in which time is identified before quantization; (II) schemes in which time is identified after quantization; and (III) schemes where time plays no fundamental role. He then concludes his article as follows [1]: In responding to this situation the main task is to decide whether ‘time’ should preserve the basic role it plays in classical general relativity - something that is most naturally achieved by incorporating it into the quantum formalism by the application of a quantization algorithm to the classical theory - or if it is a concept that should emerge phenomenologicaly from a theoretical framework based on something very different from ‘quantising’ classical general relativity. If the former is true, which suggests a type I approach to the problem, the best bet could be some ‘natural’ choice of internal time dictated by the technical requirements of mathematical consistency in a quantisation scheme; for example the programme currently being pursued by Abhay Ashtekar and collaborators. If the latter is true, two key questions arise: (i) what is this new framework?, and (ii) how, if at all, does it relate to the existing approaches to quantum gravity, especially the semi-classical scheme? In particular, how does the framework yield conventional quantum theory and our normal ideas of space and time in their appropriate domains? The most widely-studied scheme of this sort is superstring theory but, in its current manifestation, this is not well-suited for addressing these basic questions. The idea of strings moving in a spacetime already presupposes a great deal about the structure of space and time; and the quantisation techniques employed presuppose most 2

of structure of standard quantum theory, particularly at a conceptual level. It may well be that a new, non-perturbative approach to superstring theory will involve a radical reappraisal of the ideas of space, time and quantum theory; but this remains a task for the future. Perhaps the answer is to find a superstring version of Ashtekars formalism (or an Ashtekarisation of superstring theory), and with the conceptual aspects of quantum theory being handled by a consistent-histories formalism. A nice challenge for the next few years! C. Isham On the contrary, the present work will not address the problem of quantum gravity from that angle at all, although Isham’s paper raises legitimate philosophical questions. The anxiety wrt to the theoretical role of Time in gravity seems to be symptomatic of a theorist’s point of view whose distance from the real astrophysical or experimental situation has been overstretched. In a recent exchange earlier this year with Prof. Stanley L. Robertson [2] from Southwestern Oklahoma State University, I was asked if, to my experience, general relativist theorists seemed to be of the opinion that it is quantum field theory that must change in order to reach an accommodation with general relativity. To this question I replied that, coming from a field theorist research environment, I was raised with the general view that QFT should englobe a quantum gravity theory. Shortly thereafter, I started to analyze in greater details the work of Yilmaz, Alley and others on gravity and Quantum Field Theory [3, 4, 5, 6, 7, 8, 9]. This proved an enlightning experience to me and I soon realized that much of the work had already been done, but not recognized at its just value by the research community. To Prof. Robertson’s question, I now know that Yilmaz would have answered that both theories, gravity and quantum field theory, must meet halfway [6]. Of course, Isham’s point of view is definitely a general relativist’s or mathematical physicist’s point of view, and so he worries about the configurational space concept of time and its inclusion in quantum gravity. On the other hand, a field theorist would instead worry about the phase space concept of energy, which is much closer to the high energy experimental situation. And this is the fundamental problem of Einstein General Relativity (EGR), already at the classical level. Any attempt at quantizing EGR leads nowhere because, fundamentally, there is no localizable concept of energy in that theory. And this has already significant consequences at the classical level where several inconsistencies of the theory have been identified with real observational consequences. The Einstein General Relativity (EGR) theory has known success and fame very early after its publication in 1915. However cracks into its logical structure started to show up early as well. Schr¨ odinger [10] first noticed that the Einstein energy-momentum pseudotensor uµν for the Schwarzschild solution actually vanished in Cartesian coordinates, in contrast to an analogous calculation by Bauer [11] in flat spacetime who found a non trivial answer making use of a system of curvilinear coordinates. This meant that gravitational energy-momentum could be zero in curved spacetime and non-zero in flat spacetime, a contradiction [6]. 3

Einstein [12] explained these results by saying that this was a 1-body problem and that the result would be non-trivial for a N -body (N ≥ 2) solution. He further added that the coordinate dependence of the pseudotensor uµν could be understood by interpreting gravitational energy as being essentially not localizable [12, 6]. Years later however, Yilmaz [4] showed that in the slow motion (time independent) limit, EGR had no N -body interacting solutions, contradicting Newtonian mechanics. He also showed in fact that EGR with an energy-momentum pseudotensor actually constitutes an overconstrained theory as two basic identities, the Freud and Bianchi identities, clashed with each other and led to a structural inconsistency of the theory. As an example of the above difficulty of EGR with the N -body problem, we can mention its failure to predict gravitational attraction in the old two-slab Cavendish experiment [8, 13]. In recent years, Mitra [14, 15, 16, 17, 18] as well as Leiter and Robertson [19] additionally showed that the Schwarzschild solution was inconsistent unless the mass parameter (directly proportional to the horizon radius of the black hole) vanished identically. So EGR does not predict black holes with a finite mass. This resolves in a trivial way the Schrodinger-Bauer inconsistency about the Einstein energy-momentum pseudotensor for the Schwarzschild metric in EGR. Gravitational collapse however continue to be described by the physical solutions of Einstein’s field equations with physical matter sources, but in a way that dynamically prevents the formation of event horizons or trapped surfaces. Such solutions are called Eternally Collapsing Objects (ECOs) or Magnetospheric Eternally Collapsing Objects (MECOs), as discussed by Mitra [14, 15, 16, 18, 20, 21] as well as Robertson and Leiter [22, 23, 24, 25]. These latter authors also found good agreement between the MECO model predictions and astrophysical data for Galactic Black Hole Candidates (GBHCs), Active Galactic Nuclei (AGNs) as well as other compact objects such as Neutron Stars (NSs) [23, 24, 25, 22]. The compact GBHCs and AGNs replace black holes in astrophysics. In spite of all this, the fundamental problems of EGR remain. Its most fundamental problem, and this constitutes the reason for its non-quantizability, is the non-existence of an energy-momentum tensor for the gravitational field itself. Therefore, in this context, there can be no Hamiltonian description for the gravitational field and consequently, no physical gravitational waves carrying energy either, in contradiction with experimental results. Einstein having declared gravitational energy not localizable in EGR, he remained dissatisfied with his theory [8]: My field equations are like a house with two wings: The left-hand side is made of fine marble, but the right-hand side of perishable wood. Albert Einstein From 1958 onward however, Yilmaz [3, 4, 5, 6, 7] laid down the foundation for a fix which repaired the ailments of EGR. He showed that a true energymomentum tensor existed for the gravitational field and that it solved the inconsistency problem between the Freud and Bianchi identities [5, 6, 7, 8, 9].

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The new formulation required however an exact exponential form for the metric tensor, in accordance with Einstein old 1907 [26, 27] requirement which he unexplainably did not include in his final 1915 version of the theory [27]. The exact exponential form for the metric actually can be derived several ways, most notably from the redshift formula of special relativity for accelerating observers, together with the Principle of Equivalence (POE) in the context of the problem of a freely falling elevator in a gravitational field [8]. Yilmaz’s fix for Einstein General Relativity, called Yilmaz General Relativity (YGR) theory, was further shown to allow for interacting N -body solutions, with correct Newtonian mechanics limit. It also provided a consistent solution to the Cavendish problem. YGR is in full agreement with all experimental data regarding gravity and predicts however the non-existence of event horizons and trapped surfaces. Therefore, no black holes in YGR, as it should be if the strong POE (SPOE) is to be satisfied. Robertson and Leiter [22] also showed that the MECO model fits very well the redshift-luminosity data of collapsed astrophysical objects such as Galactic Black Hole Candidates (GBHCs) and Active Galactic Nuclei (AGNs), as well as other compact objects such as Neutron Stars (NSs). Robertson further showed [27, 28] that the compact ECOs and MECOs solutions of EGR still exist in YGR with only minor modifications, and still passing the observational tests of EGR. Therefore classical YGR, not EGR, is the correct starting point for a consistent quantization of gravity with localizable gravitational energy, thereby answering Isham’s point on the fundamental inconsistency between the basic conceptual frameworks of quantum theory and Einstein’s general relativity.

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Extreme Vetting of Current Theories

(a) Strings, p-Branes and M -Theories Today, despite tremendous consistency and calculability problems, string theories (or p-branes or even M -theory) are still regarded as serious contenders for the unification of quantum physics with Einstein General Relativity (EGR). This is done by giving up locality at the fundamental level. The Sherk-Schwarz Planck scale String Theory (ST) [29, 30] is basically nothing but an old Dual Resonance Model (DRM) with the nuclear scale replaced by the Planck scale, a rather bold assumption enabling the interpretation of the closed string spin-2 resonance as a graviton. On the other hand, the unwanted tachyon ground state of the purely bosonic string theories, an infrared instability disaster for critical theories, is again boldly gotten rid of by postulating a new fundamental symmetry between bosons and fermions, namely Supersymmetry (SUSY). New cards are then further added to the house of cards that is String Theory. Criticality of the theory in fact immediately contradicts the physical spacetime dimensionality. A compromise is then found by borrowing from the old ideas of Kaluza and Klein about extra compact dimensions, yielding additional internal symmetry indices, while phenomenologically maintaining at four the number of non compact physical spacetime dimensions.

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Clever patching of the bosonic and fermionic sectors then leads to so-called Heterotic Strings with compact dimensions symmetry groups chosen to allow for the incorporation of the standard model of particle physics. The theory is a clever construct, but in an unstable equilibrium. As soon as SUSY is broken, the house of cards that is String Theory collapses very fast. The tachyon always reappears. The tachyon problem of string theory basically tells us that a non-local QFT cannot describe stable elementary particles [31]. In other words, the quantum excitations of strings (or p-branes) are not fundamental (elementary) excitations. In fact, exactly as in the Dual Models, string theories actually describe unstable composite resonances. Therefore Planck scale string theories, like DRMs, are effective theories and so not suitable as models of elementary particle physics. Non-local means compositeness [31]. Nevertheless, strings and conformal theories remain interesting theories that will keep physicists as well as mathematicians busy for many years to come. But ST does not measure up as an acceptable solution to the Quantum Gravity (QG) problem.

(b) Wheeler-DeWitt Quantum Geometrodynamics Wheeler-DeWitt Quantum Geometrodynamics [32] is a 3-geometry theory for which the quantum states obey the so-called Wheeler-DeWitt functional equation, H(x)|Ψi = 0 , (2.1) where |Ψi is the wave functional (wavefunction) of the universe [33]. It is put together by making use of the Dirac quantization formalism with constraints, the Arnowitt-Deser-Misner (ADM) parametrization and the wave functional on the 3-geometry. However this theory has fundamental problems regarding the physical interpretation of the wave functional [34, 35, 36, 1]. In the abstract of his 1987 article entitled Interpretation of “The Wave Function of the Universe”, Wim B. Drees [34] wrote: Hawking and Hartle interpreted their wave function of the universe as giving the probability for the universe to appear from nothing. However, this is not a correct interpretation, since the normalisation presupposes a universe, not nothing. Transition probabilities require a measure on the initial state and a physical result requires a physical initial state. Let me again cite Isham [1]: In approaches of type II, all the canonical variables are quantised and the constraints are imposed at the quantum level ‘`a la Dirac’ as constraints on allowed state vectors. Unfortunately, there is no universally-agreed way of interpreting the ensuing Wheeler-DeWitt equation; certainly none of the ideas produced so far is satisfactory. However, it must be emphasised that there is no real justification for extending the Dirac approach to constraint generators that are quadratic functions of the momentum variables. Therefore, although

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it may be heretical to suggest it, the Wheeler-DeWitt equation - elegant though it be - may be completely the wrong way of formulating a quantum theory of gravity. C. Isham In a conference talk entitled A view on the problems of Quantum Gravity, T. P. Shestakova [35] wrote the following: . . . the Wheeler-DeWitt Quantum Geometrodynamics (. . . ) deals with 3-geometry being described by 3-metric gij , while the other components of metric tensor, g0µ , are missed in this consideration. They are traditionally believed to be redundant variables, whose role in General Relativity is just to fix a reference frame, and the choice of a reference frame should not affect physical phenomena. . . . on the opposite, gauge variables g0µ play an important role both at classical and quantum levels and must be taken into account in any quantization scheme. T. P. Shestakova Here again, Wheeler-DeWitt Quantum Geometrodynamics does not measure up as an acceptable theory of Quantum Gravity.

(c) Loop Quantum Gravity Loop Quantum Gravity is a quantum gravity theory created in the late eighties and describing the fundamental degrees of freedom in the absence of a fixed background. However let me again cite T. P. Shestakova [35]: An attractive point of Loop Quantum Gravity is that it aims at searching for its own way of description of Quantum Geometry and the Hilbert space of quantum states. At the same time, Loop Quantum Gravity inherits the ideas of the Wheeler-DeWitt Quantum Geometrodynamics which deals with 3-geometry being described by 3-metric gij , while the other components of metric tensor, g0µ , are missed in this consideration. They are traditionally believed to be redundant variables, whose role in General Relativity is just to fix a reference frame, and the choice of a reference frame should not affect physical phenomena. . . . on the opposite, gauge variables g0µ play an important role both at classical and quantum levels and must be taken into account in any quantization scheme. T. P. Shestakova

(d) Holographic and Emergent Gravity As explicitly shown in the context of the pure state semiclassical Leblanc-Harms theory (LHT) [31, 37, 38, 39, 40, 41, 42, 43], Bekenstein-Hawking Black Hole 7

Thermodynamics (BHTD) [44, 45, 46, 47, 48, 49, 50, 51, 52, 53] is inconsistent because it leads to negative heat capacity and thus to the impossibility of thermal equilibrium description. This further means that the interpretation of the black hole instanton period as an inverse canonical temperature is wrong and so no concept of canonical entropy can be defined for a single quantum black hole or black object. Indeed the Laplace transform of the Hawking density of states yields an ugly divergence (for all temperatures) because it grows too fast and as a result, the canonical partition function does not exist, contradicting the finiteness and the interpretation of the Hawking partition function as a canonical partition function from the semiclassical WKB formula. A single black hole cannot be a thermal object and simultaneously have a negative heat capacity. Because the partition function diverges for all temperatures, all black hole or black brane theories do not make any sense as finite temperature theories. Now since Hawking’s entropy (the area law) constitutes the fundamental basis for the so-called Holographic Principle [54], constraining the amount of information (in bits per square meter) contained in a theory of Quantum Gravity, we immediately conclude such a principle to be invalid. Another spinoff theory is Emergent Gravity [55, 56], which again requires concepts such as entropy and holography in Quantum Gravity. That too is therefore invalid. By discarding Black Hole Thermodynamics, the fundamental physical implications are that there is absolutely no link between geometry (horizon) and information (entropy) and that, therefore, the Holographic Principle as well as the Emergent Gravity theory spinoffs cannot be included in the foundation of a consistent theory of Quantum Gravity. In fact, it would be strikingly weird for statistical physics in the form of Black Hole Thermodynamics to become more fundamental than quantum physics at the level of the Planck scale. This is even more so since such a theory yields the inconsistent result of a thermal description with negative heat capacity. Under certain circumstances, a negative heat capacity can occur in physics, but never with systems in thermal equilibrium. Statistical Mechanics does not constitute a fundamental theory of matter. It is by construction a mesoscopic or macroscopic scale theory aiming at providing a physically sensible bulk description of many-body quantum systems and their phase transitions. It does not have the ambition to explain the fundamental nature of the quantum world. For physical quantum states, neither entropy nor temperature are good quantum numbers! It would violate the laws of quantum physics. In spite of all this, there is still a vast number of people believing in the usefulness of the concept of entropy (the area law) in Quantum Gravity, even though entropy arises solely from a sick theory with negative canonical heat capacity. This can only be appropriately called fake physics, ruling an alternative world where negative heat capacity systems can be described by thermal equilibrium. Firewalls and the more recent black hole complexity stories also fall into this category. Of course, the Holographic Principle [54] has absolutely nothing to do with the so-called AdS/CFT holographic duality (correspondance) [57].

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(e) Stochastic Electrodynamics (SED) and QG Stochastic Electrodynamics (SED) is an extension of the de Broglie-Bohm [58, 59] interpretation of Quantum Mechanics (QM) and describes the non-linear and chaotic motion of particles immersed in the stochastic, fluctuating zero-point (vacuum) of the electromagnetic field. This is a deterministic and non-local hidden-variable theory. Calculations by T. H. Boyer [60] in the seventies and eighties in electrodynamics seemed to display classical generation of the (quantum) Planckian distribution for certain shapes of the stochastic classical background. Borrowing from earlier proposal by Sakharov, these ideas have also been applied in the context of gravity [61] and they were thought to lead to a novel path toward Quantum Gravity. Such theories however are not mature enough, both conceptually and analytically, to provide a well defined vision toward Quantum Gravity. But they are interesting and their development is worth pursuing.

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Quantum Gravidynamics (QGD)

(a) Quantizability of YGR We have seen in our introductory remarks that quantizability of General Relativity requires the existence of localizable energy leading to a proper gravitational Hamiltonian, itself depending upon the existence of a gravitational energy-momentum tensor. We have also seen that Einstein General Relativity (EGR) cannot be quantized precisely because the theory does not satisfy this requirement. Yilmaz General Relativity (YGR) on the other hand, fully repairs EGR and provides a consistent energy-momentum tensor for the physical gravitational field, which describes a spin-2 generalization of Newton’s gravitational potential. The metric tensor is then constructed as an exponential functional of such a field, enabling N -body interacting solutions with the correct Newtonian mechanics limit. This follows from Doppler redshift considerations and which has been experimentally verified to high accuracy [62, 4], s g00 (r) . (3.1) ω = ω0 g00 (r0 ) This form strongly suggests a metric with the multiplicative group property of an exponential [4]. The metric is then a functional of a covariant generalization φˆ ≡ φµν of Newton’s gravitational potential with trace φ ≡ φµµ . The additional requirement of Special Relativity (SR) as a boundary condition in the limit of no gravity then leads to the following general form for the metric, ˆ

ˆ

gµν = [ η e2(φI−2φ) ]µν ,

(3.2)

where η = diag(1, −1, −1, −1) and Iˆ is the unit matrix. In the time independent (low velocity) limit (φµν → φ00 → φ), we get the following line element (interval), 2

2

ds2 = e−2φ(r)/c c2 dt2 − e2φ(r)/c d~x2 . 9

(3.3)

In addition to the matter energy-momentum tensor τ µν , a gravitational energy-momentum tensor tµν , unlike in Einstein’s theory, is then found to contribute to the curving of spacetime (gravity gravitates) and so must be added to the rhs of the traditional Einstein field equations. They are then replaced by the following Yilmaz field equations, Gµν = −

8πG µν ( τ + tµν ) , c4

(3.4)

The tensor field φµν is the gravitational analogue of the gauge potential Aν in electrodynamics. In the Lorentz (harmonic) gauge ∂µ g µν = ∂µ φµν = 0, the gravitational field φµν is a solution of the d’Alembert equation,  φµν = τµν ,

(3.5)

with the matter energy-momentum tensor as a source. Again, in the low velocity (time independent) limit, one finds, φ(r) =

GM , c2 r

(3.6)

which yields the Newton’s gravitational potential solution. For multiple sources, the following N-body solution is also easily found, φ(~x) =

G X Mi + constant . c2 i |~x − ~xi |

(3.7)

In the low velocity limit, the gravitational field energy-momentum tensor is given as, 1 τµν = − ∂µ φ∂ ν φ + δµν ∂λ φ∂ λ φ . (3.8) 2 Outside the matter source, Eq. (3.5) becomes the equation for gravitational waves in the vacuum, with arbitrary strength, unlike in EGR. Yilmaz General Relativity (YGR) resolves all the ailments of EGR and is charaterized by the following list of physical results [9]: - Newtonian correspondance in second order - Local correspondance to Special Relativity - Strong Principle of Equivalence - Localized and conserved energy-momentum of the gravitational field - Exact gravitational waves of arbitrary strength - Consistency between Freud and Bianchi identities - Complete agreement with all experimental results and observations

Yilmaz’s QFT of gravity, which is called Quantum Gravidynamics (QGD) [6], is then built from 3 basic rules. First, the construction of the Lagrangian density must satisfy the following 2 rules [6], a requirement generally valid for any gauge theory: 10

Rule no.1 The total Lagrangian of a dynamical system of fields is the sum of the Lagrangians of its individual members plus their interactions. Rule no.2 The interaction term can be absorbed into local gauge covariant derivatives in terms of which the full system appears free. Yilmaz’s QFT of gravity then relies on the Feynman path integral quantization formalism of the field φµν . However, as in the naive quantization of Einstein’s General Relativity (EGR) and because of the dimensionality of Newton’s coupling constant, Yilmaz’s Quantum Gravidynamics (QGD) is not a renormalizable theory. There is absolutely no way around it. It is simply impossible to find a renormalizable local QFT with dimensionless gravitational coupling. So we are faced with a unique option, namely going ahead and recognize non-renormalizability as a physical attribute of local Quantum Gravity. But then, as in naive quantum EGR, we strongly expect ugly divergences to occur in the evaluation of the various Feynman diagrams of the theory. It is here that Yilmaz’s 3rd and final rule of QGD intervenes. It is the most important for the consistency of the theory and is explained in greater details in the following discussion.

(b) Non-Renormalizability of QGD The acceptance of non-renormalizability as a physical attribute of the theory is a pill much easier to swallow when one realizes that the theory can actually be made finite by the same universal substraction scheme used for renormalizable theories and satisfying Bogoliubov’s recursive formula as well. Yilmaz’s fundamental third rule for the unambiguous extraction of finite expressions from the bare Feynman amplitudes is then given as follows [6]: Rule no.3 The divergent Feynman loop amplitudes AF (s, a) are not physical. The finite and unambiguous physical loop amplitudes Aphys (s, a; a0 ) can be obtained from the Feynman amplitudes by substituting the divergent loop momentum integrals as follows, Z Z n hX i−n π 2 h u is+1 du G(u) . (3.9) d4 k Ai (ki , k)xi ⇒ Γ(n) u0 i=1 where the loop 4-momentum k defines an average kav such that k − kav is the effective loop momentum, G(u) = u−1 is the effective loop propagator with u ≡ [m2 − (k − kav )2 ]av , n is the number of the loop internal lines, t is the degree of the trace and s = 2 − n + 2t ≥ 0 is the loopPindex. The averages arePover normalized Feynman n n parameters ( i=1 xi = 1) and so kav = i=1 xi ki . The above generic integral is evaluated as follows [6], hZ u is+1 K(s, u; u0 ) ≡ du G(u) u0

=

X us  u  ln + Bj u0s−j (uj − uj0 ) . s! u0 j 11

(3.10)

wich is a homogeneous function of u and u0 , K(s, λu; λu0 ) = λs K(s, u; u0 ) .

(3.11)

As noted by Wightman and Woo [6], Yilmaz’s method (YM) of finite evaluation of Feynman diagrams is very closely related to the method used by Appelquist [63] for renormalizable theories, and more particularly to the Berg`ere and Zuber method (BZM) [64], also using parametric (Schwinger) integral representations and generalizing Appelquist’s result to any theory, renormalizable or not. Both YM and BZM are applicable to general meromorphic functions. Yilmaz’s formula however is slightly more general as it involves a finite lower bound u0 , hZ u is+1 ∂ s+1 1 f (u) . (3.12) du f (u) − f (u0 ) − . . . − (u − u0 )s f (s) (u0 ) = s! ∂us+1 u0 for YM and, f (u) − f (0) − . . . −

1 s (s) u f (0) = s!

Z

1

dζ 0

1 ∂ s+1 (1 − ζ)s s+1 f (ζu) . s! ∂ζ

(3.13)

for BZM. In the case of Feynman diagrams, we have, ∂ s+1 1 f (u) = , u ∂us+1

(3.14)

and so Eq. (3.12) justifies the previous expression given by Eq. (3.9) in Rule no.3. The Berg`ere and Zuber method [64] (and therefore Yilmaz’s method [6]) is more general than that of Appelquist because the general formula is completely independent of the topology of the various Feynman graphs (except for the number of internal lines) and actually remains valid for non-renormalizable theories. It has been shown to be equivalent to Zimmermann’s R-operation of BPHZ-theory for renormalizable scalar field theories as well as theories with higher spins, modulo a finite renormalization [64]. Rule no.3 of Yilmaz’s theory therefore constitutes a universal substraction procedure to extract unambiguously the finite part of Feynman amplitudes, without bothering about the regularization methods or the renormalization of the physical parameters. It is equivalent to the Berg`ere and Zuber formula, and consequently actually works for all renormalizable and non-renormalizable theories. It is a powerful generalization of renormalization theory. We need to come to terms with the idea that renormalizability should constitute a fundamental physical requirement for acceptable QFTs. On the contrary, abandoning this requirement already led us to great progress in many-body theory, such as in the superconductivity model in condensed matter theory or the shell model in nuclear theory. These are not renormalizable theories, yet they are good physical models. The relaxation of the requirement of renormalizability in high energy physics enables us to include at last finite quantum effects from the gravitational interactions with the other physical fields.

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4

Conclusions

Recalling Prof. Robertson’s question [2] and its effective answer from the Yilmaz theory, one sees that to obtain the consistent theory of Quantum Gravidynamics (QGD), the ailments of Einstein’s General Relativity (EGR) had to be repaired first at the classical level, which led to Yilmaz’s General Relativity (YGR). In the Yilmaz form, General Relativity has become quantizable because the gravitational energy is then localizable. But it remains non-renormalizable because of the dimensionality of Newton’s constant. However, by relaxing the general requirement of renormalizability in physical QFT, considerable freedom is obtained. In particular, thanks to the YM or BZM unambiguous infinity removal procedures, equivalent to the R-operation in BPHZ renormalization theory, amplitudes of all local QFTs are guaranteed to be finite without recourse to regularization and renormalization. Such procedures actually lead to new findings such as the vanishing of anomaly graphs in perturbation theory [6]. The π o → 2γ decay should then be accounted for by other processes elsewhere from hadronic physics [6]. Of course, YM or BZM must be understood much more deeply from the physical point of view. In the case of non-renormalizable theories (and renormalizable theories as well), Yilmaz [6] suggested a physical re-interpretation of the loop themselves seen as fluctuations in energy-momentum instead of internal propagation, a re-interpretation linked to dynamical causality [6] inside given Feynman diagrams written in a parametric integral representation. So interesting work remains for the future. In conclusion, Quantum Gravidynamics (QGD) has done it. It is good and ready to predict or verify quantitatively any experimental results regarding Quantum Gravity. This theory reigns over the dead enders, the immature, the slow moving, the Lucy in the sky or the otherwise fake physics programmes. Finally, gauge fixing issues need to be resolved in both YGR and QGD as well. Regarding classical YGR theory in particular, it is important to understand better the structure of the theory beyond the Lorentz (harmonic) gauge so as to allow for a wider resolution of physical problems such as the interior of stars [2] as well as cosmological problems with high density or pressure.

Acknowledgements I sincerely thank Prof. Stanley L. Robertson for sharing his enlightning expertise and refreshingly original viewpoints on special and general relativity, and most importantly on the Yilmaz theory of gravity.

References [1] C. J. Isham, “Canonical Quantum Gravity and the Problem of Time”, Lecture at NATO Summer School (Salamanca, Spain, June 1992). Imperial/TP/91-92/25 (1992); arXiv:gr-qc/9210011. [2] Stanley L. Robertson, Private communication (April 2018). [3] Huseyin Yilmaz, “New Approach to General Relativity”, Phys. Rev. 111, 1417 (1958); doi:10.1103/PhysRev.111.1417. 13

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