About GRT limit of TGD

TGD should have General Relativity type theory as an appropriate limit. Therefore it is interesting to see what one obtains when one applies TGD picture by replacing space-times as 4-surfaces with abstract geometries as in Einstein's theory and assumes holography in the sense that space-times satisfy besides Einstein-Maxwell equations also conditions guaranteeing Bohr orbit like property. The resulting picture could be also regarded as quantized GRT type limit of quantum TGD obtained by dropping the condition that space-times are surfaces. This limit could also provide totally new insights to the quantization of GRT.

Several pleasant surprises were in store.

1. Essentially the same formalism could apply to GRT limit of TGD as TGD itself meaning that Einstein-Maxwell system can be described as almost topological QFT with holography implying that action reduces to 3-D Chern-Simons action with a metric dependent constraint term expressing the weak form of electric-magnetic duality and quantizing electric charge.

2. The existence of this limit gives valuable information also about TGD itself. In particular, the interpretation of the weak form of electric-magnetic duality is sharpened. The space-time regions with Minkowskian signature would be those in which only electromagnetic and gravitational interactions make themselves visible and regions with Euclidian signature would be the interiors of generalized Feynman graphs in which electroweak and color interactions become manifest. In particular, Weinberg angle should vanish in the Minkowskian phase so that electromagnetic field reduces to induced Kähler field identifiable as Maxwell field of Einstein-Maxwell system. This conforms with the finding that Kähler coupling strength equals to fine structure constant within the very tight constraints available.

3. The limit also suggests how one could understand the extremely small value of cosmological constant characterizing the cosmology according to GRT in terms of CP₂ geometry providing idealization for the space-time region with Euclidian signature of metric representing generalized Feynman graphs also in GRT framework.

4. Non-Euclidian regions could correspond also to blackhole like regions in TGD framework, where only part of the interior of black hole is imbeddable. Black holes would naturally correspond to gigantic values of gravitational Planck constant implying that the Compton length of black-hole is of order of Schwartschild radius. Black hole would be elementary parton with very large fermion and antifermion numbers and large Planck constant and consist of dark matter in TGD sense. This picture is mathematically consistent since at event horizon the determinant of four-metric vanishes so that it is light-like just as it is at wormhole throats. Consistency with experimental factors is also achieved: about the interiors of blackholes we know nothing so that nothing prevents from assuming that it has Euclidian signature of metric: especially so if this explains the mysterious cosmological constant and standard model quantum numbers.

GRT is a more general theory than TGD in the sense that much more general space-times are allowed than in TGD - this leads also to difficulties - and one could also argue that the mathematical existence of WCW Kähler geometry actually forces the restriction of these geometries to those imbeddable in M⁴× CP₂ so that the quantization of GRT type theory would lead to TGD.

1. The conceptual framework of TGD

There are several reasons to expect that something analogous to thermodynamics results from quantum TGD. The following summarizes the basic picture, which will be applied to a proposal about how to quantize (or rather de-quantize!) Einstein-Maxwell system with quantum states identified as the modes of classical WCW spinor field with spinors identifiable in terms of Clifford algebra of WCW generated by second quantized induced spinor fields of H.

1. In TGD framework quantum theory can be regarded as a "complex square root" of thermodynamics in the sense that zero energy states can be described in terms of what I call M-matrices which are products of hermitian square roots of density matrices and unitary S-matrix so that the moduli squared gives rise to a density matrix. The mutually orthogonal Hermitian square roots of density matrices span a Lie algebra of a subgroup of the unitary group and the M-matrices define a Kac-Moody type algebra with generators proportional to powers of S assuming that they commute with S. Therefore this algebra acts as symmetries of the theory.

   What is nice that this algebra consists of generators multi-local with respect to partonic 2-surfaces and represents therefore a generalization of Yangian algebra. The algebra of M-matrices makes sense if causal diamonds (double light-cones) have sizes coming as integer multiples of CP₂ size. U-matrix has as its rows the M-matrices. One can look how much of this structure could make sense in GRT framework.

2. In TGD framework one is forced to geometrize WCW consisting of 3-surfaces to which one can assign a unique space-time surfaces as analogs of Bohr orbits and identified as preferred extremals of Kähler action (Maxwell action essentially). The 3-surfaces could be identified as the intersections space-time surface with the future and past light-like boundaries causal diamond (CDs analogous to Penrose diagrams). The preferred extremals associated with the preferred 3-surfaces allow to realize General Coordinate Invariance (GCI) and its natural to assign quantum states with these.

   GCI in strong sense implies even stronger form of holography. Space-time regions with Euclidian signature of metric are unavoidable in TGD framework and have interpretation as particle like structure and are identified as lines of generalized Feynman diagrams. The light-like 3-surfaces at which the signature of the induced metric changes define equally good candidates for 3-surfaces with which to assign quantum numbers. If one accepts both identifications then the intersections of the ends of space-time surfaces with these light-like surfaces should code for physics. In other words, partonic 2-surfaces plus their 4-D tangent space-data would be enough and holography would be more or less what the holography of ordinary visual perception is!

   In the sequel the 3-surfaces at the ends of space-time and and light-like 3-surfaces with degenerate 4-metric will be referred to as preferred 3-surfaces.

3. WCW spinor fields are proportional to a real exponent of Kähler function of WCW defined as Kähler action for a preferred extremal so that one has indeed square root of thermodynamics also in this sense with Kähler essential one half of Hamiltonian and Kähler coupling strength playing the role of dimensionless temperature in "vibrational" degrees of freedom. One should be able to identify the counterpart of Kähler function also in General Relativity and if one has Einstein-Maxwell system one could hope that the Kähler function is just the Maxwell action for a preferred extremal and therefore formally identical with the Kähler function in TGD framework.

   Fermionic degrees of freedom correspond to spinor degrees of freedom and are representable in terms of oscillator operators for second quantized induced spinor fields. This means geometrization of fermionic statistics. There is no quantization at WCW level and everything is classical so that one has "quantum without quantum" as far as quantum states are considered.

4. The dynamics of the theory must be consistent with holography. This means that the Kähler action for preferred extremal must reduce to an integral over 3-surface. Kähler action density decomposes to a sum of two terms. The first term is j^αA_α and second term a boundary term reducing to integral over light-like 3-surfaces and ends of the space-time surface. The first term must vanish and this is achieved if the Kähler current j^α is proportional to Abelian instanton current

   j^α ∝ *j^α=ε^{α β γ δ}A_βJ_{γ δ}

   since the contraction involves A_α twice. This is at least part of the definition of preferred extremal property but not quite enough. Note that in Einstein-Maxwell system without matter j^α vanishes identically so that the action reduces automatically to a surface term.

5. The action would reduce reduce to terms which should make sense at light-like 3-surfaces. This means that only Abelian Chern-Simons term is allowed. This is guaranteed if the weak form of electric-magnetic duality stating

   *F^nβ=kF^nβ

   at preferred at light-like throats with degenerate four-metric and at the ends of space-time surface. These conditions reduce the action to Chern-Simons action with a constraint term realizing what I call weak form of electric-magnetic duality. One obtains almost topological QFT since the constraint term depends on metric. This is of course what one wants.

   Here the constant k is integer multiple of basic value which is proportional to g_K² from the quantization of Kähler electric charge which corresponds to U(1) part of electromagnetic charge. Fractional charges for quarks require k=ng_K²/3. Physical particles correspond to several Kähler magnetically charged wormhole throats with vanishing net magnetic charge but with non-vanishing Kähler electric proportional to the sum ∑_i ε_i k_iQ_m,i, with ε_i=+/- 1 determined by the direction of the normal component of the magnetic flux for i:th throat.

   The first guess is that the length of magnetic flux tube associated with the particle is of order Compton length or perhaps corresponds to weak length scale as was the original proposal. The screening of weak isospin can be understood as magnetic confinement such that neutrino pair at the second end of magnetic flux tube screens the weak charged leaving only electromagnetic charge. Also color confinement could be understood in terms of flux tubes of length of order hadronic size scales. Compton length hypothesis is enough to understand color confinement and weak screening.

   Note that 1/g_K² factor in Kähler action is compensated by the proportionality of Chern-Simons action to g_K². This need not mean the absence of non-perturbative effects coming as powers of 1/g_K² since the constraint expressing electric magnetic duality depends on g_K² and might introduce non-analytic dependence on g_K².

6. In TGD the space-like regions replace black holes and a concrete model for them is as deformations of CP₂ type vacuum extremals which are just warped imbeddings of CP₂ to M⁴× CP₂ with random light-like random curve as M⁴ projection: the light-like randomness gives Virasoro conditions. This reflects as a special case the conformal symmetries of light-like 3-surfaces and those assignable to the light-like ends of the CDs.

One could hope that this picture more or less applies for the GRT limit of quantum TGD.

2. What one wants?

What one wants is at least following.

1. Euclidian regions of the space-time should reduce to metrically deformed pieces of CP₂. Since CP₂ spinor structure does not exist without the coupling of the spinors to Kähler gauge potential of CP₂ one must have Maxwell field. CP₂ is gravitational instanton and constant curvature space so that cosmological constant is non-vanishing unless one adds a constant term to the Maxwell action, which is non-vanishing only in Euclidian regions. It is matter of taste, whether one regards V₀ as term in Maxwell action or as cosmological constant term in gravitational part of the action. CP₂ radius is determined by the value of this term so that it would define a fundamental constant.

   This raises an interesting question. Could one say that one has a small value of cosmological constant defined as the average value of cosmological constant assignable to the Euclidian regions of space-time? The average value would be proportional to the fraction of 3-space populated by Euclidian regions (particles and possibly also macroscopic Euclidian regions). The value of cosmological constant would be positive as is the observed value. In TGD framework the proposed explanation for the apparent cosmological constant is different but one must remain open minded. In fact, I have proposed the description in terms of cosmological constant also as a proper description in the approximation to TGD provided by GRT like theory. The answer to the question is far from obvious since the cosmological constant is associated with Euclidian rather than Minkowskian regions: all depends on the boundary conditions at the wormhole throats where the signature of the metric changes.

2. One can also consider the addition of Higgs term to the action in the hope that this could allow to get rid of constant term which is non-vanishing only in Euclidian regions. It turns turns out that only free action for Higgs field is possible from the condition that the sum of Higgs action and curvature scalar reduces to a surface term and that one must also now add to the action the constant term in Euclidian regions. Conformal invariance requires that Higgs is massless.

   The conceptual problem is that the surface term from Higgs does not correspond to topological action since it is expressible as as flux of Φ∇ Φ. Hence the simplest possibility is that Kähler action contains a constant term in Euclidian regions just as in TGD, where curvature scalar is however absent. Einstein-Maxwell field equations however apply that it vanishes and is effectively absent also in GRT quantized like TGD.

3. Reissner-Nordström solutions are obtained as regions exterior to CP₂ type regions. In black hole horizon the metric becomes light-like and the solution can be glued to a deformed CP₂ type region with metric becoming degenerate at the 3-surface involved. This surface corresponds to wormhole throat in TGD framework. Blackhole is replaced with CP₂ type region. In TGD black hole solutions indeed fail to be imbeddable at certain radius so that deformed CP₂ type vacuum extremal is much more natural object than black hole. In the recent framework the finite size of CP₂ means that macroscopic size for the Euclidian regions requires large deformation of CP₂ type solution.

   Remark: In TGD framework large value of hbar and space-time as 4-surface property changes the situation. The generalization of Nottale's formula for gravitational Planck constant in the case of self gravitating system gives hbar_gr= GM²/v₀, where v₀/c<1 has interpretation as velocity type parameter perhaps identifiable as a rotation velocity of matter in black hole horizon. This gives for the Compton length associated with mass M the value L_C= hbar_gr/M= GM/v₀. For v₀=c/2 one obtains Scwartschild radius as Compton length. The interpretation would be that one has CP₂ type vacuum extremal in the interior up to some macroscopic value of Minkowski distance. One can whether even the large voids containing galaxies at their boundaries could correspond to Euclidian blackhole like regions of space-time surface at the level of dark matter.

4. The geometry of CP₂ allows to understand standard model symmetries when one considers space-times as surfaces. This is not necessarily the case for GRT limit.

   1. In the recent case one has different situation color quantum numbers make sense only inside the Euclidian regions and momentum quantum numbers in Minkowskian regions. This is in conflict with the assumption that quarks can carry both momentum and color. On the other, color confinement could be used to argue that this is not a problem.

   2. One could assume that spinors are actually 8-component M⁴× CP₂ spinors but this would be somewhat ad hoc assumption in general relativistic context. Also the existence of this kind of spinor structure is not obvious for general solutions of Einstein-Maxwell equations unless one just assumes it.

   3. It is far from clear whether the symplectic transformations of CP₂ could be interpreted as isometries of WCW in general relativity like theory. These symmetries certainly act in non-trivial manner on Euclidian regions but it is highly questionable whether this could give rise to a genuine symmetry. Same applies to Kac-Moody symmetries assigned to isometries of M⁴× CP₂ in TGD framework. These symmetries are absolutely essential for the existence of WCW Kähler geometry in infinite-D context as already the uniqueness of the loop space Kähler geometries demonstrates (maximal group of isometries is required by the existence of Riemann connection).

      Note that a generalization of Equivalence Principle follows in TGD framework from the assumption that coset representations of super-conformal symplectic algebra and super Kac-Moody algebra define conformally invariant physical states. The equality of gravitational and inertial masses follows from the condition that the actions of the super-generators of two algebras are identical. This also justifies the use p-adic thermodynamics for the scaling generator of either super-conformal algebra without a loss of conformal invariance.

5. One could argue that GRT limit does not make sense since in Minkowskian regions the theory knows nothing about the color and electroweak quantum numbers: there is only metric and Maxwell field. On the other hand, in TGD one has color confinement and weak screening by magnetic confinement. If the functional integral over Euclidian regions representing generalized Feynman diagrams is enough to construct scattering amplitudes, pure Einstein-Maxwell system in Minkowskian regions might be enough. All experimental data is expressible in terms of classical em and gravitational fields. If Weinberg angle vanishes in Minkowskian regions, electromagnetic field reduces to Kähler form and the interpretation of the Maxwell field as em field should make sense. The very tight empirical constraints on the value of Kähler coupling strength α_K indeed allow its identification as fine structure constant at electron length scale.

6. One can worry about the almost total disappearance of the metric from the theory. This is not a problem in TGD framework since all elementary particles correspond to many-fermion states. For instance, gauge bosons are identified as pairs of fermion and antifermion associated with opposite throats of a wormhole connecting two space-time sheets with Minkowskian signature of the induced metric. Similar picture should make sense also now.

7. TGD possesses also approximate super-symmetries and one can argue that also these symmetries should be possessed by the GRT limit. All modes of induced spinor field generate a badly broken SUSY with rather large value of N (number of spinor modes) and right-handed neutrino and its antiparticle give rise to N=2 SUSY with R-parity breaking induced by the mixing of left- and right handed neutrinos induced by the modified Dirac equation. This picture is consistent with the existing data from LHC and there are characteristic signatures -such as the decay of super partner to partner and neutrino- allowing to test it. These super-symmetries might make sense if one replaces ordinary space-time spinors with 8-D spinors.

   Note that the possible inconsistency of Minkowskian and Euclidian 4-D spinor structures might force the use of 8-D Minkowskian spinor structure.

3. Preferred extremal property for Einstein-Maxwell system

Consider now the preferred extremal property defined to be such that the action reduces to Chern-Simons action at space-like 3-surfaces at the ends of space-time surface and at light-like wormhole throats.

1. In Maxwell-Einstein system the field equations imply

   j^α=0 .

   so that the Maxwell action for extremals reduces automatically to a surface term assignable to the preferred 3-surfaces. Note that Higgs field could in principle serve as a source of Kähler field but its presence does not look like a good idea since it is not present in the field equations of TGD and because the resulting boundary term is not topological.

2. The condition

   J=k× *J

   at preferred 3-surfaces guarantees that the surface term from Kähler action reduces to Abelian Chern-Simons term and one has hopes about almost topological QFT.

   Since CP₂ type regions carry magnetic monopole charge and since the weak form of electric-magnetic duality implies that electric charge is proportional to the magnetic charge, one has electric charge without electric charge as Wheeler would express it. The identification of elementary building blocks as magnetic monopoles leads in TGD context to the picture about particle as Kähler magnetic flux tubes having opposite magnetic charges at their ends. It is not quite clear what the length of the tubes is. One possibility is Compton length and second possibility is weak length scale and the color confinement length scale. Note that in TGD the physical charges reside at the wormhole throats and correspond to massless fermions.

3. CP₂ is constant curvature space and satisfies Einstein equations with cosmological constant. The simplest manner to realize this is to add to the action constant volume term which is non-vanishing only in Euclidian regions. This term could be also interpreted as part of Maxwell action so that it is somewhat a matter of taste whether one speaks about cosmological constant or not. In any case, this would mean that the action contains a constant potential term

   V= V₀× (1+sign(g))/2 ,

   where sign(g)=-1 holds true in Minkowskian regions and sign(g)=1 holds true in Euclidian regions.

   Note that for a piece of CP₂ V₀ term can be expressed is proportional to Maxwell action and by self-duality this is proportional to instanton action reducible to a Chern-Simons term so that V₀ is indeed harmless from the point of view of holography.

4. For Einstein-Maxwell system with similar constant potential in Euclidian regions curvature scalar vanishes automatically as a trace of energy momentum tensor so that no interior or surface term results and the only surface term corresponds to a pure Chern-Simons term for Maxwell field. This is exactly the situation also in quantum TGD. The constraint term guaranteeing the weak form of electric-magnetic duality implies that the metric couples to the dynamics and the theory does not reduce to a purely topological QFT.

5. In TGD framework a non-trivial theory is obtained only if one assumes that Kähler function corresponds apart from sign to either the Kähler action in the Euclidian regions or its negative in Minkowskian regions. This is required also by number theoretic vision. This implies a beautiful duality between field descriptions and particle descriptions.

   This also guarantees that the Kähler function reducing to Chern-Simons term is negative definite: this is essential for the existence of the functional integral and unitarity of the theory. This is due to the fact that Kähler action density as a sum of magnetic and electric energy densities is positive definite in Euclidian regions. This duality would be very much analogous to that implied by the possibility to perform Wick rotation in QFTs. Therefore it seems natural to postulate similar duality also in the proposed variant of quantized General Relativity.

6. The Kähler function of the WCW would be given by Chern-Simons term with a constraint expressing the weak form of electric-magnetic duality both in TGD and General Relativity. One should be able regard also in GRT framework WCW as a union of symmetric spaces with Kähler structure possessing therefore a maximal group of isometries. This is an absolutely essential prerequisite for the existence of WCW Kähler geometry. The symmetric spaces in the union are labelled by zero modes which do not contribute to the line element and would represent classical degrees of freedom essential for quantum measurement theory. In TGD the induced CP₂ Kähler form would represents such degrees of freedom and the quantum fluctuating degrees of freedom would correspond to symplectic group of δ M⁴_+/-× CP₂.

   The difference between TGD and GRT would be that light-like 3-surfaces for all possible space-times containing Euclidian and Minkowskian regions would be considered for GRT type theory. In TGD these space-times are representable as surfaces of M⁴× CP₂. In TGD framework the imbeddability assumption is crucial for the mathematical existence of the theory since it eliminates space-times with non-physical characteristics. The problem posed by arbitrarily large values of cosmological constants is one of the basic problems solved by this assumption. Also mass density is sub-critical for cosmologies with infinite duration and critical cosmologies are unique apart from their duration and quantum critical cosmologies replace inflationary cosmologies.

7. Note that one could consider assigning the gravitational analog of Chern-Simons term with the preferred 3-surfaces: this kind of term is discussed by Witten in this classic work about Jones polynomial. This term is a non-abelian version of Chern-Simons term and one must replace curvature tensor with its contraction with sigma matrices so that 4-D spinor structure is necessarily involved. The objection is that this term contains second derivatives. In TGD spinor structure is induced from that of M⁴× CP₂ and this kind of term need not make sense as such since gamma matrices are expressed in terms of imbedding space gamma matrices: among other things this resolves the problems caused by the non-existence of spinor structure for generic 4-geometries. The coupling to the metric however results from the constraint term expressing weak form of electric-magnetic duality.

   The difference between TGD and GRT would be basically due to the factor of scattering amplitudes coming from the duality expressing electric-magnetic duality and due to the fact that induced metric in terms of H-coordinates and Maxwell potential is expressible in terms of CP₂ coordinates. The latter implies topological field quantization and many-sheeted space-time crucial for the interpretation of quantum TGD.

4. Could ZEO and the notion of CD make sense in GRT framework?

The notion of CD is crucial in ZEO and one can ask whether the notion generalizes to GRT context. In the previous arguments related to EG the notion of ZEO plays a fundamental role since it allows to replace S-matrix with M-matrix defining "complex square root" of density matrix.

1. In TGD framework CDs are Cartesian products of Minkowskian causal diamonds of M⁴ with CP₂. The existence of double light-cones in curved space-time would be required and its is not clear whether this makes sense generally. TGD suggest that the scales of these diamonds defined in terms of the proper time distance between the tips are integer multiples of CP₂ scale defined in terms of the fundamental constant V₀ (the more restrictive assumption allowing only 2ⁿ multiples would explain p-adic length scale hypothesis but would not allow the generalization of Kac-Moody algebra spanned by M-matrices). The difference between boundaries of GRT CDs and wormhole throats would be that four-metric would not be degenerate at CDs.

2. The conformal symmetries of light-cone boundary and light-like wormhole throats generalize also now since they are due to the metric 2-dimensionality of light-like 3-surfaces. It is however far from clear whether one can have anything something analogous to conformal variants of symplectic algebra of δ M⁴_+/-× CP₂ and isometry algebra of M⁴× CP₂.

   Could one perhaps identify four-momenta as parameters associated with the representations of the conformal algebras involved? This hope might be unrealistic in TGD framework: the basic idea behind TGD indeed is that Poincare invariance lost in GRT is retained if space-times are surfaces in H=M⁴× CP₂. The reason is that that super-Kac-Moody symmetries correspond to localized isometries of H whereas the super-conformal algebra associated with the symplectic group is assignable to the light-like boundaries δ M⁴_+/-× CP₂ of CD of H rather than space-time surface.

3. One could of course argue that some physical conditions on GRT -most naturally just the highly non-trivial mathematical existence of WCW Kähler geometry and spinor structure- could force the representability of physically acceptable 4-geometries as surfaces M⁴× CP₂. If so, then also CDs would the same CDs as in TGD and quantization of GRT would lead to TGD and all the huge symmetries would emerge from quantum GRT alone.

   The first objection is that the induced spinor structure in TGD is not consistent with that natural in GRT. Second objection is that in TGD framework Einstein-Maxwell equations are not true in general and Einstein's equations can be assumed only in long length scales for the vacuum extremals of Kähler action. The Einstein tensor would characterize the energy momentum tensor assignable to the topologically condensed matter around these vacuum extremals and neither geometrically nor topologically visible in the resolution defined by very long length scale. If Maxwell field corresponds to em field in Minkowskian regions, the vacuum extremal property would make sense in scales where matter is electromagnetic neutral and em radiation is absent.

5. What can one conclude?

The previous considerations suggest that a surprisingly large piece of TGD can be applied also in GRT framework and raise the possibility about quantization of Einstein-Maxwell system in terms of Kähler geometry of WCW consisting of 3-geometries instead of 3-surfaces. One can even consider a new manner to understand TGD as resulting from the quantization of GRT in terms of WCW Kähler geometry in the space of 3-metrics realizing holography and making classical theory an exact part of quantum theory. Since the space-times allowed by TGD define a subset of those allowed by GRT one can ask whether the quantization of GRT leads to TGD or at least sub-theory of TGD. The arguments represented above however suggest that this is not the case.

The generalization of S-matrix to a complex of U-matrix, S-matrix and algebra of M-matrices forced by ZEO gives a natural justification for the modification of EG allowing gravitons and giving up the rather nebulous idea about emergent space-time. Whether ZEO crucial for EG makes sense in GRT picture is not clear. A promising signal is that the generalization of EG to all interactions in TGD framework leads to a concrete interpretation of gravitational entropy and temperature, to a more precise view about how the arrow of geometric time emerges, to a more concrete realization of the old idea that matter antimatter asymmetry could be due to different arrows of geometric time for matter and antimatter, and to the idea that the small value of cosmological constant could correspond to the small fraction of non-Euclidian regions of space-time with cosmological constant characterized by CP₂ size scale.

The above considerations were inspired by the attempt to understand what is good and what is bad in the entropic gravity scenario of Verlinde in TGD framework with the basic idea being that quantum TGD as a square root of thermodynamics must predict something analogous to thermalization of the lines of generalize Feynman graphs. The above interpretation for the lines of Feynman graphs as analogs of blackholes indeed allows to understand blackhole temperature and entropy as a manifestation of this underlying thermodynamics. The generalization of blackhole thermodynamics implies that both virtual gravitons and gauge bosons are thermalized. For details see the article TGD inspired vision about entropic gravity.