Journal of Modern Physics, 2019, 10, *-* http://www.scirp.org/journal/jmp ISSN Online: 2153-120X ISSN Print: 2153-1196
The Solution Cosmological Constant Problem
Jaykov Foukzon1, Elena Men’kova2, Alexander Potapov3
1Department of Mathematics, Israel Institute of Technology, Haifa, Israel 2All-Russian Research Institute for Optical and Physical Measurement, Moscow, Russia 3Kotel’nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Moscow, Russia Email: [email protected], [email protected], [email protected]
How to cite this paper: Author 1, Author Abstract 2 and Author 3 (2019) Paper Title. Journal of Modern Physics, 10, *-*. The cosmological constant problem arises because the magnitude of vacuum https://doi.org/10.4236/***.2019.***** energy density predicted by the Quantum Field Theory is about 120 orders of magnitude larger then the value implied by cosmological observations of ac- Received: **** **, *** Accepted: **** **, *** celerating cosmic expansion. We pointed out that the fractal nature of the Published: **** **, *** quantum space-time with negative Hausdorff-Colombeau dimensions can resolve this tension. The canonical Quantum Field Theory is widely believed Copyright © 2019 by author(s) and to break down at some fundamental high-energy cutoff Λ∗ and therefore Scientific Research Publishing Inc. This work is licensed under the Creative the quantum fluctuations in the vacuum can be treated classically seriously Commons Attribution International only up to this high-energy cutoff. In this paper we argue that the Quantum License (CC BY 4.0). Field Theory in fractal space-time with negative Hausdorff-Colombeau di- http://creativecommons.org/licenses/by/4.0/ mensions gives high-energy cutoff on natural way. We argue that there exists Open Access hidden physical mechanism which cancels divergences in canonical
QED44, QCD , Higher-Derivative-Quantum gravity, etc. In fact we argue that corresponding supermassive Pauli-Villars ghost fields really exist. It means that there exists the ghost-driven acceleration of the universe hidden in cosmological constant. In order to obtain the desired physical result we ap-
ply the canonical Pauli-Villars regularization up to Λ∗ . This would fit in the observed value of the dark energy needed to explain the accelerated ex- pansion of the universe if we choose highly symmetric masses distribution
between standard matter and ghost matter below the scale Λ∗ , i.e.,
fsm. (µ) − fgm. ( µµ),, = mc µ ≤ µeff , µ eff c <Λ∗ The small value of the cos- mological constant is explained by tiny violation of the symmetry between standard matter and ghost matter. Dark matter nature is also explained using a common origin of the dark energy and dark matter phenomena.
Keywords Cosmological Constant Problem, Quantum Field Theory, Vacuum Energy Density, Quantum Space-Time, Hausdorff-Colombeau Dimension, Quantum
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Fluctuations, High-Energy Cutoff, Canonical Pauli-Villars Regularization, Universe
1. Introduction 1.1. The Formulation of the Cosmoloigical Constant Problem
The cosmological constant problem arises at the intersection between general relativity and quantum field theory, and is regarded as a fundamental unsolved problem in modern physics. A peculiar and truly quantum mechanical feature of the quantum fields is reminded that they exhibit zero-point fluctuations every- where in space, even in regions which are otherwise “empty” (i.e. devoid of mat- ter and radiation). This vacuum energy density is believed to act as a contribu- tion to the cosmological constant λ appearing in Einstein’s field equations from 1917, 18πG Rµν−= gR µν Tµν′ (1) 2 c4
where Rµν and R refer to the curvature of space-time, gµν is the metric, Tµν′ is the the energy-momentum tensor, 10 00 4 c λ 0− 100 TTµν′ = µν + (2) 8πG 0 0− 10 00 01
where Tµν is the energy-momentum tensor of matter. Thus TT00′ = 00 + ελ ,
TTαβ′ = αβ +δ αβ P λ , where
4 ελλλ=−=Pc8π G. (3)
Reminding that under Lorentz transformations (ελλ,P) →→ εε λ′′ ,,( λλ PP) λ
the quantities ελ and Pλ are changes by the law 22 ελλ++ βPP λλβε ελλ′′= ,.P = (4) 11−−ββ22
Thus for the quantities ελ and Pλ Lorentz invariance holds by Equation (3) [1]. In modern cosmology it is assumed that the observable universe was initially vacuumlike, i.e., the cosmological medium was non-singular and Lorentz inva- riant. In the earlier, non-singular Friedmann cosmology, the Friedmann un- iverse comes into being during the phase transition of an initial vacuumlike state to the state of “ordinary” matter [2] [3]. The Friedmann equations start with the simplifying assumption that the universe is spatially homogeneous and isotropic, i.e. the cosmological principle; empirically, this is justified on scales larger than 126, ~100 Mpc. The cosmo- logical principle implies that the metric of the universe must be of the form Ro-
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bertson-Walker metric [2]. Robertson-Walker metric reads dr 2 2=− 22 ++2θ 2 2 θϕ 2 ds d t at( ) 2 r(d sin d) . (5) 1− kr
For such a metric, the Ricci curvature scalar is Rk= −6 and it is said that space has the curvature k. The scaling factor at( ) rescales this curvature for a given time t, producing a curvature kt( ) = kat( ) . The scaling factor at( ) is given by two independent Friedmann equations for modeling a homogeneous, isotropic universe reads GG a 22=εε a −=−+ ka,3 ( p) (6) 36 and the equation of state pp= (ε ), (7)
where p is pressure and ε is a density of the cosmological medium. For the case of the vacuumlike cosmological medium equation of state reads [2] [3] [4]: p = −ε. (8)
By virtue of Friedman’s Equations (1.1.6) in the Universe filled with a va- cuum-like medium, the density of the medium is preserved, i.e. ε = const , but the scale factor at( ) grows exponentially. By virtue of continuity, it can be as- sumed that the admixture of a substance does not change the nature of the growth of the latter, and the density of the medium hardly changes. This growth, interpreted by analogy with the Friedmann models as an expansion of the un- iverse, but almost without changing the density of the medium! was named in- flation. The idea of inflation is the basis of inflation scenarios [2]. Non-singular cosmology [2] [4] suggests that the initial state of the observable universe was vacuum-like, but unstable with respect to the phase transition to the ordinary non-Lorentz-invariant medium. This, for example, takes place if, by virtue of the equations of state of the medium, a fluctuation decrease in its den- sity d violates the condition of vacuum-like degeneration, p = −ε or, which is the same, 3p +=−<εε 20, replacing it with −2εε < 3p +< 0. (9)
According to Friedman’s equations, it corresponds to an accelerated expan- sion of the cosmological medium, accompanied by a drop in its density, which makes the process irreversible [2]. The impulse for expansion in this scenario, the vacuum-like environment, is not reported to itself (bloating), but to the emerging Friedmann environment. In review [5], Weinberg indicates that the first published discussion of the contribution of quantum fluctuations to the cosmological constant was a 1967 paper by Zel’dovich [6]. In his article [1] Zel’dovich emphasizes that zeropoint energies of particle physics theories cannot be ignored when gravitation is taken into account, and since he explicitly discusses the discrepancy between estimates of vacuum energy and observations, he is clearly pointing to a cosmological con-
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stant problem. As well known zeropoint energy density of scalar quantum field, etc. is divergent 2πc ∞ ε =2 +=∞ 22 2 vac (m) 3 ∫ p mc pd. p (10) (2π) 0
In order to avoid difficulties mentioned above, in article [1] Zel’dovich has applied canonical Pauli-Villars regularization [7] [8] and formally has obtained a finite result (his formulas [1], Eqs. (VIII.12)-(VIII.13) p. 228) 4 1 ∞ c λ ε =−=pf(µµ) 4 (ln µ) d µ= , (11) vac vac 88∫0 πG where ∞∞ ∞ ff(µ)ddd0 µ= ( µµ) 24 µ= f( µµ) µ= . (12) ∫∫00 ∫ 0
Remark 1.1.1. Unfortunately, Equation (11) and Equation (12) give nothing in order to obtain desired numerical values of the zero-point energy density ε . In his paper [1], Zel’dovich arrives at a zero-point energy (his formula (IX.1)) 3 mc 17 3 −−10 2 ελvac = m~10 gcm, ~10 cm , (13) where m (the ultra-violet cut-of) is taken equal to the proton mass. Zel’dovich notes that since this estimate exceeds observational bounds by 46 orders of mag- nitude it is clear that “... such an estimate has nothing in common with reality”. In his paper [1], Zel’dovich wrote: Recently A. D. Sakharov proposed a theory of gravitation, or, more precisely, a justification GR equation based on consider- ation of vacuum fluctuations. In this theory, the essential assumption is that there is some elementary length L or the corresponding limiting momentum
pL0 = . Shorter lengths or for large impulses theory is not applicable. Sakha-
rov gets the expression of gravitational constant G through L or p0 (his for- mula (IX.6)) cL32 c 3 = = G 2 . (14) p0 This expression has been known since the days of Planck, but it was read
“from right to left”: gravity determines the length L and the momentum p0 .
According to Sakharov, L and p0 are primary. Substitute (IX. 6) in the expres- sion (IX. 4), we get mc65 mc67 ρε= = vac 23,.vac 23 (15) pp00 That is expressions that the first members (in the formulas (VIII.10), (VIII.
11)) which are vanishes (with p0 →∞). Thus, we can suggest the following in- terpretation of the cosmological constant: there is a theory of elementary par- ticles, which would give (according to the mechanism that has not been revealed at the present time) identically zero vacuum energy, if this theory is applicable
infinitely, up to arbitrarily large momentum; there is a momentum p0 , beyond
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which the theory is non applicable; along with other implications, modifying the theory gives different from zero vacuum energy; general considerations make it −2 likely that the effect is portional p0 . Clarification of the question of the exis- tence and magnitude of the cosmological constant will also be of fundamental importance for the theory of elementary particles. Nonclassical Assumptions (I) In contrast with Zel’dovich paper [1] we assume that Poincaré group is
deformed at some fundamental high-energy cutoff Λ∗ [9] [10] [11] in accor- dance with the basis of the following deformed Poisson brackets {xxµ, ν} =−= −10( xµηη ν x νµ 0),{ pp µ , ν} 0, (16) {xpµ, ν} =−+ηη µν −10µp ν
µν where µν,= 0,1,2,3 , η =+−−−( 1, 1, 1, 1) and is a parameter identified as the
ratio between the high-energy cutoff Λ∗ and the light speed. The correspond- ing to (16) momentum transformation reads [11]
2 γ ( p− up ) γ ( px − up0 c ) ′′= 0 x = pp0 −−11,,x 11+(c) (γγ −−) p00 upxx 11 +(c ) (γγ −−) p up (17) p p ′′= y = z ppyz−−11,, 11+(c) (γγ −−) p00 upxx 11 +(c ) (γγ −−) p up
and coordinate transformation reads [11] 2 γ (t− ux c ) γ ( x− ut) ′′= = tx−−11,, 11+(c) (γγ −−) p up 11 +( c ) (γγ −−) p up 00xx (18) yz ′′= = yz−−11,, 11+(c) (γγ −−) p00 upxx 11 +( c ) (γγ −−) p up where γ =1 − uc22. It is easy to check that the energy Ec= , identified as
the high-energy cutoff Λ∗ , is an invariant as it is also the case for the funda- = = mental length lΛ∗ cE . Remark 1.1.2. Note that the transformation (17) defined in p-space and the transformation (1.1.18) defined in x-space becomes Lorentz for small energies −1 and momenta and defines a large invariant energy lΛ∗ . The high-energy cutoff
Λ∗ is preserved by the modified action of the Lorentz group [9] [10]. This meant that the canonical concept of metric as quadratic invariant col- lapses at high energies, being replaced by the non-quadratic invariant [9]: ab 2 η pp p = ab, (19) 1+ lp ( Λ∗ 0 )
or by the non-quadratic invariant ab 2 η pp p = ab, (20) 1− lp ( Λ∗ 0 ) =Λ=−1 where lΛ∗∗ , ab , 0,1,2,3 .
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Remark 1.1.3. Note that: 1) the invariant (16) is infinite for the new negative invariant energy scale of Λ=−−1 the theory ∗Λl ∗ , and it’s not quadratic for energies close or above and 2) the invariant (17) is infinite for the new positive invariant energy scale of Λ=−1 the theory ∗Λl ∗ . Remark 1.1.4. It is also clear from Equation (16) and Equation (17) that the symmetry of positive and negative values of the energy is broken. The two theo-
ries with the two signs of lΛ obviously are physically distinct; and we know of
no theoretical argument which fixes the sign of lΛ 2 The massive particles have a positive invariant p > 0 which can be identi- 2 fied with the square of the mass pm= 2 , ( c = 1 ). Thus in the case of the in- variant (16) we obtain pp22− 0 =21 ∈−− ∞ 2 mp,,0 ( lΛ∗ ) (21) + (1 lpΛ∗ 0 )
From Equation (18) we obtain
2 42 mlΛΛ∗∗1 ml 22 p0 = + ++pm. (22) 22 22 22 ( ) 11−−mlΛΛ− ml ∗∗1 mlΛ∗
In the case of the invariant (17) we obtain pp22− 0 =21 ∈ −∞ − 2 mp,0 ( ,. lΛ∗ ) (23) − (1 lpΛ∗ 0 )
From Equation (20) we obtain
2 42 mlΛΛ∗∗1 ml 22 p0 =− − ++pm (24) 22 22 22 ( ) 11−−mlΛΛ− ml ∗∗1 mlΛ∗
The action for a scalar field ϕ must be invariant under the deformed Lo- rentz transformations. The invariant action reads [10]
1 ηab (∂∂ ϕϕ)( ) m2 Sx= ∫ d.42ab+ ϕ (25) 221+∂l ϕ Λ∗ 0
Thus there is no linear field equation.
Remark 1.1.5.Throughout this paper, we use below high-energy cutoff Λ∗ the perturbative expansion 1 m2 S=d.42 xηab ∂ ϕϕ ∂+ ϕ +Ol ∫ ( ab)( ) ( Λ∗ ) (26) 22 and dealing in Lorentz invariant approximation 1 m2 42ηab ∂ ϕϕ ∂+ ϕ Sx ∫ d.( ab)( ) (27) 22
since for lΛ∗ 1 the expansion (26) holds.
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(II) The canonical concept of Minkowski space-time collapses at a small dis- = Λ−1 tance lΛ∗∗ to fractal space-time with Hausdorff-Colombeau negative di- mension and therefore the canonical Lebesgue measure d4 x being replaced by the Colombeau-Stieltjes measure with negative Hausdorff-Colombeau dimen- sion D− :
d,ηε( x) = v sx( ) d4 x , (28) ( )ε ( ε ( ) )ε
− −1 D µ where (vε ( sx( ))) = sx( ) + ε and sx( ) = xxµ , see Section 3 and ε ε [12]. (III) The canonical concept of momentum space collapses at fundamental
high-energy cutoff Λ∗ to fractal momentum space with Hausdorff-Colombeau negative dimension and therefore the canonical Lebesgue measure d3k , where
k = (kkkxyz,,) being replaced by the Hausdorff-Colombeau measure
−DD+++ −−1 +− ∆(D)ddk ∆∆( D) ( Dp) p d,DD, k = (29) DD−− k + ε p + ε ε ε
±±D± 2 where ∆=(DD) 2π Γ( 2) and p=k = kkkxyz ++ and where DD+−− ≤−6 , see Section 3 and ref. [9]. Hausdorff-Colombeau measure (29)
avoids classical divergence (10) of the zeropoint energy ε vac (m) and instead Equation (10) one obtains
22 p∗ ∞ pm+ ε p,d m=3 p p 22 + m +∆ D+− ∆ Dd pp2 p4. vac ( ∗∗) ∫∫( ) ( ) − (30) 0 p∗ D p + ε ε
See Section 5 and ref. [12]. Remark 1.1.6. If we take the Planck scale (i.e. the Planck mass) as a cut-off, 121 the vacuum energy density ε vac ( pm∗ , ) is 10 times larger than the observed
dark energy density εde . Several possible approaches to the problem of vacuum energy have been discussed in the contemporary literature, for the review see [5], [12]. They can be roughly devided into four different groups: 1) Modification of gravity on large scales. 2) Anthropic principle.
3) Symmetry leading to ε vac = 0 . 4) Adjustment mechanism, see. 5) Hidden nonstandard dark matter sector and corresponding hidden symmetry leading to
ε vac 0 , see [12]. (IV) We assume that there exists the nonstandard dark matter sector formed by ghost particles, see [12].
1.2. Zel’dovich Approach by Using Pauli-Villars Regularization Revisited. Ghosts as Physical Dark Matter
Remind that vacuum energy density for free scalar quantum field is
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1 c ∞∞ εµ=2 +=µ 22 2 +=µ 22 µ ( ) 3 ∫∫4π p ppd K p pp d, KI( ) (31) 2 (2π) 00
where µ = mc0 . From Equation (31) one obtains [1]
4 K∞ ppd p(µµ) = ∫ = KF ( ). (32) 3 0 p22+ µ
For fermionic quantum field one obtains εµ( ) = KI( µ), p( µ) = − 4. KF ( µ) (33)
Thus free vacuum energy density ε and corresponding pressure p is
εµ= ∑∑CIii( ),. P= CF i( µ i) (34) ii
From Equation (34) by using Pauli-Willars regularization [7] [8] in general case one obtains [1] ε= ∫∫fI( µ) ( µµ)d, PfF= ( µ) ( µµ)d. (35)
In order to obtain asymptotical expansion on the parameter p0 of the quan- p0 tity ε ( pm,d) =3 pp 22 + m let us evaluate now the following integral vac 0 ∫0
pp00pµ µ=22 += µ 2 22 ++ µµ 2 22 + 2 I( , p0 ) ∫∫∫ pp d p pp dd p pp p 00pµ (36) pp µµµµ22p0 =22 +=µ 2 3 + +3 + ∫ppd p ∫∫ p 1d22 p p 1d p 0 ppµµpp
and
p 1d1d1dpp00pp4µ pp 44 pp Fp(µ, 0 ) = = + ∫∫∫22 22 22 33300p+µ pp ++ µµpµ p 1µ pp43 d1p0 pp d (37) = + , ∫∫22 2 330 p + µµpµ 1+ p2
where pµ = rrµµ, > 1, p << 1 r 1 . Note that
µ211 µµ 24 1 µ 6 11+=+− + + p22 pp 24 8 16 p 6 (38) µ2 1 11µµ46 pp22+=µµ 2 p 31 + =+ p3 2 p − + + pp232 8p 16
By inserting Equation (38) into Equations (36) one obtains 11 1p 1µ 6 µ= µ4 ++ 4 µµ 22 − 4 0 − +−5µ 8 I( , p01) C p 0 p 0 ln 2 pO0( ), (39) 4 4 8 µ 32 p0
pµ where Cµµ4= pp 22 + 2d p. Note that 1 ∫0
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−1 µ213 µµ 24 5 µ 6 11+ =−+− + (40) 2 24 6 p2 pp 8 16 p
By inserting Equation (40) into Equation (37) one obtains 11 1p 5µ 6 µ=+− µ44224 µµ +0 + +−58µ F( , p02) C p 0 p 0 ln 2 pO0( ). (41) 12 12 8 µ 32 p0 By inserting Equation (39) and Equation (41) into Equations (35) one obtains µµ µ 11eff eff 1eff ε=4 µ µ+ 22 µµ µ+− µµ4 µ pf0∫∫( )d pf0( ) d C10 ln p ∫ f( ) d 4400 80 µ µµ 1eff 11eff eff +−fµµ4ln µ d µ fµµ6d µ + Of µµ85 p− , ∫( ) ( ) 2 ∫∫( ) ( ) 0 8 0 p0 32 00 µµ µ(42) 11eff eff 1eff = 4µ µ− 22µµ µ++ µµ4 µ p pf0∫∫( )d p0 f( ) d C20ln p ∫ f( ) d 12 0012 8 0 µ µµ 1 eff 51eff eff − fµµ4ln µ d µ++fµµ6d µ Of µµ85 p− . ∫( ) ( ) 2 ∫∫( ) ( ) 0 8 0 p0 32 00
We choose now
µµeff eff µeff ∫∫ff(µ)ddd0 µ= ( µµ) 24 µ= ∫ f( µµ) µ= . (43) 00 0 By inserting Equation (43) into. Equations (42) one obtains µ 1 eff εµ= µµ42 µ µ+ − ( eff ) ∫ f( ) (ln) dOp( 0 ) , 8 0 µ (44) 1 eff µ =−+µµ42 µ µ − p( eff ) ∫ f( ) (ln) dOp( 0 ) . 8 0
Taking the limit p →∞ in Equation (44) gives µ 1 eff εµ= µµ4 µ µ ( eff ) ∫ f ( ) (ln) d , 8 0 µ (45) 1 eff µ = − µµ4 µ µ pf( eff ) ∫ ( ) (ln) d . 8 0
Thus finally we obtain [3] µ 1 eff c4Λ εµ=−= µ µµ4 µ µ= ( eff ) pf( eff ) ∫ ( ) (ln) d . (46) 880 πG Remark 1.2.1. Remind that Pauli-Villars regularization consists of introduc- ing a fictitious mass term. For example, we would replace a propagator 22 1 (kmi−+0 ) , by the regulated propagator
NNaa1 ∆=2 ii= − (k ) ∑∑ii=0022 22 = 22, (47) kmi−+ii kmi −+0 kmi −+
where a0 = 1 and mii ,= 1, 2, , N can be thought of as the mass of a fictitious heavy particle, whose contribution is subtracted from that of an ordinary particle.
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22 Assume that mki < 1, if we expand each term of this sum (46) as a power se- ries in ki2 + we get
2 2 NNai amii N1 ∆=(kO) ∑∑= += + ∑= . (48) ii00ki2+ 22 23 i 0 (ki++) ( ki) For a renormalizable theory the maximum supercriticial power of divergence of any integral is quadratic, so that the Ok(1 6 ) terms are ultraviolet finite. The finiteness of the regulated integral is then guaranteed by requiring that NN= 2 = ∑∑ii=00ai 0, = amii 0. (49) Remark 1.2.2. Note that in order to apply Pauli-Villars regularization to QFT
with Lagrangian L (ϕψ,,∂∂µµ ϕ , ψ) we would replace the Lagrangian
L (ϕψ,,∂∂µµ ϕ , ψ) by Lagrangian L (ϕψ,,∂∂µµ ϕ , ψ) , where [7]: ϕ= ϕ+= ϕ µψ22 ψ+ ψ ( x) ( x) ∑∑nn bxnn( ,, n) ( x) ( x) cn n( x,, n) (50) where commutator for ϕn and anticommutator for ψ n reads ϕ µ22 ϕ′′ µ=−∆− ρ µδ2 m( x,, m) n( x , n ) in ( xx,,n) mn
22′′2 {ψψm( x,, m) n( x , n )} =−− i εn Sx( x,. n) δ mn
From Equations (50)-Equations (51) one obtains ϕϕ ′=N ρ22 ∆−′ µ ( x),( x) i∑n=0 nn b( xx,,n) (52) ψψ ′=−−N ε ′ 2 ( x),( x) i∑n=0 nnn ccS( x x,. n)
Assume now that NNρ2= ρµ22= N ε= N ε 2= ∑∑nn=00nnb 0, = nnb n 0, ∑ n= 0nnncc 0, ∑ n= 0nnncc n 0. (53) From Equations (53) it follows directly that QFT with Lagrangian
L (ϕψ,,∂∂µµ ϕ , ψ) is finite QFT with indefinite metric [4], see Remark 1.2.1. Remark 1.2.3. Note that “bad ghosts” represent general meaning of the word “ghost” in theoretical physics: states of negative norm [7] or fields with the wrong sign of the kinetic term, such as Pauli--Villars ghosts ϕ , whose existence allows the probabilities to be negative thus violating unitarity. The quadratic la- 2 grangian Lϕ for ϕ begins with a wrong sign kinetic term [in ( +−−−) sig- nature]
2 1 µ Lϕµ=−∂∂ϕϕ + (54) 2 Remark 1.2.4. Note that in order to obtain Equations (44), the standard quantum fields do not need to couple directly to the ghost sector. In this paper the ghost sector is considered as physical mechanism which acts only on a func- tion f (µ ) in Equations (43). It means that there exists the ghost-driven acce- leration of the universe hidden in cosmological constant Λ . Remark 1.2.5. As pointed out in paper [13] even if the standard model fields have no direct couplings to the ghost sector, they will indirectly interact with it
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through gravity, and the propagation of gravity through the ghost condensate gives rise to a fascinating modification of gravity in the IR. However, no modifi- cations of gravity can be seen directly, and no cosmological experiment can dis- tinguish the ghost-driven acceleration from a cosmological constant. Remark 1.2.6. In order to obtain desired physical result from Equations (45), i.e.,
−−29 3 −47 4 3 5 ε vac =×⋅=×0.7 10 g cm 2.8 10 GeV c (55) we assume that
ff(µµµ) =sm..( ) + fgm ..( ), (56)
where fsm..(µ ) corresponds to standard matter and where fgm..(µ ) corres- ponds to a physical ghost matter. Remark 1.2.7. We assume now that −n On(µ),1>≤ µµeff f (µ ) = (57) 0 µµ> eff
From Equation (57) and Equation (45) it follows directly that
µ 1 eff µ= εµ = µµ45 µ µ≤ µ−+n µ p( eff ) ( eff ) ∫ fO( ) (ln) d ( effln eff ) . (58) 8 0
Remark 1.2.8. However serious problem arises from non-renormalizability of canonical quantum gravity with Einstein-Hilbert action 1 S = d.4 x− gR (59) EH 16πG ∫ For example taking Λ3 particles of energy a per unit volume gives the gravi- 6 6 tational self-energy density of order Λ , i.e., the density ε Λ diverges as Λ
6 ε Λ GΛ , (60)
where Λ is a high-energy cutoff [5]. In order to avoid these difficulties we apply instead Einstein-Hilbert action (59) the gravitational action which includes terms quadratic in the curvature tensor
4 µν 22− ℑ=−∫ dx − g(α RRµν − βκ R +2, R) (61) Remark 1.2.9. Gravitational actions (61) which include terms quadratic in the curvature tensor are renormalizable [14]. The requirement that the graviton propagator behaves like p−4 for large momenta makes it necessary to choose the indefinite-metric vector space over the negative-energy states. These nega- tive-norm states cannot be excluded from the physical sector of the vector space without destroying the unitarity of the S matrix, however, for their unphysical
behavior may be restricted to arbitrarily large energy scales Λ∗ by an appropri-
ate limitation on the renormalized masses m2 and m0 .
Remark 1.2.10. We assume that mc0µµ eff, mc 2 eff . Remark 1.2.11. The canonical Quantum Field Theory is widely believed to
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break down at some fundamental high-energy cutoff Λ∗ and therefore the quantum fluctuations in the vacuum can be treated classically seriously only up to this high-energy cutoff, see for example [15]. In this paper we argue that Quantum Field Theory in fractal space-time with negative Hausdorff-Colombeau dimensions [12] gives high-energy cutoff on natural way.
2. Ghosts as Physical Dark Matter 2.1. Paulu-Villars Ghosts As Physical Dark Matter
Before explaining the role of PV ghosts, etc. as physical dark matter remind the idea of PV regularization as a conventional UV regularization. We consider, as an example, the scalar field theory with the interaction λϕ 4 . Lagrangian density of this theory reads 2 1 µ m0 24 L =∂∂ϕµ ϕ − ϕ + λϕ . (62) 22 This theory requires UV regularization (e.g. in (2+1) and (3+1) dimensions). Let us show that it is sufficient to introduce N extra fields with large mass play- ing the role of the regularization parameter. Lagrangian density can be rewritten as follows
2 N i 1 m = − ∂∂µϕ ϕ −i ϕ24 + λϕ L ∑i=0 ( 1) µ i : :, 22 (63) ϕ= ϕ += ϕNN ϕϕ = ϕ 0 ∑∑ii=00i,.= aii
Here the symbol “::” means that in perturbation theory we drop Feynman di-
agrams with loops containing only one vertex. The ϕ0 is usual field with mass
m0 and the ϕi ,iN= 1, , is the extra field with mass mii ,= 1, , N. It can be shown that in (3+1)-dimensional theory the introduction of one PV field is suf- ficient for the ultraviolet regularization of perturbation theory in λ . One can show that momentum space Feynman diagrams in the original theory with La- grangian density (62) diverge no more than quadratically [16] [17] [18] (beside of vacuum diagrams) shown in Figure 1. If we consider now Feynman diagrams in the theory with Lagrangian density (63) we see that propagators of fields ϕ0 and ϕ sum up in corresponding di- agrams so that we obtain the following expression which plays the role of regu- larized propagator
NNaajj1 ∆=2 = − (k ) ∑∑jj=0022 22 = 22 , (64) kmi−+jj00 kmi −+0 kmi −+ 0
2 2222 where k=−++ kkkk01 23. Integral corresponding to vacuum diagram is
44 ddkkN a j ℑ= ∆2 = ∫∫4(k ) 4∑ j=0 22. (65) (2π) (2π) kmi−+j 0
To do this integral, since it is convergent, we can Wick rotate.
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Figure 1. One-loop massive va- cuum diagram.
Remark 2.1.1. All the integrals in quantum field theory are written in Min- kowski space, however, the ultraviolet divergence appears for large values of modulus of momentum and it is useful to regularise it in Euclidean space [17]. Transition to Euclidean space can be achieved by replacing thr zeroth compo-
nent of momentum k04→ ik , where the integration over the fourth component of momenta goes along the imaginary axis. To go to the integration along the real axis, one has to perform the (Wick) rotation of the integration contour by 90˚ (see Figure 2). This is possible since the integral over the big circle vanishes and during the transformation of the contour it does not cross the poles. Then we get 3 i ∞ N akjE ℑ= d.k (66) EE2∫0 ∑ j=0 22 8π kmEj+
To do this integral, since it is convergent, we can deal with regularized integral 3 i Λ N akjE ℑε,d Λ= k , ( ) 2∫ε E ∑ j=0 22 (67) 8π kmEj+
where ε 0,Λ∞, i.e. ℑ(ε, Λ) ≈ℑE . We assume now that Pauli-Villars con- ditions given by Equations (48) holds. Let us consider now the quantity
3 i Λ N akjE ℑ ℑε,d Λ= k , ηη ( ) 2∫ε E ∑ j=0 22 (68) 8π kmEj+η
where η ∈(0,1] , and therefore from Equation (68) we obtain
iiΛΛNN ℑ=η dkE∑∑ ak jE= aj k Ed k E≡ 0, (69) η =0 8π22∫∫εεjj=008π =
since Equations (48) holds. From Equation (68) by differentiation we obtain
23 d i Λ N amkj jE ℑ= η 22∫ d,kE ∑ j=0 (70) dη 8π ε 22 (kmEj+η )
and therefore from Equation (39) we obtain
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Figure 2. The Wick rotation of the integration contour.
23 d i Λ N amkj jE ℑ= η 22dkE ∑ = ∫ε j 0 22 dη η = 8π 0 (kmEj+η ) η =0 (71)
Λ i N 21− = ∑ amjj k Ed k E≡ 0, 8π2 j=0 ∫ε since Equations (48) holds. From Equation (70) by differentiation we obtain
2 43 d NNi Λ amkj jE ℑ= ℜη = 223η ∑∑jj=00jE( ) ∫ d,k = d4η π ε 22 (kmEj+η ) (72) 4 3 ia m Λ k ℜ=η jj E jE( ) 23∫ d.k 4π ε 22 (kmEj+η )
Note that 43 42 iamjj∞ k iamjj −i amjj ℜ=η E = jE( ) 2∫ d.k 3 22 2 (73) 4π 0 22 4π 4ηηm 16π (kmEj+η ) j
Thus 2 d NN1 amjj ℑ=η ∑∑ ℜj (ηη)d = lnη (74) dη jj=00∫0 = 16π2
and 2 N amjj ℑ=η ∑ (ηln ηη−) , (75) j=0 16π2 Therefore 2 N amjj ℑ(ε, Λ=ℑ) η =−∑ ≡0, (76) η =1 j=0 16π2 since Equations (48) holds. Thus integral (65) corresponding to vacuum diagram by using Pauli-Villars renormalization identically equal zero, i.e. 44 ddkkN a j ℑ= ∆2 = ≡ Ren PV ( ) 4(k ) 4∑ j=0 22 0. (77) (2π) (2π) kmi−+j 0
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Let us consider now how this method works in the case of the simplest scalar diagram shown in Figure 3. The corresponding Feinman integral has the form 1d4k ℑ=2 ( p ) 4 ∫ . (78) 2π 2−+ 2 22 − −+ 2 ( ) (kmi0000) ( pk) mi
Regularized Feinman integral (78) reads
4 1 N akj d ℑ=2 reg ( p ) 4 ∫∑ j=0 , (79) 2π 2−+ 2 22 − −+ 2 ( ) (kmijj00) ( pk) mi
where N = 1 . To do this integral, since it is convergent, we can Wick rotate. Then we get 4 i N akj d ℑ=2 reg ( p ) 4 ∫∑ j=0 . (80) 2π 2+ 2 22 −+ 2 ( ) (kmjj) ( pk) m
The integral (80) can be written as
1 4 2 i N akj d ℑ=reg ( px) d ∑ = 42∫∫ j 0 22 2 (2π) 0 + −+ k px(1 x) mj (81) 1 3 i N akjEd k E = d.x ∑ = 22∫∫ j 0 22 2 8π 0 + −+ kEj px(1 x) m
To do this integral, since it is convergent, we can deal with regularized integral
1 3 Λ 2 i N akjEd k E ℑreg ( px,,ε Λ=) d ∑ = . (82) 22∫∫ε j 0 22 2 8π 0 + −+ kEj px(1 x) m
Let us consider now the quantity
1 3 Λ 2 i N akjEd k E ℑη ( px,,ε Λ=) d ∑ = . (83) 22∫∫ε j 0 22 2 8π 0 + −+η kEj px(1 x) m
2 where η ∈(0,1] , and therefore from Equation (83) we obtain ℑ0 ( p ,,ε Λ≡) 0, since Equations (48) holds. From Equation (83) by differentiation we obtain
Figure 3. The simplest scalar diagram.
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1 23 Λ d 2 i N amkj jEd k E ℑη ( px,,ε Λ=−) d ∑ = 23∫∫ε j 0 22 2 dη 4π 0 + −+η kEj px(1 x) m
i N 22 −∑ amjj ℜΛ j( p,,ηε ,) , 4π2 j=0 1 3 (84) 2 kkEEd ℜΛj ( px,,ηε ,) d ∫∫ 22 23 0 + −+η kEj px(1 x) m 1d1 x = ∫ 22. 4 0 px(1−+ x) η mj
From Equation (84) we obtain
d 2 i N 22 ℑη ( p,,ε Λ−) ∑ amjj ℜ j( p,,,ηε Λ) dη 4π2 j=0 1 (85) ixN d = − 2∑ j=0 a j ∫ −22 . 16π 0 mj px(1−+ x) η
From Equation (85) we obtain 11 i N dη ℑ=−2 reg ( p) 2∑ j=0 axj ∫∫d.−22 (86) 16π 00mj px(1−+ x) η
Note that
1 η 1 d −−22 22 − =mjj px(1 −+ x) ηη ln m px( 1−+ x) −1 ∫ 22 0 0 mj px(1−+ x) η =−−22 −+ 22 −+ mjj px(11ln11 x) m px( x) (87) −−−22 − 22 −− mjj px(1 x) ln m px( 1 x) 1.
Thus 11 i N =1 dη ℑ=−2 reg ( p) 2∑ j=0 axj ∫∫d −22 16π 00mj px(1−+ x) η 1 i N =1 =− −−22 −+ 22 −+ 2 ∑ j=0 ajj∫ d x{ m px( 11ln11 x) mj px( x) 16π 0
i N =1 −−−22 −22 −+ mjj px(1 x) ln m px( 1 x) } 2 ∑ j=0 aj 16π 1 i N =1 =− −−22 −+ 22 −+ 2 ∑ j=0 ajj∫ d x{ m px( 11ln11 x) mj px( x) 16π 0 −−−−22 22 − mjj px(1 x) ln m px( 1 x) } i 1 =− −−22 −+ 22 −+ 2 ∫ dx{ m00 px( 11ln11 x) m px( x) 16π 0 −−−−22 22 − m00 px(1 x) ln m px( 1 x) } (88) i 1 + −−22 −+ 22 −+ 2 ∫ dx{ m11 px( 11ln11 x) m px( x) 16π 0 −−−−22 22 − m11 px(1 x) ln m px( 1 x) } .
From Equation (88) we obtain
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i 1 ℑ2 =− −−22 −+ 22 −+ reg ( p) 2 ∫ dx{ m00 px( 11ln11 x) m px( x) 16π 0 −−−−22 22 − m00 px(1 x) ln m px( 1 x) } (89) i 1 + −−22 −+ 22 −+ 2 ∫ dx{ m11 px( 11ln11 x) m px( x) 16π 0 −−−−22 22 − m11 px(1 x) ln m px( 1 x) } .
−22 We assume now that mp1 1 and from Equation (89) finally we obtain i 1 ℑ2 =− −−22 −+ 22 −+ reg ( p) 2 ∫ dx{ m00 px( 11ln11 x) m px( x) 16π 0 (90) −−−22 −22 −+ −22 mpx00(1 x) ln mpx( 1 x) } Omp( 1).
Remark 2.1.2. The simple renormalizable models with finite masses
mii ,= 1, , N which we have considered in the section many years regarded only as constructs for a study of the ultraviolet problem of QFT. The difficulties with unitarity appear to preclude their direct acceptability as canonical physical theories in locally Minkowski space-time. However, for their unphysical beha-
vior may be restricted to arbitrarily large energy scales Λ∗ mentioned above by
an appropriate limitation on the finite masses mi .
2.2. Renormalizability of Higher Derivative Quantum Gravity
Gravitational actions which include terms quadratic in the curvature tensor are renormalizable. The necessary Slavnov identities are derived from Bec- chi-Rouet-Stora (BRS) transformations of the gravitational and Faddeev-Popov ghost fields. In general, non-gauge-invariant divergences do arise, but they may be absorbed by nonlinear renormalizations of the gravitational and ghost fields and of the BRS transformations [14]. The geneic expression of the action reads =−−4 αµν −+ βκ22− Isym ∫ d x g( RRµν R2, R) (91)
λλ where the curvature tensor and the Ricci is defined by Rµαν=∂Γ ν µα and λ 2 RRµν= µλν correspondingly, κ = 32πG . The convenient definition of the gra- vitational field variable in terms of the contravariant metric density reads κηhgµν= µν −− gµν . (92)
Analysis of the linearized radiation shows that there are eight dynamical de- grees of freedom in the field. Two of these excitations correspond to the familiar massless spin-2 graviton. Five more correspond to a massive spin-2 particle with
mass m2 . The eighth corresponds to a massive scalar particle with mass m0 . Although the linearized field energy of the massless spin-2 and massive scalar excitations is positive definite, the linearized energy of the massive spin-2 excita- tions is negative definite. This feature is characteristic of higher-derivative mod- els, and poses the major obstacle to their physical interpretation. In the quantum theory, there is an alternative problem which may be substi- tuted for the negative energy. It is possible to recast the theory so that the mas-
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sive spin-2 eigenstates of the free-fieid Hamiltonian have positive-definite ener- gy, but also negative norm in the state vector space. These negative-norm states cannot be excluded from the physical sector of the vector space without destroying the unitarity of the S matrix. The requirement that the graviton propagator behaves like p−4 for large momenta makes it ne- cessary to choose the indefinite-metric vector space over the negative-energy states. The presence of massive quantum states of negative norm which cancel some of the divergences due to the massless states is analogous to the Pauli-Villars re- gularization of other field theories. For quantum gravity, however, the resulting improvement in the ultraviolet behavior of the theory is sufficient only to make it renormalizable, but not finite. The gauge choice which we adopt in order to define the quantum theory is the µν canonical harmonic gauge: ∂=ν h 0 . Corresponding Green’s functions are then given by a generating functional
ZT= Nddd hµν C σ Cδ 4 Fτ ( µν ) ∫ ∏µν≤ τ ( ) (93) ×+44τ µν α +κ µν expi( Isym ∫∫ d xCτ F µν D α Cd xTµν h ) .
τ τ µν τ r Here F= Fhµν, F µν=δ µ ∂ ν and the arrow indicates the direction in which the derivative acts. N is a normalization constant. Cσ is the Faddeev-Popov σ ghost field, and Cτ is the antighost field. Notice that both C and Cτ are µν anticommuting quantities. Dα is the operator which generates gauge trans- α formations in hµν , given an arbitrary spacetime-dependent vector ξ ( x) cor- ′ responding to xxµµ= +κξ µ and where
µν α µ ν ν µ µν α Dxααξ( ) =∂+∂−∂ ξ ξη ξ µ αν ν αµ α µν α µν (94) +κξ( ∂αhhhh +∂ α ξ − ξ ∂ αα −∂ ξ )
In the functional integral (93), we have written the metric for the gravitational dhµν gg= det field as ∏µν≤ without any local factors of ( µν ) . Such factors do not contribute to the Feynman rules because their effect is to introduce terms proportional to δ 44(0) ∫ dxg ln (− ) into the effective action and δ 4 (0) is set equal to zero in dimensional regularization. In calculating the generating functional (93) by using the loop expansion, one may represent the δ function which fixes the gauge as the limit of a Gaussian, discarding an infinite normalization constant
4 ττ1 −14 δ (F ) lim expi∆ ∫ d xFτ F . (95) ∆→0 2
In this expression, the index τ has been lowered using the flat-space metric
tensor ηµν . For the remainder of this paper, we shall adopt the standard ap- proach to the covariant quantization of gravity, in which only Lorentz tensors occur, and all raising and lowering of indices is done with respect to flat space.
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1 −14 τ The graviton propagator may be calculated from I +∆ d xFτ F in the sym 2 ∫
1 −14 τ usual fashion, letting ∆→0 after inverting. The expression ∆ d xFτ F 2 ∫ contains only two derivatives. Consequently, there are parts of the graviton propagator which behave like p−2 for large momenta. Specifically, the p−2 terms consist of everything but those parts of the propagator which are trans- verse in all indices. These terms give rise to unpleasant infinities already at the one-loop order. For example, the graviton self-energy diagram shown in Figure 4 2 has a divergent part with the general structure (∂4h) . Such divergences do cancel when they are connected to tree diagrams whose outermost lines are on the mass shell, as they must if the S matrix is to be made finite without introducing counterterms for them. However, they greatly complicate the renormalization of Green’s functions. We may attempt to extricate ourselves from the situation described in the last paragraph by picking a different weighting functional. Keeping in mind that we want no part of the graviton propagator to fall off slower than p−4 for large momenta, we now choose the weighting functional [12] 1 ω ττ= ∆−14 2 4 (e) exp i∫ d xeτ e , (96) 2
where eτ is any four-vector function. The corresponding gauge-fixing term in the effective action is
1 21− 4 2τ −∆κ d.xFτ F (97) 2 ∫ The graviton propagator resulting from the gauge-fixing term (97) is derived
in [13]. For most values of the parameters α and β in Isym it satisfies the requirement that all its leading parts fall off like p−4 for large momenta. There are, however, specific choices of these parameters which must be avoided. If α = 0 , the massive spin-2 excitations disappear, and inspection of the graviton propagator shows that some terms then behave like k −2 . Likewise, if 30βα−=, the massive scalar excitation disappears, and there are again terms in the propagator which behave like p−2 . However, even if we avoid the special cases α = 0 and 30βα−=, and if we use the propagator derived from (97), we still do not obtain a clean renormalization of the Green’s functions. We now turn to the implications of gauge invariance. Before we write down the BRS transformations for gravity, let us first establish the commutation relation for gravitational gauge transformations, which reveals the group structure of the theory. Take the gauge transformation (94) of hµν , generated by ξ µ and
Figure 4. The one-loop graviton self-energy diagram.
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perform a second gauge transformation, generated by η µ , on the hµν fields appearing there. Then antisymmetrize in ξ µ and η µ . The result is µν δ Dα ρσαβ αβ µν λα αλ ρσ DDβ(ξη− ηξ) = κλα( ∂ ξη −∂ α ξη ), (98) δ h where the repeated indices denote both summation over the discrete values of the indices and integration over the spacetime arguments of the functions or operators indexed. The BRS transformations for gravity appropriate for the gauge-fixing term (96) are [13] µν µν α (a,) δBRSh= κ DCα δλ α 2 αβ (b,) δBRSC=−∂ κβ CC δλ (99) 3− 12 (c,) δBRSCFττ=−∆ κ δλ
where δλ is an infinitesimal anticommuting constant parameter. The impor- tance of these transformations resides in the quantities which they leave inva- riant. Note that σβ δBRS (∂=β CC ) 0 (100)
and µν α δBRS (DCα ) = 0. (101)
As a result of Equation (101), the only part of the ghost action which varies
under the BRS transformations is the antighost Cτ . Accordingly, the transfor- mation (99c) has been chosen to make the variation of the ghost action just can- cel the variation of the gauge-fixing term. Therefore, the entire effective action is BRS invariant:
1 21− 2τ τ µν α δκBRS Isim −∆ Fτ F + CFτ µν D α C =0. (102) 2 Equations (99), (100), and (102) now enable us to write the Slavnov identities in an economical way. In order to carry out the renormalization program, we will need to have Slavnov identities for the proper vertices. A) Slavnov identities for Green’s functions First consider the Slavnov identities for Green’s functions. τ ZT( µν,,,ββ σ K µν , L σ ) = Nddd hµν CC σ ∫ ∏µν≤ τ (103) ×Σ µν σ βσ + βσ ++ βκ τ µν expi( h , C , CKτ , µν , L σ , σ C) σ C C τ Thµν . Anticommuting sources have been included for the ghost and antighost fields, and the effective action Σ has been enlarged by the inclusion of BRS invariant
couplings of the ghosts and gravitons to some external fields Kµν (anticom-
muting) and Lσ (commuting),
1 21− 2τ τ µν α µν 2 σ β Σ =−∆Iκ Fτ F + CFτ µν D α C +κκ Kµν D α + Lσ ∂ β C C . (104) sim 2
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Σ is BRS invariant by virtue of Equation (99), Equation (100), and Equation (102). We may use the new couplings to write this invariance as
δΣ δ Σ δδ ΣΣ 3− 12 δΣ µν +σ +∆κ Fτ . (105) δδKLµν δδhCσ δ Cτ
In this equation, and throughout this subsection, we use left variational deriv- atives with respect to anticommuting quantities: δfC( στ) = δδδ C f C τ. Equ- ation (105) may be simplified by rewriting it in terms of a reduced effective ac- tion,
1 21− 2τ Σ=Σ +κ ∆ FFτ . (106) 2
Substitution of (106) into (105) gives δΣ δ Σ δδ ΣΣ µν +=σ 0, (107) δδKLµν δδhCσ
where we have used the relation
−1 τ δδΣΣ κ Fµν −=0. (108) δδKCµν τ
Note that a measure
dddhµν CC σ ∏µν≤ τ (109)
is BRS invariant since for infinitesimal transformations, the Jacobian is 1, be- cause of the trace relations δ 2Σ = (a) (µν ) 0, δδKh(µν ) (110) δ 2Σ (b) σ = 0, δδCLσ
4 α both of which follow from ∫ d0xC∂=α . The parentheses surrounding the in- dices in (110a) indicate that the summation is to be carried out only for µν≤ . Remark 2.2.1. Note that the Slavnov identity for the generating functional of Green’s functions is obtained by performing the BRS transformations (99) on the integration variables in the generating functional (103). This transformation does not change the value of the generating functional and therefore we obtain
Nddd hµν C σ Cκ22 T Dµν −∂ κβ CCσ β ∫ ∏µν≤ τ( µν α σ β (111) +∆κβ31− τ 2 µν Σ +κµν + β σ + β τ = Fτµν h)exp i( Thµν σ C C τ ) 0.
Another identity which we shall need is the ghost equation of motion. To de-
rive this equation, we shift the antighost integration variable Cτ to CCττ+δ , again with no resulting change in the value of the generating functional:
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δΣ Nddd hµν CC σ + β τ ∫ ∏µν≤ τ σ δC (112) ×Σ +κββµν ++ σ τ exp i( Thµν σ C C τ ) We define now the generating functional of connected Green’s functions as the logarithm of the functional (103), ττ WTµν,,,ββ σ KL µν , σ = − iZT lnµν ,,,ββ σ KL µν , σ . (113)
and make use of the couplings to the external fields Kµν and Lσ to rewrite (112) in terms of W
δδWW21− τ 2 δW κTFµν − βσ +∆ κβ τµν =0. (114) δδKLµν σ δ Tµν Similarly, we get the ghost equation of motion:
−1 ττδW κβFµν +=0. (115) δ Kµν B) Proper vertices A Legendre transformation takes us from the generating functional of con- nected Green’s functions (113) to the generating functional of proper vertices. First, we define the expectation values of the gravitational, ghost, and antighost τ fields in the presence of the sources Tµν, β σ , and β and the external fields
Kµν and Lσ δW (a,) hxµν ( ) = κδTxµν ( ) δW (b,) Cxσ ( ) = (116) δβσ ( x) δW (c.) Cxτ ( ) = δβ τ ( x) We have chosen to denote the expectation values of the fields by the same symbols which were used for the fields in the effective action (104). The Legendre transformation can now be performed, giving us the generating functional of proper vertices as a functional of the new variables (116) and the
external fields Kµν and Lσ µν σ Γ h, C ,, CKτ µν , L σ (117) τ µν σ τ =WTµν,,,ββ σ K µν , L σ− κ Th µν −− βσ C C τ β. τ In this equation, the quantities Tµν , βσ , and β are given implicitly in terms µν σ of h, C ,, CKτ µν , and Lσ by Equation (116). The relations dual to (116) are δ Γ (a,) κTxµν ( ) = − δ hxµν ( ) δ Γ (b,) βσ ( x) = (118) δCxσ ( ) δ Γ (c.) βτ ( x) = − δCxτ ( )
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Since the external fields Kµν and Lσ do not participate in the Legendre transformation (116), for them we have the relations
δδΓ W (a,) = δδKxµν ( ) Kxµν ( ) (119) δδΓ W (b.) = δδLxσσ( ) Lx( )
Finally, the Slavnov identity for the generating functional of proper vertices is obtained by transcribing (114) using the relations (116), (118), and (119) δΓ δ Γ δδ ΓΓ 3− 12 µν δΓ µν +σ +∆κ Fhτµν σ =0. (120) δδKLµν δδhCσ δ C
We also have the ghost equation of motion, −1 τ δδΓΓ κ Fµν −=σ 0. (121) δ Kµν δC
Since Equation (120) has exactly the same form as (105), we follow the exam- ple set by (106) and define a reduced generating functional of the proper vertic- es, 1 21− µν 2 τ ρσ Γ=Γ+κ ∆ (Fτµν h) ( Fhρσ ). (122) 2 Substituting this into (120) and (121), the Slavnov identity becomes δΓ δ Γ δδ ΓΓ µν +=σ 0. (123) δδKLµν δδhCσ
and the ghost equation of motion becomes
−1 τ δδΓΓ κ Fµν −=0. (124) δδKCµν τ
Equations (123) and (124) are of exactly the same form as (107) and (108). This is as it should be, since at the zero-loop order (0) Γ=Σ. (125) C) Structure of the divergences and renormalization equation The Slavnov identity (123) is quadratic in the functional Γ . This nonlinearity is reflected in the fact that the renormalization of the effective action generally also involves the renormalization of the BRS transformations which must leave the effective action invariant. The canonical approach uses the Slavnov identity for the generating functional of proper vertices to derive a linear equation for the divergent parts of the proper vertices. This equation is then solved to display the structure of the di- vergences. From this structure, it can be seen how to renormalize the effective action so that it remains invariant under a renormalized set of BRS transfor- mations [14]. Suppose that we have successfully renormalized the reduced effective action
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up to n −1 loop order; that is, suppose we have constructed a quantum exten- sion of Σ which satisfies Equations (107) and (108) exactly, and which leads to finite proper vertices when calculated up to order n −1. We will denote this re- (n−1) normalized quantity by Σ . In general, it contains terms of many different orders in the loop expansion, including orders greater than n −1. The n −1 loop part of the reduced generating functional of proper vertices will be denoted (n−1) by Γ . (n) When we proceed to calculate Γ , we find that it contains divergences. Some of these come from n-loop Feynman integrals. Since all the subintegrals of an n-loop Feynman integral contain less than w loops, they are finite by assumption. Therefore, the divergences which arise from w-Ioop Feynman integrals come only from the overall divergences of the integrals, so the corresponding parts of (n) Γ are local in structure. In the dimensional regularization procedure, these −1 divergences are of order −1 =(d − 4) , where d is the dimensionality of space- time in the Feynman integrals. (n) There may also be divergent parts of Γ which do not arise from loop inte- grals, and which contain higher-order poles in the regulating parameter . Such (n−1) divergences come from n-loop order parts of Σ which are necessary to en- sure that (107) is satisfied. Consequently, they too have a local structure. We (n) may separate the divergent and finite parts of Γ :
(n) ( nn) ( ) Γ =Γdiv +Γ finite . (126) If we insert this breakup into Equation (123), and keep only the terms of the equation which are of n-loop order, we get
(n) (0) ( 0) (nn) ( ) (00) ( ) (n) δΓdiv δΓ δ Γ δδ ΓΓdiv div δδ ΓΓ δ Γdiv µν +µν ++σ σ δKµν δh δ Kµν δδδ h δδ LLσ CCσ (127) δδΓΓ(ni−−) ( i) δδ ΓΓ( ni) ( i) =−+n finite finite finite finite ∑i=0 µν σ . δδKLµν δδhCσ
Since each term on the right-hand side of (127) remains finite as → 0 , while each term on the left-hand side contains a factor with at least a simple pole in e, each side of the equation must vanish separately. Remembering the Equa- tion (125), we can write the following equation, called the renormalization equa- tion:
(n) ℜΓdiv = 0, (128) where δδΣ δδδδδδ ΣΣ Σ ℜ=µν +σ +µν + σ . (129) δhδKµν δ C δδ LKσ µν δδ hC δ Lσ
Similarly by collecting the n-loop order divergences in the ghost equation of motion (124) we get (nn) ( ) −1 τ δδΓΓdiv div κ Fµν −=0. (130) δδKCµν τ
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In order to construct local solutions to Equations. (128) and (130) remind that the operator ℜ defined in (129) is nilpotent [14]: ℜ=2 0. (131) Equation (131) gives us the local solutions to Equation (128) of the form Γ(n) =ℑµν +ℜµν σ div (h) Xh( , C ,, Cτ K µν , L σ ) , (132) where ℑ is an arbitrary gauge-invariant local functional of hµν and its deriv- µν σ atives, and X is an arbitrary local functional of h,,, CCKσ τ µν and L and their derivatives. In order to satisfy the ghost equation of motion (130) we re- quire that (nn) ( ) µν σ −1 τ Γ=Γdiv div (h,, C Kµν−κ CF ττ µν,. L σ ) (133)
D) Ghost number and power counting Structure of the effective action (104) shows that we may define the following conserved quantity, called ghost number [14]: µν ==+=− Nghost h0, NCghost [ στ] 1, NCghost 1, (134) =−=− NKghost µν 1, NLghost [ σ ] 2. From Equations (134) follows that
NNghost [Σ=] ghost [ Γ=] 0. (135)
Since
Nghost [ℜ=+] 1, (136)
we require of the functional X (⋅) that
NXghost [ ] = −1. (137)
(n) In order to complete analysis of the structure of Γdiv , we must supplement the symmetry Equations (132), (133), and (137) with the constraints on the di- vergences which arise from power counting. Accordingly, we introduce the fol- lowing notations:
nE = number of graviton vertices with two derivatives,
nG = number of antighost-graviton-ghost vertices,
nK = number of K-graviton-ghost vertices,
nL = number of L-ghost-ghost vertices,
IG = number of internal-ghost propagators,
EC = number of external ghosts,
EC = number of external antighosts. Since graviton propagators behave like p−4 , and ghost propagators like p−2 , we are led by standard power counting to the degree of divergence of an arbi- trary diagram,
D=−+−42 nEGGLK 2 I 2 n −−− 3 n 3 nEC . (138)
The last term in (2.2.48) arises because each external antighost line carries with it a factor of external momentum. We can make use of the topological rela- tion
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22IG− n G = 2 nn LK +−− EE C C (139)
to write the degree of divergence as
D=−42 nnEL −− 2 nE K − C − 2 EC . (140)
Together with conservation of ghost number, Equation (140) enables us to catalog three different types of divergent structures involving ghosts. These are il-
lustrated in Figure 5. Each of the three types has degree of divergence Dn=12 − E .
Consequently, all the divergences which involve ghosts have nE = 0 . Since the (n) degree of divergence is then 1, the associated divergent structures in Γdiv have an extra derivative appearing on one of the fields. Diagrams whose external lines
are all gravitons have degree of divergence Dn=42 − E . Combining (140) with (n) (137), (133), and (132), we can finally write the most general expression for Γdiv which satisfies all the constraints of symmetries and power counting: Γ(n) =ℑµν +ℜ −κ −1 τ µν αβ + σ αβ τ div (h) ( Kµν CF τ µν ) P( h) LQσ τ ( h) C , (141)
µν αβ σ αβ where Ph( ) and Qhτ ( ) are arbitrary Lorentz-covariant functions of the gravitational field hµν , but not of its derivatives, at a single spacetime point. ℑ(hµν ) is a local gauge-invariant functional of hµν containing terms with four, two, and zero derivatives. Expanding (141), we obtain an array of possible divergent structures:
ρσ (n) µν δ Isym µν τ δ Dα α µν Γ=ℑdiv (h) +µν P +(κ Kρσ − CF τ ρσ )µν C P δδhh ρσ τ δ P µν α τ µν σ ε −−(κκKρσ CF τ ρσ ) µν Dα C−−( Kµν CF τ µν) D σ( QC ε ) δ h (142) 22στ β σ βτ −∂κκLσβ( QC τ ) C −∂ Lσβ CQC τ σ δQτ τ µν α 2 σ τ β −κκLσ CDα C+∂ LQστ β CC . δ hµν
is determined only up to a term of the form [13] [14]
Figure 5. The three types of divergent diagram which involve external ghost lines. Arbitrarily many gravitons may emerge from each of the central regions,(a) Ghost action type,(b) K type, (c) L type.
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The breakup between the gauge-invariant divergences S and the rest of (142) δ 4 µν µν Isym d,xh(ηκ+ ) µν (143) ∫ κδ h µν which can be generated by adding to P a term proportional to ηκµν+=h µν ggµν . The profusion of divergences allowed by (142) appears to make the task of renormalizing the effective action rather complicated. Although the many divergent structures do pose a considerable nuisance for practical cal- culations, the situation is still reminiscent in principle of the renormalization of Yang-Mills theories. There, the non-gauge-invariant divergences may be elimi- nated by a number of field renormalizations. We shall find the same to be true here, but because the gravitational field hµν carries no weight in the power counting, there is nothing to prevent the field renormalizations from being non- linear, or from mixing the gravitational and ghost fields. The corresponding re- normalizations procedure considered in [13] [14]. Remark 2.2.2. We assume now that: 1) The local Poincaré group of momentum space is deformed at some funda-
mental high-energy cutoff Λ∗ [9] [10]. 2 ab 2) The canonical quadratic invariant p=η ppab collapses at high-energy
cutoff Λ∗ and being replaced by the non-quadratic invariant: ab 2 η pp p = ab. (144) + (1 lpΛ∗ 0 ) 3) The canonical concept of Minkowski space-time collapses at a small dis- = Λ−1 tance lΛ∗∗ to fractal space-time with Hausdorff-Colombeau negative di- mension and therefore the canonical Lebesgue measure d4 x being replaced by the Colombeau-Stieltjes measure d,ηε( x) = v sx( ) d4 x , (145) ( )ε ( ε ( ) )ε
where
−1 D− (vε ( sx( ))) = sx( ) + ε , ε ε (146) µ sx( ) = xxµ ,
see subsection IV.2. 4) The canonical concept of local momentum space collapses at fundamental
high-energy cutoff Λ∗ to fractal momentum space with Hausdorff-Colombeau negative dimension and therefore the canonical Lebesgue measure d3k , where
k = (kkkxyz,,) being replaced by the Hausdorff-Colombeau measure
−DD+++ −−1 +− ∆(D)ddk ∆∆( D) ( Dp) p d,DD, k = (147) DD−− k + ε p + ε ε ε
+− see Subsection 3.4. Note that the integral over measure dDD, k is given by
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formula (185). Remark 2.2.3. (I) The renormalizable models which we have considered in this section many years regarded only as constructs for a study of the ultraviolet problem of quantum gravity. The difficulties with unitarity appear to preclude their direct acceptability as canonical physical theories in locally Minkowski space-time. In canonical case they do have only some promise as phenomenological models. (II) However, for their unphysical behavior may be restricted to arbitrarily
large energy scales Λ∗ mentioned above by an appropriate limitation on the
renormalized masses m2 and m0 . Actually, it is only the massive spin-two ex- citations of the field which give the trouble with unitarity and thus require a very
large mass. The limit on the mass m0 is determined only by the observational constraints on the static field. 3. Hausdorff-Colombeau Measure and Associated Negative Hausdorff-Colombeau Dimension 3.1. Fractional Integration in Negative Dimensions
D+ n Let µH be a Hausdorff measure [19] and X ⊂ is measurable set. Let sx( ) be a function sX: → such that is symmetric with respect to some
centre xX0 ∈ , i.e. sx( ) = constant for all x satisfying d( xx, 0 ) = r for arbi- trary values of r. Then the integral in respect to Hausdorff measure over n-dimensional metric space X is then given by [19]: D 2 2π + ∞ sx( )dµ DD++= srr( ) −1d. r (148) ∫∫X H 0 Γ(D+ 2) The integral in RHS of the Equation (148) is known in the theory of the Weyl fractional calculus where, the Weyl fractional integral WfxD ( ) , is given by
1 ∞ D−1 WD fx( ) =( t − x) ft( )d. t (149) Γ(D) ∫0
Remark 3.1.1. In order to extend the Weyl fractional integral (148) in nega- tive dimensions we apply the Colombeau generalized functions [20] and Co- lombeau generalized numbers [21]. Recall that Colombeau algebras (Ω) of the Colombeau generalized func- n tions defined as follows. Let Ω be an open subset of . Throughout this paper, (0,1] for elements of the space C ∞ (Ω) of sequences of smooth functions indexed ε ∈ ζ by (0,1] we shall use the canonical notations ( ε ( x))ε and (uε )ε so ∞ uCε ∈Ω( ) , ε ∈(0,1] .
Definition 3.1.1. We set (Ω=) M ( Ω) / ( Ω) , where
(0,1] Ω =uC ∈∞ Ω ∀ K ⊂⊂ Ω, ∀α ∈n ∃p ∈ with M ( ) {( ε )ε ( ) − p supxK∈ uxε ( ) = O(εε) as→ 0} , (150) (0,1] Ω =uC ∈∞ Ω ∀ K ⊂⊂ Ω, ∀α ∈n ∀q ∈ ( ) {( ε )ε ( ) q supxK∈ uxε ( ) = O(εε) as → 0} .
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Notice that (Ω) is a differential algebra. Equivalence classes of sequences (uε )ε will be denoted by cl (uε )ε is a differential algebra containing ∞ D′(Ω) as a linear subspace and C (Ω) as subalgebra. − D − Definition 3.1.2. Weyl fractional integral Wε fx( ) in negative dimen- − − ε sions D < 0 , D≠−0, 1, , − nn , , ∈ is given by
− 1 ∞ D− −1 WD fx( ) = (t− x) ft( )d t − (∫ε ) Γ(D ) ε or (151) − ∞ D− 11 Wε fx( ) = − ft( )d, t ( )ε − ∫0 D +1 Γ(D ) ε +− (tx) ε
∞ D−− where ε ∈(0,1] and ft( )d t<∞. Note that Wε fx( ) ∈ () . Thus ∫0 ( )ε in order to obtain appropriate extension of the Weyl fractional integral D+ W fx( ) on the negative dimensions D− < 0 the notion of the Colombeau generalized functions is essentially important. Remark 3.1.2. Thus in negative dimensions from Equation (148) we formally obtain
D− 2 −− 2π ∞ sr( ) sx( )dµ D = d,r= ID− (152) (∫∫X HC,εε) − 0 − ( )ε ε Γ D 2 D +1 ( ) ε + r ε
− D− where ε ∈(0,1] and D≠−0, 2, , − 2 nn , , ∈ and where µHC,ε is ap- ( )ε propriate generalized Colombeau outer measure. Namely Hausdorff-Colombeau outer measure. DD+−, Remark 3.1.3. Note that: if s(00) ≠ the quantity Iε takes infinite ( )ε DD+−, large value in sense of Colombeau generalized numbers, i.e., Iε = ∞ , ( )ε see Definition 3.3.2 and Definition 3.3.3. Remark 3.1.4. We apply through this paper more general definition then (3.1.4):
+ DD+−22D −1 +− 4π π ∞ r sr( ) + sx( )dµ DD, = d,r= IDD, − (153) (∫∫X HC,εε) +−0 − ( ) ε ΓΓDD22D ε ( ) ( ) ε + r ε
+ − where ε ∈(0,1] and D ≥ 1 , D≠−0, 2, , − 2 nn , , ∈ and where DD+−, µHC,ε is appropriate generalized Colombeau outer measure. Namely ( )ε Hausdorff-Colombeau outer measure. In Subsection 3.3 we pointed out that DD+−, there exists Colombeau generalized measure dµHC,ε and therefore Equa- ( )ε tion (151) gives appropriate extension of the Equation (148) on the negative Hausdorff-Colombeau dimensions.
3.2. Hausdorff Measure and Associated Positive Hausdorff Di- mension
Recall that the classical Hausdorff measure [19] [22] originate in Caratheodory’s construction, which is defined as follows: for each metric space X, each set FE= E ζ + E { i }i∈ of subsets i of X, and each positive function ( ) , such that
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+ + 0 ≤ζ (Ei ) ≤∞ whenever EFi ∈ , a preliminary measure φδ (E) can be + constructed corresponding to 0 <δ ≤ +∞ , and then a final measure µ (E) , as + follows: for every subset EX⊂ , the preliminary measure φδ (E) is defined by ++ φζδ (E) =inf { (Ei) | E ⊂≤ E ii, diam( E ) δ}. (154) {E } ∑i∈ i∈ i i∈ φφ++EE≥ 0 <δδ < ≤ +∞ Since δδ12( ) ( ) for 12 , the limit + ++ µ(E) =lim φφδδ( EE) = sup ( ) (155) δ →0+ δ >0
+ exists for all EX⊂ . In this context, µ (E) can be called the result of Cara- + + theodory’s construction from ζ (E) on F. φδ (E) can be referred to as the ++ size δ approximating positive measure. Let ζ (Edi , ) be for example + ++ + d + ζ (Eii, d) =Θ( d) diam( E ) ,0<Θ(d ) , (156)
+ for non-empty subsets Eii , ∈ of X. Where Θ(d ) is some geometrical fac-
tor, depends on the geometry of the sets Ei , used for covering. When F is the ++ set of all non-empty subsets of X, the resulting measure µH (Ed, ) is called the d+-dimensional Hausdorff measure over X; in particular, when F is the set of all (closed or open) balls in X,
+ d + ++ −+ d Θ(dd) Ω( ) =π 2 (2d ) Γ+ 1. (157) 2
n Consider a measurable metric space ( X,,,,µH ( dX)) d∈( −∞ +∞) . The elements of X are denoted by xyz,,, , and represented by n-tuples of real
numbers x= ( xx12,,, xn )
The metric d( xy, ) is a function dX: ×→ X R+ is defined in n dimensions by
12 =δ −− d( xy, ) ∑ ij( y i x i)( y j x j ) (158) ij
and the diameter of a subset EX⊂ is defined by
diam( E) = sup{ d( x , y) | x , y∈ E} . (159)
++ Definition 3.2.1. The Hausdorff measure µH (ED, ) of a subset EX⊂ + with the associated Hausdorff positive dimension D ∈ + is defined by ca- nonical way
++ + + µζH(ED,) = lim inf (Ei, D) | E⊂∀ E ii , i( diam( E ) < δ) , δ →0 {E } {∑i∈ i } i i∈ (160) + + + ++ D( E) =sup{ d ∈+ | d > 0,µH ( Ed ,) = +∞} .
Definition 3.2.2. Remind that a function fX: → defined in a measura- ble space ( X ,,Σ µ ) , is called a simple function if there is a finite disjoint set of
sets {EE1,, , n } of measurable sets and a finite set {αα1,, n } of real num-
bers such that fx( ) = αi if xE∈ i and fx( ) = 0 if xE∉ i . Thus
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n fx= αχ x χ x E ( ) ∑i=1 iEi ( ) , where Ei ( ) is the characteristic function of i . A
simple function f on a measurable space ( X ,,Σ µ ) is integrable if µ (Ei ) <∞
for every index i for which αi ≠ 0 . The Lebesgue-Stieltjes integral of f is defined by µ= n αµ ∫ fEd.∑i=1 ii( ) (161)
A continuous function is a function for which lim xy→ fx( ) = fy( ) whenev-
er limxy→ d( xy ,) = 0 . The Lebesgue-Stieltjes integral over continuous functions can be defined as the limit of infinitesimal covering diameter: when {E } is a disjoined cover- i i∈
ing and xEii∈ by definition (3.2.12) one obtains f( x)d,µ ++ xD ∫X H ( ) (162) ++ = lim ∑∑fx( i ) inf ζ EDEij,.( ij ) diam( E )→0 EXi = ⊃ j ( ) i {Eij } with j EEij i From now on, X is assumed metrically unbounded, i.e. for every xX∈ and r > 0 there exists a point y such that d( xy, ) > r. The standard assumption + that D is uniquely defined in all subsets E of X requires X to be regular (ho- mogeneous, uniform) with respect to the measure, i.e. + ++ + µµHr(BxD( ),,) = Hr( ByD( ) ) for all elements xy, ∈ X and (convex)
balls Bxr ( ) and Byr ( ) of the form Br>0 ( x) ={ z| d( xz ,) ≤∈ rxz ;, X} . In
the limit diam( Ei ) → 0 , the infimum is satisfied by the requirement that the
variation overall coverings {Eij } is replaced by one single covering Ei , such ij∈ E→ Ex that j ij i i . Hence µζ++= + + ∫ fx( )dH( xD ,) lim ∑ fx( ii) ( ED,.) (163) X diam( Ei )→0 EXi = The range of integration X may be parametrized by polar coordinates with
r= dx( ,0) and angle Ω . {Er ,Ω } can be thought of as spherically symme- iii∈ tric covering around a centre at the origin. In the limit, the function ++ ζ (EDr,Ω , ) defined by Equation (156) is given by + + + +DD++ −−11 dµζHr( xD ,) = lim(E,Ω , D) = d Ω r d r . (164) diam( Er,ω )→0 Let us assume now for simplification that fx( ) = f( x) = fr( ) and + limfr( ) = 0 . The integral over a D -dimensional metric space X is then given r→∞ by
D+ ++2π 2 ∞ µ ++= DD= −1 ∫∫fx( )dH ( xD , ) fx( )d x + ∫frr( ) d. r (165) XXD 0 Γ+1 2 The integral defined in (163) satisfies the following conditions. 1) Linearity: af( x) + a f( x) d,µ ++ xD ∫X 11 2 2 H ( ) (166) = a f( x)d,µµ++ xD+ a f( x) d,.++ xD 11∫∫XXHH( ) 2 2 ( )
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2) Translational invariance: f( x+= x)d,µµ++ xD f( x) d,++ xD (167) ∫∫XX0 HH( ) ( )
+ + ++ since dµµHH( x−= x0 , D) d,( xD ) . 3) Scaling property: + f( ax)d,µµ++− xD= aD f( x)d, ++ xD (168) ∫∫XXHH( ) ( )
+ +−+D+ + since dµµHH( xaD ,) = a d,( xD ) . + 4) The generalized δ D ( x) function: + The generalized δ D ( x) function for sets with non-integer Hausdorff di- mension exists and can be defined by formula + fx( )δµD ( x−= x)d,++ xD fx( ) . (169) ∫X 00H ( )
3.3. Hausdorff-Colombeau Measure and Associated Negative Hausdorff-Colombeau Dimensions
During last 20 years the notion of negative dimension in geometry was many developed, see [12] [23] [24] [25] [26] [27]. Remind that canonical definitions of noninteger positive dimension always equipped with a measure. Hausdorff-Besicovich dimension equipped with ++ Hausdorff measure d,µH ( xD ) . Let us consider example of a space of noninteger positive dimension equipped with the Haar measure. On the closed interval 01≤≤x there is a scale 01≤≤σ of Cantor dust with the Haar measure equal to xσ for any interval (0, x) similar to the entire given set of the Cantor dust. The direct product of this scale by the Euclidean cube of dimension k −1 gives the entire scale k +σ , where k ∈ and σ ∈(0,1) [24]. In this subsection we define generalized Hausdorff-Colombeau measure. In subsection III.4 we will prove that negative dimensions of fractal equipped with the Hausdorff-Colombeau measure in natural way. n n Let Ω be an open subset of , let X be metric space X and let F be a set FE= { } of subsets E of X. Let ζ (Exx,,) be a function i i∈ i ∞ ζ : F ×Ω×Ω→ . Let CF (Ω) be a set of the all functions ζ (Ex, ) such that ∞ ζ (Ex, )∈Ω C ( ) whenever EF∈ . Throughout this paper, for elements of the ∞ (0,1] space CF (Ω) of sequences of smooth functions indexed by ε ∈(0,1] we ζ ζ ζ ∈Ω∞ shall use the canonical notations ( ε (Ex, ))ε and ( ε )ε so ε CF ( ) , ε ∈(0,1] .
Definition 3.3.1. We set FM(Ω=) (FF,/ Ω) ( , Ω) , where
(0,1] F,Ω =ζα ∈ CK∞ Ω ∀ ⊂⊂ Ω, ∀ ∈n ∃ p ∈ with MF( ) {( ε )ε ( )
− p supE∈∈ Fx; K ζε (Ex, ) = O( εε) as→ 0} , (170) (0,1] F,,Ω −ζα ∈ CK∞ Ω ∀ ⊂⊂ Ω ∀ ∈n ∀ q ∈ ( ) {( ε )ε F ( )
q supE∈∈ Fx; K ζε (Ex, ) = O( εε) as → 0} .
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Notice that F (Ω) is a differential algebra. Equivalence classes of sequences ζ ζ ζ ( ε )ε will be denoted by cl ( ε )ε or simply ( ε )ε . Definition 3.3.2. We denote by the ring of real, Colombeau generalized numbers. Recall that by definition = M () / ( ) [21], where = ∈(0,1] ∃∈α ∃ ε ∈ ∀≤ εε ≤ εα M () {( xxεε)ε ( )( 00(0,1]) }, (171) = ∈(0,1] ∀∈α ∃ ε ∈ ∀≤ εε ≤ εα () {( xxεε)ε ( )( 00(0,1]) }.
Notice that the ring arises naturally as the ring of constants of the Co- r : lombeau algebras (Ω) . Recall that there exists natural embedding ∈ = ≡ such that for all r , rr ( ε )ε where rrε for all ε ∈(0,1] . We say that r is standard number and abbreviate r ∈ for short. The ring can be en- dowed with the structure of a partially ordered ring: for rs, ∈ ηη∈≤+ ,1 we abbreviate rs≤ or simply rs≤ if and only if there are representa- ,η ≤ tives (rε )ε and (sε )ε with rsεε for all εη∈(0, ] . Colombeau generalized ∈ number r with representative (rε )ε we abbreviate cl (rε )ε . Definition 3.3.3. 1) Let δ ∈ . We say that δ is infinite small Colombeau δ ≈ δ generalized number and abbreviate 0 if there exists representative ( ε )ε q and some q ∈ such that δεε = O( ) as ε → 0 . 2) Let ∆∈ . We say that ∆ ∆= ∞ is infinite large Colombeau generalized number and abbreviate if ∆≈−1 ∞ Θ∈ 0 . 3) Let ∞ be { } We say that ∞ is infinite Colombeau Θ= ∞ generalized number and abbreviate if there exists representative Θ Θ=∞ ( ε )ε where ε for all ε ∈(0,1] . Here we set = Θ = = MM(∞ ) ( ) {( ε )ε } , (∞ ) ( ) and ∞M ( ∞∞) / ( ) . Definition 3.3.4. The singular Hausdorff-Colombeau measure originate in Colombeau generalization of canonical Caratheodory’s construction, which is defined as follows: for each metric space X, each set FE= { } of subsets E i i∈ i ζ of X, and each Colombeau generalized function ( ε (Exx,,))ε , such that: 1) ≤ ζ ζ = ∞ ∈ 0( ε (Exx ,,))ε , 2) ( ε (Exx,,))ε , whenever EF, a preliminary φε Colombeau measure ( δ (Exx,,,))ε can be constructed corresponding to <δ ≤ +∞ µ 0 , and then a final Colombeau measure ( ε (Exx,,))ε , as follows: for ⊂ φε every subset EX, the preliminary Colombeau measure ( δ (Exx,,,))ε is defined by φεδε(E,,, x x ) = sup ζ(E ,, x x) | E ⊂≤ E , diam( E ) δ. (172) {∑i∈ i i∈ ii} {E } i i∈ −− ε ∈ 0,1 φExx,,, εφ≥ Exx ,,, ε 0 <δδ < ≤ +∞ Since for all ( ] : δδ12( ) ( ) for 12 , the limit (µε(Exx,,,)) = lim φεδδ(Exx ,,,) = inf φε(Exx ,,,) (173) ε (δδ→>00) ( ) + ε ε
⊂ µε exists for all EX. In this context, ( (Exx,,,))ε can be called the result of ζ φε Caratheodory’s construction from ( ε (Exx,,))ε on F and ( δ (Exx,,,))ε can be referred to as the size δ approximating Colombeau measure.
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+− Definition 3.3.5. Let ζ ε Ei , D , D ,, xx be ( ( ))ε D+ ΘΘ+− 12(D) ( D) diam( Ei ) +− − if xE∈ i D ζ ε Ei , D , D ,, xx = (174) ( ( ))ε d( xx, ) + ε ε 0 if xE∉ i
+− where ε ∈Θ(0,1] ,12(DD) , Θ( ) > 0 . In particular, when F is the set of all (closed or open) balls in X,
D+ + −D 2 + 2 π Θ=1 (D ) (175) D+ Γ+1 2
and
D− − −D 2 − 2 π Θ=2 (D ) , D− Γ+1 (176) 2 Dn− ≠−−−2, 4, 6, , − 2( + 1) ,
Definition 3.3.6. The Hausdorff-Colombeau singular measure +− µεH ED, , D ,,, xx of a subset EX⊂ with the associated ( ( ))ε +− − Hausdorff-Colombeau dimension DD( )∈∈++, D is defined by +− µεHC ED, , D ,,, xx ( ( ))ε +− =lim sup ζδε E, D , D , x , x | E⊂∀ E , i diam( E ) < , {∑i∈ ( ( i )) i ii( )} δ →0 {E } ε i i∈ ε + + +− D=sup D>= 0 |µεHC ED , , D , xx , ,∞ , { ( ( ))ε } (177) The Colombeau-Lebesgue-Stieltjes integral over continuous functions fX: → can be evaluated similarly as in Subsection III.3, (but using the limit in sense of Colombeau generalized functions) of infinitesimal covering diameter when {E } is a disjoined covering and xE∈ : i i∈ ii +− f( x)dµεHC ED , , D ,,, xx (∫X ( ))ε (178) +− = lim fx( i) inf ζ ε EDii,,,,. D xx → ∑∑ EXi = j ( ) diam( Ei ) 0 {Eij } with EEij⊃ i j ε
We assume now that X is metrically unbounded, i.e. for every xX∈ and r > 0 there exists a point y such that d( xy, ) > r. The standard assumption + − that D and D is uniquely defined in all subsets E of X requires X to be reg- ular (homogeneous, uniform) with respect to the measure, i.e. − +− − +−′ (µHC(BxDDxx r ( ),,,,,εµ)) = ( HC( ByDDxy r ( ),,,,,ε)) , where εε dxx( ,,) = dxy( ′ ) for all elements xyxx,,,′∈ X and convex balls Bxr ( ) and Byr ( ) of the form Br ( x) ={ z| d( xz ,) ≤∈ rxz ;, X} and
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Br ( y) ={ z| d( yz ,) ≤∈ ryz ;, X} . In the limit diam( Ei ) → 0 , the infimum is
satisfied by the requirement that the variation over all coverings {Eij } is re- ij∈ E E→ Ex placed by one single covering i , such that j ij i i . Therefore +− f( x)dµεHC ED , , D ,,, xx (∫X ( ))ε (179) +− = lim fx( ii)ζ ε ED,,,,. D xx i → ∑ EXi = ( ) diam( Ei ) 0 ε
Assume that fx( ) = fr( ), r = r. The range of integration X may be para- r= dx,0 ω E metrized by polar coordinates with ( ) and angle . { rii,ω } can be thought of as spherically symmetric covering around a centre at the origin. Thus
+− f( r)dµεHC ED , , D ,,, xx (∫X ( ))ε (180) +− = lim fr( ii)ζ ε ED,,,,. D xx i → ∑ EXi = ( ) diam( Ei ) 0 ε
n Notice that the metric set X ⊂ can be tesselated into regular polyhedra; n in particular it is always possible to divide into parallelepipeds of the form Π =x = x,, x ∈ Xx | = i − 1 ∆ x +γγ ,0 ≤ ≤∆x , j = 1,,. n ii1,, n { ( 1n) j( j) j j j j } (181)
n = 2 Π For the polyhedra ii12, is shown in Figure 6. Since X is uniform
+− +− dµxD , , D ,,, xx εζ= limΠ ,D , D ,, xx ( HC ( )) ε ( i1,, in ) ε diam Π ( ii1,, n ) ε D+ n n ∆x j = lim (182) ∏ j=1D− diam(Πii,, ) 1 n xx−+ε jj ε D+ n n d x j . ∏ j=1 D+ − D n xxjj−+ε ε
Notice that the range of integration X may also be parametrized by polar
coordinates with r= dx( ,0) and angle Ω . Er,Ω can be thought of as spheri- cally symmetric covering around a centre at the origin (see Figure 7 for the two-dimensional case). In the limit, the Colombeau generalized function +− ζ ε Er,Ω , DDr , , ,0 is given by ( ( ))ε
+− dµεHC r ,,Ω DD , , ( ( ))ε ++ ddΩDD−−11rr = ζ +− Ω (183) limε (Er,Ω , DD , ,{ r ,} ,0) − diam Π D ( ii1,, n ) ε r + ε ε
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Figure 6. The polyhedra covering for n = 2 .
Figure 7. The spherical covering Er,Ω .
When fx( ) is symmetric with respect to some centre xX∈ , i.e. fx( ) = constant for all x satisfying d( xx, ) = r for arbitrary values of r, then change of the variable x→=− z xx (184) can be performed to shift the centre of symmetry to the origin (since X is not a linear space, (184) need not be a map of X onto itself and (184) is measure pre- suming). The integral over metric space X is then given by formula +− f( x)dµεHC ED , , D ,,, xx (∫X ( ))ε + DD+−22D −1 4π π ∞ r fr( ) (185) = +−∫ − d.r ΓΓDD220 D ( ) ( ) ε + r ε
3.4. Main Properties of the Hausdorff-Colombeau Metric Meas- ures with Associated Negative Hausdorff-Colombeau Dimen- sions
n Definition 3.4.1. An outer Colombeau metric measure on a set X is a φ E ∈Ω Colombeau generalized function ( ε ( ))ε F ( ) (see Definition 3.3.1) defined on all subsets of X satisfies the following properties. 1) Null empty set: The empty set has zero Colombeau outer measure
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φ ∅=0. ( ε ( ))ε (186)
2) Monotonicity: For any two subsets A and B of X ABφφ( A) ≤ ( B) . (187) ( εε)εε ( )
3) Countable subadditivity: For any sequence {Aj } of subsets of X pairwise disjoint or not ∞ ∞ φφAA≤ . εεj=1 jj (∑ j=1 ( )) (188) ( ( ))ε ε
4) Whenever d( AB,) = inf{ dn ( xy ,) | x∈ Ay , ∈> B} 0 φAB = φφ A+ B , ( ε( ))ε ( εε( )) εε( ( )) (189)
2 where dn ( xy, ) is the usual Euclidean metric: dn( xy, ) =∑( xii − y) . Definition 3.4.2. We say that outer Colombeau metric measure µε∈ n ( ε )ε ,( 0,1] is a Colombeau measure on σ-algebra of subests of X if µ ( ε )ε satisfies the following property:
∞ ∞ φφAA= . εεj=1 jj(∑ j=1 ( )) (190) ( ( ))ε ε
Definition 3.4.3. If U is any non-empty subset of n-dimensional Euclidean n space, , the diamater U of U is defined as U= sup{ d( xy ,) | xy , ∈ U} (191)
If F ⊆ n , and a collection {U } satisfies the following conditions: i i∈ 1) U < δ for all i ∈ , 2) FU⊆ , then we say the collection i i∈ i {U } is a δ-cover of F. i i∈ n Definition 3.4.4. If F ⊆ and sq,,δ > 0, we define Hausdorff-Colombeau content: s ∞ U sq, ε = i (HFδ ( ,)) inf ∑i=1 q (192) ε x + ε i ε
where the infimum is taken over all δ-covers of F and where
xxi= ( i,1,, x in , )∈ U i for all i ∈ and x is the usual Euclidean norm: = n 2 xx∑ j=1 j .
Note that for 0<<<δδ12 1, ε ∈( 0,1] we have HFsq,,,,εε≥ HFsq δδ12( ) ( ) (193) δ δ HFsq, ,ε since any 1 cover of F is also a 2 cover of F, i.e. δ1 ( ) is increasing as δ decreases. Definition 3.4.5. We define the (sq, ) -dimensional Hausdorff-Colombeau (outer) measure as:
sq,,sq (HF( ,εε)) = lim HFδ ( ,) . (194) ε (δ →0 )ε
Theorem 3.4.1. For any δ-cover, {U } of F, and for any ε ∈(0,1] , ts> : i i∈
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− HFtq,,( ,εδ) ≤ t s HF sq( ,. ε) (195)
ts− Proof. Consider any δ-cover {U } of F. Then each U ≤ δ ts− since i i∈ i
Ui ≤ δ , so:
t ts− s ts− s Ui= UU ii ≤ δ U i. (196)
From (196) it follows that
ts UUδ ts− ii≤ qq (197) xxii++εε
and summing (196) over all i ∈ we obtain
ts ∞∞UU ii≤ δ ts− ∑∑ii=11qq= . (198) xxii++εε
Thus (195) follows by taking the infimum. Theorem 3.4.2. 1) If HFsq, ( ,ε ) <∞, and if ts> , then ( )ε HFtq, ( ,0ε ) = . ( )ε 2) If 0,< HFsq, ( ε ) , and if ts< , then HFtq, ( ,ε ) = ∞ . ( )ε ( )ε Proof. 1) The result follows from (195) after taking limits, since ∀∈ε (0,1] by definitions follows that HFsq, ( ,ε ) <∞. 2) From (3.4.10) ∀∈εδ(0,1] , ∀ > 0 follows that 1 HFsq,,( ,εε) ≤ HFtq( ,.) (199) δ st−
After taking limit δ → 0 , we obtain HFtq, ( ,ε ) = ∞ , since − st− 1 sq,,sq limδ →0 (δ ) = ∞ and limδδ→0 HF( ,εε) = HF( ,) > 0 . Definition 3.4.6. We define now the Hausdorff-Colombeau dimension
dimHC (Fq , ) of a set F (relative to q > 0 ) as
dim(Fq ,) = sup s = sq( ) | Hsq, ( F ,ε ) = ∞ HC { ( )ε } (200) = infs = sq( ) | Hsq, ( F ,ε ) = 0 . { ( )ε } Remark 3.4.1. From theorem 3.4.2 follows that for any fixed qq= : sq, ε = sq, ε = ∞ (HF( ,0)) or (HF( , )) everywhere except at a unique ε ε value s, where this value may be finite. As a function of s, HFsq, ( ,ε ) is de- creasing function. Therefore, the graph of HFsq, ( ,ε ) will have a unique value where it jumps from ∞ to 0. sq, Remark 3.4.2. Note that the graph of HF( ,ε ) for a fixed qq= is ( )ε ∞> ifs dimHC ( Fq , ) sq, ε = > (HF( ,)) 0 ifs dimHC ( Fq , ) (201) ε undetermined ifs= dimHC ( Fq , )
n Definition 3.4.7. We say that fractal F has a negative dimension rela-
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tive to q > 0
if dimHC (Fq ,) −< q 0 .
4. Scalar Quantum Field Theory in Spacetime with Hausdorff-Colombeau Negative Dimensions 4.1. Equation of motion and Hamiltonian
Scalar quantum field theory and quantum gravity in spacetime with noninteger positive Hausdorff dimensions developed in papers [28] [29] [30] [31]. Quan- tum mechanics in negative dimensions developed in papers [32] [33] Scalar quantum field theory and quantum gravity in spacetime with Haus- dorff-Colombeau negative dimensions originally developed in paper [12]. In this section only free scalar quantum field in spacetime with negative dimensions briefly is considered. A negative-dimensional spacetime structure is a desirable feature of superre- normalizable spacetime models of quantum gravity, and the most simply way to obtain it is to let the effective dimensionality of the multifractal universe to change at different scales. A simple realization of this feature is via suitable ex- tended fractional calculus and the definition of a fractional action. Note that be- low we use canonical isotropic scaling such that:
µ =−=µ − xD1, 0,1, ,t 1 (202)
while replacing the standard measure with a nontrivial Colombeau-Stieltjes measure, dDt xx→= dDf ( d,ηε( x)) , ε (203) [η] =Dt ⋅ αα, ∈[ 1, −∞) .
Here Dt is the topological (positive integer) dimension of embedding space- time and α is a parameter. Any Colombeau integrals on net multifractals can be approximated by the left-sided Colombeau-Riemann--Liouville complex mil- ti-fractional integral of a function L (t) :
zti ( )−1 tt{zt( )} m (tt−+) iε dηε( x ,)LL( t) ∝ Ii (tt)d, (∫∫0 ) ( t ,ε ) ∑i=1 ε ε ε Γ( zti ( )) ε (204) zt( ) zti ( ) i t−( tt −+) iε (ηε(t,,)) = ε Γ+( zti ( ) 1) ε
where ε ∈(0,1] , t is fixed and the order zt( ) is (related to) the complex Hausdorff-Colombeau dimensions of the set. In particular if
zii ∈=, 1,2, , m is a complex parameter an integral on net multifractals can be approximated by finite sum of the left-sided Colombeau-Riemann-Liouville complex fractional integral of a function L (t)
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t m {zi } d,ηε( xt)L ( ) ∝ Ii=1 (∫0 ) t ,ε ε ε
mm1 t zi −1 = zi −+ε ∑∑= (It ,ε ) = d,(tt) iL ( t) (205) ii11ε Γ z ∫ε ( i ) ε z tzi −( tt −+) iε i ηε = m ( (t,.))ε ∑i=1 Γ+( zi 1) ε Note that a change of variables t→− tt transforms Equation (205) into the form zi −1 ttm [ti+ ε ] d,ηε( x)LL( t) = dt( tt− ) . (206) (∫∫00) ∑i=1 ε Γ( zt( )) ε The Colombeau-Riemann-Liouville multifractional integral (206) can be mapped onto a Colombeau-Weyl multifractional integral in the formal limit
t → +∞ . We assume otherwise, so that there exists limt →+∞ zt( ) and
limt →+∞ LL(t−= t) qt( ), qt ( ) . In particular if z ∈ is a complex parame- ter a change of variables t→− tt transforms Equation (206) into the form
zi −1 t ti+ ε mmzi [ ] Iε = dt L qt( ) ,. qt ( ) (207) ∑∑ii=11( t , )ε = ∫ε Γ( zi ) ε This form will be the most convenient for defining a Colombeau-Stieltjes field
theory action. In Dt dimensions, we consider now the action
(Sε) = d,ηε( x)L ϕε( xx) ,∂ µε ϕ( ) , (208) ε (∫M )ε ϕϕ∂ ϕ where L , µ is the Lagrangian density of the scalar field ( ε ( x))ε and where m D −1 dηεx , = t fx,ε d, xfxµ ,ε : M→ , ( ( ))ε ∑i=1∏µ =0 ( µµ,,ii( ))εε( ( )) (209) ηε is some Colombeau-Stieltjes measure. We denote with pair (Mx,d( ( , ))ε ) the metric spacetime M equipped with Colombeau-Stieltjes measure ηε (d,( x ))ε . The former can be taken to be the canonical scalar field Lagran- gian, 1 µ (L ϕε( xx),,∂ µε ϕ( ) ) =−∂( µεϕϕ ∂ ε) −(V( ϕε)) (210) ε 2 ε ε where V (ϕ ) is a potential and contraction of Lorentz indices is done via the
Minkowski metric ηµν =( −+,, , +)µν . As for the Colombeau-Stieltjes measure, we make the multifractal spacetime isotropic choice
fµε =( fi,tε ) ,µ = 1, , Di −= 1; 1, , m . (211) ( ( ,,i) )ε ε Hence the scalar field action (208) reads
(Sε) = d,η( xxx εϕ) ε( ) ,∂ µε ϕ( ) ε (∫M )ε (212) m 1 = Dt ∂∂ϕϕµ + ϕ ∑ j=1∫ d,xvε, j ( x) µε εV ( ε) 2 ε
where (vxε ( ))ε is a coordinate-dependent
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Lorentz scalar 1 (vxε , j ( )) = . (213) ε Dt ( α −1 ) sx( ) + ε j ε
We define now the Dirac distribution as Colombeau generalized function by equation
m (Df , j ) ∑ = d,ηj ( x εδ) ( xm , ε) = . (214) j 1(∫ {v j } ) ε
In particular for the case m = 1
(D ) dη( xx , εδ) f ( , ε) = 1. (215) (∫ {v} )ε
Invariance of the action under the infinitesimal shift ϕ( xxx) →+ ϕ( ) δϕ ( ) vi,= 1, , m gives the equation of motion for a generic weight ( i,ε )ε : ∂∂LLm ∂µεvi, d −+ =0. (216) ∑i=1µ ∂ϕεεvi, dx ∂∂µεϕ ε ( ) ε
In particular for the case m = 1 we obtain ∂∂LL∂µεv d −+ =0. (217) µ ∂ϕεεv dx ∂∂µεϕ ε ( ) ε
From Equation (212) and Equation (216) we obtain m ∂µεvi, µ d (ϕ) + ∂− ϕϕV ( ) =0. (218) εε ∑i=1 εε v εεdϕ i, ε ε
µ where =∂∂µ . In particular for the case m = 1 we obtain
∂µεv µ d (ϕε) + ∂− ϕϕ εεV ( ) =0. (219) ε v dϕ εεε ε
4.2. Propagator in Configuration Space with Negative-Dimensions
We define the canonical vacuum-to-vacuum amplitude by
m ZJ,ε= D ϕexpi d ηϕL + J , ( [ ])ε ∫∫ε∑ j=1 j,εε( ) ) (220) ( ε
where J is a source. Integration by parts in the exponent leads to the Lagrangian density for a free field as 11m ∂µεv j, (L ) =ϕ + ∂−µ m2 ϕ =( ϕϕ ℑ ) , (221) εεεε∑ j=1 εεεε 22v ε j, ε
where ∂ v ℑ= +m µεj, ∂−µ 2 = ε ∑ j=1 mj; 1, , m . (222) v j,ε In particular for the case m = 1 we obtain
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∂µεv µ 2 ℑ=ε + ∂−m . (223) vε
The propagator is the Green function (Gxε ( ))ε solving the equation D− (ℑ=εεGx( )) δεv ( x,,) (224) ε ( )ε
− where DD=t (α −<10) . By virtue of Lorentz covariance, the Green function 22µ i Gxε ( ) must depend only on the Lorentz interval s= xµ x = xxi − t , where xt0 = and iD=1, , − 1. In particular, (v) = v sx( ) with the correct t εεε ( ( ))ε scaling property is
− −1 D µ (vεµ( sx( ))) =+= sx( ) ε ,.sx( ) xx (225) ε ε
Note that
xµ 2 Dt −1 ∂µ = ∂ss,. =∂+ ∂ s (226) (ss++εε)εε( )
Hence the inhomogeneous Equation (224) reads
− 22Dtα −1 D ∂+ss ∂−m( Gxε ( )) =δεv( x,.) (227) + ε ε ( )ε (s )ε
i 2 We first consider the Euclidean propagator and denote with r= xxi + t the Wick-rotated Lorentz invariant. In the massless case, the solution of the homogeneous equations for any α < 0 is
2β 2 + Dt α (Gε ( r)) = Cr ,.β = (228) ε 2 Let us now consider the massive case.The solution of the homogeneous equa- ℑ= α < tion ( εεGr( ))ε 0 for any 0 is
2+⋅Dt α 2 r (Gε ( r)) = C12KI22+⋅DDαα( mr) + C+⋅ ( mr) , (229) ε m −−tt 22
where CC12, are constants and K λ and Iλ are the modified Bessel func-
tions. The function Iν ( z) is ν +2k ∞ ( z 2) Iν ( z) = ∑ . (230) k =0 kk!1Γ(ν ++)
Formula (230) is valid providing ν ≠−−−1, 2, 3,
−+ν 2k ∞ ( z 2) I−ν ( z) = ∑ (231) k =0 kk!1Γ−( ν + +)
Formula (231) is obtained by replacing ν in (232) with a −ν . π K( z) =−−II( zz) ( ) . (232) −−ν 2sin ν π νν
The modified Bessel functions I−ν ( z) and
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K −ν ( z) have the following asymptotic forms for z → 0 : −−νν 12 1z KI−−νν( zz) Γ−( νν) , ( ) ,≠−−− 1, 2, 3, (233) 22z Γ−( ν +1)
Since for small m 0 the solution must agree with the massless case (228),
we can set C2 = 0 . To find the solution of the inhomogeneous equation, one exploits the fact that the mass term does not contribute near the origin. Ex-
panding Equation (229) at mr 0 when α < 0 ( C2 = 0 ), we find
4+⋅D α 2+⋅D α − t i 2 +⋅D α t i =2 Γ− t i 2 2 (Grε ( ))ε C1 2 (r) (234) 2
which must coincide with Equation (228). This gives the coefficient C1 and the propagator reads
Dt 2+⋅D α Γ − t,ii 1 2 m 2 = − G( r) D K 2+D α (mr). (235) t D α 2r − t 2π 2 Γ−t 2 2
5. The Solution Cosmological Constant Problem 5.1. Einstein-Gliner-Zel’dovich Vacuum with Tiny Lorentz Inva- riance Violation
We assume now that: 1) Poincaré group of momentum space is deformed at some fundamental
high-energy cutoff Λ∗ [9] [10]. 2 ab 2) The canonical quadratic invariant p=η ppab collapses at high-energy Λ cutoff ∗ and being replaced by the non-quadratic invariant: ab 2 η pp p = ab. (236) + (1 lpΛ∗ 0 )
3) The canonical concept of Minkowski space-time collapses at a small dis- = Λ−1 tance lΛ∗∗ to fractal space-time with Hausdorff-Colombeau negative di- mension and therefore the canonical Lebesgue measure d4 x being replaced by the Colombeau-Stieltjes measure
d,ηε( x) = v sx( ) d4 x , (237) ( )ε ( ε ( ) )ε
where
−1 D− (vε ( sx( ))) = sx( ) + ε , ε ε (238) µ sx( ) = xxµ ,
see subsection IV.2. 4) The canonical concept of momentum space collapses at fundamental
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high-energy cutoff Λ∗ to fractal momentum space with Hausdorff-Colombeau negative dimension and therefore the canonical Lebesgue measure d3k , where
k = (kkkxyz,,) being replaced by the Hausdorff-Colombeau measure
−DD+++ −−1 +− ∆(D)ddk ∆∆( D) ( Dp) p d DD, k =, (239) DD−− k + ε p + ε ε ε
D± 2 ± 2π where ∆=(D ) and p=k = kkkxyz ++. Γ(D± 2) +− Remark 5.1.1. Note that the integral over measure dDD, k is given by for- +− mula (185). Thus vacuum energy density εµ(DD,,eff , p∗ ) for free quantum fields is
+− +− ε(DDp,, µeff ,∗) =++ εµ( eff ) εµ( eff ,pDDp ∗∗) ε( ,, µeff ,) . (240)
Here the quantity εµ( eff ) is given by formula
µ 1 eff εµ = µ µ2+ µ 23 ( eff ) 3 ∫∫ddf ( ) kk 22( π) 0 k ≤µ
µeff = K∫∫ddµµ f( ) p2+ µ 22 pp (241) 0 p≤µ
µeff µ = K∫∫ddµµ f( ) p2+ µ 22 pp 00
2π = = εµ where Kc3 ,1. The quantity ( eff , p∗ ) is given by formula (2π)
µ 1 eff εµ = µ µ 2+ µ 23 ( eff ,dpf∗ ) 3 ∫∫( ) kk d 22( π) 0 µ< +− The quantity εµ(DD,,eff , p∗ ) (since Equation (22) holds) is given by formula +− εµ(DD,,eff , p∗ ) µeff 2 42 µµllΛΛ1 2 +− =Kfdµµ( ) ∗∗+ ++kkµ2,d,DD ∫∫−−µµ22 22 22 ( ) 0 k ≥ p∗ 11llΛΛ∗∗1− µ l Λ∗ (243) 1 ′ = = where Kc3 ,1. 22( π) µµ22 42 Remark 5.1.2. We assume now that llΛΛ∗∗1, 1 and therefore from Equation (243) we obtain DOI: 10.4236/***.2019.***** 44 Journal of Modern Physics J. Foukzon et al. +− εµ(DD,,eff , p∗ ) µµeff eff +− (244) ′′2 3,D− 2 2,DD =KlΛ ∫ f(µµ) dd µ ∫k ++K ∫∫ dµ f( µ) kk µd. 00kk≥≥pp∗∗ From Equation (244) and Equation (239) we obtain +− εµ(DD,,eff , p∗ ) µµ+− +− =Kl′′eff f(µµ) 2dd µ DD ,k ++Keff dµ f( µ) kk2 µ 2,dDD Λ ∫00 ∫ ∫∫ kk≥≥pp∗∗ D+ −1 µeff ∞ ppd =∆∆Kl′ D+− D fdµµµ2 Λ ( ) ( ) ∫∫( ) − ( 0 ) p∗ D p + ε ε (245) 22D+ − 1 µeff ∞ p+ µ ppd +∆KD′ +− ∆ Ddµµ f ( ) ( ) ∫∫( ) − 0 p∗ D p + ε ε µ ∞ −+ =∆∆Kl′ D+ D − eff f(µµ) 21dd µ pDD+− p ( Λ ( ) ( ) ∫0 ) ∫p ∗ µ ∞ −+ +∆KD′ ( +) ∆( D −) eff dµµ f( ) p22+ µ pDD+− 1d. p ∫∫0 p∗ Remark 5.1.2. We assume now that: DD−++ +2 ≤− 6. (246) Note that µ ∞ −+ eff ddµµf( ) pp22+ µDD+− 1 p ∫∫0 p∗ 2 µ ∞ µ −+ = eff µµ + DD+ ∫∫d1f( ) 2 pp d 0 p∗ p µµeff ∞∞−+ 1 eff −+ = ∫∫f(µ)dd µ ppDD+ + ∫ f(µµ) 21 d µ ∫ pDD+− d p 00pp∗∗2 µ ∞ −+ −+ 1 eff 43DD+− DD+− 4 (247) −+∫∫f(µµ) dd µ p p Op∗ 8 0 p∗ ( ) DD−+++1 DD−++ ppµµeff eff = ∗∗µ µ+ µµ2 µ −+∫ ff( )dd−+∫ ( ) DD++1 002(DD+ ) DD−++−1 p µeff −+ −+∗ f(µµ) 44d. µ OpDD+− −+ ∫0 ( ∗ ) 81(DD+−) Thus finally we obtain +− εµ(DD,,eff , p∗ ) DD−+++1 Kp′ µeff = ∗ f (µµ)d DD−+++1 ∫0 DD−++ µeff p (248) ′ +− 2 ∗ +KlΛ ∆∆+( D) ( D ) 0.5 f(µµ) d µ −+ (∫0 ) DD+ DD−++−2 Kp′ µeff −+ −+∗ f(µµ) 44d. µ OpDD+− −+ ∫0 ( ∗ ) 81(DD+−) DOI: 10.4236/***.2019.***** 45 Journal of Modern Physics J. Foukzon et al. Remark 5.1.3. Note that (see Equations (42)): εµ ( eff ,,pp∗∗) = εµ( eff ) + εµ( eff ) µµ µ 11eff eff 1eff =4µ µ+ 22 µµ µ+− µµ4 µ pf∗∗∫∫( )d pf( ) d C1 ln p∗ ∫ f( ) d (249) 4400 80 µ µµ 1eff 11eff eff +−fµµ4ln µ d µ fµµ6d µ + Of µµ85 p− . ∫( ) ( ) 2 ∫∫( ) ( ) ∗ 8 0 p∗ 32 00 From Equation (240), Equation (248) and Equation (249) finally we obtain +− +− ε(DD,, µeff , p∗) =++ εµ( eff ) εµ( eff ,p ∗∗) ε( DD ,, µeff , p) µµ µ 11eff eff 1eff =4µ µ+ 22 µµ µ+− µµ4 µ pf∗∗∫∫( )d pf( ) d C1 ln p∗ ∫ f( ) d 4400 80 µ µµ(250) 1eff 11eff eff +−fµµ4ln µ d µ fµµ6d µ + Of µµ85 p− ∫( ) ( ) 2 ∫∫( ) ( ) ∗ 8 0 p∗ 32 00 −+ + DD++2 Op( ∗ ). +− The pressure pD( ,, Dµeff , p∗ ) for free scalar quantum field is +− +− pD( ,, Dµeff , p∗) =++ p( µµeff ) p( eff , p ∗∗) pD( ,, D µeff , p) . (251) Here the quantity p(µeff ) is given by formula 4 Kpµeff pf(µeff ) = d µµ( ) d. p (252) ∫∫0 22 3 p <µ p + µ The quantity pp(µeff , ∗ ) is given by formula 4 Kpµeff pp(µeff ,∗ ) = dµµ f( ) d.p (253) ∫∫0 22 3 µ≤≤pp∗ p + µ +− The quantity pD( ,, Dµeff , p∗ ) is given by formula 4 +− Kp′ µeff pD,,, Dµeff p∗ d µµf( ) d,p (254) ( ) ∫∫0 22 3 pp> ∗ p + µ 1 ′ = = where Kc3 ,1. 22( π) Remark 5.1.4. Note that (see Equations (42): ppp (µeff ,,∗∗) =( µµeff ) + pp( eff ) µµeff eff 114 22 = pf∗∗∫∫(µ)dd µ− pf(µµ) µ 12 0012 µµ 11eff eff (255) ++ µµ44 µ− µµ µ µ C2 ln pf∗ ∫∫( ) d f( ) (ln) d 8800 µµ 51eff eff ++fµµ6d. µ Of µµ85 p− 2 ∫∫( ) ( ) ∗ p∗ 32 00 From Equation (250), Equation (254) and Equation (255) similarly as above finally we get DOI: 10.4236/***.2019.***** 46 Journal of Modern Physics J. Foukzon et al. +− pD( ,, Dµeff , p∗ ) µµeff eff 114 22 = pf∗∗∫∫(µ)dd µ− pf(µµ) µ 12 0012 µµ 11eff eff (256) ++ µµ44 µ− µµ µ µ C2 ln pf∗ ∫∫( ) d f( ) (ln) d 8800 µµeff eff 51 −+ +fµµ6d. µ ++ O f µµ85 p− OpDD++2 2 ∫∫( ) ( ) ∗∗( ) p∗ 32 00 Remark 5.1.5. We assume now that: µµeff eff µeff ∫∫ff(µ)ddd0 µ= ( µµ) 24 µ= ∫ f( µµ) µ= . (257) 00 0 From Equation (250), Equation (256) and Equation (257) finally we get µ 1 eff ε ε+− µ = µµ42 µ µ+ − (D,,, Deff p∗∗) ∫ f( ) (ln) dOp( ) , 8 0 µ (258) 1 eff +−µ =−+µµ42 µ µ − p ( D,,, Deff p∗∗) ∫ f( ) (ln) dOp( ) . 8 0 Remark 5.1.6. Note that the Equation (258) can be obtained without fine-tuning (257) which was assumed in Zel’dovich paper [1]. In order to obtain Equation (5.1.23) and strictly weaker conditions we assume now that: 1) −n ff(µ) =+=sm..( µ) fgm ..( µµ) eff, (259) where n > 0 is an parameter, fsm..(µ ) corresponds to standard matter and where fgm..(µ ) corresponds to physical ghost matter, see Equation (32). 2) µeff = 4 µµ≈ Ip1 ∗ ∫ f( )d 0, 0 µeff = 22µµ µ≈ Ip2 ∗ ∫ f( ) d 0, (260) 0 µeff = µµ4 µ≈ I3 ln pf∗ ∫ ( ) d 0 0 3) µeff ++ µµ4 µ µ III123 ∫ f( ) (ln) d . (261) 0 5.2. Zeropoint Energy Density Corresponding to a Non-Singular Gliner Cosmology We assume now that µµeff eff µeff µ µ=<> µµ42 µ µµ µ µ ∫∫f( )d 0, f( ) d 0, ∫ fp( ) d 0,∗ eff . (262) 00 0 DOI: 10.4236/***.2019.***** 47 Journal of Modern Physics J. Foukzon et al. From Equation (250), Equation (256) and (262) we obtain +− εε (DD,, µeff , p∗ ) µµ 11eff eff =22µµ µ−− µµ4 µ pf∗∗∫∫( ) d C1 ln p f( ) d 4800 µµ 1eff 11eff (263) +−µµ46 µ µ µµ µ ∫∫ff( ) (ln) d 2 ( ) d 8 00p∗ 32 µeff −+ ++µµ85− DD++2 O∫ f( ) p∗∗ Op( ), 0 and +− p pD( ,, Dµeff , p∗ ) µµ 11eff eff =− 22µµ µ−+ µµ4 µ pf∗∗∫∫( ) d C2 ln p f( ) d 12 008 µµ 1eff 51eff (364) −+µµ46 µ µ µµ µ ∫∫ff( ) (ln) d 2 ( ) d 8 00p∗ 32 µeff −+ ++µµ85− DD++2 O∫ f( ) p∗∗ Op( ) 0 correspondingly. From Equation (263) and Equation (264) we obtain µµ 13eff eff +=−ε 22µµ µ− + µµ4 µ 3p pf∗∗∫∫( ) d 3 C2 ln p f( ) d 4800 µµ 3eff 53eff −+µµ46 µ µ µµ µ ∫∫ff( ) (ln) d 2 ( ) d 8 00p∗ 32 µµ 11eff eff +22µµ µ−− µµ4 µ pf∗∗∫∫( ) d C1 ln p f( ) d 4800 µµ (265) 1eff 11eff +−µµ46 µ µ µµ µ ∫∫ff( ) (ln) d 2 ( ) d 8 00p∗ 32 µµ 1 eff eff = − µµ4 µ−+ µµ4 µ ln pf∗ ∫∫( ) d (3dCC21) f( ) 4 00 µµ 1eff 51eff −+µµ46 µ µ µµ µ < ∫∫ff( ) (ln) d 2 ( ) d 0. 4 00p∗ 16 Therefore under conditions (262) the inequality −23εε corresponding to Gliner non-singular cosmology [2] [4] is satisfied. 5.3. Zeropoint Energy Density in Models with Supermassive Physical Ghost Fields We assume now that: 1) ghost fields corresponding to massive spin-2 particle with mass m2 and to massive scalar particle with mass m0 appears (see Subsection II.2) as real phys- DOI: 10.4236/***.2019.***** 48 Journal of Modern Physics J. Foukzon et al. ical fields in action () Remark 5.3.1. Note that their unphysical behavior may be restricted to arbi- trarily high-energy cutoff Λ by an appropriate limitation on the renormalized masses m2 and m0 . Actually, it is only the massive spin-two excitations of the field which give the problem with unitarity and thus require a very large mass (see Subsection II.2). 2) Poincaré group is deformed at some fundamental high-energy cutoff Λ∗ 22 Λ=Λ∗∗(m02,. m) mc 0< mc 2 (267) 2 ab The canonical quadratic invariant p=η ppab collapses at high-energy cu- toff Λ∗ and being replaced by the non-quadratic invariant ab 2 η pp p = ab. (268) + (1 lpΛ∗ 0 ) 3) The canonical concept of Minkowski space-time collapses at a small dis- tance to fractal space-time with Hausdorff-Colombeau negative dimension and therefore the canonical Lebesgue measure d4 x being replaced by the Colom- beau-Stieltjes measure d,ηε( x) = v sx( ) d4 x , (269) ( )ε ( ε ( ) )ε where − −1 D µ (vεµ( sx( ))) =+= sx( ) ε ,,sx( ) xx (270) ε ε 4) we assume that ff(µµµ) =sm..( ) + fgm ..( ), (271) where fsm..(µ ) corresponds to standard matter and where fgm..(µ ) corres- ponds to physical ghost matter. Remark 5.3.2. We assume now that −n 12 O(µ),1 n> mc0µ eff ≤≤ µµeffmc 2 f (µ ) = (272) 2 0 µµ> eff +−12 Thus vacuum energy density ε(DD,, µµeff , eff ) for free quantum fields is +−12 12 +−12 ε(DD,, µµeff , eff ) = εµµ( eff , eff ) + ε(DD ,, µµeff , eff ) . (273) 12 Here the quantity εµ( eff, µ eff ) is given by formula 2 1 µeff εµ1 µ 2 = µ µ2+ µ 23 ( eff ,deff ) 3 1 f ( ) kk d ∫∫µeff 22( π) k ≤ µ (274) µ2 eff 2 22 = K1 dµµ f( ) p+ µ ppd, ∫∫µeff p≤ µ 2π +− = = ε µµ12 where Kc3 ,1. The quantity (DD,,eff , eff ) is given by for- (2π) mula DOI: 10.4236/***.2019.***** 49 Journal of Modern Physics J. Foukzon et al. +−12 ε(DD,, µµeff , eff ) 42 2 2 µeff µ l 1 µ lΛ 2 +− =Kf′ dµµ Λ + ∗ ++kkµ2,d,DD ∫∫µ1 ( ) 22 22 ( ) eff −−µµ22 k > µ 11llΛΛ1− µ l ∗ Λ∗ (275) 1 ′ = = where Kc3 ,1. 22( π) 22 µ l < 1 Remark 5.3.2. We assume now that Λ∗ , and therefore from Equation (5.3.9) we obtain +−12 ε(DD,, µµeff , eff ) µµ22+− eff 2 3,D− eff 2 2,DD (276) Kl′′Λ 11dµµµ f( ) dk++K dµµ f( ) kk µd. ∫µµeff ∫∫∫eff kk>>µµ From Equation (276) and Equation (239) we obtain +−12 ε(DD,, µµeff , eff ) µµ22+− +− eff 2DD , eff 2 2,DD Kl′′Λ 11dddµµµ f( ) k++Kµµ f( ) kk µ d ∫µµeff ∫ ∫∫eff kk>>µµ + 2 D −1 µeff ∞ ppd =∆∆K' D+− Dldµ f µµ2 ( ) ( ) Λ ∫∫1 ( ) − µµeff D p + ε ε (277) + 2 22D − 1 µeff ∞ p+ µ ppd +∆KD′ +− ∆ Ddµµ f ( ) ( ) ∫∫1 ( ) − µµeff D p + ε ε µ2 ∞ −+ ′ + − eff 21DD+− =∆∆K( D) ( Dl) Λ 1 ddµ f( µµ) p p ∫∫µµeff µ2 ∞ −+ ′ + − eff 22DD+− 1 +∆KD( ) ∆( D) 1 dµµ f( ) p+ µ pd. p ∫∫µµeff Note that p211 pp 24 1 p 6 22+=µµ + = µ + − + + p 112 24 6 µ2 µµ 8 16 µ (278) 11pp24 1 p 6 =+−+µ + 2µ 8µµ35 16 By inserting Equation (278) into Equation (274) we get 12 εµ( eff, µ eff ) 2 24 6 µeff 11pp 1 p = µµ µ+−+ +2 K1 dd f( ) 35 pp ∫∫µeff p≤ µ 2µ 8µµ 16 µ 2 46 8 µeff 11pp 1 p = µµ µ2 +−+ + Kf∫∫1 dd( ) p 35 p µeff µ µµ 0 2 8 16 µ 2 35 7 9 µeff pp11 p 1 p = µ µµ +− + + Kf∫ 1 d ( ) 35 µeff µ µµ 3 2 5 879 16 0 DOI: 10.4236/***.2019.***** 50 Journal of Modern Physics J. Foukzon et al. 35 1 9 2 22 2 2 µeff µµµ11 1 µ = µ µµ+− + + Kf∫ 1 d ( ) 35 µeff 3 2 5µ 879µµ 16 531 1 2 − µeff 11 1 1 = µµ µ222+−+ µ µ µ 2 + Kf∫ 1 ( )d (279) µeff 3 10 56 144 531 1 µ2 − −+n 12 eff 11222 1 1 2 1 =Kf1 (µµ)d. µ+−+ µ µ µ +o µ ∫µ (( eff ) ) eff 3 10 56 144 +−12 The pressure pD( ,, D µµeff , eff ) for free quantum fields is +−12 12 +−12 pD( ,, D µµeff , eff ) = p( µµeff , eff ) + pD( ,, D µµeff , eff ) . (280) 12 Here the quantity p(µµeff, eff ) is given by formula 2 2 1 µeff k µµ12= µµ 3 pf( eff,d eff ) 3 1 ( ) dk ∫∫µeff 22 22( π) k ≤ µ k + µ (281) 2 4 Kpµeff = 1 dµµfp( ) d. ∫∫µeff 22 3 p≤ µ p + µ +−12 The quantity pD( ,, D µµeff , eff ) is given by formula 2 µ2 +− +−12 K′ eff k DD, pD( ,, D µµeff , eff ) 1 d µµf( ) dk , (282) ∫∫µeff 22 3 p > µ k + µ 1 ′ = = where Kc3 ,1. Note that 22( π) −1 1 p2 =µ −1 1 + 22 2 p + µ µ 13pp24 5 p 6 =µ −1 −+− + 1 24 6 (283) 2µµ 8 16 µ 11pp24 3 5 p 6 =−+− + µ 2µµ35 8 16 µ 7 By inserting Equation (283) into Equation (281) we get 12 p(µµeff, eff ) 2 24 6 Kµeff 11pp 3 5 p = µµ −+− +4 1 ddf( ) 35 7 pp ∫∫µeff 3 p≤ µ µ 2µµ 8 16 µ 2 4 6 8 10 Kµeff ppp13 5 p = µµ −+− + 1 ddfp( ) 35 7 ∫∫µeff 3 p≤ µ µ 2µµ 8 16 µ µ 2 5 7 9 11 Kµeff pp13 p 5 p = µµ −+− + ∫ 1 d f ( ) 35 7 µeff µ µµ µ 3 5 27 8 9 16 10 0 DOI: 10.4236/***.2019.***** 51 Journal of Modern Physics J. Foukzon et al. 5 7 9 11 2 22 2 2 K µeff µµ13 µ 5 µ = µµ−+− + ∫ 1 d f ( ) 35 7 3 µeff 5µ 27µµ 8 9 16 10 µ 31 1 3 2 −− K µeff 11 1 1 = µµ µ22−+ µ µ 2 − µ 2 + ∫ 1 d f ( ) (284) 3 µeff 5 14 24 32 31 1 3 µ2 −− −−n 12 K eff 1122 1 2 1 2 1 =1 d.µµfo( ) µ−+ µ µ − µ + µ ∫µ (( eff ) ) 3 eff 5 14 24 32 6. Discussion and Conclusion We will now briefly review the canonical assumptions that are made in the usual formulation of the cosmological constant problem. The canonical assumptions: 1) The physical dark matter. Dark matter is a hypothetical form of matter that is thought to account for approximately 85% of the matter in the universe, and about a quarter of its total energy density. The majority of dark matter is thought to be non-baryonic in nature, possibly being composed of some as-yet-undiscovered subatomic par- ticles. Its presence is implied in a variety of astrophysical observations, including gravitational effects that cannot be explained unless more matter is present than can be seen. For this reason, most experts think dark matter to be ubiquitous in the universe and to have had a strong influence on its structure and evolution. The name dark matter refers to the fact that it does not appear to interact with observable electromagnetic radiation, such as light, and is thus invisible (or ‘dark’) to the entire electromagnetic spectrum, making it extremely difficult to detect using usual astronomical equipment. Because dark matter has not yet been observed directly, it must barely interact with ordinary baryonic matter and radiation. The primary candidate for dark matter is some new kind of elementa- ry particle that has not yet been discovered, in particular, weakly-interacting massive particles (WIMPs), or gravitationally-interacting massive particles (GIMPs). Many experiments to directly detect and study dark matter particles are being actively undertaken, but none has yet succeeded. 2) The total effective cosmological constant λeff is on at least the order of magnitude of the vacuum energy density generated by zero-point fluctuations of the standard particle fields. 3) Canonical QFT is an effective field theory description of a more funda- mental theory, which becomes significant at some high-energy scale Λ∗ . 4) The vacuum energy-momentum tensor is Lorentz invariant. 5) The Moller-Rosenfeld approach [34] [35] to semiclassical gravity by using an expectation value for the energy-momentum tensor is sound. 6) The Einstein equations for the homogeneous Friedmann-Robertson-Walker metric accurately describes the large-scale evolution of the Universe. Remark 6.1.1. Note that obviously there is a strong inconsistency between Assumptions 2 and 3: the vacuum state cannot be Lorentz invariant if modes are DOI: 10.4236/***.2019.***** 52 Journal of Modern Physics J. Foukzon et al. ignored above some high-energy cutoff Λ∗ , because a mode that is high energy in one reference frame will be low energy in another appropriately boosted frame. In this paper Assumption 3 is not used and this contradiction is avoided. Remark 6.1.2. Note that also, Assumptions 1, 3, 4 and 5 is modified, which we denote as Assumptions 4 and 5 respectively. Modified assumptions 1’) The physical dark matter. 2’) The total effective cosmological constant λeff is on at least the order −+n 5 µµeff ln eff of magnitude of the renormalized vacuum energy density gen- erated by zero-point fluctuations of standard particle fields and ghost particle fields, see subsection 1.2. 3’) The vacuum energy-momentum tensor is not Lorentz invariant. 6.1. The Physical Ghost Matter and Dark Matter Nature In the contemporary quantum field theory, a ghost field, or gauge ghost is an unphysical state in a gauge theory. Ghosts are necessary to keep gauge inva- riance in theories where the local fields exceed a number of physical degrees of freedom. For example in quantum electrodynamics, in order to maintain manif- est Lorentz invariance, one uses a four-component vector potential Axµ ( ) , whereas the photon has only two polarizations. Thus, one needs a suitable me- chanism in order to get rid of the unphysical degrees of freedom. Introducing fictitious fields, the ghosts, is one way of achieving this goal. Faddeev-Popov ghosts are extraneous fields which are introduced to maintain the consistency of the path integral formulation. Faddeev-Popov ghosts are sometimes referred to as “good ghosts”. “Bad ghosts” represent another, more general meaning of the word “ghost” in theoretical physics: states of negative norm, or fields with the wrong sign of the kinetic term, such as Pauli-Villars ghosts, whose existence allows the probabili- ties to be negative thus violating unitarity. (VI.1) In contrary with standard Assumption 1 in the case of the new ap- proach introduced in this paper we assume that: (VI.1.1.a) The ghosts fields and ghosts particles with masses at a scale less then a fixed scale meff really exist in the universe and formed dark matter sec- tor of the universe, in particular: (VI.1.1.b) these ghosts fields give additive contribution to a full zero-point fluctuation (i.e. also to effective cosmological constant λeff [5], see subsection 1.2). (VI.1.1.c) Pauli-Villars renormalization of zero-point fluctuations (see sub- section 1.2) is no longer considered as an intermediate mathematical construct but obviously has rigorous physical meaning supported by assumption (I.a-b). (VI.1.2) The physical dark matter formed by ghosts particles; (VI.1.3) The standard model fields do not to couple directly to the ghost sec- tor in the ultraviolet region of energy at a scale less then a fixed large energy scale Λ∗ , in particular: DOI: 10.4236/***.2019.***** 53 Journal of Modern Physics J. Foukzon et al. (VI.1.3.a) The “bad” ghosts fields with masses at a scale less then a fixed scale 2 meff , where mceff Λ∗ , cannot appear in any effective physical lagrangian which contains also the standard particles fields. In additional though not necessary we assume that: (VI.1.4) The “bad” ghosts fields with masses at a scale m∗ , where 2 mc∗∗ Λ can appear in any effective physical lagrangian which contains also the standard particles fields, in particular: (VI.1.4.a) Pauli-Villars finite renormalization with masses of ghosts fields at a scale m∗ of the S-matrix in QFT (see Subsection 2.1-2) is no longer considered as an intermediate mathematical construct but obviously has rigorous physical meaning supported by assumption (IV). (VI.1.4.b) If the “bad” ghosts fields coupled to matter directly, it gives rise to small and controllable violetion of the unitarity condition. Remark 6.1.3. We emphasize that in universe standard matter coupled with a physical ghost matter has the equation of state [3]: µ 1 eff c4λ ε µ=−= µ µµ4 µ µ=vac vac( eff ) pf( eff ) ∫ ( ) (ln) d , (285) 880 πG where −n On(µ),1>≤ µµeff f (µ ) = (286) 0 µµ> eff and where µeff= mc eff (see subsection I.2, Equation (46)) and therefore gives rise to a de Sitter phase of the universe even if bare cosmological constant λ = 0 . (VI.1.5) In order to obtain QFT description of the dark component of matter in natural way we expand the standard model of particle physics on a sector of ghost particles, see [12], Section 2.3.2. QFT in a ghost sector developed in [12], Section 3.1-3.4 and Section 4.1-4.8. 6.2. Different Contributions to λeff . The total effective cosmological constant λeff is on at least the order of magni- tude of the vacuum energy density generated by zero-point fluctuations of stan- dard particle fields. Assumption 2 is well justified in the case of the traditional approach, because the contribution from zero-point fluctuations is on the order of 1 in Planck units and no other known contributions are as large thus, assuming no significant cancellation of terms (e.g. fine tuning of the bare cosmological constant λ ), the total λeff should be at least on the order of the largest contribution [15]. (VI.2) In contrary with standard Assumption 1 in the case of the new ap- proach introduced in this paper we assume that: (VI.2.1) For simplicity though not necessary bare cosmological constant λ = 0 . (VI.2.2) The total effective cosmological constant λeff depends only on mass distribution f (µ ) and constant µeff= mc eff but cannot depend on large DOI: 10.4236/***.2019.***** 54 Journal of Modern Physics J. Foukzon et al. energy scale Λ∗ Remark VI.2.1. Note that in subsection we pointed out that under Assump- tion VI.1 if bare cosmological constant λ = 0 the total cosmological constant −+n 5 λvac is on at least the order µeff of magnitude of the renormalized va- cuum energy density generated by zero-point fluctuations of standard particle fields and ghost particle fields µ 1 eff ε µ= µµ42 µ µ+Λ− vac( eff ) ∫ fO( ) (ln) d( ∗ ) , 8 0 µ (287) 1 eff µ =− µµ42 µ µ+Λ− pfvac( eff ) ∫ ( ) (ln) d O( ∗ ) . 8 0 6.3. Effective Field Theory and Lorentz Invariance Violation To prevent the vacuum energy density from diverging,the traditional approach also assumes that performing a high-energy cutoff is acceptable. This type of re- gularization is a common step in renormalization procedures, which aim to eventually arrive at a physical, cutoff-independent result. However, in the case of the vacuum energy density, the result is inherently cutoff dependent, scaling quartically with the cutoff Λ∗ . Remark VI.3.1. By restricting to modes with particle energy a certain cutoff energy ωk ≤Λ∗ a finite, regularized result for the energy density can be ob- 4 tained. The result is proportional to Λ∗ . Any other fields will contribute simi- larly, so that if there are nb bosonic fields and n f fermionic fields, the density 4 scales with (nnbf− 4 ) Λ∗ . Typically, the cutoff is taken to be near = 1 in Planck units (i.e.the Planck energy), so the vacuum energy gives a contribution to the cosmological constant on the order of at least unity according to Equation (6.2.4). Thus we see the extreme ne-tuning problem: the original cosmological constant λ must cancel this large vacuum energy density ε vac 1 to a preci- sion of 1 in 10120 -but not completely- to result in the observed value −120 λeff = 10 [5]. Remark VI.3.2. As it pointed out in this paper that a high-energy theory, i.e. QFT in fractal space-time with Hausdorff-Colombeau negative dimension would not display the zero-point fluctuations that are characteristic of QFT, and hence that the divergence caused by oscillations above the corresponding cutoff fre- quency is unphysical. In this case, the cutoff Λ∗ is no longer an intermediate mathematical construct, but instead a physical scale at which the smooth, conti- nuous behavior of QFT breaks down. Poincaré group of the momentum space is deformed at some fundamental 2 ab high-energy cutoff Λ∗ The canonical quadratic invariant p=η ppab col- lapses at high-energy cutoff Λ∗ and being replaced by the non-quadratic inva- riant: ab 2 η pp p = ab. (288) + (1 lpΛ∗ 0 ) Remark VI.3.3. In contrary with canonical approach the total effective cos- DOI: 10.4236/***.2019.***** 55 Journal of Modern Physics J. Foukzon et al. mological constant λeff depends only on mass distribution f (µ ) and con- stant µeff= mc eff but cannot depend on large energy scale Λ∗ . 6.4. Semiclassical Moller-Rosenfeld Gravity Assumption 5 means that it is valid to replace the right-hand side of the Einstein equation Tµν with its expectation Tµν . It requires that either gravity is not in fact quantum, and the Moller-Rosenfeld approach is a complete description of reality, or at least a valid approximation in the weak field limit. The usual argu- ment states that the vacuum state 0 should be locally Lorentz invariant so that observers agree on the vacuum state. This means that the expectation value ˆ of the energy-momentum tensor on the vacuum, 00Tµν , must be a scalar multiple of the metric tensor gµν which is the only Lorentz invariant rank (0,2) tensor. By using Moller-Rosenfeld approach the Einstein field equations of general relativity, a term representing the curvature of spacetime Rµν is re- ˆ lated to a term describing the energy-momentum of matter 00Tµν , as well as the cosmological constant λ and metric tensor gµν reads: 1 υ ˆ Rµν− Rg υ µν +=λ g µν 8π 0 Tµν 0. (299) 2 ˆ ˆ The T00 component is an energy density, we label 00Tµν = ε vac , so that the vacuum contribution to the right-hand side of Equation (6.4.1) can be writ- ten as ˆ 8π 0Tgµν 08= πε vac µν . (290) Subtracting this from the right-hand side of Equation (6.4.1) and grouping it with the cosmological constant term replaces with an “effective” cosmological constant [5]: λλεeff = + 8π vac . (291) Note that in flat spacetime, where gµν = diag ( −+++1, 1, 1, 1) , Eq.(6.4.2) im- ˆ plies ε vac= − p vac , where pTvac = 00ii for any i = 1, 2, 3 is the pressure. Obviously this implies that if the energy density is positive as is usually assumed, then the pressure must be negative, a conclusion which extends to any metric gµν with a (−+++1, 1, 1, 1) signature. Remark VI.4.1. In this paper we assume that the vacuum state 0 should be locally invariant under modified Lorentz boost (1.1.18) but not locally Lorentz invariant. Obviously this assumption violate the Equation (6.4.2). However modified Lorentz boosts (1.1.18) becomes Lorentz boosts for sufficiently small energies and therefore in IR region one obtain in a good approximation ˆ 8π 0Tgµν 08≈ πε vac µν (292) and λλεeff ≈+8π vac . (293) Thus Moller-Rosenfeld approach holds in a good approximation. DOI: 10.4236/***.2019.***** 56 Journal of Modern Physics J. Foukzon et al. 6.5. Quantum Gravity At Energy Scale Λ≤Λ∗ . Controllable Viola- tion of the Unitarity Condition Gravitational actions which include terms quadratic in the curvature tensor are renormalizable. The necessary Slavnov identities are derived from Becchi-Rouet-Stora (BRS) transformations of the gravitational and Faddeev-Popov ghost fields. In general, non-gauge-invariant divergences do arise, but they may be absorbed by nonlinear renormalizations of the gravitational and ghost fields and of the BRS transformations [14]. The geneic expression of the action reads =−−4 αµν −+ βκ22− Isym ∫ d x g( RRµν R2, R) (294) λλ where the curvature tensor and the Ricci is defined by Rµαν=∂Γ ν µα and λ 2 RRµν= µλν correspondingly, κ = 32πG . The convenient definition of the gra- vitational field variable in terms of the contravariant metric density reads κηhgµν= µν −− gµν . (295) Analysis of the linearized radiation shows that there are eight dynamical de- grees of freedom in the field. Two of these excitations correspond to the familiar massless spin-2 graviton. Five more correspond to a massive spin-2 particle with mass m2 . The eighth corresponds to a massive scalar particle with mass m0 . Although the linearized field energy of the massless spin-2 and massive scalar excitations is positive definite, the linearized energy of the massive spin-2 excita- tions is negative definite. This feature is characteristic of higher-derivative mod- els, and poses the major obstacle to their physical interpretation. In the quantum theory, there is an alternative problem which may be substi- tuted for the negative energy. It is possible to recast the theory so that the mas- sive spin-2 eigenstates of the free-fieid Hamiltonian have positive-definite ener- gy, but also negative norm in the state vector space. These negative-norm states cannot be excluded from the physical sector of the vector space without de- stroying the unitarity of the S matrix. The requirement that the graviton propagator behaves like p−4 for large momenta makes it necessary to choose the indefinite-metric vector space over the negative-energy states. The presence of massive quantum states of negative norm which cancel some of the diver- gences due to the massless states is analogous to the Pauli-Villars regularization of other field theories. For quantum gravity, however, the resulting improve- ment in the ultraviolet behavior of the theory is sufficient only to make it re- normalizable, but not finite. Remark 6.5.1. (I) The renormalizable models which we have considered in this paper many years mistakenly regarded only as constructs for a study of the ultraviolet problem of quantum gravity. The difficulties with unitarity appear to preclude their direct acceptability as canonical physical theories in locally Min- kowski space-time. In canonical case they do have only some promise as phe- nomenological models. (II) However, for their unphysical behavior may be restricted to arbitrarily large energy scales Λ∗ mentioned above by an appropriate limitation on the DOI: 10.4236/***.2019.***** 57 Journal of Modern Physics J. Foukzon et al. renormalized masses m2 and m0 . Actually, it is only the massive spin-two ex- citations of the field which give the trouble with unitarity and thus require a very large mass. The limit on the mass m0 is determined only by the observational constraints on the static field. 6.6. Pauli-Villars Masses Distribution Corresponding to Ghost Matter Sector above High Energy Cutoff Λ∗ A Common Ori- gin of the Dark Energy and Dard Matter Phenomena Dark matter is a hypothetical form of matter that is thought to account for ap- proximately 85% of the matter in the universe, and about a quarter of its total energy density. The majority of dark matter is thought to be non-baryonic in nature, possibly being composed of some as-yet undiscovered subatomic par- ticles. Its presence is implied in a variety of astrophysical observations, including gravitational effects that cannot be explained unless more matter is present than can be seen. For this reason, most experts think dark matter to be ubiquitous in the universe and to have had a strong influence on its structure and evolution. Dark matter is called dark because it does not appear to interact with observable electromagnetic radiation, such as light, and is thus invisible to the entire elec- tromagnetic spectrum, making it extremely difficult to detect using usual astro- nomical equipment [36] [37] [38]. Figure 8 Analysis of a giant new galaxy survey, made with ESO’s VLT Survey Telescope in Chile, suggests that dark matter may be less dense and more smoothly distributed throughout space than previously thought. An internation- al team used data from the Kilo Degree Survey (KiDS) to study how the light from about 15 million distant galaxies was affected by the gravitational influence of matter on the largest scales in the Universe. The results appear to be in disa- greement with earlier results from the Planck satellite. This map of dark matter in the Universe was obtained from data from the KiDS survey, using the VLT Survey Telescope at ESO’s Paranal Observatory in Chile. It reveals an expansive web of dense (light) and empty (dark) regions. This image is one out Figure 8. Dark matter map for a patch of sky based on gravitational lensing analysis [38] Hilderbrandt 16. DOI: 10.4236/***.2019.***** 58 Journal of Modern Physics J. Foukzon et al. of five patches of the sky observed by KiDS. Here the invisible dark matter is seen rendered in pink, covering an area of sky around 420 times the size of the full moon. This image reconstruction was made by analyzing the light collected from over three million distant galaxies more than 6 billion light-years away. The observed galaxy images were warped by the gravitational pull of dark matter as the light traveled through the Universe. Some small dark regions, with sharp boundaries, appear in this image. They are the locations of bright stars and other nearby objects that get in the way of the observations of more distant galaxies and are hence masked out in these maps as no weak-lensing signal can be meas- ured in these areas [38]. The luminous (light-emitting) components of the un- iverse only comprise about 0.4% of the total energy. The remaining components are dark. Of those, roughly 3.6% are identified: cold gas and dust, neutrinos, and black holes. About 23% is dark matter, and the overwhelming majority is some type of gravitationally self-repulsive dark energy. Remark 6.6.1. In order to explain physical nature of the dark matter sector we assume that the main part of dark matter, i.e., 23%−= 4.6% 18% (see Fig- 2 ure 9) formed by supermassive ghost particles with masses such that mc >Λ∗ , see ref. [12], Subsection 2.3. Remind that vacuum energy density for free scalar quantum field with a wrong statistic is: 11 ∞∞ εµ=−2 +=µ 22 ′′2 +=µ 22 µ ( ) 3 ∫∫4πcp ppd K p pp d, KI( ) (296) 2 (2π) 00 where µ = mc . From the basic definitions [1] 11 ∞ cp 1 =µ =−==2 p Txx,4 p( ) 3 ∫ upxxπ pd, pu ,,upxx up (297) 23(2π) 0 p22+ µ one obtains 4 K′ ∞ ppd p(µµ) = ∫ = KF′ ( ). (298) 3 0 p22+ µ Remark 6.6.2. Note that the integral in RHS of Equation (297) and in Equa- tion (298) is divergent and ultraviolet cutoff is needed. Thus in accordance with [1] we set εµ( ,p0) = KI′′( µ ,, p 00) p( µ , p) = KF( µ ,, p 0) (299) where 4 pp00ppd µ=+=2 µµ 22 I( , p00) ∫∫ p pd, pF( , p) , (300) 00p22+ µ where pc0 ≤Λ∗ . For fermionic quantum field with a wrong statistic, similarly one obtains εµ( ,p00) =−= 4 KI′′(µ ,, p) p( µ) − 4 KF( µ ,. p 0) (301) Thus from Equations. (300)-(301) by using formally Pauli-Villars regulariza- tion [7] [8] and regularization by high-energy cutoff the expression for DOI: 10.4236/***.2019.***** 59 Journal of Modern Physics J. Foukzon et al. Figure 9. Matter and energy distribution in the universe today. The luminous (light-emitting) components of the universe only comprise about 0.4% of the total energy. The remaining components are dark. free vacuum energy density ε reads 2M εµvac = ∑ fIii( , p0 ) (302) i=0 and the expression for pressure p reads 2M pvac = ∑ fFii(µ ,. p0 ) (303) i=0 Definition 6.6.1. We define now discrete distribution fPV : + → by formula ffPV(µ i) = i , (304) and we will call it as a full discrete Pauli-Villars masses distribution. Remark 6.6.3. We assume now that in Equations (302)-(303): 1) the quanti- sm. ties µµii=,iM = 1, 2, , are masses of physical particles corresponding to gm. standard matter and 2) the quantities µµii=,iM = + 1,2, ,2 M are masses of ghost particles with a wrong kinetic term and wrong statistics corresponding to physical dark matter. Remark 6.6.4. We recall that the Euler-Maclaurin summation formula reads 2M µ +− ∑i=1 g( 1 ( ih1) ) µ2M =f(µµ)d +− Ag12( µM ) g( µ 1) (305) ∫µ1 ′′ 2 1 +Ah22 g(µM ) −+ g( µ 1) O( h),. f( µµ) = g( ) h Let g (µ ) be an appropriate continuous function such that: 1) g(µii) = fi, = 1,2, ,2 M, 2) gg′′(µµ21M ) =0,( ) = 0 . Thus from Equations (302)-(303) and Equations (305) we obtain 2M µ ε= fI µ, p= 2M f µµ I,d p µ vac ∑ ii( 0 ) ∫µ ( ) ( 0 ) i=0 1 (306) 2 +Ah1 f(µ 2MM) I( µ 2,, p 0) −+ f( µµ 1) I( 10 p) O( h ) DOI: 10.4236/***.2019.***** 60 Journal of Modern Physics J. Foukzon et al. and 2M µ p= fFµ, p= 2M f µµ F,d p µ vac ∑ ii( 0 ) ∫µ ( ) ( 0 ) i=0 1 (307) 2 +Ah1220110 f(µMM) F( µ, p) −+ f( µµ) F( ,. p) O( h ) Definition 6.6.2. We will call the function fPV (µ ) as a full continuous Pau- li-Villars masses distribution. bgm.. Definition 6.6.3. We define now: 1) discrete distribution fPV : + → by formula bgm.. sm . fPV(µ i) = fi i , = 1, 2, , M (308) and we will call it as discrete Pauli-Villars masses distribution of the bosonic ghost matter and f.. gm 2) discrete distribution fPV : + → by formula f.. gm fPV(µ i) = fi i , = M + 1,2, ,2 M (309) and we will call it as discrete Pauli-Villars masses distribution of the fermionic ghost matter. Remark 6.6.4. We rewrite now Equations (306)-(307) in the following equiv- alent form MM2 ε= bgm.. µµ sm . bgm .. + f.. gmµ f .. gm µ f .. gm vac ∑∑fPV( i ) Ip( i ,,0 ) fPV ( ji( ) ) I( ji( ) p0 ) (310) i=1 ji( )=1 M+ and MM2 = bgm..µµ bgm .. bgm .. + f.. gmµ f .. gm µ f.. gm pfvac ∑∑PV( i ) Fp( i ,0 ) fPV ( ji( ) ) F( ji( ) ,, p0 ) (311) i=11ji( )= M + where ji( ) =+=+ i Mi, 1 1, 2, , M. bgm.. f .. gm Remark 6.6.6. We assume now that:1) µµi ≈ ji( ) , bgm..µµ bgm ..+ f .. gm f .. gm 2) ffPV( i ) PV ( ji( ) ) 1, i.e., bgm..µµ bgm .. ≈− f.. gm f .. gm ffPV( i ) PV ( ji( ) ). (312) Note that Equation (312) meant highly symmetric discrete Pauli-Villars masses distribution between bosonic ghost matter and fermionic ghost matter above that scale Λ∗. Thus from Equations (310)-(311) and Equations (312) we obtain MM2 ε= bgm.. µµ bgm .. bgm .. + f.. gmµ f .. gm µ f .. gm vac ∑∑fPV( i ) Ip( i ,,0 ) fPV ( ji( ) ) I( ji( ) p0 ) i=11ji( )= M + (313) M = fbgm..µ bgm ..+ f f .. gm µµ f .. gm Ip, ∑ PV( i ) PV ( ji( ) ) ( i 0 ) i=1 and MM2 = bgm..µµ bgm .. bgm .. + f.. gmµ f .. gm µ f.. gm pfvac ∑∑PV( i ) Fp( i ,,0 ) fPV ( ji( ) ) F( ji( ) p0 ) i=11ji( )= M + (314) M = fbgm..µ bgm ..+ f f .. gm µµ f .. gm Fp,. ∑ PV( i ) PV ( ji( ) ) ( i 0 ) i=1 DOI: 10.4236/***.2019.***** 61 Journal of Modern Physics J. Foukzon et al. From Equations (313)-(314) and Equations (305) finally we obtain M ε= fbgm.. µ bgm ..+ f f .. gm µµ f .. gm Ip, vac ∑ PV( i ) PV ( ji( ) ) ( i 0 ) i=1 (315) µ eff bgm.. f.. gm = fPV (µ) + fPV ( µµ) Ip( ,d0 ) µ ∫µ1 and M p= fbgm..µ sm .+ f f .. gm µµ f .. gm Fp, vac ∑ PV( i) PV ( ji( ) ) ( i 0 ) i=1 (316) µ eff bgm.. f.. gm = fPV (µ) + fPV ( µµ) Fp( ,0 ) d, µ ∫µ1 where obviously bgm.. f.. gm gm.. ffPV (µ) +=≈PV ( µµ) fPV ( ) 0. (317) Thus finally we obtain (2) µ gm.. (12) ( ) eff gm.. ε µµ,,p= (1) fµ Ip µ, d, µ ( eff eff 0 ) ∫µ PV ( ) ( 0 ) (318) eff and (2) µ gm.. (12) ( ) eff gm.. pµµ,, p= (1) fµ Fp µ, d, µ ( eff eff 0 ) ∫µ PV ( ) ( 0 ) (319) eff µµ(12) ( ) εgm.. µµ(12) ( ) where eff, eff p 0 . In order to calculate ( eff,, effp 0 ) and gm.. µµ(12) ( ) pp( eff,, eff 0 ) let us evaluate now the following quantities defined above by Equations (300) pp00p2 µ=22 += µµ 2 2 + I( , p0 ) ∫∫ ppd p p 1d2 p (320) 00µ and 1pp00pp4 d1 p41µ − d p Fp(µ,,0 ) = = (321) ∫∫22 2 3300pp+ µ 1+ µ 2 where p0 µ 1. Note that p211 pp 24 1 p 6 11+=+− + + µ22 µµ 24 8 16 µ 6 p2 11 pp24 1 p 6 22+=µµ 2 2 + =2 µ + − + + pp p112 p24 6 (322) µ2 µµ 8 16 µ 11pp46 1 p 8 =+−+p2 µ + 2µ 8µµ35 16 By inserting Equation (322) into Equation (320) one obtains p0 11pp46 1 p 8 µµ=2 +−+ + Ip( ,d0 ) ∫ p 35 p 0 2µ 8µµ 16 (323) 11pp57 1 1 p 9 =+−p3µ 00 + 0 + 30 10µ 7×× 8µµ35 9 16 Note that DOI: 10.4236/***.2019.***** 62 Journal of Modern Physics J. Foukzon et al. −12 p2 13 pp24 + =−++ 112 24 µ28 µµ −12 (324) p2 ppp 46813 41µ − + =−++ p 1 2 35 µµ 28 µµ By inserting Equation (324) into Equation (321) one obtains 1pp00pp41µ − d1 p4 1 p 6 3 p 8 Fp(µ,d0 ) = = −++ p ∫∫2 35 300p 3µ 28µµ 1+ µ 2 (325) p511 pp 79 =−0 00 ++ 35×µ 237 ××µµ35 89 × By inserting Equation (323) into Equation (318) one obtains εgm.. µµ(12) ( ) ( eff,, effp 0 ) (2) µ eff gm.. = (1) fµµ Ip,d µ ∫µ PV ( ) ( 0 ) eff (2) 57 9 µ eff gm.. 113 pp00 1 1 p 0 (326) =(1) fpµµ+− + +dµ ∫µ PV ( )0 35 eff 3 10µ 7×× 8µµ 9 16 3(22) 57( ) gm.. ( 2) gm.. µµµµ µµµ p0eff gm.. pp 00eff ffPV ( )ddeff PV ( ) =(11) f µµµd +−( ) ( 1) + ∫∫∫µµPV ( ) µ3 3 eff 10 eff µ 7× 8 eff µ By inserting Equation (325) into Equation (319) one obtains gm.. µµ(12) ( ) pp( eff,, eff 0 ) (2) µ eff gm.. = (1) fµµ Fp,d µ ∫µ PV ( ) ( 0 ) eff (2) 5 79 µ eff gm.. p011 pp 00 =(1) f µµ− ++d ∫µ PV ( )35 eff 35×µ 237 ××µµ 89 × 579(2) gm.. ( 22) gm.. ( ) gm.. µµµ µµµµ µµ ppp000eff fPV ( )deff ffPV ( ) ddeff PV ( ) =−(1) ( 11) ++( ) ∫µ ∫∫ µµ35 35××eff µ 237××eff µµ89 eff (327) Remark 6.6.7. We assume now that −n (1) ( 12) ( ) Onµ,7> µ ≤≤ µµ gm.. ( eff ) eff eff fPV (µ ) = (328) (2) 0 µµ> eff εgm.. µµ(12) ( ) Note that under assumption (328) the quantities ( eff,, effp 0 ) and gm.. µµ(12) ( ) pp( eff,, eff 0 ) cannot contribute in the value of the cosmological constant. 7. Conclusion We argue that a solution to the cosmological constant problem is to assume that there exists hidden physical mechanism which cancels divergences in canonical QED44, QCD , Higher-Derivative-Quantum-Gravity, etc. In fact, we argue that corresponding supermassive Pauli-Villars ghost fields, etc. really exist. New DOI: 10.4236/***.2019.***** 63 Journal of Modern Physics J. Foukzon et al. theory of elementary particles which contain hidden ghost sector is proposed. 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