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.

DOI: 10.4236/jmp.2019.107053                

Abstract

The cosmological constant problem arises because the magnitude of vacuum energy density predicted by the Quantum Field Theory is about 120 orders of magnitude larger then the value implied by cosmological observations of accelerating cosmic expansion. We pointed out that the fractal nature of the quantum space-time with negative Hausdorff-Colombeau dimensions can resolve this tension. The canonical Quantum Field Theory is widely believed to break down at some fundamental high-energy cutoff and therefore the quantum fluctuations in the vacuum can be treated classically seriously only up to this high-energy cutoff. In this paper we argue that the Quantum Field Theory in fractal space-time with negative Hausdorff-Colombeau dimensions gives high-energy cutoff on natural way. We argue that there exists hidden physical mechanism which cancels divergences in canonical QED₄, QCD₄, 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 apply the canonical Pauli-Villars regularization up to Λ_*. This would fit in the observed value of the dark energy needed to explain the accelerated expansion of the universe if we choose highly symmetric masses distribution between standard matter and ghost matter below the scale Λ_*, i.e.,

The small value of the cosmological 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 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 everywhere in space, even in regions which are otherwise “empty” (i.e. devoid of matter and radiation). This vacuum energy density is believed to act as a contribution to the cosmological constant $\lambda$ appearing in Einstein’s field equations from 1917,

$R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R=\frac{8\text{π}G}{c^4}{T^{\prime }}_{\mu \nu }$ (1)

where $R_{\mu \nu }$ and R refer to the curvature of space-time, $g_{\mu \nu }$ is the metric, ${T^{\prime }}_{\mu \nu }$ is the the energy-momentum tensor,

${T^{\prime }}_{\mu \nu }=T_{\mu \nu }+\frac{c^4\lambda }{8\text{π}G}\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right)$ (2)

where $T_{\mu \nu }$ is the energy-momentum tensor of matter. Thus ${T^{\prime }}_{00}=T_{00}+{\epsilon }_{\lambda }{T^{\prime }}_{\alpha \beta }=T_{\alpha \beta }+{\delta }_{\alpha \beta }{P}_{\lambda }$, where

${\epsilon }_{\lambda }=-P_{\lambda }=c^4\lambda /8\text{π}G\mathrm{.}$ (3)

Reminding that under Lorentz transformations $\left({\epsilon }_{\lambda }\mathrm{,}{P}_{\lambda }\right)\to {{\epsilon }^{\prime }}_{\lambda }\mathrm{,}\left({\epsilon }_{\lambda }\mathrm{,}{P}_{\lambda }\right)\to {P^{\prime }}_{\lambda }$ the quantities ${\epsilon }_{\lambda }$ and $P_{\lambda }$ are changes by the law

${{\epsilon }^{\prime }}_{\lambda }=\frac{{\epsilon }_{\lambda }+{\beta }^2{P}_{\lambda }}{1-{\beta }^2},{P^{\prime }}_{\lambda }=\frac{P_{\lambda }+{\beta }^2{\epsilon }_{\lambda }}{1-{\beta }^2}.$ (4)

Thus for the quantities ${\epsilon }_{\lambda }$ and $P_{\lambda }$ 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 invariant. In the earlier, non-singular Friedmann cosmology, the Friedmann universe 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 ~100 Mpc. The cosmological principle implies that the metric of the universe must be of the form Robertson-Walker metric [2] . Robertson-Walker metric reads

$\text{d}{s}^2=\text{d}{t}^2-a^2\left(t\right)\left[\frac{\text{d}{r}^2}{1-k{r}^2}+r^2\left(\text{d}{\theta }^2+{\sin }^2\theta \text{d}{\phi }^2\right)\right].$ (5)

For such a metric, the Ricci curvature scalar is $R=-6k$ and it is said that space has the curvature k. The scaling factor $a\left(t\right)$ rescales this curvature for a given time t, producing a curvature $k\left(t\right)=k/a\left(t\right)$. The scaling factor $a\left(t\right)$ is given by two independent Friedmann equations for modeling a homogeneous, isotropic universe reads

${\stackrel{\dot{}}{a}}^2=\frac{G}{3}\epsilon a^2-k,\ddot{a}=-\frac{G}{6}\left(\epsilon +3p\right)$ (6)

and the equation of state

$p=p\left(\epsilon \right),$ (7)

where p is pressure and $\epsilon$ is a density of the cosmological medium. For the case of the vacuumlike cosmological medium equation of state reads [2] [3] [4] :

$p=-\epsilon .$ (8)

By virtue of Friedman’s Equation (6) in the Universe filled with a vacuum-like medium, the density of the medium is preserved, i.e. $\epsilon =const$, but the scale factor $a\left(t\right)$ grows exponentially. By virtue of continuity, it can be assumed 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 universe, but almost without changing the density of the medium! was named inflation. 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 density d violates the condition of vacuum-like degeneration, $p=-\epsilon$ or, which is the same, $3p+\epsilon =-2\epsilon <0$, replacing it with

$-2\epsilon <3p+\epsilon <0.$ (9)

According to Friedman’s equations, it corresponds to an accelerated expansion 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 constant problem. As well known zeropoint energy density of scalar quantum field, etc. is divergent

${\epsilon }_{\text{vac}}\left(m\right)=\frac{2\text{π}c}{{\left(2\text{π}\hslash \right)}^3}{\displaystyle {\int }_0^{\infty }\sqrt{p^2+m^2{c}^2}{p}^2\text{d}p}=\infty \mathrm{.}$ (10)

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)

${\epsilon }_{\text{vac}}=-p_{\text{vac}}=\frac{1}{8}{\displaystyle {\int }_0^{\infty }f\left(\mu \right){\mu }^4\left(\ln \mu \right)\text{d}\mu }=\frac{c^4\lambda }{8\text{π}G},$ (11)

where

${\displaystyle {\int }_0^{\infty }f\left(\mu \right)\text{d}\mu }={\displaystyle {\int }_0^{\infty }f\left(\mu \right){\mu }^2\text{d}\mu }={\displaystyle {\int }_0^{\infty }f\left(\mu \right){\mu }^4\text{d}\mu }=0.$ (12)

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 $\epsilon$.

In his paper [1] , Zel’dovich arrives at a zero-point energy (his formula (IX.1))

${\epsilon }_{\text{vac}}=m{\left(\frac{mc}{\hslash }\right)}^3~{10}^{17}\text{g}/{\text{cm}}^{\text{3}},\lambda ~{10}^{-10}{\text{cm}}^{-2},$ (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 magnitude 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 consideration of vacuum fluctuations. In this theory, the essential assumption is that there is some elementary length L or the corresponding limiting momentum $p_0=\hslash /L$. Shorter lengths or for large impulses theory is not applicable. Sakharov gets the expression of gravitational constant G through L or $p_0$ (his formula (IX.6))

$G=\frac{c^3{L}^2}{\hslash }=\frac{\hslash c^3}{p_0^2}.$ (14)

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 $p_0$. According to Sakharov, L and $p_0$ are primary. Substitute (IX. 6) in the expression (IX. 4), we get

${\rho }_{\text{vac}}=\frac{m^6{c}^5}{p_0^2{\hslash }^3},{\epsilon }_{\text{vac}}=\frac{m^6{c}^7}{p_0^2{\hslash }^3}.$ (15)

That is expressions that the first members (in the formulas (VIII.10), (VIII. 11)) which are vanishes (with $p_0\to \infty$ ). Thus, we can suggest the following interpretation of the cosmological constant: there is a theory of elementary particles, 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 $p_0$, beyond which the theory is non applicable; along with other implications, modifying the theory gives different from zero vacuum energy; general considerations make it likely that the effect is portional $p_0^{-2}$. Clarification of the question of the existence 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 ${\Lambda }_{\ast }$ [9] [10] [11] in accordance with the basis of the following deformed Poisson brackets

$\begin{array}{l}\left\{x^{\mu }\mathrm{,}{x}^{\nu }\right\}={\varkappa }^{-1}\left(x^{\mu }{\eta }^{0\nu }-x^{\nu }{\eta }^{\mu 0}\right)\mathrm{,}\left\{p^{\mu }\mathrm{,}{p}^{\nu }\right\}=\mathrm{0,}\\ \left\{x^{\mu }\mathrm{,}{p}^{\nu }\right\}=-{\eta }^{\mu \nu }+{\varkappa }^{-1}{\eta }^{\mu 0}{p}^{\nu }\end{array}$ (16)

where $\mu ,\nu =0,1,2,3{\eta }^{\mu \nu }=\left(+1,-1,-1,-1\right)$ and is a parameter identified as the ratio between the high-energy cutoff ${\Lambda }_{\ast }$ and the light speed. The corresponding to (16) momentum transformation reads [11]

$\begin{array}{l}{p^{\prime }}_0=\frac{\gamma \left(p_0-u{p}_x\right)}{1+{\left(c\varkappa \right)}^{-1}\left[\left(\gamma -1\right)p_0-\gamma u{p}_x\right]}\mathrm{,}{p^{\prime }}_x=\frac{\gamma \left(p_x-u{p}_0/c^2\right)}{1+{\left(c\varkappa \right)}^{-1}\left[\left(\gamma -1\right)p_0-\gamma u{p}_x\right]}\mathrm{,}\\ {p^{\prime }}_y=\frac{p_y}{1+{\left(c\varkappa \right)}^{-1}\left[\left(\gamma -1\right)p_0-\gamma u{p}_x\right]}\mathrm{,}{p^{\prime }}_z=\frac{p_z}{1+{\left(c\varkappa \right)}^{-1}\left[\left(\gamma -1\right)p_0-\gamma u{p}_x\right]}\mathrm{,}\end{array}$ (17)

and coordinate transformation reads [11]

$\begin{array}{l}{t}^{\prime }=\frac{\gamma \left(t-ux/c^2\right)}{1+{\left(c\varkappa \right)}^{-1}\left[\left(\gamma -1\right)p_0-\gamma u{p}_x\right]}\mathrm{,}{x}^{\prime }=\frac{\gamma \left(x-ut\right)}{1+{\left(c\varkappa \right)}^{-1}\left[\left(\gamma -1\right)p_0-\gamma u{p}_x\right]}\mathrm{,}\\ y^{\prime }=\frac{y}{1+{\left(c\varkappa \right)}^{-1}\left[\left(\gamma -1\right)p_0-\gamma u{p}_x\right]}\mathrm{,}{z}^{\prime }=\frac{z}{1+{\left(c\varkappa \right)}^{-1}\left[\left(\gamma -1\right)p_0-\gamma u{p}_x\right]}\mathrm{,}\end{array}$ (18)

where $\gamma =\sqrt{1-u^2/c^2}$. It is easy to check that the energy $E=c\varkappa$, identified as the high-energy cutoff ${\Lambda }_{\ast }$, is an invariant as it is also the case for the fundamental length $l_{{\Lambda }_{\ast }}=\hslash c/E=\hslash /\varkappa$.

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 and momenta and defines a large invariant energy $l_{{\Lambda }_{\ast }}^{-1}$. The high-energy cutoff ${\Lambda }_{\ast }$ is preserved by the modified action of the Lorentz group [9] [10] .

This meant that the canonical concept of metric as quadratic invariant collapses at high energies, being replaced by the non-quadratic invariant [9] :

${\Vert p\Vert }^2=\frac{{\eta }^{ab}{p}_a{p}_b}{\left(1+l_{{\Lambda }_{\ast }}{p}_0\right)},$ (19)

or by the non-quadratic invariant

${\Vert p\Vert }^2=\frac{{\eta }^{ab}{p}_a{p}_b}{\left(1-l_{{\Lambda }_{\ast }}{p}_0\right)},$ (20)

where $l_{{\Lambda }_{\ast }}={\Lambda }_{\ast }^{-1},a,b=0,1,2,3$.

Remark 1.1.3. Note that:

1) the invariant (16) is infinite for the new negative invariant energy scale of the theory ${\Lambda }_{\ast }=-l_{{\Lambda }_{\ast }}^{-1}$, 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 the theory ${\Lambda }_{\ast }=l_{{\Lambda }_{\ast }}^{-1}$.

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 theories with the two signs of $l_{\Lambda }$ obviously are physically distinct; and we know of no theoretical argument which fixes the signs of $\Lambda$

The massive particles have a positive invariant ${\Vert p\Vert }^2>0$ which can be identified with the square of the mass ${\Vert p\Vert }^2=m^2$, ( $c=1$ ). Thus in the case of the invariant (16) we obtain

$\frac{p_0^2-p^2}{{\left(1+l_{{\Lambda }_{\ast }}{p}_0\right)}^2}=m^2,p_0\in \left(-l_{{\Lambda }_{\ast }}^{-1},\infty \right)$ (21)

From Equation (18) we obtain

$p_0=\frac{m^2{l}_{{\Lambda }_{\ast }}}{1-m^2{l}_{{\Lambda }_{\ast }}^2}+\frac{1}{\sqrt{1-m^2{l}_{{\Lambda }_{\ast }}^2}}\sqrt{\frac{m^4{l}_{{\Lambda }_{\ast }}^2}{1-m^2{l}_{{\Lambda }_{\ast }}^2}+\left(p^2+m^2\right)}.$ (22)

In the case of the invariant (17) we obtain

$\frac{p_0^2-p^2}{{\left(1-l_{{\Lambda }_{\ast }}{p}_0\right)}^2}=m^2,p_0\in \left(-\infty ,l_{{\Lambda }_{\ast }}^{-1}\right).$ (23)

From Equation (20) we obtain

$p_0=-\frac{m^2{l}_{{\Lambda }_{\ast }}}{1-m^2{l}_{{\Lambda }_{\ast }}^2}-\frac{1}{\sqrt{1-m^2{l}_{{\Lambda }_{\ast }}^2}}\sqrt{\frac{m^4{l}_{{\Lambda }_{\ast }}^2}{1-m^2{l}_{{\Lambda }_{\ast }}^2}+\left(p^2+m^2\right)}.$ (24)

The action for a scalar field $\phi$ must be invariant under the deformed Lorentz transformations. The invariant action reads [10]

$S=\frac{1}{2}{\displaystyle \int {\text{d}}^{4}x}\frac{{\eta }^{ab}\left({\partial }_a\phi \right)\left({\partial }_b\phi \right)}{\left[1+l_{{\Lambda }_{\ast }}{\partial }_0\phi \right]}+\frac{m^2}{2}{\phi }^2.$ (25)

Thus there is no linear field equation.

Remark 1.1.5.Throughout this paper, we use below high-energy cutoff ${\Lambda }_{\ast }$ the perturbative expansion

$S=\frac{1}{2}{\displaystyle \int {\text{d}}^{4}x}\left({\eta }^{ab}\left({\partial }_a\phi \right)\left({\partial }_b\phi \right)+\frac{m^2}{2}{\phi }^2\right)+O\left(l_{{\Lambda }_{\ast }}\right).$ (26)

and dealing in Lorentz invariant approximation

$S\simeq \frac{1}{2}{\displaystyle \int {\text{d}}^{4}x}\left({\eta }^{ab}\left({\partial }_a\phi \right)\left({\partial }_b\phi \right)+\frac{m^2}{2}{\phi }^2\right)\mathrm{.}$ (27)

since for $l_{{\Lambda }_{\ast }}\ll 1$ the expansion (26) holds.

(II) The canonical concept of Minkowski space-time collapses at a small distance $l_{{\Lambda }_{\ast }}={\Lambda }_{\ast }^{-1}$ to fractal space-time with Hausdorff-Colombeau negative dimension and therefore the canonical Lebesgue measure ${\text{d}}^{4}x$ being replaced by the Colombeau-Stieltjes measure with negative Hausdorff-Colombeau dimension $D^{-}$ :

${\left(\text{d}\eta \left(x\mathrm{,}\epsilon \right)\right)}_{\epsilon }={\left(v_{\epsilon }\left(s\left(x\right)\right){\text{d}}^{4}x\right)}_{\epsilon }\mathrm{,}$ (28)

where ${\left(v_{\epsilon }\left(s\left(x\right)\right)\right)}_{\epsilon }={\left({\left({\left|s\left(x\right)\right|}^{\left|D^{-}\right|}+\epsilon \right)}^{-1}\right)}_{\epsilon }$ and $s\left(x\right)=\sqrt{x_{\mu }{x}^{\mu }}$, see Section 3 and [12] .

(III) The canonical concept of momentum space collapses at fundamental high-energy cutoff ${\Lambda }_{\ast }$ to fractal momentum space with Hausdorff-Colombeau negative dimension and therefore the canonical Lebesgue measure ${\text{d}}^{3}k$, where $k=\left(k_x\mathrm{,}{k}_y\mathrm{,}{k}_z\right)$ being replaced by the Hausdorff-Colombeau measure

${\text{d}}^{D^{+}\mathrm{,}{D}^{-}}k\triangleq \frac{\Delta \left(D^{-}\right){\text{d}}^{D^{+}}k}{{\left({\left|k\right|}^{\left|D^{-}\right|}+\epsilon \right)}_{\epsilon }}=\frac{\Delta \left(D^{+}\right)\Delta \left(D^{-}\right)p^{D^{+}-1}\text{d}p}{{\left(p^{\left|D^{-}\right|}+\epsilon \right)}_{\epsilon }}\mathrm{,}$ (29)

where $\Delta \left(D^{\pm }\right)=2{\text{π}}^{D^{\pm }/2}/\Gamma \left(D^{\pm }/2\right)$ and $p=\left|k\right|=\sqrt{k_x+k_y+k_z}$ and where $D^{+}-\left|D^{-}\right|\le -6$, see Section 3 and ref. [9] . Hausdorff-Colombeau measure (29) avoids classical divergence (10) of the zeropoint energy ${\epsilon }_{\text{vac}}\left(m\right)$ and instead Equation (10) one obtains

${\epsilon }_{\text{vac}}\left(p_{\ast },m\right)={\displaystyle {\int }_0^{p_{\ast }}}{\text{d}}^{3}p\sqrt{p^2+m^2}+\Delta \left(D^{+}\right)\Delta \left(D^{-}\right){\displaystyle {\int }_{p_{\ast }}^{\infty }}\text{d}p{p}^2\frac{\sqrt{p^2+m^2}}{{\left(p^{\left|D^{-}\right|}+\epsilon \right)}_{\epsilon }}\simeq p_{\ast }^4\mathrm{.}$ (30)

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, the vacuum energy density ${\epsilon }_{\text{vac}}\left(p_{\ast }\mathrm{,}m\right)$ is 10¹²¹ times larger than the observed dark energy density ${\epsilon }_{\text{de}}$. 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 five different groups: 1) Modification of gravity on large scales. 2) Anthropic principle.

3) Symmetry leading to ${\epsilon }_{\text{vac}}=0$. 4) Adjustment mechanism, see. 5) Hidden nonstandard dark matter sector and corresponding hidden symmetry leading to ${\epsilon }_{\text{vac}}\simeq 0$, 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

$\epsilon \left(\mu \right)=\frac{1}{2}\frac{c}{{\left(2\text{π}\hslash \right)}^3}{\displaystyle {\int }_0^{\infty }4\text{π}\sqrt{p^2+{\mu }^2}{p}^2\text{d}p}=K{\displaystyle {\int }_0^{\infty }\sqrt{p^2+{\mu }^2}{p}^2\text{d}p}=KI\left(\mu \right)\mathrm{,}$ (31)

where $\mu =m_{0}c$. From Equation (31) one obtains [1]

$p\left(\mu \right)=\frac{K}{3}{\displaystyle {\int }_0^{\infty }\frac{p^4\text{d}p}{\sqrt{p^2+{\mu }^2}}}=KF\left(\mu \right).$ (32)

For fermionic quantum field one obtains

$\epsilon \left(\mu \right)=KI\left(\mu \right),p\left(\mu \right)=-4KF\left(\mu \right).$ (33)

Thus free vacuum energy density $\epsilon$ and corresponding pressure p is

$\epsilon ={\displaystyle \underset{i}{\sum }}{C}_{i}I\left({\mu }_i\right),P={\displaystyle \underset{i}{\sum }}{C}_{i}F\left({\mu }_i\right).$ (34)

From Equation (34) by using Pauli-Willars regularization [7] [8] in general case one obtains [1]

$\epsilon ={\displaystyle \int f\left(\mu \right)I\left(\mu \right)\text{d}\mu },P={\displaystyle \int f\left(\mu \right)F\left(\mu \right)\text{d}\mu }.$ (35)

In order to obtain asymptotical expansion on the parameter $p_0$ of the quantity ${\epsilon }_{\text{vac}}\left(p_0\mathrm{,}m\right)={\displaystyle {\int }_0^{p_0}}{\text{d}}^{3}p\sqrt{p^2+m^2}$ let us evaluate now the following integral

$\begin{array}{c}I\left(\mu ,p_0\right)={\displaystyle \underset{0}{\overset{p_0}{\int }}{p}^2\sqrt{p^2+{\mu }^2}\text{d}p}={\displaystyle \underset{0}{\overset{p_{\mu }}{\int }}{p}^2\sqrt{p^2+{\mu }^2}\text{d}p}+{\displaystyle \underset{p_{\mu }}{\overset{p_0}{\int }}{p}^2\sqrt{p^2+{\mu }^2}\text{d}p}\\ ={\displaystyle \underset{0}{\overset{p_{\mu }}{\int }}{p}^2\sqrt{p^2+{\mu }^2}\text{d}p}={\displaystyle \underset{p_{\mu }}{\overset{p_{\mu }}{\int }}{p}^3\sqrt{1+\frac{{\mu }^2}{p^2}}\text{d}p}+{\displaystyle \underset{p_{\mu }}{\overset{p_0}{\int }}{p}^3\sqrt{1+\frac{{\mu }^2}{p^2}}\text{d}p}\end{array}$ (36)

and

$\begin{array}{c}F\left(\mu ,p_0\right)=\frac{1}{3}{\displaystyle \underset{0}{\overset{p_0}{\int }}\frac{p^4\text{d}p}{\sqrt{p^2+{\mu }^2}}}=\frac{1}{3}{\displaystyle \underset{0}{\overset{p_{\mu }}{\int }}\frac{p^4\text{d}p}{\sqrt{p^2+{\mu }^2}}}+\frac{1}{3}{\displaystyle \underset{p_{\mu }}{\overset{p_0}{\int }}\frac{p^4\text{d}p}{\sqrt{p^2+{\mu }^2}}}\\ =\frac{1}{3}{\displaystyle \underset{0}{\overset{p_{\mu }}{\int }}\frac{p^4\text{d}p}{\sqrt{p^2+{\mu }^2}}}+\frac{1}{3}{\displaystyle \underset{p_{\mu }}{\overset{p_0}{\int }}\frac{p^3\text{d}p}{\sqrt{1+\frac{{\mu }^2}{p^2}}}},\end{array}$ (37)

where $p_{\mu }=r\mu ,r>1,\mu /p<1/r<1$. Note that

$\begin{array}{l}\sqrt{1+\frac{{\mu }^2}{p^2}}=1+\frac{1}{2}\frac{{\mu }^2}{p^2}-\frac{1}{8}\frac{{\mu }^4}{p^4}+\frac{1}{16}\frac{{\mu }^6}{p^6}+\cdots \\ p^2\sqrt{p^2+{\mu }^2}=p^3\sqrt{1+\frac{{\mu }^2}{p^2}}=p^3+\frac{1}{2}{\mu }^{2}p-\frac{1}{8}\frac{{\mu }^4}{p}+\frac{1}{16}\frac{{\mu }^6}{p^3}+\cdots \end{array}$ (38)

By inserting Equation (38) into Equations (36) one obtains

$I\left(\mu ,p_0\right)=C_1{\mu }^4+\frac{1}{4}{p}_0^4+\frac{1}{4}{\mu }^2{p}_0^2-\frac{1}{8}{\mu }^4\ln \left(\frac{p_0}{\mu }\right)-\frac{1}{32}\frac{{\mu }^6}{p_0^2}+p_0^{-5}O\left({\mu }^8\right),$ (39)

where $C_1{\mu }^4={\displaystyle {\int }_0^{p_{\mu }}{p}^2\sqrt{p^2+{\mu }^2}\text{d}p}$. Note that

${\left(\sqrt{1+\frac{{\mu }^2}{p^2}}\right)}^{-1}=1-\frac{1}{2}\frac{{\mu }^2}{p^2}+\frac{3}{8}\frac{{\mu }^4}{p^4}-\frac{5}{16}\frac{{\mu }^6}{p^6}+\cdots$ (40)

By inserting Equation (40) into Equation (37) one obtains

$F\left(\mu ,p_0\right)=C_2{\mu }^4+\frac{1}{12}{p}_0^4-\frac{1}{12}{\mu }^2{p}_0^2+\frac{1}{8}{\mu }^4\ln \left(\frac{p_0}{\mu }\right)+\frac{5}{32}\frac{{\mu }^6}{p_0^2}+p_0^{-5}O\left({\mu }^8\right).$ (41)

By inserting Equation (39) and Equation (41) into Equations (35) one obtains

$\begin{array}{l}\epsilon =\frac{1}{4}{p}_0^4{\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right)\text{d}\mu }+\frac{1}{4}{p}_0^2{\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right){\mu }^2\text{d}\mu }+\left(C_1-\frac{1}{8}\ln p_0\right){\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right){\mu }^4\text{d}\mu }\\ +\frac{1}{8}{\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right){\mu }^4\left(\ln \mu \right)\text{d}\mu }-\left(\frac{1}{p_0^2}\right)\frac{1}{32}{\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right){\mu }^6\text{d}\mu }+O\left({\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right){\mu }^8}\right)p_0^{-5}\mathrm{,}\\ p=\frac{1}{12}{p}_0^4{\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right)\text{d}\mu }-\frac{1}{12}{p}_0^2{\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}}f\left(\mu \right){\mu }^{2}d\mu +\left(C_2+\frac{1}{8}\ln p_0\right){\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right){\mu }^4\text{d}\mu }\\ -\frac{1}{8}{\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right){\mu }^4\left(\ln \mu \right)\text{d}\mu }+\left(\frac{5}{p_0^2}\right)\frac{1}{32}{\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right){\mu }^6\text{d}\mu }+O\left({\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right){\mu }^8}\right)p_0^{-5}\mathrm{.}\end{array}$ (42)

We choose now

${\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right)\text{d}\mu }={\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right){\mu }^2\text{d}\mu }={\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right){\mu }^4\text{d}\mu }=0.$ (43)

By inserting Equation (43) into Equations (42) one obtains

$\begin{array}{l}\epsilon \left({\mu }_{\text{eff}}\right)=\frac{1}{8}{\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right){\mu }^4\left(\ln \mu \right)\text{d}\mu }+O\left(p_0^{-2}\right)\mathrm{,}\\ p\left({\mu }_{\text{eff}}\right)=-\frac{1}{8}{\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right){\mu }^4\left(\ln \mu \right)\text{d}\mu }+O\left(p_0^{-2}\right)\mathrm{.}\end{array}$ (44)

Taking the limit $p\to \infty$ in Equation (44) gives

$\begin{array}{l}\epsilon \left({\mu }_{\text{eff}}\right)=\frac{1}{8}{\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right){\mu }^4\left(\ln \mu \right)\text{d}\mu }\mathrm{,}\\ p\left({\mu }_{\text{eff}}\right)=-\frac{1}{8}{\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right){\mu }^4\left(\ln \mu \right)\text{d}\mu }\mathrm{.}\end{array}$ (45)

Thus finally we obtain [3]

$\epsilon \left({\mu }_{\text{eff}}\right)=-p\left({\mu }_{\text{eff}}\right)=\frac{1}{8}{\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right){\mu }^4\left(\ln \mu \right)\text{d}\mu }=\frac{c^4\Lambda }{8\text{π}G}\mathrm{.}$ (46)

Remark 1.2.1. Remind that Pauli-Villars regularization consists of introducing a fictitious mass term. For example, we would replace a propagator $1/\left(k^2-m_0^2+iϵ\right)$, by the regulated propagator

$\Delta \left(k^2\right)={\displaystyle {\sum }_{i=0}^N\frac{a_i}{k^2-m_i^2+iϵ}}=\frac{1}{k^2-m_0^2+iϵ}-{\displaystyle {\sum }_{i=0}^N\frac{a_i}{k^2-m_i^2+iϵ}}\mathrm{,}$ (47)

where $a_0=1$ and $m_i,i=1,2,\cdots ,N$ can be thought of as the mass of a fictitious heavy particle, whose contribution is subtracted from that of an ordinary particle. Assume that $m_i^2/k^2<1$, if we expand each term of this sum (46) as a power series in $k^2+iϵ$ we get

$\Delta \left(k^2\right)={\displaystyle {\sum }_{i=0}^N\frac{a_i}{k^2+iϵ}}+{\displaystyle {\sum }_{i=0}^N\frac{a_i{m}_i^2}{{\left(k^2+iϵ\right)}^2}}+{\displaystyle {\sum }_{i=0}^{N}O\left(\frac{1}{{\left(k^2+iϵ\right)}^3}\right)}.$ (48)

For a renormalizable theory the maximum supercriticial power of divergence of any integral is quadratic, so that the $O\left(1/k^6\right)$ terms are ultraviolet finite. The finiteness of the regulated integral is then guaranteed by requiring that

${\displaystyle {\sum }_{i=0}^N{a}_i}=0,{\displaystyle {\sum }_{i=0}^N{a}_i{m}_i^2}=0.$ (49)

Remark 1.2.2. Note that in order to apply Pauli-Villars regularization to QFT with Lagrangian $L\left(\phi \mathrm{,}\psi \mathrm{,}{\partial }_{\mu }\phi \mathrm{,}{\partial }_{\mu }\psi \right)$ we would replace the Lagrangian $L\left(\phi \mathrm{,}\psi \mathrm{,}{\partial }_{\mu }\phi \mathrm{,}{\partial }_{\mu }\psi \right)$ by Lagrangian $\underset{\_}{L}\left(\underset{\_}{\phi }\mathrm{,}\underset{\_}{\psi }\mathrm{,}{\partial }_{\mu }\underset{\_}{\phi }\mathrm{,}{\partial }_{\mu }\underset{\_}{\psi }\right)$, where [7] :

$\underset{\_}{\phi }\left(x\right)=\phi \left(x\right)+{\displaystyle {\sum }_n{b}_n{\stackrel{⌣}{\phi }}_n\left(x\mathrm{,}{\mu }_n^2\right)}\mathrm{,}\underset{\_}{\psi }\left(x\right)=\psi \left(x\right)+{\displaystyle {\sum }_n{c}_n{\stackrel{⌣}{\psi }}_n\left(x\mathrm{,}{\varkappa }_n^2\right)}\mathrm{,}$ (50)

where commutator for ${\stackrel{⌣}{\phi }}_n$ and anticommutator for ${\stackrel{⌣}{\psi }}_n$ reads

$\begin{array}{l}\left[{\stackrel{⌣}{\phi }}_m\left(x\mathrm{,}{\mu }_m^2\right)\mathrm{,}{\stackrel{⌣}{\phi }}_n\left(x^{\prime }\mathrm{,}{\mu }_n^2\right)\right]=-i{\rho }_n\Delta \left(x-x^{\prime }\mathrm{,}{\mu }_n^2\right){\delta }_{mn}\mathrm{,}\\ \left\{{\stackrel{⌣}{\psi }}_m\left(x\mathrm{,}{\varkappa }_m^2\right)\mathrm{,}{\stackrel{⌣}{\psi }}_n\left(x^{\prime }\mathrm{,}{\varkappa }_n^2\right)\right\}=-i{\epsilon }_{n}S\left(x-x^{\prime }\mathrm{,}{\varkappa }_n^2\right){\delta }_{mn}\mathrm{.}\end{array}$

From Equations (50)-Equations (51) one obtains

$\begin{array}{l}\left[\underset{\_}{\phi }\left(x\right)\mathrm{,}\underset{\_}{\phi }\left(x^{\prime }\right)\right]=i{\displaystyle {\sum }_{n=0}^N{\rho }_n{b}_n^2\Delta \left(x-x^{\prime }\mathrm{,}{\mu }_n^2\right)}\mathrm{,}\\ \left[\underset{\_}{\psi }\left(x\right)\mathrm{,}\underset{\_}{\psi }\left(x^{\prime }\right)\right]=-i{\displaystyle {\sum }_{n=0}^N{\epsilon }_n{\bar{c}}_n{c}_{n}S\left(x-x^{\prime }\mathrm{,}{\varkappa }_n^2\right)}\mathrm{.}\end{array}$ (52)

Assume now that

${\displaystyle {\sum }_{n=0}^N{\rho }_n{b}_n^2}=\mathrm{0,}{\displaystyle {\sum }_{n=0}^N{\rho }_n{b}_n^2{\mu }_n^2}=\mathrm{0,}{\displaystyle {\sum }_{n=0}^N{\epsilon }_n{\bar{c}}_n{c}_n}=\mathrm{0,}{\displaystyle {\sum }_{n=0}^N{\epsilon }_n{\bar{c}}_n{c}_n{\varkappa }_n^2}=0.$ (53)

From Equations (53) it follows directly that QFT with Lagrangian $\underset{\_}{L}\left(\underset{\_}{\phi }\mathrm{,}\underset{\_}{\psi }\mathrm{,}{\partial }_{\mu }\underset{\_}{\phi }\mathrm{,}{\partial }_{\mu }\underset{\_}{\psi }\right)$ 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 $\phi$, whose existence allows the probabilities to be negative thus violating unitarity. The quadratic lagrangian $L_{\phi }^2$ for $\phi$ begins with a wrong sign kinetic term [in ( $+---$ ) signature]

$L_{\phi }^2=-\frac{1}{2}{\partial }^{\mu }\phi {\partial }_{\mu }\phi +\cdots$ (54)

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 function $f\left(\mu \right)$ in Equations (43). It means that there exists the ghost-driven acceleration of the universe hidden in cosmological constant $\Lambda$.

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 through gravity, and the propagation of gravity through the ghost condensate gives rise to a fascinating modification of gravity in the IR. However, no modifications of gravity can be seen directly, and no cosmological experiment can distinguish the ghost-driven acceleration from a cosmological constant.

Remark 1.2.6. In order to obtain desired physical result from Equations (45), i.e.,

${\epsilon }_{\text{vac}}=0.7\times {10}^{-29}\text{g}\cdot {\text{cm}}^{-3}=2.8\times {10}^{-47}{\text{GeV}}^4/{\hslash }^3{c}^5$ (55)

we assume that

$f\left(\mu \right)=f_{s.m.}\left(\mu \right)+f_{g.m.}\left(\mu \right),$ (56)

where $f_{s\mathrm{.}m\mathrm{.}}\left(\mu \right)$ corresponds to standard matter and where $f_{g\mathrm{.}m\mathrm{.}}\left(\mu \right)$ corresponds to a physical ghost matter.

Remark 1.2.7. We assume now that

$\left|f\left(\mu \right)\right|=\{\begin{array}{ll}O\left({\mu }^{-n}\right)\mathrm{,}n>1\hfill & \mu \le {\mu }_{\text{eff}}\hfill \\ 0\hfill & \mu >{\mu }_{\text{eff}}\hfill \end{array}$ (57)

From Equation (57) and Equation (45) it follows directly that

$\left|p\left({\mu }_{\text{eff}}\right)\right|=\left|\epsilon \left({\mu }_{\text{eff}}\right)\right|=\frac{1}{8}\left|{\displaystyle \underset{0}{\overset{{\mu }_{\text{eff}}}{\int }}f\left(\mu \right){\mu }^4\left(\ln \mu \right)\text{d}\mu }\right|\le O\left({\mu }_{\text{eff}}^{-n+5}\ln {\mu }_{\text{eff}}\right)\mathrm{.}$ (58)

Remark 1.2.8. However serious problem arises from non-renormalizability of canonical quantum gravity with Einstein-Hilbert action

$S_{EH}=\frac{1}{16\text{π}G}{\displaystyle \int {\text{d}}^{4}x}\sqrt{-g}R\mathrm{.}$ (59)

For example taking ${\Lambda }^3$ particles of energy a per unit volume gives the gravitational self-energy density of order ${\Lambda }^6$, i.e., the density ${\epsilon }_{\Lambda }$ diverges as ${\Lambda }^6$

${\epsilon }_{\Lambda }\simeq G{\Lambda }^6\mathrm{,}$ (60)

where $\Lambda$ 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

$\Im =-{\displaystyle \int {\text{d}}^{4}x}\sqrt{-g}\left(\alpha R_{\mu \nu }{R}^{\mu \nu }-\beta R^2+2{\kappa }^{-2}R\right)\mathrm{,}$ (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 negative-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 ${\Lambda }_{\ast }$ by an appropriate limitation on the renormalized masses $m_2$ and $m_0$.

Remark 1.2.10. We assume that $m_{0}c\gg {\mu }_{\text{eff}},m_{2}c\gg {\mu }_{\text{eff}}$.

Remark 1.2.11. The canonical Quantum Field Theory is widely believed to break down at some fundamental high-energy cutoff ${\Lambda }_{\ast }$ 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 $\lambda {\phi }^4$. Lagrangian density of this theory reads

$L=\frac{1}{2}{\partial }^{\mu }\phi {\partial }_{\mu }\phi -\frac{m_0^2}{2}{\phi }^2+\lambda {\phi }^4.$ (62)

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 playing the role of the regularization parameter. Lagrangian density can be rewritten as follows

$\begin{array}{l}L={\displaystyle {\sum }_{i=0}^N{\left(-1\right)}^i\left(\frac{1}{2}{\partial }^{\mu }\phi {\partial }_{\mu }\phi -\frac{m_i^2}{2}{\phi }_i^2\right)}+\lambda :{\phi }^4:,\\ \phi ={\phi }_0+\stackrel{⌣}{\phi }={\displaystyle {\sum }_{i=0}^N{\phi }_i},\stackrel{⌣}{\phi }={\displaystyle {\sum }_{i=0}^N{a}_i{\phi }_i}.\end{array}$ (63)

Here the symbol “::” means that in perturbation theory we drop Feynman diagrams with loops containing only one vertex. The ${\phi }_0$ is usual field with mass $m_0$ and the ${\phi }_i,i=1,\cdots ,N$ is the extra field with mass $m_i,i=1,\cdots ,N$. It can be shown that in (3+1)-dimensional theory the introduction of one PV field is sufficient for the ultraviolet regularization of perturbation theory in $\lambda$. One can show that momentum space Feynman diagrams in the original theory with Lagrangian 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 ${\phi }_0$ and $\stackrel{⌣}{\phi }$ sum up in corresponding diagrams so that we obtain the following expression which plays the role of regularized propagator

$\Delta \left(k^2\right)={\displaystyle {\sum }_{j=0}^N\frac{a_j}{k^2-m_j^2+i0}}=\frac{1}{k^2-m_0^2+i0}-{\displaystyle {\sum }_{j=0}^N\frac{a_j}{k^2-m_j^2+i0}},$ (64)

where $k^2=k_0^2-k_1^2+k_2^2+k_3^2$. Integral corresponding to vacuum diagram is

$\Im ={\displaystyle \int \frac{{\text{d}}^{4}k}{{\left(2\text{π}\right)}^4}\Delta \left(k^2\right)}={\displaystyle \int \frac{{\text{d}}^{4}k}{{\left(2\text{π}\right)}^4}}{\displaystyle {\sum }_{j=0}^N\frac{a_j}{k^2-m_j^2+i0}}\mathrm{.}$ (65)

To do this integral, since it is convergent, we can Wick rotate.

Figure 1. One-loop massive vacuum diagram.

Remark 2.1.1. All the integrals in quantum field theory are written in Minkowski 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 component of momentum $k_0\to i{k}_4$, 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

${\Im }_E=\frac{i}{8{\text{π}}^2}{\displaystyle {\int }_0^{\infty }\text{d}{k}_E}{\displaystyle {\sum }_{j=0}^N\frac{a_j{k}_E^3}{k_E^2+m_j^2}}\mathrm{.}$ (66)

To do this integral, since it is convergent, we can deal with regularized integral

$\Im \left(\epsilon \mathrm{,}\Lambda \right)=\frac{i}{8{\text{π}}^2}{\displaystyle {\int }_{\epsilon }^{\Lambda }\text{d}{k}_E{\displaystyle {\sum }_{j=0}^N\frac{a_j{k}_E^3}{k_E^2+m_j^2}}}\mathrm{,}$ (67)

where $\epsilon \asymp \mathrm{0,}\Lambda \asymp \infty$, i.e. $\Im \left(\epsilon \mathrm{,}\Lambda \right)\approx {\Im }_E$. We assume now that Pauli-Villars conditions given by Equations (48) holds. Let us consider now the quantity

${\Im }_{\eta }\triangleq {\Im }_{\eta }\left(\epsilon \mathrm{,}\Lambda \right)=\frac{i}{8{\text{π}}^2}{\displaystyle {\int }_{\epsilon }^{\Lambda }\text{d}{k}_E{\displaystyle {\sum }_{j=0}^N\frac{a_j{k}_E^3}{k_E^2+\eta m_j^2}}}\mathrm{,}$ (68)

where $\eta \in \left(\mathrm{0,1}\right]$, and therefore from Equation (68) we obtain

${{\Im }_{\eta }|}_{\eta =0}=\frac{i}{8{\text{π}}^2}{\displaystyle {\int }_{\epsilon }^{\Lambda }\text{d}{k}_E{\displaystyle {\sum }_{j=0}^N{a}_j{k}_E}}=\frac{i}{8{\text{π}}^2}{\displaystyle {\sum }_{j=0}^N{a}_j{\displaystyle {\int }_{\epsilon }^{\Lambda }{k}_E\text{d}{k}_E}}\equiv 0,$ (69)

since Equations (48) holds. From Equation (68) by differentiation we obtain

$\frac{\text{d}}{\text{d}\eta }{\Im }_{\eta }=\frac{i}{8{\text{π}}^2}{\displaystyle {\int }_{\epsilon }^{\Lambda }\text{d}{k}_E}{\displaystyle {\sum }_{j=0}^N\frac{a_j{m}_j^2{k}_E^3}{{\left(k_E^2+\eta m_j^2\right)}^2}},$ (70)

and therefore from Equation (39) we obtain

Figure 2. The Wick rotation of the integration contour.

$\begin{array}{c}{\frac{\text{d}}{\text{d}\eta }{\Im }_{\eta }|}_{\eta =0}=\frac{i}{8{\text{π}}^2}{{\displaystyle {\int }_{\epsilon }^{\Lambda }\text{d}{k}_E}{\displaystyle {\sum }_{j=0}^N\frac{a_j{m}_j^2{k}_E^3}{{\left(k_E^2+\eta m_j^2\right)}^2}}|}_{\eta =0}\\ =\frac{i}{8{\text{π}}^2}{\displaystyle {\sum }_{j=0}^N{a}_j{m}_j^2}{\displaystyle {\int }_{\epsilon }^{\Lambda }{k}_E^{-1}\text{d}{k}_E}\equiv \mathrm{0,}\end{array}$ (71)

since Equations (48) holds. From Equation (70) by differentiation we obtain

$\begin{array}{l}\frac{{\text{d}}^2}{\text{d}{\eta }^2}{\Im }_{\eta }={\displaystyle {\sum }_{j=0}^N{\Re }_j\left(\eta \right)}=\frac{i}{4{\text{π}}^2}{\displaystyle {\int }_{\epsilon }^{\Lambda }\text{d}{k}_E}{\displaystyle {\sum }_{j=0}^N\frac{a_j{m}_j^4{k}_E^3}{{\left(k_E^2+\eta m_j^2\right)}^3}},\\ {\Re }_j\left(\eta \right)=\frac{i{a}_j{m}_j^4}{4{\text{π}}^2}{\displaystyle {\int }_{\epsilon }^{\Lambda }\text{d}{k}_E}\frac{k_E^3}{{\left(k_E^2+\eta m_j^2\right)}^3}.\end{array}$ (72)

Note that

${\Re }_j\left(\eta \right)\simeq \frac{i{a}_j{m}_j^4}{4{\text{π}}^2}{\displaystyle {\int }_0^{\infty }\text{d}{k}_E\frac{k_E^3}{{\left(k_E^2+\eta m_j^2\right)}^3}}=\frac{i{a}_j{m}_j^4}{4{\text{π}}^2}\frac{-i}{4\eta m_j^2}=\frac{a_j{m}_j^2}{16{\text{π}}^2\eta }\mathrm{.}$ (73)

Thus

$\frac{\text{d}}{\text{d}\eta }{\Im }_{\eta }={\displaystyle {\sum }_{j=0}^N{\displaystyle {\int }_0^1{\Re }_j\left(\eta \right)\text{d}\eta }}={\displaystyle {\sum }_{j=0}^N\frac{a_j{m}_j^2}{16{\text{π}}^2}\ln \eta }$ (74)

and

${\Im }_{\eta }={\displaystyle {\sum }_{j=0}^N\frac{a_j{m}_j^2}{16{\text{π}}^2}\left(\eta \ln \eta -\eta \right)},$ (75)

Therefore

$\Im \left(\epsilon ,\Lambda \right)={{\Im }_{\eta }|}_{\eta =1}=-{\displaystyle {\sum }_{j=0}^N\frac{a_j{m}_j^2}{16{\text{π}}^2}}\equiv 0,$ (76)

since Equations (48) holds. Thus integral (65) corresponding to vacuum diagram by using Pauli-Villars renormalization identically equal zero, i.e.

${\text{Ren}}_{PV}\left(\Im \right)=\frac{{\text{d}}^{4}k}{{\left(2\text{π}\right)}^4}\Delta \left(k^2\right)=\frac{{\text{d}}^{4}k}{{\left(2\text{π}\right)}^4}{\displaystyle {\sum }_{j=0}^N\frac{a_j}{k^2-m_j^2+i0}}\equiv 0.$ (77)

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

$\Im \left(p^2\right)=\frac{1}{{\left(2\text{π}\right)}^4}{\displaystyle \int \frac{{\text{d}}^{4}k}{\left(k^2-m_0^2+i0\right)\left[\left(p^2-k^2\right)-m_0^2+i0\right]}}.$ (78)

Regularized Feinman integral (78) reads

${\Im }_{reg}\left(p^2\right)=\frac{1}{{\left(2\text{π}\right)}^4}{\displaystyle \int {\displaystyle {\sum }_{j=0}^N\frac{a_j{\text{d}}^{4}k}{\left(k^2-m_j^2+i0\right)\left[\left(p^2-k^2\right)-m_j^2+i0\right]}}},$ (79)

where $N=1$. To do this integral, since it is convergent, we can Wick rotate. Then we get

${\Im }_{reg}\left(p^2\right)=\frac{i}{{\left(2\text{π}\right)}^4}{\displaystyle \int {\displaystyle {\sum }_{j=0}^N\frac{a_j{\text{d}}^{4}k}{\left(k^2+m_j^2\right)\left[\left(p^2-k^2\right)+m_j^2\right]}}}.$ (80)

The integral (80) can be written as

$\begin{array}{c}{\Im }_{reg}\left(p^2\right)=\frac{i}{{\left(2\text{π}\right)}^4}{\displaystyle \underset{0}{\overset{1}{\int }}\text{d}x}{\displaystyle \int {\displaystyle {\sum }_{j=0}^N\frac{a_j{\text{d}}^{4}k}{{\left[k^2+p^{2}x\left(1-x\right)+m_j^2\right]}^2}}}\\ =\frac{i}{8{\text{π}}^2}{\displaystyle \underset{0}{\overset{1}{\int }}\text{d}x}{\displaystyle \int {\displaystyle {\sum }_{j=0}^N\frac{a_j{k}_E^3\text{d}{k}_E}{{\left[k_E^2+p^{2}x\left(1-x\right)+m_j^2\right]}^2}}}\mathrm{.}\end{array}$ (81)

To do this integral, since it is convergent, we can deal with regularized integral

${\Im }_{reg}\left(p^2\mathrm{,}\epsilon \mathrm{,}\Lambda \right)=\frac{i}{8{\text{π}}^2}{\displaystyle \underset{0}{\overset{1}{\int }}\text{d}x}{\displaystyle {\int }_{\epsilon }^{\Lambda }{\displaystyle {\sum }_{j=0}^N\frac{a_j{k}_E^3\text{d}{k}_E}{{\left[k_E^2+p^{2}x\left(1-x\right)+m_j^2\right]}^2}}}\mathrm{.}$ (82)

Let us consider now the quantity

${\Im }_{\eta }\left(p^2\mathrm{,}\epsilon \mathrm{,}\Lambda \right)=\frac{i}{8{\text{π}}^2}{\displaystyle \underset{0}{\overset{1}{\int }}\text{d}x}{\displaystyle {\int }_{\epsilon }^{\Lambda }{\displaystyle {\sum }_{j=0}^N\frac{a_j{k}_E^3\text{d}{k}_E}{{\left[k_E^2+p^{2}x\left(1-x\right)+\eta m_j^2\right]}^2}}}\mathrm{.}$ (83)

where $\eta \in \left(\mathrm{0,1}\right]$, and therefore from Equation (83) we obtain ${\Im }_0\left(p^2\mathrm{,}\epsilon \mathrm{,}\Lambda \right)\equiv 0$, since Equations (48) holds. From Equation (83) by differentiation we obtain

Figure 3. The simplest scalar diagram.

$\begin{array}{l}\frac{\text{d}}{\text{d}\eta }{\Im }_{\eta }\left(p^2\mathrm{,}\epsilon \mathrm{,}\Lambda \right)=-\frac{i}{4{\text{π}}^2}{\displaystyle \underset{0}{\overset{1}{\int }}\text{d}x}{\displaystyle {\int }_{\epsilon }^{\Lambda }{\displaystyle {\sum }_{j=0}^N\frac{a_j{m}_j^2{k}_E^3\text{d}{k}_E}{{\left[k_E^2+p^{2}x\left(1-x\right)+\eta m_j^2\right]}^3}}}\\ \simeq -\frac{i}{4{\text{π}}^2}{\displaystyle {\sum }_{j=0}^N{a}_j{m}_j^2{\Re }_j\left(p^2\mathrm{,}\eta \mathrm{,}\Lambda \mathrm{,}\epsilon \right)}\mathrm{,}\\ {\Re }_j\left(p^2\mathrm{,}\eta \mathrm{,}\Lambda \mathrm{,}\epsilon \right)\simeq {\displaystyle \underset{0}{\overset{1}{\int }}\text{d}x}{\displaystyle \int \frac{k_E^3\text{d}{k}_E}{{\left[k_E^2+p^{2}x\left(1-x\right)+\eta m_j^2\right]}^3}}\\ =\frac{1}{4}{\displaystyle \underset{0}{\overset{1}{\int }}\frac{\text{d}x}{p^{2}x\left(1-x\right)+\eta m_j^2}}\mathrm{.}\end{array}$ (84)

From Equation (84) we obtain

$\begin{array}{c}\frac{\text{d}}{\text{d}\eta }{\Im }_{\eta }\left(p^2\mathrm{,}\epsilon \mathrm{,}\Lambda \right)\simeq -\frac{i}{4{\text{π}}^2}{\displaystyle {\sum }_{j=0}^N{a}_j{m}_j^2{\Re }_j\left(p^2\mathrm{,}\eta \mathrm{,}\epsilon \mathrm{,}\Lambda \right)}\\ =-\frac{i}{16{\text{π}}^2}{\displaystyle {\sum }_{j=0}^N{a}_j}{\displaystyle \underset{0}{\overset{1}{\int }}\frac{\text{d}x}{m_j^{-2}{p}^{2}x\left(1-x\right)+\eta }}\mathrm{.}\end{array}$ (85)

From Equation (85) we obtain

${\Im }_{reg}\left(p^2\right)=-\frac{i}{16{\text{π}}^2}{\displaystyle {\sum }_{j=0}^N{a}_j}{\displaystyle \underset{0}{\overset{1}{\int }}\text{d}x}{\displaystyle \underset{0}{\overset{1}{\int }}\frac{\text{d}\eta }{m_j^{-2}{p}^{2}x\left(1-x\right)+\eta }}\mathrm{.}$ (86)

Note that

$\begin{array}{c}{\displaystyle \underset{0}{\overset{1}{\int }}\frac{\text{d}\eta }{m_j^{-2}{p}^{2}x\left(1-x\right)+\eta }}={\left[m_j^{-2}{p}^{2}x\left(1-x\right)+\eta \right]\ln \left[m_j^{-2}{p}^{2}x\left(1-x\right)+\eta \right]|}_0^1-1\\ =\left[m_j^{-2}{p}^{2}x\left(1-x\right)+1\right]\ln \left[m_j^{-2}{p}^{2}x\left(1-x\right)+1\right]\\ -\left[m_j^{-2}{p}^{2}x\left(1-x\right)\right]\ln \left[m_j^{-2}{p}^{2}x\left(1-x\right)\right]-1.\end{array}$ (87)

Thus

$\begin{array}{c}{\Im }_{reg}\left(p^2\right)=-\frac{i}{16{\text{π}}^2}{\displaystyle {\sum }_{j=0}^{N=1}{a}_j}{\displaystyle \underset{0}{\overset{1}{\int }}\text{d}x}{\displaystyle \underset{0}{\overset{1}{\int }}\frac{\text{d}\eta }{m_j^{-2}{p}^{2}x\left(1-x\right)+\eta }}\\ =-\frac{i}{16{\text{π}}^2}{\displaystyle {\sum }_{j=0}^{N=1}{a}_j}{\displaystyle \underset{0}{\overset{1}{\int }}\text{d}x}\{\left[m_j^{-2}{p}^{2}x\left(1-x\right)+1\right]\ln \left[m_j^{-2}{p}^{2}x\left(1-x\right)+1\right]\\ -\left[m_j^{-2}{p}^{2}x\left(1-x\right)\right]\ln \left[m_j^{-2}{p}^{2}x\left(1-x\right)\right]\}+\frac{i}{16{\text{π}}^2}{\displaystyle {\sum }_{j=0}^{N=1}{a}_j}\\ =-\frac{i}{16{\text{π}}^2}{\displaystyle {\sum }_{j=0}^{N=1}{a}_j}{\displaystyle \underset{0}{\overset{1}{\int }}\text{d}x}\{\left[m_j^{-2}{p}^{2}x\left(1-x\right)+1\right]\ln \left[m_j^{-2}{p}^{2}x\left(1-x\right)+1\right]\\ -\left[m_j^{-2}{p}^{2}x\left(1-x\right)\right]\ln \left[m_j^{-2}{p}^{2}x\left(1-x\right)\right]\}\end{array}$

$\begin{array}{l}=-\frac{i}{16{\text{π}}^2}{\displaystyle \underset{0}{\overset{1}{\int }}\text{d}x}\{\left[m_0^{-2}{p}^{2}x\left(1-x\right)+1\right]\ln \left[m_0^{-2}{p}^{2}x\left(1-x\right)+1\right]\\ -\left[m_0^{-2}{p}^{2}x\left(1-x\right)\right]\ln \left[m_0^{-2}{p}^{2}x\left(1-x\right)\right]\}\\ +\frac{i}{16{\text{π}}^2}{\displaystyle \underset{0}{\overset{1}{\int }}\text{d}x}\{\left[m_1^{-2}{p}^{2}x\left(1-x\right)+1\right]\ln \left[m_1^{-2}{p}^{2}x\left(1-x\right)+1\right]\\ -\left[m_1^{-2}{p}^{2}x\left(1-x\right)\right]\ln \left[m_1^{-2}{p}^{2}x\left(1-x\right)\right]\}\mathrm{.}\end{array}$ (88)

From Equation (88) we obtain

$\begin{array}{c}{\Im }_{reg}\left(p^2\right)=-\frac{i}{16{\text{π}}^2}{\displaystyle \underset{0}{\overset{1}{\int }}\text{d}x}\{\left[m_0^{-2}{p}^{2}x\left(1-x\right)+1\right]\ln \left[m_0^{-2}{p}^{2}x\left(1-x\right)+1\right]\\ -\left[m_0^{-2}{p}^{2}x\left(1-x\right)\right]\ln \left[m_0^{-2}{p}^{2}x\left(1-x\right)\right]\}\\ +\frac{i}{16{\text{π}}^2}{\displaystyle \underset{0}{\overset{1}{\int }}\text{d}x}\{\left[m_1^{-2}{p}^{2}x\left(1-x\right)+1\right]\ln \left[m_1^{-2}{p}^{2}x\left(1-x\right)+1\right]\\ -\left[m_1^{-2}{p}^{2}x\left(1-x\right)\right]\ln \left[m_1^{-2}{p}^{2}x\left(1-x\right)\right]\}\mathrm{.}\end{array}$ (89)

We assume now that $m_1^{-2}{p}^2\ll 1$ and from Equation (89) finally we obtain

$\begin{array}{c}{\Im }_{reg}\left(p^2\right)=-\frac{i}{16{\text{π}}^2}{\displaystyle \underset{0}{\overset{1}{\int }}\text{d}x}\{\left[m_0^{-2}{p}^{2}x\left(1-x\right)+1\right]\ln \left[m_0^{-2}{p}^{2}x\left(1-x\right)+1\right]\\ -\left[m_0^{-2}{p}^{2}x\left(1-x\right)\right]\ln \left[m_0^{-2}{p}^{2}x\left(1-x\right)\right]\}+O\left(m_1^{-2}{p}^2\right)\mathrm{.}\end{array}$ (90)

Remark 2.1.2. The simple renormalizable models with finite masses $m_i,i=1,\cdots ,N$ 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 behavior may be restricted to arbitrarily large energy scales ${\Lambda }_{\ast }$ mentioned above by an appropriate limitation on the finite masses $m_i$.

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 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

$I_{sym}=-{\displaystyle \int {\text{d}}^{4}x}\sqrt{-g}\left(\alpha R_{\mu \nu }{R}^{\mu \nu }-\beta R^2+2{\kappa }^{-2}R\right),$ (91)

where the curvature tensor and the Ricci is defined by $R_{\mu \alpha \nu }^{\lambda }={\partial }_{\nu }{\Gamma }_{\mu \alpha }^{\lambda }$ and $R_{\mu \nu }=R_{\mu \lambda \nu }^{\lambda }$ correspondingly, ${\kappa }^2=32\text{π}G$. The convenient definition of the gravitational field variable in terms of the contravariant metric density reads

$\kappa h^{\mu \nu }=g^{\mu \nu }\sqrt{-g}-{\eta }^{\mu \nu }.$ (92)

Analysis of the linearized radiation shows that there are eight dynamical degrees 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 $m_2$. The eighth corresponds to a massive scalar particle with mass $m_0$. 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 excitations is negative definite. This feature is characteristic of higher-derivative models, and poses the major obstacle to their physical interpretation.

In the quantum theory, there is an alternative problem which may be substituted for the negative energy. It is possible to recast the theory so that the massive spin-2 eigenstates of the free-fieid Hamiltonian have positive-definite energy, 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 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 divergences due to the massless states is analogous to the Pauli-Villars regularization 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 canonical harmonic gauge: ${\partial }_{\nu }{h}^{\mu \nu }=0$. Corresponding Green’s functions are then given by a generating functional

$\begin{array}{c}Z\left(T_{\mu \nu }\right)=N{\displaystyle \int \left[{\displaystyle {\prod }_{\mu \le \nu }\text{d}{h}^{\mu \nu }}\right]\left[\text{d}{C}^{\sigma }\right]\left[\text{d}{\bar{C}}_{\tau }\right]{\delta }^4\left(F^{\tau }\right)}\\ \times \exp \left[i\left(I_{sym}+{\displaystyle \int {\text{d}}^{4}x}{\bar{C}}_{\tau }{\stackrel{\to }{F}}_{\mu \nu }^{\tau }{D}_{\alpha }^{\mu \nu }{C}^{\alpha }+\kappa {\displaystyle \int {\text{d}}^{4}x}{T}_{\mu \nu }{h}^{\mu \nu }\right)\right]\mathrm{.}\end{array}$ (93)

Here $F^{\tau }={\stackrel{\to }{F}}_{\mu \nu }^{\tau }{h}^{\mu \nu },{\stackrel{\to }{F}}_{\mu \nu }^{\tau }={\delta }_{\mu }^r{\stackrel{\to }{\partial }}_{\nu }$ and the arrow indicates the direction in which the derivative acts. N is a normalization constant. $C^{\sigma }$ is the Faddeev-Popov ghost field, and ${\bar{C}}_{\tau }$ is the antighost field. Notice that both $C^{\sigma }$ and ${\bar{C}}_{\tau }$ are anticommuting quantities. $D_{\alpha }^{\mu \nu }$ is the operator which generates gauge transformations in $h^{\mu \nu }$, given an arbitrary spacetime-dependent vector ${\xi }^{\alpha }\left(x\right)$ corresponding to $x^{{\mu }^{\prime }}=x^{\mu }+\kappa {\xi }^{\mu }$ and where

$\begin{array}{c}{D}_{\alpha }^{\mu \nu }{\xi }^{\alpha }\left(x\right)={\partial }^{\mu }{\xi }^{\nu }+{\partial }^{\nu }{\xi }^{\mu }-{\eta }^{\mu \nu }{\partial }_{\alpha }{\xi }^{\alpha }\\ +\kappa \left({\partial }_{\alpha }{\xi }^{\mu }{h}^{\alpha \nu }+{\partial }_{\alpha }{\xi }^{\nu }{h}^{\alpha \mu }-{\xi }^{\alpha }{\partial }_{\alpha }{h}^{\mu \nu }-{\partial }_{\alpha }{\xi }^{\alpha }{h}^{\mu \nu }\right)\end{array}$ (94)

In the functional integral (93), we have written the metric for the gravitational field as $\left[{\displaystyle {\prod }_{\mu \le \nu }\text{d}{h}^{\mu \nu }}\right]$ without any local factors of $g=\det \left(g_{\mu \nu }\right)$. Such factors do not contribute to the Feynman rules because their effect is to introduce terms proportional to ${\delta }^4\left(0\right){\displaystyle \int {\text{d}}^{4}x\ln \left(-g\right)}$ into the effective action and ${\delta }^4\left(0\right)$ is set equal to zero in dimensional regularization.

In calculating the generating functional (93) by using the loop expansion, one may represent the $\delta$ function which fixes the gauge as the limit of a Gaussian, discarding an infinite normalization constant

${\delta }^4\left(F^{\tau }\right)\sim \underset{\Delta \to 0}{\lim }\exp \left[i\left(\frac{1}{2}{\Delta }^{-1}{\displaystyle \int {\text{d}}^{4}x{F}_{\tau }{F}^{\tau }}\right)\right]\mathrm{.}$ (95)

In this expression, the index $\tau$ has been lowered using the flat-space metric tensor ${\eta }_{\mu \nu }$. For the remainder of this paper, we shall adopt the standard approach 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. The graviton propagator may be calculated from $I_{sym}+\frac{1}{2}{\Delta }^{-1}{\displaystyle \int {\text{d}}^{4}x{F}_{\tau }{F}^{\tau }}$ in the usual fashion, letting $\Delta \to 0$ after inverting. The expression $\frac{1}{2}{\Delta }^{-1}{\displaystyle \int {\text{d}}^{4}x{F}_{\tau }{F}^{\tau }}$ 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 transverse 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 has a divergent part with the general structure ${\left({\partial }^{4}h\right)}^2$. 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 [14]

${\omega }_4\left(e^{\tau }\right)=\exp \left[i\left(\frac{1}{2}{\Delta }^{-1}{\displaystyle \int {\text{d}}^{4}x{e}_{\tau }{\square }^2{e}^{\tau }}\right)\right]\mathrm{,}$ (96)

where $e^{\tau }$ is any four-vector function. The corresponding gauge-fixing term in the effective action is

$-\frac{1}{2}{\kappa }^2{\Delta }^{-1}{\displaystyle \int {\text{d}}^{4}x{F}_{\tau }{\square }^2{F}^{\tau }}\mathrm{.}$ (97)

The graviton propagator resulting from the gauge-fixing term (97) is derived in [14] . For most values of the parameters $\alpha$ and $\beta$ in $I_{sym}$ 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 $\alpha =0$, the massive spin-2 excitations disappear, and inspection of the graviton propagator shows that some terms then behave like $k^{-2}$. Likewise, if $3\beta -\alpha =0$, 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 $\alpha =0$ and $3\beta -\alpha =0$, 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 $h^{\mu \nu }$, generated by ${\xi }^{\mu }$ and

Figure 4. The one-loop graviton self-energy diagram.

perform a second gauge transformation, generated by ${\eta }^{\mu }$, on the $h^{\mu \nu }$ fields appearing there. Then antisymmetrize in ${\xi }^{\mu }$ and ${\eta }^{\mu }$. The result is

$\frac{\delta D_{\alpha }^{\mu \nu }}{\delta h^{\rho \sigma }}{D}_{\beta }^{\rho \sigma }\left({\xi }^{\alpha }{\eta }^{\beta }-{\eta }^{\alpha }{\xi }^{\beta }\right)=\kappa D_{\lambda }^{\mu \nu }\left({\partial }_{\alpha }{\xi }^{\lambda }{\eta }^{\alpha }-{\partial }_{\alpha }{\xi }^{\alpha }{\eta }^{\lambda }\right),$ (98)

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]

$\begin{array}{l}\left(\text{a}\right){\delta }_{\text{BRS}}{h}^{\mu \nu }=\kappa D_{\alpha }^{\mu \nu }{C}^{\alpha }\delta \lambda \mathrm{,}\\ \left(\text{b}\right){\delta }_{\text{BRS}}{C}^{\alpha }=-{\kappa }^2{\partial }_{\beta }{C}^{\alpha }{C}^{\beta }\delta \lambda \mathrm{,}\\ \left(\text{c}\right){\delta }_{\text{BRS}}{\bar{C}}_{\tau }=-{\kappa }^3{\Delta }^{-1}{\square }^2{F}_{\tau }\delta \lambda \mathrm{,}\end{array}$ (99)

where $\delta \lambda$ is an infinitesimal anticommuting constant parameter. The importance of these transformations resides in the quantities which they leave invariant. Note that

${\delta }_{\text{BRS}}\left({\partial }_{\beta }{C}^{\sigma }{C}^{\beta }\right)=0$ (100)

and

${\delta }_{\text{BRS}}\left(D_{\alpha }^{\mu \nu }{C}^{\alpha }\right)=0.$ (101)

As a result of Equation (101), the only part of the ghost action which varies under the BRS transformations is the antighost ${\bar{C}}_{\tau }$. Accordingly, the transformation (99c) has been chosen to make the variation of the ghost action just cancel the variation of the gauge-fixing term. Therefore, the entire effective action is BRS invariant:

${\delta }_{\text{BRS}}\left(I_{sim}-\frac{1}{2}{\kappa }^2{\Delta }^{-1}{F}_{\tau }{\square }^2{F}^{\tau }+{\bar{C}}_{\tau }{F}_{\mu \nu }^{\tau }{D}_{\alpha }^{\mu \nu }{C}^{\alpha }\right)=0.$ (102)

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.

$\begin{array}{l}Z\left(T_{\mu \nu }\mathrm{,}{\bar{\beta }}_{\sigma }\mathrm{,}{\beta }^{\tau }\mathrm{,}{K}_{\mu \nu }\mathrm{,}{L}_{\sigma }\right)\\ =N{\displaystyle \int \left[{\displaystyle {\prod }_{\mu \le \nu }\text{d}{h}^{\mu \nu }}\right]\left[\text{d}{C}^{\sigma }\right]\left[\text{d}{\bar{C}}_{\tau }\right]}\\ \times \exp \left[i\tilde{\Sigma }\left(h^{\mu \nu }\mathrm{,}{C}^{\sigma }\mathrm{,}{\bar{C}}_{\tau }\mathrm{,}{K}_{\mu \nu }\mathrm{,}{L}_{\sigma }\mathrm{,}{\bar{\beta }}_{\sigma }{C}^{\sigma }\right)+{\bar{\beta }}_{\sigma }{C}^{\sigma }+{\bar{C}}_{\tau }{\beta }^{\tau }+\kappa T_{\mu \nu }{h}^{\mu \nu }\right]\mathrm{.}\end{array}$ (103)

Anticommuting sources have been included for the ghost and antighost fields, and the effective action $\tilde{\Sigma }$ has been enlarged by the inclusion of BRS invariant couplings of the ghosts and gravitons to some external fields $K_{\mu \nu }$ (anticommuting) and $L_{\sigma }$ (commuting),

$\tilde{\Sigma }=I_{sim}-\frac{1}{2}{\kappa }^2{\Delta }^{-1}{F}_{\tau }{\square }^2{F}^{\tau }+{\bar{C}}_{\tau }{\stackrel{\to }{F}}_{\mu \nu }^{\tau }{D}_{\alpha }^{\mu \nu }{C}^{\alpha }+\kappa K_{\mu \nu }{D}_{\alpha }^{\mu \nu }+{\kappa }^2{L}_{\sigma }{\partial }_{\beta }{C}^{\sigma }{C}^{\beta }\mathrm{.}$ (104)

$\tilde{\Sigma }$ is BRS invariant by virtue of Equation (99), Equation (100), and Equation (102). We may use the new couplings to write this invariance as

$\frac{\delta \tilde{\Sigma }}{\delta K_{\mu \nu }}\frac{\delta \tilde{\Sigma }}{\delta h^{\mu \nu }}+\frac{\delta \tilde{\Sigma }}{\delta L_{\sigma }}\frac{\delta \tilde{\Sigma }}{\delta C^{\sigma }}+{\kappa }^3{\Delta }^{-1}{\square }^2{F}_{\tau }\frac{\delta \tilde{\Sigma }}{\delta {\bar{C}}_{\tau }}\mathrm{.}$ (105)

In this equation, and throughout this subsection, we use left variational derivatives with respect to anticommuting quantities: $\delta f\left(C^{\sigma }\right)=\delta C^{\tau }\delta f/\delta C^{\tau }$. Equation (105) may be simplified by rewriting it in terms of a reduced effective action,

$\Sigma =\tilde{\Sigma }+\frac{1}{2}{\kappa }^2{\Delta }^{-1}{F}_{\tau }{\square }^2{F}^{\tau }\mathrm{.}$ (106)

Substitution of (106) into (105) gives

$\frac{\delta \Sigma }{\delta K_{\mu \nu }}\frac{\delta \Sigma }{\delta h^{\mu \nu }}+\frac{\delta \Sigma }{\delta L_{\sigma }}\frac{\delta \Sigma }{\delta C^{\sigma }}=\mathrm{0,}$ (107)

where we have used the relation

${\kappa }^{-1}{\stackrel{\to }{F}}_{\mu \nu }^{\tau }\frac{\delta \Sigma }{\delta K_{\mu \nu }}-\frac{\delta \Sigma }{\delta {\bar{C}}_{\tau }}=0.$ (108)

Note that a measure

$\left[{\displaystyle {\prod }_{\mu \le \nu }\text{d}{h}^{\mu \nu }}\right]\left[\text{d}{C}^{\sigma }\right]\left[\text{d}{\bar{C}}_{\tau }\right]$ (109)

is BRS invariant since for infinitesimal transformations, the Jacobian is 1, because of the trace relations

$\begin{array}{l}\left(\text{a}\right)\frac{{\delta }^2\tilde{\Sigma }}{\delta K_{\left(\mu \nu \right)}\delta h^{\left(\mu \nu \right)}}=\mathrm{0,}\\ \left(\text{b}\right)\frac{{\delta }^2\tilde{\Sigma }}{\delta C^{\sigma }\delta L_{\sigma }}=\mathrm{0,}\end{array}$ (110)

both of which follow from ${\displaystyle \int {\text{d}}^{4}x{\partial }_{\alpha }{C}^{\alpha }}=0$. The parentheses surrounding the indices in (110a) indicate that the summation is to be carried out only for $\mu \le \nu$.

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

$\begin{array}{l}N{\displaystyle \int \left[{\displaystyle {\prod }_{\mu \le \nu }\text{d}{h}^{\mu \nu }}\right]\left[\text{d}{C}^{\sigma }\right]\left[\text{d}{\bar{C}}_{\tau }\right]}({\kappa }^2{T}_{\mu \nu }{D}_{\alpha }^{\mu \nu }-{\kappa }^2{\bar{\beta }}_{\sigma }{\partial }_{\beta }{C}^{\sigma }{C}^{\beta }\\ +{\kappa }^3{\Delta }^{-1}{\beta }^{\tau }{\square }^2{\stackrel{\to }{F}}_{\tau \mu \nu }{h}^{\mu \nu })\exp \left[i\left(\tilde{\Sigma }+\kappa T_{\mu \nu }{h}^{\mu \nu }+{\bar{\beta }}_{\sigma }{C}^{\sigma }+{\bar{C}}_{\tau }{\beta }^{\tau }\right)\right]=0.\end{array}$ (111)

Another identity which we shall need is the ghost equation of motion. To derive this equation, we shift the antighost integration variable ${\bar{C}}_{\tau }$ to ${\bar{C}}_{\tau }+\delta {\bar{C}}_{\tau }$, again with no resulting change in the value of the generating functional:

$\begin{array}{l}N{\displaystyle \int \left[{\displaystyle {\prod }_{\mu \le \nu }\text{d}{h}^{\mu \nu }}\right]\left[\text{d}{C}^{\sigma }\right]\left[\text{d}{\bar{C}}_{\tau }\right]}\left(\frac{\delta \tilde{\Sigma }}{\delta C^{\sigma }}+{\beta }^{\tau }\right)\\ \times \exp \left[i\left(\tilde{\Sigma }+\kappa T_{\mu \nu }{h}^{\mu \nu }+{\bar{\beta }}_{\sigma }{C}^{\sigma }+{\bar{C}}_{\tau }{\beta }^{\tau }\right)\right]\end{array}$ (112)

We define now the generating functional of connected Green’s functions as the logarithm of the functional (103),

$W\left[T_{\mu \nu }\mathrm{,}{\bar{\beta }}_{\sigma }\mathrm{,}{\beta }^{\tau }\mathrm{,}{K}_{\mu \nu }\mathrm{,}{L}_{\sigma }\right]=-i\ln Z\left[T_{\mu \nu }\mathrm{,}{\bar{\beta }}_{\sigma }\mathrm{,}{\beta }^{\tau }\mathrm{,}{K}_{\mu \nu }\mathrm{,}{L}_{\sigma }\right]\mathrm{.}$ (113)

and make use of the couplings to the external fields $K_{\mu \nu }$ and $L_{\sigma }$ to rewrite (112) in terms of W

$\kappa T_{\mu \nu }\frac{\delta W}{\delta K_{\mu \nu }}-{\bar{\beta }}_{\sigma }\frac{\delta W}{\delta L_{\sigma }}+{\kappa }^2{\Delta }^{-1}{\beta }^{\tau }{\square }^2{\stackrel{\to }{F}}_{\tau \mu \nu }\frac{\delta W}{\delta T_{\mu \nu }}=0.$ (114)

Similarly, we get the ghost equation of motion:

${\kappa }^{-1}{\stackrel{\to }{F}}_{\mu \nu }^{\tau }\frac{\delta W}{\delta K_{\mu \nu }}+{\beta }^{\tau }=0.$ (115)

B) Proper vertices

A Legendre transformation takes us from the generating functional of connected Green’s functions (113) to the generating functional of proper vertices. First, we define the expectation values of the gravitational, ghost, and antighost fields in the presence of the sources $T_{\mu \nu },{\bar{\beta }}_{\sigma }$, and ${\beta }^{\tau }$ and the external fields $K_{\mu \nu }$ and $L_{\sigma }$

$\begin{array}{l}\left(\text{a}\right)h^{\mu \nu }\left(x\right)=\frac{\delta W}{\kappa \delta T_{\mu \nu }\left(x\right)},\\ \left(\text{b}\right)C^{\sigma }\left(x\right)=\frac{\delta W}{\delta {\bar{\beta }}_{\sigma }\left(x\right)},\\ \left(\text{c}\right){\bar{C}}_{\tau }\left(x\right)=\frac{\delta W}{\delta {\beta }^{\tau }\left(x\right)}.\end{array}$ (116)

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 $K_{\mu \nu }$ and $L_{\sigma }$

$\begin{array}{l}\tilde{\Gamma }\left[h^{\mu \nu },C^{\sigma },{\bar{C}}_{\tau },K_{\mu \nu },L_{\sigma }\right]\\ =W\left[T_{\mu \nu },{\bar{\beta }}_{\sigma },{\beta }^{\tau },K_{\mu \nu },L_{\sigma }\right]-\kappa T_{\mu \nu }{h}^{\mu \nu }-{\bar{\beta }}_{\sigma }{C}^{\sigma }-{\bar{C}}_{\tau }{\beta }^{\tau }.\end{array}$ (117)

In this equation, the quantities $T_{\mu \nu }{\bar{\beta }}_{\sigma }$, and ${\beta }^{\tau }$ are given implicitly in terms of $h^{\mu \nu }\mathrm{,}{C}^{\sigma }\mathrm{,}{\bar{C}}_{\tau }\mathrm{,}{K}_{\mu \nu }$, and $L_{\sigma }$ by Equation (116). The relations dual to (116) are

$\begin{array}{l}\left(\text{a}\right)\kappa T_{\mu \nu }\left(x\right)=-\frac{\delta \tilde{\Gamma }}{\delta h^{\mu \nu }\left(x\right)},\\ \left(\text{b}\right){\bar{\beta }}_{\sigma }\left(x\right)=\frac{\delta \tilde{\Gamma }}{\delta C^{\sigma }\left(x\right)},\\ \left(\text{c}\right){\beta }_{\tau }\left(x\right)=-\frac{\delta \tilde{\Gamma }}{\delta {\bar{C}}_{\tau }\left(x\right)}.\end{array}$ (118)

Since the external fields $K_{\mu \nu }$ and $L_{\sigma }$ do not participate in the Legendre transformation (116), for them we have the relations

$\begin{array}{l}\left(\text{a}\right)\frac{\delta \tilde{\Gamma }}{\delta K_{\mu \nu }\left(x\right)}=\frac{\delta W}{\delta K_{\mu \nu }\left(x\right)},\\ \left(\text{b}\right)\frac{\delta \tilde{\Gamma }}{\delta L_{\sigma }\left(x\right)}=\frac{\delta W}{\delta L_{\sigma }\left(x\right)}.\end{array}$ (119)

Finally, the Slavnov identity for the generating functional of proper vertices is obtained by transcribing (114) using the relations (116), (118), and (119)

$\frac{\delta \tilde{\Gamma }}{\delta K_{\mu \nu }}\frac{\delta \tilde{\Gamma }}{\delta h^{\mu \nu }}+\frac{\delta \tilde{\Gamma }}{\delta L_{\sigma }}\frac{\delta \tilde{\Gamma }}{\delta C^{\sigma }}+{\kappa }^3{\Delta }^{-1}{\square }^2{\stackrel{\to }{F}}_{\tau \mu \nu }{h}^{\mu \nu }\frac{\delta \tilde{\Gamma }}{\delta C^{\sigma }}=0.$ (120)

We also have the ghost equation of motion,

${\kappa }^{-1}{\stackrel{\to }{F}}_{\mu \nu }^{\tau }\frac{\delta \tilde{\Gamma }}{\delta K_{\mu \nu }}-\frac{\delta \tilde{\Gamma }}{\delta C^{\sigma }}=0.$ (121)

Since Equation (120) has exactly the same form as (105), we follow the example set by (106) and define a reduced generating functional of the proper vertices,

$\Gamma =\tilde{\Gamma }+\frac{1}{2}{\kappa }^2{\Delta }^{-1}\left({\stackrel{\to }{F}}_{\tau \mu \nu }{h}^{\mu \nu }\right){\square }^2\left({\stackrel{\to }{F}}_{\rho \sigma }^{\tau }{h}^{\rho \sigma }\right)\mathrm{.}$ (122)

Substituting this into (120) and (121), the Slavnov identity becomes

$\frac{\delta \Gamma }{\delta K_{\mu \nu }}\frac{\delta \Gamma }{\delta h^{\mu \nu }}+\frac{\delta \Gamma }{\delta L_{\sigma }}\frac{\delta \Gamma }{\delta C^{\sigma }}=0.$ (123)

and the ghost equation of motion becomes

${\kappa }^{-1}{\stackrel{\to }{F}}_{\mu \nu }^{\tau }\frac{\delta \Gamma }{\delta K_{\mu \nu }}-\frac{\delta \Gamma }{\delta {\bar{C}}_{\tau }}=0.$ (124)

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

${\Gamma }^{\left(0\right)}=\Sigma .$ (125)

C) Structure of the divergences and renormalization equation

The Slavnov identity (123) is quadratic in the functional $\Gamma$. 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 divergences. From this structure, it can be seen how to renormalize the effective action so that it remains invariant under a renormalized set of BRS transformations [14] .

Suppose that we have successfully renormalized the reduced effective action up to $n-1$ loop order; that is, suppose we have constructed a quantum extension of $\Sigma$ 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 renormalized quantity by ${\Sigma }^{\left(n-1\right)}$. 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 by ${\Gamma }^{\left(n-1\right)}$.

When we proceed to calculate ${\Gamma }^{\left(n\right)}$, 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 ${\Gamma }^{\left(n\right)}$ are local in structure. In the dimensional regularization procedure, these divergences are of order ${ϵ}^{-1}={\left(d-4\right)}^{-1}$, where d is the dimensionality of spacetime in the Feynman integrals.

There may also be divergent parts of ${\Gamma }^{\left(n\right)}$ which do not arise from loop integrals, and which contain higher-order poles in the regulating parameter $ϵ$. Such divergences come from n-loop order parts of ${\Sigma }^{\left(n-1\right)}$ which are necessary to ensure that (107) is satisfied. Consequently, they too have a local structure. We may separate the divergent and finite parts of ${\Gamma }^{\left(n\right)}$ :

${\Gamma }^{\left(n\right)}={\Gamma }_{\text{div}}^{\left(n\right)}+{\Gamma }_{\text{finite}}^{\left(n\right)}\mathrm{.}$ (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

$\begin{array}{l}\frac{\delta {\Gamma }_{\text{div}}^{\left(n\right)}}{\delta K_{\mu \nu }}\frac{\delta {\Gamma }^{\left(0\right)}}{\delta h^{\mu \nu }}+\frac{\delta {\Gamma }^{\left(0\right)}}{\delta K_{\mu \nu }}\frac{\delta {\Gamma }_{\text{div}}^{\left(n\right)}}{\delta h^{\mu \nu }}+\frac{\delta {\Gamma }_{\text{div}}^{\left(n\right)}}{\delta L_{\sigma }}\frac{\delta {\Gamma }^{\left(0\right)}}{\delta C^{\sigma }}+\frac{\delta {\Gamma }^{\left(0\right)}}{\delta L_{\sigma }}\frac{\delta {\Gamma }_{\text{div}}^{\left(n\right)}}{\delta C^{\sigma }}\\ =-{\displaystyle {\sum }_{i\mathrm{<mo>\; =\; </mo>0}}^n\left[\frac{\delta {\Gamma }_{\text{finite}}^{\left(n-i\right)}}{\delta K_{\mu \nu }}\frac{\delta {\Gamma }_{\text{finite}}^{\left(i\right)}}{\delta h^{\mu \nu }}+\frac{\delta {\Gamma }_{\text{finite}}^{\left(n-i\right)}}{\delta L_{\sigma }}\frac{\delta {\Gamma }_{\text{finite}}^{\left(i\right)}}{\delta C^{\sigma }}\right]}\mathrm{.}\end{array}$ (127)

Since each term on the right-hand side of (127) remains finite as $ϵ\to 0$, 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 Equation (125), we can write the following equation, called the renormalization equation:

$\Re {\Gamma }_{\text{div}}^{\left(n\right)}=\mathrm{0,}$ (128)

where

$\Re =\frac{\delta \Sigma }{\delta h^{\mu \nu }}\frac{\delta }{\delta K_{\mu \nu }}+\frac{\delta \Sigma }{\delta C^{\sigma }}\frac{\delta }{\delta L_{\sigma }}+\frac{\delta \Sigma }{\delta K_{\mu \nu }}\frac{\delta }{\delta h^{\mu \nu }}+\frac{\delta \Sigma }{\delta L_{\sigma }}\frac{\delta }{\delta C^{\sigma }}\mathrm{.}$ (129)

Similarly by collecting the n-loop order divergences in the ghost equation of motion (124) we get

${\kappa }^{-1}{\stackrel{\to }{F}}_{\mu \nu }^{\tau }\frac{\delta {\Gamma }_{\text{div}}^{\left(n\right)}}{\delta K_{\mu \nu }}-\frac{\delta {\Gamma }_{\text{div}}^{\left(n\right)}}{\delta {\bar{C}}_{\tau }}=0.$ (130)

In order to construct local solutions to Equations. (128) and (130) remind that the operator $\Re$ defined in (129) is nilpotent [14] :

${\Re }^2=0.$ (131)

Equation (131) gives us the local solutions to Equation (128) of the form

${\Gamma }_{\text{div}}^{\left(n\right)}=\Im \left(h^{\mu \nu }\right)+\Re \left[X\left(h^{\mu \nu }\mathrm{,}{C}^{\sigma }\mathrm{,}{\bar{C}}_{\tau }\mathrm{,}{K}_{\mu \nu }\mathrm{,}{L}_{\sigma }\right)\right]\mathrm{,}$ (132)

where $\Im$ is an arbitrary gauge-invariant local functional of $h^{\mu \nu }$ and its derivatives, and X is an arbitrary local functional of $h^{\mu \nu }\mathrm{,}{C}_{\sigma }\mathrm{,}{\bar{C}}_{\tau }\mathrm{,}{K}_{\mu \nu }$ and $L^{\sigma }$ and their derivatives. In order to satisfy the ghost equation of motion (130) we require that

${\Gamma }_{\text{div}}^{\left(n\right)}={\Gamma }_{\text{div}}^{\left(n\right)}\left(h^{\mu \nu }\mathrm{,}{C}^{\sigma }\mathrm{,}{K}_{\mu \nu }-{\kappa }^{-1}{\bar{C}}_{\tau }{}_{\tau }{\stackrel{\to }{F}}_{\mu \nu }^{\tau }\mathrm{,}{L}_{\sigma }\right)\mathrm{.}$ (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] :

$\begin{array}{l}{N}_{\text{ghost}}\left[h^{\mu \nu }\right]=\mathrm{0,}{N}_{\text{ghost}}\left[C_{\sigma }\right]=+\mathrm{1,}{N}_{\text{ghost}}\left[{\bar{C}}_{\tau }\right]=-\mathrm{1,}\\ N_{\text{ghost}}\left[K_{\mu \nu }\right]=-\mathrm{1,}{N}_{\text{ghost}}\left[L_{\sigma }\right]=-2.\end{array}$ (134)

From Equations (134) follows that

$N_{\text{ghost}}\left[\Sigma \right]=N_{\text{ghost}}\left[\Gamma \right]=0.$ (135)

Since

$N_{\text{ghost}}\left[\Re \right]=+\mathrm{1,}$ (136)

we require of the functional $X(\cdot )$ that

$N_{\text{ghost}}\left[X\right]=-1.$ (137)

In order to complete analysis of the structure of ${\Gamma }_{\text{div}}^{\left(n\right)}$, we must supplement the symmetry Equations (132), (133), and (137) with the constraints on the divergences which arise from power counting. Accordingly, we introduce the following notations:

$n_E$ = number of graviton vertices with two derivatives,

$n_G$ = number of antighost-graviton-ghost vertices,

$n_K$ = number of K-graviton-ghost vertices,

$n_L$ = number of L-ghost-ghost vertices,

$I_G$ = number of internal-ghost propagators,

$E_C$ = number of external ghosts,

$E_{\bar{C}}$ = 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 arbitrary diagram,

$D=4-2n_E+2I_G-2n_G-3n_L-3n_K-E_{\bar{C}}.$ (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 relation

$2I_G-2n_G=2n_L+n_K-E_C-E_{\bar{C}}$ (139)

to write the degree of divergence as

$D=4-2n_E-n_L-2n_K-E_C-2E_{\bar{C}}.$ (140)

Together with conservation of ghost number, Equation (140) enables us to catalog three different types of divergent structures involving ghosts. These are illustrated in Figure 5. Each of the three types has degree of divergence $D=1-2n_E$. Consequently, all the divergences which involve ghosts have $n_E=0$. Since the degree of divergence is then 1, the associated divergent structures in ${\Gamma }_{\text{div}}^{\left(n\right)}$ have an extra derivative appearing on one of the fields. Diagrams whose external lines are all gravitons have degree of divergence $D=4-2n_E$. Combining (140) with (137), (133), and (132), we can finally write the most general expression for ${\Gamma }_{\text{div}}^{\left(n\right)}$ which satisfies all the constraints of symmetries and power counting:

${\Gamma }_{\text{div}}^{\left(n\right)}=\Im \left(h^{\mu \nu }\right)+\Re \left[\left(K_{\mu \nu }-{\kappa }^{-1}{\bar{C}}_{\tau }{\stackrel{\to }{F}}_{\mu \nu }^{\tau }\right)P^{\mu \nu }\left(h^{\alpha \beta }\right)+L_{\sigma }{Q}_{\tau }^{\sigma }\left(h^{\alpha \beta }\right)C^{\tau }\right]\mathrm{,}$ (141)

where $P^{\mu \nu }\left(h^{\alpha \beta }\right)$ and $Q_{\tau }^{\sigma }\left(h^{\alpha \beta }\right)$ are arbitrary Lorentz-covariant functions of the gravitational field $h^{\mu \nu }$, but not of its derivatives, at a single spacetime point. $\Im \left(h^{\mu \nu }\right)$ is a local gauge-invariant functional of $h^{\mu \nu }$ containing terms with four, two, and zero derivatives. Expanding (141), we obtain an array of possible divergent structures:

$\begin{array}{c}{\Gamma }_{\text{div}}^{\left(n\right)}=\Im \left(h^{\mu \nu }\right)+\frac{\delta I_{sym}}{\delta h^{\mu \nu }}{P}^{\mu \nu }+\left(\kappa K_{\rho \sigma }-{\bar{C}}_{\tau }{\stackrel{\to }{F}}_{\rho \sigma }^{\tau }\right)\left(\frac{\delta D_{\alpha }^{\rho \sigma }}{\delta h^{\mu \nu }}{C}^{\alpha }\right)P^{\mu \nu }\\ -\left(\kappa K_{\rho \sigma }-{\bar{C}}_{\tau }{\stackrel{\to }{F}}_{\rho \sigma }^{\tau }\right)\frac{\delta P^{\rho \sigma }}{\delta h^{\mu \nu }}{D}_{\alpha }^{\mu \nu }{C}^{\alpha }-\left(\kappa K_{\mu \nu }-{\bar{C}}_{\tau }{\stackrel{\to }{F}}_{\mu \nu }^{\tau }\right)D_{\sigma }^{\mu \nu }\left(Q_{\epsilon }^{\sigma }{C}^{\epsilon }\right)\\ -{\kappa }^2{L}_{\sigma }{\partial }_{\beta }\left(Q_{\tau }^{\sigma }{C}^{\tau }\right)C^{\beta }-{\kappa }^2{L}_{\sigma }{\partial }_{\beta }{C}^{\sigma }{Q}_{\tau }^{\beta }{C}^{\tau }\\ -\kappa L_{\sigma }\frac{\delta Q_{\tau }^{\sigma }}{\delta h^{\mu \nu }}{C}^{\tau }{D}_{\alpha }^{\mu \nu }{C}^{\alpha }+{\kappa }^2{L}_{\sigma }{Q}_{\tau }^{\sigma }{\partial }_{\beta }{C}^{\tau }{C}^{\beta }\mathrm{.}\end{array}$ (142)

The breakup between the gauge-invariant divergences S and the rest (142) is determined only up to a term of the form [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.

The breakup between the gauge-invariant divergences S and the rest of (142)

${\displaystyle \int {\text{d}}^{4}x}\left({\eta }^{\mu \nu }+\kappa h^{\mu \nu }\right)\frac{\delta I_{sym}}{\kappa \delta h^{\mu \nu }}\mathrm{,}$ (143)

which can be generated by adding to $P^{\mu \nu }$ a term proportional to ${\eta }^{\mu \nu }+\kappa h^{\mu \nu }=\sqrt{g}{g}^{\mu \nu }$. 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 calculations, the situation is still reminiscent in principle of the renormalization of Yang-Mills theories. There, the non-gauge-invariant divergences may be eliminated by a number of field renormalizations. We shall find the same to be true here, but because the gravitational field $h^{\mu \nu }$ carries no weight in the power counting, there is nothing to prevent the field renormalizations from being nonlinear, or from mixing the gravitational and ghost fields. The corresponding renormalizations procedure considered in [14] .

Remark 2.2.2. We assume now that:

1) The local Poincaré group of momentum space is deformed at some fundamental high-energy cutoff ${\Lambda }_{\ast }$ [9] [10] .

2) The canonical quadratic invariant ${\Vert p\Vert }^2={\eta }^{ab}{p}_a{p}_b$ collapses at high-energy cutoff ${\Lambda }_{\ast }$ and being replaced by the non-quadratic invariant:

${\Vert p\Vert }^2=\frac{{\eta }^{ab}{p}_a{p}_b}{\left(1+l_{{\Lambda }_{\ast }}{p}_0\right)}.$ (144)

3) The canonical concept of Minkowski space-time collapses at a small distance $l_{{\Lambda }_{\ast }}={\Lambda }_{\ast }^{-1}$ to fractal space-time with Hausdorff-Colombeau negative dimension and therefore the canonical Lebesgue measure ${\text{d}}^{4}x$ being replaced by the Colombeau-Stieltjes

measure

${\left(\text{d}\eta \left(x\mathrm{,}\epsilon \right)\right)}_{\epsilon }={\left(v_{\epsilon }\left(s\left(x\right)\right){\text{d}}^{4}x\right)}_{\epsilon }\mathrm{,}$ (145)

where

$\begin{array}{l}{\left(v_{\epsilon }\left(s\left(x\right)\right)\right)}_{\epsilon }={\left({\left({\left|s\left(x\right)\right|}^{\left|D^{-}\right|}+\epsilon \right)}^{-1}\right)}_{\epsilon }\mathrm{,}\\ s\left(x\right)=\sqrt{x_{\mu }{x}^{\mu }}\mathrm{,}\end{array}$ (146)

see subsection IV.2.

4) The canonical concept of local momentum space collapses at fundamental high-energy cutoff ${\Lambda }_{\ast }$ to fractal momentum space with Hausdorff-Colombeau negative dimension and therefore the canonical Lebesgue measure ${\text{d}}^{3}k$, where $k=\left(k_x\mathrm{,}{k}_y\mathrm{,}{k}_z\right)$ being replaced by the Hausdorff-Colombeau measure

${\text{d}}^{D^{+}\mathrm{,}{D}^{-}}k\triangleq \frac{\Delta \left(D^{-}\right){\text{d}}^{D^{+}}k}{{\left({\left|k\right|}^{\left|D^{-}\right|}+\epsilon \right)}_{\epsilon }}=\frac{\Delta \left(D^{+}\right)\Delta \left(D^{-}\right)p^{D^{+}-1}\text{d}p}{{\left(p^{\left|D^{-}\right|}+\epsilon \right)}_{\epsilon }}\mathrm{,}$ (147)

see Subsection 3.4. Note that the integral over measure ${\text{d}}^{D^{+}\mathrm{,}{D}^{-}}k$ is given by 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 ${\Lambda }_{\ast }$ mentioned above by an appropriate limitation on the renormalized masses $m_2$ and $m_0$. Actually, it is only the massive spin-two excitations of the field which give the trouble with unitarity and thus require a very large mass. The limit on the mass $m_0$ 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

Let ${\mu }_H^{D_{+}}$ be a Hausdorff measure [19] and $X\subset {ℝ}^n$ is measurable set. Let $s\left(x\right)$ be a function $s\mathrm{<mo>\; :\; </mo>}X\to ℝ$ such that is symmetric with respect to some centre $x_0\in X$, i.e. $s\left(x\right)$ = constant for all x satisfying $d\left(x\mathrm{,}{x}_0\right)=r$ for arbitrary values of r. Then the integral in respect to Hausdorff measure over n-dimensional metric space X is then given by [19] :

${\displaystyle {\int }_{X}s\left(x\right)\text{d}{\mu }_H^{D_{+}}}=\frac{2{\text{π}}^{D_{+}/2}}{\Gamma \left(D_{+}/2\right)}{\displaystyle {\int }_0^{\infty }s\left(r\right)r^{D_{+}-1}\text{d}r}\mathrm{.}$ (148)

The integral in RHS of the Equation (148) is known in the theory of the Weyl fractional calculus where, the Weyl fractional integral $W^{D}f\left(x\right)$, is given by

$W^{D}f\left(x\right)=\frac{1}{\Gamma \left(D\right)}{\displaystyle {\int }_0^{\infty }{\left(t-x\right)}^{D-1}f\left(t\right)\text{d}t}\mathrm{.}$ (149)

Remark 3.1.1. In order to extend the Weyl fractional integral (148) in negative dimensions we apply the Colombeau generalized functions [20] and Colombeau generalized numbers [21] .

Recall that Colombeau algebras $\mathcal{G}\left(\Omega \right)$ of the Colombeau generalized functions defined as follows. Let $\Omega$ be an open subset of ${ℝ}^n$. Throughout this paper, for elements of the space $C^{\infty }{\left(\Omega \right)}^{\left(\mathrm{0,1}\right]}$ of sequences of smooth functions indexed by $\epsilon \in \left(\mathrm{0,1}\right]$ we shall use the canonical notations ${\left({\zeta }_{\epsilon }\left(x\right)\right)}_{\epsilon }$ and ${\left(u_{\epsilon }\right)}_{\epsilon }$ so $u_{\epsilon }\in C^{\infty }\left(\Omega \right)\epsilon \in \left(\mathrm{0,1}\right]$.

Definition 3.1.1. We set $\mathcal{G}\left(\Omega \right)={\mathcal{E}}_M\left(\Omega \right)\mathrm{<mo>\; /\; </mo>}\mathcal{N}\left(\Omega \right)$, where

$\begin{array}{l}{\mathcal{E}}_M\left(\Omega \right)=\{{\left(u_{\epsilon }\right)}_{\epsilon }\in C^{\infty }{\left(\Omega \right)}^{\left(\mathrm{0,1}\right]}|\forall K\subset \subset \Omega \mathrm{,}\forall \alpha \in {\mathbb{N}}^n\exists p\in \mathbb{N}\text{with}\\ {\sup }_{x\in K}\left|u_{\epsilon }\left(x\right)\right|=O\left({\epsilon }^{-p}\right)\text{as}\epsilon \to 0\}\mathrm{,}\\ \mathcal{N}\left(\Omega \right)=\{{\left(u_{\epsilon }\right)}_{\epsilon }\in C^{\infty }{\left(\Omega \right)}^{\left(\mathrm{0,1}\right]}|\forall K\subset \subset \Omega \mathrm{,}\forall \alpha \in {\mathbb{N}}^n\forall q\in \mathbb{N}\\ {\sup }_{x\in K}\left|u_{\epsilon }\left(x\right)\right|=O\left({\epsilon }^q\right)\text{as}\epsilon \to 0\}\mathrm{.}\end{array}$ (150)

Notice that $\mathcal{G}\left(\Omega \right)$ is a differential algebra. Equivalence classes of sequences ${\left(u_{\epsilon }\right)}_{\epsilon }$ will be denoted by $\text{cl}\left[{\left(u_{\epsilon }\right)}_{\epsilon }\right]$ is a differential algebra containing $D^{\prime }\left(\Omega \right)$ as a linear subspace and $C^{\infty }\left(\Omega \right)$ as subalgebra.

Definition 3.1.2. Weyl fractional integral ${\left(W_{\epsilon }^{D^{{}^{-}-}}f\left(x\right)\right)}_{\epsilon }$ in negative dimensions $D^{-}<0D^{-}\ne \mathrm{0,}-\mathrm{1,}\cdots \mathrm{,}-n\mathrm{,}\cdots \mathrm{,}n\in \mathbb{N}$ is given by

$\begin{array}{l}{W}^{D^{-}}f\left(x\right)=\frac{1}{\Gamma \left(D^{-}\right)}{\left({\displaystyle {\int }_{\epsilon }^{\infty }{\left(t-x\right)}^{D^{-}-1}f\left(t\right)\text{d}t}\right)}_{\epsilon }\\ \text{or}\\ {\left(W_{\epsilon }^{D_{-}^{-}}f\left(x\right)\right)}_{\epsilon }=\frac{1}{\Gamma \left(D^{-}\right)}{\left({\displaystyle {\int }_0^{\infty }\frac{1}{\epsilon +{\left(t-x\right)}^{\left|D^{-}\right|+1}}f\left(t\right)\text{d}t}\right)}_{\epsilon }\mathrm{,}\end{array}$ (151)

where $\epsilon \in \left(\mathrm{0,1}\right]$ and ${\displaystyle {\int }_0^{\infty }}\left|f\left(t\right)\text{d}t\right|<\infty$. Note that ${\left(W_{\epsilon }^{D^{--}}f\left(x\right)\right)}_{\epsilon }\in \mathcal{G}\left(ℝ\right)$. Thus in order to obtain appropriate extension of the Weyl fractional integral $W^{D^{+}}f\left(x\right)$ 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

${\left({\displaystyle {\int }_{X}s\left(x\right)\text{d}{\mu }_{HC\mathrm{,}\epsilon }^{D^{-}{}^{-}}}\right)}_{\epsilon }=\frac{2{\text{π}}^{D^{-}/2}}{\Gamma \left(D^{-}/2\right)}{\left({\displaystyle {\int }_0^{\infty }\frac{s\left(r\right)}{\epsilon +r^{\left|D^{-}\right|+1}}\text{d}r}\right)}_{\epsilon }={\left(I_{\epsilon }^{D_{-}}\right)}_{\epsilon }\mathrm{,}$ (152)

where $\epsilon \in \left(\mathrm{0,1}\right]$ and $D^{-}\ne \mathrm{0,}-\mathrm{2,}\cdots \mathrm{,}-2n\mathrm{,}\cdots \mathrm{,}n\in \mathbb{N}$ and where ${\left({\mu }_{HC\mathrm{,}\epsilon }^{D^{-}}\right)}_{\epsilon }$ is appropriate generalized Colombeau outer measure. Namely Hausdorff-Colombeau outer measure.

Remark 3.1.3. Note that: if $s\left(0\right)\ne 0$ the quantity ${\left(I_{\epsilon }^{D^{+}\mathrm{,}{D}^{-}}\right)}_{\epsilon }$ takes infinite large value in sense of Colombeau generalized numbers, i.e., ${\left(I_{\epsilon }^{D^{+},D^{-}}\right)}_{\epsilon }{=}_{\widetilde{ℝ}}\tilde{\infty }$, 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):

${\left({\displaystyle {\int }_{X}s\left(x\right)\text{d}{\mu }_{HC\mathrm{,}\epsilon }^{D^{+}\mathrm{,}{D}^{-}}}\right)}_{\epsilon }=\frac{4{\text{π}}^{D^{+}/2}{\text{π}}^{D^{-}/2}}{\Gamma \left(D^{+}/2\right)\Gamma \left(D^{-}/2\right)}{\left({\displaystyle {\int }_0^{\infty }\frac{r^{D^{+}-1}s\left(r\right)}{\epsilon +r^{\left|D^{-}\right|}}\text{d}r}\right)}_{\epsilon }={\left(I_{\epsilon }^{D^{+}\mathrm{,}{D}_{-}}\right)}_{\epsilon }\mathrm{,}$ (153)

where $\epsilon \in \left(\mathrm{0,1}\right]$ and $D^{+}\ge 1D^{-}\ne \mathrm{0,}-\mathrm{2,}\cdots \mathrm{,}-2n\mathrm{,}\cdots \mathrm{,}n\in \mathbb{N}$ and where ${\left({\mu }_{HC\mathrm{,}\epsilon }^{D^{+}\mathrm{,}{D}^{-}}\right)}_{\epsilon }$ is appropriate generalized Colombeau outer measure. Namely Hausdorff-Colombeau outer measure. In Subsection 3.3 we pointed out that there exists Colombeau generalized measure ${\left(\text{d}{\mu }_{HC\mathrm{,}\epsilon }^{D^{+}\mathrm{,}{D}^{-}}\right)}_{\epsilon }$ and therefore Equation (151) gives appropriate extension of the Equation (148) on the negative Hausdorff-Colombeau dimensions.

3.2. Hausdorff Measure and Associated Positive Hausdorff Dimension

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 $F={\left\{E_i\right\}}_{i\in \mathbb{N}}$ of subsets $E_i$ of X, and each positive function ${\zeta }^{+}\left(E\right)$, such that $0\le {\zeta }^{+}\left(E_i\right)\le \infty$ whenever $E_i\in F$, a preliminary measure ${\varphi }_{\delta }^{+}\left(E\right)$ can be constructed corresponding to $0<\delta \le +\infty$, and then a final measure ${\mu }^{+}\left(E\right)$, as follows: for every subset $E\subset X$, the preliminary measure ${\varphi }_{\delta }^{+}\left(E\right)$ is defined by

${\varphi }_{\delta }^{+}\left(E\right)=\underset{{\left\{E_i\right\}}_{i\in \mathbb{N}}}{\inf }\left\{{\displaystyle {\sum }_{i\in \mathbb{N}}{\zeta }^{+}\left(E_i\right)}\mathrm{<mo>\; |\; </mo>}E\subset {\displaystyle {\cup }_{i\in \mathbb{N}}{E}_i}\mathrm{,}diam\left(E_i\right)\le \delta \right\}\mathrm{.}$ (154)

Since ${\varphi }_{{\delta }_1}^{+}\left(E\right)\ge {\varphi }_{{\delta }_2}^{+}\left(E\right)$ for $0<{\delta }_1<{\delta }_2\le +\infty$, the limit

${\mu }^{+}\left(E\right)=\underset{\delta \to 0_{+}}{\lim }{\varphi }_{\delta }^{+}\left(E\right)=\underset{\delta >0}{\sup }{\varphi }_{\delta }^{+}\left(E\right)$ (155)

exists for all $E\subset X$. In this context, ${\mu }^{+}\left(E\right)$ can be called the result of Caratheodory’s construction from ${\zeta }^{+}\left(E\right)$ on F. ${\varphi }_{\delta }^{+}\left(E\right)$ can be referred to as the size $\delta$ approximating positive measure. Let ${\zeta }^{+}\left(E_i\mathrm{,}{d}^{+}\right)$ be for example

${\zeta }^{+}\left(E_i,d^{+}\right)=\Theta \left(d^{+}\right){\left[diam\left(E_i\right)\right]}^{d^{+}},0<\Theta \left(d^{+}\right),$ (156)

for non-empty subsets $E_i\mathrm{,}i\in \mathbb{N}$ of X. Where $\Theta \left(d^{+}\right)$ is some geometrical factor, depends on the geometry of the sets $E_i$, used for covering. When F is the set of all non-empty subsets of X, the resulting measure ${\mu }_H^{+}\left(E\mathrm{,}{d}^{+}\right)$ is called the d⁺-dimensional Hausdorff measure over X; in particular, when F is the set of all (closed or open) balls in X,

$\Theta \left(d^{+}\right)\triangleq \Omega \left(d^{+}\right)={\text{π}}^{\frac{d^{+}}{2}}\left(2^{-d^{+}}\right)\Gamma \left(1+\frac{d^{+}}{2}\right)\mathrm{.}$ (157)

Consider a measurable metric space $\left(X\mathrm{,}{\mu }_H\left(d\right)\right)\mathrm{,}X\subseteq {ℝ}^n\mathrm{,}d\in \left(-\infty \mathrm{,}+\infty \right)$. The elements of X are denoted by $x\mathrm{,}y\mathrm{,}z\mathrm{,}\cdots$, and represented by n-tuples of real numbers $x=\left(x_1,x_2,\cdots ,x_n\right)$

The metric $d\left(x\mathrm{,}y\right)$ is a function $d\mathrm{<mo>\; :\; </mo>}X\times X\to R_{+}$ is defined in n dimensions by

$d\left(x,y\right)={\displaystyle \underset{ij}{\sum }}{\left[{\delta }_{ij}\left(y_i-x_i\right)\left(y_j-x_j\right)\right]}^{1/2}$ (158)

and the diameter of a subset $E\subset X$ is defined by

$diam\left(E\right)=\sup \left\{d\left(x\mathrm{,}y\right)\mathrm{<mo>\; |\; </mo>}x\mathrm{,}y\in E\right\}\mathrm{.}$ (159)

Definition 3.2.1. The Hausdorff measure ${\mu }_H^{+}\left(E\mathrm{,}{D}^{+}\right)$ of a subset $E\subset X$ with the associated Hausdorff positive dimension $D^{+}\in {ℝ}_{+}$ is defined by canonical way

$\begin{array}{l}{\mu }_H^{+}\left(E\mathrm{,}{D}^{+}\right)=\underset{\delta \to 0}{\lim }\left[\underset{{\left\{E_i\right\}}_{i\in \mathbb{N}}}{\inf }\left\{{\displaystyle {\sum }_{i\in \mathbb{N}}{\zeta }^{+}\left(E_i\mathrm{,}{D}^{+}\right)}\mathrm{<mo>\; |\; </mo>}E\subset {\displaystyle {\cup }_i{E}_i}\mathrm{,}\forall i\left(diam\left(E_i\right)<\delta \right)\right\}\right]\mathrm{,}\\ D^{+}\left(E\right)=\sup \left\{d^{+}\in {ℝ}_{+}\mathrm{<mo>\; |\; </mo>}{d}^{+}>\mathrm{0,}{\mu }_H^{+}\left(E\mathrm{,}{d}^{+}\right)=+\infty \right\}\mathrm{.}\end{array}$ (160)

Definition 3.2.2. Remind that a function $f\mathrm{<mo>\; :\; </mo>}X\to ℝ$ defined in a measurable space $\left(X\mathrm{,}\Sigma \mathrm{,}\mu \right)$, is called a simple function if there is a finite disjoint set of sets $\left\{E_1\mathrm{,,}\cdots \mathrm{,}{E}_n\right\}$ of measurable sets and a finite set $\left\{{\alpha }_1\mathrm{,}\cdots \mathrm{,}{\alpha }_n\right\}$ of real numbers such that $f\left(x\right)={\alpha }_i$ if $x\in E_i$ and $f\left(x\right)=0$ if $x\notin E_i$. Thus $f\left(x\right)={\displaystyle {\sum }_{i=1}^n{\alpha }_i{\chi }_{E_i}\left(x\right)}$, where ${\chi }_{E_i}\left(x\right)$ is the characteristic function of $E_i$. A simple function f on a measurable space $\left(X\mathrm{,}\Sigma \mathrm{,}\mu \right)$ is integrable if $\mu \left(E_i\right)<\infty$ for every index i for which ${\alpha }_i\ne 0$. The Lebesgue-Stieltjes integral of f is defined by

${\displaystyle \int f\text{d}\mu }={\displaystyle {\sum }_{i\mathrm{<mo>\; =\; </mo>1}}^n{\alpha }_i\mu \left(E_i\right)}\mathrm{.}$ (161)

A continuous function is a function for which ${\lim }_{x\to y}f\left(x\right)=f\left(y\right)$ whenever ${\lim }_{x\to y}d\left(x,y\right)=0$.

The Lebesgue-Stieltjes integral over continuous functions can be defined as the limit of infinitesimal covering diameter: when ${\left\{E_i\right\}}_{i\in \mathbb{N}}$ is a disjoined covering and $x_i\in E_i$ by definition (3.2.12) one obtains

$\begin{array}{l}{\displaystyle {\int }_{X}f\left(x\right)\text{d}{\mu }_H^{+}\left(x\mathrm{,}{D}^{+}\right)}\\ =\underset{diam\left(E_i\right)\to 0}{\lim }\left[{\displaystyle {\sum }_{{\displaystyle \cup E_i}\mathrm{<mo>\; =\; </mo>}X}f\left(x_i\right)}\underset{\left\{E_{ij}\right\}\text{with}{\displaystyle {\cup }_j}{E}_{ij}\supset E_i}{\inf }{\displaystyle {\sum }_j{\zeta }^{+}\left(E_{ij}\mathrm{,}{D}^{+}\left(E_{ij}\right)\right)}\right]\mathrm{.}\end{array}$ (162)

From now on, X is assumed metrically unbounded, i.e. for every $x\in X$ and $r>0$ there exists a point y such that $d\left(x,y\right)>r$. The standard assumption that $D^{+}$ is uniquely defined in all subsets E of X requires X to be regular (homogeneous, uniform) with respect to the measure, i.e. ${\mu }_H^{+}\left(B_r\left(x\right),D^{+}\right)={\mu }_H^{+}\left(B_r\left(y\right),D^{+}\right)$ for all elements $x\mathrm{,}y\in X$ and (convex) balls $B_r\left(x\right)$ and $B_r\left(y\right)$ of the form $B_{r>0}\left(x\right)=\left\{z|d\left(x,z\right)\le r;x,z\in X\right\}$. In the limit $diam\left(E_i\right)\to 0$, the infimum is satisfied by the requirement that the variation overall coverings ${\left\{E_{ij}\right\}}_{ij\in \mathbb{N}}$ is replaced by one single covering $E_i$, such that ${\displaystyle {\cup }_j{E}_{ij}}\to E_i\ni x_i$. Hence

${\displaystyle {\int }_{X}f\left(x\right)\text{d}{\mu }_H^{+}\left(x\mathrm{,}{D}^{+}\right)}=\underset{diam\left(E_i\right)\to 0}{\lim }{\displaystyle \underset{{\displaystyle \cup E_i}=X}{\sum }}f\left(x_i\right){\zeta }^{+}\left(E_i\mathrm{,}{D}^{+}\right)\mathrm{.}$ (163)

The range of integration X may be parametrized by polar coordinates with $r=d\left(x,0\right)$ and angle $\Omega$. ${\left\{E_{r_i\mathrm{,}{\Omega }_i}\right\}}_{i\in \mathbb{N}}$ can be thought of as spherically symmetric covering around a centre at the origin. In the limit, the function ${\zeta }^{+}\left(E_{r\mathrm{,}\Omega }\mathrm{,}{D}^{+}\right)$ defined by Equation (156) is given by

$\text{d}{\mu }_H^{+}\left(x\mathrm{,}{D}^{+}\right)=\underset{diam\left(E_{r\mathrm{,}\omega }\right)\to 0}{\lim }{\zeta }^{+}\left(E_{r\mathrm{,}\Omega }\mathrm{,}{D}^{+}\right)=\text{d}{\Omega }^{D^{+}-1}{r}^{D^{+}-1}\text{d}r\mathrm{.}$ (164)

Let us assume now for simplification that $f\left(x\right)=f\left(\Vert x\Vert \right)=f\left(r\right)$ and $\underset{r\to \infty }{\lim }f\left(r\right)=0$. The integral over a $D^{+}$ -dimensional metric space X is then given by

${\displaystyle {\int }_{X}f\left(x\right)\text{d}{\mu }_H^{+}\left(x\mathrm{,}{D}^{+}\right)}={\displaystyle {\int }_{X}f\left(x\right){\text{d}}^{D^{+}}x}=\frac{2{\text{π}}^{\frac{D^{+}}{2}}}{\Gamma \left(1+\frac{D^{+}}{2}\right)}{\displaystyle {\int }_0^{\infty }}f\left(r\right)r^{D^{+}-1}\text{d}r\mathrm{.}$ (165)

The integral defined in (163) satisfies the following conditions.

1) Linearity:

$\begin{array}{l}{\displaystyle {\int }_X}\left[a_1{f}_1\left(x\right)+a_2{f}_2\left(x\right)\right]\text{d}{\mu }_H^{+}\left(x\mathrm{,}{D}^{+}\right)\\ =a_1{\displaystyle {\int }_X}{f}_1\left(x\right)\text{d}{\mu }_H^{+}\left(x\mathrm{,}{D}^{+}\right)+a_2{\displaystyle {\int }_X}{f}_2\left(x\right)\text{d}{\mu }_H^{+}\left(x\mathrm{,}{D}^{+}\right)\mathrm{.}\end{array}$ (166)

2) Translational invariance:

${\displaystyle {\int }_X}f\left(x+x_0\right)\text{d}{\mu }_H^{+}\left(x\mathrm{,}{D}^{+}\right)={\displaystyle {\int }_X}f\left(x\right)\text{d}{\mu }_H^{+}\left(x\mathrm{,}{D}^{+}\right)$ (167)

since $\text{d}{\mu }_H^{+}\left(x-x_0\mathrm{,}{D}^{+}\right)=\text{d}{\mu }_H^{+}\left(x\mathrm{,}{D}^{+}\right)$.

3) Scaling property:

${\displaystyle {\int }_X}f\left(ax\right)\text{d}{\mu }_H^{+}\left(x\mathrm{,}{D}^{+}\right)=a^{-D^{+}}{\displaystyle {\int }_X}f\left(x\right)\text{d}{\mu }_H^{+}\left(x\mathrm{,}{D}^{+}\right)$ (168)

since $\text{d}{\mu }_H^{+}\left(x/a\mathrm{,}{D}^{+}\right)=a^{-D^{+}}\text{d}{\mu }_H^{+}\left(x\mathrm{,}{D}^{+}\right)$.

4) The generalized ${\delta }^{D^{+}}\left(x\right)$ function:

The generalized ${\delta }^{D^{+}}\left(x\right)$ function for sets with non-integer Hausdorff dimension exists and can be defined by formula

${\displaystyle {\int }_X}f\left(x\right){\delta }^{D^{+}}\left(x-x_0\right)\text{d}{\mu }_H^{+}\left(x\mathrm{,}{D}^{+}\right)=f\left(x_0\right)\mathrm{.}$ (169)

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 $\text{d}{\mu }_H^{+}\left(x\mathrm{,}{D}^{+}\right)$.

Let us consider example of a space of noninteger positive dimension equipped with the Haar measure. On the closed interval $0\le x\le 1$ there is a scale $0\le \sigma \le 1$ of Cantor dust with the Haar measure equal to $x^{\sigma }$ for any interval $\left(\mathrm{0,}x\right)$ 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 $k+\sigma$, where $k\in \mathbb{Z}$ and $\sigma \in \left(\mathrm{0,1}\right)$ [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.

Let $\Omega$ be an open subset of ${ℝ}^n$, let X be metric space $X\subseteq {ℝ}^n$ and let F be a set $F={\left\{E_i\right\}}_{i\in \mathbb{N}}$ of subsets $E_i$ of X. Let $\zeta \left(E\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\right)$ be a function $\zeta \mathrm{<mo>\; :\; </mo>}F\times \Omega \times \Omega \to ℝ$. Let $C_F^{\infty }\left(\Omega \right)$ be a set of the all functions $\zeta \left(E\mathrm{,}x\right)$ such that $\zeta \left(E\mathrm{,}x\right)\in C^{\infty }\left(\Omega \right)$ whenever $E\in F$. Throughout this paper, for elements of the space $C_F^{\infty }{\left(\Omega \right)}^{\left(\mathrm{0,1}\right]}$ of sequences of smooth functions indexed by $\epsilon \in \left(\mathrm{0,1}\right]$ we shall use the canonical notations ${\left({\zeta }_{\epsilon }\left(E\mathrm{,}x\right)\right)}_{\epsilon }$ and ${\left({\zeta }_{\epsilon }\right)}_{\epsilon }$ so ${\zeta }_{\epsilon }\in C_F^{\infty }\left(\Omega \right)\epsilon \in \left(\mathrm{0,1}\right]$.

Definition 3.3.1. We set ${\mathcal{G}}_F\left(\Omega \right)={\mathcal{E}}_M\left(F\mathrm{,}\Omega \right)\mathrm{<mo>\; /\; </mo>}\mathcal{N}\left(F\mathrm{,}\Omega \right)$, where

$\begin{array}{l}{\mathcal{E}}_M\left(F\mathrm{,}\Omega \right)=\{{\left({\zeta }_{\epsilon }\right)}_{\epsilon }\in C_F^{\infty }{\left(\Omega \right)}^{\left(\mathrm{0,1}\right]}|\forall K\subset \subset \Omega \mathrm{,}\forall \alpha \in {\mathbb{N}}^n\exists p\in \mathbb{N}\text{with}\\ {\sup }_{E\in F\mathrm{<mo>\; ;\; </mo>}x\in K}\left|{\zeta }_{\epsilon }\left(E\mathrm{,}x\right)\right|=O\left({\epsilon }^{-p}\right)\text{as}\epsilon \to 0\}\mathrm{,}\\ \mathcal{N}\left(F\mathrm{,}\Omega \right)-\{{\left({\zeta }_{\epsilon }\right)}_{\epsilon }\in C_F^{\infty }{\left(\Omega \right)}^{\left(\mathrm{0,1}\right]}|\forall K\subset \subset \Omega \mathrm{,}\forall \alpha \in {\mathbb{N}}^n\forall q\in \mathbb{N}\\ {\sup }_{E\in F\mathrm{<mo>\; ;\; </mo>}x\in K}\left|{\zeta }_{\epsilon }\left(E\mathrm{,}x\right)\right|=O\left({\epsilon }^q\right)\text{as}\epsilon \to 0\}\mathrm{.}\end{array}$ (170)

Notice that ${\mathcal{G}}_F\left(\Omega \right)$ is a differential algebra. Equivalence classes of sequences ${\left({\zeta }_{\epsilon }\right)}_{\epsilon }$ will be denoted by $\text{cl}\left[{\left({\zeta }_{\epsilon }\right)}_{\epsilon }\right]$ or simply $\left[{\left({\zeta }_{\epsilon }\right)}_{\epsilon }\right]$.

Definition 3.3.2. We denote by $\widetilde{ℝ}$ the ring of real, Colombeau generalized numbers. Recall that by definition $\widetilde{ℝ}={\mathcal{E}}_M\left(ℝ\right)\mathrm{<mo>\; /\; </mo>}\mathcal{N}\left(ℝ\right)$ [21] , where

$\begin{array}{l}{\mathcal{E}}_M\left(ℝ\right)=\left\{{\left(x_{\epsilon }\right)}_{\epsilon }\in {ℝ}^{\left(\mathrm{0,1}\right]}|\left(\exists \alpha \in ℝ\right)\left(\exists {\epsilon }_0\in \left(\mathrm{0,1}\right]\right)\forall \epsilon \le {\epsilon }_0\left[\left|x_{\epsilon }\right|\le {\epsilon }^{\alpha }\right]\right\}\mathrm{,}\\ \mathcal{N}\left(ℝ\right)=\left\{{\left(x_{\epsilon }\right)}_{\epsilon }\in {ℝ}^{\left(\mathrm{0,1}\right]}|\left(\forall \alpha \in ℝ\right)\left(\exists {\epsilon }_0\in \left(\mathrm{0,1}\right]\right)\forall \epsilon \le {\epsilon }_0\left[\left|x_{\epsilon }\right|\le {\epsilon }^{\alpha }\right]\right\}\mathrm{.}\end{array}$ (171)

Notice that the ring $\widetilde{ℝ}$ arises naturally as the ring of constants of the Colombeau algebras $\mathcal{G}\left(\Omega \right)$. Recall that there exists natural embedding $$ such that for all $r\in ℝ\tilde{r}={\left(r_{\epsilon }\right)}_{\epsilon }$ where $r_{\epsilon }\equiv r$ for all $\epsilon \in \left(\mathrm{0,1}\right]$. We say that r is standard number and abbreviate $r\in ℝ$ for short. The ring $\widetilde{ℝ}$ can be endowed with the structure of a partially ordered ring: for $r\mathrm{,}s\in \widetilde{ℝ}$ $\eta \in {ℝ}_{+}\mathrm{,}\eta \le 1$ we abbreviate $r{\le }_{\widetilde{ℝ}\mathrm{,}\eta }s$ or simply $r{\le }_{\widetilde{ℝ}}s$ if and only if there are representatives ${\left(r_{\epsilon }\right)}_{\epsilon }$ and ${\left(s_{\epsilon }\right)}_{\epsilon }$ with $r_{\epsilon }\le s_{\epsilon }$ for all $\epsilon \in \left(\mathrm{0,}\eta \right]$. Colombeau generalized number $r\in \widetilde{ℝ}$ with representative ${\left(r_{\epsilon }\right)}_{\epsilon }$ we abbreviate $\text{cl}\left[{\left(r_{\epsilon }\right)}_{\epsilon }\right]$.

Definition 3.3.3. 1) Let $\delta \in \widetilde{ℝ}$. We say that $\delta$ is infinite small Colombeau generalized number and abbreviate $\delta {\approx }_{\widetilde{ℝ}}\tilde{0}$ if there exists representative ${\left({\delta }_{\epsilon }\right)}_{\epsilon }$ and some $q\in \mathbb{N}$ such that $\left|{\delta }_{\epsilon }\right|=O\left({\epsilon }^q\right)$ as $\epsilon \to 0$. 2) Let $\Delta \in \widetilde{ℝ}$. We say that $\Delta$ is infinite large Colombeau generalized number and abbreviate $\Delta {=}_{\widetilde{ℝ}}\tilde{\infty }$ if ${\Delta }_{\widetilde{ℝ}}^{-1}{\approx }_{\widetilde{ℝ}}\tilde{0}$. 3) Let ${ℝ}_{\infty }$ be $ℝ\cup \left\{\infty \right\}$ We say that $\Theta \in {\widetilde{ℝ}}_{\infty }$ is infinite Colombeau generalized number and abbreviate $\Theta {=}_{\widetilde{ℝ}}{\infty }_{\widetilde{ℝ}}$ if there exists representative ${\left({\Theta }_{\epsilon }\right)}_{\epsilon }$ where ${\Theta }_{\epsilon }=\infty$ for all $\epsilon \in \left(\mathrm{0,1}\right]$. Here we set ${\mathcal{E}}_M\left({ℝ}_{\infty }\right)={\mathcal{E}}_M\left(ℝ\right)\cup \left\{{\left({\Theta }_{\epsilon }\right)}_{\epsilon }\right\}\mathcal{N}\left({ℝ}_{\infty }\right)=\mathcal{N}\left(ℝ\right)$ and ${\widetilde{ℝ}}_{\infty }={\mathcal{E}}_M\left({ℝ}_{\infty }\right)\mathrm{<mo>\; /\; </mo>}\mathcal{N}\left({ℝ}_{\infty }\right)$.

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 $F={\left\{E_i\right\}}_{i\in \mathbb{N}}$ of subsets $E_i$ of X, and each Colombeau generalized function ${\left({\zeta }_{\epsilon }\left(E\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\right)\right)}_{\epsilon }$, such that: 1) $0\le {\left({\zeta }_{\epsilon }\left(E\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\right)\right)}_{\epsilon }$, 2) ${\left({\zeta }_{\epsilon }\left(E\mathrm{,}\stackrel{⌣}{x}\mathrm{,}\stackrel{⌣}{x}\right)\right)}_{\epsilon }{=}_{\widetilde{ℝ}}\tilde{\infty }$, whenever $E\in F$, a preliminary Colombeau measure ${\left({\varphi }_{\delta }\left(E\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\mathrm{,}\epsilon \right)\right)}_{\epsilon }$ can be constructed corresponding to $0<\delta \le +\infty$, and then a final Colombeau measure ${\left({\mu }_{\epsilon }\left(E\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\right)\right)}_{\epsilon }$, as follows: for every subset $E\subset X$, the preliminary Colombeau measure ${\left({\varphi }_{\delta }\left(E\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\mathrm{,}\epsilon \right)\right)}_{\epsilon }$ is defined by

${\varphi }_{\delta }\left(E\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\mathrm{,}\epsilon \right)=\underset{{\left\{E_i\right\}}_{i\in \mathbb{N}}}{\sup }\left\{{\displaystyle {\sum }_{i\in \mathbb{N}}{\zeta }_{\epsilon }\left(E_i\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\right)}\mathrm{<mo>\; |\; </mo>}E\subset {\displaystyle {\cup }_{i\in \mathbb{N}}{E}_i}\mathrm{,}diam\left(E_i\right)\le \delta \right\}\mathrm{.}$ (172)

Since for all $\epsilon \in \left(\mathrm{0,1}\right]$ : ${\varphi }_{{\delta }_1}^{-}\left(E\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\mathrm{,}\epsilon \right)\ge {\varphi }_{{\delta }_2}^{-}\left(E\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\mathrm{,}\epsilon \right)$ for $0<{\delta }_1<{\delta }_2\le +\infty$, the limit

${\left(\mu \left(E,x,\stackrel{⌣}{x},\epsilon \right)\right)}_{\epsilon }={\left(\underset{\delta \to 0_{+}}{\lim }{\varphi }_{\delta }\left(E,x,\stackrel{⌣}{x},\epsilon \right)\right)}_{\epsilon }={\left(\underset{\delta >0}{\inf }{\varphi }_{\delta }\left(E,x,\stackrel{⌣}{x},\epsilon \right)\right)}_{\epsilon }$ (173)

exists for all $E\subset X$. In this context, ${\left(\mu \left(E\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\mathrm{,}\epsilon \right)\right)}_{\epsilon }$ can be called the result of Caratheodory’s construction from ${\left({\zeta }_{\epsilon }\left(E\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\right)\right)}_{\epsilon }$ on F and ${\left({\varphi }_{\delta }\left(E\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\mathrm{,}\epsilon \right)\right)}_{\epsilon }$ can be referred to as the size $\delta$ approximating Colombeau measure.

Definition 3.3.5. Let ${\left({\zeta }_{\epsilon }\left(E_i\mathrm{,}{D}^{+}\mathrm{,}{D}^{-}\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\right)\right)}_{\epsilon }$ be

${\left({\zeta }_{\epsilon }\left(E_i\mathrm{,}{D}^{+}\mathrm{,}{D}^{-}\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\right)\right)}_{\epsilon }=\{\begin{array}{ll}{\left(\frac{{\Theta }_1\left(D^{+}\right){\Theta }_2\left(D^{-}\right){\left[diam\left(E_i\right)\right]}^{D^{+}}}{{\left[d\left(x\mathrm{,}\stackrel{⌣}{x}\right)\right]}^{\left|D^{-}\right|}+\epsilon }\right)}_{\epsilon }\hfill & \text{if}x\in E_i\hfill \\ 0\hfill & \text{if}x\notin E_i\hfill \end{array}$ (174)

where $\epsilon \in \left(0,1\right],{\Theta }_1\left(D^{+}\right),{\Theta }_2\left(D^{-}\right)>0$. In particular, when F is the set of all (closed or open) balls in X,

${\Theta }_1\left(D^{+}\right)=\frac{2^{-D^{+}}{\text{π}}^{\frac{D^{+}}{2}}}{\Gamma \left(1+\frac{D^{+}}{2}\right)}$ (175)

and

$\begin{array}{l}{\Theta }_2\left(D^{-}\right)=\frac{2^{-D^{-}}{\text{π}}^{\frac{D^{-}}{2}}}{\left|\Gamma \left(1+\frac{D^{-}}{2}\right)\right|},\\ D^{-}\ne -2,-4,-6,\cdots ,-2\left(n+1\right),\cdots \end{array}$ (176)

Definition 3.3.6. The Hausdorff-Colombeau singular measure ${\left({\mu }_H\left(E\mathrm{,}{D}^{+}\mathrm{,}{D}^{-}\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\mathrm{,}\epsilon \right)\right)}_{\epsilon }$ of a subset $E\subset X$ with the associated Hausdorff-Colombeau dimension ${\stackrel{⌣}{D}}^{+}\left(D^{-}\right)\in {ℝ}_{+}\mathrm{,}{D}^{-}\in {ℝ}_{+}$ is defined by

$\begin{array}{l}{\left({\mu }_{HC}\left(E\mathrm{,}{\stackrel{⌣}{D}}^{+}\mathrm{,}{D}^{-}\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\mathrm{,}\epsilon \right)\right)}_{\epsilon }\\ ={\left(\underset{\delta \to 0}{\lim }\left[\underset{{\left\{E_i\right\}}_{i\in \mathbb{N}}}{\sup }\left\{{\displaystyle {\sum }_{i\in \mathbb{N}}{\left({\zeta }_{\epsilon }\left(E_i\mathrm{,}{\stackrel{⌣}{D}}^{+}\mathrm{,}{D}^{-}\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\right)\right)}_{\epsilon }}\mathrm{<mo>\; |\; </mo>}E\subset {\displaystyle {\cup }_i{E}_i}\mathrm{,}\forall i\left(diam\left(E_i\right)<\delta \right)\right\}\right]\right)}_{\epsilon }\mathrm{,}\\ {\stackrel{⌣}{D}}^{+}=\sup \left\{D^{+}>\mathrm{0\; <mo>\; |\; </mo>}{\left({\mu }_{HC}\left(E\mathrm{,}{D}^{+}\mathrm{,}{D}^{-}\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\mathrm{,}\epsilon \right)\right)}_{\epsilon }={\infty }_{\widetilde{ℝ}}\right\}\mathrm{,}\end{array}$ (177)

The Colombeau-Lebesgue-Stieltjes integral over continuous functions $f\mathrm{<mo>\; :\; </mo>}X\to ℝ$ can be evaluated similarly as in Subsection III.3, (but using the limit in sense of Colombeau generalized functions) of infinitesimal covering diameter when ${\left\{E_i\right\}}_{i\in \mathbb{N}}$ is a disjoined covering and $x_i\in E_i$ :

$\begin{array}{l}{\left({\displaystyle {\int }_X}f\left(x\right)\text{d}{\mu }_{HC}\left(E\mathrm{,}{D}^{+}\mathrm{,}{D}^{-}\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\mathrm{,}\epsilon \right)\right)}_{\epsilon }\\ ={\left(\underset{diam\left(E_i\right)\to 0}{\lim }\left[{\displaystyle {\sum }_{{\displaystyle \cup E_i}=X}f\left(x_i\right)}\underset{\left\{E_{ij}\right\}\text{with}{\displaystyle {\cup }_j{E}_{ij}}\supset E_i}{\inf }{\displaystyle {\sum }_j{\zeta }_{\epsilon }\left(E_i\mathrm{,}{D}^{+}\mathrm{,}{D}^{-}\mathrm{,}{x}_i\mathrm{,}\stackrel{⌣}{x}\right)}\right]\right)}_{\epsilon }\mathrm{.}\end{array}$ (178)

We assume now that X is metrically unbounded, i.e. for every $x\in X$ and $r>0$ there exists a point y such that $d\left(x,y\right)>r$. The standard assumption that ${\stackrel{⌣}{D}}^{+}$ and ${\stackrel{⌣}{D}}^{-}$ is uniquely defined in all subsets E of X requires X to be regular (homogeneous, uniform) with respect to the measure, i.e. ${\left({\mu }_{HC}^{-}\left(B_r\left(\stackrel{⌣}{x}\right)\mathrm{,}{\stackrel{⌣}{D}}^{+}\mathrm{,}{\stackrel{⌣}{D}}^{-}\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\mathrm{,}\epsilon \right)\right)}_{\epsilon }={\left({\mu }_{HC}^{-}\left(B_r\left(\stackrel{⌣}{y}\right)\mathrm{,}{\stackrel{⌣}{D}}^{+}\mathrm{,}{\stackrel{⌣}{D}}^{-}\mathrm{,}{x}^{\prime }\mathrm{,}\stackrel{⌣}{y}\mathrm{,}\epsilon \right)\right)}_{\epsilon }$, where $d\left(x\mathrm{,}\stackrel{⌣}{x}\right)=d\left(x^{\prime }\mathrm{,}\stackrel{⌣}{y}\right)$ for all elements $\stackrel{⌣}{x}\mathrm{,}\stackrel{⌣}{y}\mathrm{,}x\mathrm{,}{x}^{\prime }\in X$ and convex balls $B_r\left(\stackrel{⌣}{x}\right)$ and $B_r\left(\stackrel{⌣}{y}\right)$ of the form $B_r\left(\stackrel{⌣}{x}\right)=\left\{z\mathrm{<mo>\; |\; </mo>}d\left(\stackrel{⌣}{x}\mathrm{,}z\right)\le r\mathrm{<mo>\; ;\; </mo>}\stackrel{⌣}{x}\mathrm{,}z\in X\right\}$ and $B_r\left(\stackrel{⌣}{y}\right)=\left\{z\mathrm{<mo>\; |\; </mo>}d\left(\stackrel{⌣}{y}\mathrm{,}z\right)\le r\mathrm{<mo>\; ;\; </mo>}\stackrel{⌣}{y}\mathrm{,}z\in X\right\}$. In the limit $diam\left(E_i\right)\to 0$, the infimum is satisfied by the requirement that the variation over all coverings ${\left\{E_{ij}\right\}}_{ij\in \mathbb{N}}$ is replaced by one single covering $E_i$, such that ${\displaystyle {\cup }_j{E}_{ij}}\to E_i\ni x_i$. Therefore

$\begin{array}{l}{\left({\displaystyle {\int }_X}f\left(x\right)\text{d}{\mu }_{HC}\left(E\mathrm{,}{\stackrel{⌣}{D}}^{+}\mathrm{,}{\stackrel{⌣}{D}}^{-}\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\mathrm{,}\epsilon \right)\right)}_{\epsilon }\\ ={\left(\underset{diam\left(E_i\right)\to 0}{\lim }{\displaystyle {\sum }_{{\displaystyle \cup E_i}=X}f\left(x_i\right){\zeta }_{\epsilon }\left(E_i\mathrm{,}{\stackrel{⌣}{D}}^{+}\mathrm{,}{\stackrel{⌣}{D}}^{-}\mathrm{,}{x}_i\mathrm{,}\stackrel{⌣}{x}\right)}\right)}_{\epsilon }\mathrm{.}\end{array}$ (179)

Assume that $f\left(x\right)=f\left(r\right),r=\Vert r\Vert$. The range of integration X may be parametrized by polar coordinates with $r=d\left(x,0\right)$ and angle $\omega$. $\left\{E_{r_i\mathrm{,}{\omega }_i}\right\}$ can be thought of as spherically symmetric covering around a centre at the origin. Thus

$\begin{array}{l}{\left({\displaystyle {\int }_X}f\left(r\right)\text{d}{\mu }_{HC}\left(E\mathrm{,}{\stackrel{⌣}{D}}^{+}\mathrm{,}{\stackrel{⌣}{D}}^{-}\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\mathrm{,}\epsilon \right)\right)}_{\epsilon }\\ ={\left(\underset{diam\left(E_i\right)\to 0}{\lim }{\displaystyle {\sum }_{{\displaystyle \cup E_i}=X}f\left(r_i\right){\zeta }_{\epsilon }\left(E_i\mathrm{,}{\stackrel{⌣}{D}}^{+}\mathrm{,}{\stackrel{⌣}{D}}^{-}\mathrm{,}{x}_i\mathrm{,}\stackrel{⌣}{x}\right)}\right)}_{\epsilon }\mathrm{.}\end{array}$ (180)

Notice that the metric set $X\subset {ℝ}^n$ can be tesselated into regular polyhedra; in particular it is always possible to divide ${ℝ}^n$ into parallelepipeds of the form

${\Pi }_{i_1,\cdots ,i_n}=\left\{x=\left(x_1,\cdots ,x_n\right)\in X|x_j=\left(i_j-1\right)\Delta x_j+{\gamma }_j,0\le {\gamma }_j\le \Delta x_j,j=1,\cdots ,n\right\}.$ (181)

For $n=2$ the polyhedra ${\Pi }_{i_1\mathrm{,}{i}_2}$ is shown in Figure 6. Since X is uniform

$\begin{array}{c}{\left(\text{d}{\mu }_{HC}\left(x\mathrm{,}{\stackrel{⌣}{D}}^{+}\mathrm{,}{\stackrel{⌣}{D}}^{-}\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\mathrm{,}\epsilon \right)\right)}_{\epsilon }={\left(\underset{diam\left({\Pi }_{i_1\mathrm{,}\cdots \mathrm{,}{i}_n}\right)}{\lim }{\zeta }_{\epsilon }\left({\Pi }_{i_1\mathrm{,}\cdots \mathrm{,}{i}_n}\mathrm{,}{\stackrel{⌣}{D}}^{+}\mathrm{,}{\stackrel{⌣}{D}}^{-}\mathrm{,}x\mathrm{,}\stackrel{⌣}{x}\right)\right)}_{\epsilon }\\ ={\left(\underset{diam\left({\Pi }_{i_1\mathrm{,}\cdots \mathrm{,}{i}_n}\right)}{\lim }{\displaystyle {\prod }_{j=1}^n{\left(\frac{\Delta x_j}{{\left|x_j-{\stackrel{⌣}{x}}_j\right|}^{\left|{\stackrel{⌣}{D}}^{-}\right|}+\epsilon }\right)}^{\frac{{\stackrel{⌣}{D}}^{+}}{n}}}\right)}_{\epsilon }\\ \triangleq {\left({\displaystyle {\prod }_{j=1}^n\frac{{\text{d}}^{\frac{{\stackrel{⌣}{D}}^{+}}{n}}{x}_j}{{\left({\left|x_j-{\stackrel{⌣}{x}}_j\right|}^{\left|{\stackrel{⌣}{D}}^{-}\right|}+\epsilon \right)}^{\frac{{\stackrel{⌣}{D}}^{+}}{n}}}}\right)}_{\epsilon }\mathrm{.}\end{array}$ (182)

Notice that the range of integration X may also be parametrized by polar coordinates with $r=d\left(x,0\right)$ and angle $\Omega$. $E_{r\mathrm{,}\Omega }$ can be thought of as spherically symmetric covering around a centre at the origin (see Figure 7 for the two-dimensional case). In the limit, the Colombeau generalized function ${\left({\zeta }_{\epsilon }\left(E_{r\mathrm{,}\Omega }\mathrm{,}{\stackrel{⌣}{D}}^{+}\mathrm{,}{\stackrel{⌣}{D}}^{-}\mathrm{,}r\mathrm{,0}\right)\right)}_{\epsilon }$ is given by

$\begin{array}{l}{\left(\text{d}{\mu }_{HC}\left(r\mathrm{,}\Omega \mathrm{,}{\stackrel{⌣}{D}}^{+}\mathrm{,}{\stackrel{⌣}{D}}^{-}\mathrm{,}\epsilon \right)\right)}_{\epsilon }\\ ={\left(\underset{diam\left({\Pi }_{i_1\mathrm{,}\cdots \mathrm{,}{i}_n}\right)}{\lim }{\zeta }_{\epsilon }\left(E_{r\mathrm{,}\Omega }\mathrm{,}{\stackrel{⌣}{D}}^{+}\mathrm{,}{\stackrel{⌣}{D}}^{-}\mathrm{,}\left\{r\mathrm{,}\Omega \right\}\mathrm{,0}\right)\right)}_{\epsilon }\triangleq \frac{\text{d}{\Omega }^{{\stackrel{⌣}{D}}^{+}-1}{r}^{{\stackrel{⌣}{D}}^{+}-1}\text{d}r}{{\left(r^{\left|{\stackrel{⌣}{D}}^{-}\right|}+\epsilon \right)}_{\epsilon }}\end{array}$ (183)

Figure 6. The polyhedra covering for $n=2$.

Figure 7. The spherical covering $E_{r\mathrm{,}\Omega }$.

When $f\left(x\right)$ is symmetric with respect to some centre $\stackrel{⌣}{x}\in X$, i.e. $f\left(x\right)$ = constant for all x satisfying $d\left(x\mathrm{,}\stackrel{⌣}{x}\right)=r$ for arbitrary values of r, then change of the variable

$x\to z=x-\stackrel{⌣}{x}$ (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 presuming). The integral over metric space X is then given by formula

$\begin{array}{l}{\left({\displaystyle {\int }_X}f\left(x\right)\text{d}{\mu }_{HC}\left(E,{\stackrel{⌣}{D}}^{+},{\stackrel{⌣}{D}}^{-},x,\stackrel{⌣}{x},\epsilon \right)\right)}_{\epsilon }\\ =\frac{4{\text{π}}^{D^{+}/2}{\text{π}}^{D^{-}/2}}{\Gamma \left(D^{+}/2\right)\Gamma \left(D^{-}/2\right)}{\left({\displaystyle {\int }_0^{\infty }\frac{r^{D^{+}-1}f\left(r\right)}{\epsilon +r^{\left|D^{-}\right|}}\text{d}r}\right)}_{\epsilon }.\end{array}$ (185)

3.4. Main Properties of the Hausdorff-Colombeau Metric Measures with Associated Negative Hausdorff-Colombeau Dimensions

Definition 3.4.1. An outer Colombeau metric measure on a set $X\subseteq {ℝ}^n$ is a Colombeau generalized function $\left[{\left({\varphi }_{\epsilon }\left(E\right)\right)}_{\epsilon }\right]\in {\mathcal{G}}_F\left(\Omega \right)$ (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

$\left[{\left({\varphi }_{\epsilon }(\varnothing )\right)}_{\epsilon }\right]=0.$ (186)

2) Monotonicity: For any two subsets A and B of X

$A\subseteq B\left[{\left({\varphi }_{\epsilon }\left(A\right)\right)}_{\epsilon }\right]{\le }_{\widetilde{ℝ}}\left[{\left({\varphi }_{\epsilon }\left(B\right)\right)}_{\epsilon }\right]\mathrm{.}$ (187)

3) Countable subadditivity: For any sequence $\left\{A_j\right\}$ of subsets of X pairwise disjoint or not

$\left[{\left({\varphi }_{\epsilon }\left({\displaystyle {\cup }_{j=1}^{\infty }{A}_j}\right)\right)}_{\epsilon }\right]{\le }_{\widetilde{ℝ}}\left[{\left({\displaystyle {\sum }_{j=1}^{\infty }{\varphi }_{\epsilon }\left(A_j\right)}\right)}_{\epsilon }\right]\mathrm{.}$ (188)

4) Whenever $d\left(A\mathrm{,}B\right)=\inf \left\{d_n\left(x\mathrm{,}y\right)\mathrm{<mo>\; |\; </mo>}x\in A\mathrm{,}y\in B\right\}>0$

$\left[{\left({\varphi }_{\epsilon }\left(A\cup B\right)\right)}_{\epsilon }\right]=\left[{\left({\varphi }_{\epsilon }\left(A\right)\right)}_{\epsilon }\right]+\left[{\left({\varphi }_{\epsilon }\left(B\right)\right)}_{\epsilon }\right]\mathrm{,}$ (189)

where $d_n\left(x\mathrm{,}y\right)$ is the usual Euclidean metric: $d_n\left(x\mathrm{,}y\right)=\sqrt{{\displaystyle \sum }{\left(x_i-y_i\right)}^2}$.

Definition 3.4.2. We say that outer Colombeau metric measure ${\left({\mu }_{\epsilon }\right)}_{\epsilon }\mathrm{,}\epsilon \in \left(\mathrm{0,1}\right]$ is a Colombeau measure on σ-algebra of subests of $X\subseteq {ℝ}^n$ if ${\left({\mu }_{\epsilon }\right)}_{\epsilon }$ satisfies the following property:

$\left[{\left({\varphi }_{\epsilon }\left({\displaystyle {\cup }_{j=1}^{\infty }{A}_j}\right)\right)}_{\epsilon }\right]=\left[{\left({\displaystyle {\sum }_{j=1}^{\infty }{\varphi }_{\epsilon }\left(A_j\right)}\right)}_{\epsilon }\right]\mathrm{.}$ (190)

Definition 3.4.3. If U is any non-empty subset of n-dimensional Euclidean space, ${ℝ}^n$, the diamater $\left|U\right|$ of U is defined as

$\left|U\right|=\sup \left\{d\left(x,y\right)|x,y\in U\right\}$ (191)

If $F\subseteq {ℝ}^n$, and a collection ${\left\{U_i\right\}}_{i\in \mathbb{N}}$ satisfies the following conditions:

1) $\left|U_i\right|<\delta$ for all $i\in \mathbb{N}$, 2) $F\subseteq {\displaystyle {\cup }_{i\in \mathbb{N}}{U}_i}$, then we say the collection ${\left\{U_i\right\}}_{i\in \mathbb{N}}$ is a δ-cover of F.

Definition 3.4.4. If $F\subseteq {ℝ}^n$ and $s,q,\delta >0$, we define Hausdorff-Colombeau content:

${\left(H_{\delta }^{s\mathrm{,}q}\left(F\mathrm{,}\epsilon \right)\right)}_{\epsilon }={\left(\inf \left\{{\displaystyle {\sum }_{i=1}^{\infty }\frac{{\left|U_i\right|}^s}{{\Vert x_i\Vert }^q+\epsilon }}\right\}\right)}_{\epsilon }$ (192)

where the infimum is taken over all δ-covers of F and where $x_i=\left(x_{i,1},\cdots ,x_{i,n}\right)\in U_i$ for all $i\in \mathbb{N}$ and $\Vert x\Vert$ is the usual Euclidean norm: $\Vert x\Vert =\sqrt{{\displaystyle {\sum }_{j=1}^n}{x}_j^2}$.

Note that for $0<{\delta }_1<{\delta }_2<1,\epsilon \in \left(0,1\right]$ we have

$H_{{\delta }_1}^{s\mathrm{,}q}\left(F\mathrm{,}\epsilon \right)\ge H_{{\delta }_2}^{s\mathrm{,}q}\left(F\mathrm{,}\epsilon \right)$ (193)

since any ${\delta }_1$ cover of F is also a ${\delta }_2$ cover of F, i.e. $H_{{\delta }_1}^{s\mathrm{,}q}\left(F\mathrm{,}\epsilon \right)$ is increasing as $\delta$ decreases.

Definition 3.4.5. We define the $\left(s\mathrm{,}q\right)$ -dimensional Hausdorff-Colombeau (outer) measure as:

${\left(H^{s,q}\left(F,\epsilon \right)\right)}_{\epsilon }={\left(\underset{\delta \to 0}{\lim }{H}_{\delta }^{s,q}\left(F,\epsilon \right)\right)}_{\epsilon }.$ (194)

Theorem 3.4.1. For any δ-cover, ${\left\{U_i\right\}}_{i\in \mathbb{N}}$ of F, and for any $\epsilon \in \left(\mathrm{0,1}\right]t>s$ :

$H^{t\mathrm{,}q}\left(F\mathrm{,}\epsilon \right)\le {\delta }^{t-s}{H}^{s\mathrm{,}q}\left(F\mathrm{,}\epsilon \right)\mathrm{.}$ (195)

Proof. Consider any δ-cover ${\left\{U_i\right\}}_{i\in \mathbb{N}}$ of F. Then each ${\left|U_i\right|}^{t-s}\le {\delta }^{t-s}$ since $\left|U_i\right|\le \delta$, so:

${\left|U_i\right|}^t={\left|U_i\right|}^{t-s}{\left|U_i\right|}^s\le {\delta }^{t-s}{\left|U_i\right|}^s.$ (196)

From (196) it follows that

$\frac{{\left|U_i\right|}^t}{{\Vert x_i\Vert }^q+\epsilon }\le \frac{{\delta }^{t-s}{\left|U_i\right|}^s}{{\Vert x_i\Vert }^q+\epsilon }$ (197)

and summing (196) over all $i\in \mathbb{N}$ we obtain

${\displaystyle {\sum }_{i=1}^{\infty }\frac{{\left|U_i\right|}^t}{{\Vert x_i\Vert }^q+\epsilon }}\le {\delta }^{t-s}{\displaystyle {\sum }_{i=1}^{\infty }\frac{{\left|U_i\right|}^s}{{\Vert x_i\Vert }^q+\epsilon }}.$ (198)

Thus (195) follows by taking the infimum.

Theorem 3.4.2. 1) If ${\left(H^{s,q}\left(F,\epsilon \right)\right)}_{\epsilon }{<}_{\widetilde{ℝ}}{\infty }_{\widetilde{ℝ}}$, and if $t>s$, then ${\left(H^{t,q}\left(F,\epsilon \right)\right)}_{\epsilon }=0_{\widetilde{ℝ}}$.

2) If $0_{\widetilde{ℝ}}{<}_{\widetilde{ℝ}}{\left(H^{s,q}\left(F,\epsilon \right)\right)}_{\epsilon }$, and if $t<s$, then ${\left(H^{t,q}\left(F,\epsilon \right)\right)}_{\epsilon }={\infty }_{\widetilde{ℝ}}$.

Proof. 1) The result follows from (195) after taking limits, since $\forall \epsilon \in \left(\mathrm{0,1}\right]$ by definitions follows that $H^{s,q}\left(F,\epsilon \right)<\infty$.

2) From (3.4.10) $\forall \epsilon \in \left(0,1\right],\forall \delta >0$ follows that

$\frac{1}{{\delta }^{s-t}}{H}^{s\mathrm{,}q}\left(F\mathrm{,}\epsilon \right)\le H^{t\mathrm{,}q}\left(F\mathrm{,}\epsilon \right)\mathrm{.}$ (199)

After taking limit $\delta \to 0$, we obtain $H^{t,q}\left(F,\epsilon \right)=\infty$, since ${\lim }_{\delta \to 0}{\left({\delta }^{s-t}\right)}^{-1}=\infty$ and ${\lim }_{\delta \to 0}{H}_{\delta }^{s,q}\left(F,\epsilon \right)=H^{s,q}\left(F,\epsilon \right)>0$.

Definition 3.4.6. We define now the Hausdorff-Colombeau dimension ${\dim }_{HC}\left(F\mathrm{,}q\right)$ of a set F (relative to $q>0$ ) as

$\begin{array}{c}{\dim }_{HC}\left(F\mathrm{,}q\right)=\sup \left\{s=s\left(q\right)\mathrm{<mo>\; |\; </mo>}{\left(H^{s\mathrm{,}q}\left(F\mathrm{,}\epsilon \right)\right)}_{\epsilon }={\infty }_{\widetilde{ℝ}}\right\}\\ =\inf \left\{s\mathrm{<mo>\; =\; </mo>}s\left(q\right)\mathrm{<mo>\; |\; </mo>}{\left(H^{s\mathrm{,}q}\left(F\mathrm{,}\epsilon \right)\right)}_{\epsilon }=0_{\widetilde{ℝ}}\right\}\mathrm{.}\end{array}$ (200)

Remark 3.4.1. From theorem 3.4.2 it follows that for any fixed $q=\stackrel{⌣}{q}$ :

${\left(H^{s,\stackrel{⌣}{q}}\left(F,\epsilon \right)\right)}_{\epsilon }=0_{\widetilde{ℝ}}$ or ${\left(H^{s,\stackrel{⌣}{q}}\left(F,\epsilon \right)\right)}_{\epsilon }={\infty }_{\widetilde{ℝ}}$ everywhere except at a unique value s, where this value may be finite. As a function of s, $H^{s\mathrm{,}\stackrel{⌣}{q}}\left(F\mathrm{,}\epsilon \right)$ is decreasing function. Therefore, the graph of $H^{s\mathrm{,}\stackrel{⌣}{q}}\left(F\mathrm{,}\epsilon \right)$ will have a unique value where it jumps from $\infty$ to 0.

Remark 3.4.2. Note that the graph of ${\left(H^{s\mathrm{,}\stackrel{⌣}{q}}\left(F\mathrm{,}\epsilon \right)\right)}_{\epsilon }$ for a fixed $q=\stackrel{⌣}{q}$ is

${\left(H^{s\mathrm{,}\stackrel{⌣}{q}}\left(F\mathrm{,}\epsilon \right)\right)}_{\epsilon }=\{\begin{array}{ll}{\infty }_{\widetilde{ℝ}}\hfill & \text{if}s>{\dim }_{HC}\left(F\mathrm{,}\stackrel{⌣}{q}\right)\hfill \\ 0_{\widetilde{ℝ}}\hfill & \text{if}s>{\dim }_{HC}\left(F\mathrm{,}\stackrel{⌣}{q}\right)\hfill \\ \text{undetermined}\hfill & \text{if}s={\dim }_{HC}\left(F\mathrm{,}\stackrel{⌣}{q}\right)\hfill \end{array}$ (201)

Definition 3.4.7. We say that fractal $F\subseteq {ℝ}^n$ has a negative dimension relative to $q>0$

if ${\dim }_{HC}\left(F\mathrm{,}q\right)-q<0$.

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] . Quantum mechanics in negative dimensions developed in papers [32] [33] Scalar quantum field theory and quantum gravity in spacetime with Hausdorff-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 superrenormalizable 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 extended fractional calculus and the definition of a fractional action. Note that below we use canonical isotropic scaling such that:

$\left[x^{\mu }\right]=-\mathrm{1,}\mu =\mathrm{0,1,}\cdots \mathrm{,}{D}_t-1$ (202)

while replacing the standard measure with a nontrivial Colombeau-Stieltjes measure,

$\begin{array}{l}{\text{d}}^{D_{\text{t}}}x\to {\text{d}}^{D_{\text{f}}}x={\left(\text{d}\eta \left(x\mathrm{,}\epsilon \right)\right)}_{\epsilon }\mathrm{,}\\ \left[\eta \right]=D_{\text{t}}\cdot \alpha \mathrm{,}\alpha \in \left[\mathrm{1,}-\infty \right)\mathrm{.}\end{array}$ (203)

Here $D_{\text{t}}$ is the topological (positive integer) dimension of embedding spacetime and $\alpha$ is a parameter. Any Colombeau integrals on net multifractals can be approximated by the left-sided Colombeau-Riemann--Liouville complex milti-fractional integral of a function $L\left(t\right)$ :

$\begin{array}{l}{\left({\displaystyle {\int }_0^{\bar{t}}}\text{d}\eta \left(x\mathrm{,}\epsilon \right)L\left(t\right)\right)}_{\epsilon }\propto {\left(I_{\bar{t}\mathrm{,}\epsilon }^{\left\{z_i\left(\bar{t}\right)\right\}}\right)}_{\epsilon }\triangleq {\left({\displaystyle {\sum }_{i=1}^m{\displaystyle {\int }_{\epsilon }^{\bar{t}}}\frac{{\left[\left(\bar{t}-t\right)+i\epsilon \right]}^{z_i\left(\bar{t}\right)-1}}{\Gamma \left(z_i\left(\bar{t}\right)\right)}L\left(t\right)\text{d}t}\right)}_{\epsilon }\mathrm{,}\\ {\left(\eta \left(t\mathrm{,}\epsilon \right)\right)}_{\epsilon }={\left(\frac{{\bar{t}}^{z_i\left(\bar{t}\right)}-{\left[\left(\bar{t}-t\right)+i\epsilon \right]}^{z_i\left(\bar{t}\right)}}{\Gamma \left(z_i\left(\bar{t}\right)+1\right)}\right)}_{\epsilon }\mathrm{,}\end{array}$ (204)

where $\epsilon \in \left(\mathrm{0,1}\right]\bar{t}$ is fixed and the order $z\left(\bar{t}\right)$ is (related to) the complex Hausdorff-Colombeau dimensions of the set. In particular if $z_i\in ℂ\mathrm{,}i=\mathrm{1,2,}\cdots \mathrm{,}m$ 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 $L(t)$

$\begin{array}{l}{\left({\displaystyle {\int }_0^{\bar{t}}}\text{d}\eta \left(x\mathrm{,}\epsilon \right)L\left(t\right)\right)}_{\epsilon }\propto {\left(I_{\bar{t}\mathrm{,}\epsilon }^{{\left\{z_i\right\}}_{i=1}^m}\right)}_{\epsilon }\\ ={\displaystyle {\sum }_{i=1}^m{\left(I_{\bar{t}\mathrm{,}\epsilon }^{z_i}\right)}_{\epsilon }}\triangleq {\displaystyle {\sum }_{i=1}^m{\left(\frac{1}{\Gamma \left(z_i\right)}{\displaystyle {\int }_{\epsilon }^{\bar{t}}}\text{d}{\left[\left(\bar{t}-t\right)+i\epsilon \right]}^{z_i-1}L\left(t\right)\right)}_{\epsilon }}\mathrm{,}\\ {\left(\eta \left(t\mathrm{,}\epsilon \right)\right)}_{\epsilon }={\displaystyle {\sum }_{i=1}^m{\left(\frac{{\bar{t}}^{z_i}-{\left[\left(\bar{t}-t\right)+i\epsilon \right]}^{z_i}}{\Gamma \left(z_i+1\right)}\right)}_{\epsilon }}\mathrm{.}\end{array}$ (205)

Note that a change of variables $t\to \bar{t}-t$ transforms Equation (205) into the form

${\left({\displaystyle {\int }_0^{\bar{t}}}\text{d}\eta \left(x\mathrm{,}\epsilon \right)L\left(t\right)\right)}_{\epsilon }={\displaystyle {\sum }_{i=1}^m{\left({\displaystyle {\int }_0^{\bar{t}}}\text{d}t\frac{{\left[t+i\epsilon \right]}^{z_i-1}}{\Gamma \left(z\left(\bar{t}\right)\right)}L\left(\bar{t}-t\right)\right)}_{\epsilon }}\mathrm{.}$ (206)

The Colombeau-Riemann-Liouville multifractional integral (206) can be mapped onto a Colombeau-Weyl multifractional integral in the formal limit $\bar{t}\to +\infty$. We assume otherwise, so that there exists ${\lim }_{\bar{t}\to +\infty }z\left(\bar{t}\right)$ and ${\lim }_{\bar{t}\to +\infty }L\left(\bar{t}-t\right)=L\left[q\left(t\right)\mathrm{,}\stackrel{\dot{}}{q}\left(t\right)\right]$. In particular if $z\in ℂ$ is a complex parameter a change of variables $t\to \bar{t}-t$ transforms Equation (206) into the form

${\displaystyle {\sum }_{i=1}^m{\left(I_{\bar{t}\mathrm{,}\epsilon }^{z_i}\right)}_{\epsilon }}={\displaystyle {\sum }_{i=1}^m{\left({\displaystyle {\int }_{\epsilon }^{\bar{t}}}\text{d}t\frac{{\left[t+i\epsilon \right]}^{z_i-1}}{\Gamma \left(z_i\right)}L\left[q\left(t\right)\mathrm{,}\stackrel{\dot{}}{q}\left(t\right)\right]\right)}_{\epsilon }}\mathrm{.}$ (207)

This form will be the most convenient for defining a Colombeau-Stieltjes field theory action. In $D_{\text{t}}$ dimensions, we consider now the action

${\left(S_{\epsilon }\right)}_{\epsilon }={\left({\displaystyle {\int }_M}\text{d}\eta \left(x\mathrm{,}\epsilon \right)L\left[{\phi }_{\epsilon }\left(x\right)\mathrm{,}{\partial }_{\mu }{\phi }_{\epsilon }\left(x\right)\right]\right)}_{\epsilon }\mathrm{,}$ (208)

where $L\left[\phi \mathrm{,}{\partial }_{\mu }\phi \right]$ is the Lagrangian density of the scalar field ${\left({\phi }_{\epsilon }\left(x\right)\right)}_{\epsilon }$ and where

${\left(\text{d}\eta \left(x\mathrm{,}\epsilon \right)\right)}_{\epsilon }={\displaystyle {\sum }_{i=1}^m{\displaystyle {\prod }_{\mu =0}^{D_{\text{t}}-1}{\left(f_{\mu \mathrm{,}i}\left(x\mathrm{,}\epsilon \right)\right)}_{\epsilon }\text{d}{x}^{\mu }}}\mathrm{,}{\left(f_{\mu \mathrm{,}i}\left(x\mathrm{,}\epsilon \right)\right)}_{\epsilon }\mathrm{<mo>\; :\; </mo>}M\to \widetilde{ℝ}\mathrm{,}$ (209)

is some Colombeau-Stieltjes measure. We denote with pair $\left(M\mathrm{,}{\left(\text{d}\eta \left(x\mathrm{,}\epsilon \right)\right)}_{\epsilon }\right)$ the metric spacetime M equipped with Colombeau-Stieltjes measure ${\left(\text{d}\eta \left(x\mathrm{,}\epsilon \right)\right)}_{\epsilon }$. The former can be taken to be the canonical scalar field Lagrangian,

${\left(L\left[{\phi }_{\epsilon }\left(x\right)\mathrm{,}{\partial }_{\mu }{\phi }_{\epsilon }\left(x\right)\right]\right)}_{\epsilon }=-\frac{1}{2}{\left({\partial }_{\mu }{\phi }_{\epsilon }{\partial }^{\mu }{\phi }_{\epsilon }\right)}_{\epsilon }-{\left(V\left({\phi }_{\epsilon }\right)\right)}_{\epsilon }\mathrm{,}$ (210)

where $V\left(\phi \right)$ is a potential and contraction of Lorentz indices is done via the Minkowski metric ${\eta }_{\mu \nu }={\left(-\mathrm{,}+\mathrm{,}\cdots ,+\right)}_{\mu \nu }$. As for the Colombeau-Stieltjes measure, we make the multifractal spacetime isotropic choice

${\left(f_{\left(\mu ,i\right),\epsilon }\right)}_{\epsilon }={\left(f_{i,\epsilon }\right)}_{\epsilon },\mu =1,\cdots ,D_{\text{t}}-1;i=1,\cdots ,m.$ (211)

Hence the scalar field action (208) reads

$\begin{array}{c}{\left(S_{\epsilon }\right)}_{\epsilon }={\left({\displaystyle {\int }_M}\text{d}\eta \left(x\mathrm{,}\epsilon \right)\left[{\phi }_{\epsilon }\left(x\right)\mathrm{,}{\partial }_{\mu }{\phi }_{\epsilon }\left(x\right)\right]\right)}_{\epsilon }\\ ={\displaystyle {\sum }_{j=1}^m{\left({\displaystyle \int {\text{d}}^{D_{\text{t}}}x{v}_{\epsilon \mathrm{,}j}\left(x\right)}\left[\frac{1}{2}{\partial }_{\mu }{\phi }_{\epsilon }{\partial }^{\mu }{\phi }_{\epsilon }+V\left({\phi }_{\epsilon }\right)\right]\right)}_{\epsilon }}\mathrm{,}\end{array}$ (212)

where ${\left(v_{\epsilon }\left(x\right)\right)}_{\epsilon }$ is a coordinate-dependent

Lorentz scalar

${\left(v_{\epsilon \mathrm{,}j}\left(x\right)\right)}_{\epsilon }={\left(\frac{1}{{\left[s_j\left(x\right)\right]}^{D_{\text{t}}\left(\left|\alpha -1\right|\right)}+\epsilon }\right)}_{\epsilon }\mathrm{.}$ (213)

We define now the Dirac distribution as Colombeau generalized function by equation

${\displaystyle {\sum }_{j=1}^m{\left({\displaystyle \int \text{d}{\eta }_j\left(x\mathrm{,}\epsilon \right){\delta }_{\left\{v_j\right\}}^{\left(D_{\text{f},j}\right)}\left(x\mathrm{,}\epsilon \right)}\right)}_{\epsilon }}=m\mathrm{.}$ (214)

In particular for the case $m=1$

${\left({\displaystyle \int \text{d}\eta \left(x\mathrm{,}\epsilon \right){\delta }_{\left\{v\right\}}^{\left(D_{\text{f}}\right)}\left(x\mathrm{,}\epsilon \right)}\right)}_{\epsilon }=1.$ (215)

Invariance of the action under the infinitesimal shift $\phi \left(x\right)\to \phi \left(x\right)+\delta \phi \left(x\right)$ gives the equation of motion for a generic weight ${\left(v_{i\mathrm{,}\epsilon }\right)}_{\epsilon }\mathrm{,}i=\mathrm{1,}\cdots \mathrm{,}m$ :

${\left(\frac{\partial L}{\partial {\phi }_{\epsilon }}\right)}_{\epsilon }-{\displaystyle {\sum }_{i=1}^m{\left(\left[\left(\frac{{\partial }_{\mu }{v}_{i\mathrm{,}\epsilon }}{v_{i\mathrm{,}\epsilon }}\right)+\frac{\text{d}}{\text{d}{x}^{\mu }}\right]\frac{\partial L}{\partial \left({\partial }_{\mu }{\phi }_{\epsilon }\right)}\right)}_{\epsilon }}=0.$ (216)

In particular for the case $m=1$ we obtain

${\left(\frac{\partial L}{\partial {\phi }_{\epsilon }}\right)}_{\epsilon }-{\left(\left[\frac{{\partial }_{\mu }{v}_{\epsilon }}{v_{\epsilon }}+\frac{\text{d}}{\text{d}{x}^{\mu }}\right]\frac{\partial L}{\partial \left({\partial }_{\mu }{\phi }_{\epsilon }\right)}\right)}_{\epsilon }=0.$ (217)

From Equation (212) and Equation (216) we obtain

${\left(\square {\phi }_{\epsilon }\right)}_{\epsilon }+{\displaystyle {\sum }_{i=1}^m\left\{{\left(\left[\frac{{\partial }_{\mu }{v}_{i\mathrm{,}\epsilon }}{v_{i\mathrm{,}\epsilon }}\right]{\partial }^{\mu }{\phi }_{\epsilon }\right)}_{\epsilon }\right\}}-{\left(\frac{\text{d}}{\text{d}{\phi }_{\epsilon }}V\left({\phi }_{\epsilon }\right)\right)}_{\epsilon }=0.$ (218)

where $\square ={\partial }_{\mu }{\partial }^{\mu }$. In particular for the case $m=1$ we obtain

${\left(\square {\phi }_{\epsilon }\right)}_{\epsilon }+{\left(\left[\frac{{\partial }_{\mu }{v}_{\epsilon }}{v_{\epsilon }}\right]{\partial }^{\mu }{\phi }_{\epsilon }\right)}_{\epsilon }-{\left(\frac{\text{d}}{\text{d}{\phi }_{\epsilon }}V\left({\phi }_{\epsilon }\right)\right)}_{\epsilon }=0.$ (219)

4.2. Propagator in Configuration Space with Negative-Dimensions

We define the canonical vacuum-to-vacuum amplitude by

${\left(Z\left[J,\epsilon \right]\right)}_{\epsilon }={\left({\displaystyle \int D{\phi }_{\epsilon }\exp \left[i{\displaystyle {\sum }_{j=1}^m{\displaystyle \int \text{d}{\eta }_{j\mathrm{,}\epsilon }\left(L+{\phi }_{\epsilon }J\right)}}\right]}\right)}_{\epsilon }\mathrm{,}$ (220)

where J is a source. Integration by parts in the exponent leads to the Lagrangian density for a free field as

${\left(L_{\epsilon }\right)}_{\epsilon }=\frac{1}{2}{\left({\phi }_{\epsilon }\left(\square +{\displaystyle {\sum }_{j=1}^m\frac{{\partial }_{\mu }{v}_{j\mathrm{,}\epsilon }}{v_{j\mathrm{,}\epsilon }}{\partial }^{\mu }}-m^2\right){\phi }_{\epsilon }\right)}_{\epsilon }=\frac{1}{2}{\left({\phi }_{\epsilon }{\Im }_{\epsilon }{\phi }_{\epsilon }\right)}_{\epsilon }\mathrm{,}$ (221)

where

${\Im }_{\epsilon }=\square +{\displaystyle {\sum }_{j=1}^m\frac{{\partial }_{\mu }{v}_{j\mathrm{,}\epsilon }}{v_{j\mathrm{,}\epsilon }}{\partial }^{\mu }}-m^2\mathrm{<mo>\; ;\; </mo>}j=\mathrm{1,}\cdots \mathrm{,}m\mathrm{.}$ (222)

In particular for the case $m=1$ we obtain

${\Im }_{\epsilon }=\square +\frac{{\partial }_{\mu }{v}_{\epsilon }}{v_{\epsilon }}{\partial }^{\mu }-m^2\mathrm{.}$ (223)

The propagator is the Green function ${\left(G_{\epsilon }\left(x\right)\right)}_{\epsilon }$ solving the equation

${\left({\Im }_{\epsilon }{G}_{\epsilon }\left(x\right)\right)}_{\epsilon }={\left({\delta }_v^{D^{-}}\left(x\mathrm{,}\epsilon \right)\right)}_{\epsilon }\mathrm{,}$ (224)

where $D^{-}=D_{\text{t}}\left(\alpha -1\right)<0$. By virtue of Lorentz covariance, the Green function $G_{\epsilon }\left(x\right)$ must depend only on the Lorentz interval $s^2=x_{\mu }{x}^{\mu }=x_i{x}^i-t^2$, where $x^0=t$ and $i=\mathrm{1,}\cdots \mathrm{,}{D}_{\text{t}}-1$. In particular, ${\left(v_{\epsilon }\right)}_{\epsilon }={\left(v_{\epsilon }\left(s\left(x\right)\right)\right)}_{\epsilon }$ with the correct scaling property is

${\left(v_{\epsilon }\left(s\left(x\right)\right)\right)}_{\epsilon }={\left({\left({\left|s\left(x\right)\right|}^{\left|D^{-}\right|}+\epsilon \right)}^{-1}\right)}_{\epsilon }\mathrm{,}s(x$

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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