$$ (\zeta_{\varepsilon}(E_i, D^+, D^-, x, \tilde{x}))_{\varepsilon} = \begin{cases} \left( \frac{\Theta_1(D^+) \Theta_2(D^-) [diam(E_i)]^{D^+}}{[d(x, \tilde{x})]^{D^-} + \varepsilon} \right)_{\varepsilon} & \text{if } x \in E_i \\ 0 & \text{if } x \notin E_i \end{cases} \quad (174) $$

where $\varepsilon \in (0,1], \Theta_1(D^+), \Theta_2(D^-) > 0$. In particular, when $F$ is the set of all (closed or open) balls in $X$,

$$ \Theta_1(D^+) = \frac{2^{-D^+} \pi^{\frac{D^+}{2}}}{\Gamma\left(1 + \frac{D^+}{2}\right)} \quad (175) $$

and

$$ \Theta_2(D^-) = \frac{2^{-D^-} \pi^{\frac{D^-}{2}}}{\left| \Gamma\left(1 + \frac{D^-}{2}\right) \right|}, \quad (176) $$

$$ D^- \neq -2, -4, -6, \dots, -2(n+1), \dots $$

**Definition 3.3.6.** The Hausdorff-Colombeau singular measure $(\mu_H(E, D^+, D^-, x, \tilde{x}, \varepsilon))_{\varepsilon}$ of a subset $E \subset X$ with the associated Hausdorff-Colombeau dimension $\tilde{D}^+(D^-) \in \mathbb{R}_+, D^- \in \mathbb{R}_+$ is defined by

$$ \begin{aligned} & (\mu_{HC}(E, \tilde{D}^+, D^-, x, \tilde{x}, \varepsilon))_{\varepsilon} \\ &= \left( \lim_{\delta \to 0} \left[ \sup_{\{E_i\}_{i \in \mathbb{N}}} \left\{ \sum_{i \in \mathbb{N}} (\zeta_{\varepsilon}(E_i, \tilde{D}^+, D^-, x, \tilde{x}))_{\varepsilon} \mid E_i \subset \bigcup_j E_i, \forall i (diam(E_i) < \delta) \right\} \right] \right)_{\varepsilon}, \\ & \tilde{D}^+ = \sup\{D^+ > 0 \mid (\mu_{HC}(E, D^+, D^-, x, \tilde{x}, \varepsilon))_{\varepsilon} = \infty\}_{\mathbb{R}}, \end{aligned} \quad (177) $$

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

$$ \begin{aligned} & \left( \int_X f(x) d\mu_{HC}(E, D^+, D^-, x, \tilde{x}, \varepsilon) \right)_{\varepsilon} \\ &= \left( \lim_{diam(E_i) \to 0} \left[ \sum_{\bigcup_{E_i=X} f(x_i)} \inf_{\{E_{ij}\}_{i \in \mathbb{N}}} \sum_j \zeta_{\varepsilon}(E_i, D^+, D^-, x_i, \tilde{x}) \right] \right)_{\varepsilon}. \end{aligned} \quad (178) $$

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 $d(x, y) > r$. The standard assumption that $\tilde{D}^+$ and $\tilde{D}^-$ is uniquely defined in all subsets $E$ of $X$ requires $X$ to be regular (homogeneous, uniform) with respect to the measure, i.e.

$$ \begin{aligned} & (\mu_{HC}^{-}(B_r(\tilde{x}), \tilde{D}^+, \tilde{D}^-, x, \tilde{x}, \varepsilon))_{\varepsilon} = (\mu_{HC}^{-}(B_r(\tilde{y}), \tilde{D}^+, \tilde{D}^-, x', \tilde{y}, \varepsilon))_{\varepsilon}, \text{ where} \\ & d(x, \tilde{x}) = d(x', \tilde{y}) \text{ for all elements } \tilde{x}, \tilde{y}, x, x' \in X \text{ and convex balls } B_r(\tilde{x}) \\ & \text{and } B_r(\tilde{y}) \text{ of the form } B_r(\tilde{x}) = \{z \mid d(\tilde{x}, z) \le r; \tilde{x}, z \in X\} \text{ and} \\ & B_r(\tilde{y}) = \{z \mid d(\tilde{y}, z) \le r; \tilde{y}, z \in X\}. \text{ In the limit } diam(E_i) \to 0, \text{ the infimum is} \end{aligned} $$
satisfied by the requirement that the variation over all coverings {$E_{ij}$}$_{ij}$$_{∈}$N is re-placed by one single covering $E_i$, such that $\bigcup_j E_{ij} \to E_i \ni x_i$. Therefore

$$ \left( \int_X f(x) d\mu_{HC} (E, \tilde{D}^+, \tilde{D}^-, x, \tilde{x}, \varepsilon) \right)_\varepsilon \\ = \left( \lim_{diam(E_i) \to 0} \sum_{\bigcup E_i = X} f(x_i) \zeta_\varepsilon (E_i, \tilde{D}^+, \tilde{D}^-, x_i, \tilde{x}) \right)_\varepsilon. \quad (179) $$

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

$$ \left( \int_X f(r) d\mu_{HC} (E, \tilde{D}^+, \tilde{D}^-, x, \tilde{x}, \varepsilon) \right)_\varepsilon \\ = \left( \lim_{diam(E_i) \to 0} \sum_{\bigcup E_i = X} f(r_i) \zeta_\varepsilon (E_i, \tilde{D}^+, \tilde{D}^-, x_i, \tilde{x}) \right)_\varepsilon. \quad (180) $$

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

$$ \Pi_{\tilde{h}_1, \dots, \tilde{h}_n} = \{ x = (x_1, \dots, x_n) \in X \mid x_j = (i_j - 1)\Delta x_j + \gamma_j, 0 \le \gamma_j \le \Delta x_j, j = 1, \dots, n \}. \quad (181) $$

For $n=2$ the polyhedra $\Pi_{\tilde{h}_1, \tilde{h}_2}$ is shown in Figure 6. Since $X$ is uniform

$$ \begin{aligned} \left( d\mu_{HC} (x, \tilde{D}^+, \tilde{D}^-, x, \tilde{x}, \varepsilon) \right)_\varepsilon &= \left( \lim_{diam(\Pi_{\tilde{h}_1, \dots, \tilde{h}_n})} \zeta_\varepsilon (\Pi_{\tilde{h}_1, \dots, \tilde{h}_n}, \tilde{D}^+, \tilde{D}^-, x, \tilde{x}) \right)_\varepsilon \\ &= \left( \lim_{diam(\Pi_{\tilde{h}_1, \dots, \tilde{h}_n})} \prod_{j=1}^n \left( \frac{\Delta x_j}{|x_j - \tilde{x}_j|^{\|\tilde{D}^j\|} + \varepsilon} \right)^{\frac{\tilde{D}^j}{n}} \right)_\varepsilon \\ &\triangleq \left( \prod_{j=1}^n \frac{d^{\frac{\tilde{D}^j}{n}} x_j}{(|x_j - \tilde{x}_j|^{\|\tilde{D}^j\|} + \varepsilon)^{\frac{\tilde{D}^j}{n}}} \right)_\varepsilon. \end{aligned} \quad (182) $$

Notice that the range of integration $X$ may also be parametrized by polar coordinates with $r = d(x, 0)$ and angle $\Omega$. $E_{r, \Omega}$ 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 $(\zeta_\varepsilon(E_{r, \Omega}, \tilde{D}^+, \tilde{D}^-, r, 0))_\varepsilon$ is given by

$$ \left( d\mu_{HC} (r, \Omega, \tilde{D}^+, \tilde{D}^-, \varepsilon) \right)_\varepsilon = \left( \lim_{diam(\Pi_{\tilde{h}_1, \dots, \tilde{h}_n})} \zeta_\varepsilon (E_{r, \Omega}, \tilde{D}^+, \tilde{D}^-, \{r, \Omega\}, 0) \right)_\varepsilon \triangleq \frac{d\Omega^{\tilde{D}^+-1} r^{\tilde{D}^+-1} dr}{\left( r^{\|\tilde{D}^j\|} + \varepsilon \right)_\varepsilon} \quad (183) $$
satisfied by the requirement that the variation over all coverings {$E_{ij}$}$_{ij}$$_{∈}$N is re-placed by one single covering $E_i$, such that $\bigcup_j E_{ij} \to E_i \ni x_i$. Therefore

$$ \left( \int_X f(x) d\mu_{HC} (E, \tilde{D}^+, \tilde{D}^-, x, \tilde{x}, \varepsilon) \right)_\varepsilon = \left( \lim_{diam(E_i) \to 0} \sum_{\cup E_i = X} f(x_i) \zeta_\varepsilon (E_i, \tilde{D}^+, \tilde{D}^-, x_i, \tilde{x}) \right)_\varepsilon. \quad (179) $$

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

$$ \left( \int_X f(r) d\mu_{HC} (E, \tilde{D}^+, \tilde{D}^-, x, \tilde{x}, \varepsilon) \right)_\varepsilon = \left( \lim_{diam(E_i) \to 0} \sum_{\cup E_i = X} f(r_i) \zeta_\varepsilon (E_i, \tilde{D}^+, \tilde{D}^-, x_i, \tilde{x}) \right)_\varepsilon. \quad (180) $$

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

$$ \Pi_{\tilde{h}_1, \dots, \tilde{h}_n} = \{ x = (x_1, \dots, x_n) \in X \mid x_j = (i_j - 1) \Delta x_j + \gamma_j, 0 \le \gamma_j \le \Delta x_j, j = 1, \dots, n \}. \quad (181) $$

For $n=2$ the polyhedra $\Pi_{\tilde{h}_1, \tilde{h}_2}$ is shown in Figure 6. Since $X$ is uniform

$$ \begin{aligned} \left( d\mu_{HC} (x, \tilde{D}^+, \tilde{D}^-, x, \tilde{x}, \varepsilon) \right)_\varepsilon &= \left( \lim_{diam(\Pi_{\tilde{h}_1, \dots, \tilde{h}_n})} \zeta_\varepsilon (\Pi_{\tilde{h}_1, \dots, \tilde{h}_n}, \tilde{D}^+, \tilde{D}^-, x, \tilde{x}) \right)_\varepsilon \\ &= \left( \lim_{diam(\Pi_{\tilde{h}_1, \dots, \tilde{h}_n})} \prod_{j=1}^n \left( \frac{\Delta x_j}{|x_j - \tilde{x}_j|^{\|\tilde{D}^j\|} + \varepsilon} \right)^{\frac{\tilde{D}^j}{n}} \right)_\varepsilon \\ &\triangleq \left( \prod_{j=1}^n \frac{d^{\frac{\tilde{D}^j}{n}} x_j}{(|x_j - \tilde{x}_j|^{\|\tilde{D}^j\|} + \varepsilon)^{\frac{\tilde{D}^j}{n}}} \right)_\varepsilon. \end{aligned} \quad (182) $$

Notice that the range of integration $X$ may also be parametrized by polar coordinates with $r = d(x, 0)$ and angle $\Omega$. $E_{r, \Omega}$ 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 $(\zeta_\varepsilon(E_{r, \Omega}, \tilde{D}^+, \tilde{D}^-, r, 0))_\varepsilon$ is given by

$$ \left( d\mu_{HC} (r, \Omega, \tilde{D}^+, \tilde{D}^-, \varepsilon) \right)_\varepsilon = \left( \lim_{diam(\Pi_{\tilde{h}_1, \dots, \tilde{h}_n})} \zeta_\varepsilon (E_{r, \Omega}, \tilde{D}^+, \tilde{D}^-, \{r, \Omega\}, 0) \right)_\varepsilon \triangleq \frac{d\Omega^{\tilde{D}^+-1} r^{\tilde{D}^+-1} dr}{(r^{\|\tilde{D}^j\|} + \varepsilon)_\varepsilon} \quad (183) $$
<figure><img src="image_8.png" /><figcaption>Figure 6. The polyhedra covering for $n = 2$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 7. The spherical covering $E_{r,\Omega}$.</figcaption></figure>

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

$$x \to z = x - \tilde{x} \qquad (184)$$

can be performed to shift the centre of symmetry to the origin (since $X$ is not a linear space, (184) need not be a map of $X$ onto itself and (184) is measure pre- suming). The integral over metric space $X$ is then given by formula

$$\left(\int_X f(x) d\mu_{HC}(E, \tilde{D}^+, \tilde{D}^-, x, \tilde{x}, \varepsilon)\right) = \frac{4\pi^{D^+/2}\pi^{D^-/2}}{\Gamma(D^+/2)\Gamma(D^-/2)} \left(\int_0^\infty \frac{r^{D^+-1} f(r)}{\varepsilon + r^{|D^-|}} dr\right)_\varepsilon. \qquad (185)$$

### 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 $X \subseteq \mathbb{R}^n$ is a Colombeau generalized function $[(\phi_\varepsilon(E))_\varepsilon] \in \mathcal{G}_F(\Omega)$ (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
<figure><img src="image_8.png" /><figcaption>Figure 6. The polyhedra covering for $n = 2$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 7. The spherical covering $E_{r,\Omega}$.</figcaption></figure>

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

$$x \to z = x - \tilde{x} \qquad (184)$$

can be performed to shift the centre of symmetry to the origin (since $X$ is not a linear space, (184) need not be a map of $X$ onto itself and (184) is measure pre- suming). The integral over metric space $X$ is then given by formula

$$\left( \int_X f(x) d\mu_{HC}(E, \tilde{D}^+, \tilde{D}^-, x, \tilde{x}, \varepsilon) \right)_{\varepsilon} = \frac{4\pi^{D^+/2} \pi^{D^-/2}}{\Gamma(D^+/2) \Gamma(D^-/2)} \left( \int_0^{\infty} \frac{r^{D^+-1} f(r)}{\varepsilon + r^{|D^-|}} dr \right)_{\varepsilon}. \qquad (185)$$

### 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 $X \subseteq \mathbb{R}^n$ is a Colombeau generalized function $[(\phi_\varepsilon(E))_\varepsilon] \in \mathcal{G}_F(\Omega)$ (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
$$ [ (\phi_{\varepsilon}(\emptyset))_{\varepsilon} ] = 0. \qquad (186) $$

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

$$ A \subseteq B [ (\phi_{\varepsilon}(A))_{\varepsilon} ] \leq \bar{R} [ (\phi_{\varepsilon}(B))_{\varepsilon} ]. \qquad (187) $$

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

$$ [ (\phi_{\varepsilon}(\bigcup_{j=1}^{\infty} A_j))_{\varepsilon} ] \leq \bar{R} [ (\sum_{j=1}^{\infty} \phi_{\varepsilon}(A_j))_{\varepsilon} ]. \qquad (188) $$

4) Whenever $d(A, B) = \inf \{d_n(x, y) | x \in A, y \in B\} > 0$

$$ [ (\phi_{\varepsilon}(A \cup B))_{\varepsilon} ] = [ (\phi_{\varepsilon}(A))_{\varepsilon} ] + [ (\phi_{\varepsilon}(B))_{\varepsilon} ], \qquad (189) $$

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

**Definition 3.4.2.** We say that outer Colombeau metric measure $(\mu_{\varepsilon})_{\varepsilon} \in (0,1]$ is a Colombeau measure on $\sigma$-algebra of subests of $X \subseteq \mathbb{R}^n$ if $(\mu_{\varepsilon})_{\varepsilon}$ satisfies the following property:

$$ [ (\phi_{\varepsilon}(\bigcup_{j=1}^{\infty} A_j))_{\varepsilon} ] = [ (\sum_{j=1}^{\infty} \phi_{\varepsilon}(A_j))_{\varepsilon} ]. \qquad (190) $$

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

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

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

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

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

$$ (H_{\delta}^{s,q}(F, \varepsilon))_{\varepsilon} = \left( \inf \left\{ \sum_{i=1}^{\infty} \frac{|U_i|^s}{\|x_i\|^q + \varepsilon} \right\} \right)_{\varepsilon} \qquad (192) $$

where the infimum is taken over all $\delta$-covers of $F$ and where

$x_i = (x_{i1}, \cdots, x_{in}) \in U_i$ for all $i \in \mathbb{N}$ and $\|x\|$ is the usual Euclidean norm: $\|x\| = \sqrt{\sum_{j=1}^{n} x_j^2}$.

Note that for $0 < \delta_1 < \delta_2 < 1$, $\varepsilon \in (0,1]$ we have

$$ H_{\delta_1}^{s,q}(F, \varepsilon) \geq H_{\delta_2}^{s,q}(F, \varepsilon) \qquad (193) $$

since any $\delta_1$ cover of $F$ is also a $\delta_2$ cover of $F$, i.e. $H_{\delta_1}^{s,q}(F, \varepsilon)$ is increasing as $\delta$ decreases.

**Definition 3.4.5.** We define the $(s, q)$-dimensional Hausdorff-Colombeau (outer) measure as:

$$ (H^{s,q}(F, \varepsilon))_{\varepsilon} = \left( \lim_{\delta \to 0} H_{\delta}^{s,q}(F, \varepsilon) \right)_{\varepsilon}. \qquad (194) $$

**Theorem 3.4.1.** For any $\delta$-cover, $\{U_i\}_{i \in \mathbb{N}}$ of $F$, and for any $\varepsilon \in (0,1]$, $t > s$:
$$ \left[ \left( \phi_{\varepsilon}(\emptyset) \right)_{\varepsilon} \right] = 0. \qquad (186) $$

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

$$ A \subseteq B \left[ \left( \phi_{\varepsilon}(A) \right)_{\varepsilon} \right] \leq \bar{R} \left[ \left( \phi_{\varepsilon}(B) \right)_{\varepsilon} \right]. \qquad (187) $$

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

$$ \left[ \left( \phi_{\varepsilon} \left( \bigcup_{j=1}^{\infty} A_j \right) \right)_{\varepsilon} \right] \leq \bar{R} \left[ \left( \sum_{j=1}^{\infty} \phi_{\varepsilon}(A_j) \right)_{\varepsilon} \right]. \qquad (188) $$

4) Whenever $d(A, B) = \inf \{d_n(x, y) | x \in A, y \in B\} > 0$

$$ \left[ \left( \phi_{\varepsilon}(A \cup B) \right)_{\varepsilon} \right] = \left[ \left( \phi_{\varepsilon}(A) \right)_{\varepsilon} \right] + \left[ \left( \phi_{\varepsilon}(B) \right)_{\varepsilon} \right], \qquad (189) $$

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

**Definition 3.4.2.** We say that outer Colombeau metric measure $(\mu_{\varepsilon})_{\varepsilon} \in (0,1]$ is a Colombeau measure on $\sigma$-algebra of subests of $X \subseteq \mathbb{R}^n$ if $(\mu_{\varepsilon})_{\varepsilon}$ satisfies the following property:

$$ \left[ \left( \phi_{\varepsilon} \left( \bigcup_{j=1}^{\infty} A_j \right) \right)_{\varepsilon} \right] = \left[ \left( \sum_{j=1}^{\infty} \phi_{\varepsilon}(A_j) \right)_{\varepsilon} \right]. \qquad (190) $$

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

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

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

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

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

$$ (H_{\delta}^{s,q}(F, \varepsilon))_{\varepsilon} = \left( \inf \left\{ \sum_{i=1}^{\infty} \frac{|U_i|^s}{\|x_i\|^q + \varepsilon} \right\} \right)_{\varepsilon} \qquad (192) $$

where the infimum is taken over all $\delta$-covers of F and where

$x_i = (x_{i1}, \cdots, x_{in}) \in U_i$ for all $i \in \mathbb{N}$ and $\|x\|$ is the usual Euclidean norm: $\|x\| = \sqrt{\sum_{j=1}^{n} x_j^2}$.

Note that for $0 < \delta_1 < \delta_2 < 1$, $\varepsilon \in (0,1]$ we have

$$ H_{\delta_1}^{s,q}(F, \varepsilon) \geq H_{\delta_2}^{s,q}(F, \varepsilon) \qquad (193) $$

since any $\delta_1$ cover of F is also a $\delta_2$ cover of F, i.e. $H_{\delta_1}^{s,q}(F, \varepsilon)$ is increasing as $\delta$ decreases.

**Definition 3.4.5.** We define the $(s, q)$-dimensional Hausdorff-Colombeau (outer) measure as:

$$ (H^{s,q}(F, \varepsilon))_{\varepsilon} = \left( \lim_{\delta \to 0} H_{\delta}^{s,q}(F, \varepsilon) \right)_{\varepsilon}. \qquad (194) $$

**Theorem 3.4.1.** For any $\delta$-cover, {$U_i$}$_{i \in \mathbb{N}}$ of F, and for any $\varepsilon \in (0,1]$, $t > s$:
$$H^{t,q}(F, \varepsilon) \le \delta^{t-s} H^{s,q}(F, \varepsilon). \qquad (195)$$

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

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

From (196) it follows that

$$\frac{|U_i|^t}{\|x_i\|^q + \varepsilon} \le \frac{\delta^{t-s} |U_i|^s}{\|x_i\|^q + \varepsilon} \qquad (197)$$

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

$$\sum_{i=1}^{\infty} \frac{|U_i|^t}{\|x_i\|^q + \varepsilon} \le \delta^{t-s} \sum_{i=1}^{\infty} \frac{|U_i|^s}{\|x_i\|^q + \varepsilon}. \qquad (198)$$

Thus (195) follows by taking the infimum.

**Theorem 3.4.2.** 1) If $(H^{s,q}(F, \varepsilon))_{\varepsilon} <_{\mathbb{R}} \infty_{\mathbb{R}}$, and if $t > s$, then $(H^{t,q}(F, \varepsilon))_{\varepsilon} = 0_{\mathbb{R}}$.

2) If $0_{\mathbb{R}} <_{\mathbb{R}} (H^{s,q}(F, \varepsilon))_{\varepsilon}$, and if $t < s$, then $(H^{t,q}(F, \varepsilon))_{\varepsilon} = \infty_{\mathbb{R}}$.

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

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

$$\frac{1}{\delta^{s-t}} H^{s,q}(F, \varepsilon) \le H^{t,q}(F, \varepsilon). \qquad (199)$$

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

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

$$\dim_{HC}(F, q) = \sup \{s = s(q) | (H^{s,q}(F, \varepsilon))_{\varepsilon} = \infty_{\mathbb{R}}\} \\ = \inf \{s = s(q) | (H^{s,q}(F, \varepsilon))_{\varepsilon} = 0_{\mathbb{R}}\}. \qquad (200)$$

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

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

**Remark 3.4.2.** Note that the graph of $(H^{s,\tilde{q}}(F, \varepsilon))_{\varepsilon}$ for a fixed $q = \tilde{q}$ is

$$(H^{s,\tilde{q}}(F, \varepsilon))_{\varepsilon} = \begin{cases} \infty_{\mathbb{R}} & \text{if } s > \dim_{HC}(F, \tilde{q}) \\ 0_{\mathbb{R}} & \text{if } s > \dim_{HC}(F, \tilde{q}) \\ \text{undetermined} & \text{if } s = \dim_{HC}(F, \tilde{q}) \end{cases} \qquad (201)$$

**Definition 3.4.7.** We say that fractal $\mathcal{F} \subseteq \mathbb{R}^n$ has a negative dimension rela-
$$H^{t,q}(F, \varepsilon) \leq \delta^{t-s} H^{s,q}(F, \varepsilon). \qquad (195)$$

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

$$|U_i|^t = |U_i|^{t-s} |U_i|^s \leq \delta^{t-s} |U_i|^s. \qquad (196)$$

From (196) it follows that

$$\frac{|U_i|^t}{\|x_i\|^q + \varepsilon} \leq \frac{\delta^{t-s} |U_i|^s}{\|x_i\|^q + \varepsilon} \qquad (197)$$

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

$$\sum_{i=1}^{\infty} \frac{|U_i|^t}{\|x_i\|^q + \varepsilon} \leq \delta^{t-s} \sum_{i=1}^{\infty} \frac{|U_i|^s}{\|x_i\|^q + \varepsilon}. \qquad (198)$$

Thus (195) follows by taking the infimum.

**Theorem 3.4.2.** 1) If $(H^{s,q}(F, \varepsilon))_{\varepsilon} <_{\mathbb{R}} \infty_{\mathbb{R}}$, and if $t > s$, then $(H^{t,q}(F, \varepsilon))_{\varepsilon} = 0_{\mathbb{R}}$.

2) If $0_{\mathbb{R}} <_{\mathbb{R}} (H^{s,q}(F, \varepsilon))_{\varepsilon}$, and if $t < s$, then $(H^{t,q}(F, \varepsilon))_{\varepsilon} = \infty_{\mathbb{R}}$.

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

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

$$\frac{1}{\delta^{s-t}} H^{s,q}(F, \varepsilon) \leq H^{t,q}(F, \varepsilon). \qquad (199)$$

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

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

$$\dim_{HC}(F, q) = \sup \{s = s(q) | (H^{s,q}(F, \varepsilon))_{\varepsilon} = \infty_{\mathbb{R}}\} \\ = \inf \{s = s(q) | (H^{s,q}(F, \varepsilon))_{\varepsilon} = 0_{\mathbb{R}}\}. \qquad (200)$$

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

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

**Remark 3.4.2.** Note that the graph of $(H^{s,\tilde{q}}(F, \varepsilon))_{\varepsilon}$ for a fixed $q = \tilde{q}$ is

$$(H^{s,\tilde{q}}(F, \varepsilon))_{\varepsilon} = \begin{cases} \infty_{\mathbb{R}} & \text{if } s > \dim_{HC}(F, \tilde{q}) \\ 0_{\mathbb{R}} & \text{if } s > \dim_{HC}(F, \tilde{q}) \\ \text{undetermined} & \text{if } s = \dim_{HC}(F, \tilde{q}) \end{cases} \qquad (201)$$

**Definition 3.4.7.** We say that fractal $\mathcal{F} \subseteq \mathbb{R}^n$ has a negative dimension rela-
tive to $q>0$
if $\dim_{HC}(F,q)-q<0$.

# 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 simple 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:

$$[x^{\mu}] = -1, \mu = 0,1,\dots,D_t-1 \qquad (202)$$

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

$$\begin{gathered} d^{D_t} x \to d^{D_t} x = (d\eta(x, \varepsilon))_{\varepsilon}, \qquad (203) \\ [\eta] = D_t \cdot \alpha, \alpha \in [1, -\infty). \end{gathered}$$

Here $D_t$ 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 multi-fractional integral of a function $\mathcal{L}(t)$:

$$\begin{gathered} \left(\int_{0}^{\bar{t}} d\eta(x, \varepsilon) \mathcal{L}(t)\right)_{\varepsilon} \propto \left(I_{\frac{z_i(\bar{t})}{\varepsilon}}\right)_{\varepsilon} \triangleq \left(\sum_{i=1}^{m} \int_{\varepsilon}^{\bar{t}} \frac{[(\bar{t}-t)+i\varepsilon]^{z_i(\bar{t})-1}}{\Gamma(z_i(\bar{t}))} \mathcal{L}(t) dt\right)_{\varepsilon}, \qquad (204) \\ (\eta(t, \varepsilon))_{\varepsilon} = \left(\frac{\bar{t}^{z_i(\bar{t})} - [(\bar{t}-t)+i\varepsilon]^{z_i(\bar{t})}}{\Gamma(z_i(\bar{t})+1)}\right)_{\varepsilon}, \end{gathered}$$

where $\varepsilon \in (0,1]$, $\bar{t}$ is fixed and the order $z_i(\bar{t})$ is (related to) the complex Hausdorff-Colombeau dimensions of the set. In particular if

$z_i \in \mathbb{C}, i=1,2,\dots,m$ 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 $\mathcal{L}(t)$
tive to $q > 0$
if $\dim_{HC}(F, q) - q < 0$.

# 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 simple 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:

$$[x^\mu] = -1, \mu = 0, 1, \dots, D_t - 1 \qquad (202)$$

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

$$\begin{gathered} d^{D_t} x \to d^{D_t} x = (d\eta(x, \varepsilon))_\varepsilon, \\ [\eta] = D_t \cdot \alpha, \alpha \in [1, -\infty). \end{gathered} \qquad (203)$$

Here $D_t$ 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 multi-fractional integral of a function $\mathcal{L}(t)$:

$$\begin{gathered} \left(\int_0^{\bar{t}} d\eta(x, \varepsilon) \mathcal{L}(t)\right)_\varepsilon \propto \left(I_{\frac{z_i(\bar{t})}{\varepsilon}}\right)_\varepsilon \triangleq \left(\sum_{i=1}^m \int_\varepsilon^{\bar{t}} \frac{[(\bar{t}-t)+i\varepsilon]^{z_i(\bar{t})-1}}{\Gamma(z_i(\bar{t}))} \mathcal{L}(t) dt\right)_\varepsilon, \\ (\eta(t, \varepsilon))_\varepsilon = \left(\frac{\bar{t}^{z_i(\bar{t})} - [(\bar{t}-t)+i\varepsilon]^{z_i(\bar{t})}}{\Gamma(z_i(\bar{t})+1)}\right)_\varepsilon, \end{gathered} \qquad (204)$$

where $\varepsilon \in (0,1]$, $\bar{t}$ is fixed and the order $z_i(\bar{t})$ is (related to) the complex Hausdorff-Colombeau dimensions of the set. In particular if

$z_i \in \mathbb{C}, i = 1, 2, \dots, m$ 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 $\mathcal{L}(t)$
$$ 
\begin{aligned}
& \left( \int_0^t d\eta(x, \varepsilon) \mathcal{L}(t) \right)_\varepsilon \propto \left( I_{\bar{t}, \varepsilon}^{[z_i]_{i=1}^m} \right)_\varepsilon \\
& = \sum_{i=1}^m (I_{\bar{t}, \varepsilon}^{z_i})_\varepsilon \triangleq \sum_{i=1}^m \left( \frac{1}{\Gamma(z_i)} \int_\varepsilon^t d[\bar{t}-t] + i\varepsilon \right)^{z_i-1} \mathcal{L}(t) \right)_\varepsilon, \quad (205) \\
& (\eta(t, \varepsilon))_\varepsilon = \sum_{i=1}^m \left( \frac{\bar{t}^{z_i} - [\bar{t}-t] + i\varepsilon}{\Gamma(z_i+1)} \right)_\varepsilon.
\end{aligned}
$$

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

$$ \left( \int_0^t d\eta(x, \varepsilon) \mathcal{L}(t) \right)_\varepsilon = \sum_{i=1}^m \left( \int_0^t dt \frac{[t+i\varepsilon]^{z_i-1}}{\Gamma(z_i)} \mathcal{L}(\bar{t}-t) \right)_\varepsilon. \quad (206) $$

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 $\lim_{\bar{t} \to +\infty} \mathcal{L}(\bar{t}-t) = \mathcal{L}[q(t), \dot{q}(t)]$. In particular if $z \in \mathbb{C}$ is a complex parameter a change of variables $t \to \bar{t}-t$ transforms Equation (206) into the form

$$ \sum_{i=1}^{m} (I_{\bar{t}, \varepsilon}^{z_i})_{\varepsilon} = \sum_{i=1}^{m} \left( \int_{\varepsilon}^{\bar{t}} dt \frac{[t + i\varepsilon]^{z_i-1}}{\Gamma(z_i)} \mathcal{L}[q(t), \dot{q}(t)] \right)_{\varepsilon}. \quad (207) $$

This form will be the most convenient for defining a Colombeau-Stieltjes field theory action. In $D_t$ dimensions, we consider now the action

$$ (S_{\varepsilon})_{\varepsilon} = \left( \int_{M} d\eta(x, \varepsilon) \mathcal{L}[\varphi_{\varepsilon}(x), \partial_{\mu}\varphi_{\varepsilon}(x)] \right)_{\varepsilon}, \quad (208) $$

where $\mathcal{L}[\varphi, \partial_{\mu}\varphi]$ is the Lagrangian density of the scalar field $(\varphi_\varepsilon(x))_\varepsilon$ and where

$$ (d\eta(x, \varepsilon))_{\varepsilon} = \sum_{i=1}^{m} \prod_{\mu=0}^{D_t-1} (f_{\mu,i}(x, \varepsilon))_{\varepsilon} dx^{\mu}, \quad (f_{\mu,i}(x, \varepsilon))_{\varepsilon}: M \to \tilde{\mathbb{R}}, \quad (209) $$

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

$$ (\mathcal{L}[\varphi_{\varepsilon}(x), \partial_{\mu}\varphi_{\varepsilon}(x)])_{\varepsilon} = -\frac{1}{2}(\partial_{\mu}\varphi_{\varepsilon}\partial^{\mu}\varphi_{\varepsilon})_{\varepsilon} - (V(\varphi_{\varepsilon}))_{\varepsilon}, \quad (210) $$

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

$$ (f_{(\mu,i),\varepsilon})_{\varepsilon} = (f_{i,\varepsilon})_{\varepsilon}, \quad \mu = 1, \cdots, D_t-1; i = 1, \cdots, m. \quad (211) $$

Hence the scalar field action (208) reads

$$ 
\begin{aligned}
(S_{\varepsilon})_{\varepsilon} &= \left( \int_{M} d\eta(x, \varepsilon) [\varphi_{\varepsilon}(x), \partial_{\mu}\varphi_{\varepsilon}(x)] \right)_{\varepsilon} \\
&= \sum_{j=1}^{m} \left( \int d^{D_t}x v_{\varepsilon,j}(x) \left[ \frac{1}{2} \partial_{\mu}\varphi_{\varepsilon} \partial^{\mu}\varphi_{\varepsilon} + V(\varphi_{\varepsilon}) \right] \right)_{\varepsilon},
\end{aligned}
\quad (212) $$

where $(v_\varepsilon(x))_\varepsilon$ is a coordinate-dependent
$$ \begin{aligned} & \left( \int_0^t d\eta(x, \varepsilon) \mathcal{L}(t) \right)_\varepsilon \propto \left( I_{\bar{t}, \varepsilon}^{[z_i]_{i=1}^m} \right)_\varepsilon \\ &= \sum_{i=1}^m (I_{\bar{t}, \varepsilon}^{z_i})_\varepsilon \triangleq \sum_{i=1}^m \left( \frac{1}{\Gamma(z_i)} \int_\varepsilon^t d[\bar{t}-t] + i\varepsilon \right)^{z_i-1} \mathcal{L}(t) \right)_\varepsilon, \quad (205) \\ & (\eta(t, \varepsilon))_\varepsilon = \sum_{i=1}^m \left( \frac{\bar{t}^{z_i} - [(\bar{t}-t)+i\varepsilon]^{z_i}}{\Gamma(z_i+1)} \right)_\varepsilon. \end{aligned} $$

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

$$ \left( \int_0^t d\eta(x, \varepsilon) \mathcal{L}(t) \right)_\varepsilon = \sum_{i=1}^m \left( \int_0^t dt \frac{[t+i\varepsilon]^{z_i-1}}{\Gamma(z_i)} \mathcal{L}(\bar{t}-t) \right)_\varepsilon. \quad (206) $$

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 $\lim_{\bar{t} \to +\infty} \mathcal{L}(\bar{t}-t) = \mathcal{L}[q(t), \dot{q}(t)]$. In particular if $z \in \mathbb{C}$ is a complex parameter a change of variables $t \to \bar{t}-t$ transforms Equation (206) into the form

$$ \sum_{i=1}^{m} (I_{\bar{t}, \varepsilon}^{z_i})_{\varepsilon} = \sum_{i=1}^{m} \left( \int_{\varepsilon}^{\bar{t}} dt \frac{[t + i\varepsilon]^{z_i-1}}{\Gamma(z_i)} \mathcal{L}[q(t), \dot{q}(t)] \right)_{\varepsilon}. \quad (207) $$

This form will be the most convenient for defining a Colombeau-Stieltjes field theory action. In $D_t$ dimensions, we consider now the action

$$ (S_{\varepsilon})_{\varepsilon} = \left( \int_{M} d\eta(x, \varepsilon) \mathcal{L}[\varphi_{\varepsilon}(x), \partial_{\mu}\varphi_{\varepsilon}(x)] \right)_{\varepsilon}, \quad (208) $$

where $\mathcal{L}[\varphi, \partial_{\mu}\varphi]$ is the Lagrangian density of the scalar field $(\varphi_\varepsilon(x))_\varepsilon$ and where

$$ (d\eta(x, \varepsilon))_{\varepsilon} = \sum_{i=1}^{m} \prod_{\mu=0}^{D_t-1} (f_{\mu,i}(x, \varepsilon))_{\varepsilon} dx^{\mu}, \quad (f_{\mu,i}(x, \varepsilon))_{\varepsilon}: M \to \tilde{\mathbb{R}}, \quad (209) $$

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

$$ \left( \mathcal{L}[\varphi_{\varepsilon}(x), \partial_{\mu}\varphi_{\varepsilon}(x)] \right)_{\varepsilon} = -\frac{1}{2} \left( \partial_{\mu}\varphi_{\varepsilon}\partial^{\mu}\varphi_{\varepsilon} \right)_{\varepsilon} - (V(\varphi_{\varepsilon}))_{\varepsilon}, \quad (210) $$

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

$$ (f_{(\mu,i),\varepsilon})_{\varepsilon} = (f_{i,\varepsilon})_{\varepsilon}, \quad \mu = 1, \cdots, D_t-1; i = 1, \cdots, m. \quad (211) $$

Hence the scalar field action (208) reads

$$ \begin{aligned} (S_{\varepsilon})_{\varepsilon} &= \left( \int_{M} d\eta(x, \varepsilon) [\varphi_{\varepsilon}(x), \partial_{\mu}\varphi_{\varepsilon}(x)] \right)_{\varepsilon} \\ &= \sum_{j=1}^{m} \left( \int d^{D_t}x v_{\varepsilon,j}(x) \left[ \frac{1}{2} \partial_{\mu}\varphi_{\varepsilon} \partial^{\mu}\varphi_{\varepsilon} + V(\varphi_{\varepsilon}) \right] \right)_{\varepsilon}, \end{aligned} \quad (212) $$

where $(v_\varepsilon(x))_\varepsilon$ is a coordinate-dependent
Lorentz scalar

$$ (v_{\varepsilon,j}(x))_{\varepsilon} = \left( \frac{1}{[s_j(x)]^{D_j(|\alpha|-1)} + \varepsilon} \right)_{\varepsilon}. \quad (213) $$

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

$$ \sum_{j=1}^{m} \left( \int d\eta_j(x, \varepsilon) \delta_{\{v_j\}}^{(D_j)}(x, \varepsilon) \right)_{\varepsilon} = m. \quad (214) $$

In particular for the case $m=1$

$$ \left( \int d\eta(x, \varepsilon) \delta_{\{v\}}^{(D)}(x, \varepsilon) \right)_{\varepsilon} = 1. \quad (215) $$

Invariance of the action under the infinitesimal shift $\varphi(x) \to \varphi(x) + \delta\varphi(x)$ gives the equation of motion for a generic weight $(v_{i,\varepsilon})_{\varepsilon}, i=1, \dots, m$:

$$ \left( \frac{\partial \mathcal{L}}{\partial \varphi_{\varepsilon}} \right)_{\varepsilon} - \sum_{i=1}^{m} \left( \left[ \left( \frac{\partial_{\mu} v_{i, \varepsilon}}{v_{i, \varepsilon}} \right) + \frac{d}{dx^{\mu}} \right] \frac{\partial \mathcal{L}}{\partial \left( \partial_{\mu} \varphi_{\varepsilon} \right)} \right)_{\varepsilon} = 0. \quad (216) $$

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

$$ \left( \frac{\partial \mathcal{L}}{\partial \varphi_{\varepsilon}} \right)_{\varepsilon} - \left( \left[ \frac{\partial_{\mu} v_{\varepsilon}}{v_{\varepsilon}} + \frac{d}{dx^{\mu}} \right] \frac{\partial \mathcal{L}}{\partial \left( \partial_{\mu} \varphi_{\varepsilon} \right)} \right)_{\varepsilon} = 0. \quad (217) $$

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

$$ (\Box \varphi_{\varepsilon})_{\varepsilon} + \sum_{i=1}^{m} \left\{ \left[ \left( \frac{\partial_{\mu} v_{i, \varepsilon}}{v_{i, \varepsilon}} \right) \partial^{\mu} \varphi_{\varepsilon} \right]_{\varepsilon} \right\} - \left( \frac{d}{d\varphi_{\varepsilon}} V(\varphi_{\varepsilon}) \right)_{\varepsilon} = 0. \quad (218) $$

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

$$ (\Box \varphi_{\varepsilon})_{\varepsilon} + \left( \left[ \frac{\partial_{\mu} v_{\varepsilon}}{v_{\varepsilon}} \right] \partial^{\mu} \varphi_{\varepsilon} \right)_{\varepsilon} - \left( \frac{d}{d\varphi_{\varepsilon}} V(\varphi_{\varepsilon}) \right)_{\varepsilon} = 0. \quad (219) $$

## 4.2. Propagator in Configuration Space with Negative-Dimensions

We define the canonical vacuum-to-vacuum amplitude by

$$ (Z[J, \varepsilon])_{\varepsilon} = \left( \int \mathbf{D}\varphi_{\varepsilon} \exp \left[ i \sum_{j=1}^{m} \int d\eta_{j, \varepsilon} (\mathcal{L} + \varphi_{\varepsilon} J) \right] \right)_{\varepsilon}, \quad (220) $$

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

$$ (\mathcal{L}_{\varepsilon})_{\varepsilon} = \frac{1}{2} \left( \varphi_{\varepsilon} \left( \Box + \sum_{j=1}^{m} \frac{\partial_{\mu} v_{j, \varepsilon}}{v_{j, \varepsilon}} \partial^{\mu} - m^2 \right) \varphi_{\varepsilon} \right)_{\varepsilon} = \frac{1}{2} (\varphi_{\varepsilon} \Im \varphi_{\varepsilon})_{\varepsilon}, \quad (221) $$

where

$$ \mathfrak{I}_{\varepsilon} = \Box + \sum_{j=1}^{m} \frac{\partial_{\mu} v_{j, \varepsilon}}{v_{j, \varepsilon}} \partial^{\mu} - m^2; \quad j=1, \dots, m. \quad (222) $$

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

$$ (v_{\varepsilon,j}(x))_{\varepsilon} = \left( \frac{1}{[s_j(x)]^{D_j(|\alpha|-1)} + \varepsilon} \right)_{\varepsilon}. \qquad (213) $$

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

$$ \sum_{j=1}^{m} \left( \int d\eta_j(x, \varepsilon) \delta_{\{v_j\}}^{(D_j)}(x, \varepsilon) \right)_{\varepsilon} = m. \qquad (214) $$

In particular for the case $m=1$

$$ \left( \int d\eta(x, \varepsilon) \delta_{\{v\}}^{(D)}(x, \varepsilon) \right)_{\varepsilon} = 1. \qquad (215) $$

Invariance of the action under the infinitesimal shift $\varphi(x) \to \varphi(x) + \delta\varphi(x)$ gives the equation of motion for a generic weight $(v_{i,\varepsilon})_{\varepsilon}, i=1, \dots, m$:

$$ \left( \frac{\partial \mathcal{L}}{\partial \varphi_{\varepsilon}} \right)_{\varepsilon} - \sum_{i=1}^{m} \left( \left[ \left( \frac{\partial_{\mu} v_{i, \varepsilon}}{v_{i, \varepsilon}} \right) + \frac{d}{dx^{\mu}} \right] \frac{\partial \mathcal{L}}{\partial \left( \partial_{\mu} \varphi_{\varepsilon} \right)} \right)_{\varepsilon} = 0. \qquad (216) $$

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

$$ \left( \frac{\partial \mathcal{L}}{\partial \varphi_{\varepsilon}} \right)_{\varepsilon} - \left( \left[ \frac{\partial_{\mu} v_{\varepsilon}}{v_{\varepsilon}} + \frac{d}{dx^{\mu}} \right] \frac{\partial \mathcal{L}}{\partial \left( \partial_{\mu} \varphi_{\varepsilon} \right)} \right)_{\varepsilon} = 0. \qquad (217) $$

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

$$ (\Box \varphi_{\varepsilon})_{\varepsilon} + \sum_{i=1}^{m} \left\{ \left[ \left( \frac{\partial_{\mu} v_{i, \varepsilon}}{v_{i, \varepsilon}} \right) \partial^{\mu} \varphi_{\varepsilon} \right]_{\varepsilon} \right\} - \left( \frac{d}{d\varphi_{\varepsilon}} V(\varphi_{\varepsilon}) \right)_{\varepsilon} = 0. \qquad (218) $$

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

$$ (\Box \varphi_{\varepsilon})_{\varepsilon} + \left( \left[ \frac{\partial_{\mu} v_{\varepsilon}}{v_{\varepsilon}} \right] \partial^{\mu} \varphi_{\varepsilon} \right)_{\varepsilon} - \left( \frac{d}{d\varphi_{\varepsilon}} V(\varphi_{\varepsilon}) \right)_{\varepsilon} = 0. \qquad (219) $$

## 4.2. Propagator in Configuration Space with Negative-Dimensions

We define the canonical vacuum-to-vacuum amplitude by

$$ (Z[J, \varepsilon])_{\varepsilon} = \left( \int \mathbf{D}\varphi_{\varepsilon} \exp \left[ i \sum_{j=1}^{m} \int d\eta_{j, \varepsilon} (\mathcal{L} + \varphi_{\varepsilon} J) \right] \right)_{\varepsilon}, \qquad (220) $$

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

$$ (\mathcal{L}_{\varepsilon})_{\varepsilon} = \frac{1}{2} \left( \varphi_{\varepsilon} \left( \Box + \sum_{j=1}^{m} \frac{\partial_{\mu} v_{j, \varepsilon}}{v_{j, \varepsilon}} \partial^{\mu} - m^2 \right) \varphi_{\varepsilon} \right)_{\varepsilon} = \frac{1}{2} (\varphi_{\varepsilon} \Im \varphi_{\varepsilon})_{\varepsilon}, \qquad (221) $$

where

$$ \Im \varphi_{\varepsilon} = \Box + \sum_{j=1}^{m} \frac{\partial_{\mu} v_{j, \varepsilon}}{v_{j, \varepsilon}} \partial^{\mu} - m^2; \quad j=1, \dots, m. \qquad (222) $$

In particular for the case $m=1$ we obtain
$$ \mathfrak{I}_{\varepsilon} = \Box + \frac{\partial_{\mu} v_{\varepsilon}}{v_{\varepsilon}} \partial^{\mu} - m^2. \qquad (223) $$

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

$$ (\mathfrak{I}_{\varepsilon} G_{\varepsilon}(x))_{\varepsilon} = (\delta_{\nu}^{D^{-}}(x, \varepsilon))_{\varepsilon}, \qquad (224) $$

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

$$ (v_{\varepsilon}(s(x)))_{\varepsilon} = \left( \left( |s(x)|^{D^{-}} + \varepsilon \right)^{-1} \right)_{\varepsilon}, \quad s(x) = \sqrt{x_{\mu}x^{\mu}}. \qquad (225) $$

Note that

$$ \partial_{\mu} = \frac{x_{\mu}}{(s + \varepsilon)_{\varepsilon}} \partial_{s}, \quad \Box = \partial_{s}^{2} + \frac{D_{t} - 1}{(s + \varepsilon)_{\varepsilon}} \partial_{s}. \qquad (226) $$

Hence the inhomogeneous Equation (224) reads

$$ \left( \partial_s^2 + \frac{D_t \alpha - 1}{(s + \varepsilon)_{\varepsilon}} \partial_s - m^2 \right) (G_{\varepsilon}(x))_{\varepsilon} = (\delta_{\nu}^{D^{-}}(x, \varepsilon))_{\varepsilon}. \qquad (227) $$

We first consider the Euclidean propagator and denote with $r = \sqrt{x_i x^i + t^2}$ the Wick-rotated Lorentz invariant. In the massless case, the solution of the homogeneous equations for any $\alpha < 0$ is

$$ (G_{\varepsilon}(r))_{\varepsilon} = Cr^{2\beta}, \quad \beta = \frac{2 + D_{t} |\alpha|}{2}. \qquad (228) $$

Let us now consider the massive case. The solution of the homogeneous equation $(\mathfrak{I}_{\varepsilon} G_{\varepsilon}(r))_{\varepsilon} = 0$ for any $\alpha < 0$ is

$$ (G_{\varepsilon}(r))_{\varepsilon} = \left(\frac{r}{m}\right)^{\frac{2+D_{t}|\alpha|}{2}} \left[ C_{1}\mathbf{K}_{\frac{2+D_{t}|\alpha|}{2}}(mr) + C_{2}\mathbf{I}_{\frac{2+D_{t}|\alpha|}{2}}(mr) \right], \qquad (229) $$

where $C_1, C_2$ are constants and $\mathbf{K}_{\lambda}$ and $\mathbf{I}_{\lambda}$ are the modified Bessel functions. The function $\mathbf{I}_{\nu}(z)$ is

$$ \mathbf{I}_{\nu}(z) = \sum_{k=0}^{\infty} \frac{(z/2)^{\nu+2k}}{k!\Gamma(\nu+k+1)}. \qquad (230) $$

Formula (230) is valid providing $\nu \neq -1, -2, -3, \dots$

$$ \mathbf{I}_{-|\nu|}(z) = \sum_{k=0}^{\infty} \frac{(z/2)^{-|\nu|+2k}}{k!\Gamma(-|\nu|+k+1)}. \qquad (231) $$

Formula (231) is obtained by replacing $\nu$ in (232) with a $-\nu$.

$$ \mathbf{K}_{-|\nu|}(z) = - \frac{\pi}{2 \sin |\nu| \pi} \left[ \mathbf{I}_{|\nu|}(z) - \mathbf{I}_{-|\nu|}(z) \right]. \qquad (232) $$

The modified Bessel functions $\mathbf{I}_{-|\nu|}(z)$ and
$$ \mathfrak{I}_{\varepsilon} = \Box + \frac{\partial_{\mu} v_{\varepsilon}}{v_{\varepsilon}} \partial^{\mu} - m^2. \qquad (223) $$

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

$$ (\mathfrak{I}_{\varepsilon} G_{\varepsilon}(x))_{\varepsilon} = (\delta_{\nu}^{D^{-}}(x, \varepsilon))_{\varepsilon}, \qquad (224) $$

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

$$ (v_{\varepsilon}(s(x)))_{\varepsilon} = \left( \left( |s(x)|^{D^{-}} + \varepsilon \right)^{-1} \right)_{\varepsilon}, \quad s(x) = \sqrt{x_{\mu}x^{\mu}}. \qquad (225) $$

Note that

$$ \partial_{\mu} = \frac{x_{\mu}}{(s + \varepsilon)_{\varepsilon}} \partial_{s}, \quad \Box = \partial_{s}^{2} + \frac{D_{t} - 1}{(s + \varepsilon)_{\varepsilon}} \partial_{s}. \qquad (226) $$

Hence the inhomogeneous Equation (224) reads

$$ \left( \partial_s^2 + \frac{D_t \alpha - 1}{(s + \varepsilon)_{\varepsilon}} \partial_s - m^2 \right) (G_{\varepsilon}(x))_{\varepsilon} = (\delta_{\nu}^{D^{-}}(x, \varepsilon))_{\varepsilon}. \qquad (227) $$

We first consider the Euclidean propagator and denote with $r = \sqrt{x_i x^i + t^2}$ the Wick-rotated Lorentz invariant. In the massless case, the solution of the homogeneous equations for any $\alpha < 0$ is

$$ (G_{\varepsilon}(r))_{\varepsilon} = Cr^{2\beta}, \quad \beta = \frac{2 + D_t |\alpha|}{2}. \qquad (228) $$

Let us now consider the massive case. The solution of the homogeneous equation $(\mathfrak{I}_{\varepsilon} G_{\varepsilon}(r))_{\varepsilon} = 0$ for any $\alpha < 0$ is

$$ (G_{\varepsilon}(r))_{\varepsilon} = \left(\frac{r}{m}\right)^{\frac{2+D_t|\alpha|}{2}} \left[ C_1 \mathbf{K}_{\frac{2+D_t|\alpha|}{2}}(mr) + C_2 \mathbf{I}_{\frac{2+D_t|\alpha|}{2}}(mr) \right], \qquad (229) $$

where $C_1, C_2$ are constants and $\mathbf{K}_{\lambda}$ and $\mathbf{I}_{\lambda}$ are the modified Bessel functions. The function $\mathbf{I}_{\nu}(z)$ is

$$ \mathbf{I}_{\nu}(z) = \sum_{k=0}^{\infty} \frac{(z/2)^{v+2k}}{k!\Gamma(\nu + k + 1)}. \qquad (230) $$

Formula (230) is valid providing $\nu \neq -1, -2, -3, \dots$

$$ \mathbf{I}_{-|\nu|}(z) = \sum_{k=0}^{\infty} \frac{(z/2)^{-|\nu|+2k}}{k!\Gamma(-|\nu|+k+1)}. \qquad (231) $$

Formula (231) is obtained by replacing $\nu$ in (232) with a $-\nu$.

$$ \mathbf{K}_{-|\nu|}(z) = - \frac{\pi}{2 \sin|\nu| \pi} \left[ \mathbf{I}_{|\nu|}(z) - \mathbf{I}_{-|\nu|}(z) \right]. \qquad (232) $$

The modified Bessel functions $\mathbf{I}_{-|\nu|}(z)$ and
$K_{-|v|}(z)$ have the following asymptotic forms for $z \to 0$:

$$K_{-|v|}(z) \approx \frac{1}{2} \Gamma(-|v|) \left(\frac{2}{z}\right)^{-|v|}, \quad \mathbf{I}_{-|v|}(z) \approx \frac{1}{\Gamma(-|v|+1)} \left(\frac{z}{2}\right)^{-|v|}, \quad v \neq -1, -2, -3, \dots \quad (233)$$

Since for small $m \approx 0$ the solution must agree with the massless case (228), we can set $C_2 = 0$. To find the solution of the inhomogeneous equation, one exploits the fact that the mass term does not contribute near the origin. Expanding Equation (229) at $mr \approx 0$ when $\alpha < 0$ ($C_2 = 0$), we find

$$(G_{\varepsilon}(r))_{\varepsilon} = C_1 2^{-\frac{4+D_t|\alpha_t|}{2}} \Gamma\left(-\frac{2+D_t|\alpha_t|}{2}\right) (r^2)^{-\frac{2+D_t|\alpha_t|}{2}} \quad (234)$$

which must coincide with Equation (228). This gives the coefficient $C_1$ and the propagator reads

$$G(r) = -\frac{1}{2\pi^2} \frac{\Gamma\left(\frac{D_t}{2}\right)}{\Gamma\left(-\frac{D_t}{2}\right)} \left(\frac{m}{2r}\right)^{-\frac{2+D_t|\alpha_t|}{2}} \mathbf{K}_{-\frac{2+D_t|\alpha_t|}{2}}(mr). \quad (235)$$

# 5. The Solution Cosmological Constant Problem

## 5.1. Einstein-Gliner-Zel'dovich Vacuum with Tiny Lorentz Invariance Violation

We assume now that:

1) Poincaré group of momentum space is deformed at some fundamental high-energy cutoff $\Lambda_*$ [9] [10].

2) The canonical quadratic invariant $\|p\|^2 = \eta^{ab} p_a p_b$ collapses at high-energy cutoff $\Lambda_*$ and being replaced by the non-quadratic invariant:

$$\|p\|^2 = \frac{\eta^{ab} p_a p_b}{(1 + l_{\Lambda_*} p_0)}. \quad (236)$$

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

$$(d\eta(x, \varepsilon))_{\varepsilon} = (v_{\varepsilon}(s(x))d^4x)_{\varepsilon}, \quad (237)$$

where

$$(v_{\varepsilon}(s(x)))_{\varepsilon} = \left( \left( |s(x)|^{D^{-}} + \varepsilon \right)^{-1} \right)_{\varepsilon}, \quad (238)$$
$$s(x) = \sqrt{x_{\mu}x^{\mu}},$$

see subsection IV.2.

4) The canonical concept of momentum space collapses at fundamental
$\mathbf{K}_{-|\nu|}(z)$ have the following asymptotic forms for $z \to 0$:

$$ \mathbf{K}_{-|\nu|}(z) \approx \frac{1}{2} \Gamma(-|\nu|) \left(\frac{2}{z}\right)^{-|\nu|}, \quad \mathbf{I}_{-|\nu|}(z) \approx \frac{1}{\Gamma(-|\nu|+1)} \left(\frac{z}{2}\right)^{-|\nu|}, \quad \nu \neq -1, -2, -3, \dots \quad (233) $$

Since for small $m \approx 0$ the solution must agree with the massless case (228), we can set $C_2 = 0$. To find the solution of the inhomogeneous equation, one exploits the fact that the mass term does not contribute near the origin. Expanding Equation (229) at $mr \approx 0$ when $\alpha < 0$ ($C_2 = 0$), we find

$$ (G_{\varepsilon}(r))_{\varepsilon} = C_1 2^{-\frac{4+D_t|\alpha_t|}{2}} \Gamma\left(-\frac{2+D_t|\alpha_t|}{2}\right) (r^2)^{\frac{2+D_t|\alpha_t|}{2}} \quad (234) $$

which must coincide with Equation (228). This gives the coefficient $C_1$ and the propagator reads

$$ G(r) = - \frac{\Gamma\left(\frac{D_t}{2}\right)}{2\pi^2 \Gamma\left(-\frac{D_t}{2}\right)} \left(\frac{m}{2r}\right)^{-\frac{2+D_t|\alpha_t|}{2}} \mathbf{K}_{-\frac{2+D_t|\alpha_t|}{2}}(mr). \quad (235) $$

# 5. The Solution Cosmological Constant Problem

## 5.1. Einstein-Gliner-Zel'dovich Vacuum with Tiny Lorentz Invariance Violation

We assume now that:

1) Poincaré group of momentum space is deformed at some fundamental high-energy cutoff $\Lambda_*$ [9] [10].

2) The canonical quadratic invariant $\|p\|^2 = \eta^{ab} p_a p_b$ collapses at high-energy cutoff $\Lambda_*$ and being replaced by the non-quadratic invariant:

$$ \|p\|^2 = \frac{\eta^{ab} p_a p_b}{(1 + l_{\Lambda_*} p_0)}. \quad (236) $$

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

$$ (d\eta(x, \varepsilon))_{\varepsilon} = (v_{\varepsilon}(s(x))d^4x)_{\varepsilon}, \quad (237) $$

where

$$ \begin{gathered} (v_{\varepsilon}(s(x)))_{\varepsilon} = \left( \left( |s(x)|^{D^{-}} + \varepsilon \right)^{-1} \right)_{\varepsilon}, \quad (238) \\ s(x) = \sqrt{x_{\mu}x^{\mu}}, \end{gathered} $$

see subsection IV.2.

4) The canonical concept of momentum space collapses at fundamental
high-energy cutoff $\Lambda_*$ to fractal momentum space with Hausdorff-Colombeau negative dimension and therefore the canonical Lebesgue measure $d^3\mathbf{k}$, where $\mathbf{k} = (k_x, k_y, k_z)$ being replaced by the Hausdorff-Colombeau measure

$$d^{D^+,D^-} \mathbf{k} \triangleq \frac{\Delta(D^-) d^{D^+} \mathbf{k}}{\left(\left|\mathbf{k}\right|^{D^-} + \varepsilon\right)_\varepsilon} = \frac{\Delta(D^+) \Delta(D^-) p^{D^+ - 1} dp}{\left(p^{|D^-|} + \varepsilon\right)_\varepsilon}, \quad (239)$$

$$\text{where } \Delta(D^{\pm}) = \frac{2\pi^{D^{\pm}/2}}{\Gamma(D^{\pm}/2)} \text{ and } p = |\mathbf{k}| = \sqrt{k_x + k_y + k_z}.$$

**Remark 5.1.1.** Note that the integral over measure $d^{D^+,D^-} \mathbf{k}$ is given by formula (185). Thus vacuum energy density $\varepsilon(D^+,D^-,\mu_{\text{eff}},p_*)$ for free quantum fields is

$$\varepsilon(D^+, D^-, \mu_{\text{eff}}, p_*) = \varepsilon(\mu_{\text{eff}}) + \varepsilon(\mu_{\text{eff}}, p_*) + \bar{\varepsilon}(D^+, D^-, \mu_{\text{eff}}, p_*) \quad (240)$$

Here the quantity $\varepsilon(\mu_{\text{eff}})$ is given by formula

$$\begin{aligned} \varepsilon(\mu_{\text{eff}}) &= \frac{1}{2(2\pi\hbar)^3} \int_0^{\mu_{\text{eff}}} d\mu f(\mu) \int_{\|\mathbf{k}\| \le \mu} \sqrt{\mathbf{k}^2 + \mu^2} d^3\mathbf{k} \\ &= K \int_0^{\mu_{\text{eff}}} d\mu f(\mu) \int_{p \le \mu} \sqrt{p^2 + \mu^2} p^2 dp \quad (241) \\ &= K \int_0^{\mu_{\text{eff}}} d\mu f(\mu) \int_0^{\mu} \sqrt{p^2 + \mu^2} p^2 dp \end{aligned}$$

where $K = \frac{2\pi}{(2\pi\hbar)^3}, c = 1$. The quantity $\varepsilon(\mu_{\text{eff}}, p_*)$ is given by formula

$$\begin{aligned} \varepsilon(\mu_{\text{eff}}, p_*) &= \frac{1}{2(2\pi\hbar)^3} \int_0^{\mu_{\text{eff}}} d\mu f(\mu) \int_{\mu < \|\mathbf{k}\| < p_*} \sqrt{\mathbf{k}^2 + \mu^2} d^3\mathbf{k} \\ &= K \int_0^{\mu_{\text{eff}}} d\mu f(\mu) \int_{\mu < \|\mathbf{k}\| < p_*} \sqrt{p^2 + \mu^2} p^2 dp. \end{aligned} \quad (242)$$

The quantity $\bar{\varepsilon}(D^+,D^-,\mu_{\text{eff}},p_*)$ (since Equation (22) holds) is given by formula

$$\bar{\varepsilon}(D^+, D^-, \mu_{\text{eff}}, p_*) = K \int_0^{\mu_{\text{eff}}} d\mu f(\mu) \int_{\|\mathbf{k}\| \ge p_*} \left[ \frac{\mu^2 l_{\Lambda_*}}{1 - \mu^2 l_{\Lambda_*}^2} + \frac{1}{\sqrt{1 - \mu^2 l_{\Lambda_*}^2}} \sqrt{\frac{\mu^4 l_{\Lambda_*}^2}{1 - \mu^2 l_{\Lambda_*}^2} + (|\mathbf{k}|^2 + \mu^2)} \right] d^{D^+, D^-} \mathbf{k}, \quad (243)$$

$$\text{where } K' = \frac{1}{2(2\pi\hbar)^3}, c = 1.$$

**Remark 5.1.2.** We assume now that $\mu^2 l_{\Lambda_*}^2 \ll 1, \mu^4 l_{\Lambda_*}^2 \ll 1$ and therefore from Equation (243) we obtain
high-energy cutoff $\Lambda_*$ to fractal momentum space with Hausdorff-Colombeau negative dimension and therefore the canonical Lebesgue measure $d^3\mathbf{k}$, where $\mathbf{k} = (k_x, k_y, k_z)$ being replaced by the Hausdorff-Colombeau measure

$$d^{D^+,D^-} \mathbf{k} \triangleq \frac{\Delta(D^-) d^{D^+} \mathbf{k}}{(|\mathbf{k}|^{D^+} + \varepsilon)_\varepsilon} = \frac{\Delta(D^+) \Delta(D^-) p^{D^+ - 1} dp}{(p^{|D^-|} + \varepsilon)_\varepsilon}, \quad (239)$$

where $\Delta(D^+) = \frac{2\pi^{D^+/2}}{\Gamma(D^+/2)}$ and $p = |\mathbf{k}| = \sqrt{k_x + k_y + k_z}$.

**Remark 5.1.1.** Note that the integral over measure $d^{D^+,D^-} \mathbf{k}$ is given by formula (185). Thus vacuum energy density $\varepsilon(D^+, D^-, \mu_{\text{eff}}, p_*)$ for free quantum fields is

$$\varepsilon(D^+, D^-, \mu_{\text{eff}}, p_*) = \varepsilon(\mu_{\text{eff}}) + \varepsilon(\mu_{\text{eff}}, p_*) + \bar{\varepsilon}(D^+, D^-, \mu_{\text{eff}}, p_*) \quad (240)$$

Here the quantity $\varepsilon(\mu_{\text{eff}})$ is given by formula

$$\begin{aligned} \varepsilon(\mu_{\text{eff}}) &= \frac{1}{2(2\pi\hbar)^3} \int_0^{\mu_{\text{eff}}} d\mu f(\mu) \int_{||\mathbf{k}|| \le \mu} \sqrt{\mathbf{k}^2 + \mu^2} d^3\mathbf{k} \\ &= K \int_0^{\mu_{\text{eff}}} d\mu f(\mu) \int_{p \le \mu} \sqrt{p^2 + \mu^2} p^2 dp \quad (241) \\ &= K \int_0^{\mu_{\text{eff}}} d\mu f(\mu) \int_0^{\mu} \sqrt{p^2 + \mu^2} p^2 dp \end{aligned}$$

where $K = \frac{2\pi}{(2\pi\hbar)^3}, c = 1$. The quantity $\varepsilon(\mu_{\text{eff}}, p_*)$ is given by formula

$$\begin{aligned} \varepsilon(\mu_{\text{eff}}, p_*) &= \frac{1}{2(2\pi\hbar)^3} \int_0^{\mu_{\text{eff}}} d\mu f(\mu) \int_{\mu < \|\mathbf{k}\| < p_*} \sqrt{\mathbf{k}^2 + \mu^2} d^3\mathbf{k} \\ &= K \int_0^{\mu_{\text{eff}}} d\mu f(\mu) \int_{\mu < \|\mathbf{k}\| < p_*} \sqrt{p^2 + \mu^2} p^2 dp. \end{aligned} \quad (242)$$

The quantity $\bar{\varepsilon}(D^+, D^-, \mu_{\text{eff}}, p_*)$ (since Equation (22) holds) is given by formula

$$\bar{\varepsilon}(D^+, D^-, \mu_{\text{eff}}, p_*) = K \int_0^{\mu_{\text{eff}}} d\mu f(\mu) \int_{||\mathbf{k}|| \ge p_*} \left[ \frac{\mu^2 l_{\Lambda_*}}{1 - \mu^2 l_{\Lambda_*}^2} + \frac{1}{\sqrt{1 - \mu^2 l_{\Lambda_*}^2}} \sqrt{\frac{\mu^4 l_{\Lambda_*}^2}{1 - \mu^2 l_{\Lambda_*}^2} + (|\mathbf{k}|^2 + \mu^2)} \right] d^{D^+, D^-} \mathbf{k}, \quad (243)$$

where $K' = \frac{1}{2(2\pi\hbar)^3}, c = 1$.

**Remark 5.1.2.** We assume now that $\mu^2 l_{\Lambda_*}^2 \ll 1, \mu^4 l_{\Lambda_*}^2 \ll 1$ and therefore from Equation (243) we obtain
$$ \begin{aligned} & \varepsilon(D^+, D^-, \mu_{\text{eff}}, p_*) \\ &= K' \eta_{\Lambda} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu \int_{\|\mathbf{k}\| \ge p_*} d^{3,D} \mathbf{k} + K' \int_{0}^{\mu_{\text{eff}}} d\mu f(\mu) \int_{\|\mathbf{k}\| \ge p_*} \sqrt{\mathbf{k}^2 + \mu^2} d^{D^+, D^-} \mathbf{k}. \end{aligned} \quad (244) $$

From Equation (244) and Equation (239) we obtain

$$ \begin{aligned} & \varepsilon(D^+, D^-, \mu_{\text{eff}}, p_*) \\ &= K' \eta_{\Lambda} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu \int_{\|\mathbf{k}\| \ge p_*} d^{D^+, D^-} \mathbf{k} + K' \int_{0}^{\mu_{\text{eff}}} d\mu f(\mu) \int_{\|\mathbf{k}\| \ge p_*} \sqrt{\mathbf{k}^2 + \mu^2} d^{D^+, D^-} \mathbf{k} \\ &= (K' \eta_{\Lambda} \Delta(D^+) \Delta(D^-) \int_{0}^{\mu_{\text{eff}}} f d\mu(\mu) \mu^2) \int_{p_*}^{\infty} \frac{p^{D^+-1} dp}{(p^{|D^-|} + \varepsilon)_{\varepsilon}} \\ &\quad + K' \Delta(D^+) \Delta(D^-) \int_{0}^{\mu_{\text{eff}}} d\mu f(\mu) \int_{p_*}^{\infty} \frac{\sqrt{p^2 + \mu^2} p^{D^+-1} dp}{(p^{|D^-|} + \varepsilon)_{\varepsilon}} \\ &= (K' \eta_{\Lambda} \Delta(D^+) \Delta(D^-) \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu) \int_{p_*}^{\infty} p^{D^-+D^+-1} dp \\ &\quad + K' \Delta(D^+) \Delta(D^-) \int_{0}^{\mu_{\text{eff}}} d\mu f(\mu) \int_{p_*}^{\infty} \sqrt{p^2 + \mu^2} p^{D^-+D^+-1} dp. \end{aligned} \quad (245) $$

**Remark 5.1.2.** We assume now that:

$$ D^{-} + D^{+} + 2 \le -6. \quad (246) $$

Note that

$$ \begin{aligned} & \int_{0}^{\mu_{\text{eff}}} d\mu f(\mu) \int_{p_*}^{\infty} \sqrt{p^2 + \mu^2} p^{D^{-}+D^{+}-1} dp \\ &= \int_{0}^{\mu_{\text{eff}}} d\mu f(\mu) \int_{p_*}^{\infty} \sqrt{1 + \frac{\mu^2}{p^2}} p^{D^{-}+D^{+}} dp \\ &= \int_{0}^{\mu_{\text{eff}}} f(\mu) d\mu \int_{p_*}^{\infty} p^{D^{-}+D^{+}} dp + \frac{1}{2} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu \int_{p_*}^{\infty} p^{D^{-}+D^{+}-1} dp \\ &\quad - \frac{1}{8} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu \int_{p_*}^{\infty} p^{D^{-}+D^{+}-3} dp + O(p_*^{D^{-}+D^{+}-4}) \\ &= \frac{p_*^{D^{-}+D^{+}+1}}{D^{-}+D^{+}+1} \int_{0}^{\mu_{\text{eff}}} f(\mu) d\mu + \frac{p_*^{D^{-}+D^{+}}}{2(D^{-}+D^{+})} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu \\ &\quad - \frac{p_*^{D^{-}+D^{+}-1}}{8(D^{-}+D^{+}-1)} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu + O(p_*^{D^{-}+D^{+}-4}). \end{aligned} \quad (247) $$

Thus finally we obtain

$$ \begin{aligned} & \varepsilon(D^+, D^-, \mu_{\text{eff}}, p_*) \\ &= \frac{K' p_*^{D^-+D^+ + 1}}{D^- + D^+ + 1} \int_0^{\mu_{\text{eff}}} f(\mu) d\mu \\ &\quad + \left( \left[ K' \eta_{\Lambda} \Delta(D^+) \Delta(D^-) + 0.5 \right] \int_0^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu \right) \frac{p_*^{D^-+D^+}}{D^- + D^+} \\ &\quad - \frac{K' p_*^{D^-+D^+ - 2}}{8(D^- + D^+ - 1)} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu + O(p_*^{D^-+D^+ - 4}). \end{aligned} \quad (248) $$
$$ \begin{aligned} & \varepsilon(D^+, D^-, \mu_{\text{eff}}, p_*) \\ &= K' \eta_{\Lambda} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu \int_{\|\mathbf{k}\| \ge p_*} d^{3,D} \mathbf{k} + K' \int_{0}^{\mu_{\text{eff}}} d\mu f(\mu) \int_{\|\mathbf{k}\| \ge p_*} \sqrt{\mathbf{k}^2 + \mu^2} d^{D^+, D^-} \mathbf{k}. \end{aligned} \quad (244) $$

From Equation (244) and Equation (239) we obtain

$$ \begin{aligned} & \varepsilon(D^+, D^-, \mu_{\text{eff}}, p_*) \\ &= K' \eta_{\Lambda} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu \int_{\|\mathbf{k}\| \ge p_*} d^{D^+, D^-} \mathbf{k} + K' \int_{0}^{\mu_{\text{eff}}} d\mu f(\mu) \int_{\|\mathbf{k}\| \ge p_*} \sqrt{\mathbf{k}^2 + \mu^2} d^{D^+, D^-} \mathbf{k} \\ &= (K' \eta_{\Lambda} \Delta(D^+) \Delta(D^-) \int_{0}^{\mu_{\text{eff}}} f d\mu(\mu) \mu^2) \int_{p_*}^{\infty} \frac{p^{D^+ - 1} dp}{(p^{|D^-|} + \varepsilon)_{\varepsilon}} \\ &\quad + K' \Delta(D^+) \Delta(D^-) \int_{0}^{\mu_{\text{eff}}} d\mu f(\mu) \int_{p_*}^{\infty} \frac{\sqrt{p^2 + \mu^2} p^{D^+ - 1} dp}{(p^{|D^-|} + \varepsilon)_{\varepsilon}} \end{aligned} \quad (245) \\ &= (K' \eta_{\Lambda} \Delta(D^+) \Delta(D^-) \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu) \int_{p_*}^{\infty} p^{D^- + D^+ - 1} dp \\ &\quad + K' \Delta(D^+) \Delta(D^-) \int_{0}^{\mu_{\text{eff}}} d\mu f(\mu) \int_{p_*}^{\infty} \sqrt{p^2 + \mu^2} p^{D^- + D^+ - 1} dp. \end{aligned} $$

**Remark 5.1.2.** We assume now that:

$$ D^{-} + D^{+} + 2 \le -6. \quad (246) $$

Note that

$$ \begin{aligned} & \int_{0}^{\mu_{\text{eff}}} d\mu f(\mu) \int_{p_*}^{\infty} \sqrt{p^2 + \mu^2} p^{D^{-} + D^{+} - 1} dp \\ &= \int_{0}^{\mu_{\text{eff}}} d\mu f(\mu) \int_{p_*}^{\infty} \sqrt{1 + \frac{\mu^2}{p^2}} p^{D^{-} + D^{+}} dp \\ &= \int_{0}^{\mu_{\text{eff}}} f(\mu) d\mu \int_{p_*}^{\infty} p^{D^{-} + D^{+}} dp + \frac{1}{2} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu \int_{p_*}^{\infty} p^{D^{-} + D^{+} - 1} dp \\ &\quad - \frac{1}{8} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu \int_{p_*}^{\infty} p^{D^{-} + D^{+} - 3} dp + O(p_*^{D^{-} + D^{+} - 4}) \end{aligned} \quad (247) \\ &= \frac{p_*^{D^{-} + D^{+} + 1}}{D^{-} + D^{+} + 1} \int_{0}^{\mu_{\text{eff}}} f(\mu) d\mu + \frac{p_*^{D^{-} + D^{+}}}{2(D^{-} + D^{+})} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu \\ &\quad - \frac{p_*^{D^{-} + D^{+} - 1}}{8(D^{-} + D^{+} - 1)} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu + O(p_*^{D^{-} + D^{+} - 4}). \end{aligned} $$

Thus finally we obtain

$$ \begin{aligned} & \varepsilon(D^+, D^-, \mu_{\text{eff}}, p_*) \\ &= \frac{K' p_*^{D^{-} + D^{+} + 1}}{D^{-} + D^{+} + 1} \int_{0}^{\mu_{\text{eff}}} f(\mu) d\mu \\ &\quad + \left( \left[ K' \eta_{\Lambda} \Delta(D^+) \Delta(D^-) + 0.5 \right] \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu \right) \frac{p_*^{D^{-} + D^{+}}}{D^{-} + D^{+}} \end{aligned} \quad (248) \\ &\quad - \frac{K' p_*^{D^{-} + D^{+} - 2}}{8(D^{-} + D^{+} - 1)} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu + O(p_*^{D^{-} + D^{+} - 4}). \end{aligned} $$
**Remark 5.1.3.** Note that (see Equations (42)):

$$ 
\begin{aligned}
\tilde{\varepsilon}(\mu_{\text{eff}}, p_{*}) &= \varepsilon(\mu_{\text{eff}}) + \varepsilon(\mu_{\text{eff}}, p_{*}) \\
&= \frac{1}{4} p_{*}^{4} \int_{0}^{\mu_{\text{eff}}} f(\mu) d\mu + \frac{1}{4} p_{*}^{2} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{2} d\mu + \left(C_{1} - \frac{1}{8} \ln p_{*}\right) \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} d\mu \\
&\quad + \frac{1}{8} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} (\ln \mu) d\mu - \left(\frac{1}{p_{*}^{2}}\right) \frac{1}{32} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{6} d\mu + O\left(\int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{8}\right) p_{*}^{-5}.
\end{aligned}
\quad (249) $$

From Equation (240), Equation (248) and Equation (249) finally we obtain

$$ 
\begin{aligned}
\varepsilon(D^{+}, D^{-}, \mu_{\text{eff}}, p_{*}) &= \varepsilon(\mu_{\text{eff}}) + \varepsilon(\mu_{\text{eff}}, p_{*}) + \tilde{\varepsilon}(D^{+}, D^{-}, \mu_{\text{eff}}, p_{*}) \\
&= \frac{1}{4} p_{*}^{4} \int_{0}^{\mu_{\text{eff}}} f(\mu) d\mu + \frac{1}{4} p_{*}^{2} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{2} d\mu + \left(C_{1} - \frac{1}{8} \ln p_{*}\right) \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} d\mu \\
&\quad + \frac{1}{8} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} (\ln \mu) d\mu - \left(\frac{1}{p_{*}^{2}}\right) \frac{1}{32} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{6} d\mu + O\left(\int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{8}\right) p_{*}^{-5} \\
&\quad + O\left(p_{*}^{D^{-}+D^{+}+2}\right).
\end{aligned}
\quad (250) $$

The pressure $p(D^+, D^-, \mu_{\text{eff}}, p_*)$ for free scalar quantum field is

$$ p(D^{+}, D^{-}, \mu_{\text{eff}}, p_{*}) = p(\mu_{\text{eff}}) + p(\mu_{\text{eff}}, p_{*}) + \tilde{p}(D^{+}, D^{-}, \mu_{\text{eff}}, p_{*}). \quad (251) $$

Here the quantity $p(\mu_{\text{eff}})$ is given by formula

$$ p(\mu_{\text{eff}}) = \frac{K}{3} \int_{0}^{\mu_{\text{eff}}} d\mu f(\mu) \int_{\|p\|<\mu} \frac{p^4}{\sqrt{p^2 + \mu^2}} dp. \quad (252) $$

The quantity $p(\mu_{\text{eff}}, p_*)$ is given by formula

$$ p(\mu_{\text{eff}}, p_{*}) = \frac{K}{3} \int_{0}^{\mu_{\text{eff}}} d\mu f(\mu) \int_{\mu \le \|p\| \le p_{*}} \frac{p^4}{\sqrt{p^2 + \mu^2}} dp. \quad (253) $$

The quantity $\tilde{p}(D^+, D^-, \mu_{\text{eff}}, p_*)$ is given by formula

$$ \tilde{p}(D^{+}, D^{-}, \mu_{\text{eff}}, p_{*}) \approx \frac{K'}{3} \int_{0}^{\mu_{\text{eff}}} d\mu \int_{\|p\| > p_{*}} f(\mu) \frac{p^4}{\sqrt{p^2 + \mu^2}} dp, \quad (254) $$

where $K' = \frac{1}{2(2\pi\hbar)^3}, c = 1$.

**Remark 5.1.4.** Note that (see Equations (42)):

$$ 
\begin{aligned}
\tilde{p}(\mu_{\text{eff}}, p_{*}) &= p(\mu_{\text{eff}}) + p(\mu_{\text{eff}}, p_{*}) \\
&= \frac{1}{12} p_{*}^{4} \int_{0}^{\mu_{\text{eff}}} f(\mu) d\mu - \frac{1}{12} p_{*}^{2} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{2} d\mu \\
&\quad + \left(C_{2} + \frac{1}{8} \ln p_{*}\right) \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} d\mu - \frac{1}{8} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} (\ln \mu) d\mu \\
&\quad + \left(\frac{5}{p_{*}^{2}}\right) \frac{1}{32} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{6} d\mu + O\left(\int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{8}\right) p_{*}^{-5}.
\end{aligned}
\quad (255) $$

From Equation (250), Equation (254) and Equation (255) similarly as above
finally we get
**Remark 5.1.3.** Note that (see Equations (42)):

$$ 
\begin{aligned}
\tilde{\varepsilon}(\mu_{\text{eff}}, p_*) &= \varepsilon(\mu_{\text{eff}}) + \varepsilon(\mu_{\text{eff}}, p_*) \\
&= \frac{1}{4} p_*^4 \int_0^{\mu_{\text{eff}}} f(\mu) d\mu + \frac{1}{4} p_*^2 \int_0^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu + \left(C_1 - \frac{1}{8} \ln p_*\right) \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu \\
&\quad + \frac{1}{8} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 (\ln \mu) d\mu - \left(\frac{1}{p_*^2}\right) \frac{1}{32} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^6 d\mu + O\left(\int_0^{\mu_{\text{eff}}} f(\mu) \mu^8\right) p_*^{-5}.
\end{aligned}
\quad (249) $$

From Equation (240), Equation (248) and Equation (249) finally we obtain

$$ 
\begin{aligned}
\varepsilon(D^+, D^-, \mu_{\text{eff}}, p_*) &= \varepsilon(\mu_{\text{eff}}) + \varepsilon(\mu_{\text{eff}}, p_*) + \tilde{\varepsilon}(D^+, D^-, \mu_{\text{eff}}, p_*) \\
&= \frac{1}{4} p_*^4 \int_0^{\mu_{\text{eff}}} f(\mu) d\mu + \frac{1}{4} p_*^2 \int_0^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu + \left(C_1 - \frac{1}{8} \ln p_*\right) \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu \\
&\quad + \frac{1}{8} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 (\ln \mu) d\mu - \left(\frac{1}{p_*^2}\right) \frac{1}{32} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^6 d\mu + O\left(\int_0^{\mu_{\text{eff}}} f(\mu) \mu^8\right) p_*^{-5} \\
&\quad + O\left(p_*^{D^- + D^+ + 2}\right).
\end{aligned}
\quad (250) $$

The pressure $p(D^+, D^-, \mu_{\text{eff}}, p_*)$ for free scalar quantum field is

$$ p(D^+, D^-, \mu_{\text{eff}}, p_*) = p(\mu_{\text{eff}}) + p(\mu_{\text{eff}}, p_*) + \tilde{p}(D^+, D^-, \mu_{\text{eff}}, p_*) \quad (251) $$

Here the quantity $p(\mu_{\text{eff}})$ is given by formula

$$ p(\mu_{\text{eff}}) = \frac{K}{3} \int_{0}^{\mu_{\text{eff}}} d\mu f(\mu) \int_{\|p\|<\mu} \frac{p^4}{\sqrt{p^2 + \mu^2}} dp. \quad (252) $$

The quantity $p(\mu_{\text{eff}}, p_*)$ is given by formula

$$ p(\mu_{\text{eff}}, p_*) = \frac{K}{3} \int_{0}^{\mu_{\text{eff}}} d\mu f(\mu) \int_{\mu \le \|p\| \le p_*} \frac{p^4}{\sqrt{p^2 + \mu^2}} dp. \quad (253) $$

The quantity $\tilde{p}(D^+, D^-, \mu_{\text{eff}}, p_*)$ is given by formula

$$ \tilde{p}(D^+, D^-, \mu_{\text{eff}}, p_*) \approx \frac{K'}{3} \int_0^{\mu_{\text{eff}}} d\mu \int_{\|p\| > p_*} f(\mu) \frac{p^4}{\sqrt{p^2 + \mu^2}} dp, \quad (254) $$

where $K' = \frac{1}{2(2\pi\hbar)^3}, c = 1$.

**Remark 5.1.4.** Note that (see Equations (42)):

$$ 
\begin{aligned}
\tilde{p}(\mu_{\text{eff}}, p_*) &= p(\mu_{\text{eff}}) + p(\mu_{\text{eff}}, p_*) \\
&= \frac{1}{12} p_*^4 \int_0^{\mu_{\text{eff}}} f(\mu) d\mu - \frac{1}{12} p_*^2 \int_0^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu \\
&\quad + \left(C_2 + \frac{1}{8} \ln p_*\right) \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu - \frac{1}{8} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 (\ln \mu) d\mu \\
&\quad + \left(\frac{5}{p_*^2}\right) \frac{1}{32} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^6 d\mu + O\left(\int_0^{\mu_{\text{eff}}} f(\mu) \mu^8\right) p_*^{-5}.
\end{aligned}
\quad (255) $$

From Equation (250), Equation (254) and Equation (255) similarly as above
finally we get
$$
\begin{align}
p(D^+, D^-, \mu_{\text{eff}}, p_*) & \\
&= \frac{1}{12} p_*^4 \int_0^{\mu_{\text{eff}}} f(\mu) \, d\mu - \frac{1}{12} p_*^2 \int_0^{\mu_{\text{eff}}} f(\mu) \, \mu^2 \, d\mu \nonumber \\
&\quad + \left(C_2 + \frac{1}{8} \ln p_*\right) \int_0^{\mu_{\text{eff}}} f(\mu) \, \mu^4 \, d\mu - \frac{1}{8} \int_0^{\mu_{\text{eff}}} f(\mu) \, \mu^4 (\ln \mu) \, d\mu \nonumber \\
&\quad + \left(\frac{5}{p_*^2}\right) \frac{1}{32} \int_0^{\mu_{\text{eff}}} f(\mu) \, \mu^6 \, d\mu + O\left(\int_0^{\mu_{\text{eff}}} f(\mu) \, \mu^8\right) p_*^{-5} + O\left(p_*^{D^-+D^++2}\right). \tag{256}
\end{align}
$$

**Remark 5.1.5.** We assume now that:

$$ \int_0^{\mu_{\text{eff}}} f(\mu) d\mu = \int_0^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu = \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu = 0. \quad (257) $$

From Equation (250), Equation (256) and Equation (257) finally we get

$$
\begin{align}
\varepsilon \triangleq \varepsilon(D^+, D^-, \mu_{\text{eff}}, p_*) &= \frac{1}{8} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 (\ln \mu) d\mu + O(p_*^{-2}), \tag{258} \\
p \triangleq (D^+, D^-, \mu_{\text{eff}}, p_*) &= -\frac{1}{8} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 (\ln \mu) d\mu + O(p_*^{-2}). \nonumber
\end{align}
$$

**Remark 5.1.6.** Note that the Equation (258) can be obtained without fine-tuning (257) which was assumed in Zel'dovich paper [1].

In order to obtain Equation (5.1.23) and strictly weaker conditions we assume now that:

1)

$$ |f(\mu)| = |f_{s.m.}(\mu) + f_{g.m.}(\mu)| = \mu_{\text{eff}}^{-n}, \quad (259) $$

where $n > 0$ is an parameter, $f_{s.m.}(\mu)$ corresponds to standard matter and where $f_{g.m.}(\mu)$ corresponds to physical ghost matter, see Equation (32).

2)

$$
\begin{align}
I_1 &= p_*^4 \int_0^{\mu_{\text{eff}}} f(\mu) \, d\mu \approx 0, \nonumber \\
I_2 &= p_*^2 \int_0^{\mu_{\text{eff}}} f(\mu) \, \mu^2 \, d\mu \approx 0, \tag{260} \\
I_3 &= \ln p_* \int_0^{\mu_{\text{eff}}} f(\mu) \, \mu^4 \, d\mu \approx 0 \nonumber
\end{align}
$$

3)

$$ |I_1 + I_2 + I_3| \ll \left| \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 (\ln \mu) d\mu \right|. \quad (261) $$

## 5.2. Zeropoint Energy Density Corresponding to a Non-Singular Gliner Cosmology

We assume now that

$$ \int_0^{\mu_{\text{eff}}} f(\mu) d\mu = 0, \quad \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu < 0, \quad \int_0^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu > 0, \quad p_* \gg \mu_{\text{eff}}. \quad (262) $$
$$ 
\begin{aligned}
p(D^+, D^-, \mu_{\text{eff}}, p_*) &= \frac{1}{12} p_*^4 \int_0^{\mu_{\text{eff}}} f(\mu) d\mu - \frac{1}{12} p_*^2 \int_0^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu \\
&\quad + \left(C_2 + \frac{1}{8} \ln p_*\right) \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu - \frac{1}{8} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 (\ln \mu) d\mu \\
&\quad + \left(\frac{5}{p_*^2}\right) \frac{1}{32} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^6 d\mu + O\left(\int_0^{\mu_{\text{eff}}} f(\mu) \mu^8\right) p_*^{-5} + O\left(p_*^{D^-+D^++2}\right).
\end{aligned} 
\quad (256) $$

**Remark 5.1.5.** We assume now that:

$$ \int_0^{\mu_{\text{eff}}} f(\mu) d\mu = \int_0^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu = \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu = 0. \quad (257) $$

From Equation (250), Equation (256) and Equation (257) finally we get

$$ 
\begin{aligned}
\varepsilon \triangleq \varepsilon(D^+, D^-, \mu_{\text{eff}}, p_*) &= \frac{1}{8} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 (\ln \mu) d\mu + O(p_*^{-2}), \\
p \triangleq (D^+, D^-, \mu_{\text{eff}}, p_*) &= -\frac{1}{8} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 (\ln \mu) d\mu + O(p_*^{-2}).
\end{aligned} 
\quad (258) $$

**Remark 5.1.6.** Note that the Equation (258) can be obtained without fine-tuning (257) which was assumed in Zel'dovich paper [1].

In order to obtain Equation (5.1.23) and strictly weaker conditions we assume now that:

1)

$$ |f(\mu)| = |f_{s.m.}(\mu) + f_{g.m.}(\mu)| = \mu_{\text{eff}}^{-n}, \quad (259) $$

where $n > 0$ is an parameter, $f_{s.m.}(\mu)$ corresponds to standard matter and where $f_{g.m.}(\mu)$ corresponds to physical ghost matter, see Equation (32).

2)

$$ 
\begin{aligned}
I_1 &= p_*^4 \int_0^{\mu_{\text{eff}}} f(\mu) d\mu \approx 0, \\
I_2 &= p_*^2 \int_0^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu \approx 0, \\
I_3 &= \ln p_* \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu \approx 0
\end{aligned} 
\quad (260) $$

3)

$$ |I_1 + I_2 + I_3| \ll \left| \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 (\ln \mu) d\mu \right|. \quad (261) $$

## 5.2. Zeropoint Energy Density Corresponding to a Non-Singular Gliner Cosmology

We assume now that

$$ \int_0^{\mu_{\text{eff}}} f(\mu) d\mu = 0, \quad \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu < 0, \quad \int_0^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu > 0, \quad p_* \gg \mu_{\text{eff}}. \quad (262) $$
From Equation (250), Equation (256) and (262) we obtain

$$
\begin{aligned}
\varepsilon \triangleq \varepsilon(D^{+}, D^{-}, \mu_{\text{eff}}, p_{*}) \\
&= \frac{1}{4} p_{*}^{2} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{2} d\mu - \left(C_{1} - \frac{1}{8} \ln p_{*}\right) \left|\int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} d\mu\right| \\
&\quad + \frac{1}{8} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} (\ln \mu) d\mu - \left(\frac{1}{p_{*}^{2}}\right) \frac{1}{32} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{6} d\mu \\
&\quad + O\left(\int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{8}\right) p_{*}^{-5} + O\left(p_{*}^{D^{-}+D^{+}+2}\right),
\end{aligned}
\qquad (263)
$$

and

$$
\begin{aligned}
p \triangleq p(D^{+}, D^{-}, \mu_{\text{eff}}, p_{*}) \\
&= -\frac{1}{12} p_{*}^{2} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{2} d\mu - \left(C_{2} + \frac{1}{8} \ln p_{*}\right) \left|\int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} d\mu\right| \\
&\quad -\frac{1}{8} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} (\ln \mu) d\mu + \left(\frac{5}{p_{*}^{2}}\right) \frac{1}{32} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{6} d\mu \\
&\quad + O\left(\int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{8}\right) p_{*}^{-5} + O\left(p_{*}^{D^{-}+D^{+}+2}\right)
\end{aligned}
\qquad (364)
$$

correspondingly. From Equation (263) and Equation (264) we obtain

$$
\begin{aligned}
3p + \varepsilon &= -\frac{1}{4} p_*^2 \left| \int_0^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu - \left(3C_2 + \frac{3}{8} \ln p_*\right) \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu \right| \\
&\quad -\frac{3}{8} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 (\ln \mu) d\mu + \left(\frac{5}{p_*^2}\right) \frac{3}{32} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^6 d\mu \\
&\quad +\frac{1}{4} p_*^2 \left| \int_0^{\mu_{\text{eff}}} f(\mu) \mu^2 d\mu - \left(C_1 - \frac{1}{8} \ln p_*\right) \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu \right| \\
&\quad +\frac{1}{8} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 (\ln \mu) d\mu - \left(\frac{1}{p_*^2}\right) \frac{1}{32} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^6 d\mu \\
&= -\frac{1}{4} \ln p_* \left| \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu \right| - \left(3C_2 + C_1\right) \left| \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 d\mu \right| \\
&\quad -\frac{1}{4} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^4 (\ln \mu) d\mu + \left(\frac{5}{p_*^2}\right) \frac{1}{16} \int_0^{\mu_{\text{eff}}} f(\mu) \mu^6 d\mu < 0.
\end{aligned}
\qquad (265)
$$

Therefore under conditions (262) the inequality

$$ -2\varepsilon < 3p + \varepsilon < 0 \qquad (266) $$

corresponding to Gliner non-singular cosmology [2] [4] is satisfied.

## 5.3. Zeropoint Energy Density in Models with Supermassive Physical Ghost Fields

We assume now that:

1) ghost fields corresponding to massive spin-2 particle with mass $m_2$ and to massive scalar particle with mass $m_0$ appears (see Subsection 2.2) as real phys-
From Equation (250), Equation (256) and (262) we obtain

$$ 
\begin{aligned}
\varepsilon \triangleq \varepsilon(D^{+}, D^{-}, \mu_{\text{eff}}, p_{*}) \\
&= \frac{1}{4} p_{*}^{2} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{2} d\mu - \left(C_{1} - \frac{1}{8} \ln p_{*}\right) \left|\int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} d\mu\right| \\
&\quad + \frac{1}{8} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} (\ln \mu) d\mu - \left(\frac{1}{p_{*}^{2}}\right) \frac{1}{32} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{6} d\mu \\
&\quad + O\left(\int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{8}\right) p_{*}^{-5} + O\left(p_{*}^{D^{-}+D^{+}+2}\right),
\end{aligned}
\quad (263) $$

and

$$ 
\begin{aligned}
p \triangleq p(D^{+}, D^{-}, \mu_{\text{eff}}, p_{*}) \\
&= -\frac{1}{12} p_{*}^{2} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{2} d\mu - \left(C_{2} + \frac{1}{8} \ln p_{*}\right) \left|\int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} d\mu\right| \\
&\quad -\frac{1}{8} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} (\ln \mu) d\mu + \left(\frac{5}{p_{*}^{2}}\right) \frac{1}{32} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{6} d\mu \\
&\quad + O\left(\int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{8}\right) p_{*}^{-5} + O\left(p_{*}^{D^{-}+D^{+}+2}\right)
\end{aligned}
\quad (364) $$

correspondingly. From Equation (263) and Equation (264) we obtain

$$ 
\begin{aligned}
3p + \varepsilon = & -\frac{1}{4} p_{*}^{2} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{2} d\mu - \left(3C_{2} + \frac{3}{8} \ln p_{*}\right) \left|\int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} d\mu\right| \\
& -\frac{3}{8} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} (\ln \mu) d\mu + \left(\frac{5}{p_{*}^{2}}\right) \frac{3}{32} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{6} d\mu \\
& +\frac{1}{4} p_{*}^{2} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{2} d\mu - \left(C_{1} - \frac{1}{8} \ln p_{*}\right) \left|\int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} d\mu\right| \\
& +\frac{1}{8} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} (\ln \mu) d\mu - \left(\frac{1}{p_{*}^{2}}\right) \frac{1}{32} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{6} d\mu \\
& = -\frac{1}{4} \ln p_{*} \left|\int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} d\mu\right| - \left(3C_{2} + C_{1}\right) \left|\int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} d\mu\right| \\
& -\frac{1}{4} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} (\ln \mu) d\mu + \left(\frac{5}{p_{*}^{2}}\right) \frac{1}{16} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{6} d\mu < 0.
\end{aligned}
\quad (265) $$

Therefore under conditions (262) the inequality

$$ -2\varepsilon < 3p + \varepsilon < 0 \quad (266) $$

corresponding to Gliner non-singular cosmology [2] [4] is satisfied.

## 5.3. Zeropoint Energy Density in Models with Supermassive Physical Ghost Fields

We assume now that:

1) ghost fields corresponding to massive spin-2 particle with mass $m_2$ and to massive scalar particle with mass $m_0$ appears (see Subsection 2.2) as real phys-
ical fields in action (91).

**Remark 5.3.1.** Note that their unphysical behavior may be restricted to arbitrarily high-energy cutoff $\Lambda$ 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 problem with unitarity and thus require a very large mass (see Subsection II.2).

2) Poincaré group is deformed at some fundamental high-energy cutoff $\Lambda_*$

$$ \Lambda_* = \Lambda_*(m_0, m_2) \ll m_0 c^2 < m_2 c^2. \qquad (267) $$

The canonical quadratic invariant $\|p\|^2 = \eta^{ab} p_a p_b$ collapses at high-energy cutoff $\Lambda_*$ and being replaced by the non-quadratic invariant

$$ \|p\|^2 = \frac{\eta^{ab} p_a p_b}{(1 + l_{\Lambda_*} p_0)}. \qquad (268) $$

3) The canonical concept of Minkowski space-time collapses at a small distance to fractal space-time with Hausdorff-Colombeau negative dimension and therefore the canonical Lebesgue measure $d^4x$ being replaced by the Colombeau-Stieltjes measure

$$ (d\eta(x, \varepsilon))_{\varepsilon} = (v_{\varepsilon}(s(x))d^4x)_{\varepsilon}, \qquad (269) $$

where

$$ (v_{\varepsilon}(s(x)))_{\varepsilon} = \left( \left( |s(x)|^{D^{-}} + \varepsilon \right)^{-1} \right)_{\varepsilon}, \quad s(x) = \sqrt{x_{\mu} x^{\mu}}, \qquad (270) $$

4) we assume that

$$ f(\mu) = f_{s.m.}(\mu) + f_{g.m.}(\mu), \qquad (271) $$

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

**Remark 5.3.2.** We assume now that

$$ |f(\mu)| = \begin{cases} O(\mu^{-n}), & n > 1 \quad m_0 c \ll \mu_{\text{eff}}^1 \le \mu \le \mu_{\text{eff}}^2 \ll m_2 c \\ 0, & \mu > \mu_{\text{eff}}^2 \end{cases} \qquad (272) $$

Thus vacuum energy density $\varepsilon(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2)$ for free quantum fields is

$$ \varepsilon(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2) = \varepsilon(\mu_{\text{eff}}^1, \mu_{\text{eff}}^2) + \bar{\varepsilon}(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2). \qquad (273) $$

Here the quantity $\varepsilon(\mu_{\text{eff}}^1, \mu_{\text{eff}}^2)$ is given by formula

$$ \begin{aligned} \varepsilon(\mu_{\text{eff}}^1, \mu_{\text{eff}}^2) &= \frac{1}{2(2\pi\hbar)^3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{\|\mathbf{k}\| \le \sqrt{\mu}} \sqrt{\mathbf{k}^2 + \mu^2} d^3\mathbf{k} \\ &= K \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{p \le \sqrt{\mu}} \sqrt{p^2 + \mu^2} p^2 dp, \end{aligned} \qquad (274) $$

where $K = \frac{2\pi}{(2\pi\hbar)^3}, c = 1$. The quantity $\bar{\varepsilon}(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2)$ is given by formula
ical fields in action (91).

**Remark 5.3.1.** Note that their unphysical behavior may be restricted to arbitrarily high-energy cutoff $\Lambda$ 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 problem with unitarity and thus require a very large mass (see Subsection II.2).

2) Poincaré group is deformed at some fundamental high-energy cutoff $\Lambda_*$

$$ \Lambda_* = \Lambda_*(m_0, m_2) \ll m_0 c^2 < m_2 c^2. \qquad (267) $$

The canonical quadratic invariant $\|p\|^2 = \eta^{ab} p_a p_b$ collapses at high-energy cutoff $\Lambda_*$ and being replaced by the non-quadratic invariant

$$ \|p\|^2 = \frac{\eta^{ab} p_a p_b}{(1 + l_{\Lambda_*} p_0)}. \qquad (268) $$

3) The canonical concept of Minkowski space-time collapses at a small distance to fractal space-time with Hausdorff-Colombeau negative dimension and therefore the canonical Lebesgue measure $d^4x$ being replaced by the Colombeau-Stieltjes measure

$$ (d\eta(x, \varepsilon))_{\varepsilon} = (v_{\varepsilon}(s(x))d^4x)_{\varepsilon}, \qquad (269) $$

where

$$ (v_{\varepsilon}(s(x)))_{\varepsilon} = \left( \left( |s(x)|^{D^{-}} + \varepsilon \right)^{-1} \right)_{\varepsilon}, \quad s(x) = \sqrt{x_{\mu} x^{\mu}}, \qquad (270) $$

4) we assume that

$$ f(\mu) = f_{s.m.}(\mu) + f_{g.m.}(\mu), \qquad (271) $$

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

**Remark 5.3.2.** We assume now that

$$ |f(\mu)| = \begin{cases} O(\mu^{-n}), & n > 1 \quad m_0 c \ll \mu_{\text{eff}}^1 \le \mu \le \mu_{\text{eff}}^2 \ll m_2 c \\ 0, & \mu > \mu_{\text{eff}}^2 \end{cases} \qquad (272) $$

Thus vacuum energy density $\varepsilon(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2)$ for free quantum fields is

$$ \varepsilon(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2) = \varepsilon(\mu_{\text{eff}}^1, \mu_{\text{eff}}^2) + \bar{\varepsilon}(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2). \qquad (273) $$

Here the quantity $\varepsilon(\mu_{\text{eff}}^1, \mu_{\text{eff}}^2)$ is given by formula

$$ \begin{aligned} \varepsilon(\mu_{\text{eff}}^1, \mu_{\text{eff}}^2) &= \frac{1}{2(2\pi\hbar)^3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{\|\mathbf{k}\| \le \sqrt{\mu}} \sqrt{\mathbf{k}^2 + \mu^2} d^3\mathbf{k} \\ &= K \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{p \le \sqrt{\mu}} \sqrt{p^2 + \mu^2} p^2 dp, \end{aligned} \qquad (274) $$

where $K = \frac{2\pi}{(2\pi\hbar)^3}, c = 1$. The quantity $\bar{\varepsilon}(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2)$ is given by formula
$$
\begin{aligned}
& \bar{\epsilon}(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2) \\
& = K' \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{\|\mathbf{k}\| > \sqrt{\mu}} \left[ \frac{\mu^2 l_{\Lambda}}{1 - \mu^2 l_{\Lambda}^2} + \frac{1}{\sqrt{1 - \mu^2 l_{\Lambda}^2}} \sqrt{\frac{\mu^4 l_{\Lambda}^2}{1 - \mu^2 l_{\Lambda}^2} + (|\mathbf{k}|^2 + \mu^2)} \right] d^{D^+, D^-} \mathbf{k},
\end{aligned}
\qquad (275)
$$

where $K' = \frac{1}{2(2\pi\hbar)^3}$, $c = 1$.

**Remark 5.3.2.** We assume now that $\mu^2 l_{\Lambda_*}^2 < 1$, and therefore from Equation (5.3.9) we obtain

$$
\begin{aligned}
& \epsilon(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2) \\
& \approx K' \eta_{\Lambda} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \mu^2 \int_{\|\mathbf{k}\| > \sqrt{\mu}} d^{3,D_{-}} \mathbf{k} + K' \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{\|\mathbf{k}\| > \sqrt{\mu}} \sqrt{\mathbf{k}^2 + \mu^2} d^{D^+, D^-} \mathbf{k}.
\end{aligned}
\qquad (276)
$$

From Equation (276) and Equation (239) we obtain

$$
\begin{aligned}
& \epsilon(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2) \\
& \approx K' \eta_{\Lambda} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \mu^2 \int_{\|\mathbf{k}\| > \sqrt{\mu}} d^{D^+, D^-} \mathbf{k} + K' \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{\|\mathbf{k}\| > \sqrt{\mu}} \sqrt{\mathbf{k}^2 + \mu^2} d^{D^+, D^-} \mathbf{k} \\
& = K' \Delta(D^+) \Delta(D^-) l_{\Lambda} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \mu^2 \left[ \int_{\sqrt{\mu}}^{\infty} \frac{p^{D^-+1} dp}{(p^{|D^-|} + \epsilon)_{\epsilon}} \right] \\
& \qquad + K' \Delta(D^+) \Delta(D^-) \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \left[ \int_{\sqrt{\mu}}^{\infty} \frac{\sqrt{p^2 + \mu^2} p^{D^-+1} dp}{(p^{|D^-|} + \epsilon)_{\epsilon}} \right] \\
& = K' \Delta(D^+) \Delta(D^-) l_{\Lambda} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \mu^2 \left[ \int_{\sqrt{\mu}}^{\infty} p^{D^-+D^+-1} dp \right] \\
& \qquad + K' \Delta(D^+) \Delta(D^-) \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \left[ \int_{\sqrt{\mu}}^{\infty} \sqrt{p^2 + \mu^2} p^{D^-+D^+-1} dp \right].
\end{aligned}
\qquad (277)
$$

Note that

$$
\begin{aligned}
\sqrt{p^2 + \mu^2} &= \mu \sqrt{1 + \frac{p^2}{\mu^2}} = \mu \left( 1 + \frac{1}{2} \frac{p^2}{\mu^2} - \frac{1}{8} \frac{p^4}{\mu^4} + \frac{1}{16} \frac{p^6}{\mu^6} + \dots \right) \\
&= \mu + \frac{1}{2} \frac{p^2}{\mu} - \frac{1}{8} \frac{p^4}{\mu^3} + \frac{1}{16} \frac{p^6}{\mu^5} + \dots
\end{aligned}
\qquad (278)
$$

By inserting Equation (278) into Equation (274) we get

$$
\begin{aligned}
& \epsilon(\mu_{\text{eff}}^1, \mu_{\text{eff}}^2) \\
& = K \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{p \le \sqrt{\mu}} \left( \mu + \frac{1}{2} \frac{p^2}{\mu} - \frac{1}{8} \frac{p^4}{\mu^3} + \frac{1}{16} \frac{p^6}{\mu^5} + \dots \right) p^2 dp \\
& = K \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \left[ \int_{0}^{\sqrt{\mu}} \left( \mu p^2 + \frac{1}{2} \frac{p^4}{\mu} - \frac{1}{8} \frac{p^6}{\mu^3} + \frac{1}{16} \frac{p^8}{\mu^5} + \dots \right) dp \right] \\
& = K \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \left[ \mu \frac{p^3}{3} + \frac{1}{2} \frac{p^5}{5\mu} - \frac{1}{8} \frac{p^7}{7\mu^3} + \frac{1}{16} \frac{p^9}{9\mu^5} + \dots \right]_{0}^{\sqrt{\mu}}
\end{aligned}
$$
$$
\begin{align}
& \bar{\epsilon}(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2) \nonumber \\
& = K' \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{\|\mathbf{k}\| > \sqrt{\mu}} \left[ \frac{\mu^2 l_{\Lambda}}{1 - \mu^2 l_{\Lambda}^2} + \frac{1}{\sqrt{1 - \mu^2 l_{\Lambda}^2}} \sqrt{\frac{\mu^4 l_{\Lambda}^2}{1 - \mu^2 l_{\Lambda}^2} + (|\mathbf{k}|^2 + \mu^2)} \right] d^{D^+, D^-} \mathbf{k}, \tag{275}
\end{align}
$$

where $K' = \frac{1}{2(2\pi\hbar)^3}, c = 1$.

**Remark 5.3.2.** We assume now that $\mu^2 l_{\Lambda_*}^2 < 1$, and therefore from Equation (5.3.9) we obtain

$$
\begin{align}
& \epsilon(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2) \nonumber \\
& \approx K' \eta_{\Lambda} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \mu^2 \int_{\|\mathbf{k}\| > \sqrt{\mu}} d^{3,D_{-}} \mathbf{k} + K' \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{\|\mathbf{k}\| > \sqrt{\mu}} \sqrt{\mathbf{k}^2 + \mu^2} d^{D^+, D^-} \mathbf{k}. \tag{276}
\end{align}
$$

From Equation (276) and Equation (239) we obtain

$$
\begin{align}
& \epsilon(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2) \nonumber \\
& \approx K' \eta_{\Lambda} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \mu^2 \int_{\|\mathbf{k}\| > \sqrt{\mu}} d^{D^+, D^-} \mathbf{k} + K' \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{\|\mathbf{k}\| > \sqrt{\mu}} \sqrt{\mathbf{k}^2 + \mu^2} d^{D^+, D^-} \mathbf{k} \nonumber \\
& = K' \Delta(D^+) \Delta(D^-) l_{\Lambda} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \mu^2 \left[ \int_{\sqrt{\mu}}^{\infty} \frac{p^{D^-+1} dp}{(p^{|D^-|} + \epsilon)_{\epsilon}} \right] \tag{277} \\
& \quad + K' \Delta(D^+) \Delta(D^-) \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \left[ \int_{\sqrt{\mu}}^{\infty} \frac{\sqrt{p^2 + \mu^2} p^{D^-+1} dp}{(p^{|D^-|} + \epsilon)_{\epsilon}} \right] \nonumber \\
& = K' \Delta(D^+) \Delta(D^-) l_{\Lambda} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \mu^2 \left[ \int_{\sqrt{\mu}}^{\infty} p^{D^-+D^+-1} dp \right] \nonumber \\
& \quad + K' \Delta(D^+) \Delta(D^-) \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \left[ \int_{\sqrt{\mu}}^{\infty} \sqrt{p^2 + \mu^2} p^{D^-+D^+-1} dp \right]. \nonumber
\end{align}
$$

Note that

$$
\begin{align}
\sqrt{p^2 + \mu^2} &= \mu \sqrt{1 + \frac{p^2}{\mu^2}} = \mu \left( 1 + \frac{1}{2} \frac{p^2}{\mu^2} - \frac{1}{8} \frac{p^4}{\mu^4} + \frac{1}{16} \frac{p^6}{\mu^6} + \dots \right) \tag{278} \\
&= \mu + \frac{1}{2} \frac{p^2}{\mu} - \frac{1}{8} \frac{p^4}{\mu^3} + \frac{1}{16} \frac{p^6}{\mu^5} + \dots \nonumber
\end{align}
$$

By inserting Equation (278) into Equation (274) we get

$$
\begin{align*}
& \epsilon(\mu_{\text{eff}}^1, \mu_{\text{eff}}^2) \\
&= K \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{p \le \sqrt{\mu}} \left( \mu + \frac{1}{2} \frac{p^2}{\mu} - \frac{1}{8} \frac{p^4}{\mu^3} + \frac{1}{16} \frac{p^6}{\mu^5} + \dots \right) p^2 dp \\
&= K \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \left[ \int_0^{\sqrt{\mu}} \left( \mu p^2 + \frac{1}{2} \frac{p^4}{\mu} - \frac{1}{8} \frac{p^6}{\mu^3} + \frac{1}{16} \frac{p^8}{\mu^5} + \dots \right) dp \right] \\
&= K \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \left[ \mu \frac{p^3}{3} + \frac{1}{2} \frac{p^5}{5\mu} - \frac{1}{8} \frac{p^7}{7\mu^3} + \frac{1}{16} \frac{p^9}{9\mu^5} + \dots \right] \Big|_0^{\sqrt{\mu}}
\end{align*}
$$
$$
\begin{align}
&= K \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \left[ \mu \frac{\mu^2}{3} + \frac{1}{2} \frac{\mu^2}{5\mu} - \frac{1}{8} \frac{\mu^2}{7\mu^3} + \frac{1}{16} \frac{\mu^2}{9\mu^5} + \dots \right] \nonumber \\
&= K \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} f(\mu) d\mu \left[ \frac{1}{3} \frac{\mu^2}{5} + \frac{1}{10} \frac{\mu^2}{3} - \frac{1}{56} \frac{\mu^2}{10} + \frac{1}{144} \mu^{-\frac{1}{2}} + \dots \right] \tag{279} \\
&= K \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} f(\mu) d\mu \left[ \frac{1}{3} \frac{\mu^2}{5} + \frac{1}{10} \frac{\mu^2}{3} - \frac{1}{56} \frac{\mu^2}{10} + \frac{1}{144} \mu^{-\frac{1}{2}} \right] + o\left( (\mu_{\text{eff}}^1)^{-n+1/2} \right). \nonumber
\end{align}
$$

The pressure $p(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2)$ for free quantum fields is

$$p(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2) = p(\mu_{\text{eff}}^1, \mu_{\text{eff}}^2) + \bar{p}(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2). \quad (280)$$

Here the quantity $p(\mu_{\text{eff}}^1, \mu_{\text{eff}}^2)$ is given by formula

$$
\begin{align}
p(\mu_{\text{eff}}^1, \mu_{\text{eff}}^2) &= \frac{1}{2(2\pi\hbar)^3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{\|\mathbf{k}\| \le \sqrt{\mu}} \frac{\|\mathbf{k}\|^2}{\sqrt{\mathbf{k}^2 + \mu^2}} d^3\mathbf{k} \tag{281} \\
&= \frac{K}{3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{p \le \sqrt{\mu}} \frac{p^4}{\sqrt{p^2 + \mu^2}} dp. \nonumber
\end{align}
$$

The quantity $\tilde{p}(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2)$ is given by formula

$$\tilde{p}(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2) \approx \frac{K'}{3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{\|p\| > \sqrt{\mu}} \frac{\|\mathbf{k}\|^2}{\sqrt{\mathbf{k}^2 + \mu^2}} d^{D^+, D^-} \mathbf{k}, \quad (282)$$

where $K' = \frac{1}{2(2\pi\hbar)^3}, c = 1$. Note that

$$
\begin{align}
\frac{1}{\sqrt{p^2 + \mu^2}} &= \mu^{-1} \left( \sqrt{1 + \frac{p^2}{\mu^2}} \right)^{-1} \nonumber \\
&= \mu^{-1} \left( 1 - \frac{1}{2} \frac{p^2}{\mu^2} + \frac{3}{8} \frac{p^4}{\mu^4} - \frac{5}{16} \frac{p^6}{\mu^6} + \dots \right) \tag{283} \\
&= \frac{1}{\mu} - \frac{1}{2} \frac{p^2}{\mu^3} + \frac{3}{8} \frac{p^4}{\mu^5} - \frac{5}{16} \frac{p^6}{\mu^7} + \dots \nonumber
\end{align}
$$

By inserting Equation (283) into Equation (281) we get

$$
\begin{align*}
p(\mu_{\text{eff}}^1, \mu_{\text{eff}}^2) &= \frac{K}{3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{p \le \sqrt{\mu}} \left[ \frac{1}{\mu} - \frac{1}{2} \frac{p^2}{\mu^3} + \frac{3}{8} \frac{p^4}{\mu^5} - \frac{5}{16} \frac{p^6}{\mu^7} + \dots \right] p^4 dp \\
&= \frac{K}{3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{p \le \sqrt{\mu}} \left[ \frac{p^4}{\mu} - \frac{1}{2} \frac{p^6}{\mu^3} + \frac{3}{8} \frac{p^8}{\mu^5} - \frac{5}{16} \frac{p^{10}}{\mu^7} + \dots \right] dp \\
&= \frac{K}{3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \left[ \frac{p^5}{5\mu} - \frac{1}{2} \frac{p^7}{7\mu^3} + \frac{3}{8} \frac{p^9}{9\mu^5} - \frac{5}{16} \frac{p^{11}}{10\mu^7} + \dots \right] \Big|_{0}^{\sqrt{\mu}}
\end{align*}
$$
$$
\begin{align}
&= K \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \left[ \mu \frac{\mu^2}{3} + \frac{1}{2} \frac{\mu^2}{5\mu} - \frac{1}{8} \frac{\mu^2}{7\mu^3} + \frac{1}{16} \frac{\mu^2}{9\mu^5} + \dots \right] \nonumber \\
&= K \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} f(\mu) d\mu \left[ \frac{1}{3} \frac{\mu^2}{5} + \frac{1}{10} \frac{\mu^2}{3} - \frac{1}{56} \frac{\mu^2}{10} + \frac{1}{144} \mu^{-\frac{1}{2}} + \dots \right] \tag{279} \\
&= K \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} f(\mu) d\mu \left[ \frac{1}{3} \frac{\mu^2}{5} + \frac{1}{10} \frac{\mu^2}{3} - \frac{1}{56} \frac{\mu^2}{10} + \frac{1}{144} \mu^{-\frac{1}{2}} \right] + o\left( (\mu_{\text{eff}}^1)^{-n+1/2} \right). \nonumber
\end{align}
$$

The pressure $p(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2)$ for free quantum fields is

$$p(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2) = p(\mu_{\text{eff}}^1, \mu_{\text{eff}}^2) + \bar{p}(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2). \quad (280)$$

Here the quantity $p(\mu_{\text{eff}}^1, \mu_{\text{eff}}^2)$ is given by formula

$$
\begin{align}
p(\mu_{\text{eff}}^1, \mu_{\text{eff}}^2) &= \frac{1}{2(2\pi\hbar)^3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{\|\mathbf{k}\| \le \sqrt{\mu}} \frac{\|\mathbf{k}\|^2}{\sqrt{\mathbf{k}^2 + \mu^2}} d^3\mathbf{k} \tag{281} \\
&= \frac{K}{3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{p \le \sqrt{\mu}} \frac{p^4}{\sqrt{p^2 + \mu^2}} dp. \nonumber
\end{align}
$$

The quantity $\tilde{p}(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2)$ is given by formula

$$\tilde{p}(D^+, D^-, \mu_{\text{eff}}^1, \mu_{\text{eff}}^2) \approx \frac{K'}{3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{\|p\| > \sqrt{\mu}} \frac{\|\mathbf{k}\|^2}{\sqrt{\mathbf{k}^2 + \mu^2}} d^{D^+, D^-} \mathbf{k}, \quad (282)$$

where $K' = \frac{1}{2(2\pi\hbar)^3}, c = 1$. Note that

$$
\begin{align}
\frac{1}{\sqrt{p^2 + \mu^2}} &= \mu^{-1} \left( \sqrt{1 + \frac{p^2}{\mu^2}} \right)^{-1} \nonumber \\
&= \mu^{-1} \left( 1 - \frac{1}{2} \frac{p^2}{\mu^2} + \frac{3}{8} \frac{p^4}{\mu^4} - \frac{5}{16} \frac{p^6}{\mu^6} + \dots \right) \tag{283} \\
&= \frac{1}{\mu} - \frac{1}{2} \frac{p^2}{\mu^3} + \frac{3}{8} \frac{p^4}{\mu^5} - \frac{5}{16} \frac{p^6}{\mu^7} + \dots \nonumber
\end{align}
$$

By inserting Equation (283) into Equation (281) we get

$$
\begin{align*}
p(\mu_{\text{eff}}^1, \mu_{\text{eff}}^2) &= \frac{K}{3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{p \le \sqrt{\mu}} \left[ \frac{1}{\mu} - \frac{1}{2} \frac{p^2}{\mu^3} + \frac{3}{8} \frac{p^4}{\mu^5} - \frac{5}{16} \frac{p^6}{\mu^7} + \dots \right] p^4 dp \\
&= \frac{K}{3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \int_{p \le \sqrt{\mu}} \left[ \frac{p^4}{\mu} - \frac{1}{2} \frac{p^6}{\mu^3} + \frac{3}{8} \frac{p^8}{\mu^5} - \frac{5}{16} \frac{p^{10}}{\mu^7} + \dots \right] dp \\
&= \frac{K}{3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \left[ \frac{p^5}{5\mu} - \frac{1}{2} \frac{p^7}{7\mu^3} + \frac{3}{8} \frac{p^9}{9\mu^5} - \frac{5}{16} \frac{p^{11}}{10\mu^7} + \dots \right] \Big|_{0}^{\sqrt{\mu}}
\end{align*}
$$
$$ = \frac{K}{3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \left[ \frac{\mu^2}{5\mu} - \frac{1}{2}\frac{\mu^2}{7\mu^3} + \frac{3}{8}\frac{\mu^2}{9\mu^5} - \frac{5}{16}\frac{\mu^2}{10\mu^7} + \dots \right] \\ = \frac{K}{3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \left[ \frac{1}{5}\mu^2 - \frac{1}{14}\mu^2 + \frac{1}{24}\mu^2 - \frac{1}{32}\mu^2 + \dots \right] \quad (284) \\ = \frac{K}{3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \left[ \frac{1}{5}\mu^2 - \frac{1}{14}\mu^2 + \frac{1}{24}\mu^2 - \frac{1}{32}\mu^2 \right] + o\left( (\mu_{\text{eff}}^1)^{-n-1/2} \right). $$

## 6. Discussion and Conclusion

We will now briefly review the canonical assumptions that are made in the usual formulation of the cosmological constant problem.

**The canonical assumptions:**

**1) The physical dark matter.**

Dark matter is a hypothetical form of matter that is thought to account for approximately 85% of the matter in the universe, and about a quarter of its total energy density. The majority of dark matter is thought to be non-baryonic in nature, possibly being composed of some as-yet-undiscovered subatomic particles. Its presence is implied in a variety of astrophysical observations, including gravitational effects that cannot be explained unless more matter is present than can be seen. For this reason, most experts think dark matter to be ubiquitous in the universe and to have had a strong influence on its structure and evolution. The name dark matter refers to the fact that it does not appear to interact with observable electromagnetic radiation, such as light, and is thus invisible (or ‘dark’) to the entire electromagnetic spectrum, making it extremely difficult to detect using usual astronomical equipment. Because dark matter has not yet been observed directly, it must barely interact with ordinary baryonic matter and radiation. The primary candidate for dark matter is some new kind of elementary particle that has not yet been discovered, in particular, weakly-interacting massive particles (WIMPs), or gravitationally-interacting massive particles (GIMPs). Many experiments to directly detect and study dark matter particles are being actively undertaken, but none has yet succeeded.

2) The total effective cosmological constant $\lambda_{\text{eff}}$ is on at least the order of magnitude of the vacuum energy density generated by zero-point fluctuations of the standard particle fields.

3) Canonical QFT is an effective field theory description of a more fundamental theory, which becomes significant at some high-energy scale $\Lambda_{*}$.

4) The vacuum energy-momentum tensor is Lorentz invariant.

5) The Moller-Rosenfeld approach [34] [35] to semiclassical gravity by using an expectation value for the energy-momentum tensor is sound.

6) The Einstein equations for the homogeneous Friedmann-Robertson-Walker metric accurately describes the large-scale evolution of the Universe.

**Remark 6.1.1.** Note that obviously there is a strong inconsistency between Assumptions 2 and 3: the vacuum state cannot be Lorentz invariant if modes are
$$ = \frac{K}{3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \left[ \frac{\mu^2}{5\mu} - \frac{1}{2}\frac{\mu^2}{7\mu^3} + \frac{3}{8}\frac{\mu^2}{9\mu^5} - \frac{5}{16}\frac{\mu^2}{10\mu^7} + \dots \right] \\ = \frac{K}{3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \left[ \frac{1}{5}\mu^2 - \frac{1}{14}\mu^2 + \frac{1}{24}\mu^2 - \frac{1}{32}\mu^2 + \dots \right] \quad (284) \\ = \frac{K}{3} \int_{\mu_{\text{eff}}^1}^{\mu_{\text{eff}}^2} d\mu f(\mu) \left[ \frac{1}{5}\mu^2 - \frac{1}{14}\mu^2 + \frac{1}{24}\mu^2 - \frac{1}{32}\mu^2 \right] + o\left( (\mu_{\text{eff}}^1)^{-n-1/2} \right). $$

# 6. Discussion and Conclusion

We will now briefly review the canonical assumptions that are made in the usual formulation of the cosmological constant problem.

**The canonical assumptions:**

## 1) The physical dark matter.

Dark matter is a hypothetical form of matter that is thought to account for approximately 85% of the matter in the universe, and about a quarter of its total energy density. The majority of dark matter is thought to be non-baryonic in nature, possibly being composed of some as-yet-undiscovered subatomic particles. Its presence is implied in a variety of astrophysical observations, including gravitational effects that cannot be explained unless more matter is present than can be seen. For this reason, most experts think dark matter to be ubiquitous in the universe and to have had a strong influence on its structure and evolution. The name dark matter refers to the fact that it does not appear to interact with observable electromagnetic radiation, such as light, and is thus invisible (or ‘dark’) to the entire electromagnetic spectrum, making it extremely difficult to detect using usual astronomical equipment. Because dark matter has not yet been observed directly, it must barely interact with ordinary baryonic matter and radiation. The primary candidate for dark matter is some new kind of elementary particle that has not yet been discovered, in particular, weakly-interacting massive particles (WIMPs), or gravitationally-interacting massive particles (GIMPs). Many experiments to directly detect and study dark matter particles are being actively undertaken, but none has yet succeeded.

2) The total effective cosmological constant $\lambda_{\text{eff}}$ is on at least the order of magnitude of the vacuum energy density generated by zero-point fluctuations of the standard particle fields.

3) Canonical QFT is an effective field theory description of a more fundamental theory, which becomes significant at some high-energy scale $\Lambda_*$.

4) The vacuum energy-momentum tensor is Lorentz invariant.

5) The Moller-Rosenfeld approach [34] [35] to semiclassical gravity by using an expectation value for the energy-momentum tensor is sound.

6) The Einstein equations for the homogeneous Friedmann-Robertson-Walker metric accurately describes the large-scale evolution of the Universe.

**Remark 6.1.1.** Note that obviously there is a strong inconsistency between Assumptions 2 and 3: the vacuum state cannot be Lorentz invariant if modes are
ignored above some high-energy cutoff $\Lambda_*$, because a mode that is high energy in one reference frame will be low energy in another appropriately boosted frame. In this paper Assumption 3 is not used and this contradiction is avoided.

**Remark 6.1.2.** Note that also, Assumptions 1, 3, 4 and 5 is modified, which we denote as Assumptions 4 and 5 respectively.

**Modified assumptions**

1') The physical dark matter.

2') The total effective cosmological constant $\lambda_{\text{eff}}$ is on at least the order $|\mu_{\text{eff}}|^{-n+5} \ln|\mu_{\text{eff}}|$ of magnitude of the renormalized vacuum energy density generated by zero-point fluctuations of standard particle fields and ghost particle fields, see subsection 1.2.

3') The vacuum energy-momentum tensor is not Lorentz invariant.

**6.1. The Physical Ghost Matter and Dark Matter Nature**

In the contemporary quantum field theory, a ghost field, or gauge ghost is an unphysical state in a gauge theory. Ghosts are necessary to keep gauge inva-
riance in theories where the local fields exceed a number of physical degrees of
freedom. For example in quantum electrodynamics, in order to maintain manif-
est Lorentz invariance, one uses a four-component vector potential $A_\mu(x)$,
whereas the photon has only two polarizations. Thus, one needs a suitable me-
chanism in order to get rid of the unphysical degrees of freedom. Introducing
fictitious fields, the ghosts, is one way of achieving this goal. Faddeev-Popov
ghosts are extraneous fields which are introduced to maintain the consistency of
the path integral formulation. Faddeev-Popov ghosts are sometimes referred to
as “good ghosts”.

“Bad ghosts” represent another, more general meaning of the word “ghost” in
theoretical physics: states of negative norm, or fields with the wrong sign of the
kinetic term, such as Pauli-Villars ghosts, whose existence allows the probabili-
ties to be negative thus violating unitarity.

(VI.1) In contrary with standard Assumption 1 in the case of the new ap-
proach introduced in this paper we assume that:

(VI.1.1.a) The ghosts fields and ghosts particles with masses at a scale less
then a fixed scale $m_{\text{eff}}$ really exist in the universe and formed dark matter sec-
tor of the universe, in particular:

(VI.1.1.b) these ghosts fields give additive contribution to a full zero-point
fluctuation (i.e. also to effective cosmological constant $\lambda_{\text{eff}}$ [5], see subsection
1.2).

(VI.1.1.c) Pauli-Villars renormalization of zero-point fluctuations (see sub-
section 1.2) is no longer considered as an intermediate mathematical construct but
obviously has rigorous physical meaning supported by assumption (VI.1.1.a-b).

(VI.1.2) The physical dark matter formed by ghosts particles;

(VI.1.3) The standard model fields do not to couple directly to the ghost sec-
tor in the ultraviolet region of energy at a scale less then a fixed large energy
scale $\Lambda_*$, in particular:
ignored above some high-energy cutoff $\Lambda_*$, because a mode that is high energy in one reference frame will be low energy in another appropriately boosted frame. In this paper Assumption 3 is not used and this contradiction is avoided.

**Remark 6.1.2.** Note that also, Assumptions 1, 3, 4 and 5 is modified, which we denote as Assumptions 4 and 5 respectively.

**Modified assumptions**

1') The physical dark matter.

2') The total effective cosmological constant $\lambda_{\text{eff}}$ is on at least the order $|\mu_{\text{eff}}|^{-n+5} \ln|\mu_{\text{eff}}|$ of magnitude of the renormalized vacuum energy density generated by zero-point fluctuations of standard particle fields and ghost particle fields, see subsection 1.2.

3') The vacuum energy-momentum tensor is not Lorentz invariant.

**6.1. The Physical Ghost Matter and Dark Matter Nature**

In the contemporary quantum field theory, a ghost field, or gauge ghost is an unphysical state in a gauge theory. Ghosts are necessary to keep gauge inva-
riance in theories where the local fields exceed a number of physical degrees of
freedom. For example in quantum electrodynamics, in order to maintain manif-
est Lorentz invariance, one uses a four-component vector potential $A_\mu(x)$,
whereas the photon has only two polarizations. Thus, one needs a suitable me-
chanism in order to get rid of the unphysical degrees of freedom. Introducing
fictitious fields, the ghosts, is one way of achieving this goal. Faddeev-Popov
ghosts are extraneous fields which are introduced to maintain the consistency of
the path integral formulation. Faddeev-Popov ghosts are sometimes referred to
as “good ghosts”.

“Bad ghosts” represent another, more general meaning of the word “ghost” in theoretical physics: states of negative norm, or fields with the wrong sign of the kinetic term, such as Pauli-Villars ghosts, whose existence allows the probabilities to be negative thus violating unitarity.

(VI.1) In contrary with standard Assumption 1 in the case of the new ap-
proach introduced in this paper we assume that:

(VI.1.1.a) The ghosts fields and ghosts particles with masses at a scale less
than a fixed scale $m_{\text{eff}}$ really exist in the universe and formed dark matter sec-
tor of the universe, in particular:

(VI.1.1.b) these ghosts fields give additive contribution to a full zero-point fluctuation (i.e. also to effective cosmological constant $\lambda_{\text{eff}}$ [5], see subsection 1.2).

(VI.1.1.c) Pauli-Villars renormalization of zero-point fluctuations (see sub-section 1.2) is no longer considered as an intermediate mathematical construct but obviously has rigorous physical meaning supported by assumption (VI.1.1.a-b).

(VI.1.2) The physical dark matter formed by ghosts particles;

(VI.1.3) The standard model fields do not to couple directly to the ghost sec-
tor in the ultraviolet region of energy at a scale less then a fixed large energy
scale $\Lambda_*$, in particular:
(VI.1.3.a) The “bad” ghosts fields with masses at a scale less then a fixed scale $m_{\text{eff}}$, where $m_{\text{eff}}c^2 \ll \Lambda_*$, cannot appear in any effective physical lagrangian which contains also the standard particles fields.

In additional though not necessary we assume that:

(VI.1.4) The “bad” ghosts fields with masses at a scale $m_*$, where $m_*c^2 \gg \Lambda_*$ can appear in any effective physical lagrangian which contains also the standard particles fields, in particular:

(VI.1.4.a) Pauli-Villars finite renormalization with masses of ghosts fields at a scale $m_*$ of the S-matrix in QFT (see Subsection 2.1-2) is no longer considered as an intermediate mathematical construct but obviously has rigorous physical meaning supported by assumption (IV1.4).

(VI.1.4.b) If the “bad” ghosts fields coupled to matter directly, it gives rise to small and controllable violation of the unitarity condition.

**Remark 6.1.3.** We emphasize that in universe standard matter coupled with a physical ghost matter has the equation of state [3]:

$$ \varepsilon_{\text{vac}}(\mu_{\text{eff}}) = -p(\mu_{\text{eff}}) = \frac{1}{8} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^4 (\ln \mu) d\mu = \frac{c^4 \lambda_{\text{vac}}}{8\pi G}, \quad (285) $$

where

$$ |f(\mu)| = \begin{cases} O(\mu^{-n}), & n > 1 \quad \mu \le \mu_{\text{eff}} \\ 0, & \mu > \mu_{\text{eff}} \end{cases} \quad (286) $$

and where $\mu_{\text{eff}} = m_{\text{eff}}c$ (see subsection I.2, Equation (46)) and therefore gives rise to a de Sitter phase of the universe even if bare cosmological constant $\lambda = 0$.

(VI.1.5) In order to obtain QFT description of the dark component of matter in natural way we expand the standard model of particle physics on a sector of ghost particles, see [12], Section 2.3.2. QFT in a ghost sector developed in [12], Section 3.1-3.4 and Section 4.1-4.8.

## 6.2. Different Contributions to $\lambda_{\text{eff}}$.

The total effective cosmological constant $\lambda_{\text{eff}}$ is on at least the order of magnitude of the vacuum energy density generated by zero-point fluctuations of standard particle fields.

Assumption 2 is well justified in the case of the traditional approach, because the contribution from zero-point fluctuations is on the order of 1 in Planck units and no other known contributions are as large thus, assuming no significant cancellation of terms (e.g. fine tuning of the bare cosmological constant $\lambda$), the total $\lambda_{\text{eff}}$ should be at least on the order of the largest contribution [15].

(VI.2) In contrary with standard Assumption 1 in the case of the new approach introduced in this paper we assume that:

(VI.2.1) For simplicity though not necessary bare cosmological constant $\lambda = 0$.

(VI.2.2) The total effective cosmological constant $\lambda_{\text{eff}}$ depends only on mass distribution $f(\mu)$ and constant $\mu_{\text{eff}} = m_{\text{eff}}c$ but cannot depend on large
(VI.1.3.a) The “bad” ghosts fields with masses at a scale less then a fixed scale $m_{\text{eff}}$, where $m_{\text{eff}}c^2 \ll \Lambda_*$, cannot appear in any effective physical lagrangian which contains also the standard particles fields.

In additional though not necessary we assume that:

(VI.1.4) The “bad” ghosts fields with masses at a scale $m_*$, where $m_*c^2 \gg \Lambda_*$ can appear in any effective physical lagrangian which contains also the standard particles fields, in particular:

(VI.1.4.a) Pauli-Villars finite renormalization with masses of ghosts fields at a scale $m_*$ of the S-matrix in QFT (see Subsection 2.1-2) is no longer considered as an intermediate mathematical construct but obviously has rigorous physical meaning supported by assumption (IV1.4).

(VI.1.4.b) If the “bad” ghosts fields coupled to matter directly, it gives rise to small and controllable violation of the unitarity condition.

**Remark 6.1.3.** We emphasize that in universe standard matter coupled with a physical ghost matter has the equation of state [3]:

$$ \varepsilon_{\text{vac}}(\mu_{\text{eff}}) = -p(\mu_{\text{eff}}) = \frac{1}{8} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^4 (\ln \mu) d\mu = \frac{c^4 \lambda_{\text{vac}}}{8\pi G}, \quad (285) $$

where

$$ |f(\mu)| = \begin{cases} O(\mu^{-n}), & n > 1 \quad \mu \le \mu_{\text{eff}} \\ 0, & \mu > \mu_{\text{eff}} \end{cases} \quad (286) $$

and where $\mu_{\text{eff}} = m_{\text{eff}}c$ (see subsection I.2, Equation (46)) and therefore gives rise to a de Sitter phase of the universe even if bare cosmological constant $\lambda = 0$.

(VI.1.5) In order to obtain QFT description of the dark component of matter in natural way we expand the standard model of particle physics on a sector of ghost particles, see [12], Section 2.3.2. QFT in a ghost sector developed in [12], Section 3.1-3.4 and Section 4.1-4.8.

## 6.2. Different Contributions to $\lambda_{\text{eff}}$.

The total effective cosmological constant $\lambda_{\text{eff}}$ is on at least the order of magnitude of the vacuum energy density generated by zero-point fluctuations of standard particle fields.

Assumption 2 is well justified in the case of the traditional approach, because the contribution from zero-point fluctuations is on the order of 1 in Planck units and no other known contributions are as large thus, assuming no significant cancellation of terms (e.g. fine tuning of the bare cosmological constant $\lambda$), the total $\lambda_{\text{eff}}$ should be at least on the order of the largest contribution [15].

(VI.2) In contrary with standard Assumption 1 in the case of the new approach introduced in this paper we assume that:

(VI.2.1) For simplicity though not necessary bare cosmological constant $\lambda = 0$.

(VI.2.2) The total effective cosmological constant $\lambda_{\text{eff}}$ depends only on mass distribution $f(\mu)$ and constant $\mu_{\text{eff}} = m_{\text{eff}}c$ but cannot depend on large
energy scale ~ $\Lambda_*$

**Remark 6.2.1.** Note that in subsection we pointed out that under Assumption VI.1 if bare cosmological constant $\lambda = 0$ the total cosmological constant $\lambda_{vac}$ is on at least the order $|\mu_{eff}|^{-n+5}$ of magnitude of the renormalized vacuum energy density generated by zero-point fluctuations of standard particle fields and ghost particle fields

$$ 
\begin{aligned}
\varepsilon_{\text{vac}}(\mu_{\text{eff}}) &= \frac{1}{8} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} (\ln \mu) d\mu + O(\Lambda_{*}^{-2}), \\
p_{\text{vac}}(\mu_{\text{eff}}) &= -\frac{1}{8} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} (\ln \mu) d\mu + O(\Lambda_{*}^{-2}).
\end{aligned}
\qquad (287) $$

## 6.3. Effective Field Theory and Lorentz Invariance Violation

To prevent the vacuum energy density from diverging, the traditional approach also assumes that performing a high-energy cutoff is acceptable. This type of regularization is a common step in renormalization procedures, which aim to eventually arrive at a physical, cutoff-independent result. However, in the case of the vacuum energy density, the result is inherently cutoff dependent, scaling quartically with the cutoff $\Lambda_*$.

**Remark 6.3.1.** By restricting to modes with particle energy a certain cutoff energy $\omega_k \le \Lambda_*$ a finite, regularized result for the energy density can be obtained. The result is proportional to $\Lambda_*^4$. Any other fields will contribute similarly, so that if there are $n_b$ bosonic fields and $n_f$ fermionic fields, the density scales with $(n_b - 4n_f) \Lambda_*^4$. Typically, the cutoff is taken to be near = 1 in Planck units (i.e. the Planck energy), so the vacuum energy gives a contribution to the cosmological constant on the order of at least unity according to Equation (6.2.4). Thus we see the extreme ne-tuning problem: the original cosmological constant $\lambda$ must cancel this large vacuum energy density $\varepsilon_{vac} \simeq 1$ to a precision of 1 in $10^{120}$—but not completely—to result in the observed value $\lambda_{eff} = 10^{-120}$ [5].

**Remark 6.3.2.** As it pointed out in this paper that a high-energy theory, i.e. QFT in fractal space-time with Hausdorff-Colombeau negative dimension would not display the zero-point fluctuations that are characteristic of QFT, and hence that the divergence caused by oscillations above the corresponding cutoff frequency is unphysical. In this case, the cutoff $\Lambda_*$ is no longer an intermediate mathematical construct, but instead a physical scale at which the smooth, continuous behavior of QFT breaks down.

Poincaré group of the momentum space is deformed at some fundamental high-energy cutoff $\Lambda_*$. The canonical quadratic invariant $\|p\|^2 = \eta^{ab} p_a p_b$ collapses at high-energy cutoff $\Lambda_*$ and being replaced by the non-quadratic invariant:

$$ \|p\|^2 = \frac{\eta^{ab} p_a p_b}{(1 + l_{\Lambda_*} p_0)}. \qquad (288) $$

**Remark 6.3.3.** In contrary with canonical approach the total effective cosmological
energy scale ~ $\Lambda_*$

**Remark 6.2.1.** Note that in subsection we pointed out that under Assumption VI.1 if bare cosmological constant $\lambda = 0$ the total cosmological constant $\lambda_{vac}$ is on at least the order $|\mu_{eff}|^{-n+5}$ of magnitude of the renormalized vacuum energy density generated by zero-point fluctuations of standard particle fields and ghost particle fields

$$ 
\begin{aligned}
\varepsilon_{\text{vac}}(\mu_{\text{eff}}) &= \frac{1}{8} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} (\ln \mu) d\mu + O(\Lambda_{*}^{-2}), \\
p_{\text{vac}}(\mu_{\text{eff}}) &= -\frac{1}{8} \int_{0}^{\mu_{\text{eff}}} f(\mu) \mu^{4} (\ln \mu) d\mu + O(\Lambda_{*}^{-2}).
\end{aligned}
\qquad (287) $$

## 6.3. Effective Field Theory and Lorentz Invariance Violation

To prevent the vacuum energy density from diverging, the traditional approach also assumes that performing a high-energy cutoff is acceptable. This type of regularization is a common step in renormalization procedures, which aim to eventually arrive at a physical, cutoff-independent result. However, in the case of the vacuum energy density, the result is inherently cutoff dependent, scaling quartically with the cutoff $\Lambda_*$.

**Remark 6.3.1.** By restricting to modes with particle energy a certain cutoff energy $\omega_k \le \Lambda_*$ a finite, regularized result for the energy density can be obtained. The result is proportional to $\Lambda_*^4$. Any other fields will contribute similarly, so that if there are $n_b$ bosonic fields and $n_f$ fermionic fields, the density scales with $(n_b - 4n_f) \Lambda_*^4$. Typically, the cutoff is taken to be near = 1 in Planck units (i.e. the Planck energy), so the vacuum energy gives a contribution to the cosmological constant on the order of at least unity according to Equation (6.2.4). Thus we see the extreme ne-tuning problem: the original cosmological constant $\lambda$ must cancel this large vacuum energy density $\varepsilon_{vac} \simeq 1$ to a precision of 1 in $10^{120}$—but not completely—to result in the observed value $\lambda_{eff} = 10^{-120}$ [5].

**Remark 6.3.2.** As it pointed out in this paper that a high-energy theory, i.e. QFT in fractal space-time with Hausdorff-Colombeau negative dimension would not display the zero-point fluctuations that are characteristic of QFT, and hence that the divergence caused by oscillations above the corresponding cutoff frequency is unphysical. In this case, the cutoff $\Lambda_*$ is no longer an intermediate mathematical construct, but instead a physical scale at which the smooth, continuous behavior of QFT breaks down.

Poincaré group of the momentum space is deformed at some fundamental high-energy cutoff $\Lambda_*$. The canonical quadratic invariant $\|p\|^2 = \eta^{ab} p_a p_b$ collapses at high-energy cutoff $\Lambda_*$ and being replaced by the non-quadratic invariant:

$$ \|p\|^2 = \frac{\eta^{ab} p_a p_b}{(1 + l_{\Lambda_*} p_0)}. \qquad (288) $$

**Remark 6.3.3.** In contrary with canonical approach the total effective cosmo-
logical constant $\lambda_{\text{eff}}$ depends only on mass distribution $f(\mu)$ and constant $\mu_{\text{eff}} = m_{\text{eff}}c$ but cannot depend on large energy scale $\sim \Lambda_{*}$.

## 6.4. Semiclassical Möller-Rosenfeld Gravity

Assumption 5 means that it is valid to replace the right-hand side of the Einstein equation $T_{\mu\nu}$ with its expectation $\langle T_{\mu\nu} \rangle$. It requires that either gravity is not in fact quantum, and the Moller-Rosenfeld approach is a complete description of reality, or at least a valid approximation in the weak field limit. The usual argument states that the vacuum state $|0\rangle$ should be locally Lorentz invariant so that observers agree on the vacuum state. This means that the expectation value of the energy-momentum tensor on the vacuum, $\langle 0|\hat{T}_{\mu\nu}|0\rangle$, must be a scalar multiple of the metric tensor $g_{\mu\nu}$ which is the only Lorentz invariant rank (0,2) tensor. By using Moller-Rosenfeld approach the Einstein field equations of general relativity, a term representing the curvature of spacetime $R_{\mu\nu}$ is related to a term describing the energy-momentum of matter $\langle 0|\hat{T}_{\mu\nu}|0\rangle$, as well as the cosmological constant $\lambda$ and metric tensor $g_{\mu\nu}$ reads:

$$R_{\mu\nu} - \frac{1}{2} R_{\mu\nu}^{\nu} g_{\mu\nu} + \lambda g_{\mu\nu} = 8\pi \langle 0 | \hat{T}_{\mu\nu} | 0 \rangle. \quad (289)$$

The $\hat{T}_{00}$ component is an energy density, we label $\langle 0|\hat{T}_{\mu\nu}|0\rangle = \varepsilon_{\text{vac}}$, so that the vacuum contribution to the right-hand side of Equation (289) can be written as

$$8\pi \langle 0 | \hat{T}_{\mu\nu} | 0 \rangle = 8\pi \varepsilon_{\text{vac}} g_{\mu\nu}. \quad (290)$$

Subtracting this from the right-hand side of Equation (289) and grouping it with the cosmological constant term replaces with an “effective” cosmological constant [5]:

$$\lambda_{\text{eff}} = \lambda + 8\pi\varepsilon_{\text{vac}}. \quad (291)$$

Note that in flat spacetime, where $g_{\mu\nu} = \text{diag}(-1, +1, +1, +1)$, Eq. (290) implies $\varepsilon_{\text{vac}} = -p_{\text{vac}}$, where $p_{\text{vac}} = \langle 0|\hat{T}_{ii}|0\rangle$ for any $i=1,2,3$ is the pressure. Obviously this implies that if the energy density is positive as is usually assumed, then the pressure must be negative, a conclusion which extends to any metric $g_{\mu\nu}$ with a $(-1, +1, +1, +1)$ signature.

**Remark 6.4.1.** In this paper we assume that the vacuum state $|0\rangle$ should be locally invariant under modified Lorentz boost (17)-(18) but not locally Lorentz invariant. Obviously this assumption violate the Equation (290). However modified Lorentz boosts (17)-(18) becomes Lorentz boosts for sufficiently small energies and therefore in IR region one obtain in a good approximation

$$8\pi \langle 0 | \hat{T}_{\mu\nu} | 0 \rangle \approx 8\pi \varepsilon_{\text{vac}} g_{\mu\nu} \quad (292)$$

and

$$\lambda_{\text{eff}} \approx \lambda + 8\pi\varepsilon_{\text{vac}}. \quad (293)$$

Thus Möller-Rosenfeld approach holds in a good approximation.
logical constant $\lambda_{\text{eff}}$ depends only on mass distribution $f(\mu)$ and constant $\mu_{\text{eff}} = m_{\text{eff}}c$ but cannot depend on large energy scale $\sim \Lambda_{*}$.

## 6.4. Semiclassical Möller-Rosenfeld Gravity

Assumption 5 means that it is valid to replace the right-hand side of the Einstein equation $T_{\mu\nu}$ with its expectation $\langle T_{\mu\nu} \rangle$. It requires that either gravity is not in fact quantum, and the Moller-Rosenfeld approach is a complete description of reality, or at least a valid approximation in the weak field limit. The usual argument states that the vacuum state $|0\rangle$ should be locally Lorentz invariant so that observers agree on the vacuum state. This means that the expectation value of the energy-momentum tensor on the vacuum, $\langle 0|\hat{T}_{\mu\nu}|0\rangle$, must be a scalar multiple of the metric tensor $g_{\mu\nu}$ which is the only Lorentz invariant rank (0,2) tensor. By using Moller-Rosenfeld approach the Einstein field equations of general relativity, a term representing the curvature of spacetime $R_{\mu\nu}$ is related to a term describing the energy-momentum of matter $\langle 0|\hat{T}_{\mu\nu}|0\rangle$, as well as the cosmological constant $\lambda$ and metric tensor $g_{\mu\nu}$ reads:

$$R_{\mu\nu} - \frac{1}{2} R_{\mu\nu}^{\nu} g_{\mu\nu} + \lambda g_{\mu\nu} = 8\pi \langle 0 | \hat{T}_{\mu\nu} | 0 \rangle. \qquad (289)$$

The $\hat{T}_{00}$ component is an energy density, we label $\langle 0|\hat{T}_{\mu\nu}|0\rangle = \varepsilon_{\text{vac}}$, so that the vacuum contribution to the right-hand side of Equation (289) can be written as

$$8\pi \langle 0 | \hat{T}_{\mu\nu} | 0 \rangle = 8\pi \varepsilon_{\text{vac}} g_{\mu\nu}. \qquad (290)$$

Subtracting this from the right-hand side of Equation (289) and grouping it with the cosmological constant term replaces with an “effective” cosmological constant [5]:

$$\lambda_{\text{eff}} = \lambda + 8\pi\varepsilon_{\text{vac}}. \qquad (291)$$

Note that in flat spacetime, where $g_{\mu\nu} = \text{diag}(-1, +1, +1, +1)$, Eq. (290) implies $\varepsilon_{\text{vac}} = -p_{\text{vac}}$, where $p_{\text{vac}} = \langle 0|\hat{T}_{ii}|0\rangle$ for any $i=1,2,3$ is the pressure. Obviously this implies that if the energy density is positive as is usually assumed, then the pressure must be negative, a conclusion which extends to any metric $g_{\mu\nu}$ with a $(-1, +1, +1, +1)$ signature.

**Remark 6.4.1.** In this paper we assume that the vacuum state $|0\rangle$ should be locally invariant under modified Lorentz boost (17)-(18) but not locally Lorentz invariant. Obviously this assumption violate the Equation (290). However modified Lorentz boosts (17)-(18) becomes Lorentz boosts for sufficiently small energies and therefore in IR region one obtain in a good approximation

$$8\pi \langle 0 | \hat{T}_{\mu\nu} | 0 \rangle \approx 8\pi \varepsilon_{\text{vac}} g_{\mu\nu} \qquad (292)$$

and

$$\lambda_{\text{eff}} \approx \lambda + 8\pi\varepsilon_{\text{vac}}. \qquad (293)$$

Thus Möller-Rosenfeld approach holds in a good approximation.
## 6.5. Quantum Gravity at Energy Scale $\Lambda \le \Lambda_*$ . Controllable Violation of the Unitarity Condition

Gravitational actions which include terms quadratic in the curvature tensor are renormalizable. The necessary Slavnov identities are derived from

Becchi-Rouet-Stora (BRS) transformations of the gravitational and Faddeev-Popov ghost fields. In general, non-gauge-invariant divergences do arise, but they may be absorbed by nonlinear renormalizations of the gravitational and ghost fields and of the BRS transformations [14]. The geneic expression of the action reads

$$I_{\text{sym}} = \int -d^4 x \sqrt{-g} (\alpha R_{\mu\nu} R^{\mu\nu} - \beta R^2 + 2\kappa^{-2} R), \quad (294)$$

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

$$\kappa h^{\mu\nu} = g^{\mu\nu} \sqrt{-g} - \eta^{\mu\nu}. \quad (295)$$

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-field 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.

**Remark 6.5.1. (I)** The renormalizable models which we have considered in this paper many years mistakenly regarded only as constructs for a study of the ultraviolet problem of quantum gravity. The difficulties with unitarity appear to preclude their direct acceptability as canonical physical theories in locally Minowski 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 $\Lambda_*$ mentioned above by an appropriate limitation on the
## 6.5. Quantum Gravity at Energy Scale $Λ ≤ Λ_*$ . Controllable Violation of the Unitarity Condition

Gravitational actions which include terms quadratic in the curvature tensor are renormalizable. The necessary Slavnov identities are derived from

Becchi-Rouet-Stora (BRS) transformations of the gravitational and Faddeev-Popov ghost fields. In general, non-gauge-invariant divergences do arise, but they may be absorbed by nonlinear renormalizations of the gravitational and ghost fields and of the BRS transformations [14]. The geneic expression of the action reads

$$I_{\text{sym}} = \int -d^4 x \sqrt{-g} (\alpha R_{\mu\nu} R^{\mu\nu} - \beta R^2 + 2\kappa^{-2} R), \quad (294)$$

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

$$\kappa h^{\mu\nu} = g^{\mu\nu} \sqrt{-g} - \eta^{\mu\nu}. \quad (295)$$

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-field 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.

**Remark 6.5.1.** (I) The renormalizable models which we have considered in this paper many years mistakenly regarded only as constructs for a study of the ultraviolet problem of quantum gravity. The difficulties with unitarity appear to preclude their direct acceptability as canonical physical theories in locally Minowski space-time. In canonical case they do have only some promise as phenomenological models.

(II) However, for their unphysical behavior may be restricted to arbitrarily large energy scales $Λ_*$ mentioned above by an appropriate limitation on the
renormalized masses $m_2$ and $m_0$. Actually, it is only the massive spin-two ex-
citations of the field which give the trouble with unitarity and thus require a very
large mass. The limit on the mass $m_0$ is determined only by the observational
constraints on the static field.

6.6. Pauli-Villars Masses Distribution Corresponding to Ghost Matter Sector above High Energy Cutoff $\Lambda_{*}$ A Common Origin of the Dark Energy and Dard Matter Phenomena

Dark matter is a hypothetical form of matter that is thought to account for ap-
proximately 85% of the matter in the universe, and about a quarter of its total
energy density. The majority of dark matter is thought to be non-baryonic in
nature, possibly being composed of some as-yet undiscovered subatomic par-
ticles. Its presence is implied in a variety of astrophysical observations, including
gravitational effects that cannot be explained unless more matter is present than
can be seen. For this reason, most experts think dark matter to be ubiquitous in
the universe and to have had a strong influence on its structure and evolution.
Dark matter is called dark because it does not appear to interact with observable
electromagnetic radiation, such as light, and is thus invisible to the entire elec-
tromagnetic spectrum, making it extremely difficult to detect using usual astro-
nomical equipment [36] [37] [38].

**Figure 8** Analysis of a giant new galaxy survey, made with ESO's VLT Survey Telescope in Chile, suggests that dark matter may be less dense and more smoothly distributed throughout space than previously thought. An international team used data from the Kilo Degree Survey (KiDS) to study how the light from about 15 million distant galaxies was affected by the gravitational influence of matter on the largest scales in the Universe. The results appear to be in disagreement with earlier results from the Planck satellite. This map of dark matter in the Universe was obtained from data from the KiDS survey, using the VLT Survey Telescope at ESO's Paranal Observatory in Chile. It reveals an expansive web of dense (light) and empty (dark) regions. This image is one out

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Dark matter map for a patch of sky based on gravitational lensing analysis [38] Hilderbrandt 16.</figcaption></figure>
renormalized masses $m_2$ and $m_0$. Actually, it is only the massive spin-two ex-
citations of the field which give the trouble with unitarity and thus require a very
large mass. The limit on the mass $m_0$ is determined only by the observational
constraints on the static field.

6.6. Pauli-Villars Masses Distribution Corresponding to Ghost Matter Sector above High Energy Cutoff $\Lambda_{*}$ A Common Origin of the Dark Energy and Dard Matter Phenomena

Dark matter is a hypothetical form of matter that is thought to account for ap-
proximately 85% of the matter in the universe, and about a quarter of its total
energy density. The majority of dark matter is thought to be non-baryonic in
nature, possibly being composed of some as-yet undiscovered subatomic par-
ticles. Its presence is implied in a variety of astrophysical observations, including
gravitational effects that cannot be explained unless more matter is present than
can be seen. For this reason, most experts think dark matter to be ubiquitous in
the universe and to have had a strong influence on its structure and evolution.
Dark matter is called dark because it does not appear to interact with observable
electromagnetic radiation, such as light, and is thus invisible to the entire elec-
tromagnetic spectrum, making it extremely difficult to detect using usual astro-
nomical equipment [36] [37] [38].

**Figure 8** Analysis of a giant new galaxy survey, made with ESO's VLT Survey Telescope in Chile, suggests that dark matter may be less dense and more smoothly distributed throughout space than previously thought. An international team used data from the Kilo Degree Survey (KiDS) to study how the light from about 15 million distant galaxies was affected by the gravitational influence of matter on the largest scales in the Universe. The results appear to be in disagreement with earlier results from the Planck satellite. This map of dark matter in the Universe was obtained from data from the KiDS survey, using the VLT Survey Telescope at ESO's Paranal Observatory in Chile. It reveals an expansive web of dense (light) and empty (dark) regions. This image is one out

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Dark matter map for a patch of sky based on gravitational lensing analysis [38] Hilderbrandt 16.</figcaption></figure>
of five patches of the sky observed by KiDS. Here the invisible dark matter is seen rendered in pink, covering an area of sky around 420 times the size of the full moon. This image reconstruction was made by analyzing the light collected from over three million distant galaxies more than 6 billion light-years away. The observed galaxy images were warped by the gravitational pull of dark matter as the light traveled through the Universe. Some small dark regions, with sharp boundaries, appear in this image. They are the locations of bright stars and other nearby objects that get in the way of the observations of more distant galaxies and are hence masked out in these maps as no weak-lensing signal can be measured in these areas [38]. The luminous (light-emitting) components of the universe only comprise about 0.4% of the total energy. The remaining components are dark. Of those, roughly 3.6% are identified: cold gas and dust, neutrinos, and black holes. About 23% is dark matter, and the overwhelming majority is some type of gravitationally self-repulsive dark energy.

**Remark 6.6.1.** In order to explain physical nature of the dark matter sector we assume that the main part of dark matter, i.e., ≃ 23% – 4.6% = 18% (see **Figure 9**) formed by supermassive ghost particles with masses such that $mc^2 > \Lambda_*$, see ref. [13], Subsection 2.3.

Remind that vacuum energy density for free scalar quantum field with a wrong statistic is:

$$ \varepsilon(\mu) = -\frac{1}{2} \frac{1}{(2\pi\hbar)^3} \int_0^\infty 4\pi c \sqrt{p^2 + \mu^2} p^2 dp = K' \int_0^\infty \sqrt{p^2 + \mu^2} p^2 dp = K' I(\mu), \quad (296) $$

where $\mu = mc$. From the basic definitions [1]

$$ p = T_{xx}, p(\mu) = -\frac{1}{2} \frac{1}{(2\pi\hbar)^3} \int_0^\infty u_x p_x 4\pi p^2 dp, \mathbf{u} = \frac{cp}{\sqrt{p^2 + \mu^2}}, \overline{u_x p_x} = \frac{1}{3} \langle \mathbf{u}, \mathbf{p} \rangle \quad (297) $$

one obtains

$$ p(\mu) = \frac{K'}{3} \int_0^\infty \frac{p^4 dp}{\sqrt{p^2 + \mu^2}} = K'F(\mu). \quad (298) $$

**Remark 6.6.2.** Note that the integral in RHS of Equation (297) and in Equation (298) is divergent and ultraviolet cutoff is needed. Thus in accordance with [1] we set

$$ \varepsilon(\mu, p_0) = K' I(\mu, p_0), p(\mu, p_0) = K' F(\mu, p_0), \quad (299) $$

where

$$ I(\mu, p_0) = \int_0^{p_0} \sqrt{p^2 + \mu^2} p^2 dp, F(\mu, p_0) = \int_0^{p_0} \frac{p^4 dp}{\sqrt{p^2 + \mu^2}}, \quad (300) $$

where $p_0 \le \Lambda_*/c$. For fermionic quantum field with a wrong statistic, similarly one obtains

$$ \varepsilon(\mu, p_0) = -4K' I(\mu, p_0), p(\mu, p_0) = -4K' F(\mu, p_0). \quad (301) $$

Thus from Equations. (300)-(301) by using formally Pauli-Villars regularization [7] [8] and regularization by high-energy cutoff the expression for
of five patches of the sky observed by KiDS. Here the invisible dark matter is seen rendered in pink, covering an area of sky around 420 times the size of the full moon. This image reconstruction was made by analyzing the light collected from over three million distant galaxies more than 6 billion light-years away. The observed galaxy images were warped by the gravitational pull of dark matter as the light traveled through the Universe. Some small dark regions, with sharp boundaries, appear in this image. They are the locations of bright stars and other nearby objects that get in the way of the observations of more distant galaxies and are hence masked out in these maps as no weak-lensing signal can be measured in these areas [38]. The luminous (light-emitting) components of the universe only comprise about 0.4% of the total energy. The remaining components are dark. Of those, roughly 3.6% are identified: cold gas and dust, neutrinos, and black holes. About 23% is dark matter, and the overwhelming majority is some type of gravitationally self-repulsive dark energy.

**Remark 6.6.1.** In order to explain physical nature of the dark matter sector we assume that the main part of dark matter, i.e., ≃ 23% – 4.6% = 18% (see **Figure 9**) formed by supermassive ghost particles with masses such that $mc^2 > \Lambda_*$, see ref. [13], Subsection 2.3.

Remind that vacuum energy density for free scalar quantum field with a wrong statistic is:

$$ \varepsilon(\mu) = -\frac{1}{2} \frac{1}{(2\pi\hbar)^3} \int_0^\infty 4\pi c \sqrt{p^2 + \mu^2} p^2 dp = K' \int_0^\infty \sqrt{p^2 + \mu^2} p^2 dp = K'I(\mu), \quad (296) $$

where $\mu = mc$. From the basic definitions [1]

$$ p = T_{xx}, p(\mu) = -\frac{1}{2} \frac{1}{(2\pi\hbar)^3} \int_0^\infty u_x p_x 4\pi p^2 dp, \mathbf{u} = \frac{c\mathbf{p}}{\sqrt{p^2 + \mu^2}}, \overline{u_x p_x} = \frac{1}{3} \langle \mathbf{u}, \mathbf{p} \rangle \quad (297) $$

one obtains

$$ p(\mu) = \frac{K'}{3} \int_0^\infty \frac{p^4 dp}{\sqrt{p^2 + \mu^2}} = K'F(\mu). \quad (298) $$

**Remark 6.6.2.** Note that the integral in RHS of Equation (297) and in Equation (298) is divergent and ultraviolet cutoff is needed. Thus in accordance with [1] we set

$$ \varepsilon(\mu, p_0) = K'I(\mu, p_0), p(\mu, p_0) = K'F(\mu, p_0), \quad (299) $$

where

$$ I(\mu, p_0) = \int_0^{p_0} \sqrt{p^2 + \mu^2} p^2 dp, F(\mu, p_0) = \int_0^{p_0} \frac{p^4 dp}{\sqrt{p^2 + \mu^2}}, \quad (300) $$

where $p_0 \le \Lambda_*/c$. For fermionic quantum field with a wrong statistic, similarly one obtains

$$ \varepsilon(\mu, p_0) = -4K'I(\mu, p_0), p(\mu) = -4K'F(\mu, p_0). \quad (301) $$

Thus from Equations. (300)-(301) by using formally Pauli-Villars regularization [7] [8] and regularization by high-energy cutoff the expression for
<figure><img src="image_11.png" /><figcaption>Figure 9. Matter and energy distribution in the universe today. The luminous (light-emitting) components of the universe only comprise about 0.4% of the total energy. The remaining components are dark.</figcaption></figure>

free vacuum energy density $\epsilon$ reads

$$ \epsilon_{\text{vac}} = \sum_{i=0}^{2M} f_i I(\mu_i, p_0) \qquad (302) $$

and the expression for pressure $p$ reads

$$ p_{\text{vac}} = \sum_{i=0}^{2M} f_i F(\mu_i, p_0). \qquad (303) $$

**Definition 6.6.1.** We define now discrete distribution $f_{PV}: \mathbb{R}_+ \to \mathbb{R}$ by formula

$$ f_{PV}(\mu_i) = f_i, \qquad (304) $$

and we will call it as a full discrete Pauli-Villars masses distribution.

**Remark 6.6.3.** We assume now that in Equations (302)-(303): 1) the quantities $\mu_i^{s,m} = \mu_i, i = 1, 2, \dots, M$ are masses of physical particles corresponding to standard matter and 2) the quantities $\mu_i^{g,m} = \mu_i, i = M+1, 2, \dots, 2M$ are masses of ghost particles with a wrong kinetic term and wrong statistics corresponding to physical dark matter.

**Remark 6.6.5.** We recall that the Euler-Maclaurin summation formula reads

$$ \begin{aligned} & \sum_{i=1}^{2M} g(\mu_i + (i-1)h) \\ & = \int_{\mu_1}^{\mu_2} f(\mu) d\mu + A_1 [g(\mu_{2M}) - g(\mu_1)] \\ & \quad + A_2 h [g'(\mu_{2M}) - g'(\mu_1)] + O(h^2), \quad f(\mu) = \frac{1}{h} g(\mu). \end{aligned} \qquad (305) $$

Let $g(\mu)$ be an appropriate continuous function such that: 1)

$$ g(\mu_i) = f_i, i = 1, 2, \dots, 2M, $$

$$ 2) g'(\mu_{2M}) = 0, g'(\mu_1) = 0. $$

Thus from Equations (302)-(303) and Equations (305) we obtain

$$ \begin{aligned} \epsilon_{\text{vac}} = \sum_{i=0}^{2M} f_i I(\mu_i, p_0) = \int_{\mu_1}^{\mu_{2M}} & f(\mu) I(\mu, p_0) d\mu \\ & + A_1 h [f(\mu_{2M}) I(\mu_{2M}, p_0) - f(\mu_1) I(\mu_1, p_0)] + O(h^2) \end{aligned} \qquad (306) $$
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Matter and energy distribution in the universe today. The luminous (light-emitting) components of the universe only comprise about 0.4% of the total energy. The remaining components are dark.</figcaption></figure>

free vacuum energy density $\varepsilon$ reads

$$ \varepsilon_{\text{vac}} = \sum_{i=0}^{2M} f_i I(\mu_i, p_0) \qquad (302) $$

and the expression for pressure $p$ reads

$$ p_{\text{vac}} = \sum_{i=0}^{2M} f_i F(\mu_i, p_0). \qquad (303) $$

**Definition 6.6.1.** We define now discrete distribution $f_{PV}: \mathbb{R}_+ \to \mathbb{R}$ by formula

$$ f_{PV}(\mu_i) = f_i, \qquad (304) $$

and we will call it as a full discrete Pauli-Villars masses distribution.

**Remark 6.6.3.** We assume now that in Equations (302)-(303): 1) the quantities $\mu_i^{s,m} = \mu_i, i = 1, 2, \dots, M$ are masses of physical particles corresponding to standard matter and 2) the quantities $\mu_i^{g,m} = \mu_i, i = M+1, 2, \dots, 2M$ are masses of ghost particles with a wrong kinetic term and wrong statistics corresponding to physical dark matter.

**Remark 6.6.5.** We recall that the Euler-Maclaurin summation formula reads

$$ \begin{aligned} & \sum_{i=1}^{2M} g(\mu_i + (i-1)h) \\ &= \int_{\mu_1}^{\mu_2} f(\mu) d\mu + A_1 [g(\mu_{2M}) - g(\mu_1)] \\ & \qquad + A_2 h[g'(\mu_{2M}) - g'(\mu_1)] + O(h^2), \quad f(\mu) = \frac{1}{h} g(\mu). \end{aligned} \qquad (305) $$

Let $g(\mu)$ be an appropriate continuous function such that: 1)

$$ g(\mu_i) = f_i, i = 1, 2, \dots, 2M, $$

$$ 2) \quad g'(\mu_{2M}) = 0, g'(\mu_1) = 0. $$

Thus from Equations (302)-(303) and Equations (305) we obtain

$$ \begin{aligned} \varepsilon_{\text{vac}} = \sum_{i=0}^{2M} f_i I(\mu_i, p_0) = \int_{\mu_1}^{\mu_{2M}} f(\mu) I(\mu, p_0) d\mu \\ + A_1 h[f(\mu_{2M}) I(\mu_{2M}, p_0) - f(\mu_1) I(\mu_1, p_0)] + O(h^2) \end{aligned} \qquad (306) $$
and

$$p_{\text{vac}} = \sum_{i=0}^{2M} f_i F(\mu_i, p_0) = \int_{\mu_1}^{\mu_{2M}} f(\mu) F(\mu, p_0) d\mu + A_1 h[f(\mu_{2M}) F(\mu_{2M}, p_0) - f(\mu_1) F(\mu_1, p_0)] + O(h^2). \quad (307)$$

**Definition 6.6.2.** We will call the function $f_{PV}(\mu)$ as a full continuous Pauli-Villars masses distribution.

**Definition 6.6.3.** We define now: 1) discrete distribution $f_{PV}^{b,g,m}: \mathbb{R}_+ \to \mathbb{R}$ by formula

$$f_{PV}^{b,g,m}(\mu_i^{s,m}) = f_i, i = 1, 2, \dots, M \quad (308)$$

and we will call it as discrete Pauli-Villars masses distribution of the bosonic ghost matter and

2) discrete distribution $f_{PV}^{f,g,m}: \mathbb{R}_+ \to \mathbb{R}$ by formula

$$f_{PV}^{f,g,m}(\mu_i) = f_i, i = M+1, 2, \dots, 2M \quad (309)$$

and we will call it as discrete Pauli-Villars masses distribution of the fermionic ghost matter.

**Remark 6.6.4.** We rewrite now Equations (306)-(307) in the following equivalent form

$$\epsilon_{\text{vac}} = \sum_{i=1}^{M} f_{PV}^{b,g,m}(\mu_i^{s,m}) I(\mu_i^{b,g,m}, p_0) + \sum_{j(i)=M+1}^{2M} f_{PV}^{f,g,m}(\mu_{j(i)}^{f,g,m}) I(\mu_{j(i)}^{f,g,m}, p_0) \quad (310)$$

and

$$p_{\text{vac}} = \sum_{i=1}^{M} f_{PV}^{b,g,m}(\mu_i^{b,g,m}) F(\mu_i^{b,g,m}, p_0) + \sum_{j(i)=M+1}^{2M} f_{PV}^{f,g,m}(\mu_{j(i)}^{f,g,m}) F(\mu_{j(i)}^{f,g,m}, p_0), \quad (311)$$

where $j(i) = i + M, i = 1, 2, \dots, M$.

**Remark 6.6.6.** We assume now that:1) $\mu_i^{b,g,m} \approx \mu_{j(i)}^{f,g,m}$,

2) $|f_{PV}^{b,g,m}(\mu_i^{b,g,m}) + f_{PV}^{f,g,m}(\mu_{j(i)}^{f,g,m})| \ll 1, \text{ i.e.,}$

$$f_{PV}^{b,g,m}(\mu_i^{b,g,m}) \approx -f_{PV}^{f,g,m}(\mu_{j(i)}^{f,g,m}). \quad (312)$$

Note that Equation (312) meant highly symmetric discrete Pauli-Villars masses distribution between bosonic ghost matter and fermionic ghost matter above that scale $\Lambda_*$.

Thus from Equations (310)-(311) and Equations (312) we obtain

$$\begin{aligned} \epsilon_{\text{vac}} &= \sum_{i=1}^{M} f_{PV}^{b,g,m}(\mu_i^{b,g,m}) I(\mu_i^{b,g,m}, p_0) + \sum_{j(i)=M+1}^{2M} f_{PV}^{f,g,m}(\mu_{j(i)}^{f,g,m}) I(\mu_{j(i)}^{f,g,m}, p_0) \\ &= \sum_{i=1}^{M} \left[ f_{PV}^{b,g,m}(\mu_i^{b,g,m}) + f_{PV}^{f,g,m}(\mu_{j(i)}^{f,g,m}) \right] I(\mu_i, p_0) \end{aligned} \quad (313)$$

and

$$\begin{aligned} p_{\text{vac}} &= \sum_{i=1}^{M} f_{PV}^{b,g,m}(\mu_i^{b,g,m}) F(\mu_i^{b,g,m}, p_0) + \sum_{j(i)=M+1}^{2M} f_{PV}^{f,g,m}(\mu_{j(i)}^{f,g,m}) F(\mu_{j(i)}^{f,g,m}, p_0) \\ &= \sum_{i=1}^{M} \left[ f_{PV}^{b,g,m}(\mu_i^{b,g,m}) + f_{PV}^{f,g,m}(\mu_{j(i)}^{f,g,m}) \right] F(\mu_i, p_0). \end{aligned} \quad (314)$$
and

$$p_{\text{vac}} = \sum_{i=0}^{2M} f_i F(\mu_i, p_0) = \int_{\mu_1}^{\mu_{2M}} f(\mu) F(\mu, p_0) d\mu + A_1 h[f(\mu_{2M}) F(\mu_{2M}, p_0) - f(\mu_1) F(\mu_1, p_0)] + O(h^2). \quad (307)$$

**Definition 6.6.2.** We will call the function $f_{PV}(\mu)$ as a full continuous Pauli-Villars masses distribution.

**Definition 6.6.3.** We define now: 1) discrete distribution $f_{PV}^{b,g,m}: \mathbb{R}_+ \to \mathbb{R}$ by formula

$$f_{PV}^{b,g,m}(\mu_i^{s,m}) = f_i, \quad i=1,2,\dots,M \quad (308)$$

and we will call it as discrete Pauli-Villars masses distribution of the bosonic ghost matter and

2) discrete distribution $f_{PV}^{f,g,m}: \mathbb{R}_+ \to \mathbb{R}$ by formula

$$f_{PV}^{f,g,m}(\mu_i) = f_i, \quad i = M+1,2,\dots,2M \quad (309)$$

and we will call it as discrete Pauli-Villars masses distribution of the fermionic ghost matter.

**Remark 6.6.4.** We rewrite now Equations (306)-(307) in the following equivalent form

$$\varepsilon_{\text{vac}} = \sum_{i=1}^{M} f_{PV}^{b,g,m}(\mu_i^{s,m}) I(\mu_i^{b,g,m}, p_0) + \sum_{j(i)=M+1}^{2M} f_{PV}^{f,g,m}(\mu_{j(i)}^{f,g,m}) I(\mu_{j(i)}^{f,g,m}, p_0) \quad (310)$$

and

$$p_{\text{vac}} = \sum_{i=1}^{M} f_{PV}^{b,g,m}(\mu_i^{b,g,m}) F(\mu_i^{b,g,m}, p_0) + \sum_{j(i)=M+1}^{2M} f_{PV}^{f,g,m}(\mu_{j(i)}^{f,g,m}) F(\mu_{j(i)}^{f,g,m}, p_0), \quad (311)$$

where $j(i) = i + M, i = 1, 2, \dots, M$.

**Remark 6.6.6.** We assume now that:1) $\mu_i^{b,g,m} \approx \mu_{j(i)}^{f,g,m}$,

2) $|f_{PV}^{b,g,m}(\mu_i^{b,g,m}) + f_{PV}^{f,g,m}(\mu_{j(i)}^{f,g,m})| \ll 1$, i.e.,

$$f_{PV}^{b,g,m}(\mu_i^{b,g,m}) \approx -f_{PV}^{f,g,m}(\mu_{j(i)}^{f,g,m}). \quad (312)$$

Note that Equation (312) meant highly symmetric discrete Pauli-Villars masses distribution between bosonic ghost matter and fermionic ghost matter above that scale $\Lambda_*$.

Thus from Equations (310)-(311) and Equations (312) we obtain

$$\begin{aligned} \varepsilon_{\text{vac}} &= \sum_{i=1}^{M} f_{PV}^{b,g,m}(\mu_i^{b,g,m}) I(\mu_i^{b,g,m}, p_0) + \sum_{j(i)=M+1}^{2M} f_{PV}^{f,g,m}(\mu_{j(i)}^{f,g,m}) I(\mu_{j(i)}^{f,g,m}, p_0) \\ &= \sum_{i=1}^{M} \left[ f_{PV}^{b,g,m}(\mu_i^{b,g,m}) + f_{PV}^{f,g,m}(\mu_{j(i)}^{f,g,m}) \right] I(\mu_i, p_0) \end{aligned} \quad (313)$$

and

$$\begin{aligned} p_{\text{vac}} &= \sum_{i=1}^{M} f_{PV}^{b,g,m}(\mu_i^{b,g,m}) F(\mu_i^{b,g,m}, p_0) + \sum_{j(i)=M+1}^{2M} f_{PV}^{f,g,m}(\mu_{j(i)}^{f,g,m}) F(\mu_{j(i)}^{f,g,m}, p_0) \\ &= \sum_{i=1}^{M} \left[ f_{PV}^{b,g,m}(\mu_i^{b,g,m}) + f_{PV}^{f,g,m}(\mu_{j(i)}^{f,g,m}) \right] F(\mu_i, p_0). \end{aligned} \quad (314)$$
From Equations (313)-(314) and Equations (305) finally we obtain

$$ 
\begin{aligned}
\epsilon_{\text{vac}} &= \sum_{i=1}^{M} \left[ f_{\text{PV}}^{\text{b.g.m.}}(\mu_{i}^{\text{b.g.m.}}) + f_{\text{PV}}^{\text{f.g.m.}}(\mu_{j(i)}^{\text{f.g.m.}}) \right] I(\mu_{i}, p_{0}) \\
&= \int_{\mu_{1}}^{\mu_{\text{eff}}} \left[ f_{\text{PV}}^{\text{b.g.m.}}(\mu) + f_{\text{PV}}^{\text{f.g.m.}}(\mu) \right] I(\mu, p_{0}) d\mu
\end{aligned}
\quad (315) $$

and

$$ 
\begin{aligned}
P_{\text{vac}} &= \sum_{i=1}^{M} \left[ f_{\text{PV}}^{\text{b.g.m.}}(\mu_{i}^{\text{s.m.}}) + f_{\text{PV}}^{\text{f.g.m.}}(\mu_{j(i)}^{\text{f.g.m.}}) \right] F(\mu, p_{0}) \\
&= \int_{\mu_{1}}^{\mu_{\text{eff}}} \left[ f_{\text{PV}}^{\text{b.g.m.}}(\mu) + f_{\text{PV}}^{\text{f.g.m.}}(\mu) \right] F(\mu, p_{0}) d\mu,
\end{aligned}
\quad (316) $$

where obviously

$$ f_{\text{PV}}^{\text{b.g.m.}}(\mu) + f_{\text{PV}}^{\text{f.g.m.}}(\mu) = f_{\text{PV}}^{\text{g.m.}}(\mu) \approx 0. \quad (317) $$

Thus finally we obtain

$$ \epsilon^{\text{g.m.}}(\mu_{\text{eff}}^{(1)}, \mu_{\text{eff}}^{(2)}, p_{0}) = \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} f_{\text{PV}}^{\text{g.m.}}(\mu) I(\mu, p_{0}) d\mu, \quad (318) $$

and

$$ p^{\text{g.m.}}(\mu_{\text{eff}}^{(1)}, \mu_{\text{eff}}^{(2)}, p_{0}) = \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} f_{\text{PV}}^{\text{g.m.}}(\mu) F(\mu, p_{0}) d\mu, \quad (319) $$

where $\mu_{\text{eff}}^{(1)}, \mu_{\text{eff}}^{(2)} \gg p_0$. In order to calculate $\epsilon^{\text{g.m.}}(\mu_{\text{eff}}^{(1)}, \mu_{\text{eff}}^{(2)}, p_0)$ and $p^{\text{g.m.}}(\mu_{\text{eff}}^{(1)}, \mu_{\text{eff}}^{(2)}, p_0)$ let us evaluate now the following quantities defined above by Equations (300)

$$ I(\mu, p_0) = \int_0^{p_0} p^2 \sqrt{p^2 + \mu^2} dp = \int_0^{p_0} \mu p^2 \sqrt{1 + \frac{p^2}{\mu^2}} dp \quad (320) $$

and

$$ F(\mu, p_0) = \frac{1}{3} \int_0^{p_0} \frac{p^4 dp}{\sqrt{p^2 + \mu^2}} = \frac{1}{3} \int_0^{p_0} \frac{p^4 \mu^{-1} dp}{\sqrt{1 + \frac{p^2}{\mu^2}}} \quad (321) $$

where $p_0/\mu \ll 1$. Note that

$$ 
\begin{aligned}
\sqrt{1 + \frac{p^2}{\mu^2}} &= 1 + \frac{1}{2} \frac{p^2}{\mu^2} - \frac{1}{8} \frac{p^4}{\mu^4} + \frac{1}{16} \frac{p^6}{\mu^6} + \dots \\
p^2 \sqrt{p^2 + \mu^2} &= p^2 \mu \sqrt{1 + \frac{p^2}{\mu^2}} = p^2 \mu \left( 1 + \frac{1}{2} \frac{p^2}{\mu^2} - \frac{1}{8} \frac{p^4}{\mu^4} + \frac{1}{16} \frac{p^6}{\mu^6} + \dots \right) \quad (322) \\
&= p^2 \mu + \frac{1}{2} \frac{p^4}{\mu} - \frac{1}{8} \frac{p^6}{\mu^3} + \frac{1}{16} \frac{p^8}{\mu^5} + \dots
\end{aligned}
$$

By inserting Equation (322) into Equation (320) one obtains

$$ 
\begin{aligned}
I(\mu, p_0) &= \int_0^{p_0} \left( p^2 \mu + \frac{1}{2} \frac{p^4}{\mu} - \frac{1}{8} \frac{p^6}{\mu^3} + \frac{1}{16} \frac{p^8}{\mu^5} + \dots \right) dp \\
&= \frac{1}{3} p_0^3 \mu + \frac{1}{10} \frac{p_0^5}{\mu} - \frac{1}{7 \times 8} \frac{p_0^7}{\mu^3} + \frac{1}{9 \times 16} \frac{p_0^9}{\mu^5} + \dots
\end{aligned}
\quad (323) $$

Note that
From Equations (313)-(314) and Equations (305) finally we obtain

$$ 
\begin{aligned}
\epsilon_{\text{vac}} &= \sum_{i=1}^{M} \left[ f_{\text{PV}}^{\text{b.g.m.}}(\mu_i^{\text{b.g.m.}}) + f_{\text{PV}}^{\text{f.g.m.}}(\mu_{j(i)}^{\text{f.g.m.}}) \right] I(\mu_i, p_0) \\
&= \int_{\mu_1}^{\mu_{\text{eff}}} \left[ f_{\text{PV}}^{\text{b.g.m.}}(\mu) + f_{\text{PV}}^{\text{f.g.m.}}(\mu) \right] I(\mu, p_0) d\mu
\end{aligned} 
\quad (315) $$

and

$$ 
\begin{aligned}
P_{\text{vac}} &= \sum_{i=1}^{M} \left[ f_{\text{PV}}^{\text{b.g.m.}}(\mu_i^{\text{s.m.}}) + f_{\text{PV}}^{\text{f.g.m.}}(\mu_{j(i)}^{\text{f.g.m.}}) \right] F(\mu_i, p_0) \\
&= \int_{\mu_1}^{\mu_{\text{eff}}} \left[ f_{\text{PV}}^{\text{b.g.m.}}(\mu) + f_{\text{PV}}^{\text{f.g.m.}}(\mu) \right] F(\mu, p_0) d\mu,
\end{aligned} 
\quad (316) $$

where obviously

$$ f_{\text{PV}}^{\text{b.g.m.}}(\mu) + f_{\text{PV}}^{\text{f.g.m.}}(\mu) = f_{\text{PV}}^{\text{g.m.}}(\mu) \approx 0. \quad (317) $$

Thus finally we obtain

$$ \epsilon^{\text{g.m.}}(\mu_{\text{eff}}^{(1)}, \mu_{\text{eff}}^{(2)}, p_0) = \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} f_{\text{PV}}^{\text{g.m.}}(\mu) I(\mu, p_0) d\mu, \quad (318) $$

and

$$ p^{\text{g.m.}}(\mu_{\text{eff}}^{(1)}, \mu_{\text{eff}}^{(2)}, p_0) = \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} f_{\text{PV}}^{\text{g.m.}}(\mu) F(\mu, p_0) d\mu, \quad (319) $$

where $\mu_{\text{eff}}^{(1)}, \mu_{\text{eff}}^{(2)} \gg p_0$. In order to calculate $\epsilon^{\text{g.m.}}(\mu_{\text{eff}}^{(1)}, \mu_{\text{eff}}^{(2)}, p_0)$ and $p^{\text{g.m.}}(\mu_{\text{eff}}^{(1)}, \mu_{\text{eff}}^{(2)}, p_0)$ let us evaluate now the following quantities defined above by Equations (300)

$$ I(\mu, p_0) = \int_0^{p_0} p^2 \sqrt{p^2 + \mu^2} dp = \int_0^{p_0} \mu p^2 \sqrt{1 + \frac{p^2}{\mu^2}} dp \quad (320) $$

and

$$ F(\mu, p_0) = \frac{1}{3} \int_{0}^{p_0} \frac{p^4 dp}{\sqrt{p^2 + \mu^2}} = \frac{1}{3} \int_{0}^{p_0} \frac{p^4 \mu^{-1} dp}{\sqrt{1 + \frac{p^2}{\mu^2}}} \quad (321) $$

where $p_0/\mu \ll 1$. Note that

$$ 
\begin{aligned}
\sqrt{1 + \frac{p^2}{\mu^2}} &= 1 + \frac{1}{2} \frac{p^2}{\mu^2} - \frac{1}{8} \frac{p^4}{\mu^4} + \frac{1}{16} \frac{p^6}{\mu^6} + \dots \\
p^2 \sqrt{p^2 + \mu^2} &= p^2 \mu \sqrt{1 + \frac{p^2}{\mu^2}} = p^2 \mu \left( 1 + \frac{1}{2} \frac{p^2}{\mu^2} - \frac{1}{8} \frac{p^4}{\mu^4} + \frac{1}{16} \frac{p^6}{\mu^6} + \dots \right) \quad (322) \\
&= p^2 \mu + \frac{1}{2} \frac{p^4}{\mu} - \frac{1}{8} \frac{p^6}{\mu^3} + \frac{1}{16} \frac{p^8}{\mu^5} + \dots
\end{aligned} $$

By inserting Equation (322) into Equation (320) one obtains

$$ 
\begin{aligned}
I(\mu, p_0) &= \int_0^{p_0} \left( p^2 \mu + \frac{1}{2} \frac{p^4}{\mu} - \frac{1}{8} \frac{p^6}{\mu^3} + \frac{1}{16} \frac{p^8}{\mu^5} + \dots \right) dp \\
&= \frac{1}{3} p_0^3 \mu + \frac{1}{10} \frac{p_0^5}{\mu} - \frac{1}{7 \times 8} \frac{p_0^7}{\mu^3} + \frac{1}{9 \times 16} \frac{p_0^9}{\mu^5} + \dots
\end{aligned} 
\quad (323) $$

Note that
$$ \left(1 + \frac{p^2}{\mu^2}\right)^{-1/2} = 1 - \frac{1}{2}\frac{p^2}{\mu^2} + \frac{3}{8}\frac{p^4}{\mu^4} + \cdots \tag{324} $$
$$ p^4 \mu^{-1} \left(1 + \frac{p^2}{\mu^2}\right)^{-1/2} = \frac{p^4}{\mu} - \frac{1}{2}\frac{p^6}{\mu^3} + \frac{3}{8}\frac{p^8}{\mu^5} + \cdots $$

By inserting Equation (324) into Equation (321) one obtains

$$ F(\mu, p_0) = \frac{1}{3} \int_0^{p_0} \frac{p^4 \mu^{-1} dp}{\sqrt{1 + \frac{p^2}{\mu^2}}} = \frac{1}{3} \int_0^{p_0} \left( \frac{p^4}{\mu} - \frac{1}{2} \frac{p^6}{\mu^3} + \frac{3}{8} \frac{p^8}{\mu^5} + \cdots \right) dp \tag{325} $$
$$ = \frac{p_0^5}{3 \times 5 \mu} - \frac{1}{2 \times 3 \times 7} \frac{p_0^7}{\mu^3} + \frac{1}{8 \times 9} \frac{p_0^9}{\mu^5} + \cdots $$

By inserting Equation (323) into Equation (318) one obtains

$$ \begin{aligned} & \varepsilon^{g.m.} (\mu_{\text{eff}}^{(1)}, \mu_{\text{eff}}^{(2)}, p_0) \\ &= \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} f_{\text{PV}}^{g.m.}(\mu) I(\mu, p_0) d\mu \\ &= \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} f_{\text{PV}}^{g.m.}(\mu) \left( \frac{1}{3} p_0^3 \mu + \frac{1}{10} \frac{p_0^5}{\mu} - \frac{1}{7 \times 8} \frac{p_0^7}{\mu^3} + \frac{1}{9 \times 16} \frac{p_0^9}{\mu^5} + \cdots \right) d\mu \tag{326} \\ &= \frac{p_0^3}{3} \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} f_{\text{PV}}^{g.m.}(\mu) \mu d\mu + \frac{p_0^5}{10} \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} \frac{f_{\text{PV}}^{g.m.}(\mu) d\mu}{\mu} - \frac{p_0^7}{7 \times 8} \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} \frac{f_{\text{PV}}^{g.m.}(\mu) d\mu}{\mu^3} + \cdots \end{aligned} $$

By inserting Equation (325) into Equation (319) one obtains

$$ \begin{aligned} & p^{g.m.} (\mu_{\text{eff}}^{(1)}, \mu_{\text{eff}}^{(2)}, p_0) \\ &= \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} f_{\text{PV}}^{g.m.}(\mu) F(\mu, p_0) d\mu \\ &= \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} f_{\text{PV}}^{g.m.}(\mu) \left( \frac{p_0^5}{3 \times 5 \mu} - \frac{1}{2 \times 3 \times 7} \frac{p_0^7}{\mu^3} + \frac{1}{8 \times 9} \frac{p_0^9}{\mu^5} + \cdots \right) d\mu \\ &= \frac{p_0^5}{3 \times 5} \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} \frac{f_{\text{PV}}^{g.m.}(\mu) d\mu}{\mu} - \frac{p_0^7}{2 \times 3 \times 7} \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} \frac{f_{\text{PV}}^{g.m.}(\mu) d\mu}{\mu^3} + \frac{p_0^9}{8 \times 9} \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} \frac{f_{\text{PV}}^{g.m.}(\mu) d\mu}{\mu^5} + \cdots \end{aligned} \tag{327} $$

Remark 6.6.7. We assume now that

$$ |f_{\text{PV}}^{g.m.}(\mu)| = \begin{cases} O\left( (\mu_{\text{eff}}^{(1)})^{-n} \right), & n > 7 \quad \mu_{\text{eff}}^{(1)} \le \mu \le \mu_{\text{eff}}^{(2)} \\ 0 & \mu > \mu_{\text{eff}}^{(2)} \end{cases} \tag{328} $$

Note that under assumption (328) the quantities $\varepsilon^{g.m.} (\mu_{\text{eff}}^{(1)}, \mu_{\text{eff}}^{(2)}, p_0)$ and $p^{g.m.} (\mu_{\text{eff}}^{(1)}, \mu_{\text{eff}}^{(2)}, p_0)$ cannot contribute in the value of the cosmological constant.

## 7. Conclusion

We argue that a solution to the cosmological constant problem is to assume that there exists hidden physical mechanism which cancels divergences in canonical $QED_4$, $QCD_4$, Higher-Derivative-Quantum-Gravity, etc. In fact, we argue that corresponding supermassive Pauli-Villars ghost fields, etc. really exist. New
$$ \left(1 + \frac{p^2}{\mu^2}\right)^{-1/2} = 1 - \frac{1}{2} \frac{p^2}{\mu^2} + \frac{3}{8} \frac{p^4}{\mu^4} + \cdots \qquad (324) $$ $$ p^4 \mu^{-1} \left(1 + \frac{p^2}{\mu^2}\right)^{-1/2} = \frac{p^4}{\mu} - \frac{1}{2} \frac{p^6}{\mu^3} + \frac{3}{8} \frac{p^8}{\mu^5} + \cdots $$

By inserting Equation (324) into Equation (321) one obtains

$$ F(\mu, p_0) = \frac{1}{3} \int_{0}^{p_0} \frac{p^4 \mu^{-1} dp}{\sqrt{1 + \frac{p^2}{\mu^2}}} = \frac{1}{3} \int_{0}^{p_0} \left( \frac{p^4}{\mu} - \frac{1}{2} \frac{p^6}{\mu^3} + \frac{3}{8} \frac{p^8}{\mu^5} + \cdots \right) dp \qquad (325) $$ $$ = \frac{p_0^5}{3 \times 5 \mu} - \frac{1}{2 \times 3 \times 7} \frac{p_0^7}{\mu^3} + \frac{1}{8 \times 9} \frac{p_0^9}{\mu^5} + \cdots $$

By inserting Equation (323) into Equation (318) one obtains

$$ \begin{aligned} & \varepsilon^{g.m.} (\mu_{\text{eff}}^{(1)}, \mu_{\text{eff}}^{(2)}, p_0) \\ &= \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} f_{\text{PV}}^{g.m.}(\mu) I(\mu, p_0) d\mu \\ &= \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} f_{\text{PV}}^{g.m.}(\mu) \left( \frac{1}{3} p_0^3 \mu + \frac{1}{10} \frac{p_0^5}{\mu} - \frac{1}{7 \times 8} \frac{p_0^7}{\mu^3} + \frac{1}{9 \times 16} \frac{p_0^9}{\mu^5} + \cdots \right) d\mu \qquad (326) \\ &= \frac{p_0^3}{3} \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} f_{\text{PV}}^{g.m.}(\mu) \mu d\mu + \frac{p_0^5}{10} \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} \frac{f_{\text{PV}}^{g.m.}(\mu) d\mu}{\mu} - \frac{p_0^7}{7 \times 8} \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} \frac{f_{\text{PV}}^{g.m.}(\mu) d\mu}{\mu^3} + \cdots \end{aligned} $$

By inserting Equation (325) into Equation (319) one obtains

$$ \begin{aligned} & p^{g.m.} (\mu_{\text{eff}}^{(1)}, \mu_{\text{eff}}^{(2)}, p_0) \\ &= \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} f_{\text{PV}}^{g.m.}(\mu) F(\mu, p_0) d\mu \\ &= \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} f_{\text{PV}}^{g.m.}(\mu) \left( \frac{p_0^5}{3 \times 5 \mu} - \frac{1}{2 \times 3 \times 7} \frac{p_0^7}{\mu^3} + \frac{1}{8 \times 9} \frac{p_0^9}{\mu^5} + \cdots \right) d\mu \\ &= \frac{p_0^5}{3 \times 5} \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} \frac{f_{\text{PV}}^{g.m.}(\mu) d\mu}{\mu} - \frac{p_0^7}{2 \times 3 \times 7} \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} \frac{f_{\text{PV}}^{g.m.}(\mu) d\mu}{\mu^3} + \frac{p_0^9}{8 \times 9} \int_{\mu_{\text{eff}}^{(1)}}^{\mu_{\text{eff}}^{(2)}} \frac{f_{\text{PV}}^{g.m.}(\mu) d\mu}{\mu^5} + \cdots \end{aligned} \qquad (327) $$

Remark 6.6.7. We assume now that

$$ |f_{\text{PV}}^{g.m.}(\mu)| = \begin{cases} O\left( (\mu_{\text{eff}}^{(1)})^{-n} \right), & n > 7 \quad \mu_{\text{eff}}^{(1)} \le \mu \le \mu_{\text{eff}}^{(2)} \\ 0 & \mu > \mu_{\text{eff}}^{(2)} \end{cases} \qquad (328) $$

Note that under assumption (328) the quantities $\varepsilon^{g.m.} (\mu_{\text{eff}}^{(1)}, \mu_{\text{eff}}^{(2)}, p_0)$ and $p^{g.m.} (\mu_{\text{eff}}^{(1)}, \mu_{\text{eff}}^{(2)}, p_0)$ cannot contribute in the value of the cosmological constant.

## 7. Conclusion

We argue that a solution to the cosmological constant problem is to assume that there exists hidden physical mechanism which cancels divergences in canonical $QED_4$, $QCD_4$, Higher-Derivative-Quantum-Gravity, etc. In fact, we argue that corresponding supermassive Pauli-Villars ghost fields, etc. really exist. New
theory of elementary particles which contain hidden ghost sector is proposed. In accordance with Zel'dovich hypothesis [1] we suggest that physics of elementary particles is separated into low/high energy ones, the standard notion of smooth spacetime is assumed to be altered at a high energy cutoff scale $Λ_*$ and a new treatment based on QFT in a fractal spacetime with negative dimension is used above that scale. 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 below that scale $Λ_*$, i.e.,

$$f_{s,m}(\mu) \approx f_{g,m}(\mu), \mu \le \mu_{\text{eff}}, \mu_{\text{eff}} c^2 < \Lambda_*$$

## Acknowledgements

We thank the reviewers for their comments.

## Conflicts of Interest

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

## References

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https://doi.org/10.1070/PU1968v011n03ABEH003927

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[11] Bouda, A. and Foughali, T. (2012) On the Fock Transformation in Nonlinear Relativity. *Modern Physics Letters A*, **27**, Article ID: 1250036.
theory of elementary particles which contain hidden ghost sector is proposed. In accordance with Zel'dovich hypothesis [1] we suggest that physics of elementary particles is separated into low/high energy ones, the standard notion of smooth spacetime is assumed to be altered at a high energy cutoff scale $\Lambda_*$ and a new treatment based on QFT in a fractal spacetime with negative dimension is used above that scale. 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 below that scale $\Lambda_*$, i.e.,

$$f_{s,m}(\mu) \approx f_{g,m}(\mu), \mu \le \mu_{\text{eff}}, \mu_{\text{eff}} c^2 < \Lambda_*$$

## Acknowledgements

We thank the reviewers for their comments.

## Conflicts of Interest

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

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https://doi.org/10.1070/PU1968v011n03ABEH003927

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https://doi.org/10.1070/PU2002v045n02ABEH001108

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