## 2 Coulomb Problem in \(D\) Dimensions  
We have the following expression for the solution of the Poisson equation with a point-like source in \(D\)-dimensional space [1, 7] (see Appendix A):  
\[\Delta\varphi=e\delta^{D}(x),\] \[\varphi(D,r)=-\frac{\Gamma(D/2)}{2(D-2)\pi^{D/2}}er^{2-D},\] \[V(D,r)=e\varphi(D,r)=-\alpha(D)r^{2-D},\] \[\alpha(D)=\frac{e^{2}\Gamma(D/2)}{2(D-2)\pi^{D/2}},\ D\
eq 2,\ r=|x| \tag{2}\]  
and, for \(D=2\),  
\[\varphi(2,r)=\frac{e}{2\pi}\ln r,\ V(2,r)=\frac{e^{2}}{2\pi}\ln r, \tag{3}\]  
In particular, for spatial dimensions 1, 3 and 4, we have from (2):  
\[V(1,r)=\frac{e^{2}r}{2},\ V(3,r)=-\frac{e^{2}}{4\pi r},\ V(4,r)=-\frac{e^{2}} {4\pi^{2}r^{2}}\,. \tag{4}\]Now, let us consider [PERSON]'s equation in \(D\) dimensions with a general source,  
\[\Delta V(x)=e^{2}\rho(x)\, \tag{5}\]  
where \(V(x)\) is the generalized \(D\)-dimensional potential, \(\rho(x)\) is some source function. We find that for the special case \(\rho(x)=\delta(x)\) we obtain the fundamental solution (2) (for textbook derivations using generalized functions, see, e.g., [8]):  
\[E(x)=-\frac{\alpha(D)}{r^{D-2}}. \tag{6}\]  
For a distribution \(\rho(x)\) of delta functions over a volume \(v\), the corresponding value of \(V(x)\) is  
\[V(x)=E*\rho=-\alpha_{D}\int_{v}\frac{\rho(y)d^{D}y}{|x-y|^{D-2}} \tag{7}\]  
where \(*\) denotes convolution. As defined so far, the coupling constant has a mass dimension \(d_{e}=(D-3)/2=-\varepsilon\). To work with a dimensionless coupling constant \(e\), we introduce the mass scale \(\mu\). Then, the potential energy (2) takes the following form  
\[V(D,r)=-\frac{\Gamma(D/2)}{2(D-2)\pi^{D/2}}e^{2}\mu^{2r}\mu^{2-D}=-\alpha(D)( \mu r)^{2\varepsilon}/r=-\alpha(D)(x)^{2-D}\mu. \tag{8}\]  
Let us relate the point charge problem to quarkonium and derive the Coulomb coefficient \(k=4/3\alpha_{s}\) in the Cornell potential (1). The static potential between two heavy quarks belongs to the fundamental quantities of QCD. In lowest order, in coordinate space (see, e.g., [9])  
\[\varphi(D,r)=-\frac{\Gamma(D/2)}{2(D-2)\pi^{D/2}}er^{2-D},\]0  
with \(C_{1}(N_{c})=C_{F}=(N_{c}-N_{c}^{-1})/2>0\) for the colour-singlet and \(C_{8}(N_{c})=C_{F}-C_{A}/2=-1/2N_{c}<0\) for the colour-octet state. \(N_{c}\) denotes the number of colors. In QCD, \(N_{c}=3\). Thus \(C_{1}(3)=(3-1/3)/2=4/3\), as we wanted to show. **3. Fractal Calculus of Quantum Field Theory in Examples**  
Let us solve the point charge problem another way--using fractal calculus. The solution in case of a non-integer dimension is, by definition, a fractal (for concise introduction in fractal calculus with some applications see, e.g., [10]). Matrix calculus in QFT perturbation theory [11] can be interpreted as operator fractal calculus. Indeed, with the following definitions  
\[\varphi(D,r)=-\frac{\Gamma(D/2)}{2(D-2)\pi^{D/2}}er^{2-D},\]1 \[\varphi(D,r)=-\frac{\Gamma(D/2)}{2(D-2)\pi^{D/2}}er^{2-D},\]2  
for two point function we have (see Appendix B),  
\[\varphi(D,r)=-\frac{\Gamma(D/2)}{2(D-2)\pi^{D/2}}er^{2-D},\]3 \[\varphi(D,r)=-\frac{\Gamma(D/2)}{2(D-2)\pi^{D/2}}er^{2-D},\]4  
As an example, consider the Coulomb potential, the solution of the equation for potential of a point source  
\[\varphi(D,r)=-\frac{\Gamma(D/2)}{2(D-2)\pi^{D/2}}er^{2-D},\]5Note that, \(\Delta=-\hat{p}^{2}\),  
\[\varphi(D,r)=-\frac{\Gamma(D/2)}{2(D-2)\pi^{D/2}}er^{2-D},\]6  
Using \(\Gamma(D/2-1)=\Gamma(D/2)/(D/2-1)\), Formula (13) is easily reduced to (2). As a side note, let us mention another application of fractal calculus--we take the following integral, for composition of two point functions, useful for diagram calculations in Quantum field theories (QFT),  
\[\varphi(D,r)=-\frac{\Gamma(D/2)}{2(D-2)\pi^{D/2}}er^{2-D},\]7 \[\varphi(D,r)=-\frac{\Gamma(D/2)}{2(D-2)\pi^{D/2}}er^{2-D},\]8 \[\varphi(D,r)=-\frac{\Gamma(D/2)}{2(D-2)\pi^{D/2}}er^{2-D},\]9 \[V(D,r)=e\varphi(D,r)=-\alpha(D)r^{2-D},\]0  
The constructions of this section can be translated in terms of the theory of generalized functions [8].