which is known as the radial basis function kernel (RBF kernel) and has applications in machine learning and statistics.
Interestingly, $k(a,b)$ can be shown to be the inner product $\langle a,b\rangle$ to an infinite dimensional Hilbert space, specifically the Hilbert Sphere:
$$
S_\infty = \{ x \in H \mid k(x,x) = \langle x,x\rangle = 1 \}.
$$
By solving the last equality for $F(a,b)$, we can define the force in any Hilbert sphere with inner product $k(a,b)$ for $a \neq b$ as:
$$
F(a,b) := - \frac{m_N(a) \cdot m_N(b) }{ \log( k(a,b) )},
$$
where $m_N(x)$ denotes the mass of $x$.
Now, if we consider only kernels that satisfy the axioms of force that implement Mach's principle (A. K. Assis' axioms), we could create something I would like to name Weber-Assis-MachWeber–Assis–Mach-space (WAM-space for short) that adds more dynamics to the Hilbert sphere. The axioms from A. Assis are as follows:
However, I am unsure if the Weber law for gravity, as described by Assis, allows one to define such a kernel. Additionally, from a physics standpoint, the geodesic distance in Hilbert sphere is given by $\arccos(k(a,b))$, which results in $$ d(a,b) = \arccos \exp\left( -\frac{|x_a-x_b|^2}{G} \right) = \arccos\exp\left( -\frac{ m_a \cdot m_b }{ F(a,b)} \right). $$ I am imagining a so called WAM-space, where you can define objects having $a, b, c, d, \ldots$$a, b, c, d, \dotsc$ living in a Hilbert sphere with inner product $k(a,b)$ having a certain 'mass' $m_N(a), m_N(b), \ldots$$m_N(a), m_N(b), \dotsc$ which are non-zero real numbers, and between two of these objects, there is a force: $$ F(a,b) := - \frac{m_N(a) \cdot m_N(b) }{ \log( k(a,b) )}. $$ subject to the axioms of A. Assis, which can be found in his book here in his bookRelational Mechanics and Implementation of Mach’s Principle with Weber’s Gravitational Force.
This approach described above, might maybe be used to find a Hilbert sphere, where the forces of NewtonsNewton's gravity and CoulombsCoulomb's electricity are combined to one force. Indeed, suppose on the material body $a$ act two forces of different quality such as the gravity force of Newton and the electric force of Coulomb. In physics, if they act upon the same body, then the sum of the forces does again by Newtons law, corresponds to a new force, hence:
$$
F(a,b) = F_N(a,b) + F_C(a,b) \label{1}\tag{$\ast$}
$$
where $F_N$ is the Newton gravity force, and $F_C$ is the electric Coulomb force. The "masses" or let us better say, "quantities" are given by:
$$
m(a) = \big(m_N(a),m_C(a)\big),\quad
m(b) = \big(m_N(b),m_C(b)\big),
$$
where $m_N(x)$ denotes, as above, the mass of $x$ while $m_C(x)$ denotes the electric charge of $x$. Hence $m(x)$ is a two dimensional vector.
Let us define in this case:
$$
F(a,b) = \frac{\langle m(a) , m(b)\rangle}{ \log( k(a,b) )}.
$$
where we want to determine $k(a,b)$ in such a way, that \eqref{1} is fulfilled
$$
\begin{split}
F_N(a,b) & = \frac{\langle m_N(a),m_N(b)\rangle}{\log(k_N(a,b))},\\ \langle m_N(a),m_N(b)\rangle &:= m_N(a)\cdot m_N(b),\\
k_N(a,b) &:= \exp\left(-\frac{|x_a-x_b|^2}{G}\right)
\end{split}
$$
where $G$ is the Gravity constant and
$$
\begin{split}
F_C(a,b) &= \frac{\langle m_C(a),m_C(b)\rangle}{\log(k_C(a,b))},\\ \langle m_C(a),m_C(b)\rangle &:= m_C(a)\cdot m_C(b),\\
k_C(a,b) &:= \exp\left(-\frac{|x_a-x_b|^2}{C}\right)
\end{split}
$$
where $C$ is CoulombsCoulomb's constant.
Using \eqref{1} and solving for $k(a,b)$ we find that for $a\neq b$:
$$
k(a,b) = \exp\left( \frac{\langle m(a),m(b)\rangle}{ F_C(a,b)+F_N(a,b) } \right)
$$
and $k(a,a) := 1$.
Why should the kernel defined by k(a,b)$k(a,b)$ as above, be positive definite?
