Inverse square-law as a positive definite kernel?

Question

Newtons law for gravity states that:

$$F_{12} = \frac{G m_1 m_2} {|x_1-x_2|^2}$$

The function :

$$k(x,y):=\exp(-| x-y|^2)$$

is known to be a positive definite function, called the RBF-kernel.

It follows that

$$k(x_1,x_2) := \exp( -|x_1-x_2|^2) = \exp(-\frac{G m_1 m_2}{F_{12}})$$

is a positive definite , symmetric function. My question, is if you know of any other situations, where terms like

$$\exp(-\frac{m_1 m_2 G}{F_{12}})$$

are shown to be positive definite kernels and what is the explanation for this?

(By this I mean: Do you know of situations where one proves that functions like:

$$f(a,b) = \exp(-\frac{m_a m_b G}{F_{ab}})$$

are positive definite - and how is it proved? Here $m_x$ denotes numbers $>0$ and $F_{xy}$ is a function of $x,y$.)

Thanks for your help.

Answer 1

Please look pages 141-147 of my book:

S. Saitoh and Y. Sawano, Theory of Reproducing Kernels and Applications, Developments in Mathematics 44, Springer (2016). 2023.5.4.20:36　https://doi.org/10.1007/978-981-10-0530-5

Answer 2

The answer to this question, can be given a meaning with the Schoenberg criterion from which the following explanation is borrowed:

The function $\Psi \colon X \times X \to \mathbb{R}$ is said to be a kernel of conditionally negative type if

1. $\Psi(x,x) = 0$ for all $x \in X$.
2. $\Psi(x,y) = \Psi(y,x)$ for all $x,y \in X$.
3. For all $n \in \mathbb{N}$, all $x_1,\dots,x_n \in X$ and all $c_1,\dots,c_n \in \mathbb{R}$ such that $\sum_{i=1}^n c_i = 0$ the inequality $$ \sum_{i,j=1}^n c_i c_j \Psi(x_i,x_j) \leq 0 $$ holds.

Theorem. For a symmetric function $\Psi\colon X \times X \to \mathbb{R}$ with $\Psi(x,x) = 0$ for all $x$ the following are equivalent:

1. $\Psi$ is a kernel of conditionally negative type.
2. The function $K(x,y) = \exp(-\gamma \Psi(x,y))$ is a positive semidefinite kernel for all $\gamma \geq 0$.

If we set:

$$\Psi(a,b) := \frac{G m_a m_b}{F_{ab}}$$

and by physics argument where we are considering only one material body, or by the Division by Zero Calculus, we observe that:

$$\Psi(a,a) = 0 \forall a$$

then by Schoenbergs criterion, since we already know that the function

$$\exp(-\Psi(a,b)) $$

is positive definite, it must follow that

$$\Psi(a,b) = \frac{G m_a m_b}{F_{ab}}$$

is a kernel of conditionally negative type. I have not thought about any physics application of this yet, if it has any.

I will remark the following, which shows, that this observation is not limited to Newton's gravity law, but can be combined also with Colulombs law:

First let us look at the situation where two material bodies $a,b$ which might posses mass, electric charge and magnetic charge, which exert force on each other, by the Newtons gravity law and Columb's law for electric charges and magnetic charges. By Newton's law, we can build the sum of these forces:

$$F(a,b) \cdot r/|r| = (F_N(a,b) + F_C(a,b) + F_M(a,b) ) r/|r|$$

which is equal to:

$$F(a,b) \cdot r/|r| = (\mu_N(a)\mu_N(b)G+\mu_C(a)\mu_C(b)k_e+\mu_M(a)\mu_M(b)k_m)/|x_a-x_b|^2)\cdot r/|r|$$

and this might be written as:

$$F(a,b) \cdot r/|r| = \frac{\left < \mu(a),\mu(b)\right>}{|x_a-x_b|^2} \cdot r/|r|$$

which again might be written as:

$$F(a,b) = -\frac{\left < \mu(a),\mu(b)\right>}{\log(k(a,b))}$$

where we have put:

$$\mu(x) = (\mu_N(x),\mu_C(x),\mu_M(x))$$

and here $\mu_N(x),\mu_C(x),\mu_M(x) \equiv $ mass, electric charge, magnetic charge, $k(a,b) := \exp( - |x_a-x_b|^2)$ and $$\left < \mu(a),\mu(b) \right> := \mu_N(a)\mu_N(b)G+\mu_C(a)\mu_C(b)k_e+\mu_M(a)\mu_M(b)k_m$$ is an inner product on the measurable quantities.

You can find a proof of this as Theorem C.3.2 on page 370 of the book Kazhdan's Property $(T)$ by Bekka, de la Harpe, and Valette, (link goes to Bekka's homepage). The rest of this appendix might have some useful information, too.

---

Added: Schoenberg's original article: Metric spaces and positive definite functions, Trans. Amer. Math. Soc. 44 (1938), 522-536.