The Norm on the forward mass-shell — reference request and previous use in Physics?

Question

The shortest distance Norm on the forward mass-shell of mass $m$ is

$$d(k,k')=m\ln\left[\frac{k\cdot k'}{m^2}+\sqrt{\frac{(k\cdot k')^2}{m^4}-1}\right],$$ which extends to a projective Norm for 4–vectors within the forward light–cone, $$d_0(k,k')=\ln\left[\frac{k\cdot k'}{mm'}+\sqrt{\frac{(k\cdot k')^2}{m^2{m'}^2}-1}\right],$$ where $k\cdot k=m^2,k'\cdot k'={m'}^2$.

I'd like to know of previous uses in Physics of either of these Norms and a citable reference in either the Physics or Mathematics literature.

Comments

I have a reference: Michel Marie Deza & Elena Deza, The Encyclopedia of Distance, Springer-Verlag Berlin Heidelberg 2009, p. 114, where the hyperbolic metric is in effect given as

$$\mathrm{arccosh}\!\left[\frac{k\cdot k'}{mm'}\right].$$ I'm still interested in any previous uses in Physics.

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This is nice, what do you use it for? Is there a reasonable $m \to 0^\pm$ extension? I would be interested in seeing the preprint once you finish it. (Sorry for OT, the concept is quite natural and will probably be constructed somewhere but i haven't really seen it before)

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On the light–cone, you would have a semi–norm at best because distance along light–rays is zero. If we can construct a semi–norm, taking the quotient set of light–rays would give a norm, but I can't immediately see a covariant presentation. If for some reason we can fix a time–like direction, there is a norm on the light–rays that is just the norm on the unit sphere, which to me makes a covariant norm on the set of light–rays look unlikely.

As natural as this is on the mass–shell, I worry considerably about tractability if I use it. $d(k,k')$ is an infinite–dimensional distance matrix. I'd be most likely to use the positive semi–definite matrix $\mathrm{e}^{-\frac{\alpha}{m} d(k,k')}$, which is just $\frac{1}{\left(\frac{k\cdot k'}{m^2}+\sqrt{\frac{(k\cdot k')^2}{m^4}-1}\right)^\alpha}$, aiming to satisfy a variant of the Wightman axioms in a way that would satisfy the Haag–Kastler axioms (or some very close variant thereof). One can't usually use this construction because translation invariance requires the use of $(2\pi)^4\delta^4(k-k')$, diagonal in $k,k'$, for the one–particle commutation relation.

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The forward cone is a symmetric space of rank 1, and there are general formulas from differential geometry for the geodetic distance on any of them. I believe these can be found, e.g., in Helgason's book ''Differential geometry and symmetric spaces". I haven't seen any use in physics, but this doesn't mean much - the literature is too vast to see much more than one is looking for.

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@Void $\lim\limits_{m\rightarrow 0} f(m)d(k,k')$ either doesn't exist or is trivial for any multiple $f(m)$, but nonetheless $\lim\limits_{m\rightarrow 0}\left(\frac{2}{m^2}\right)^\alpha\mathrm{e}^{-\frac{\alpha}{m}d(k,k')}=(k\cdot k')^{-\alpha}$, which might perhaps be useful for my purposes.