• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,345 answers , 22,721 comments
1,470 users with positive rep
818 active unimported users
More ...

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

+ 4 like - 0 dislike

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.

asked May 2, 2015 in Mathematics by Peter Morgan (1,230 points) [ no revision ]

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.

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)

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.

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.

@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.

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights