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In Quantum Field Theory in a (1, D - 1) space-time, to calculate transition amplitudes, we are using Feynman diagrams, where internal lines (internal propagators) corresponds to momenta which are said to be off-shell : $p^2 \neq 0$.

Is it posssible to interpret these momenta as on-shell, but within (2) space-time(s), with a supplementary dimension ?

For instance, consider a massless (1,3) space-time Quantum Field Theory, like pure Yang-Mills theory (gluons only).

External legs corresponds to on-shell momenta $p^2 = (p_o)^2 - (p_1)^2 - (p_2)^2 - (p_3)^2 = 0$

Internal lines correspond to (1, 3) off-shell momenta, but there are 2 possibilites, $p^2 >= 0$ and $p^2 <= 0$.

For $p^2 >= 0$, a (1, 4) on-shell condition could be written :

$$ (p_o)^2 - (p_1)^2 - (p_2)^2 - (p_3)^2 - (p_4)^2 = 0$$

For $p^2 <= 0$, a (2, 3) on-shell condition could be written :

$$ (p_{-1})^2 + (p_o)^2 - (p_1)^2 - (p_2)^2 - (p_3)^2 = 0$$

So, we could consider than free particles (on-shell external legs) live in a (1, 3) space-time, but than "interactive" particles (off-shell internal lines), live in 2 "interaction" space-times, which have signatures (1, 4) and (2, 3).

So we could say, that in fact, everything is on-shell, but interaction make one supplementary dimension appears (depending on the value of $p^2$, this is a time-like or space-like dimension).

Another way to say it, is that free particles could live on a space-time which could be the intersection of 2 "interaction" space-times.

A even more interesting model would be that free particles live in a space-time which would be the boundary of some "interaction" space-time.

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