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What is the meaning of 'on-shell' and 'off-shell'?
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This applies only to relativistic particles.
A particle of mass m is on-shell if its momentum p satisfies
p^2 (= p_0^2-p_1^2-p_2^2-p_3^2) = m^2,
and off-shell otherwise. The 'mass shell' is the manifold of
momenta p with p^2=m^2.
In vacuum, observable (i.e., physical) particles are asymptotic states
(scattering states) described (modulo unresolved mathematical
difficulties) by free fields based on the dispersion relation p^2=m^2,
and hence are necessarily on-shell. Off-shell particles only
arise in intermediate perturbative calculations; they are necessarily
'virtual'.
The situation is muddled by the fact that one has to distinguish
(formal) bare mass and (physical) dressed mass; the above is valid
only for the dressed mass.
Note that the mass shell loses its meaning in external fields
or dense media, where instead a so-called 'gap equation' appears.
In particular, in matter, particles can be off-shell in the different
sense of ''not on any mass shell'' -- due to their interaction with the
background matter; in fact, due to ''collisional broadening'',
they no longer have a well-defined mass and energy but must be
characterized by a space- and time-dependent spectral function.