# Where does this term "shell" with prefix "on-/off-" come from?

Is there some historical reasons or is there a specific reason behind it?

This question is connected to: Why on-shell vs. off-shell matters?

• In your linked Physics.SE question, there's a link to this WP page. Are you looking for something more in-depth?
– BMS
Apr 15 '14 at 21:55

A particle is said to be on-shell if it satisfies the relativistic dispersion relation,

$$E^2 = p^2 +m^2$$

in units wherein $c=\hbar=1$. If you graph it, you obtain a parabolic surface for massive particles, and a cone for massless particles, like a photon. This is known as the mass shell, it is quite literally a shell or surface. The momentum of a real particle can be represented as a vector on the surface, hence the expression on shell. Virtual particles do not have these on the surface, hence they are off shell.  Source: Perimeter Institute, A Deeper Dive: On Shell and Off Shell

The term "shell" originally derives from the non-relativistic version of the answer by @JamalS. In a non-relativistic theory, a free particle satisfies the following dispersion relation $$E = \frac{ {\bf p}^2 }{ 2m }$$ For a fixed energy a particle satisfies $${\bf p}_x^2 + {\bf p}_y^2 + {\bf p}_z^2 = 2 m E$$ In momentum space, this is precisely the equation of a spherical "shell" with radius $\sqrt{2mE}$. (See a plot here)

Now, in momentum space, the 3-momentum of a particle is described by a point. If $E$ is fixed, then the 3-momentum of the particle is only allowed to lie on the spherical shell described above, and is therefore said to be on-shell.

In quantum mechanics, a particle is allowed to not satisfy the dispersion relation. Thus, the momentum of a particle in quantum mechanics is allowed to not lie on the shell, and can be off-shell.

In a relativistic theory, the dispersion relation changes to $$E^2 = {\bf p}^2 + m^2 \implies {\bf p}_x^2 + {\bf p}_y^2 + {\bf p}_z^2 = E^2 - m^2$$ The corresponding constant $E$ surface is again a spherical shell, (this time of radius $\sqrt{ E^2 - m^2 }$)