On shell and off shell

In physics, particularly in quantum field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell.

In quantum field theory, virtual particles are termed off shell (mass-shell in this case) because they don't satisfy the Einstein energy-momentum relationship; real exchange particles do satisfy this relation and are termed on shell (mass-shell). In classical mechanics for instance, in the action formulation, extremal solutions to the variational principle are on shell and the Euler–Lagrange equations give the on shell equations. Noether's theorem is another on shell theorem.

The term is a synonym for mass hyperboloid, meaning the hyperboloid in energy–momentum space describing the solutions to the equation:

which gives the energy E in terms of the momentum ${\displaystyle {\vec {p}}}$ and the rest mass m of a particle in classical special relativity. The equation for the mass shell is also often written in terms of the four-momentum; in Einstein notation with metric signature (+,–,–,–) and units where the speed of light c = 1, as ${\displaystyle p^{\mu }p_{\mu }\equiv p^{2}=m^{2}}$. In the literature, one may also encounter ${\displaystyle p^{\mu }p_{\mu }=-m^{2}}$ if the metric signature used is (–,+,+,+).

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