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.

For instance, in classical mechanics in the action formulation, extremal solutions to the variational principle are on shell and the Euler-Lagrange equations are on shell equations (i.e., they do not hold off shell). Noether's theorem is also another on shell theorem.

Mass shell

The term comes from the phrase mass shell, which is a synonym for mass hyperboloid, meaning the hyperboloid in energy-momentum space describing the solutions to the equation

$E^2 - |\vec{p} \,|^2 c^2 = m^2 c^4$

describing combinations of energy E and momentum p allowed by classical special relativity for a particle of mass m; where c is the speed of light. The equation for the mass shell is also often written in terms of the four-momentum, in Einstein notation and units where c = 1, as p^μp_μ = m² or simply as p² = m².

Virtual particles corresponding to internal propagators in a Feynman diagram are in general allowed to be off shell, but the amplitude for the process will diminish depending on how far off shell they are; the propagator typically has singularities on the mass shell.

When speaking of the propagator, negative values for E that satisfy the equation are thought of as being on shell, though the classical theory does not allow negative values for the energy of a particle. This is because the propagator incorporates into one expression the cases in which the particle carries energy in one direction, and in which its antiparticle carries energy in the other direction; negative and positive on-shell E then simply represent opposing flows of positive energy.