On shell and off shell
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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.
[edit] 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
describing combinations of energy E and momentum p allowed by classical special relativity for a particle of rest mass m; c here 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 pμpμ = 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.
A common misconception about off shell sets is that they violate energy conservation law while in fact they do not – because energy can not be mathematically defined over an arbitrary small time period (see uncertainty principle). The longer the time within which energy has to be defined, the more accurately it can be defined. Therefore, the energy of a virtual particle is an arbitrary value allowed by the uncertainty principle.
(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.)
