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. This is because the -dependence of the propagator is determined by the four-momenta of the incoming and outgoing particles. 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.
Now, consider an infinitesimal spacetime translation. The Lagrangian density is a scalar, and so will infinitesimally transform as under the infinitesimal transformation. On the other hand, by Taylor expansion, we have in general
Substituting for and noting that (since the variations are independent at each point in spacetime):
Since this has to hold for independent translations , we may "divide" by and write:
This is an example of equation that holds off shell, since it is true for any fields configuration regardless of whether it respects the equations of motion (in this case, the Euler-Lagrage equation given above). However, we can derive an on shell equation by simply substituting the Euler-Lagrange equation:
We can write this as:
And if we define the quantity in brackets as , we have: