Severian said:
One should keep in mind that all particles are in fact virtual.
I've been working on rewriting QM to the density marix formalism and one of the things that comes out of this is that all particles have to be treated as if they were virtual. That is, one ends up rewriting Feyman diagram amplitudes so that the initial and final states are treated the same as the intermediate states.
A short way of stating this is that in the usual state vector formalism an amplitude looks like:
\langle I | M | F\rangle
where I and F are the initial and final state vectors and M is some complicated thing. In rewriting this in density operator form, M is easy to convert as it is already in operator form. And the conversion for I and F is to turn them into density operators and rewrite the above amplitude into operator form:
|I \rangle \langle I | M | F\rangle\langle F|
But the internal (virtual) propagators already were in density matrix |\psi\rangle\langle \psi| form, so written in density matrix form, everything looks to be in virtual form.
What happens in the above is that a complex number gets replaced with what turns out to be a complex multiple of an operator. You can then get the squared magnitude (i.e. a real number) by the usual technique, but it ends up coming out multiplied by a density operator (which, unlike the state vectors, do not have arbitrary complex phases).
Carl
