Mass of the particle and gravitational field

Question

Which mass of the particle is the source of gravitational field? If we define mass as a pole of the propagator, and calculate loop corrections to the pole we get infinities. Now the way we get rid of these infinities defines what is our renormalized propagator and what is its pole. What is the mass of the particle is dependent on which renormalization scheme we choose. Now, gravity field knows about everything that happens to the particle (as any energy gravitates), all contributions of whatever virtual particles contribute to the mass of particle (up to very short distances).

Answer 1

Classically, the source of the gravitational field is the energy momentum tensor $T_{\mu\nu}$. In the absence of a full quantum theory of gravity, a semiclassical approach is often taken, whereby the source of the gravitational field is the expectation value $\langle T_{\mu\nu}\rangle$. The procedure is quite tricky to carry out though, because $T_{\mu\nu}$ generally contains products of fields, and in the quantum case we would then be evaluating an expectation value of operators at the same spacetime point, with the obvious problem that causes. There are various schemes for handling this, discussed, for example here.

Edit: Here is an online reference talking about the semiclassical approach.

As these references point out, even the definition of a particle is not trivial in curved spacetime.

Answer 2

The correct answer is "observable mass".

Renormalization can be used to get rid of infinities, but it provides no answer about the values of observable masses. And the problem is not in using different renormalization schemes. One can use the same scheme for electron and muon, and it is known from experiment that the result must be different.