vanesch said:
I was going to say "no that's not what I mean", but now I'm not so sure actually. I was of course thinking of the *classical* argument of electron mass by the EM field energy (argument which is quantitatively erroneous, but of which I thought the principle was still valid).
Aren't you mixing up the sum over bubble diagrams on one hand - the vacuum energy or the zero point energy of a sourceless field - on one hand, and the "mass correction terms" terms on the other hand ?
But I don't even know whether this includes the field energy of a field with source.
The term 'source' in QFT has a special meaning. For example, the EM field is described by the 4-potential A^{\mu}(x). Then, we introduce an artifical current density J^{\mu}(x) and include an extra interaction term in the Lagrangian:
<br />
\mathcal{L}(A, J) = \mathcal{L}_{0}(A) - J_{\mu} \, A^{\mu}<br />
Using this Lagrangian, one calcualates the partition function:
<br />
Z[J] = \exp{\left[i \, \int{d^{d}x \, \mathcal{L}(A, J)} \right]} = Z_{J = 0} \, \exp{\left[\frac{i}{2} \, \int{d^{d}y \, d^{d}z \, J_{\mu}(y) \, D^{\mu \nu}(y - z) \, J_{\nu}(z)}\right]}<br />
In fact, it is by taking (functional) derivatives w.r.t. to these external currents that we get the propagators:
<br />
\frac{1}{i} \, D^{\mu \, \nu}(x - y) = \langle T A^{\mu}(x) \, A^{\nu}(y) \rangle = \frac{1}{i} \, \frac{\delta}{\delta J_{\mu}(x)} \, \frac{1}{i} \, \frac{\delta}{\delta J_{\nu}(y)} \, \left. Z[J] \right|_{J = 0}<br />
At the end of the day we always set the sources equal to zero!
There is another famous calculation of the total energy of the field for a given external source distribution. This energy in fact diverges for a point source. However, in QFT, this divergence is meaningless, since it is an artifice of the 'weird apparatus' the experimenter was using to probe the field.
Another thing is when the current that couples to the field is a Noether current of another field that is conserved due to the intrinsic symmetry of the other field (for QED it is U(1)). Then, one understands the EM field as a gauge field that is necessary to be introduced in the 'gauge covariant derivative' in order to preserve the
local gauge invariance.
However, now the field that creates the (microscopic) current is itself a quantum object and one cannot treat it as a parameter over which we can differentiate. It has its own dynamics described by its own Lagrangian and we arrive at an interacting field theory. In this way, loop corrections to the propagators play role. And, in fact, the integrals involved in these corrections actually diverge even at first order. The procedure to make these integrals convergent in a controlled manner is called regularization and the procedure of canceling out divergences order by order in perturbation theory through the redefinition of the bare parameters (mass, charge) is called renormalization.
