Is there a good reference for the distinction between projection operators in QFT, with an eigenvalue spectrum of $\{1,0\}$, representing yes/no measurements, the prototype of which is the Vacuum Projection Operator $\left|0\right>\left<0\right|$, which allows the elementary construction of a panoply of projection operators such as $$\frac{\hat\phi_f\left|0\right>\left<0\right|\hat\phi_f^\dagger}{\left<0\right|\hat\phi_f^\dagger\hat\phi_f\left|0\right>},\qquad \frac{\hat\phi_f\hat\phi_g\left|0\right>\left<0\right|\hat\phi_g^\dagger\hat\phi_f^\dagger}{\left<0\right|\hat\phi_g^\dagger\hat\phi_f^\dagger\hat\phi_f\hat\phi_g\left|0\right>},$$ or of higher degree; in contrast to (smeared) field operators such as $\hat\phi_f$, which have a continuous spectrum of eigenvalues? I see this as effectively the distinction between, respectively, the S-matrix and the Wightman field as observables.
I'm particularly interested in anything that considers the operational difference between these different classes of QFT observables in detail. It's obvious that the projection operators are nonlocal, insofar as they clearly don't satisfy microcausality, in contrast to the requirement of microcausality for the field operators. It also seems that the field operators cannot be used on their own to construct models for the detection of a particle, which is a yes/no event, without introducing the vacuum projection operator (but is there a way of constructing projection operators without introducing the vacuum projection operator? EDIT: yes, obviously enough, "is the observed value in the range $[a..b]$" is a yes/no observable, etc., etc., ….)
Best Answer
It is difficult to answer your question as it has no clear focus.
Projection operators play hardly any role in QFT, as the measurement problem is extremely rarely discussed in this context. Your projection operators are even more special as they are all rank 1. To construct projection operators, take any observable (e.g., a smeared Hermitian field), and integrate its spectral projection measure over an interval. Apart from that, the interpretation of projection operators is identical to the case of quantum mechanics.
The S-matrix (though the main observable object in QFT) is not an observable in the sense this term is used in QM, as it is unitary rather than Hermitian. Moreover, measuring a scattering cross section derived from the S-matrix has nothing to do with the kind of measurement discussed in the foundations of QM.
The same holds for other measurable consequences of QFT such as the Lamb shift, field expectations (which lead to hydrodynamical equations), or field correlations (which lead to kinetic equations).
Thus QFT has very little use for discussions about measuring ''observables'' in the textbook sense.