- #1
Peter Morgan
Gold Member
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For free quantum fields, there are two types of observables indexed by wave-number, [tex]\tilde{\hat{\phi}}(k)[/tex], the Fourier transform of the local field, which can be written as [tex]a(-k)+a^\dagger(k)[/tex], and projection operators such as [tex]a(k)^\dagger\left|0\right>\left<0\right|a(k)[/tex], [tex]a(k_1)^\dagger a(k_2)^\dagger\left|0\right>\left<0\right|a(k_1)a(k_2)[/tex], etc. [strictly speaking, none of these are operators acting on the Fock space, they're operator-valued distributions, but there are well-established ways to finesse this distinction].
In quantum optics, the latter construction is always used to model measurements, the local field is not used, as far as I know. Notably, the projection operator form of observables are not local if we smear them with test functions.
Additionally, the projection form of observable gives a zero result in the vacuum state, which corresponds to the actual response of most experimental apparatuses to the vacuum state, whereas no local observable can give a zero result in the vacuum state.
So, to rephrase the question, is there a well-established operational distinction between these two quite distinct types of self-adjoint operator? How does one go about measuring the field observable [tex]\tilde{\hat{\phi}}(k)[/tex] instead of the projection operator observable [tex]a(k)^\dagger\left|0\right>\left<0\right|a(k)[/tex]?
In quantum optics, the latter construction is always used to model measurements, the local field is not used, as far as I know. Notably, the projection operator form of observables are not local if we smear them with test functions.
Additionally, the projection form of observable gives a zero result in the vacuum state, which corresponds to the actual response of most experimental apparatuses to the vacuum state, whereas no local observable can give a zero result in the vacuum state.
So, to rephrase the question, is there a well-established operational distinction between these two quite distinct types of self-adjoint operator? How does one go about measuring the field observable [tex]\tilde{\hat{\phi}}(k)[/tex] instead of the projection operator observable [tex]a(k)^\dagger\left|0\right>\left<0\right|a(k)[/tex]?