Is $hat{phi}^{-}hat{phi}^{+}$ a well defined observable in the Quantum field theory of a scalar field?

Is the Hermitian operator $\hat{\mathcal{O}}=\hat{\phi}^{-}(x)\hat{\phi}^{+}(x)$, where $\hat{\phi}^{+}(x)$ is positive frequency part of the scalar field operator, a well defined observable in QFT?

Some background:

1. 
   

   In Phys. Rev. 130, 2529 (1963), Glauber argues that square-law detectors measure the average value of the product $\hat{E}^{-}(x)\hat{E}^{+}(x)$, i.e., $\langle\Psi|\hat{E}^{-}(x)\hat{E}^{+}(x)|\Psi\rangle$ where $\hat{E}$ is the electric field. This is the basis of quantum optics which is an extremely successful empirically tested theory. I'm viewing $\hat{\mathcal{O}}$ as a generalisation of this theory to the scalar field.

   
   


2. 
   

   Lubos Motl answers here: In what sense is a scalar field observable in QFT?:

   
   
   
   
   • "Every observable in the technical or mathematical sense (linear Hermitian operator on the Hilbert space) is, in principle, observable in the physical operational sense, too."
   
   
   
   

   Since $\hat{\mathcal{O}}$ a well-defined (modulo smearing by test functions etc) linear Hermitian operator, it would be according to this definition physically observable in principle.

   
   



3. 
   

   However, the commutator $[\hat{\mathcal{O}}(x),\hat{\mathcal{O}}(y)]$ is non-zero for x-y space like separated distances. One usually demands such a condition, for example on the observables in Wightman axioms, see for example the (micro)locality Axiom 4, http://www.maths.ed.ac.uk/~jthomas7/GeomQuant/Wightman-Axioms.pdf.

   
   

   I've tried to find out why such a condition is necessary. Ron Maimon gives the following nice physical reason, Is microcausality *necessary* for no-signaling?:

   
   
   
   
   • "when you have arbitrarily tiny external agents capable of measuring any bosonic field in an arbitrarily tiny region, then microcausality is obviously necessary for no signaling, since if you have two noncommuting operators A and B associated with two tiny spacelike separated regions, and two external agents wants to transmit information from A's region to B's, the agent can either measure A repeatedly or not, while another agent measures B a few times to see if A is being measured. The B measurements will have a probability of giving different answers, which will inform the B agent about the A measurement."
   
   
   
   

   But this assumes that the measurement of A is a projective measurement, and leaves the state of the field in an eigenstate of A (i.e., so that the B measurement will pull the state out of the A eigenstate and hence signal to the A agent superluminally). In the Glauber theory, one never knows the state of the field. The detector interacts with the field by absorbing particles. This doesn't leave the state of the field in an eigenstate of $\hat{\mathcal{O}}$. Nevertheless, the observable $\langle \hat{\mathcal{O}}\rangle$ is observed from the particle detection intensity.

   
   

One might try to rule out the Glauber theory by absurdity: design a gedanken experiment using the Glauber detector on two space like separated positions, and show that they can be used to signal faster than the speed of light. I've tried but not been able to establish such an experiment. I would accept any answer that achieves to do this.

Another possibility is that the micro-locality conditions in axiomatic quantum field theory too strong? Can generalised (non-projective) measurements be used to obtain expectation values of micro-local violating operators?

Any other comments on why my question is incorrectly posed or based on any misunderstandings would be appreciated.

Answer

Microcausality holds for observables that obey microcausality; it doesn't hold for observables that don't obey microcausality. The previous sentence is a tautology but I had to write it in this way because it seems to me that the question implicitly tries to disagree with this tautology.

There is no general condition that "all observables" have to commute at spacelike separation. Instead, what (local or relativistic) quantum field theory demands is that "there exist" elementary observables that obey this condition, like the "elementary fields" or the "stress-energy tensor" field, and so on. This is needed for special relativity or Lorentz symmetry to hold.

The field operators $\hat \phi^\pm$ clearly don't obey this condition (and, consequently, their general functions don't obey this condition) because they may be expressed (it's their definition) as linear superpositions of the microcausality-obeying fields $\phi(x)$ at "almost all" spacetime points (although the points where both arguments are nearby dominate in some sense). But that doesn't mean that they are not observables. It only means that the apparatuses that may measure these observables should be thought of as "intrinsically extended", not pointlike, gadgets.