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nrqed

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First, P&S discuss causality in QFT in the first chapter of the book and, after showing that [itex] <0| \phi(x)\phi(y)|0> [/itex] does not vanish for spacelike intervals, they say "to really discuss causality, however, we should ask not whether particles can propagate over spacelike intervals but whether a measurement performed at one point....."

What do they mean by this? I have my own interpretation but it's nontrivial and it may be wrong (they mention that so casually that it seems obvious to them). Of course, classically, it would make no sense to say that a a particle could propagate over spacelike intervals. In the quantum (and relativistic) world, the only way for me to make sense of what they are saying is that measuring the position of a particle precisely involves energies that necessarily lead to the creation of more particles than the ones already present in the system. Therefore, the very idea of checking whether a single particle that was at a point "x" at t=0 is now located at a point "y" at a time t is impossible in QFT.

Second, how is a measurement actually defined in QFT?

In NRQM, it's pretty clear. For a given hermitian operator, one finds it's eigenvalues and eigenstates, and so on. How is this defined in QFT?

For example, consider the field [itex] \phi(x) [/itex] itself. Now, I always thought that this is not in itself an observable, so it does not make sense to talk about *measuring* phi. And yet, P&S talk about measurements of phi in the first chapter. Is that an abuse of language?

One problem I find with QFT books is that there is no effort devoted to making the connection with NRQM. This is strange, it's the equivalent of teaching GR and never talking about how one may recover Newton's gravitation. One should be able to treat a simple sysytem (let's say the infinite square well!) and show in what way one may recover the NRQM result, within some limit! Does anyone know of a book that does that type of connection?

Another question is about the classical limit of QED. People mention that a coherent state of photons correspond to the classical limit of classical E&M. But how does that work, exactly? I know that one can then replace the creation and annihilation photon operators by their expectation values, but the field obtained is still imaginary so it's not the classical field. There has to be much more work (with many subtleties involved, no doubt) before connecting with the classical treatment of E&M.