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In non-relativistic QM, the position of a particle is an observable. In QFT, fields are the observables. However, particles must have some sort of position, otherwise we wouldn't see pictures like the ones below. What is the (linear, Hermitian) operator that describes what we are observing?

I've tried considering making a "local" version of $N = \int dk\, a^+ a^- = \int dx \, \phi^+(x) \phi^-(x)$

by replacing this with $N_R =\int dx \, f(x) \phi^+ \phi^-$ where $f$ is some function that is concentrated in the localized region $R$. However, this is not Hermitian!

enter image description here

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    $\begingroup$ You might like to look at M. Hawton, "Photon position measure" and other works by Margaret Hawton at mhawton.lakeheadu.ca. In these works the distinction is made between an observable and a more general (weaker) concept of positive operator valued measure. These later operators need not Hermitian and are particularly applicable to the detection of photons when you need a theoretical abstraction of the photon detection process (which is more often than not destructive). My understanding is that when you talk ... $\endgroup$ – WetSavannaAnimal Oct 24 '13 at 23:50
  • $\begingroup$ ...about detection as opposed to "collapsing to a position eigenstate" the particle concerned effectively "vanishes" from your theoretical considerations (i.e. is "absorbed" by the detector) and so its state evolution is non unitary in the simplified description, although if your quantum description is big enough to model all the details of the detection process, its evolution is unitary. I have heard the description that a POVM is to a projection valued measure what the density matrix is to a pure state. $\endgroup$ – WetSavannaAnimal Oct 24 '13 at 23:50
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    $\begingroup$ Maybe the Photodetection article of Wikipedia will interest you. $\endgroup$ – Trimok Oct 29 '13 at 5:39
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The observation of particle tracks is in terms of a sequence of ionizations of atoms, which are subsequently magnified by an appropriate mechanism.

How particle tracks arise in quantum mechanics is described in the following famous old paper:

N.F. Mott, The Wave Mechanics of alpha-Ray Tracks, Proc. Royal Soc. London A 126 (1929), 79-84. http://rspa.royalsocietypublishing.org/content/126/800/79.full.pdf

He shows that once some atom isionized in a track chamber, the following ionizations will be overwhelmingly in the same direction, thus explaining the track. The detectors, in principle atoms, are reduced to essentially just single valence electron, and only two states are of interest: ionized or not. But since we do not know in advance where a particle will be recorded first, one needs a large array of such electrons.

The Hamiltonian can thus be considered to be that of an array of localized 2-state systems, each initially in the same state, with an interaction that allows a transition of this state to an ionized state.

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