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If two particles approach each other, they can [provided that their properties add to those of other particle(s)] interact and go from, say, $$e + \bar e \to \gamma + \gamma$$

My question is how would one estimate the range of this? What distance is needed between $e$ and $\bar e$, is it less than or equal to some expectation value of an operator? Or do they particles just have to be delocalised enough to overlap, and then there's a finite probability of them interacting as a function of that overlap? If two particles like these interact, is there any restriction on where the two photons propagate from [i.e. the same point, or just anywhere in the overlap etc]?

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    $\begingroup$ You are using a strangely nonstandard mechanistic model for the process. If you appreciate the flux x cross-section language, e.g. eqn(40) here you can answer the basic questions on "range" as a Compton wavelength of the electron. $\endgroup$ – Cosmas Zachos Sep 19 '17 at 13:19
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A hand waving estimate can come from using the so called wave-particle duality ,

deBroglie

The deBroglie wavelength gives "dimensions" to scattering particles, useful for experimental estimates, without having to go through the scattering formalism.

in the equivalent form it can include photons:

debroglie

The wavelengths give an estimate in classical terms of the size needed in focusing of the two beams so that the probability of scattering would be high. Scattering theory is used on the data to get the interaction crossection.

The real construction of the interaction region is a complicated process.

If two particles like these interact, is there any restriction on where the two photons propagate from [i.e. the same point, or just anywhere in the overlap etc]?

There is a definite mathematical system for calculating interactions of elementary particles , it is called quantum field theory,( quantum electrodynamics) for your interaction) and the calculations are done using Feynman diagrams to calculate the crossection in a perturbative expansion. The first order diagrams of e+e- to γγ (page 16)e_e-gamma

e+e-gammagamma

In the standard model of particle physics the interactions of the point particles happen at points called vertices , the impulse carrier is called a virtual particle, in this case a virtual electron. In the diagram time is in the y direction, and space the x. The photons appear at separate vertices. As it is functions under an integral in the x direction, it is not possible to define a unique distance or duration.

The Heisenberg uncertainty principle for the interaction will give a ΔpΔx region whence the photons may come.

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  • $\begingroup$ In your 2nd to last paragraph you state it's not possible to define a Unique distance or duration, do you just end up with a family of functions? What are the domain and range? Also, if it's impossible to define a unique distance or duration, how do you evaluate how two systems with identical particle numbers but different configurations in position change? Sorry if I've asked another poorly formed question, thanks a lot for the good answer though! $\endgroup$ – user95137 Sep 20 '17 at 11:08
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    $\begingroup$ You end up with definite integrals, with variables that have limits the physical situation. The matrix element above will be integrated, so it is afunction containing a propagator ( en.wikipedia.org/wiki/… ), which will have the mass of the electron in the denominator and will be integrated over the available phase space. If you have two systems they are macroscopic, you need different solutions, for crystals for example, there are wavefunctions which have the locations of the ions and electrons ( within the Heisenberf uncertainty) $\endgroup$ – anna v Sep 20 '17 at 11:44
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My understanding is as follows: "in" and "out" states are fully delocalized states with exact momentum (plane waves) whose history starts at time $-\infty$ and ends in $+\infty$. Everything in between is not a state but so-called "S-matrix evolution". So: how is you question truly answered in nature? My answer: I do not know. How is it answered in our perturbation-based quantum field description: there is no "typical range". Particle do not feel each other only in times $\pm \infty$ when they become states from the free theory (on which perturbations are applied). In times in between they feel each other always. Actually the mere existence of "asymptotic states" for long-range interactions (such as electrodynamics) is questioned.

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