# Timescale of virtual photon interaction in particle scattering

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• Lux137
Overall, the key concept to understand is that the uncertainty principle limits the precision with which we can measure certain physical quantities, and this has important implications for the time-scales of interactions between particles.

#### Lux137

TL;DR Summary
Why is the time-scale of the virtual photon exchange τ~1/Q in two particle scattering?
I'm studying the electron-proton deep inelastic scattering. In the notes that I'm studying from the author states that the time-scale for a virtual photon to interact with a proton is ##τ\sim\frac{1}{Q}##, where ##Q## is the momentum transfer with ##Q\gg M##, which is the mass of the proton. I assume by ##Q## he means the Minkowski norm of the 4-momentum transfer. And then he compares it with the time-scale of the quarks interaction in a proton ##τ\sim\frac{1}{M}##. All this in a reference frame where the proton is at rest.
Later on delving more into time-scales he writed that the typical time of interaction of an electron with a virtual photon is also ##\frac{1}{Q}## which is to be compared with the time the electron emits and reabsorb its own virtual photons ##\frac{1}{q}##

I'm having a very hard time justifying all these time-scale estimations by myself. I tried to apply the time evolution on a state with an electron and a photon, as eigenstates of the free hamiltonian ##H_0## (not the full ##H=H_0+H_i##) in the interaction picture, in order to visualize what happens with virtual particles, and what i got is that the amplitudes for scattered states reach its max at ##t\sim\frac{1}{ΔE}## where ##ΔE## is the difference between the virtual photon energy and the energy it would have if it were on-shell. Which is still different from the statements above.

I can't seem to get past this and i feel like it's something silly that i should have already grasped, since in all the textbooks i have checked the topic of interaction time-scale is always barely mentioned or completely skipped. Does anyone know a derivation of these results or maybe some sources where i can find this topic discussed in detail?