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  • Are the force carrying photons really something physical?

  • Do they really occur during electromagnetic interactions?

  • Does this mean that in different mediums where the speed of light differ, the interaction speed of the electric fields also differ?

  • Bonus: if two electrons interacted via force carrying photon, is there a possibility that the electron and force carrying photon cause a Compton scattering?

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The concepts defining the context of quantum mechanics, come from classical physics which is relevant to the dimensions we live in and most macroscopic observations and experiments.

The concept of force carrying photons

The relevance of the concept of force to the quantum mechanical level which is ruled by different postulates (a probabilistic theory, well supported by the experimental observations) comes from the form for the force as $dp/dt$, the change in momentum.

Are the force carrying photons really something physical?

Force is "carried" in quantum mechanical frames , not only by photons, but by all virtual particles depicted a the Feynman diagrams.

Here is an example of electron electron scattering in a first order Feynman diagram

e-e-

Only lines entering or leaving the diagram represent observable particles. Here two electrons enter, exchange a photon, and then exit. The time and space axes are usually not indicated. The vertical direction indicates the progress of time upward, but the horizontal spacing does not give the distance between the particles.

All internal lines in these diagrams are force carrying. The photon, the gluon the Z and W are the gauge bosons of the forces accepted as fundamental in quantum mechanics, and appear in the lowest order diagrams, but all internal lines carry a $dp/dt$ variable during the integration, and are force carriers.

The real particles are the ones inputted, and outputted, their four vector length equal to the mass of the electron. The wavy line is a place holder for the quantum numbers exchanged and represents, in this case, the quantum numbers of a photon, but the four vector exchanged is under the integral indicated by the Feynman diagram and gives a mass different than zero in each step process of integration. The photon is in the formula as the propagator. A different particle would also have its real mass in the propagator ( in the photon propagator the mass is zero) , but the kinematics within the integral have the representative four vector off the mass shell.

You ask

Do they really occur during electromagnetic interactions?

Is mathematics real? Or is it a tool that one uses to model observations? The name given to mathematical objects is just a useful mnemonic, useful in keeping count of quantum numbers in this case.

Does this mean that in different mediums where the speed of light differ, the interaction speed of the electric fields also differ?

Yes. The boundary conditions are different in different media.

Bonus: if two electrons interacted via force carrying photon, is there a possibility that the electron and force carrying photon cause a Compton scattering?

The internal lines could have a Compton scattering topology in a Feynman diagram, but as they are under an integral it is not possible to separate them, although their mathematical effect will affect the probability of the total interaction.

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