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In quantum field theory, the concept of a force is not explicitly present, and we speak of interactions. I guess we could say that a force is an emergent phenomenon.

Interactions manifest themselves in (at least) three important ways: particle decay, scattering and forces.

In (introductory) QFT, typically the first two, particle decay and scattering processes, are treated extensively and computations are made, but forces, e.g. the repulsion between two fermions, or bound systems, are usually not studied. My questions:

  • What is the reason for that? One possible reason I could think of is that for forces we need the concept of a localized particle, whereas in QFT it seems that we mostly work in the momentum representation. Maybe it is just much more complex (for this reason or others) to study forces than decay and scattering.

  • Is it possible to see, not necessarily very formally, how an attractive or repulsive force could emerge from an interaction Lagrangian?

doetoe
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1 Answers1

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In quantum mechanics forces are emergent. They can arise when you try to keep track on the correlations through time of your quantum system. You may try to measure over and over a position of a quantum particle and see if the correlations obey some dynamical law (Newton's law).

In quantum field theories, because locality, forces are always produced by interactions, i.e. correlations made by virtual particle exchange, or field fluctuations if you prefer. So, a repulsion between particles are actually a probabilistic phenomena. The probability of two electrons be close together decreases as they get close. This is so because they have a probability to exchange virtual photons that affects the probability of each particle, producing this kind of correlation. Everything in quantum mechanics are probabilistic phenomena.

You can see all this applied to the Coulomb force here

Also, you can define force in quantum mechanics through:

$$ F\leftarrow i\left[H,\,P\right]/\hbar $$ that is, how momentum operator behaves under infinitesimal evolution on time. You can see more here. But all this is yet a probabilistic phenomena.

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Nogueira
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