# Are there non interacting particles?

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Thank you for expressing your pros and contras about non-interacting particles. The question about existence of non-interacting particles is not stupid, vague, low-level, or rhetorical. Note, I specified that non-interacting particles are those, which are described with the free Lagrangians.

Let us remember that in Classical Mechanics (CM) we write the equations of motion containing an external force $\mathbf{F}_{\rm{ext}}$, which changes the kinetic energy of the particle. In CM the observability of a particle is encoded in the particle equation form, which is experimentally "dictated". The very notions of particle coordinate implies observability. In CM observation is continuous and such that does not change the particle behavior in an external force. In other words, exchange of energy with the observer is relatively small.

Now, what we say when the external force becomes relatively small $|\Delta \mathbf{p}|/ \mathbf{p}|\ll 1$?  The equation is simplified and results in a free equation. The particle is still as observable as in the presence of an external force. The Lagrangian contains a free part and an interaction term with physical, measured constants. When the external force is negligible, we may write a free Lagrangian, which still contains the observable constants. $E=\sqrt{\mathbf{p}^2+m^2}$ is observable.

The same is implied in QM and QFT. Free Lagrangians describe so called non-interacting particles with physical constants because adding an external force potential yields good equations tested experimentally. "Non-interacting" means non-interacting with an external force due to relative weakness of the latter and nothing else. There are many problems in QFT books solved in this approximation and used in practice. Thus, the stories about non-interacting particles being non-observable are not convincing. In Physics we do not deal with non-observable particles, we do not write equations for their dynamics because it is senseless.

The equations with an external field are approximate, though. We want to develop our theory to get exact equations. But when we try to introduce and additional interaction, by analogy with the external one, as an addendum to the free Lagrangian, the results obtained are discouraging. Precisely at this moment people appeal to the notion of non-observable (bare) particles, whose Lagrangians/equations "happily" coincide with free Lagrangians/equations of physical particles, but involve bare constants. The latter are nothing but counter-terms to subtract the wrong part of our additional interaction (self-action) added by us to physical Lagrangians/equations. If we could fulfil this subtraction in the Lagrangian, we would deal with physical Lagrangian, without mentioning bare stuff. Alas, we could not. It means we just have not succeeded in writing exact equations, mainly, because we lack understanding how it should be done without spoiling the previous good equations. It is an open problem, not a noise. It is an open, fundamental problem in CED as well as in QED. Those who disagree must ask themselves why the Lagrangians/equations with external fields work so well and to what they correspond in the "exact" theory.

asked Feb 6, 2015 in Chat
edited Feb 6, 2015

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