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After reading many questions, like this and this, I wonder:

Is it possible to consider the other fundamental forces too, the electroweak interaction and the strong interaction or ultimately the unification of these, to be fictitious forces like gravity in the framework of general relativity?

If we want a final unification of all fundamental forces, hasn't this feature of gravity to become a feature of the other forces as well?

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    $\begingroup$ In principle, and given a broad enough definition of "fictitious force", yes. But so far, all actual models based on this idea have suffered from serious problems. $\endgroup$ – Harry Johnston Nov 22 '14 at 3:09
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The classical theory of electrodynamics can indeed be written as a geometrical theory in a similar way to general relativity. As it happens there is a question and answer addressing just this, but it's in the Maths SE: Electrodynamics in general spacetime.

Classical electrodynamics is an example of a class of theories called classical Yang-Mills gauge theories, though Maxwell didn't realise this as the Yang-Mills theories were first described in 1954. These are geometric theories like general relativity, though note that GR is not a Yang-Mills theory - if it was we'd probably have quantised it by now. There are various introductions to Yang-Mills theory around, and a quick Google found this introduction (350KB PDF) that seems pretty good. The theories use a curvature tensor, though this is different to the Riemann tensor used in GR.

Quantising the Yang-Mills classical theory gives quantum electrodynamics i.e. the quantum field theory describing electrodynamics. The weak and strong forces are also quantum Yang-Mills theories, though in these two cases there is no useful classical theory.

Christoph points out in a comment that there is an alternative route to a geometric theory of electrodynamics. In 1919 Theodor Kaluza pointed out that if general relativity was formulated in 5 dimensions (4 space and 1 time) the theory incorporated electrodynamics. This approach was built upon by Oskar Klein and is now known as Kaluza-Klein theory. However the theory requires there to be extra dimensions of space, and in any case the electrodynamic bit of the theory is really a Yang-Mills theory in disguise.

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    $\begingroup$ wouldn't a link to something about Kaluza-Klein theory be more appropriate instead of classical YM? $\endgroup$ – Christoph Nov 21 '14 at 15:49
  • $\begingroup$ @Christoph: yes, but only if you're prepared to accept extra dimensions. $\endgroup$ – John Rennie Nov 21 '14 at 15:55
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    $\begingroup$ Why can't one quantize this description? $\endgroup$ – Ruslan Nov 21 '14 at 17:09
  • $\begingroup$ @JohnRennie Yes, but since that's how General Relativity does it (I think?) it would seem appropriate to this question... $\endgroup$ – RBarryYoung Nov 21 '14 at 17:36
  • $\begingroup$ Though I don't disagree with the explicit statements made here, it is important to note that even though Yang-Mills happens to have geometry in it, it's a very different kind of geometry than GR. In particular, you can only possibly think of particles as following straight lines in a curved spacetime if the particles obey the equivalence principle. But the equivalence principle is not true for any forces we know of besides gravity. For example, a proton will accelerate in an electric field differently than an electron. $\endgroup$ – knzhou May 23 '19 at 16:33
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On the quantum level, force is not acceleration. The concept of "fictitious force" makes no sense on a QFT level, because forces are interactions between quantum states, not the classical forces you might imagine. Quantum forces are not vector fields in space.

The notion of "fictitious force" would mean that, e.g., the strong force is something influencing the motion of a particle that goes away when transforming into a particular reference frame. But the strong force doesn't influence "motion", because there is no "motion" of particles, only the interaction of states. Do not think about the strong or the weak force as forces to which you could ever apply classical thinking in this way.

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    $\begingroup$ but also keep in mind that there are classical forces corresponding to quantum YM fields (cf Wong's equations) - they just aren't terribly useful $\endgroup$ – Christoph Nov 21 '14 at 15:51
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is it possible to consider also the other fundamental forces [...] to be fictitious forces like gravity in the framework of general relativity?

No, because the equivalence principle only holds for gravity.

If we want a final unification of all fundamental forces, hasn't this feature of gravity to become a feature of the other forces as well?

The other approach is making gravity less special to begin with, which is the approach Luboš takes in his answers to the linked questions. There are also classical approaches to general relativity like teleparallel gravity (where gravity is a proper force and the equivalence principle needs not necessarily hold) or bimetric theories (which more or less mirrors how gravity in thought of in the context of string theory).

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