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Is it somehow possible to theoretically unify the electric charge in electromagnetic interaction with the colour charge in the strong interaction and how these charges give rise to the respective forces? Is there a description of this in the GUT?

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Yes, subtly. See this.

A GUT, such as SU(5), proposes unifying color (strong) interactions with electroweak SU(2)xU(1) interactions, which at lower energies SSBreak to electromagnetic interactions, so $$ SU(5)\to SU(3)\times SU(2)\times U(1)\to SU(3)\times U(1), $$ where the second breaking below a quarter of a TeV is the standard model EW breaking, with a subtle Weinberg angle rearrangement. The first breaking was meant to be at more than a dozen orders of magnitude higher energies. The resulting EM charge of the U(1) and the color charges of SU(3) thus started out as siblings in the same liter. Any review of GUTs would detail that.

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