Why magnetic forces MUST exist besides electric forces? The discovery of electric forces resultet in the discovery of the force mediated by the electric field $\vec{E}$ (this is $\vec{F} = e \vec{E}$) at first. But later, it was observed that charges also exert magnetic forces. My question is:
WHY electrically charged particles exert both electric forces and magnetic forces and not ONLY the electric force?
Lies the answer in gauge theory in quantum mechanics or are there other theories for the existence of both forces?
 A: 
WHY electrically charged particles exert both electric forces and magnetic forces and not ONLY the electric force? Lies the answer in gauge theory in quantum mechanics or are there other theories for the existence of both forces?

To search the answer of how something is in a theory seems to be misleading. A theory can explain something but could not be the reason for something.
The question itself is an important one. The question underlines that today it is obvious that electrons as well as protons and their antiparticles have both an electric charge and a magnetic dipole moment.
At the time of Maxwell the magnetic force seems to come from moving charges. Today the magnetic field can be explained by the alignment of the magnetic dipole moments of the moving charges. For permanent magnets it is clear that the alignment of the dipole moments is the reason for the magnetic field. Why not take this explanation for moving charges too?
To get a deeper understanding of the phenomenon of the magnetic field of moving charges one has to keep in mind that the magnetic dipole moment and the intrinsic spin are related. When ever the particle is accelerated the spins of the involved particles will be aligned (gyroscopic effect) and by this the magnetic dipole moments will be aligned too and this forms the macroscopic magnetic field.
Remarks after Ruslans and bobs comments
@Ruslan


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*A straight wire is not straight at its ends. Where are always bended regions between the source and the straight wire. In every curved connection it will be induced a magnetic field. Since we know how self inductance works it is naturally that in the straight wire there will be a magnetic field too.

*As I mentioned in my answer how somebody will measure a magnetic field without inducing in this field a magnetic field? Once again since the electrons are flowing they will be influenced by the measurement instrument and will be alaigned after this by selfinductance.

*Take an antenna rod and measure the magnetic field from accelerated electrons. Than connect the ends of the rod to a DC source and let through the same average current. Will the strength of the magnetic field will be the same as in the first case?

*Spin-orbit coupling does not mean that the magnetic dipole moments of the involved electrons vanishes. The main point about induced magnetic fields is that electrons have the intrinsic properties of magnetic dipole moment and intrinsic spin and they are parallel (or anti-parallel for other particles) oriented. To say that a not moving in a frame electron has no magnetic field is a simplification or a relict from the time this two intrinsic properties where unknown.


@rob
Cooper pairs are a possible explanation for superconductivity. But the concept of ever flowing electrons without any energy loss is a strange concept. Fact is that it works to infinity and it is possible to use this magnetic field to lent temporarily energy from it. But not to much, overwise the field crashes.


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*Is it allowed to understand the induced magnetic field as a "frozend" and by this comparable with permanent magnets?

*The point is that Cooper found that materials with special configurations of the spins are usable for superconductivity. Instead of spin it is equal to talk about their magnetic dipole moments. It is not the spin coupling of ghostlike (not involved from the atoms of the wires) moving electron pairs that allow superconductivity. is simple the spezial configuration of the electron shell (orbitals) that get aligned with there magnetic dipole moments and than frozen by low temperatures. BTW superconductivity magnets are not the strongest, they are only smaller. More powerful magnetic fields one will induce with traditional coils.


See this beautiful answer from Steve B:

In reality, electricity and magnetism are equally fundamental parts of physics. Special relativity unites electricity and magnetism into electromagnetism, in exactly the same way that it unites space and time into spacetime. Time does not cause space, space does not cause time, and SR causes neither space nor time. SR merely reveals the relatedness of space and time. Similarly, electricity does not cause magnetism, magnetism does not cause electricity, and SR causes neither electricity nor magnetism. SR merely reveals the relatedness of electricity and magnetism.

