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From what I read in Penrose's “Road to Reality”, I think that an electron, being a spinorial object, will turn into a positron when it is rotated around $2\pi$. Is that really true?

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What? Positron? Does he really say that? Electrons and positrons have opposite electrical charge. It can't be true since $[Q,R]=0$, charge commutes with spatial rotations. – Michael Brown Feb 21 at 12:56
If the original poster is quoting right, it seems like Penrose is implying a rotation in spinorial space, not in physical space. – zakk Feb 21 at 13:34

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No, you probably misread Penrose.

After a $2\pi$ rotation it is the sign of the wavefunction of the electron that changes. (This is hard to detect as measure as observables are typically quadratic in $\Psi$.) The total charge remains the same, and the electron does not turn into a positron.

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Just to expand upon the other answers, there is the Coleman-Mandula theorem which states, roughly, that a relativistic quantum field theory can not have any conserved quantities, apart from those of the Lorentz group (energy, momentum, angular momentum and the boost one that nobody uses), that change under Lorentz transformations. So for instance the electric charge, which is conserved but not on the approved list above, must be a scalar. Therefore any Lorentz transformation will leave the electric charge of the system invariant.

There is a loophole in the Coleman-Mandula theorem: there can be anticommuting spinorial charges which are conserved. This gives a generalisation of ordinary symmetry: supersymmetry. The Coleman-Mandula theorem extends to another theorem in the supersymmetric case. However, it is still the case that only a few conserved charges are allowed to transform under the Lorentz group: just the Lorentz ones and the new spinorial charges. Charges associated to "internal symmetries" - i.e., symmetries which don't have to do with spacetime transformations - are still scalars.

The short answer is a resounding no - rotations cannot change the charge of a particle.

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Just for fun: a neutral wire with electric current $I$ "looks" as a positively charged from a reference frame moving with $V$ and as a negatively charged from a reference frame moving with $-V$ along the wire. – Vladimir Kalitvianski Feb 21 at 13:52
@Michael Brown See it from symmetry point of view. The fact is that, because the electron has spin 1/2, you need a rotation by 4$\pi$ to bring it back to its adentical configuration. A rotation by $2\pi$ will leave the electron's wave function with a global phase factor which has no measurable effects on any of the electron properties. So the electron remains an electron. – JKL Feb 21 at 14:48
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Yes, it's true.

Normally, when we speak of "rotations" in QM, we do not mean physical rotation, we mean a calculation rule for recalculating the results of observation from one reference system into another one. In a 3D physical space there are only $2\pi$ "angular possibilities" for differently oriented reference frames.

EDIT: Sorry, I misread "positron" as "position". No, electron does not turn into positron due to change of the reference frame. The wave function sign changes, but it is not the electron charge.

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