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Things that oscillate have a phase. Does it allow two electrons in string theory to be different from each other (different phases, while everything else except position is the same)? Would that translate into something experimentally measurable?

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I think this is a good question but it isn't unique to string theory. A classical harmonic oscillator also has a phase. Think about how this is represented in quantum mechanics. We usually consider energy eigenstates of the harmonic oscillator. These eigenstates do not change in time even though the classical harmonic oscillator does. If we want a quantum mechanical object that describes the classical phase, we need to consider something called coherent states, which are a sum of all different energy eigenstates.

Now the quantum harmonic oscillator has more of a connection to string theory than you might think. A relativistic version of the ordinary loop of string you have in mind in your question is the bosonic string. To talk about fermions we would need to make it supersymmetric, which is an unnecessary complication for your question. The bosonic string is closely related to a theory of free fields in two spacetime dimensions, and free fields are closely related to a harmonic oscillator.

The problem is the energy eigenstates of the bosonic string are what we are thinking of as distinct particles. The analogue of the ground state of a harmonic oscillator is the tachyon in bosonic string theory. The analogue of the first exicited state is the graviton (and some other massless particles with different spin like the dilaton). But to see something like a classical phase of a mode on a string we would need to consider a superposition of different energy eigenvalues and that would correspond to a superposition of different particle types.

So no, the phase of oscillation on a string doesn't give a new quantum number to any given particle, but yes, it is something you can talk about in quantum mechanics.

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