Q:

"distinguishability" of 1D identical particles

Recently when I deal with 1D electron system, it occurred to my mind that since these electrons are not able to bypass each other during the scattering processes, we can actually label them as the 1st, 2nd,..., Nth electron. As a result, it seems that these electrons now become distinguishable. 
so my questions is: does this kind of distinguishability have any deep physical consequences? For instance, for 3D identical particles the wave function has to be either symmetric or anti-symmetric, whereas in 2D case we have the interesting anyons that obey a different statistics. Then what about the 1D case? Further more, what kind of distribution functions should we use (i.e. Fermi-Dirac or Bose-Einstein)? I do remember that in undergraduate condensed matter modules people deal with 1D electron gas using Fermi-Dirac distribution, but now it seems not so natural to me.

A:

Your intuition is exactly correct. In 1D, fermions and "hard-core" bosons (i.e. bosons with strong on-site repulsion that prohibits putting two bosons on the same site) are exactly dual to each other and produce the same energy spectrum for any given Hamiltonian. This (nonlocal) duality is easy to construct: a system of fermions is dual to a spin-1/2 chain by the (nonlocal) Jordan-Wigner transformation, and a system of hard-core bosons is dual to a spin-1/2 chain by the (local) Holstein-Primakoff transformation. By composing the two dualities together and "passing through" the intermediate spin-1/2 chain, you get get a duality between fermions and hard-core bosons.

A:

One can also label the electrons in an atom by the energy in its Hartree-Fock approximation, and thus makes them distinguishable. This has physical consequences, for example one can speak unambiguously about the outer electron of a Lithium atom.
For a 1D quantum system one may have nonstandard statistics related to the braid group. Instead of Bose or Fermi statistics one has exchange relations satisfying Yang-Baxter equations. There is a nearly endless literature about this and the related quantum groups. 
In 1D there is also no spin-statistic theorem, and one can describe bosons by fermionic fields and conversely.

