1+1D Bosonization on a line segment or a compact ring

Question

I have been informed that 1+1D Bosonization/Fermionization on a line segment or 1+1D Bosonization/Fermionization a compact ring are different -

Although I know that Bosonization can rewrite fermions in the non-local expression of bosons. But:

bosons and fermions are fundamentally different for the case of on a 1D compact ring.

Is this true? How is the Bosonization/Fermionization different on a line segment or a compact ring? Does it matter whether the line segment is finite $x\in[a,b]$ or infinite $x\in(-\infty,\infty)$? Why? Can someone explain it physically? Thanks!

This post imported from StackExchange Physics at 2014-06-04 11:37 (UCT), posted by SE-user Idear

Comments

I think, when people say line segment, they often mean an interval $[a,b]$. Regarding your question, my low-brow understanding is that the difference has to do with braiding. You cannot braid two (hard-core) bosons or fermions on a line segment, but you can on a ring. On a slightly more technical level, an example would be the Jordan-Wigner transformation for hard-core bosons. We need to be careful about boundary conditions when we do JW transformation on a ring.

This post imported from StackExchange Physics at 2014-06-04 11:37 (UCT), posted by SE-user Isidore Seville

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Very good, Isidore, it totally makes sense.

This post imported from StackExchange Physics at 2014-06-04 11:37 (UCT), posted by SE-user Idear