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

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!

• 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. – Isidore Seville Mar 9 '14 at 1:54
• Very good, Isidore, it totally makes sense. – wonderich Mar 9 '14 at 2:09