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

+ 3 like - 0 dislike
149 views

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
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.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverf$\varnothing$owThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.