237 reputation
18
bio website sbseminar.wordpress.com
location New York, NY
age 34
visits member for 3 years, 1 month
seen Aug 19 at 1:16

I'm an assistant professor in mathematics at Indiana and an early adopter of Math Overflow. I was the first user at mathematics.SE.


Feb
24
awarded  Commentator
Feb
24
comment Why does the bathroom become hot after a bath?
Even if the water is heated by a water heater, it could be heated by the sun depending on the location of the pipes.
Jul
13
awarded  Scholar
Jul
13
accepted Why are telescopes on top of Mauna Kea instead of Mauna Loa?
Jul
12
awarded  Yearling
Jul
12
comment Why are telescopes on top of Mauna Kea instead of Mauna Loa?
@CrazyBuddy: I spent at least 15 minutes trying to find this on google and failing.
Jul
12
comment Why are telescopes on top of Mauna Kea instead of Mauna Loa?
Sounds reasonable, but it'd be great to have something more specific (Is the peak really threatened by lava? Would a nearby eruption ruin visibility?) and some evidence (I can think of several other plausible explanations).
Jul
11
awarded  Student
Jul
11
asked Why are telescopes on top of Mauna Kea instead of Mauna Loa?
Jul
26
awarded  Critic
May
4
awarded  Teacher
Oct
16
comment direct sum of anyons?
@Heidar You didn't take your nitpick far enough, as $X^{\otimes n}$ can also contains summands which don't appear in $X \otimes X$. I think the formula in my answer is right: you want to sum over all particle types.
Oct
16
comment direct sum of anyons?
I totally rewrote the answer based on the above discussion. Hopefully it's less wrong now.
Oct
16
comment direct sum of anyons?
@JoeFitzsimons Good point. I somehow got thrown off by 5 being a Fibonacci number as well. So the Hilbert space for an n-particle system is $\mathrm{Hom}(\phi^{\otimes n}, \phi) \oplus \mathrm{Hom}(\phi^{\otimes n}, 1)$?
Oct
16
comment direct sum of anyons?
Ways of fusing down to a single particle is $\mathrm{Hom}(\phi \otimes \phi, \phi)$ which is indeed a vector space (unlike $\phi$ itself). This is the point I was trying to make: it's the Hom spaces that are important physically, more than the objects themselves.
Oct
16
comment direct sum of anyons?
Look at the first two sentences of chapter 3 of Pachos's lecture notes and the last sentence on page 10.
Oct
16
comment direct sum of anyons?
I think that equation says that the Hilbert space for a 3-particle system is 5 dimensional (which is roughly what it should be since each particle has golden ratio internal degrees of freedom). One of these states is the state that you'd get if annihilated then created them all from the vacuum. The other four correspond to ways of fusing them down to a single particle and then unfusing them back up (there are 2 ways of doing each, so 4 total).
Oct
16
comment direct sum of anyons?
In this comment thread people seem to be referring to $\phi$ as though it were a Hilbert space. In all interesting examples it is not even a vector space. The Hilbert spaces are $\mathrm{End}(\phi)$ and $\mathrm{End}(\phi \oplus \phi)$ (which are 1 and 4 dimensional respectively).
Oct
16
answered direct sum of anyons?
Oct
5
awarded  Citizen Patrol