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| ``Fermions are quaternionic''
by asar on 2006-06-10 09:30:22 |
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| Symplectic, Quaternionic, Fermionic
John Baez
September 7, 2000
ttp://math.ucr.edu/home/baez/symplectic.html
"So, the reals have to do with bosons and the quaternions with fermions?"
And I replied: "I guess that's what the math gods are trying to tell us!"
Let's say a unitary rep H of a group G is "real" if it has a conjugate-linear intertwiner j: H -> H with j2 = 1, and let's say it's "quaternionic" if it has one with j2 = -1.
By this definition, it's clear that if we tensor two quaternionic representations of a group we get a real one. Tensoring two real reps also gives a real rep. On the other hand, tensoring a real rep and a quaternionic rep gives a quaternionic rep.
Every integer-spin rep of SU(2) sits inside an even tensor power of the spin-1/2 rep, while every half-integer rep sits inside an odd tensor power of the spin-1/2 rep.
Presto! Fermions are quaternionic, bosons real. |
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