Supersymmetry as a quantum group
The concept in theoretical physics of supersymmetry can be reinterpreted in the language of noncommutative geometry and quantum groups. In particular, it involves a mild form of noncommutativity, namely supercommutativity.
Unitary (–1)F operator
Following is the essence of supersymmetry, which is encapsulated within the following minimal quantum group. We have the two dimensional Hopf algebra generated by (-1)F subject to
- [math]\displaystyle{ {(-1)^F}^2=1 }[/math]
with the counit
- [math]\displaystyle{ \epsilon((-1)^F)=1 }[/math]
and the coproduct
- [math]\displaystyle{ \Delta (-1)^F=(-1)^F \otimes (-1)^F }[/math]
and the antipode
- [math]\displaystyle{ S(-1)^F=(-1)^F }[/math]
Thus far, there is nothing supersymmetric about this Hopf algebra at all; it is isomorphic to the Hopf algebra of the two element group [math]\displaystyle{ \mathbb{Z}_2 }[/math]. Supersymmetry comes in when introducing the nontrivial quasitriangular structure
- [math]\displaystyle{ \mathcal{R}=\frac{1}{2}\left[ 1 \otimes 1 + (-1)^F \otimes 1 + 1 \otimes (-1)^F - (-1)^F \otimes (-1)^F\right] }[/math]
where +1 eigenstates of (-1)F are called bosons and -1 eigenstates are called fermions.
This describes a fermionic braiding; don't pick up a phase factor when interchanging two bosons or a boson and a fermion, but multiply by -1 when interchanging two fermions. This provides the essence of the boson/fermion distinction.
Fermionic operators
The previous analysis only introduced the concept of fermions, and is not actual supersymmetry. The Hopf algebra is [math]\displaystyle{ \mathbb{Z}_2 }[/math] graded and contains even and odd elements. Even elements commute with (-1)F; odd ones anticommute. The subalgebra not containing (-1)^F is supercommutative.
Let's say we are dealing with a super Lie algebra with even generators x and odd generators y.
Then,
- [math]\displaystyle{ \Delta x = x \otimes 1 + 1 \otimes x }[/math]
- [math]\displaystyle{ \Delta y = y \otimes 1 + (-1)^F \otimes y }[/math]
This is compatible with [math]\displaystyle{ \mathcal{R} }[/math].
Supersymmetry is the symmetry over systems where interchanging two fermions attain a minus sign.
See also
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