“Supersymmetry as a quantum group”的意思、由来-开放百科全书

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Supersymmetry as a quantum group

释义

Unitary (–1)^F operator
Fermionic operators
See also

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

with the counit

and the coproduct

and the antipode

Thus far, there is nothing supersymmetric about this Hopf algebra at all; it is isomorphic to the Hopf algebra of the two element group . Supersymmetry comes in when introducing the nontrivial quasitriangular structure

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 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,

This is compatible with .

Supersymmetry is the symmetry over systems where interchanging two fermions attain a minus sign.

Unitary (–1)F operator

Fermionic operators

See also

Unitary (–1)^F operator