Susy introduction paper superspace

[Barmecides]

Hello,

I have read in a susy introduction paper that if we call Q the charge which links fermions and bosons, then we have the following anticommutation relation :
{Q_a, Q_b} = 2 sigma_ab P
where P is 4-momentum.
So, is this relation only due to Coleman-Mandula theorem which force the result of the anticommutation relation to be P ? Or is there any other deeper reason ?
Because it seems this is this relation which explains the building of a superspace.

 

[Eisenhorn]

Greetings,

Actually Coleman-Mandula says that there can't be any symmetry linked in a nontrivial way with the Poincare-group. The way around is the Haag-Lopuszanski-Sohnius-Theorem (the second name may be misspelled). If you want a proof for this theorem, try Weinberg Vol 3.
The basic idea is that there are no other operators transforming the way Q_a Q_b has to, so the anticommutator can only be proportional to P.

Eisenhorn.

 

[Entropia]

Hello,

Thank you for bringing up this interesting topic. The anticommutation relation you mentioned is indeed a fundamental aspect of supersymmetry (SUSY) and is known as the SUSY algebra. It is a result of the Coleman-Mandula theorem, which states that in a theory with both bosonic and fermionic symmetries, the only possible way for them to coexist is through a direct product, as seen in the SUSY algebra.

However, there are also deeper reasons for this relation, which are rooted in the concept of superspace. Superspace is a mathematical framework that extends the traditional spacetime to include both bosonic and fermionic coordinates, allowing for the unification of bosonic and fermionic fields in a single description. The SUSY algebra arises naturally in this framework as a consequence of the Grassmann nature of the fermionic coordinates.

Furthermore, the SUSY algebra is a crucial component in the construction of supersymmetric theories, as it allows for the cancellation of divergences that would otherwise plague these theories. This is known as the "supersymmetry miracle" and is a powerful tool in addressing the hierarchy problem in particle physics.

In summary, while the anticommutation relation in the SUSY algebra is a result of the Coleman-Mandula theorem, it also has deep connections to the concept of superspace and plays a crucial role in the construction and understanding of supersymmetric theories. I hope this helps clarify the link between the SUSY algebra and superspace.