Supercharge

For other uses, see Supercharge (disambiguation).

In theoretical physics, a supercharge is a generator of supersymmetry transformations. It is an example of the general notion of a charge in physics.

Supercharge, denoted by the symbol Q, is an operator which transforms bosons into fermions, and vice versa. Since the supercharge operator changes a particle with spin one-half to a particle with spin one or zero, the supercharge itself is a spinor that carries one half unit of spin.^[1][2]

Depending on the context, supercharges may also be called Grassmann variables or Grassmann directions; they are generators of the exterior algebra of anti-commuting numbers, the Grassmann numbers. All these various usages are essentially synonymous; they refer to the ${\displaystyle \mathbb {Z} _{2}}$ grading between bosons and fermions, or equivalently, the grading between c-numbers and a-numbers. Calling it a charge emphasizes the notion of a symmetry at work.

Commutation

Supercharge is described by the super-Poincaré algebra.

Supercharge commutes with the Hamiltonian operator:

[ Q , H ] = 0

So does its adjoint.

See also

• R-symmetry

References

1. "Supersymmetry to the rescue?". The Official String Theory Web Site. Archived from the original on 2018-11-02.
2. von Hippel, Matthew. "Supersymmetry, to the Rescue!". 4 Gravitons The trials and tribulations of four gravitons and a postdoc. Retrieved 18 April 2021.