The SUSY QM Superalgebra
In fundamental quantum mechanics, we learn that an algebra of operators is defined by commutation relations among those operators. For example, the canonical operators of position and momentum have the commutator =i. (Here, we use "natural units" where Planck's constant is set equal to 1.) A more intricate case is the algebra of angular momentum operators; these quantities are closely connected to the rotational symmetries of three-dimensional space. To generalize this concept, we define an anticommutator, which relates operators the same way as an ordinary commutator, but with the opposite sign:
If operators are related by anticommutators as well as commutators, we say they are part of a Lie superalgebra. Let's say we have a quantum system described by a Hamiltonian and a set of N self-adjoint operators Qi. We shall call this system supersymmetric if the following anticommutation relation is valid for all :
If this is the case, then we call Qi the system's supercharges.
Read more about this topic: Supersymmetric Quantum Mechanics