Editing Supersymmetric quantum mechanics (section)

== SUSY QM superalgebra ==

In fundamental quantum mechanics, we learn that an algebra of operators is defined by [[commutator|commutation]] relations among those operators.  For example, the canonical operators of position and momentum have the commutator <math>[x,p]=i</math>. (Here, we use "[[natural unit]]s" where the [[Planck 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:
: <math>\{A,B\} = AB + BA.</math>

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 <math>\mathcal{H}</math> and a set of <math>N</math> operators <math>Q_i</math>.  We shall call this system ''supersymmetric'' if the following anticommutation relation is valid for all <math>i,j = 1,\ldots,N</math>:
: <math>\{Q_i,Q^\dagger_j\} = \mathcal{H}\delta_{ij}.</math>

If this is the case, then we call ''<math>Q_i</math>'' the system's ''supercharges''.