Quantum mechanics uses the commutator

as a definitive utility. Define angular momentum as the pseudo-vector quantity

that satisfies

,

where

is a scale factor with dimensions of action and

is the Levi–Civita tensor of rank three. The commutator for the

operators also defines the Lie algebra

[2].

The

algebra describes the gauge symmetry of the 2D quantum harmonic oscillator (QHO) and admits

as a subalgebra, so it is possible to write the angular momentum operators in terms of the Pauli matrices

and bilinear combinations of the

creation/annihilation operators

. Specifically,

,

,

The simple

commutation relations

,

lead to a simple set of basis vectors, say

, with

, which are also the eigenfunctions of the operator

,

the Hamiltonian of the 2D QHO in frequency dimensions. After we know how the

variables act on the basis vectors

, it is possible to determine the matrix elements of any function

in the basis provided by

so long as the function

can be expanded in a power series of the noncommuting variables. Fortunately, the

operators have only quadratic terms, which makes explicit calculation of matrix elements easy.

The angular momentum operators

have another, more natural set of quantum numbers:

,

. Each integer or half-integer

indexes an irreducible representation of the angular momentum algebra. Then we construct the irreducible representations

by taking the block diagonal subspace spanned by the basis vectors

that also have quantum number

. In the usual representation where

J_{3} is diagonal, the

eigenvalues of representation

range from

to

.