Jordan map

The map utilizes several creation and annihilation operators ${\displaystyle a_i^{†}}$ and ${\displaystyle a_i^{}}$ of routine use in quantum field theories and many-body problems, each pair representing a quantum harmonic oscillator. The commutation relations of creation and annihilation operators in a multiple-boson system are,

${\displaystyle [a_i^{},a_j^{†}]\equiv a_i^{}{a}_j^{†}-a_j^{†}{a}_i^{}={\delta }_{ij},}$

${\displaystyle [a_i^{†},a_j^{†}]=[a_i^{},a_j^{}]=0,}$

where ${\displaystyle [,]}$ is the commutator and ${\displaystyle {\delta }_{ij}}$ is the Kronecker delta.

These operators change the eigenvalues of the number operator,

${\displaystyle N=\sum _i{n}_i=\sum _i{a}_i^{†}{a}_i^{}}$,

by one, as for multidimensional quantum harmonic oscillators.

The Jordan map from a set of matrices M_ij to Fock space bilinear operators M,

${\displaystyle \mathbf{M}⟼M\equiv \sum _{i,j}{a}_i^{†}{\mathbf{M}}_{ij}{a}_j,}$

is clearly a Lie algebra isomorphism, i.e. the operators M satisfy the same commutation relations as the matrices M.

The example of angular momentum

For example, the image of the Pauli matrices of SU(2) in this map,

${\displaystyle \overrightarrow{J}\equiv {\mathbf{a}}^{†}\cdot \frac{\overrightarrow{\sigma }}{2}\cdot \mathbf{a},}$

for two-vector a^†s, and as satisfy the same commutation relations of SU(2) as well, and moreover, by reliance on the completeness relation for Pauli matrices,

${\displaystyle J^2\equiv \overrightarrow{J}\cdot \overrightarrow{J}=\frac{N}{2}\left(\frac{N}{2}+1\right).}$

This is the starting point of Schwinger’s treatment of the theory of quantum angular momentum, predicated on the action of these operators on Fock states built of arbitrary higher powers of such operators. For instance, acting on an (unnormalized) Fock eigenstate,

${\displaystyle J^2{a}_1^{†k}{a}_2^{†n}|0⟩=\frac{k+n}{2}\left(\frac{k+n}{2}+1\right)a_1^{†k}{a}_2^{†n}|0⟩,}$

while

${\displaystyle J_z{a}_1^{†k}{a}_2^{†n}|0⟩=\frac{1}{2}\left(k-n\right)a_1^{†k}{a}_2^{†n}|0⟩,}$

so that, for j = (k+n)/2, m = (k−n)/2, this is proportional to the eigenstate |j,m⟩,

${\displaystyle |j,m⟩=\frac{a_1^{†k}{a}_2^{†n}}{\sqrt{k!n!}}|0⟩=\frac{a_1^{†(j+m)}{a}_2^{†(j-m)}}{\sqrt{(j+m)!(j-m)!}}|0⟩.}$

Observe ${\displaystyle J_{+}=a_1^{†}{a}_2}$ and ${\displaystyle J_{-}=a_2^{†}{a}_1}$, as well as ${\displaystyle J_z=(a_1^{†}{a}_1-a_2^{†}{a}_2)/2}$.