Jordan map

In theoretical physics, the Jordan map, often also called the Jordan–Schwinger map is a map from matrices M_ij to bilinear expressions of quantum oscillators which expedites computation of representations of Lie algebras occurring in physics. It was introduced by Pascual Jordan in 1935^[1] and was utilized by Julian Schwinger^[2] in 1952 to re-work out the theory of quantum angular momentum efficiently, given that map’s ease of organizing the (symmetric) representations of su(2) in Fock space.

The map utilizes several creation and annihilation operators ${\displaystyle a_{i}^{\dagger }}$ 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,

${\displaystyle [a_{i}^{\,},a_{j}^{\dagger }]\equiv a_{i}^{\,}a_{j}^{\dagger }-a_{j}^{\dagger }a_{i}^{\,}=\delta _{ij},}$

${\displaystyle [a_{i}^{\dagger },a_{j}^{\dagger }]=[a_{i}^{\,},a_{j}^{\,}]=0,}$

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

These operators change the eigenvalues of the number operator,

${\displaystyle N=\sum _{i}n_{i}=\sum _{i}a_{i}^{\dagger }a_{i}^{\,}}$,

by one, as for multidimensional quantum harmonic oscillators.

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

${\displaystyle {\mathbf {M} }\qquad \longmapsto \qquad M\equiv \sum _{i,j}a_{i}^{\dagger }{\mathbf {M} }_{ij}a_{j}~,}$

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