Jordan map

For example, the image of the Pauli matrices of SU(2) in this map, :{\vec J} \equiv {\mathbf a}^\dagger \cdot\frac{ \vec \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, :J^2\equiv {\vec J} \cdot {\vec 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, :J^2~ a^{\dagger k}_1 a^{\dagger n}_2 |0\rangle = \frac{k+n}{2} \left ( \frac{k+n}{2}+1\right ) ~ a^{\dagger k}_1 a^{\dagger n}_2 |0\rangle ~, while :J_z ~ a^{\dagger k}_1 a^{\dagger n}_2 |0\rangle = \frac{1}{2} \left ( k-n\right ) a^{\dagger k}_1 a^{\dagger n}_2 |0\rangle ~, so that, for , this is proportional to the eigenstate , {{Equation box 1 Observe J_+ = a_1^\dagger a_2 and J_- = a_2^\dagger a_1 , as well as J_z = (a_1^\dagger a_1 - a_2^\dagger a_2 )/2 . ==Fermions==

Antisymmetric representations of Lie algebras can further be accommodated by use of the fermionic operators b^\dagger_i and b^{\,}_i, as also suggested by Jordan. For fermions, the commutator is replaced by the anticommutator \{\ \ , \ \ \}, : \{b^{\,}_i, b^\dagger_j\} \equiv b^{\,}_i b^\dagger_j +b^\dagger_j b^{\,}_i = \delta_{i j}, : \{b^\dagger_i, b^\dagger_j\} = \{b^{\,}_i, b^{\,}_j\} = 0. Therefore, exchanging disjoint (i.e. i \ne j) operators in a product of creation of annihilation operators will reverse the sign in fermion systems, but not in boson systems. This formalism has been used by A. A. Abrikosov in the theory of the Kondo effect to represent the localized spin-1/2, and is called Abrikosov fermions in the solid-state physics literature. ==See also==