# Jordan map

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In theoretical physics, the '''Jordan map''', often also called the '''Jordan–Schwinger map''' is a map from matrices {{math|'''M'''<sub>''ij''</sub>}} to bilinear expressions of quantum oscillators which expedites computation of  representations of [Lie algebra](/source/Lie_algebra)s occurring in physics.  It was introduced by [Pascual Jordan](/source/Pascual_Jordan) in 1935<ref>Jordan, Pascual (1935). "Der Zusammenhang der symmetrischen und linearen Gruppen und das Mehrkörperproblem", ''Zeitschrift für Physik'' '''94''', Issue 7-8, 531-535</ref> and was utilized by [Julian Schwinger](/source/Julian_Schwinger)<ref>Schwinger, J. (1952). [https://www.ifi.unicamp.br/%7Ecabrera/teaching/paper_schwinger.pdf  "On Angular Momentum"], Unpublished Report, Harvard University, Nuclear Development Associates, Inc., [United States Department of Energy](/source/United_States_Department_of_Energy) (through predecessor agency the [Atomic Energy Commission](/source/United_States_Atomic_Energy_Commission)), Report Number  NYO-3071 (January 26, 1952).</ref> in 1952 to re-work out the theory of [quantum angular momentum](/source/Angular_momentum) efficiently, given that map’s ease of organizing the (symmetric) [representations](/source/Lie_algebra_representation) of  [su(2)](/source/Special_unitary_group) in [Fock space](/source/Fock_space).

The map utilizes several [creation and annihilation operators](/source/Creation_and_annihilation_operators)  
<math>a^\dagger_i</math> and <math>a^{\,}_i</math> of routine use in [quantum field theories](/source/Quantum_field_theory) and [many-body problem](/source/many-body_problem)s, each pair representing a [quantum harmonic oscillator](/source/quantum_harmonic_oscillator).
The commutation relations of creation and annihilation operators in a multiple-[boson](/source/boson) system are,
: <math>[a^{\,}_i, a^\dagger_j] \equiv a^{\,}_i a^\dagger_j - a^\dagger_ja^{\,}_i = \delta_{i j},</math>
: <math>[a^\dagger_i, a^\dagger_j] = [a^{\,}_i, a^{\,}_j] = 0,</math>
where <math>[\ \ , \ \ ]</math> is the [commutator](/source/commutator) and <math>\delta_{i j}</math> is the [Kronecker delta](/source/Kronecker_delta).

These operators change the eigenvalues of the [number operator](/source/number_operator),
: <math>N = \sum_i n_i = \sum_i a^\dagger_i a^{\,}_i</math>,
by one, as for [multidimensional  quantum  harmonic oscillators](/source/Quantum_harmonic_oscillator).

The Jordan map from a set of matrices {{math|'''M'''<sub>''ij''</sub>}} to Fock space bilinear operators {{math|''M''}},
:<math>{\mathbf M}  \qquad \longmapsto \qquad   M \equiv  \sum_{i,j}  a^\dagger_i  {\mathbf M}_{ij}    a_j  ~,</math>  
is clearly a [Lie algebra](/source/Lie_algebra) isomorphism, i.e. the operators {{math|''M''}} satisfy the same commutation relations as  the matrices {{math|'''M'''}}.

==The example of angular momentum==
For example, the image of the [Pauli matrices](/source/Pauli_matrices) of [SU(2)](/source/Special_unitary_group) in this map, 
:<math>{\vec J} \equiv {\mathbf a}^\dagger \cdot\frac{ \vec \sigma } {2} \cdot {\mathbf a} ~,</math>
for two-vector '''a'''<sup>†</sup>s, and '''a'''s satisfy the same commutation relations of SU(2) as well, and moreover, by reliance on the [completeness relation for Pauli matrices](/source/Pauli_matrices), 
:<math>J^2\equiv {\vec J} \cdot {\vec J} = \frac{N}{2} \left ( \frac{N}{2}+1\right ) . </math>

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, 
:<math>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 ~,</math> 
while 
:<math>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 ~,</math> 
so that, for {{math|1= ''j'' = (''k+n'')/2, ''m'' = (''k−n'')/2}}, this is proportional to the eigenstate {{math|{{ket|''j'',''m''}}}},<ref>{{cite book | last1=Sakurai | first1=J. J. | last2=Napolitano | first2=Jim | title=[Modern Quantum Mechanics](/source/Modern_Quantum_Mechanics) | date=2011 | publisher=Addison-Wesley | isbn=978-0-8053-8291-4 | edition=2nd | location=Boston | oclc=641998678}}</ref>
{{Equation box 1
|indent =:
|equation = <math>|j,m\rangle= \frac{a_1^{\dagger ~k} a_2^{\dagger ~ n} }{\sqrt{k!~n!}} |0\rangle = \frac{a_1^{\dagger ~(j+m)} a_2^{\dagger ~ (j-m)} }{\sqrt{(j+m)!~(j-m)!}} |0\rangle ~ .</math>
|cellpadding= 6
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Observe <math>J_+ = a_1^\dagger a_2</math> and <math>J_- = a_2^\dagger a_1 </math>, as well as <math>J_z = (a_1^\dagger a_1 - a_2^\dagger a_2 )/2 </math>.

==Fermions==
Antisymmetric representations of Lie algebras can further be accommodated by use of the fermionic operators  
<math>b^\dagger_i</math> and <math>b^{\,}_i</math>,  as also suggested by Jordan. For [fermion](/source/fermion)s, the commutator is replaced by the [anticommutator](/source/anticommutator) <math>\{\ \ , \ \ \}</math>,
: <math>\{b^{\,}_i, b^\dagger_j\} \equiv b^{\,}_i b^\dagger_j +b^\dagger_j b^{\,}_i = \delta_{i j},</math>
: <math>\{b^\dagger_i, b^\dagger_j\} = \{b^{\,}_i, b^{\,}_j\} = 0.</math>
Therefore, exchanging disjoint (i.e. <math>i \ne j</math>) 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<ref>{{Cite journal|last=Abrikosov|first=A. A.|date=1965-09-01|title=Electron scattering on magnetic impurities in metals and anomalous resistivity effects|url=https://link.aps.org/doi/10.1103/PhysicsPhysiqueFizika.2.5|journal=Physics Physique Fizika|language=en|volume=2|issue=1|pages=5–20|doi=10.1103/PhysicsPhysiqueFizika.2.5|issn=0554-128X|doi-access=free}}</ref> by [A. A. Abrikosov](/source/Alexei_Abrikosov_(physicist)) in the theory of the [Kondo effect](/source/Kondo_effect) to represent the localized spin-1/2, and is called ''Abrikosov fermions'' in the solid-state physics literature. 

==See also==
* [Borel-Weil-Bott Theorem](/source/Borel%E2%80%93Weil%E2%80%93Bott_theorem)
* [Current algebra](/source/Current_algebra)
* [Angular momentum operator](/source/Angular_momentum_operator)
* [Klein transformation](/source/Klein_transformation)
* [Bogoliubov transformation](/source/Bogoliubov_transformation)
* [Holstein–Primakoff transformation](/source/Holstein%E2%80%93Primakoff_transformation)
* [Jordan–Wigner transformation](/source/Jordan%E2%80%93Wigner_transformation)
*[Clebsch–Gordan coefficients for SU(3)#Symmetry group of the 3D oscillator Hamiltonian operator](/source/Clebsch%E2%80%93Gordan_coefficients_for_SU(3))

==References==
{{reflist}}

Category:Representation theory of Lie algebras
Category:Mathematical physics

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Adapted from the Wikipedia article [Jordan map](https://en.wikipedia.org/wiki/Jordan_map) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Jordan_map?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.