Jordan–Schwinger Transformation
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In theoretical physics, the Jordan map, often also called the Jordan–Schwinger map is a map from matrices to bilinear expressions of quantum oscillators which expedites computation of representations of
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
s occurring in physics. It was introduced by
Pascual Jordan Ernst Pascual Jordan (; 18 October 1902 – 31 July 1980) was a German theoretical and mathematical physicist who made significant contributions to quantum mechanics and quantum field theory. He contributed much to the mathematical form of matri ...
in 1935 and was utilized by
Julian Schwinger Julian Seymour Schwinger (; February 12, 1918 – July 16, 1994) was a Nobel Prize-winning American theoretical physicist. He is best known for his work on quantum electrodynamics (QED), in particular for developing a relativistically invariant ...
in 1952 to re-work out the theory of quantum angular momentum efficiently, given that map’s ease of organizing the (symmetric)
representations ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
of
su(2) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 ...
in
Fock space The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after V. A. Fock who first intro ...
. The map utilizes several
creation and annihilation operators Creation operators and annihilation operators are Operator (mathematics), mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilatio ...
a^\dagger_i and a^_i of routine use in
quantum field theories In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct physical models of subatom ...
and
many-body problem The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. Terminology ''Microscopic'' here implies that quantum mechanics has to be ...
s, each pair representing a
quantum harmonic oscillator The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
. The commutation relations of creation and annihilation operators in a multiple-
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...
system are, : ^_i, a^\dagger_j\equiv a^_i a^\dagger_j - a^\dagger_ja^_i = \delta_, : ^\dagger_i, a^\dagger_j=
^_i, a^_j Caret () is the name used familiarly for the character provided on most QWERTY keyboards by typing . The symbol has a variety of uses in programming and mathematics. The name "caret" arose from its visual similarity to the original proofre ...
= 0, where \ , \ \ /math> is the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
and \delta_ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
. These operators change the eigenvalues of the number operator, : N = \sum_i n_i = \sum_i a^\dagger_i a^_i, by one, as for multidimensional quantum harmonic oscillators. The Jordan map from a set of matrices to Fock space bilinear operators , : \qquad \longmapsto \qquad M \equiv \sum_ a^\dagger_i _ a_j ~, is clearly a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
isomorphism, i.e. the operators satisfy the same commutation relations as the matrices .


The example of angular momentum

For example, the image of the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
of
SU(2) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 ...
in this map, : \equiv ^\dagger \cdot\frac \cdot ~, for two-vector as, 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 \cdot = \frac \left ( \frac+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^_1 a^_2 , 0\rangle = \frac \left ( \frac+1\right ) ~ a^_1 a^_2 , 0\rangle ~, while :J_z ~ a^_1 a^_2 , 0\rangle = \frac \left ( k-n\right ) a^_1 a^_2 , 0\rangle ~, so that, for , this is proportional to the eigenstate , 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
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s, the commutator is replaced by the
anticommutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
\, : \ \equiv b^_i b^\dagger_j +b^\dagger_j b^_i = \delta_, : \ = \ = 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 In physics, the Kondo effect describes the scattering of conduction electrons in a metal due to magnetic impurities, resulting in a characteristic change i.e. a minimum in electrical resistivity with temperature. The cause of the effect was firs ...
to represent the localized spin-1/2, and is called ''Abrikosov fermions'' in the solid-state physics literature.


See also

* Borel-Weil-Bott Theorem *
Current algebra Certain commutation relations among the current density operators in quantum field theories define an infinite-dimensional Lie algebra called a current algebra. Mathematically these are Lie algebras consisting of smooth maps from a manifold into a ...
*
Angular momentum operator In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum pro ...
* Klein transformation * Bogoliubov transformation * Holstein–Primakoff transformation *
Jordan–Wigner transformation The Jordan–Wigner transformation is a transformation that maps spin operators onto fermionic creation and annihilation operators. It was proposed by Pascual Jordan and Eugene Wigner for one-dimensional lattice models, but now two-dimensional a ...
* Clebsch–Gordan coefficients for SU(3)#Symmetry group of the 3D oscillator Hamiltonian operator


References

{{reflist Representation theory of Lie algebras Mathematical physics