Jordan map

In theoretical physics, the Jordan map, often also called the Jordan–Schwinger map is a map from matrices Mij 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 a i † {\displaystyle a_{i}^{\dagger }} and a i {\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,

[ a i , a j † ] ≡ a i a j † − a j † a i = δ i j , {\displaystyle [a_{i}^{\,},a_{j}^{\dagger }]\equiv a_{i}^{\,}a_{j}^{\dagger }-a_{j}^{\dagger }a_{i}^{\,}=\delta _{ij},}
[ a i † , a j † ] = [ a i , a j ] = 0 , {\displaystyle [a_{i}^{\dagger },a_{j}^{\dagger }]=[a_{i}^{\,},a_{j}^{\,}]=0,}

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

These operators change the eigenvalues of the number operator,

N = ∑ i n i = ∑ i a i † a i {\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 Mij to Fock space bilinear operators M,

M ⟼ M ≡ ∑ i , j a i † M i j a j   , {\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.

The example of angular momentum

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

J → ≡ a † ⋅ σ → 2 ⋅ a   , {\displaystyle {\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 ≡ J → ⋅ J → = N 2 ( N 2 + 1 ) . {\displaystyle 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 1 † k a 2 † n | 0 ⟩ = k + n 2 ( k + n 2 + 1 )   a 1 † k a 2 † n | 0 ⟩   , {\displaystyle J^{2}~a_{1}^{\dagger k}a_{2}^{\dagger n}|0\rangle ={\frac {k+n}{2}}\left({\frac {k+n}{2}}+1\right)~a_{1}^{\dagger k}a_{2}^{\dagger n}|0\rangle ~,}

while

J z   a 1 † k a 2 † n | 0 ⟩ = 1 2 ( k − n ) a 1 † k a 2 † n | 0 ⟩   , {\displaystyle J_{z}~a_{1}^{\dagger k}a_{2}^{\dagger n}|0\rangle ={\frac {1}{2}}\left(k-n\right)a_{1}^{\dagger k}a_{2}^{\dagger n}|0\rangle ~,}

so that, for j = (k+n)/2, m = (k−n)/2, this is proportional to the eigenstate |j,m⟩, [3]

| j , m ⟩ = a 1 †   k a 2 †   n k !   n ! | 0 ⟩ = a 1 †   ( j + m ) a 2 †   ( j − m ) ( j + m ) !   ( j − m ) ! | 0 ⟩   . {\displaystyle |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 ~.}

Observe J + = a 1 † a 2 {\displaystyle J_{+}=a_{1}^{\dagger }a_{2}} and J − = a 2 † a 1 {\displaystyle J_{-}=a_{2}^{\dagger }a_{1}} , as well as J z = ( a 1 † a 1 − a 2 † a 2 ) / 2 {\displaystyle 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 i † {\displaystyle b_{i}^{\dagger }} and b i {\displaystyle b_{i}^{\,}} , as also suggested by Jordan. For fermions, the commutator is replaced by the anticommutator {     ,     } {\displaystyle \{\ \ ,\ \ \}} ,

{ b i , b j † } ≡ b i b j † + b j † b i = δ i j , {\displaystyle \{b_{i}^{\,},b_{j}^{\dagger }\}\equiv b_{i}^{\,}b_{j}^{\dagger }+b_{j}^{\dagger }b_{i}^{\,}=\delta _{ij},}
{ b i † , b j † } = { b i , b j } = 0. {\displaystyle \{b_{i}^{\dagger },b_{j}^{\dagger }\}=\{b_{i}^{\,},b_{j}^{\,}\}=0.}

Therefore, exchanging disjoint (i.e. i ≠ j {\displaystyle i\neq 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[4] 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

References

  1. ^ Jordan, Pascual (1935). "Der Zusammenhang der symmetrischen und linearen Gruppen und das Mehrkörperproblem", Zeitschrift für Physik 94, Issue 7-8, 531-535
  2. ^ Schwinger, J. (1952). "On Angular Momentum", Unpublished Report, Harvard University, Nuclear Development Associates, Inc., United States Department of Energy (through predecessor agency the Atomic Energy Commission), Report Number NYO-3071 (January 26, 1952).
  3. ^ Sakurai, J. J.; Napolitano, Jim (2011). Modern Quantum Mechanics (2nd ed.). Boston: Addison-Wesley. ISBN 978-0-8053-8291-4. OCLC 641998678.
  4. ^ Abrikosov, A. A. (1965-09-01). "Electron scattering on magnetic impurities in metals and anomalous resistivity effects". Physics Physique Fizika. 2 (1): 5–20. doi:10.1103/PhysicsPhysiqueFizika.2.5. ISSN 0554-128X.