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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Jordan map ： ウィキペディア英語版 Jordan map 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 Algebras occurring in physics. It was introduced by Pascual Jordan in 1935〔Jordan, Pascual (1935). "Der Zusammenhang der symmetrischen und linearen Gruppen und das Mehrkörperproblem", ''Zeitschrift für Physik'' 94, Issue 7-8, 531-535〕 and was utilized by Julian Schwinger〔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 ("On Angular Momentum" ), Unpublished Report, Harvard University, Nuclear Development Associates, Inc., United States Department of Energy (through predecessor agency the Atomic Energy Commission),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).〕 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 $a^\backslash dagger\_i$ and $a^\_i$ 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, : $(a^\backslash dagger\_j)\; \backslash equiv\; a^\_i\; a^\backslash dagger\_j\; -\; a^\backslash dagger\_ja^\_i\; =\; \backslash delta\_,$ : $(a^\backslash dagger\_j)\; =\; (a^\_j)\; =\; 0,$ where $(\backslash \; ,\; \backslash \; \backslash )$ is the commutator and $\backslash delta\_$ is the Kronecker delta. These operators change the eigenvalues of the number operator, : $N\; =\; \backslash sum\_i\; n\_i\; =\; \backslash sum\_i\; a^\backslash dagger\_i\; a^\_i$, by one, as for multidimensional quantum harmonic oscillators. The Jordan map from a set of matrices to Fock space bilinear operators , :$\backslash qquad\; \backslash longmapsto\; \backslash qquad\; M\; \backslash equiv\; \backslash sum\_\; a^\backslash dagger\_i\; \_\; a\_j\; ~,$ is clearly a Lie Algebra isomorphism, i.e. the operators satisfy the same commutation relations as the matrices . For example, the image of the Pauli matrices of SU(2) in this map, :$\backslash equiv\; ^\backslash dagger\; \backslash cdot\backslash frac\; \backslash cdot\; ^\; ~,$ for two-vector ''a''†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, :$J^2\backslash equiv\; \backslash cdot\; =\; \backslash frac\; \backslash left\; (\; \backslash frac+1\backslash right\; )~.$ 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, :$J^2~\; a^\_1\; a^\_2\; |0\backslash rangle=\; \backslash frac\; \backslash left\; (\; \backslash frac+1\backslash right\; )\; ~\; a^\_1\; a^\_2\; |0\backslash rangle\; ~,$ while :$J\_z\; ~\; a^\_1\; a^\_2\; |0\backslash rangle=\; \backslash frac\; \backslash left\; (\; k-n\backslash right\; )\; ~\; a^\_1\; a^\_2\; |0\backslash rangle\; ~,$ so that, for , this is proportional to the eigenstate . Antisymmetric representations of Lie algebras can further be accommodated by use of the fermionic operators $b^\backslash dagger\_i$ and $b^\_i$, as also suggested by Jordan. For fermions, the commutator is replaced by the anticommutator $\backslash$, : $\backslash \; \backslash equiv\; b^\_i\; b^\backslash dagger\_j\; +b^\backslash dagger\_j\; b^\_i\; =\; \backslash delta\_,$ : $\backslash \; =\; \backslash \_j\backslash \}\; =\; 0.$ Therefore, exchanging disjoint (i.e. $i\; \backslash ne\; j$) operators in a product of creation of annihilation operators will reverse the sign in fermion systems, but not in boson systems. ==References== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Jordan map」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.