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In quantum mechanics (physics), the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, : between the position operator and momentum operator in the direction of a point particle in one dimension, where is the commutator of and , is the imaginary unit, and is the reduced Planck's constant . In general, position and momentum are vectors of operators and their commutation relation between different components of position and momentum can be expressed as :. where is the Kronecker delta. This relation is attributed to Max Born (1925), who called it a "quantum condition" serving as a postulate of the theory; it was noted by E. Kennard (1927) to imply the Heisenberg uncertainty principle. == Relation to classical mechanics == By contrast, in classical physics, all observables commute and the commutator would be zero. However, an analogous relation exists, which is obtained by replacing the commutator with the Poisson bracket multiplied by : : This observation led Dirac to propose that the quantum counterparts , of classical observables , satisfy : In 1946, Hip Groenewold demonstrated that a ''general systematic correspondence'' between quantum commutators and Poisson brackets could not hold consistently. However, he did appreciate that such a systematic correspondence does, in fact, exist between the quantum commutator and a ''deformation'' of the Poisson bracket, the Moyal bracket, and, in general, quantum operators and classical observables and distributions in phase space. He thus finally elucidated the correspondence mechanism, the Wigner–Weyl transform, that underlies an alternate equivalent mathematical representation of quantum mechanics known as deformation quantization.〔 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「Canonical commutation relation」の詳細全文を読む スポンサード リンク
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