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北星論集（短） 第 16 号（通巻第54号） March 2018

正準交換関係の導出について

内 山   智

Satoshi U

CHIYAMA

研究ノート

目 次 ［Abstract］

On Derivations of Canonical Commutation Relations

 A mathematical model of a classical mechanical system that reproduces quantum−mechanical probabilities is considered. The basic idea is that a quantum−mechanical state corresponds to a periodic trajectory in a classical−mechanical phase space. The quantities in this model that correspond to amplitudes of wave functions are considered as values of a functional of a periodic trajectory. Then an observable is realized as a linear mapping of a linear space spanned by these functionals. The linear mapping is induced from an infinitesimal transformation of a periodic trajectory; the infinitesimal transformation of a periodic trajectory is induced from the Hamiltonian vector field with the observable physical quantity. It is shown that the observables obtained from Hamiltonian vector fields with the position and the momentum satisfy the canonical commutation relations by choosing values of parameters of the model appropriately.

２. 周期的軌道上の対象の測定の理論

１. はじめに

1. はじめに 2. 周期的軌道上の対象の測定の 理論 2. 1 復元可能な変形 2. 2. 単一の測定の文脈におけ る測定 2. 3. 別の測定の文脈を経由す る測定 3. オブザーバブルの交換関係 3. 1. 物理量の作用 3. 2. 軌道の汎関数 3. 3. 軌道の汎関数の線形写像 による正準交換関係 4. Lie 代数を保存する対応 5. 結 論

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─ 16 ─ 2. 3. 別の測定の文脈を経由する測定

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正準交換関係の導出について

３. オブザーバブルの交換関係

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正準交換関係の導出について

3. 3. 軌道の汎関数の線形写像による正準交 換関係

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正準交換関係の導出について

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５. 結 論

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参照

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