非对易微积分与离散物理
Non-commutative Calculus and Discrete Physics
摘要：
行动的顺序不能互换。例如，时钟的滴答声和位置的测量值彼此不互换，因为位置将在滴答声之后移动到下一个位置。我们采用非交换微积分，导数由微分器表示。在开始时，不同的导数彼此不对易，提供了曲率形式，因此规范场的曲率形式几乎在微积分定义之后就出现了。这提供了费曼-戴森从电磁形式主义推导出电磁形式主义的背景，以及包括从雅可比恒等式推导出的列维-奇维塔连接形式的早期出现在内的推广。在这个版本的非对易物理学中，裸量子力学（它的对易关系）似乎是所有其他构造的平坦背景。上升到经典物理学是通过用满足莱布尼茨规则的泊松括号代替迭代器来获得的。附录矩阵代数从离散的观点（迭代）提供。这篇论文将出现在2002年夏天在英国剑桥举行的ANPA会议上。
英文摘要：
Sequences of actions do not commute.. For example, the tick of a clock and the measurement of a position do not commute with one another, since the position will have moved to the next position after the tick. We adopt non-commutative calculus, with derivatives represented by commutators. In the beginning distinct derivatives do not commute with one another, providing curvature formalism so that the form of the curvature of a gauge field appears almost as soon as the calculus is defined. This provides context for the Feynman-Dyson derivation of electromagnetic formalism from commutators, and generalizations including the early appearance of the form of the Levi-Civita connection dervived from the Jacobi identity. In this version of non-commutative physics bare quantum mechanics (its commutation relations) appears as the flat background for all other constructions. Ascent to classical physics is obtained by replacing commutators with Poisson brackets that satisfy the Leibniz rule. An appendix on matrix algebra from a discrete point of view (Iterants) is provided. This paper will appear in the proceedings of the ANPA conference held in Cambridge, England in the summer of 2002.
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