Room P4.35, Mathematics Building
This talk shows how discrete measurement leads to commutators and how discrete derivatives are naturally represented by commutators in a non-commutative extension of the calculus in which they originally occurred. We show how the square root of minus one () arises naturally as a time-sensitive observable for an elementary oscillator. In this sense the square root of minus one is a clock and/or a clock/observer. This sheds new light on Wick rotation, which replaces (temporal quantity) by . In this view, the Wick rotation replaces numerical time with elementary temporal observation. The relationship of this remark with the Heisenberg commutator is explained. We discuss iterants - a generalization of the complex numbers as described above. This generalization includes all of matrix algebra in a temporal interpretation. We then give a generalization of the Feynman-Dyson derivation of electromagnetism in the context of non-commutative worlds. This generalization depends upon the definitions of derivatives via commutators and upon the way the non-commutative calculus mimics standard calculus. We examine constraints that link standard and non-commutative calculus and show how asking for these constraints to be satisfied leads to some possibly new physics.