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Room P4.35, Mathematics Building

Louis H. Kauffman, Univ. of Illinois at Chicago

Non-Commutative Worlds and Classical Constraints

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 ($i$)
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 $t$ (temporal quantity) by $it$. In this view, the
Wick rotation replaces numerical time with elementary temporal
observation. The relationship of this remark with the Heisenberg
commutator $[P,Q]=i\hslash $ 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.

Note also another seminar session by the same speaker on Friday 30th November