M Theory Lesson 118
Duality is everywhere, whether $T$ or $S$, categorical or Fourier. The process of integration hinges on a pairing between dual entities. For a simple monomial, the rule of integration $x^{n} \rightarrow \frac{1}{n + 1} x^{n + 1}$ looks a bit like the operator for (left) multiplication by $x$, except for the coefficient depending on $n$. Similarly, differentiation is like (left) multiplication by a factor of $\frac{1}{x}$.
An ordinary momentum operator $p$ is also like differentiation, up to a factor of $i \hbar$, which is more or less an integer $m$ when $\hbar$ belongs to a hierarchy. So if monomials in $x$ represent quantized paths in one dimension, then $p$ is a kind of dual to the operator that increases the path length by 1. That is, $p$ decreases the path length by 1. We could take care of the annoying factors by rescaling $x$ appropriately, as an operation is performed. Then for large values of $n$ (long paths) a momentum operation rescales $x$ almost by infinity!
What about noncommutative polynomials in two variables or more? Now one must specify whether left or right multiplication is being performed, but these operators are easy enough to define. For two variables $x$ and $y$ there would be four such operators for integration: $L_x$, $L_y$, $R_x$ and $R_y$. These operators also have simple multiplicative inverses, but a path may only be decreased by a step if the polynomial happens to start with the right letter. For example, the monomial $xyx^{3}$ can be multiplied by $\frac{1}{x}$ on the left, but not by $\frac{1}{y}$. That is, unless we allow negative components in the paths, but we didn't allow that in one dimension, where $\frac{d}{dx}$ applied to a constant just gives zero.
A duality between expansion and contraction may remind M theorists of either $T$ duality, which interchanges large and small scales, or the $S$ duality which turned up in our musings of the plane of Space and Time.
An ordinary momentum operator $p$ is also like differentiation, up to a factor of $i \hbar$, which is more or less an integer $m$ when $\hbar$ belongs to a hierarchy. So if monomials in $x$ represent quantized paths in one dimension, then $p$ is a kind of dual to the operator that increases the path length by 1. That is, $p$ decreases the path length by 1. We could take care of the annoying factors by rescaling $x$ appropriately, as an operation is performed. Then for large values of $n$ (long paths) a momentum operation rescales $x$ almost by infinity!
What about noncommutative polynomials in two variables or more? Now one must specify whether left or right multiplication is being performed, but these operators are easy enough to define. For two variables $x$ and $y$ there would be four such operators for integration: $L_x$, $L_y$, $R_x$ and $R_y$. These operators also have simple multiplicative inverses, but a path may only be decreased by a step if the polynomial happens to start with the right letter. For example, the monomial $xyx^{3}$ can be multiplied by $\frac{1}{x}$ on the left, but not by $\frac{1}{y}$. That is, unless we allow negative components in the paths, but we didn't allow that in one dimension, where $\frac{d}{dx}$ applied to a constant just gives zero.
A duality between expansion and contraction may remind M theorists of either $T$ duality, which interchanges large and small scales, or the $S$ duality which turned up in our musings of the plane of Space and Time.
posted by Kea | 11:42 AM
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