occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

Friday, January 18, 2008

M Theory Lesson 147

First let us consider the relation on matrices defined by $A \simeq B$ if $[A,B] = \lambda I$ for a scalar $\lambda$. It is not necessarily transitive, except for triples satisfying rules of the form

$[A,B] + [B,C] - [A,C] = 0$

but it is reflexive (since $[A,A] = 0$) and symmetric (since $[A,B] = -[B,A]$). In the world of categories we think of transitivity as a triangle of arrows, but we might weaken this triangle by allowing 2-arrows, or even higher dimensional structure.

Under this equivalence, the usual Heisenberg rule $[X,P] = i \hbar$ is a kind of equivalence between position and momentum. If we exponentiate this expression we find that

$\textrm{exp}(XP)=\textrm{exp}(i \hbar) \textrm{exp}(PX)$

which naturally reminds us of the Weyl rule for the discrete Fourier transform underlying the mass matrices. Now we see that $\hbar$ naturally defines a root of unity, and there is no reason to assume it takes on a fixed value. Moreover, when the root of unity is specified by the dimension of the matrix, as is the case for the Fourier transform, the value of $\hbar$ is specified.