Wednesday, August 10, 2011

Theory Update 98

The magic square of number fields is a $4 \times 4$ square $M_{ij}$ with the $16$ entries given by pairs $(A(i),A(j))$ of number fields, for $A(k)$ one of the real, complex, quaternion or octonion fields. Thus the bioctonions correspond to $M_{24}$ or $M_{42}$. People who love Lie algebras will tell you that this entry gives the algebra for the group $E_6$. The fundamental representation of this group has complex dimension $27$, just like the dimension of the three qutrit path space for a $3 \times 3$ Jordan algebra.

This and other coincidences are not surprising, because qudit path spaces are responsible for the emergence of all classical spaces and their symmetries. With the octooctonions at $M_{44}$ we obtain the $248$ dimensional $E_8$ Lie algebra. There are many ways to partition the integer $248$, such as Kostant's nice way, with $248 = 31 \times 8$. When higher categories allow us to twist classical logic, a number field is not just a boring set of things that can be added and multiplied.

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