2,253 research outputs found
K-Bessel functions associated to 3-rank Jordan algebra
Using Bessel-Muirhead system, we can express the K-bessel function defined on
a Jordan algebra as linear combination of the J-solutions. We determine
explicitly the coefficients when the rank of this Jordan algebra is three after
a reduction to the rank two. The main tools are some algebraic identities
developed for the occasion
Maximum Entropy and Sufficiency
The notion of Bregman divergence and sufficiency will be defined on general
convex state spaces. It is demonstrated that only spectral sets can have a
Bregman divergence that satisfies a sufficiency condition. Positive elements
with trace 1 in a Jordan algebra are examples of spectral sets, and the most
important example is the set of density matrices with complex entries. It is
conjectured that information theoretic considerations lead directly to the
notion of Jordan algebra under some regularity conditions
Octonions, E6, and Particle Physics
In 1934, Jordan et al. gave a necessary algebraic condition, the Jordan
identity, for a sensible theory of quantum mechanics. All but one of the
algebras that satisfy this condition can be described by Hermitian matrices
over the complexes or quaternions. The remaining, exceptional Jordan algebra
can be described by 3x3 Hermitian matrices over the octonions.
We first review properties of the octonions and the exceptional Jordan
algebra, including our previous work on the octonionic Jordan eigenvalue
problem. We then examine a particular real, noncompact form of the Lie group
E6, which preserves determinants in the exceptional Jordan algebra.
Finally, we describe a possible symmetry-breaking scenario within E6: first
choose one of the octonionic directions to be special, then choose one of the
2x2 submatrices inside the 3x3 matrices to be special. Making only these two
choices, we are able to describe many properties of leptons in a natural way.
We further speculate on the ways in which quarks might be similarly encoded.Comment: 13 pages; 6 figures; TonyFest plenary talk (York 2008
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