174 research outputs found
Octonionic Cayley Spinors and E6
Attempts to extend our previous work using the octonions to describe
fundamental particles lead naturally to the consideration of a particular real,
noncompact form of the exceptional Lie group E6, and of its subgroups. We are
therefore led to a description of E6 in terms of 3x3 octonionic matrices,
generalizing previous results in the 2x2 case. Our treatment naturally includes
a description of several important subgroups of E6, notably G2, F4, and (the
double cover of) SO(9,1), An interpretation of the actions of these groups on
the squares of 3-component "Cayley spinors" is suggested.Comment: 14 pages, 1 figure, contributed talk at 2nd Mile High Conference
(Denver 2009
Octonionic Mobius Transformations
A vexing problem involving nonassociativity is resolved, allowing a
generalization of the usual complex Mobius transformations to the octonions.
This is accomplished by relating the octonionic Mobius transformations to the
Lorentz group in 10 spacetime dimensions. The result will be of particular
interest to physicists working with lightlike objects in 10 dimensions.Comment: Plain TeX, 12 pages, 1 PostScript figure included using eps
Finite Lorentz Transformations, Automorphisms, and Division Algebras
We give an explicit algebraic description of finite Lorentz transformations
of vectors in 10-dimensional Minkowski space by means of a parameterization in
terms of the octonions. The possible utility of these results for superstring
theory is mentioned. Along the way we describe automorphisms of the two highest
dimensional normed division algebras, namely the quaternions and the octonions,
in terms of conjugation maps. We use similar techniques to define and
via conjugation, via symmetric multiplication, and via
both symmetric multiplication and one-sided multiplication. The
non-commutativity and non-associativity of these division algebras plays a
crucial role in our constructions.Comment: 24 pages, Plain TeX, 2 figures on 1 page submitted separately as
uuencoded compressed tar fil
Assessing student reasoning in upper-division electricity and magnetism at Oregon State University
Standardized assessment tests that allow researchers to compare the
performance of students under various curricula are highly desirable. There are
several research-based conceptual tests that serve as instruments to assess and
identify students' difficulties in lower-division courses. At the
upper-division level, however, assessing students' difficulties is a more
challenging task. Although several research groups are currently working on
such tests, their reliability and validity are still under investigation. We
analyze the results of the Colorado Upper-Division Electrostatics diagnostic
from Oregon State University and compare it with data from University of
Colorado. In particular, we show potential shortcomings in the Oregon State
University curriculum regarding separation of variables and boundary
conditions, as well as uncover weaknesses of the rubric to the free response
version of the diagnostic. We also demonstrate that the diagnostic can be used
to obtain information about student learning during a gap in instruction. Our
work complements and extends the previous findings from the University of
Colorado by highlighting important differences in student learning that may be
related to the curriculum, illuminating difficulties with the rubric for
certain problems and verifying decay in post-test results over time.Comment: 11 pages, 12 figure
Revealing Differences Between Curricula Using the Colorado Upper-Division Electrostatics Diagnostic
The Colorado Upper-Division Electrostatics (CUE) Diagnostic is an exam
developed as part of the curriculum reform at the University of Colorado,
Boulder (CU). It was designed to assess conceptual learning within
upper-division electricity and magnetism (E&M). Using the CUE, we have been
documenting students' understanding of E&M at Oregon State University (OSU)
over a period of 5 years. Our analysis indicates that the CUE identifies
concepts that are generally difficult for students, regardless of the
curriculum. The overall pattern of OSU students' scores reproduces the pattern
reported by Chasteen et al. at CU. There are, however, some important
differences that we will address. In particular, our students struggle with the
CUE problems involving separation of variables and boundary conditions. We will
discuss the possible causes for this, as well as steps that may rectify the
situation.Comment: 4 pages, 3 figures, 1 tabl
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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