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 $SO(3)$ and $SO(7)$ via conjugation, $SO(4)$ via symmetric multiplication, and $SO(8)$ 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