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Corinne A. Manogue

Publications and source records attributed to Corinne A. Manogue.

At least 19 recordsLinked to original sources
arXiv2024

A New Division Algebra Representation of $E_7$

We decompose the Lie algebra $\mathfrak{e}_{8(-24)}$ into representations of $\mathfrak{e}_{7(-25)}\oplus\mathfrak{sl}(2,\mathbb{R})$ using our recent description of $\mathfrak{e}_8$ in terms of (generalized) $3\times3$ matrices over pairs of division algebras. Freudenthal's description of both $\mathfrak{e}_7$ and its minimal representation are therefore realized explicitly within $\mathfrak{e}_8$, with the action given by the (generalized) matrix commutator in $\mathfrak{e}_8$, and with a natural parameterization using division algebras. Along the way, we show how to implement standard operations on the Albert algebra such as trace of the Jordan product, the Freudenthal product, and the determinant, all using commutators in $\mathfrak{e}_8$.

math.GR↗
arXiv2023

A New Division Algebra Representation of $E_6$

We construct the well-known decomposition of the Lie algebra $\mathfrak{e}_8$ into representations of $\mathfrak{e}_6\oplus\mathfrak{su}(3)$ using explicit matrix representations over pairs of division algebras. The minimal representation of $\mathfrak{e}_6$, namely the Albert algebra, is thus realized explicitly within $\mathfrak{e}_8$, with the action given by the matrix commutator in $\mathfrak{e}_8$, and with a natural parameterization using division algebras. Each resulting copy of the Albert algebra consists of anti-Hermitian matrices in $\mathfrak{e}_8$, labeled by imaginary (split) octonions. Our formalism naturally extends from the Lie algebra to the Lie group $E_6\subset E_8$.

math.GR↗
arXiv2022

Octions: An $E_8$ description of the Standard Model

We interpret the elements of the exceptional Lie algebra $\mathfrak{e}_{8(-24)}$ as objects in the Standard Model, including lepton and quark spinors with the usual properties, the Standard Model Lie algebra $\mathfrak{su}(3)+\mathfrak{su}(2)+\mathfrak{u}(1)$, and the Lorentz Lie algebra $\mathfrak{so}(3,1)$. Our construction relies on identifying a complex structure on spinors and then working in the enveloping algebra. The resulting model naturally contains GUTs based on $SO(10)$ (Georgi--Glashow), $SU(5)$ (Georgi), and $SU(4)\times SU(2)\times SU(2)$ (Pati--Salam). We then briefly speculate on the role of the remaining elements of $\mathfrak{e}_8$, and propose a mechanism leading to exactly three generations of particles.

hep-ph↗
arXiv2017

Re-thinking the Rubric for Grading the CUE: The Superposition Principle

While introductory electricity and magnetism (E&M) has been investigated for decades, research at the upper-division is relatively new. The University of Colorado has developed the Colorado Upper-Division Electrostatics (CUE) Diagnostic to test students' understanding of the content of the first semester of an upper-division E&M course. While the questions on the CUE cover many learning goals in an appropriate manner, we believe the rubric for the CUE is particularly aligned to the topics and methods of teaching at the University of Colorado. We suggest that changes to the rubric would allow for better assessment of a wider range of teaching schemes. As an example, we highlight one problem from the CUE involving the superposition principle. Using data from both Oregon State University and the University of Colorado, we discuss the limitations of the current rubric, compare results using a different analysis scheme, and discuss the implications for assessing students' understanding.

physics.ed-ph↗
arXiv2017

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.

physics.ed-ph↗
arXiv2017

Paradigms in Physics 2.0

In 2016, the Department of Physics at Oregon State University began a process to revise our Paradigms in Physics curriculum for physics majors. We began with a colloquium to inform the department of our plans and request their assistance, followed by a survey of students and faculty as well as individual interviews with the faculty teaching each course. As we developed a plan to address student- and faculty-identified challenges in the curriculum, we met with each faculty member individually to explain and refine our proposal, which was unanimously approved by the faculty. Major changes include major changes to several courses (math methods, computational physics, modern physics, electronics, and classical mechanics), including the introduction of two sophomore-year courses designed specifically to help prepare students for their upper-division courses.

physics.ed-ph↗
arXiv2015

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.

physics.ed-ph↗
arXiv2014

Experts' understanding of partial derivatives using the Partial Derivative Machine

Partial derivatives are used in a variety of different ways within physics. Most notably, thermodynamics uses partial derivatives in ways that students often find confusing. As part of a collaboration with mathematics faculty, we are at the beginning of a study of the teaching of partial derivatives, a goal of better aligning the teaching of multivariable calculus with the needs of students in STEM disciplines. As a part of this project, we have performed a pilot study of expert understanding of partial derivatives across three disciplines: physics, engineering and mathematics. Our interviews made use of the Partial Derivative Machine (PDM), which is a mechanical system featuring four observable and controllable properties, of which any two are independent. Using the PDM, we probed expert understanding of partial derivatives in an experimental context in which there is not a known functional form. Through these three interviews, we found that the mathematicians exhibited a striking difference in their understanding of derivatives relative to the other groups. The physicists and engineers were quick to use measurements to find a numeric approximation for a derivative. In contrast, the mathematicians repeatedly returned to speculation as to the functional form, and although they were comfortable drawing qualitative conclusions about the system from measurements, were reluctant to approximate the derivative through measurement. This pilot study led us to further questions. How do fields differ in their experts' concept image of partial derivatives? What representations of partial derivatives are preferred by experts? We plan to address these questions by means of further interviews with a wider range of disciplinary experts.

physics.ed-ph↗
arXiv2013

A Symplectic Representation of $\mathrm{E}_7$

We explicitly construct a particular real form of the Lie algebra $\mathfrak{e}_7$ in terms of symplectic matrices over the octonions, thus justifying the identifications $\mathfrak{e}_7\cong\mathfrak{sp}(6,\mathbb{O})$ and, at the group level, $\mathrm{E}_7\cong\mathrm{Sp}(6,\mathbb{O})$. Along the way, we provide a geometric description of the minimal representation of $\mathfrak{e}_7$ in terms of rank 3 objects called cubies.

math.RA↗
arXiv2010

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.

math.RA↗
arXiv2010

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.

math.RA↗
arXiv2000

Octonionic Hermitian Matrices with Non-Real Eigenvalues

We extend previous work on the eigenvalue problem for Hermitian octonionic matrices by discussing the case where the eigenvalues are not real, giving a complete treatment of the 2x2 case, and summarizing some prelimenary results for the 3x3 case.

math.RA↗
arXiv1999

The Exceptional Jordan Eigenvalue Problem

We discuss the eigenvalue problem for 3x3 octonionic Hermitian matrices which is relevant to the Jordan formulation of quantum mechanics. In contrast to the eigenvalue problems considered in our previous work, all eigenvalues are real and solve the usual characteristic equation. We give an elementary construction of the corresponding eigenmatrices, and we further speculate on a possible application to particle physics.

math-ph↗
arXiv1999

Quaternionic Spin

We rewrite the standard 4-dimensional Dirac equation in terms of quaternionic 2-component spinors, leading to a formalism which treats both massive and massless particles on an equal footing. The resulting unified description has the correct particle spectrum to be a generation of leptons, with the correct number of spin/helicity states. Furthermore, precisely three such generations naturally combine into an octonionic description of the 10-dimensional massless Dirac equation, as discussed in previous work.

hep-th↗
arXiv1999

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.

math-ph↗
arXiv1998

Finding Octonionic Eigenvectors Using Mathematica

The eigenvalue problem for 3x3 octonionic Hermitian matrices contains some surprises, which we have reported elsewhere. In particular, the eigenvalues need not be real, there are 6 rather than 3 real eigenvalues, and the corresponding eigenvectors are not orthogonal in the usual sense. The nonassociativity of the octonions makes computations tricky, and all of these results were first obtained via brute force (but exact) Mathematica computations. Some of them, such as the computation of real eigenvalues, have subsequently been implemented more elegantly; others have not. We describe here the use of Mathematica in analyzing this problem, and in particular its use in proving a generalized orthogonality property for which no other proof is known.

math.RA↗
arXiv1998

The Octonionic Eigenvalue Problem

We discuss the eigenvalue problem for 2x2 and 3x3 octonionic Hermitian matrices. In both cases, we give the general solution for real eigenvalues, and we show there are also solutions with non-real eigenvalues.

math.RA↗