Corinne A. Manogue

Finite Lorentz transformations, automorphisms, and division algebras

An explicit algebraic description of finite Lorentz transformations of vectors in ten‐dimensional Minkowski space is given by means of a parametrization in terms of the octonions. The possible utility of these results for superstring theory is mentioned. Along the way automorphisms of the two highest dimensional normed division algebras, namely, the quaternions and the octonions, are described in terms of conjugation maps. Similar techniques are used 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 noncommutativity and nonassociativity of these division algebras plays a crucial role in our constructions.

The octonionic eigenvalue problem

We discuss the eigenvalue problem for 2×2 and 3×3 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.

Detecting the rotating quantum vacuum

We derive conditions for rotating particle detectors to respond in a variety of bounded spacetimes and compare the results with the folklore that particle detectors do not respond in the vacuum state appropriate to their motion. Applications involving possible violations of the second law of thermodynamics are briefly addressed.

Scalar field equation in the presence of signature change

We consider the (massless) scalar field on a two-dimensional manifold with metric that changes signature from Lorentzian to Euclidean. Requiring a conserved momentum in the spatially homogeneous case leads to a particular choice of propagation rule. The resulting mix of positive and negative frequencies depends only on the total (conformal) size of the spacelike regions and not on the detailed form of the metric. Reformulating the problem using junction conditions, we then show that the solutions obtained above are the unique ones which satisfy the natural distributional wave equation everywhere. We also give a variational approach, obtaining the same results from a natural Lagrangian.

Octonionic representations of Clifford algebras and triality

The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the nonassociativity and noncommutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octoninic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest Σ3×SO(8) structure in this framework.

The Exceptional Jordan Eigenvalue Problem

We discuss the eigenvalue problem for 3 ×3 octonionic Hermitian matrices which is relevant to theJordan formulation of quantum mechanics. In contrast tothe eigenvalue problems considered in our previous work, all eigenvalues are real and solve theusual characteristic equation. We give an elementaryconstruction of the corresponding eigenmatrices, and wefurther speculate on a possible application to particle physics.

Why is Ampère’s law so hard? A look at middle-division physics

Because mathematicians and physicists think differently about mathematics, they have different goals for their courses and teach different ways of thinking about the material. As a consequence, there are a number of capabilities that physics majors need in order to be successful that might not be addressed by any traditional course. The result is that the total cognitive load is too high for many students at the transition from the calculus and introductory physics sequences to upper-division courses for physics majors. We illustrate typical student difficulties in the context of an Ampere’s law problem.

The Klein paradox and superradiance

Abstract The classic Klein paradox, particle creation due to the presence of very strong electromagnetic potentials, is reconsidered. Vacuum expectation values of the current and momentum operators are calculated for both massive spin zero and massive spin one-half fields. These new results, used as a measure of the number of particles produced, are compared with the results of others for the vacuum expectation value of the number operator. The relationship between superradiance and pair creation is explained and common conceptual errors are pointed out.

Paradigms in Physics: A new upper-division curriculum

We describe a new curriculum for the final two years of a B.S. program in Physics. Case studies in the junior year provide concrete examples or Paradigms as pillars to support systematic Capstone lectures in the senior year. In each of nine three-week Paradigms, the junior progresses from a descriptive lower-division understanding to an advanced analysis of a topic defined by phenomenon rather than discipline. Students generally view the new format with favor. They are better at visualization and make important connections among physics disciplines. Independent assessment is ongoing.

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 3 × 3 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 2×2 submatrices inside the 3×3 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.