Research
A selection of research posts and project write-ups.
Layout algorithms for flexigrids
An exposition of the elastic coordinate system used in fletcher. Read more
Geometric Algebra for Special Relativity and Manifold Geometry · PDF
Master’s thesis in mathematical physics. Read more
Explicit Baker–Campbell–Hausdorff–Dynkin formula for Spacetime via Geometric Algebra · arχiv
In brief. In $≤4$ dimensions, there’s a simple formula for the bivector $σ_3$ in terms of bivectors $σ_1$ and $σ_2$ such that $e^{σ_1}e^{σ_2} = ±e^{σ_3}$.
Abstract. We present a compact Baker–Campbell–Hausdorff–Dynkin formula for the composition of Lorentz transformations $e^{σ_i}$ in the spin representation (a.k.a. Lorentz rotors) in terms of their generators $σ_i$:
$$ \ln(e^{σ_1}e^{σ_2}) = \tanh^{-1}\qty(\frac{ \tanh σ_1 + \tanh σ_2 + \frac12\qty[\tanh σ_1, \tanh σ_2] }{ 1 + \frac12\qty{\tanh σ_1, \tanh σ_2} }) $$This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension $≤ 4$, naturally generalising Rodrigues’ formula for rotations in $ℝ^3$. In particular, it applies to Lorentz rotors within the framework of Hestenes’ spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex $2×2$ matrix representation realised by the Pauli spin matrices. The formula is applied to the composition of relativistic $3$-velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle.
Read moreAn Overview of the Strong CP Problem and Axion Cosmology · PDF
Literature review supervised by Dr. Jenni Adams at the University of Canterbury.
I wanted to learn more about particle physics after my Bachelor’s, so a year of part-time study culminated in this literature review. I learned basic classical (and a little quantum) field theory, and read about the “strong $CP$ problem” of quantum chromodynamics and its solution via “axions”. The report is aimed at the mathematically-inclined graduate who is uninitiated in particle physics.
Read moreAsymptotic Structure and Symmetries of FLRW Universes · PDF
Honours project supervised by Prof. David Wiltshire at the University of Canterbury.
There is a fascinating relationship between the asymptotic symmetries of spacetime and gravitational waves and memory, sometimes referred to as The Infrared Triangle. We investigated the asymptotic structures of simple cosmological spacetimes, extending the usual analysis for flat spacetime.
You can find the full report (PDF) and also the poster (PDF).
Computing Crystal Band Structures with CP2K · PDF
Summer project about the theory behind numerical computation of the semiconductor band structure of crystals using the open source software CP2K. Full PDF.
Read moreVisualising the Freudenreich 1998 Model of the Galactic Bar and Disk · PDF
Cosmology summer project in which we reproduced an empirical model of our galaxy’s centre and wrote code to visualise it. Full PDF.
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