A little late to the party here, but Happy New Year friends! It’s been a while since the last post on here, so it’s time for shenanigans again! :D While the world has been turning a radian or two, I’ve been kept busy by life and, thankfully, the innumerable opportunities it has offered to travel, meet new people, explore music, art, hobbies, ne...

“What is particularly curious about quantum theory is that there can be actual physical effects arising from what philosophers refer to as counterfactuals - that is, things that might have happened, although they did not in fact happen.” — Roger Penrose The measurement problem Quantum measurement has remained a disputed aspect of quantum m...

In The Real Reason Nothing Travels Faster Than Light, we analytically proved that faster-than-light motion is logically impossible within the framework of special relativity. In doing so, we disproved the very kinematics of the situation \(v>c\). This, as we discussed, is logically stronger than disproving the dynamics of the situation, i.e. ...

In classical mechanics, kinetic energy encodes the tendency of a system being tracked to move through space. Since relativity is based on the geometric setting of spacetime, one would expect that a relativistic analogue of kinetic energy talks about how a system moves through space and time. In this post, we will explore these notions of system...

Field theory is fundamental to our modern understanding of the Cosmos. From classical fields to quantum fields, we see a startling complexity of incredibly wide-ranging physical phenomena emerge from relatively simpler structures and axioms. Often, the essence of these complicated mechanisms is captured by geometry, which among other (and somew...

Hi everyone, hope you’re all doing well :) In this short post, I’ll be sharing the slides from my recent presentation at SASMS (Short Attention Span Math Seminars), hosted by the Pure Math Club at the University of Waterloo. The presentation was on an informal construction of the ideas developed in some posts on this blog: Scalar Field Lagr...

In A Brief Geometric Analysis of Harmonic Oscillators: Part 2 (Tensor Algebra), we generalized the methods used in A Brief Geometric Analysis of Harmonic Oscillators to analyze the behaviour of harmonic oscillators in phase space. We began with Hooke’s law, \(\omega^{-1} \ddot{x} + \omega x = 0\) and explored the Hamiltonian flow described by i...

Statement Consider the 2 well-known properties of a linear operator \(T : \mathbb{R}^m \to \mathbb{R}^n : m, n \in \mathbb{N}\), [\begin{align} T \left( \sum_a \pmb{u}_a \right) & = \sum_a T \left( \pmb{u}_a \right) & \forall : \pmb{u}_a \in \mathbb{R}^m && \left( 1 \right) T \left( c \pmb{u} \right) & = c T \left( \pmb{u}...

In A Brief Geometric Analysis of Harmonic Oscillators, we examined how a convenient choice of phase space for the motion of a harmonic oscillator reveals its periodicity, without explicitly solving the equation of motion (Hooke’s law). Here, we will apply the same intuition but in the language of tensor algebra, introducing greater rigour as wel...

Some thoughts Hi everyone! It’s been a few months since the last post here. A lot (of good) has happened since then. Before coming to that, I would like to thank all my old readers for sticking by, and new readers for visiting this blog :) I hope you all are doing great and had a wonderful year … Happy New Year! In this period of absence, I’ve...