Dirac’s Belt Trick

Is classical physics boring? In his preface to Volume 1 of The Feynman Lectures on Physics, Richard Feynman worries about students' enthusiasm: ... They have heard a lot about how interesting and exciting physics is—the theory of relativity, quantum mechanics, and other modern ideas. By the end of two years of our previous course, many … Continue reading Dirac’s Belt Trick

Statistical Independence and Logarithms

In classical mechanics you want to understand the motion of all constituents of a system in detail. The trajectory of each 'particle' can be calculated from the forces between them and initial positions and velocities. In statistical mechanics you try to work out what can still be said about a system even though - or … Continue reading Statistical Independence and Logarithms

Integrating the Delta Function (Again) – Dirac Version

The Delta Function is, roughly speaking, shaped like an infinitely tall and infinitely thin needle. It's discovery - or invention - is commonly attributed to Paul Dirac[*]. Dirac needed a function like this to work with integrals that are common on quantum mechanics, a generalization of a matrix that has 1's in the diagonal and … Continue reading Integrating the Delta Function (Again) – Dirac Version

The Improper Function and the Poetry of Proofs

Later the Delta Function was named after their founder. Dirac himself called it an improper function. This time, the poem is not from repurposed snippets of his prose. These are just my own words to describe a proof: ~ In the limit the Lorentzian becomes the improper function. In the limit of tiny epsilons it … Continue reading The Improper Function and the Poetry of Proofs

Poetry: Dynamical Variables and Observables

The lines of the following poem are phrases selected from consecutive pages of the second chapter of Paul Dirac's Principles of Quantum Mechanics, Fourth Edition (Revised), Dynamical Variables and Observables. we may look upon the passage for the triple product We therefore make the general rule in spite of this fundamental difference which conforms with … Continue reading Poetry: Dynamical Variables and Observables

Poetry: The Principle of Superposition

The lines of the following poem are phrases selected from consecutive pages of the first chapter of Paul Dirac's Principles of Quantum Mechanics, Fourth Edition (Revised), The Principle of Superposition. ~ one would be inclined to think There must certainly be some internal motion from general philosophical grounds we cannot expect to find any causal … Continue reading Poetry: The Principle of Superposition

Entropy and Dimensions (Following Landau and Lifshitz)

Some time ago I wrote about volumes of spheres in multi-dimensional phase space - as needed in integrals in statistical mechanics. The post was primarily about the curious fact that the 'bulk of the volume' of such spheres is contained in a thin shell beneath their hyperspherical surfaces. The trick to calculate something reasonable is … Continue reading Entropy and Dimensions (Following Landau and Lifshitz)

Spheres in a Space with Trillions of Dimensions

I don't venture into speculative science writing - this is just about classical statistical mechanics; actually about a special mathematical aspect. It was one of the things I found particularly intriguing in my first encounters with statistical mechanics and thermodynamics a long time ago - a curious feature of volumes. I was mulling upon how … Continue reading Spheres in a Space with Trillions of Dimensions

Learning General Relativity

Math blogger Joseph Nebus does another A - Z series of posts, explaining technical terms in mathematics. He asked readers for their favorite pick of things to be covered in this series, and I came up with General Covariance. Which he laid out in this post - in his signature style, using neither equations nor … Continue reading Learning General Relativity