Expository notes

These notes may be written in plain text, HTML, or TeX. The TeX papers are available here in DjVu, which can be viewed on almost any operating system using DjVuLibre. Other file formats may be available upon request.

Abstract Hamiltonian mechanics

An unorthodox approach to Hamiltonian mechanics. Note in particular that time is a momentum (see Time is imaginary below). It all comes to the same thing, of course.

Black hole information paradox

Many people resolve the black hole information paradox by simply denying the existence of a paradox: They say that pure states really can evolve to mixed states. This is in the Schroedinger picture. Can this explanation work in the Heisenberg picture? I argue that it can in this post to sci.physics.research.

C* algebras

A step by step definition of C* algebras.


A definition of categories, including functors and natural transformations.

Comma categories

A quick definition of comma categories, in the most general sense, followed by the surprisingly long list of other categorial constructions that are special cases.


A list of dualities in category theory —or rather, one duality in several parts.

Functional analysis with quaternions

A heavily edited transcript of a discussion with John Baez about how to do functional analysis with quaternions. If you want to do quantum mechanics with quaternions, this would be the underlying mathematics. By the way, you should know that there are some minor errors in here.


How to use Lagrange multipliers.

Lie groups

Abstract definitions of the classical Lie groups.

Quantum measurement problem

A post from the newsgroup sci.physics.research explaining the C* algebraic approach to quantum mechanics. There's a technical error there with the infima; if you really care about it, go to the sci.physics.research archive and look up the article.

Radon Nikodym Theorem

A proof of the Radon Nikodym theorem (for σfinite measures), with application to the Lebesgue decomposition theorem, using notation that lacks the spurious ‘d’s that usually infest this subject. This is from a guest lecture in an introductory graduate-level course on measure theory.

Rotation with quaternions

How to rotate using quaternions. Just like the title says.


The link between the languages of div/grad/curl and differential forms.

Time is imaginary

The classic post to sci.physics. You wouldn't believe how many things this fits. The verse at the end is a parody of a crackpot on sci.physics.


A quick derivation of the uncertainty principle with a surprise twist.
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