Papers by Toby Bartels

These papers may be written in text, HTML, or TeX. The TeX papers are available here in PostScript and PDF. Other 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.


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 errors in here; John and I are writing a real paper on it now.


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.

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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This web page and the papers linked from it were written between 1997 and 2006 by Toby Bartels. Toby reserves no legal rights to them.