# Expository notes

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## 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.
## Categories

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.
## Duality

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.
## Lagrange

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.
## Stokes

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`.
## Uncertainty

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 2014 by Toby Bartels,
last edited on 2014 June 23.
Toby reserves no legal rights to them.

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