Poisson algebras and the classical limit

In the previous post we described the Heisenberg picture of quantum mechanics, which can be phrased quite generally as follows: given a noncommutative algebra (the algebra of observables of some quantum system) and a Hamiltonian , we obtain a derivation , which is (up to some scalar multiple) the infinitesimal generator of time evolution. This [...]

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The representation theory of SU(2)

Today we will give four proofs of the classification of the (finite-dimensional complex continuous) irreducible representations of (which you’ll recall we assumed way back in this previous post). As a first step, it turns out that the finite-dimensional representation theory of compact groups looks a lot like the finite-dimensional representation theory of finite groups, and [...]

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The quaternions and Lie algebras II

We now know what a Lie algebra is and we know they are abstractions of infinitesimal symmetries, which are given by derivations. Today we will see what we can say about associating infinitesimal symmetries to continuous symmetries: that is, given a matrix Lie group , we will describe its associated Lie algebra of infinitesimal elements [...]

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The quaternions and Lie algebras I

Someone who has just read the previous post on how exponentiating quaternions gives a nice parameterization of might object as follows: “that’s nice and all, but there has to be a general version of this construction for more general Lie groups, right? You can’t always depend on the nice properties of division algebras.” And that [...]

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SU(2) and the quaternions

The simplest compact Lie group is the circle . Part of the reason it is so simple to understand is that Euler’s formula gives an extremely nice parameterization of its elements, showing that it can be understood either in terms of the group of elements of norm in (that is, the unitary group ) or [...]

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SO(3) and SU(2)

In order to study the hydrogen atom, we’ll need to know something about the representation theory of the special orthogonal group . This post consists of a few preliminaries along the way to doing this. I’ll be somewhat vague about a few things that 1) I don’t have much experience with, and 2) that would [...]

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