Archive for the ‘So you want to be a non-commutative geometer’ Category

So you want to be a non-commutative geometer, Part I: Preliminaries

29 May 2007

OK, these are just my random notes on my studies of non-commutative geometry. I hope that someone who understands only vector calculus (and perhaps linear algebra and most certainly abstract index notation — tensors and all that jazz) alone could look through this series and become a non-commutative geometer.

So, before we begin, we should probably review: differential forms, (basic) abstract algebra, and some operator theory/Hilbert space geometry/fun stuff. This is more of a pragmatic introduction, so mathematicians will probably be shocked at this rather unorthodox introduction…but don’t worry, it’s only the preliminary material!

Differential Forms

OK, so, this is meant to be a brief review, and brief it shall be! Consider some continuous function y=f(x). A differential is defined as:

dy = f'(x)dx

where the prime denotes differentiation with respect to x here. dy is a differential, of course! We can give it a special name like the “one form” (because there is one differential).

Now that we have a 1-form and a 0-form, and so forth, we can introduce an operator which is sort of like the cross-product from vector analysis. It is called the wedge product and it works for vectors as well as differential forms. The wedge product is defined as:

\alpha\wedge\beta = \alpha^{[i}\beta^{j]} = \frac{1}{2}(\alpha^{i}\beta^{j}- \alpha^{j}\beta^{i})

or for three vectors:

\alpha\wedge\beta\wedge\gamma = \alpha^{[p}\beta^{q}\gamma^{r]} = \frac{1}{6} (\alpha^{p}\beta^{q}\gamma^{r} + \alpha^{q}\beta^{r}\gamma^{p} + \alpha^{r}\beta^{p}\gamma^{q} - \alpha^{q}\beta^{p}\gamma^{r} - \alpha^{p}\beta^{r}\gamma^{q} - \alpha^{r}\beta^{q}\gamma^{p})

And so on and so forth (note that the pattern is 1/n! times the sum of the various permutations in the n indices similar to the definition of the determinant of a matrix). So it has the following properties:

\alpha\wedge\beta = -\beta\wedge\alpha (Anti-symmetry)

\alpha\wedge\alpha = 0

One can think of it as a sort of generalization of the cross product. The geometric image that one should think of is (this is according to Penrose’s Road to Reality - what an utterly Platonic sounding title that is…) for say two vectors a and b, a \wedge b is the oriented and scaled plane-element spanned by the independent vectors. For three vectors, it represents the oriented and scaled 3-volume element spanned by the independent vectors; and so forth. (See pages 212 to 216 of the Hard Cover first edition.)

There is this operator called the exterior derivative which essentially takes in a k-form, and returns a (k+1)-form. Suppose we had some k-form \alpha = f_I dx_I where I ranges from 1 to k. We can find the exterior derivative of this:

d\alpha = \sum_{i} \frac{\partial f_{I}}{\partial x_{i}}dx_{i}\wedge dx_I

Note that there are some properties which are rather interesting:

d(d\alpha ) = 0

d(\alpha\wedge\beta ) = d\alpha\wedge\beta + (-1)^{k}\alpha\wedge d\beta where $\alpha$ is a k-form (This is the leibniz property, remniscient of the usual derivative of a product in differential calculus)

Coincidentally wikipedia has a good page on exterior derivatives and wedge products…just something to go to for reference. It’s important to understand differential forms since non-commutative geometry begins with (and, as little of it as I have studied, it is) “quantum differential forms”. If you do not understand differential forms, you cannot understand non-commutative geometry!

I’d put up practice problems, but I’m lazy and pressed for time.

“But…but I don’t know differential forms! This is all greek to me!”

No worries! Here are some lecture notes, here’s something if you don’t know (or are rusty with) vector calculus.

Abstract Algebra

Now that you’ve mastered differential forms, it’s time to take on abstract algebra. Actually this isn’t so bad if you don’t mind learning definitions. If the term “abstract algebra” is absolutely new to you, go read Abstract Algebra by David S. Dummit and Ricard M. Foote. Once you have done that (be sure to do the exercises too!), go and read Algebra by Michael Artin. Then, finally, go and read Lie Algebras by Nathan Jacobson. That should equip you with the algebraic tools necessary to understand noncommutative geometry.

But suppose you live in the middle of nowhere with no access to the library and you’d still like to be a non-commutative geometer. Luckily, there are dozens of lecture notes available online for you…I’d try a whack at explaining it, but I’m exhausted and need to finish my term paper.

Operators and Hilbert Spaces

And so forth. Well, to start with, you need to understand linear algebra (linear operators, hermitian matrices, eigenvalues, etc.)…without understanding it, Quantum Mechanics becomes the fevered dream of a madman. So here are some lecture notes on linear algebra, notes on advanced linear algebra, more linear algebra, finite difference and spectrl methods in differential equations, differential equations, more differential equations, Hilbert Space fun, Hilbert Space Methods for Partial Differential Equations, (here is a semi non mathematical introduction to Hilbert Spaces for Quantum Mechanics, just what we’d like), bounded linear operators on a Hilbert Space, and if I may highly recommend for people absolutely new to this equipped only with knowledge of basic linear algebra: please read Griffith’s Quantum Mechanics! It’s a gem.

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