Sunday, December 16, 2012

TeX macro for normal operator ordering

I've always been bothered with normal operator ordering, writing $:O(a)O(b):$ always produces bad results.

The quick fix I've been using is the following:

\def\normOrd#1{\mathop{:}\nolimits\!#1\!\mathop{:}\nolimits}

%%
% example:
% \begin{equation}
% \normOrd{a(z)b(\omega)} = a(z)_{+}b(\omega)+(-1)^{\alpha\beta}b(\omega)a(z)_{-}
% \end{equation}
%%
Which in practice looks like:

How I got this solution

I determined this solution iteratively after many different attempts, which I shall enumerate along with the problems they each had.

However, using mere colons :a(z)b(\omega): = ... produces the following:
Being clever, I asked myself "Hey, why not write :x\colon for the normal ordering?" This was clever, but wrong. Consider the following example:
g = :x\colon
Producing:
Not one to give up easily, I found a \cocolon definition on tex.stackexchange. Trying that instead:
g = \cocolon x\colon = y
Produces strange extra whitespace on the right:
After examining the co-colon code, I just determined that something along the lines of
% rough draft definition #1
\def\normOrd#1{\mathrel{:}\!#1\!\mathrel{:}}
would work. This didn't quite work, the whitespacing was strange. So instead I just use \mathop{:}\nolimits..., which produces the desired result.

Sunday, August 12, 2012

Revising my Notes on General Relativity

So I've been revising my notes on general relativity, and I've found several things worth mentioning.

1. Equivalence Principle. The equivalence principle gives us geometry. This is often poorly described (I too committed this error in my drafts).

The equivalence principle tells us neither the composition of a body nor its mass determines its trajectory in a gravitational field. So gravity determines paths, and this gives us geometry.

Moreover, there are different equivalence principles which should be mentioned. I yielded to this, and became incoherent (alas!). The trick is to stick this into a box, for the interested reader to find out more about it, but not obstruct the writing.

2. Coordinates for Black Hole. Different coordinates for the Schwarzschild solution are described beautifully in Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, and Eduard Herlt's Exact Solutions of Einstein’s Field Equations (Cambridge University Press, 2d edition, 2009).

3. Manifolds, Mathematics. I think I ought to examine Christopher Isham's Modern Differential Geometry for Physicists for a Physicist's differential geometry.

I should like to discuss the exponential map, which relates paths to geometry (as alluded in the equivalence principle discussion).

Most readers probably will agree that "Part II" of my notes (which specifically discuss differential geometry) are the toughest part of the notes.

Probably, I should mention a few examples of manifolds and explicitly study their coordinates in lecture 5.

3.1. Functions. I never discussed what it means for a function on a manifold (a) to exist, (b) to be smooth.

Really, this let us discuss curves too. Why? A curve is just a function γ:IM where I is just a closed interval, and M is the manifold.

3.2. Diffeomorphisms. This word is thrown around a lot, but never defined rigorously (or at all!). So I should re-investigate this a bit.

Friday, May 25, 2012

Vertex algebras

Think about your favorite Lie algebra Lie(G). We have a mapping on it, namely, the adjoint representation:

ad:Lie(G) → End[Lie(G)]

where "End[Lie(G)]" are the endomorphisms of the Lie algebra Lie(G).

Normally this is of the form "ad(u)v∈Lie(G)" and is shorthand for "ad(u)=[u,-]".

The Jacobi identity looks like:

ad(u)ad(v)-ad(v)ad(u)=ad(ad(u)v).

This is the most important identity. Vertex operator algebras are an algebra with a similar property.

A vertex operator algebra consists of a vector space V equipped with a mapping usually denoted

Y:V→(End V)[[x,x-1]].

In this form, it looks like left-multiplication operator...or that's the intuition anyways. So if "v∈V", we should think Y(v,x) belongs to "(End V)[[x,x-1]]" and acts on the left.

Really through currying this should be thought of as "V⊗V→V[[x,x-1]]", i.e., a sort of multiplication operator with a parameter "x". (This is related to the "state-operator correspondence" physicists speak of with conformal field theories.)

Just like a Lie algebra, the Vertex Operator algebra satisfies a Jacobi identity and it is the most important defining property for the VOA.

Lets stop and look at this structure again:

Y:V→(End V)[[x,x-1]].

What's the codomain exactly? Well, it's a formal distribution (not a mere formal power series!).

So what does one look like? Consider δ(z-1) = Σ zn where the summation ranges over n∈ℤ. This series representation is a formal distribution, and behaves in the obvious way. Lets prove this!

Desired Property: δ(z-1) vanishes almost everywhere.

Consider the geometric series f(z) = Σzn where n is any non-negative integer (n=0,1,...).

Observe that δ(z-1) = f(z) + z-1f(z-1). Lets now substitute in the resulting geometric series:

δ(z-1) = [1/(1-z)] + z-1[1/(1-z-1)]

and after some simple arithmetic we see for z≠1 we have δ(z-1)=0.

Desired Property: for any Laurent polynomial f(z) we have δ(z-1)f(z)=δ(z-1)f(1).

This turns out to be true, thanks to the magic of infinite series; but due to html formatting, I omit the proof. The proof is left as an exercise to the reader (the basic sketch is consider δ(z-1)zn, then prove linearity, and you're done).