Tuesday, February 21, 2012

Metapost and Labels

This is just a quick note to myself. When I want to write a label with a smaller font, I should use \scriptstyle...but it is tricky since it requires math mode!

So an example, consider this diagram describing an experiment for gravitational redshift:

numeric u;
u = 1pc;
  path earth;
  pair clock[];

  earth = fullcircle scaled u;

  clock[0] = (0,2u);
  clock[1] = (0,4u);

  draw (0,0)--(0,5u) dashed evenly;

  for i=0 upto 1:
    label(btex $\bullet$ etex, clock[i]);

  fill earth withcolor 0.75white;
  draw earth;
  label.rt(btex ${\scriptstyle\rm Earth}$ etex, (.5u,0));
  label.rt(btex ${\scriptstyle\rm Satellite\ 1}$ etex, clock[0]);
  label.rt(btex ${\scriptstyle\rm Satellite\ 2}$ etex, clock[1]);

Just remember to use "\ " for spaces. Otherwise it will all run together horribly!

Tuesday, February 14, 2012

Feynman Diagrams and Motives

I have been re-reading the following book:

Alain Connes and Matilde Marcolli,
Noncommutative Geometry, Quantum Fields, and Motives,
Colloquium Publications, Vol.55, American Mathematical Society, 2008.

It turns out that Dr Marcolli has taught a course on related material back in 2008! It is mostly dealing with the first chapter of the book.

Hopf Algebras and Feynman Calculations

There is a nice review of Hopf algebras used in Feynman diagram calculations:

Kurusch Ebrahimi-Fard, Dirk Kreimer,
"Hopf algebra approach to Feynman diagram calculations".
Eprint arXiv:hep-th/0510202v2, 30 pages.

For another specifically reviewing the noncommutative approach discussed in Connes and Matilde's book, see:

Herintsitohaina Ratsimbarison,
"Feynman diagrams, Hopf algebras and renormalization."
Eprint arXiv:math-ph/0512012v2, 12 pages.

What is a "Hopf algebra", anyways?

Pierre Cartier,
"A primer of Hopf algebras."
Eprint [math.osu.edu], 81 pages.

Hopf Algebras

What the deuce is a "Hopf algebra"? That's a very good question, and I'm very glad you asked it. Wikipedia has its definition, which may or may not be enlightening.

Lets consider a concrete example. Consider a finite group G, and the field of complex number ℂ. We assert the collection Hom(G,ℂ) is a Hopf algebra.

Recall we have multiplication of group elements. This is a mapping G×G→G.

Now, observe we have functoriality to give us a mapping Hom(G×G→G,ℂ) = ℂG→ℂG×ℂG. Lets call this thing Δ

Great, but what does it do? Good question!

Take some f∈Hom(G,ℂ) then what is Δ(f)?

It is a function of two variables, [Δ(f)](x,y). Functoriality demands, if we fix one of the arguments to be the identity element e∈G of the group, then [Δ(f)](e,y)=f(y) and [Δ(f)](x,e)=f(x).

It follows logically that [Δ(f)](x,y)=f(xy).

We also need to consider the antipode map S:ℂG→ℂG. We have [S(f)](x) be determined by the Hopf property, and a long story short [S(f)](x)=f(x-1).

Note that the antipode map is a generalization of the "group inverse" notion.

The other algebraic structure is a triviality, lets consider other interesting applications!

Feynman Diagrams

Now, I have written some notes [pdf] on the basic algorithm evaluating Feynman diagrams and producing a number (the "probability amplitude").

As I understand it (and I don't!!) Ebrahimi-Fard and Kreimer suggest considering the Hopf algebra of "Feynman graphs" (which are just considered as colored graphs representing physical processes).

The basic algorithm to evaluating Feynman diagrams are based on the "Feynman rules" (what we assign to each edge, vertex, etc.). So Feynman rules are linear and multiplicative maps, associating to each Feynman graph (again, seen as a collection of vertices and edges) its corresponding Feynman integral.

So these maps are the important things, which enable us to algorithmically do stuff.

Lets stop! I said "Feynman integrals" are assigned to each graph...am I drunk, or is that correct?

Yes yes, the answer is "yes" ;)

What a horrible joke...but what I mean is: the scattering process of electrons, for example, is the infinite sum taking into account all the virtual processes.

Usually we only care up to a few orders.

Of course, this is my understanding of the Hopf algebra treatment of Feynman diagrams...and I openly admit: I could be completely wrong!

So to figure it out, I'll stop rambling, and continue reading.

Monday, February 6, 2012

(Ramblings on) Writing Notes on Quantum Field Theory

So, in the long run, my aim is to write great notes on quantum field theory and quantum gravity. Since quantum gravity depends on quantum field theory, it makes sense to begin there!

For those uninterested in my rambling thought process, here's the punch line: just as integrals and derivatives are first covered symbolically in calculus, then rigorously in analysis...we likewise believe that a naive and symbolic approach first ought to be covered, then a rigorous and axiomatic approach second.

Greiner and Reinhardt's Field Quantization provides a good level for detail, at least for the naive/symbolic treatment of field theory.

What Other People Do

The basic approach other books take is "Well, here's Feynman diagrams. Quantum Field Theory just plays with these...here's how you get Feynman rules...and here's renormalization, the end."

This is not terrible. But it is lacking a certain je ne sais quoi.

So instead, perhaps I should look at it from the mathematical perspective. This has its own problems.

Depends on...

The problem I have is with dependencies! It doesn't make sense to write about quantum field theory without first writing about classical field theory, quantum mechanics, and a bit about functional analysis.

I have written a note about Relativistic Quantum Mechanics [pdf] (which may make more sense after reading my notes on Lie Groups, Lie Algebras, and their Representations [pdf]).

However, there is still more to do with quantum mechanics. Particularly, the subject of scattering theory is lacking. (Erik Koelink has some great Lecture Notes [tudelft.nl] too)

Despite my notes on Functional Techniques in Path Integral Quantization [pdf], I still feel lacking in the "path integral" department.

Perhaps I should write notes on measure theory, functional analysis, then tackle Glimm and Jaffe's Quantum physics: a functional integral point of view?

With classical field theory, the subject quickly becomes a can of worms (sadly enough).

Gauge theory, as Derek Wise notes in his blog post "The geometric role of symmetry breaking in gravity", is intimately connected to Cartan Geometry.

There are dozens of exercises/examples to consider in gauge theory: Yang-Mills Theory, Born-Infield Action, Non-linear Sigma-Model, Non-linear Electrodynamics, Chern-Simons Theory, etc.

What Outline

So far, I've been considering my obstacles...but what about an outline?

The model I am following is the treatment of integration and differentiation in mathematics. First we have the naive symbolic manipulations (as done in calculus), then later we have the formal and rigorous proof based approach (as done in analysis).

Perhaps we should begin with naive field theory, where we obtain classical field theory "naively" from a "many body" problem.

This has merit from modelling fields as densities on the intuitive level.

Canonical quantization of this scheme becomes a triviality.

The problem with this approach is: what about the treatment of gauge theories, and their quantization?

After a few miracles, I expect to end up working with path integral quantization and formal calculus.

Naive treatment on quantizing gauge systems ought to be considered a bit more closely...

So that concludes the "naive" approach, and we begin the Axiomatic Approach. We should clarify the term "Axiom" means specification (not "God given truth", as dictionaries insist!).

The axiomatic approach would be done in a "guess-and-check" manner, modifying the axioms as necessary.

We naturally begin with Wightman axioms for the canonical approach, and the Osterwalder-Schrader axioms for the path integral approach. (Quickly, we ought to prove these two are equivalent!)

Kac's Vertex Algebras for Beginners takes the Wightman axioms, then extends them to conformal field theory through some magic. Perhaps this would be a worth-while example to consider?