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?

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