Shah research blog, spring 2009

This blog will record my progress in researching aspects of nonlinear flow (and applications of what we discuss in the study group) to quantum field theory. I am using John Gibson's blog as a template for how I construct this blog, so please let me know if there is anything I can do differently in formatting this blog. E-mail: sarang.shah@gatech.edu

Is it possible to embed LaTeX directly into this wiki?
Yes, embed it within brackets

   <latex> .... </latex>

More than you want to know: plugin latex is installed on www.channelflow.org/dokuwiki

I will be updating this page over the next few months with a literature review of the problem and a brief introduction to quantum field theory.

2009-1-27 Hello all, I have come down with a stomach virus and will be unable to make it to our install-fest. I will be installing channelflow on my copy of Linux and will post my progress on this blog today. I will also review my research into nonlinear field theories (in particular in n=m nonabelian Yang-Mills fields) and quantum chaos.

Sorry to hear you're ill! Get well soon, and we'll see you next week. –John G.

2009-2-3 Installed Channelflow on my local linux installation. I will now work through the tutorial.

Finished tutorial. I am still having a problem with installing the MATLAB scripts so that I can compile a movie.

The predominant means of understanding and developing the Standard Model interactions is through the use of Gauge Field Theory. In much the same way that we exact Lorentz invariance from our solutions in developing relativistic quantum theory, we demand that our fields also remain invariant under a gauge transformation, such as that of U(1), SU(2), and SU(3). Such a requirement entails the introduction of a gauge field in order for the action (Lagrangian) to remain stationary. It is from such consideration that the electromagnetic field A_μ is naturally introduced given an invariance of the action when ψ→exp(iθ)ψ.

Yang-Mills Fields wikipedia.org/wiki/Yang-Mills_field result from an extension of ψ→exp(iθ)ψ, where ψ→exp(iθ^a T_a)ψ where a indexes the generators of a Lie Algebra. There is a Yang-Mills field A^a_μ for each generator. The nonabelian (noncommuting) nature of the generators T_a introduces further changes to the Field strength tensor. We may then assume that there is no energy flow and fix the gauge such that the Poynting vector vanishes A^a_0=0. The vacuum solution and conserved quantities lead to a partial differential equation in time and space for the field which may yield nonintegrable chaotic solutions.

Sarang, taking all of nonAbelian field theory in a term project is too much to chew on. I suggest 2-track approach.

1. Left track - keep reading QFT literature.
2. Right track: If GaTech is to maintain the world domination in the matters turbulent, we must learn how to quotient continuous symmetries, otherwise we cannot make sense of John G's periodic orbits. For plane Couette that is simpler to do than for the Yang-Mills (where we a totally clueless), the symmetries are global. If you read World in a mirror and join forces with Vaggelis, you might be able to quotient discrete symmetries from John G's periodic orbits. In the Lorenz example that leads to a very pretty picture of Lorenz flow topology, Poincaré sections and return maps. In the Kuramoto-Sivashinsky example it also leads to very pretty return maps. If you get it for plane Couette, great. If you do not, you will still learn lots of pretty group theory that you will need later on anyhow.

Thank you for the suggestion professor, I was afraid I was not making the right connections between these subjects.