Supersymmetry on Lattice : An Introduction Wednesday, Jul 4 2007

Last week, I gave a presentation titled “Supersymmetry on Lattice- An Introduction”[PDF] as a part of a course on Lattice Field theory.

It was an attempt at outlining the broad issues that arise when one tries to put Supersymmetry in Lattice. I have uploaded the presentation in the link above – Readers comments and criticisms are welcome.

A Question in Geography Sunday, Feb 18 2007

A theorem in topology says:

For every continuous map $f: S^n \rightarrow R^n$ there exists a pair of anitpodal points $x$ and $- x$ in $S^n$ such that $f(x) = f(-x)$.

Specializing to the case $n = 2$ , one might conclude that at any point of time there are two antipodal points on Earth’s surface (which is homeomorphic to $S^2$) having same, say, pressure $P$ and temperature $T$ (which together constitute $R^2$ of the theorem) .

Question: Isn’t it that the presence of $\textit{two}$ polar caps (The Arctic and The Antarctica) where the day/night variation in temperature is quite low, may then be looked upon as a kind of `consequence’ of this theorem? (ofcourse the theorem doesn’t say where $x$ is located but at almost all other regions on Earth’s surface, the antipodal points are expected to have quite different temperatures due to day/night variation.)

ps1: One might worry about the validity of the assumption of continuity of $P$ and $T$ since there are wild local fluctuations; but I feel once you coarse-grain things out, this is a reasonable assumption.

ps2: For proof of the theorem, see http://www.mi.ras.ru/~scepin/elem-proof-reduct.pdf

Classical Mechanics and Differential Geometry Friday, Feb 9 2007

After learning basic concepts of classical mechanics, a wierd question arises about mathematical approach to this field. The general approach goes like this differential manifold structure is associated with lagrangian and symplectic structure comes with hamiltonian. We tried to learn this, so i am attaching project report(more like a formula sheet) with topic “differential geometric treatment to hamiltonian mechanics”. Please go through this and post some views on this mathematical approach.

Ising model and Epsilon Expansion Monday, Jan 1 2007

As one might know that topology plays an important role in the classical O(2) model in d=2 (i.e. x-y model in two dimension) becuase one can have non-trivial excitations, namely, vortices. These excitations, while having a huge energy, can contribute to the partition function at high temperatures becuase the entropy associated with them is also proportionately large. One can easily verify that perturbative epsilon-expansion (i.e. an expansion in no. of dimensions) fails here. The reason which is often cited is that such an expansion destroys all information related to topology since atleast till now there is no homotopy theory for fractional dimensions.

Considering an even simpler case, the ising transition in d=2 could be looked upon as condensation of domain walls which are again topological objects. My question is: why does an epsilon-expansion in this case works out quite nicely? (note: here one does an expansion near the lower critical dimension which is one). Is it that notion of domain walls can be extended to even fractional dimension in some way?

Btw, one might like to visit the website http://www.math.princeton.edu/~aizenman/OpenProblems.iamp/index.html  which contains a collection of open problems in mathematical physics. Coming to a problem related to this post (though somewhat remotely), one might be surprised to find through this link that the perturbative result of O(N > 2) model in d=2 being always in a single disordered phase at all temperatures has actually not been rigourosly proved. Infact, it is even disputed by some numerical results (though I don’t claim any understanding of these). See for example, http://prola.aps.org/abstract/PRB/v54/i10/p7177_1.

Torus subgroups Saturday, Dec 2 2006

This is the report of the presentation I gave for the Group Theory Course.

It is basically a complete proof of a very important and curious result in compact connected groups which states that all maximal connected abelian subgroups( tori) are conjugate to each other and the union of all these conjugates cover the entire set.

The proof is quite long and somewhat involved but nonetheless there are some very interesting ideas that go into it. A glimpse of that can be seen in the section where I have proved some theorems that would be required in the main proof.

Geometry and Topology for Physicists Thursday, Oct 26 2006

Previous post reminded me of one link I wanted to post earlier but somehow forgot:

Geometry and Topology for Physicists

I think these are quite a gem.
As the webpage says: “No particular mathematics background is required beyond that usually expected of graduate students in physics (linear algebra, complex analysis, etc), but it will help if you have some familiarity with mathematical notation and ways of thinking”.

Books by Sternberg Wednesday, Oct 25 2006

Prereq. : Varies. But, I guess some amount of “math-maturity” and interest can take one far. 😉

Some good mathematical books written by sternberg. The books available are

* Theory of Functions of real variable (2 Meg PDF)
* Advanced Calculus (58 Meg PDF)
* Dynamical systems (1 Meg PDF)
* Lie Algebras (900 K PDF)
* Geometric Asymptotics (AMS Books online)
* Semiriemannian Geometry (1 Meg PDF)

And over here, you can see a list with names of many math books.

Supersymmetry series at TowardsTengen Friday, Jul 7 2006

Background: Nothing more than basic Quantum Mechanics, actually. Knowing a bit of QFT would be useful, but not essential.

I’m trying to write up a bit about what I’ve been doing, the first (badly written) installment is up here.  In case you check it out, let me know if there are any errors or mistakes.

Enjoy Physics …

Link Dump Friday, Jul 7 2006

This post is mainly a collection of a lot of nice links to Physics related resources. Enjoy! :

• Predrag Cvitanović:
• A nice couple of what the authour calls “webboks” are available here.

• African Summer Theory Institute:
• Online, available video lectures, on QFT, Quantum mechanics, Supersymmetry etc. It really is a wonder, why these aren’t more popular. They are available here

And the best for the last …

• John Baez:
• Baez is a mathematical physicist at the university of California. He writes a weekly column on This Week’s Finds in mathematical Physics. Wonderfully written, and a really nice source of information of all sorts. This may require a little bit of maths, but he writes in a lucid and interesting way that even if you’re in your first year, UG, you will definitely learn something from each article, and the good news is there are archives going back to around 200 weeks!

• Mark Sredniki:
• This is a nice QFT book, it’s available online now, soon, it’s going to be taken off after it is published in print, so hurry and get your copy now! [Thanks to Tarun, for pointing this on out]. I’ve seen this once earlier, and it seemed really nice, especially has a nice section on spinors, and dotted & undotted indices, unless I’m confusing it with something else.

Next Page »