Least Action

Nontrivializing triviality..and vice versa.

Los Alamos Science

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Continuing the expository theme of my last post, I want to bring to your attention a collection of beautiful, crisp and entertaining articles by Richard Slansky in a 1984 publication of Los Alamos Science. They are available at

http://www.fas.org/sgp/othergov/doe/lanl/pubs/number11.htm

and are (in my opinion) must readings for students of theoretical physics, particularly those specializing in high energy theory, string theory, etc. In case you didn’t know, Slansky is also the author of a definitive review on group theory, which is a standard resource for particle physicists. It is worth having a printout of the review at your desk (and also an online copy in your tablet, smartphone, etc.) if you are interested in doing anything serious with group theory.

I should also take this opportunity to bring a list of Physics articles that have appeared over the years in Los Alamos Science. The comprehensive list of these articles with hyperlinks to online versions is at

http://la-science.lanl.gov/cat_physics.shtml

Written by Vivek

December 9, 2014 at 18:49

Review Articles – Particle Physics, String Theory, Supersymmetry and Supergravity

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[to be updated]

A list of useful reviews on various aspects of string theory, branes, etc. is at http://theory.tifr.res.in/~iqubal/stringrev.html. There are links to TASI lectures as well as review articles by prominent string theorists.

Additionally, a list of books and useful review articles for supersymmetry and supergravity is at http://www.stringwiki.org/wiki/Supersymmetry_and_Supergravity.

A useful list of references for Collider Physics is at http://tigger.uic.edu/~keung/me/class/collider/web-docs.html.

Written by Vivek

December 3, 2014 at 15:07

An inexpensive way to connect an HDMI screen to a Macbook

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I am not a huge Mac fan, but I do happen to have access to a Macbook Air, and since I find it hard to carry two laptops, a tablet and at least 5-6 books daily, I wanted to figure out a cheap way to get my Macbook Air to work with my HP Pavilion 27xi HDMI monitor. After some research, I went ahead and purchased an AmazonBasics Mini DisplayPort (Thunderbolt) to HDMI Adapter cable for about $10. There are many more cables available on Amazon, and most of them are much more expensive. But they’re all manufactured in roughly the same areas of our planet, so I wasn’t convinced about buying a costlier adapter especially for something I wouldn’t be heavily relying on. But this adapter is very nifty and works well.

The only other problem was to get the resolutions to match. For this, click on the leftmost button on the touch-activated panel of the HP Pavilion 27xi monitor. A menu opens up. Use the third soft key from left (with a “-” symbol) to navigate down to Image Control. When Image Control is highlighted, tap on the leftmost soft key again. Now, select Custom Scaling and press the leftmost soft key. Then, select Overscan and select “Off”. This should eliminate the problem of screen contents on the HDMI monitor being chopped. In my case at least, the default setting (Auto) does not work well with the Macbook Air I am working on.

Written by Vivek

November 6, 2014 at 21:53

Posted in Electronics

Tagged with , , ,

Divergence Theorem in Complex Coordinates

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The divergence theorem in complex coordinates,

\int_R d^2{z} (\partial_z v^z + \partial_{\bar{z}}v^{\bar{z}}) = i \oint_{\partial R}(v^z d\bar{z} - v^{\bar{z}}dz)

(where the contour integral circles the region R counterclockwise) appears in the context of two dimensional conformal field theory, to derive Noether’s Theorem and the Ward Identity for a conformally invariant scalar field theory (for example), and is useful in general in 2D CFT/string theory. This is equation (2.1.9) of Polchinski’s volume 1, but a proof is not given in the book.

This is straightforward to prove by converting both sides separately to Cartesian coordinates (\sigma^1, \sigma^2), through

z = \sigma^1 + i \sigma^2
\bar{z} = \sigma^1 - i \sigma^2

\partial_z = \frac{1}{2}(\partial_1 - i \partial_2)

\partial_{\bar{z}} = \frac{1}{2}(\partial_1 + i \partial_2)

d^2 z = 2 d\sigma^1 d\sigma^2 = 2 d^2 \sigma

and using the Green’s theorem in the plane

\oint_{\partial R}(L dx + M dy) = \int \int_{R} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) dx dy

with the identifications

x \rightarrow \sigma^1, y \rightarrow \sigma^2
L \rightarrow -v^2, M \rightarrow v^1

There is perhaps a faster and more elegant way of doing this directly in the complex plane, but this particular line of reasoning makes contact with the underlying Green’s theorem in the plane, which is more familiar from real analysis.

If I were a Springer-Verlag Graduate Text in Mathematics….

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If I were a Springer-Verlag Graduate Text in Mathematics, I would be William S. Massey’s A Basic Course in Algebraic Topology.

I am intended to serve as a textbook for a course in algebraic topology at the beginning graduate level. The main topics covered are the classification of compact 2-manifolds, the fundamental group, covering spaces, singular homology theory, and singular cohomology theory. These topics are developed systematically, avoiding all unecessary definitions, terminology, and technical machinery. Wherever possible, the geometric motivation behind the various concepts is emphasized.

Which Springer GTM would you be? The Springer GTM Test

Written by Vivek

October 10, 2014 at 17:52

Posted in Uncategorized

Calligraphic symbols in LaTeX

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If you’re unhappy with the way the standard amsmath/amssymb packages display a caligraphic L, and you want a more curly/loopy L for your Lagrangian density, then here’s a way out.

1. Install the package texlive-calrsfs. I had to run “yum install texlive-calrsfs” as superuser to accomplish this on Fedora 20.

2. Declare the following headers above your document

\documentclass{article}
\usepackage{calrsfs}
\DeclareMathAlphabet{\pazocal}{OMS}{zplm}{m}{n}
\newcommand{\La}{\mathcal{L}}
\newcommand{\Lb}{\pazocal{L}}

 

3. The more loopy version of L is now mapped to \La, whereas the conventional version is \Lb.

Hope this helps!

Adapted from: http://tex.stackexchange.com/questions/69085/two-different-calligraphic-font-styles-in-math-mode

Written by Vivek

July 18, 2014 at 10:35

Posted in LaTeX, Linux

The Yang Mills Field Strength: a motivation for differential forms

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Although their importance and applications are widely recognized in theoretical physics, differential forms are not part of the standard curriculum in physics courses, except for the rare mention in a general relativity course, e.g. if you have used the book by Sean Carroll. In this blog post, we describe the construction of the field strength tensor for Yang Mills Theory, using differential forms. The derivation (which is based on the one given by Tony Zee in his QFT book) is elegant, and mathematically less tedious than the more conventional derivation based on the matrix gauge field. It also illustrates the power of the differential forms approach.

To begin with, let me define a normalized matrix gauge field A_\mu which is -i times the usual matrix gauge field. So in the equations that appear below, the covariant derivative has no lurking factors of -i. This greatly simplifies the algebra for us, as we don’t have to keep track of conjugations and sign changes associated with i. The covariant derivative is thus D_\mu = \partial_\mu + A_\mu in terms of this new gauge field. The matrix 1-form A is

A = A_\mu dx^\mu

So, A^2 = A_\mu A_\nu dx^\mu dx^\nu. But since dx^\mu dx^\nu = -dx^\nu dx^\mu, only the antisymmetric part of the product survives and hence we can write

A^2 = \frac{1}{2}[A_\mu, A_\nu] dx^\mu dx^\nu

We want to construct an appropriate 2-form F = \frac{1}{2}F_{\mu\nu}dx^\mu dx^\nu from this 1-form A. Now, if d denotes the exterior derivative, then dA is a 2-form, as is A^2. These are the only two forms we can construct from A. So, F must be a linear combination of these two forms. This is a very simple, and neat argument!

Now, the transformation law for the gauge potential is

A \rightarrow U A U^\dagger + U dU^\dagger

where U is a 0 form (so dU^\dagger = \partial_\mu U^\dagger dx^\mu). Applying d to the transformation law, we get

dA \rightarrow U dA U^\dagger + dU A U^\dagger - U A dU^\dagger + dU dU^\dagger

where the negative sign in the third term comes from moving the 1-form d past the 1-form A. Squaring the transformation law yields

A^2 \rightarrow UA^2 U^\dagger + U A dU ^\dagger + U dU^\dagger U A U^\dagger + U dU^\dagger U dU^\dagger

Now, $UU^\dagger = 1$, so applying d again to both sides we get U dU^\dagger = -dU U^\dagger. So, we can write the square transformation law as

A^2 \rightarrow U A^2 U^\dagger + U A dU^\dagger - dU A U^\dagger - dU dU^\dagger
whereas if we recall the expression for the transformation of dA, it was just
dA \rightarrow U dA U^\dagger + dU A U^\dagger - U A dU^\dagger + dU dU^\dagger

Clearly if we merely add A^2 and dA, the last 3 terms on the RHS of each cancel out, and we get

A^2 + dA \rightarrow U(A^2 + dA)U^\dagger

which is the expected transformation law for a field strength of the form F = A^2 + dA:

F \rightarrow U F U^{\dagger}

The differential form approach uses compact notation that suppresses the Lorentz index \mu as well as the group index a, and gives us a fleeting glimpse into the connection between gauge theory and fibre bundles.

For a gentle yet semi-rigorous introduction to differential forms, the reader is referred to the book on General Relativity by Sean Carroll.

Written by Vivek

June 11, 2014 at 10:53

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