Least Action

Nontrivializing triviality..and vice versa.

Archive for November 2010

Ubuntu Maverick Meerkat 10.10 Screenshot

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Written by Vivek

November 18, 2010 at 09:31

Posted in Linux, Technology

A Mathematica notebook for Angular Momentum Matrix Algebra

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This is a Mathematica notebook for angular momentum matrix algebra. I have not yet added a component to add angular momenta, which I plan to do soon.

Feel free to use and distribute the notebook. Please just add a citation and a link to this webpage (also optional, of course).

Angular Momentum Matrix Algebra Notebook

Written by Vivek

November 17, 2010 at 19:12

Constructing a vector from a spinor

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The purpose of this post is to examine why the quantity

\chi^\dagger \sigma_i \chi

transforms like a vector under rotations. Here, \chi is a 2 component spinor, and \sigma_i is one of the 3 Pauli matrices. The explanation follows essentially the arguments put forth by Sakurai in chapter 3 of his book on quantum mechanics.

Spadework

An arbitrary state ket in spin space can be written as

|\alpha\rangle = |+\rangle\langle +|\alpha\rangle + |-\rangle\langle -|\alpha\rangle = \left(\begin{array}{c}\langle +|\alpha\rangle\\ \langle -|\alpha\rangle\end{array}\right) = \left(\begin{array}{c}c_{+}\\c_{-}\end{array}\right)

where \chi_{+} = |+ \rangle = \left(\begin{array}{c}1 \\ 0 \end{array}\right) and \chi_{-} = |-\rangle = \left(\begin{array}{c}0\\1\end{array}\right) are the base “kets” (actually they are spinors, for the spin-1/2 case). This identification suggests that we can specify a general two component spinor as

\chi = c_{+}\chi_{+} + \chi_{-}\chi_{-}

Further, from Sakurai’s equations 3.2.30,

\langle \pm|S_{k}|+\rangle = \frac{\hbar}{2}(\sigma_{k})_{\pm, +}

\langle \pm|S_{k}|-\rangle = \frac{\hbar}{2}(\sigma_{k})_{\pm, -}

Therefore, we get the most important result:

\langle S_k\rangle = \frac{\hbar}{2}\chi^{\dagger}\sigma_k\chi

Physically, this equation implies that the right hand side equals the expectation value of the spin operator S_k in spinor-space (or in more precise terms, in the basis generated by the set of all possible \chi‘s). So, the quantity that we are examining is just the expectation value of the operator S_k (modulo a constant factor of \hbar/2).

The argument..

Now, Sakurai has shown that under a rotation of the state kets, the expectation value of \langle S_k\rangle transforms like the component of a classical vector. That is,

\langle S_k\rangle \longrightarrow \sum_{l}R_{kl}\langle S_l\rangle

(This is equation 3.2.11 of Sakurai)

where R_{kl} are the elements of the 3 x 3 orthogonal matrix R that specifies this rotation.

The corresponding rotation of the state kets is carried out by the operator \mathcal{D}(R) = \exp{(i\frac{\phi}{2\hbar}\hat{n}\cdot\boldsymbol{S})}.

Therefore, \langle S_k\rangle transforms like a vector, and hence so does \chi^\dagger \sigma_i \chi.

Thanks to David Angelaszek for a long, patient and extremely useful discussion, which led to the resolution of this argument. And of course, thanks to JJ Sakurai for forcing us to read between the lines 🙂

Written by Vivek

November 13, 2010 at 16:13

Posted in Uncategorized

Cadabra – a Computer Algebra System for field theory

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This seems very exciting:

Cadabra is a computer algebra system (CAS) designed specifically for the solution of problems encountered in field theory. It has extensive functionality for tensor computer algebra, tensor polynomial simplification including multi-term symmetries, fermions and anti-commuting variables, Clifford algebras and Fierz transformations, implicit coordinate dependence, multiple index types and many more. The input format is a subset of TeX. Both a command-line and a graphical interface are available.

Source: http://cadabra.phi-sci.com/

There are two interesting papers on this. The first is a semi-technical overview, and the other (hep-th/0701238) is a more comprehensive one geared towards an audience familiar with various problems in modern field theory. The abstract of the second paper reads:

Introducing Cadabra: a symbolic computer algebra system for field theory problems

(Submitted on 25 Jan 2007 (v1), last revised 14 Jun 2007 (this version, v2))

Abstract: Cadabra is a new computer algebra system designed specifically for the solution of problems encountered in field theory. It has extensive functionality for tensor polynomial simplification taking care of Bianchi and Schouten identities, for fermions and anti-commuting variables, Clifford algebras and Fierz transformations, implicit coordinate dependence, multiple index types and many other field theory related concepts. The input format is a subset of TeX and thus easy to learn. Both a command-line and a graphical interface are available. The present paper is an introduction to the program using several concrete problems from gravity, supergravity and quantum field theory.

Source = http://arxiv.org/abs/hep-th/0701238

Written by Vivek

November 8, 2010 at 00:35