## Archive for **November 2010**

## A Mathematica notebook for Angular Momentum Matrix Algebra

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).

## Constructing a vector from a spinor

The purpose of this post is to examine why the quantity

transforms like a vector under rotations. Here, is a 2 component spinor, and 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

where and 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

Further, from Sakurai’s equations 3.2.30,

Therefore, we get the most important result:

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

**The argument..**

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

(This is equation 3.2.11 of Sakurai)

where 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 .

Therefore, transforms like a vector, and hence so does .

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 ðŸ™‚

## Cadabra – a Computer Algebra System for field theory

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

Authors: Kasper Peeters(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