The First Computer Algebra System: Maxima

Setup and Operator Algebra

2024-01-04 967 words 5 minutes

Contents

Why

We learned to do proofs and derivations with pen and paper. There are two problems with this method. Firstly, there will be errors and typos. Secondly, efficient preservation of handwritten materials is hard. As a programmer, a tool that helps to overcome such problems is a combination of computer algebra system (CAS) and a proof assistant. There have been mature and feature rich CAS like Mathematica and Maple. However, I have two problems with them. Firstly, I don't like their syntax and the steep learning curve that comes with it. Less personally, they are both proprietary and close-sourced. Contrary to open-sourced software, this means I cannot hack on them and the support and evolution of them ceases to exist should the company abandons them. As the direct descendant of the ancestor of all modern CAS, Maxima comes up as a promising alternative. The goal of this post is to document how to download and start using Maxima for the operator algebra that is used frequently in condensed matter physics and quantum mechanics related topics.

Maxima

Maxima is a CAS written in Lisp. A CAS is a software that can manipulate symbolic and numerical expressions. Hence, it can perform the laborious manual computation in mathematics and physics in place of human being. Maxima was the descendant of Macsyma, the lengendary CAS developed in MIT during the 1960s. Macsyma inspired both Mathematica and Maple. Maxima was maintained by William Schelter from 1982 until 2001 when he passed away. If it weren't for his effort to converting the project to open-source and maintaining it, we would have lost a great tool in open-source CAS. Currently, Maxima is maintained by the community.

Installation

To install maxima, we simply rely on our favourite package manager. For example, on a Mac, you can do

brew install maxima

After installing maxima, you will need a lisp compiler in order to run it. You can do that by

brew install sbcl

Let's verify that we have successfully installed maxima.

maxima

Maxima 5.47.0 https://maxima.sourceforge.io using Lisp SBCL 2.4.0 Distributed under the GNU Public License. See the file COPYING. Dedicated to the memory of William Schelter. The function bug_report() provides bug reporting information. (%i1)

Looks like we are able to start the maxima repl successfully.

Configuration

I will be mainly interacting with maxima either through org-babel or the terminal directly. Therefore, I will not demonstrate how to install a GUI frontend for maxima. The interested user should find this guide helpful.

In order to run maxima in org-mode, we have to install Maxima.el package in Emacs. You may use the following code from the package repo in your config. I used use-package! because I am using doom emacs.

(use-package! maxima
  :init
  (setq org-format-latex-options (plist-put org-format-latex-options :scale 2.0)
	  maxima-display-maxima-buffer nil)
  (add-to-list 'auto-mode-alist
		 (cons "\\.mac\\'" 'maxima-mode))
  (add-to-list 'interpreter-mode-alist
		 (cons "maxima" 'maxima-mode)))

For maxima to be available to org-babel, the following configuration is necessary.

(org-babel-do-load-languages
 'org-babel-load-languages
 '((maxima . t)))

Now, we can use maxima in org-babel.

programmode: false;
tex(diff(exp(x)^2));

(linenum:0, $$2\,e^{2\,x}\,dx$$

External libraries

Maxima is already a feature rich CAS by itself. It can do differentiation, integration, solve ODE, and etc. It can be make more powerful by loading libraries written by others for more features.

Search Path Expansion

A simple way of including other's code is to include the file containing the code. Assuming the external library you would like to include is in file something.mac.

load("something.mac");

To include a larger library with multiple files, you could follow the instruction in Clifford's repo.

Firstly, clone the repo to your local machine.

cd ~/projects/
git clone git@github.com:dprodanov/clifford.git

Then, you would add the path of this repo to Maxima's search path.

file_search_demo: append (file_search_demo,["~/projects/clifford/$$$.{dem.wxm}"]);

file_search_maxima: append (file_search_maxima,["~/projects/clifford/$$$.{mac,wxm}"]);

setup_autoload ("clifford.mac", clifford);

load(clifford);
tex(clifford(dx,3));

(linenum:0, define: warning: redefining the built-in function clifford package name: clifford.mac author: Dimiter Prodanov version: v25 Recommended location: share/contrib last update: 9 May 2023 $$\left[ 1 , 1 , 1 \right] $$

Interested reader should try these examples given in the Clifford package repo.

Alternative Method

An alternative would be to directly add the files to the default place where maxima looks for code. This is at .maxima/ directory. Using the install.sh script with qinf package, you can directly move the necessary files into the desired directory.

Non-commutative Algebra

Now that we have things all setup, we could move on to the real business: having maxima carry out the tedious operator algebra for us.

Problem Statement

In condensed matter physics, canonical transformations like Holstein-Primakoff transformation are used to transform into a picture where quasi-particles emerge and fascilate the better understanding of certain phenomenon.

In the Holstein-Primakoff transformation, we define the bosonic creation and anhilation operator of magnons at site $i$ as $\hat{a}^{\dagger}_{i}$ and $\hat{a}_{i}$ respectively. The raising and lowering quantum spin operator are defined as $\hat{S_{j}}^{+} = \sqrt{2S_{j} - \hat{n_{j}}} \hat{a_{j}}$ and $\hat{S_{j}}^{-} = \hat{a_{j}}^{\dagger} \sqrt{2S_{j} - \hat{n_{j}}} $. $S_j$ is a number representing the maximum spin at site $j$. $\hat{n}_{j}$ is the number operator of magnon at site $j$. We would like to verify the commutation relation $[\hat{S_{j}}^{+},\hat{S_{j}}^{-}] = 2 \hat{S_{j}}^{z} \delta_{ij}$ where $S_{j} - \hat{n_{j}}$

Code

It does not appear to be easily doable in maxima. I need to postpone this.

Outlooks

There is a blog written in 2015 that compares the result of some real world usage of Maxima vs Mathematica. The conclusion is that Maxima will return either incorrect or no result at all for some problems.

Let's verify some of the problems still exist:

programmode: false;
tex(integrate(1/(x^2 - x^3)^(1/3),x,0,1));

$$\int_{0}^{1}{{{1}\over{\left(x^2-x^3\right)^{{{1}\over{3}}}}}\;dx}$$

It seems like at least this problem still remains. Maxima is unable to carry out this integration.