How to implement support for non-commutative algebra?

[m93a]

I'm thinking about implementing a simple module for quantum mechanics similar to that of Sympy. The most important features would be:

1. kets – abstract vectors, they aren't arrays of numbers but instead symbols with specific multiplication rules
2. operators – abstraction of matrices, again instead of multi-dimensional arrays, they are just symbols, multiplying A * B * ket(n) is equivalent to applying operator B to the ket n and then applying operator A to the result
3. bras – duals of kets, multiplying bra(m) * ket(n) is equivalent to dot product of vectors

These three types of objects have to obey specific multiplication rules, and multiplication isn't commutative for them.

In Sympy these are implemented this way: each new object is defined by a class that extends the core Expression class. The classes redefine the base class's multiply method. This way they can define custom multiplication rules without modifying the core library.

If I understand it correctly, math.js doesn't have multiply method on expressions, but instead I can define a new non-commutative multiplication that extends the core Operator class and mark it as non-commutative. Is that right?

Or is there a better way to approach this?

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This is what the workflow could look like:

import { Operator } from "math/quantum"

let H = new Operator('H', { hermitian: true });
let E3 = H.eigenket(3);
let scope = { H, E3 }

math.evaluate('dagger(H * E3) * H * E3', scope)
// <E3| H⁺ H |E3> = 3 * 3 <E3|E3> = 9

EDIT: Maybe somewhat related to #467.
EDIT 2: Obviously I'll need symbols first #1732.

[josdejong]

The symbolic computation that mathjs has so far uses nodes like SymbolNode and OperatorNode, the same as generated by the parser when parsing an expression (math.parse('...')).

See code: https://github.com/josdejong/mathjs/tree/develop/src/expression/node
See docs: https://mathjs.org/docs/expressions/expression_trees.html

Is that what you mean?