Prerequisites
If you are not familiar with this subject you may like to look at the following pages first:
Description
There are 3 types of 2 dimensional algebras described here:
- complex numbers
- doubles
- duals
Is there a general way that we can combine these in different combinations to give higher order variables.
For example, if we have a complex number whose elements are themselves complex numbers, the resulting entity will have four dimensions with the properties of a quaternion. If we then let the elements of this be a dual we get dual quaternions and so on.
How can we combine these entities in a general way so that we can generate many different algebras.
Using Cayley Tables
Since the flavor of these algebras is defined by their multiplication rules, can we combine these tables to give a table for a higher dimensional algebra?
The starting point is the following tables:
Complex:
| a*b | b.1 | b.i |
| a.1 | 1 | i |
| a.i | i | -1 |
Double:
| a*b | b.1 | b.e |
| a.1 | 1 | e |
| a.e | e | 1 |
Dual:
| a*b | b.1 | b.ε |
| a.1 | 1 | ε |
| a.ε | ε | 0 |
How can these tables be combined?
This question is investigated here, a s this explains, its not easy to do this because the individual tables don't define how the two algebras interact. It doesn't work just to put one table into each element of the other table.
Using Pauli Matrix
double - the number a + e b can be represented by the matrix:
| a | b |
| b | a |
dual - the number a + ε b can be represented by the matrix:
| a | b |
| 0 | a |
complex - the number a + i b can be represented by the matrix:
| a | -b |
| b | a |
So can we combine these matrices to get a pauli matrix for the higher order algebras?
Since we already know that a complex number whose elements are complex numbers is a quaternion, let us try to derive the result by combining two Pauli Matricies for complex numbers:
We could try letting:
| a = |
|
b= |
|
and substitute this into the original complex pauli matrix to give:
| A | -B | -C | D |
| B | A | -D | -C |
| C | -D | A | -B |
| D | C | B | A |
The usual Pauli representation of quaternions is:
| A + i B | D + i C |
| -D + i C | A - i B |
Is this equivalent?
