Maths - Pauli matrices - 2D

The basis vectors can be represented by matrices, this algebra was worked out independently by Pauli for his work on quantum mechanics, we can define the following equivalents:

e1=

0 1 1 0

e2=

1 0 0 -1

The scalar would be the identity matrix.

1=

1 0 0 1

The bivector can be calculated by multiplying the matrices:

e1e2=

0 1 1 0

*

1 0 0 -1

=

0 -1 1 0

So the complete geometric multiplication table is:

a*b	b.e	b.e1	b.e2	b.e12
a.e	1 0 0 1	0 1 1 0	1 0 0 -1	0 -1 1 0
a.e1	0 1 1 0	1 0 0 1	0 -1 1 0	1 0 0 -1
a.e2	1 0 0 -1	0 1 -1 0	1 0 0 1	0 -1 -1 0
a.e12	0 -1 1 0	-1 0 0 1	0 1 1 0	-1 0 0 -1

which is equivilant to the table derived here.

 

Any multivector can be represented by a single matrix representing a linear sum of these basis, for instance,

4 3 3 -4

= 3*

0 1 1 0

+ 4*

1 0 0 -1

= 3e1+ 4e2

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Further Reading

Other uses of Pauli Matrix:

• Pauli Matrix for 3D Multivectors
• Pauli Matrix for 4D Multivectors

Related Concepts:

• Groups

EuclideanSpace

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Generating Geometric Algebras