Maths - Pauli matrices - 4D

The basis vectors can be represented by matrices, this algebra was worked out independently by Pauli for his work on quantum mechanics. Murray Gell-Mann defined an extention of Pauli matricies to 3x3 matricies:

e1=
0 1 0
1 0 0
0 0 0
e2=
0 -i 0
i 0 0
0 0 0
e3=
1 0 0
0 -1 0
0 0 0
e4=
0 0 1
0 0 0
1 0 0
e5=
0 0 -i
0 0 0
i 0 0
e6=
0 0 0
0 0 1
0 1 0
e7=
0 0 0
0 0 -i
0 i 0
e8= 1/√ 3 *
1 0 0
0 1 0
0 0 -2

The scalar would be the identity matrix.

1=
1 0 0
0 1 0
0 0 1

 

The structure constant is antisymmetric in the three indicies and has values:

f123 = 2

f147 = f165 = f246 = f257 = f345 = f376 = 1

f458 = f678 =√ 3

The bivectors can be calculated by multiplying the matrices:

e1e2=
0 1
1 0
*
0 -i
i 0
=
i 0
0 -i
e1e3=
0 1
1 0
*
1 0
0 -1
=
0 -1
1 0
e2e3=
0 -i
i 0
*
1 0
0 -1
=
0 i
i 0

The tri-vectors are:

e1e2e3=
i 0
0 -i
*
1 0
0 -1
=
i 0
0 i

 

So the complete geometric multiplication table is:

a*b
b.e b.e1 b.e2 b.e3 b.e12 b.e31 b.e23 b.e123
a.e
1 0
0 1
0 1
1 0
0 -i
i 0
1 0
0 -1
e12 e31 e23 e123
a.e1
0 1
1 0
1 0
0 1
e12 -e31
0 -i
i 0
-1 0
0 1
e123 e23
a.e2
0 -i
i 0
-e12
1 0
0 1
e23
0 -1
-1 0
e123
1 0
0 -1
e31
a.e3
1 0
0 -1
e31 -e23
1 0
0 1
e123
0 1
1 0
0 i
-i 0
e12
a.e12 e12
0 i
-i 0
0 1
1 0
e123
-1 0
0 -1
e23 -e31
-1 0
0 1
a.e31 e31
1 0
0 -1
e123
0 -1
-1 0
-e23
-1 0
0 -1
e12
0 i
-i 0
a.e23 e23 e123
-1 0
0 1
0 -i
i 0
e31 -e12
-1 0
0 -1
0 -1
-1 0
a.e123 e123 e23 e31 e12
-1 0
0 1
0 i
-i 0
0 -1
-1 0
-1 0
0 -1

which is equivalent to the table derived here.


Further Reading

Other uses of Pauli Matrix:

Related Concepts:


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see also:

 

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics). This book is intended for mathematicians and physicists rather than programmers, it is very theoretical. It covers the algebra and calculus of multivectors of any dimension and is not specific to 3D modelling.

 

Terminology and Notation

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