Imaginary number i in Pauli matrixes

[Jonathan freeman]

Is the imaginary number i "necessary" in the pauli matrices simply because of the condition of having 3 mutually orthogonal axi?
If space were two dimensional we wouldn't need the i imaginary number?

 

[PeroK]

Is the imaginary number i "necessary" in the pauli matrices simply because of the condition of having 3 mutually orthogonal axi?
If space were two dimensional we wouldn't need the i imaginary number?

The Pauli matrices are what they are because a) they represent spin components in 3D space (in some sense); b) they apply to two-component spinors, which are complex-valued; and, c) in effect, they must have mutually orthogonal eigenvectors.

You could try to find three 2x2 real matrices that meet these criteria, but it's not difficult to show that it's not possible.