The discussion centers on the necessity of the imaginary number i in the context of Pauli matrices. The Pauli matrices are essential for representing spin components in three-dimensional space and apply to two-component spinors, which are inherently complex-valued. The requirement for mutually orthogonal eigenvectors further necessitates the inclusion of the imaginary unit. Attempts to construct three 2x2 real matrices that satisfy these conditions demonstrate the impossibility of doing so without the imaginary component.
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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.Jonathan freeman said: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?