Pauli Basis Matrices

Question

When I search for the Pauli basis matrices I find both the following sets but I wonder which one is the right one and why does the first set have an imaginary term which is absent in the second set.

First Set

Second Set

Answer 1

The first set of matrices are what is conventionally called the Pauli matrices. The identity matrix is sometimes included as a Pauli matrix $\sigma_0$. With this included, we have a correspondence between the two sets of matrices:

$S_a= \frac 1 {\sqrt 2} \sigma_0 \\S_b= \frac 1 {\sqrt 2} \sigma_3 \\S_c= \frac 1 {\sqrt 2} \sigma_1 \\S_d= \frac {-i} {\sqrt 2} \sigma_2$

Apart from the common factor of $\frac 1 {\sqrt 2}$, the only other difference is the factor of $-i$ which makes the elements of $S_d$ real. Why is this significant ? The Pauli matrices are both Hermitian i.e. $\sigma_n^\dagger = \sigma_n$ and unitary i.e. $\sigma_n^\dagger = \sigma_n^{-1}$. As a result they are involutory i.e. $\sigma_n^2=I$.

$\sqrt 2 S_d$ is unitary since

$(\sqrt 2 S_d)^\dagger = (\sqrt 2 S_d)^T = (\sqrt 2 S_d)^{-1}$

However $\sqrt 2 S_d$ is not Hermitian or involutory since

$(\sqrt 2 S_d)^\dagger = (\sqrt 2 S_d)^{-1} = -(\sqrt 2 S_d) \\ \Rightarrow (\sqrt 2 S_d)^2 = -I$

Since $(\sqrt 2 S_d)$ is not Hermitian and not involutory it cannot be a Pauli matrix.