quantum mechanics - Matrix representing the quantity - why can some matrices not be physical quantity?

In Heisenberg picture, my textbook says that the following matrix

$A = \frac{5}{3}\Sigma_1 + i\frac{4}{3}\Sigma_2$ cannot represent physical quantity.

the book says this is because $\frac{5}{3}\Sigma_1$ and $\frac{4}{3}\Sigma_2$ cannot have definite values.

At here, $\Sigma_1$ and $\Sigma_2$ represent Pauli matrices.

I am curious why this is; how can $\frac{5}{3}\Sigma_1$ and $\frac{4}{3}\Sigma_2$ cannot have definite values?

Answer

I) Recall that the observables in quantum mechanics are given by self-adjoint operators (or matrices).

The matrix

$$A ~=~ \frac{5}{3}\Sigma_1 + i\frac{4}{3}\Sigma_2$$

is not self-adjoint

$$A^{\dagger} ~=~ \frac{5}{3}\Sigma_1 - i\frac{4}{3}\Sigma_2,$$

so $A$ is not an observable in the traditional sense.

II) However, there is a caveat to the above. If we e.g. have two mutually commuting self-adjoint operators $B\!=\!B^{\dagger}$ and $C\!=\!C^{\dagger}$, we can construct a so-called "complex observable"

$$A~:=~B+iC.$$

Then we could find a complete set of simultaneous eigenstates for $B$ and $C$. And then $A$ would have complex-valued eigenvalues on these eigenstates. (Technically $A$ is then known as a normal operator. This is the quantum version of what we know from classical geometry, that we can equivalently represent a point in the 2D plane as two real coordinates $(x,y)\in\mathbb{R}^2$, or as a single complex number $z=x+iy$.)

In our case we have $B=\frac{5}{3}\Sigma_1$ and $C=\frac{4}{3}\Sigma_2$. But the commutator $[B,C]\neq 0$ is not zero, so we can not interpret $A$ as a "complex observable".

III) Conclusion: $A$ is neither a "real" (i.e. self-adjoint) nor a "complex" (i.e. normal) physical observable.