I have been studying some Lie theory recently and I came across the idea of representing complex numbers using matrices, e.g. $$1= \begin{pmatrix} 1 & 0\\ 0 & 1\\ \end{pmatrix} , i= \begin{pmatrix} 0 & -1\\ 1 & 0\\ \end{pmatrix}. $$ Then everything involving complex numbers can be converted into a problem of 2 by 2 matrices. For example, a 2 by 2 spin matrix can be converted into a 4 by 4 real matrix in the following way: $$ \begin{pmatrix} a+id & -b-ic\\ b-ic & a-id\\ \end{pmatrix} \longrightarrow \begin{pmatrix} a & -d & -b & c\\ d & a & -c & -b\\ b & c & a & d\\ -c & b & -d & a\\ \end{pmatrix} $$ The determinant is preserved using the rule for block matrices: $$ \det \begin{pmatrix} A & B\\ C & D\\ \end{pmatrix} =\det(AD-BC), $$ and the eigenvalue problem follows in a similar manner.

For me it seems a legit formulation of QM without using complex numbers, but I was always told by my professor that QM requires complex numbers. Is there anything I am missing here or is my professor wrong?

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    $\begingroup$ Does this answer your question? QM without complex numbers $\endgroup$ Commented Dec 30, 2020 at 18:17
  • $\begingroup$ Not going to check your math, but if it works, then it is isomorphic to complex number arithmetic. So, IMO, your professor maybe is being a bit narrow-minded. $\endgroup$ Commented Dec 30, 2020 at 18:20
  • $\begingroup$ Your professor is wrong, the complex numbers are only a super short way to express things, they are isomorphic to R^2. I am not sure about using matrices but it seems plausible at first sight $\endgroup$
    – user65081
    Commented Dec 30, 2020 at 18:26
  • $\begingroup$ Thank you all for the help, I believe the wave function at a point can be represented as a 2 by 2 real matrix too. $\endgroup$
    – Jieyan Zhu
    Commented Dec 30, 2020 at 18:29
  • $\begingroup$ P.S., Benefit № 1 to using complex arithmetic: The notation provides a compact and elegant way to describe periodic motion (e.g., waves). Benefit № 2: It's an extension of real arithmetic, so the notation and the algebraic rules already are familiar to students who are just learning it. $\endgroup$ Commented Dec 30, 2020 at 18:29