2 Fixed some sign errors.
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The rotation group, its representations, and their carrier spaces are fundamental parts of quantum mechanics. Every object in the universe is either a spin=0, 1/2, 1, 3/2, 2,… object. For the integer spin objects, the rotation group is O(3), and the rotation matrices contain only real numbers. However, there are half integer spin particles in the world, and matrices with complex numbers in them are needed to rotate them. The group that covers all rotations is SU(2) which has a 2 x 2 array of generators $J$. The 3 rotation angles must be encoded into the 2 x 2 array of lie group parameters $\Theta$. The group element (ie: the rotation matrix) is then $$ R=e^{\Theta^{mn} J^{nm}} $$ The special “S” in SU(2) means $det(R)=1$ which implies $Trace(\Theta)=0$. R is unitary which makes $(\Theta^{nm})^* + \Theta^{mn} = 0$. If the elements in $\Theta$ are real, then $\Theta$ is antisymmetric. So $$ \Theta = \begin{bmatrix} a & b \\ -b & -a \end{bmatrix} $$ Notice there is no way to stuff a third angle c into $\Theta$ without using $ i $. Then using $i$ the 3 rotations can be put into $\Theta$. $$ \Theta=\begin{bmatrix} i\theta_z & \theta_x + i\theta_y \\ \theta_x - i\theta_y & - i\theta_z \end{bmatrix} $$$$ \Theta=(i/2)\begin{bmatrix} -\theta_z & \theta_x + i\theta_y \\ \theta_x - i\theta_y & \theta_z \end{bmatrix} $$

Therefore, a reason complex numbers are needed in quantum mechanics is because there exists half-integer spin particles.

The rotation group, its representations, and their carrier spaces are fundamental parts of quantum mechanics. Every object in the universe is either a spin=0, 1/2, 1, 3/2, 2,… object. For the integer spin objects, the rotation group is O(3), and the rotation matrices contain only real numbers. However, there are half integer spin particles in the world, and matrices with complex numbers in them are needed to rotate them. The group that covers all rotations is SU(2) which has a 2 x 2 array of generators $J$. The 3 rotation angles must be encoded into the 2 x 2 array of lie group parameters $\Theta$. The group element (ie: the rotation matrix) is then $$ R=e^{\Theta^{mn} J^{nm}} $$ The special “S” in SU(2) means $det(R)=1$ which implies $Trace(\Theta)=0$. R is unitary which makes $(\Theta^{nm})^* + \Theta^{mn} = 0$. If the elements in $\Theta$ are real, then $\Theta$ is antisymmetric. So $$ \Theta = \begin{bmatrix} a & b \\ -b & -a \end{bmatrix} $$ Notice there is no way to stuff a third angle c into $\Theta$ without using $ i $. Then using $i$ the 3 rotations can be put into $\Theta$. $$ \Theta=\begin{bmatrix} i\theta_z & \theta_x + i\theta_y \\ \theta_x - i\theta_y & - i\theta_z \end{bmatrix} $$

Therefore, a reason complex numbers are needed in quantum mechanics is because there exists half-integer spin particles.

The rotation group, its representations, and their carrier spaces are fundamental parts of quantum mechanics. Every object in the universe is either a spin=0, 1/2, 1, 3/2, 2,… object. For the integer spin objects, the rotation group is O(3), and the rotation matrices contain only real numbers. However, there are half integer spin particles in the world, and matrices with complex numbers in them are needed to rotate them. The group that covers all rotations is SU(2) which has a 2 x 2 array of generators $J$. The 3 rotation angles must be encoded into the 2 x 2 array of lie group parameters $\Theta$. The group element (ie: the rotation matrix) is then $$ R=e^{\Theta^{mn} J^{nm}} $$ The special “S” in SU(2) means $det(R)=1$ which implies $Trace(\Theta)=0$. R is unitary which makes $(\Theta^{nm})^* + \Theta^{mn} = 0$. If the elements in $\Theta$ are real, then $\Theta$ is antisymmetric. So $$ \Theta = \begin{bmatrix} a & b \\ -b & -a \end{bmatrix} $$ Notice there is no way to stuff a third angle c into $\Theta$ without using $ i $. Then using $i$ the 3 rotations can be put into $\Theta$. $$ \Theta=(i/2)\begin{bmatrix} -\theta_z & \theta_x + i\theta_y \\ \theta_x - i\theta_y & \theta_z \end{bmatrix} $$

Therefore, a reason complex numbers are needed in quantum mechanics is because there exists half-integer spin particles.

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The rotation group, its representations, and their carrier spaces are fundamental parts of quantum mechanics. Every object in the universe is either a spin=0, 1/2, 1, 3/2, 2,… object. For the integer spin objects, the rotation group is O(3), and the rotation matrices contain only real numbers. However, there are half integer spin particles in the world, and matrices with complex numbers in them are needed to rotate them. The group that covers all rotations is SU(2) which has a 2 x 2 array of generators $J$. The 3 rotation angles must be encoded into the 2 x 2 array of lie group parameters $\Theta$. The group element (ie: the rotation matrix) is then $$ R=e^{\Theta^{mn} J^{nm}} $$ The special “S” in SU(2) means $det(R)=1$ which implies $Trace(\Theta)=0$. R is unitary which makes $(\Theta^{nm})^* + \Theta^{mn} = 0$. If the elements in $\Theta$ are real, then $\Theta$ is antisymmetric. So $$ \Theta = \begin{bmatrix} a & b \\ -b & -a \end{bmatrix} $$ Notice there is no way to stuff a third angle c into $\Theta$ without using $ i $. Then using $i$ the 3 rotations can be put into $\Theta$. $$ \Theta=\begin{bmatrix} i\theta_z & \theta_x + i\theta_y \\ \theta_x - i\theta_y & - i\theta_z \end{bmatrix} $$

Therefore, a reason complex numbers are needed in quantum mechanics is because there exists half-integer spin particles.