Disclaimer: Nothing in this note is original.

Quantum mechanics deals almost exclusively with sections of complex vector fields, so it is important to understand the mathematical structure of it.

Consider the complex plane $\mathbb{C}$ with $z=x+iy$. $\mathbb{C}$ is a one-dimensional vector space in terms of complex dimension $\text{dim}_ {\mathbb{C}}$, which is also a two dimension manifold in terms of real dimension $\text{dim}_ {\mathbb{R}}$. A complex number $z$ can be written as a

\[z= (x,y)^{T}\]

where $T$ is the transpose. With this notation, the multiplication of $i$ is translated into a $2\times 2$ matrix,

\[iz = i(x+iy) = -y+ix,\quad i: \begin{pmatrix} x \\ y \end{pmatrix} \to \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}\]

It is conventional to denote the matrix by $J$, \(J = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \quad J^{2}=-\mathbb{1}_ {2\times 2}.\)

What we did before can be trivially generalized to $\mathbb{C}^{K}$, then

\[z = (z_ {1},\dots,z_ {K}) = (x_ {1}+iy_ {1},\dots,x_ {k}+iy_ {K})\to(x_ {1},y_ {1},\dots,x_ {K},y_ {K}) =: \mathbf{x} \in \mathbb{R}^{2K}\]

and

\[J = \text{diag}(J, \dots,J)\]

where we have abused the notation a little, it is understood that $J$ in the parenthesis are two-dimensional and $J$ on the LHS is $2K$ dimensional, all in the sense of real dimensions.

Note that, given the Euclidian metric, $J$ is an isometry,

\[\left\langle x,x' \right\rangle =\left\langle Jx,Jx' \right\rangle .\]

We’ll show that, given a $2K$ dimensional real vector space $F^{2K}$, with an inner product $\left\langle -,- \right\rangle$, is there exist a matrix

\[J: F^{2K} \to F^{2K}\]

which is a linear isometry of $F$ which is also anti-involution, that is

\[J^{2}=-\mathbb{1},\]

then we can turn $F$ into a $K$-dimensional complex vector space.

Since $J^{2}=-\mathbb{1}$, the eigenvalues of $J$ is $\pm i$. Thus the determinant of $J$ is $i\times(-i)=1$, thus $J\in SO(2K)$. Then we know that $J$ can be put to $2\times 2$ block-diagonal form

\[\text{diag}(R(\theta_ {1}),\dots,R(\theta_ {K})), \quad R(\theta_ {i}) \text{ is }2\times 2 \text{ dimensional matrix.}\]

But $J$ is also anti-symmetric, since

\[\left\langle Jx,x' \right\rangle = \left\langle J^{2}x,Jx' \right\rangle = \left\langle -x,Jx' \right\rangle = \left\langle x,-Jx' \right\rangle = \left\langle x,J^{T}x' \right\rangle .\]

Then we can find a coordinate system in which $J$ is block-diagonalized by

\[\begin{pmatrix} 0&-1 \\ 1&0 \end{pmatrix}.\]

In this coordinate, write the coordinates be $(x_ {1},y_ {1},\dots,x_ {K},y_ {K})$, we can recombine them into

\[(z_ {1},\dots,z_ {K}),\quad z_ {i} = x_ {i} + i y_ {i}.\]

In particular, $\mathbb{R}^{2}$ with $J$ as earlier can be considered a complex 1-dimensional vector $\mathbb{C}^{1}=\mathbb{C}$, which can be called a complex line.