## Complex numbers $ℂ$

The complex numbers $ℂ$ is the set $ℂ= { a+bi | a,b∈ℝ }, with i2=-1,$ so that addition is given by $(a+bi) + (c+di) = (a+c) + (b+d) i ,$ and multiplication is given by $(a+bi) (c+di) = (ac-bd) + (ad+bc) i .$

Define the absolute value on $ℂ$ by $|x| : ℂ ⟶ ℝ≥0 z ⟼ |z| by |z| = a2 + b2$ if $z=a+bi$.

Define the distance on $ℂ$, $d:ℂ×ℂ →ℝ≥0 by d(x,y) =|y-x| .$

Let $ϵ\in {ℝ}_{>0}$. The $ϵ$-ball at $x$ is $ℬϵ(x) = { y∈ℂ | d(x,y) <ϵ }.$

Let $E$ be a subset of $ℂ$. The set $E$ is open if $E$ is a union of $ϵ$-balls.

Complex conjugation is the function $P‾ :ℂ→ℂ given by a+bi ‾ = a-bi .$

(a) The set $ℂ$ with the operations of addition, multiplication and open sets as in (1.1) is a topological field.
(b) If $f:ℂ\to ℂ$ is a function such that
(a) If $x,y\in ℂ$ then $f\left(x+y\right)=f\left(x\right)+f\left(y\right)$,
(b) If $x,y\in ℂ$ then $f\left(xy\right)=f\left(x\right)f\left(y\right)$, and
(c) If $x\in ℝ$ then $f\left(x\right)=x$,
then $f$ is either the identity function or complex conjugation.

## $ℂ$ as an $ℝ$-algebra

The complex numbers is the $ℝ$-algebra
 $ℂ=ℝ\text{-span}\left\{1,i\right\}=\left\{{x}_{0}+{x}_{1}i\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}{x}_{0},{x}_{1}\in ℝ\right\}$
with product and topology determined by
 ${i}^{2}=-1\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}ℂ\simeq {ℝ}^{2},\phantom{\rule{2em}{0ex}}\text{respectively.}$
The norm $N:ℂ\to {ℝ}_{\ge 0}$ and absolute value $|\phantom{x}|:ℂ\to {ℝ}_{\ge 0}$ are given by
 $N\left({x}_{0}+{x}_{1}i\right)={x}_{0}^{2}+{x}_{1}^{2}={|x|}^{2}$.
Then
 $N\left(xy\right)=N\left(x\right)N\left(y\right)$.
The conjugate $\stackrel{‾}{\phantom{P}}:ℂ\to ℂ$ is given by
 $\stackrel{‾}{{x}_{0}+{x}_{1}i}={x}_{0}-{x}_{1}i\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\stackrel{‾}{xy}=\stackrel{‾}{y}\stackrel{‾}{x}$.
The conjugate is the unique automorphism of the topological field $ℂ$ that is not the identity. Then
 $\begin{array}{cl}{\mathrm{GL}}_{1}\left(ℂ\right)& ={ℂ}^{×}=\left\{x\in ℂ\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}x\ne 0\right\}\\ \cup |& \\ {U}_{1}\left(ℂ\right)& =\left\{x\in ℂ\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}x{\stackrel{‾}{x}}^{t}=1\right\}=\left\{{x}_{0}+{x}_{1}i\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}{x}_{0}^{2}+{x}_{1}^{2}=1\right\}\end{array}$
and
 $\begin{array}{ccc}{ℂ}^{×}& \stackrel{\sim }{⟶}& {\left({ℝ}^{×}\right)}^{\circ }×{U}_{1}\left(ℂ\right)\\ x& ⟼& |x|\cdot \frac{x}{|x|}\end{array}$
Here ${\left({ℝ}^{×}\right)}^{\circ }={ℝ}_{>0}$ is the connected component of the identity in the Lie group ${\mathrm{GL}}_{1}\left(ℝ\right)={ℝ}^{×}$. This polar decomposition is an example of the Cartan decomposition $G=K\cdot \left(\mathrm{exp}𝔭\right)$ (see Segal Theorem 4.1 and/or Knapp Prop. 1.2), where $K$ is a maximal compact subgroup of $G$, and $𝔤=𝔨\oplus 𝔭$ with $𝔤=\mathrm{Lie}\left(G\right)$ and $𝔭$ orthogonal to $𝔨=\mathrm{Lie}\left(K\right)$ with respect to the Killing form.

## Notes and References

The reference [Ch. VIII § 1, BouTop] provides a thorough introduction to the complex numbers. Viewing $ℂ$ as an $ℝ$-algebra helps to make the construction of the Hamiltonians feel natural.

## References

[BouTop] N. Bourbaki, General Topology, Chapter VI, Springer-Verlag, Berlin 1989. MR?????