Reality structure

In mathematics, a reality structure on a complex vector space V is a decomposition of V into two real subspaces, called the real and imaginary parts of V:

${\displaystyle V=V_{\mathbb {R} }\oplus iV_{\mathbb {R} }.}$

Here VR is a real subspace of V, i.e. a subspace of V considered as a vector space over the real numbers. If V has complex dimension n (real dimension 2n), then VR must have real dimension n.

The standard reality structure on the vector space ${\displaystyle \mathbb {C} ^{n}}$ is the decomposition

${\displaystyle \mathbb {C} ^{n}=\mathbb {R} ^{n}\oplus i\,\mathbb {R} ^{n}.}$

In the presence of a reality structure, every vector in V has a real part and an imaginary part, each of which is a vector in VR:

${\displaystyle v=\operatorname {Re} \{v\}+i\,\operatorname {Im} \{v\}}$

In this case, the complex conjugate of a vector v is defined as follows:

${\displaystyle {\overline {v}}=\operatorname {Re} \{v\}-i\,\operatorname {Im} \{v\}}$

This map ${\displaystyle v\mapsto {\overline {v}}}$ is an antilinear involution, i.e.

${\displaystyle {\overline {\overline {v}}}=v,\quad {\overline {v+w}}={\overline {v}}+{\overline {w}},\quad {\text{and}}\quad {\overline {\alpha v}}={\overline {\alpha }}\,{\overline {v}}.}$

Conversely, given an antilinear involution ${\displaystyle v\mapsto c(v)}$ on a complex vector space V, it is possible to define a reality structure on V as follows. Let

${\displaystyle \operatorname {Re} \{v\}={\frac {1}{2}}\left(v+c(v)\right),}$

and define

${\displaystyle V_{\mathbb {R} }=\left\{\operatorname {Re} \{v\}\mid v\in V\right\}.}$

Then

${\displaystyle V=V_{\mathbb {R} }\oplus iV_{\mathbb {R} }.}$

This is actually the decomposition of V as the eigenspaces of the real linear operator c. The eigenvalues of c are +1 and −1, with eigenspaces VR and ${\displaystyle i}$ VR, respectively. Typically, the operator c itself, rather than the eigenspace decomposition it entails, is referred to as the reality structure on V.