Reality structure

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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:

V = V_\mathbb{R} \oplus i V_\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 \mathbb{C}^n is the decomposition

\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:

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

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

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

This map v \mapsto \overline v is an antilinear involution, i.e.

\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 v \mapsto c(v) on a complex vector space V, it is possible to define a reality structure on V as follows. Let

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

and define

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

Then

V = V_\mathbb{R} \oplus i V_\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 i VR, respectively. Typically, the operator c itself, rather than the eigenspace decomposition it entails, is referred to as the reality structure on V.

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