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:

V = V_\mathbb{R} \oplus i V_\mathbb{R}.

Here V_R 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 V_R 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 V_R:

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 V_R and $i$ V_R, respectively. Typically, the operator c itself, rather than the eigenspace decomposition it entails, is referred to as the reality structure on V.

References

Penrose, Roger; Rindler, Wolfgang (1986), Spinors and space-time. Vol. 2, Cambridge Monographs on Mathematical Physics, Cambridge University Press, ISBN 978-0-521-25267-6, MR 0838301

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Reality structure

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References