The standard reality structure on the vector space is the decomposition
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:
In this case, the complex conjugate of a vector v is defined as follows:
Conversely, given an antilinear involution on a complex vector space V, it is possible to define a reality structure on V as follows. Let
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 VR, respectively. Typically, the operator c itself, rather than the eigenspace decomposition it entails, is referred to as the reality structure on V.
- 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 838301
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