# Reality constraint

What is the "definition" of a reality constraint and why is it called that way?

(I mean how it is used for example in quantum field theory and string theory)

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## 1 Answer

A reality constraint typically cuts a quantity with complex degrees of freedom down to the same number of real degrees of freedom.

An example of a reality constraint is to impose that a complex $n\times n$ matrix $M\in{\rm Mat_{n\times n}(\mathbb{C})}$ should be Hermitian $M^{\dagger}=M$.

A bit more abstractly, one could also call the conditions $M^{\dagger}=-M$ (anti-Hermiticity) and $M^{\dagger}M={\bf 1}_{n\times n}$ (unitarity) for reality conditions, because they also cut the degrees of freedom in half.

In quantum theory, one for instance imposes that observables are Hermitian operators and evolution is unitary.

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Thx Qmechanic that was about what i thought it would be and answered my question. The term reality is also a little bit misleading (for me) beacuse my first idea was that it makes the quantity REAL but that's not always the case. I thought it's a good idea to make some proper reference in the internet. It is used very often without telling what it really is (and there is no information about it online)! –  ungerade Nov 9 '11 at 12:00