Reality constraint

Question

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.)

Answer 1

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.