# The Field $\mathbb{F}$ of A Hilbert Space [duplicate]

Is it always necessary for the field of some arbitrary Hilbert space I define to describe a system be a field of complex numbers only? Is it possible to have a field of naturals, or reals?

Since the vectors in the Hilbert space can be whatever I want (if I follow the two-three rules), I was wondering if the definition is agnostic to the elements in the field as well.