Then entire universe would also stay the same if we replaced “i” for “-i” for each phasor (without any change to the multipliers).
It’s complicated by i appearing in the Schrodinger equation. Which is essentially a convention. The thing close to ‘replace i with -i in states’ that would produce a real symmetry is CPT.
real numbers are not enough to describe quantum physics, but complex numbers are in some sense too much
One thing that quotients out some of this info is the density matrix |ψ⟩⟨ψ|, corresponding to ¯¯¯v⊗v in the post. If H is a Hilbert space, operators H→H have a real structure given by the Hermitian adjoint; and density matrices by definition satisfy conditions including being self-adjoint (i.e. Hermitian, i.e. ρ†=ρ). This means density matrices ‘are real’ in the relevant sense, analogous to real numbers. (Operators representing POVM observables are also self-adjoint/Hermitian.) On the other hand, the derivative of unitary time evolution U’(0) is ‘entirely imaginary’, which is related to the ‘i’ in the Schrodinger equation being conventional.
Yes this is a good approximation.
It’s complicated by i appearing in the Schrodinger equation. Which is essentially a convention. The thing close to ‘replace i with -i in states’ that would produce a real symmetry is CPT.
One thing that quotients out some of this info is the density matrix |ψ⟩⟨ψ|, corresponding to ¯¯¯v⊗v in the post. If H is a Hilbert space, operators H→H have a real structure given by the Hermitian adjoint; and density matrices by definition satisfy conditions including being self-adjoint (i.e. Hermitian, i.e. ρ†=ρ). This means density matrices ‘are real’ in the relevant sense, analogous to real numbers. (Operators representing POVM observables are also self-adjoint/Hermitian.) On the other hand, the derivative of unitary time evolution U’(0) is ‘entirely imaginary’, which is related to the ‘i’ in the Schrodinger equation being conventional.