I don’t understand most of this, but my intuition is that real numbers are not enough to describe quantum physics, but complex numbers are in some sense too much—they include some extra information we don’t actually need (but we use them anyway because we do not have a convenient intermediate data structure).
Some complex numbers are used to represent physical values (called “phasors” in the article), other complex numbers are used as multipliers (called “scalars” in the article). Multipliers are used to multiply the physical values; they do not represent values, but operations on values.
The entire universe would stay the same if we shifted a phase of each phasor by a constant value (without any change to the multipliers). The phases of phasors are only meaningful relative to each other.
Then entire universe would also stay the same if we replaced “i” for “-i” for each phasor (without any change to the multipliers). It is only the absolute value of the difference of phases that matters physically, not the direction.
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.
I don’t understand most of this, but my intuition is that real numbers are not enough to describe quantum physics, but complex numbers are in some sense too much—they include some extra information we don’t actually need (but we use them anyway because we do not have a convenient intermediate data structure).
Some complex numbers are used to represent physical values (called “phasors” in the article), other complex numbers are used as multipliers (called “scalars” in the article). Multipliers are used to multiply the physical values; they do not represent values, but operations on values.
The entire universe would stay the same if we shifted a phase of each phasor by a constant value (without any change to the multipliers). The phases of phasors are only meaningful relative to each other.
Then entire universe would also stay the same if we replaced “i” for “-i” for each phasor (without any change to the multipliers). It is only the absolute value of the difference of phases that matters physically, not the direction.
...and some other thing that I didn’t understand.
Is this a good approximation?
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.