A Schrodinger equation/Madelung equation phase transformation

[PeterDonis]

to which I said I understand it may seem illogical as you would then be introducing two more real quantities.

You did mention in post #8 that it would introduce more fields. However, once again, a number of other posts you have made in this thread made me wonder exactly what you understood.

 

[binbagsss]

No, I said the opposite, that the amplitudes must be the same. Go read that post again.

i said, with that, as in with theta being the same, the amps must be the same. whereas i asked about differing theta. so basically if i did a polar decomposition using different amplitudes and phases for both B and B*, then I would end up with two more equations when varying the action. Ignoring the fact that that I would now have four independent variables, if I were to write $B=B_1 e^{i\phi}$ and $B*=B_2 e^{i\phi_2}$ , then, from them being complex conjugates of each other, I would obtain constraints, such as $B_1 \exp^{-i\theta}=B_2 \exp^{i\phi_2}$, and $B_1 \exp^{i\theta}=B_2 \exp^{-i\phi_2}$but because of the differing amplitudes/arguments, neither the amplitudes or arguments in the exponents need to be the same?

 

[PeterDonis]

Ignoring the fact that that I would now have four independent variables

In other words, ignoring the fact that in the case you were asking about, the Proca equation, you don't. You only have two. Or more precisely, two for each 4-vector component (since the Proca equation is for a spin-1 vector field instead of a spin-0 scalar field); but all that means is that you stick a 4-vector index $\mu$ on $B$.

You need to make up your mind what you are asking about. Are you asking about the actual Proca equation? Or are you asking about some other case, which AFAIK has no physical relevance and is not discussed in the literature, where you somehow have four independent variables instead of two? You said before that you were asking about the Proca equation. Are you now changing the discussion to be about something else?