Undergrad Why are there no probability amplitudes in MWI?

[entropy1]

If I consider the MWI, one of the notions for what happens during measurement is that the initial wavefunction, if I use Dirac notation and two dimensions, $|A\rangle+|B\rangle$ undergoes the transformation $(|A\rangle+|B\rangle)|E_{before}\rangle \rightarrow |A\rangle|E_{after}\rangle+|B\rangle|E_{after}\rangle$. So would it then also be correct that we can include the probability amplitudes, to get for instance this?: $(a|A\rangle+b|B\rangle)|E_{before}\rangle \rightarrow a|A\rangle|E_{after}\rangle+b|B\rangle|E_{after}\rangle$? It seems to me the latter is not correct, because in MWI there are no amplitudes used. Why is that?

 

[PeterDonis]

in MWI there are no amplitudes used

Yes, they are. MWI uses the same math as all QM interpretations, and that math has amplitudes.

 

[entropy1]

Yes, they are. MWI uses the same math as all QM interpretations, and that math has amplitudes.

Do my example formulas in #1 agree for the large part with the math you are referring to, or are they far off?

 

[PeterDonis]

Do my example formulas in #1 agree for the large part with the math you are referring to

I'm not sure what you are trying to describe with the math in the OP. But the basic math of QM is described in any QM textbook.

 

[PeterDonis]

$|A\rangle+|B\rangle$ undergoes the transformation $(|A\rangle+|B\rangle)|E_{before}\rangle \rightarrow |A\rangle|E_{after}\rangle+|B\rangle|E_{after}\rangle$.

Did you intend the two $|E_{after}\rangle$'s to be different? If they're the same, what you're writing does not represent any interaction at all.

 

[Vanadium 50]

Why are there no probability amplitudes in MWI?

I notice that you often start threads with a statement that isn't true. I can't believe this is the most effective way of learning. As Peter said, MWI uses the same math as all QM interpretations.

 

[entropy1]

Did you intend the two $|E_{after}\rangle$'s to be different? If they're the same, what you're writing does not represent any interaction at all.

Yes, sorry, that is an error. It should be _afterA and _afterB, or _measuredA and _measuredB. Is the formula correct if these errors are corrected?

I guess I have to start with that textbook or go study something. Thanks so far.

 

[PeterDonis]

Is the formula correct if these errors are corrected?

If there is an appropriate interaction Hamiltonian between the two systems (the thing whose basis states are $A$ and $B$, and the thing whose basis states are the $E$ states), yes. And this is true regardless of what complex amplitudes you put in front of the $A$ and $B$ terms: they just get carried through the process since the process is unitary.