Doesn't the Many Worlds Interpretation violate Lorentz symmetry when the universe splits?
Yes, but so do collapse interpretations when the wave function collapses everywhere all at the same time. No matter what interpretation you use, "ordinary" quantum mechanics, the stuff you study in the first year or so of college QM, is non-relativistic and doesn't even pretend to be Lorentz invariant. You won't find a proper relativistic version of QM until you step up to quantum field theory.
If this is so.. why do we keep discussing about non-relativistic version of Copenhagen or many worlds and whether there is collapse.. why didn't the community go directly to quantum field theory and the many worlds and Copenhagen version of QFT?
For the same reasons that we still study and use Newtonian mechanics instead of going directly to relativistic mechanics:
1) There is no "go directly"; you can't learn the relativistic theory until you understand and can use the simpler non-relativistic theory.
2) There are many important problems in which the relativistic effects are negligible. Using the relativistic theory for these adds enormous amounts of mathematical complexity and obscures the underlying physics without producing better answers or additional insight into the underlying physics.
Splitting is a result of continuous unitary evolution, so splitting as such does not necessarily violate Lorentz symmetry unless unitary evolution also violates it.
Why do you say that? The Dirac and Klein-Gordon equations do not require field theory. Neither does S-matrix theory. Or Wigner's prescription for a Lorentz transformation. I understand that there are plenty of good reasons for quantum field theory, but I don't see how Lorentz invariance requires it.
The Dirac and KG equations don't require QFT, true, BUT (and this is a big BUT) interpreting the wave function in these equations in terms of Copenhagen interpretation (and its Born rule) of non specially relativistic quantum mechanics is wrong (leads to the possibility of negative probabilities). One needs therefore an interpretation of specially relativistic quantum mechanics. The currently accepted one is actually a reformulation in terms of quantum fields and their regrettable mathematical problems.
Hmm, Dirac leads to negative energy solutions and KG to negative probabilities with positive energies. Don't both lead to antiparticles?
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