When a quantum system interacts with an environment the combined system develops into an entangled state where system and environment become entangled through their interaction, resulting in the superposition of correlated states. This does not require the environment to physically duplicate. This naturally emerges from the unitary evolution governed by the Schrödinger equation.
The “branching” into multiple outcomes is a way to conceptualize the resulting entangled superposition, with each “branch” corresponding to a different outcome, with the system and environment in correlated states. Also, I thought the tensor product is bilinear (linear in each of its arguments). So for quantum states, the following would be a standard operation: (∣a⟩+∣b⟩)⊗∣e⟩=∣a⟩⊗∣e⟩+∣b⟩⊗∣e⟩ The distributive law applies due to properties of tensor products. Yes, we have nonlinear functions but linear operators. There’s potential for confusion here, as the Hamiltonian includes terms that are nonlinear functions of position and momentum. Iirc these functions are however constructed from linear operators and, when properly defined, result in linear operators themselves. Linearity for me refers to the way operators act on quantum states within a Hilbert space. Obviously this is essential for the superposition principle. I’m a tourist here, but nonlinear operators seem exotic. Linearity of QM is essential for consistency of the theory/accurate predictions. On Thursday, October 3, 2024 at 9:16:41 PM UTC+2 John Clark wrote: > On Wed, Oct 2, 2024 at 7:03 PM Brent Meeker <[email protected]> wrote: > > *>> All Many Worlds says is that everything always obeys Schrodinger's >>> Wave Equation, it never collapses,* >> >> >> * > That's right. It never says where the Born rule comes from. * >> > > *1) Many worlds is the only quantum interpretation that even tries to > derive the Born Rule, the others just assume it's true. * > > *2) Gleason's theorem mathematically proves that in dimensions of 3 or > greater and if all probabilities are required to be non-negative, and add > up to exactly 1, then the only consistent way to assign probabilities > is the squared amplitudes of the wavefunction, provided you also insist > that any combination of two valid quantum states is also a valid quantum > state. * > > *So the real question Many Worlds needs to answer is not why is > probability the squared amplitudes of the wavefunction but rather why is > probability necessary at all given the fact that Schrodinger's wave > equation is 100% deterministic? The answer is because of self locating > uncertainty. In the instant after the split but before an observer has > registered the outcome of a measurement there is only one rational way to > apportion credence as to which branch of the wave function he is on and > that is the Born Rule.* > > *Sean Carroll and Charles Sebens go into much more detail here: * > > *Many Worlds, the Born Rule, and Self-Locating Uncertainty* > <https://arxiv.org/pdf/1405.7907> > > John K Clark See what's on my new list at Extropolis > <https://groups.google.com/g/extropolis> > slu > > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/77e4d1ac-acb0-4190-ad96-2e0b15db5d8an%40googlegroups.com.
