On 11/22/2024 6:19 AM, PGC wrote:
These discussions around Bell's theorem, the Many-Worlds
Interpretation (MWI), and the challenges of deriving the Born rule
continue invoking the interplay between epistemic frameworks and
ontological commitments. A significant point of contention is whether
MWI can account for the correlations observed in entangled systems
without additional postulates, such as collapse, and how these
correlations map onto the observer accounts and global description
perspectives. There are interpretational gaps that persist.
John’s description of branching in the Many-Worlds Interpretation
(MWI) assumes that decoherence ensures each branch corresponds to a
distinct outcome of a quantum measurement. This can be expressed using
the density matrix ρ in a composite system-environment state:
ρ=∣ψ⟩⟨ψ∣,where ∣ψ⟩=i∑ci∣si⟩∣ei⟩.
Decoherence suppresses off-diagonal terms in ρ, effectively yielding a
mixed state:
ρ′=i∑∣ci∣2∣si⟩⟨si∣.
Consider the correlations in entangled systems that violate Bell's
inequality. These correlations are quantitatively expressed as
deviations from the CHSH inequality:
S=∣E(a,b)+E(a′,b)+E(a,b′)−E(a′,b′)∣≤2,
where E(a,b) represents the expectation value of measurements along
directions a and b. Experimental results consistently show that S>2,
as predicted by quantum mechanics but inconsistent with local hidden
variable theories (Bell, 1964, p.195). In MWI, these results follow
from the unitary evolution of the wavefunction. The wavefunction for
an entangled pair,
∣ψ⟩=21(∣↑⟩A∣↓⟩B−∣↓⟩A∣↑⟩B),
evolves unitarily under the Schrödinger equation. Decoherence ensures
that interference terms vanish in the density matrix describing
macroscopic observers, giving the appearance of distinct "branches."
However, Bruce keeps raising the critical challenge: how do these
branches remain correlated across spacelike separations? In MWI, the
correlations are not post-measurement artifacts but inherent to the
global wavefunction. The key is the consistency enforced by the
universal wf's structure, which ensures that for any measurement
basis, the resulting "branches" respect the original entanglement. The
reduced density matrix formalism explicitly demonstrates this:
ρA=TrB(∣ψ⟩⟨ψ∣),
yielding probabilities consistent with the Born rule. Yet, the Born
rule itself remains elusive within MWI's framework and demands further
clarification, as acknowledged by Carroll (2014, p.18).
Critics like Brent and Bruce argue that without an explicit derivation
of the Born rule, MWI fails to fully account for observed
probabilities. This is valid but reflects a broader epistemological
gap. Probabilities, as noted, have different interpretations:
frequentist, Bayesian, and, uniquely in computational contexts,
"objective" probabilities derived from "subjective probabilities"
(Everett used "subjective probabilities" iirc, and Bruno's refinement
was terming them "objective" in this sense). In this framework,
probabilities emerge not as axioms but as limits of frequency
operators over the ensemble of computations or histories:
Something akin to:
n→∞limn1i=1∑nPi≈PBorn,
where PBorn=∣⟨ψ∣ϕ⟩∣2. This connects subjective perspectives (what the
observer experiences) to 3p descriptions (what the formalism
predicts), which is insufficiently addressed/incomplete in MWI or
collapse approaches and open with Bruno's approach iirc (correct me,
if otherwise). The merit of this kind of approach is that observer
experience is no longer outside the scope of the clearest ontology.
Now, consider the Gödelian critique. All frameworks—whether MWI,
collapse postulates, or alternatives like Invariant Set Theory
(Palmer, 2009)—assume arithmetical or stronger foundations. Gödel's
incompleteness theorems (Gödel, 1931) demonstrate that within any
sufficiently rich formal system F, there exist true statements T that
are unprovable within F. Explicitly:
∃T(T∈True∧T∈/Provable in F).
Applied to quantum mechanics and ontology, this indicates that any
framework aiming for ontological finality will inevitably encounter
unprovable truths if it includes arithmetic
First, that's only relative to a fix set of axioms. Physics isn't an
axiomatic game. Second, the Goedel proposition that is true but
unprovable may be no more that "This is not provable." expressed within
the theory...not necessarily some deep truth.
or its use in its formulations. For example, the observer's role
versus the formalism's predictions remains a gap that cannot be fully
bridged within any single system.
"Observer" is a carryover from Bohr. A measurement can be performed by
a system that is diagonalized by decoherence. Decoherence isn't magic.
See the formulation of QM by Barandes that I posted.
Brent
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