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 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. Collapse postulates introduce "magic" by assuming the
wavefunction's reality only to dismiss it post-measurement, while MWI
faces the unresolved challenge of deriving probabilities without
external axioms.
The whack-a-mole nature of these discussions therefore may find an
explanation in this incompleteness. Every attempt to resolve one gap
(e.g., deriving Born within MWI) introduces others (e.g., defining the
observer). As Saibal notes, local hidden variables fail due to Bell's
theorem, but Bruce counters that this implies non-locality within
standard QM. Both points reflect the limits of purely formal reasoning
without acknowledging the epistemic/ontological split.
In conclusion, these discussions risk circularity if participants
prioritize defending their preferred interpretations over
collaborative inquiry. Recognizing the limitations imposed by Gödelian
constraints and the potential irreducibility of observer perspectives
relative to global descriptions is essential. While frameworks like
MWI or collapse postulates have epistemic value, they are better seen
as tools for exploring the boundaries of what can be explained or
inspiration for developing new problems and possible application,
rather than as definitive ontological inquiry. The quest for consensus
may remain elusive, but acknowledging these limits instead of giving
in to the whack-a-mole discourse may mitigate circularity risk. Work
has to be done from all sides. Have a great weekend, whether collapse
or in some world, or while riding computations.
On Friday, November 22, 2024 at 1:59:10 PM UTC+1 John Clark wrote:
On Thu, Nov 21, 2024 at 6:01 PM Bruce Kellett <[email protected]>
wrote:
*>> The spin of 2 electrons has been quantummechanically
entangled. One electron is given to Alice and the other to
Bob. Alice and her electron stay on earth but Bob takes
his electron and gets in a near light speed spaceship and
after 4 years is on Alpha Centauri. And after 4 years
Alice picks a direction at random, calls that "up" and
measures the spin of her electron in that direction with a
Stern Gerlach magnet.*
*At that instant the universe splits into two, in one
Alicehas the spin up electron and Bob has spin down, and
in the other universe Alice has spin down and Bob has spin
up.*
/
/
/> Bob is at a spacelike separation, and does not know either
the angle of Alice's measurement, or her result. /
*And that's why the resulting correlation is so weird, not
paradoxical but definitelyvery weird. *
> /This 4-way split, two branches for Alice and two for Bob/[...]
*That is incorrect. There is only a two-way split:
1) Alice sees up and Bob sees down.
2) Alice sees down and Bob sees up.
There is no universe in which both electrons are spin-up, and
there is no universe in which both electrons are spin-down. This
is because the laws of physics (a.k.a. Schrodinger's Quantum Wave)
forbids it. As soon as Alice measures her electron and sees what
her spin is she knows for certain that she will be in the same
universe where Bob sees that his electron has the opposite spin.
And a similar statement could be said about Bob and his electron. *
/> How does that happen, exactly? /
*Are you sure you reallywant to know _EXACTLY_?The short answer is
it happens because of the [COS (x)]^2 polarization rule, but you
said you wanted all the details about how that apparently innocent
sounding rule could lead to a violation of Bell's inequality and
put philosophers in a panic. I'm not sure you really want all the
details but about two weeks ago somebody else asked the same
question you did and I went into much more detail. I'm not going
to rephrase what I wrote then I'm just gonna repeat it because I
don't think anybody actually read it the first time:*
*== *
*If you want all the details this is going to be a long post, you
asked for it. First I'm gonna have to show that any theory (except
for super determinism which is idiotic) that is deterministic,
local and realistic cannot possibly explain the violation of
Bell's Inequality that we see in our experiments, and then show
why a theory like Many Worlds which is deterministic and local but
NOT realistic can.
*
*
*
*The hidden variable concept was Einstein's idea, he thought there
was a local reason all events happened, even quantum mechanical
events, but we just can't see what they are. It was a reasonable
guess at the time but today experiments have shown that Einstein
was wrong, to do that I'm gonna illustrate some of the details of
Bell's inequality with an example.*
*
When a photon of undetermined polarization hits a polarizing
filter there is a 50% chance it will make it through. For many
years physicists like Einstein who disliked the idea that God
played dice with the universe figured there must be a hidden
variable inside the photon that told it what to do. By "hidden
variable" they meant something different about that particular
photon that we just don't know about. They meant something
equivalent to a look-up table inside the photon that for one
reason or another we are unable to access but the photon can when
it wants to know if it should go through a filter or be stopped by
one. We now understand that is impossible. In 1964 (but not
published until 1967) John Bell showed that correlations that work
by hidden variables must be less than or equal to a certain value,
this is called Bell's Inequality. In experiment it was found that
some correlations are actually greater than that value. Quantum
Mechanics can explain this, classical physics or even classical
logic can not.
Even if Quantum Mechanics is someday proven to be untrue Bell's
argument is still valid, in fact his original paper had no Quantum
Mechanics in it and can be derived with high school algebra; his
point was that any successful theory about how the world works
must explain why his inequality is violated, and today we know for
a fact from experiments that it is indeed violated. Nature just
refuses to be sensible and doesn't work the way you'd think it
should.
I have a black box, it has a red light and a blue light on it, it
also has a rotary switch with 6 connections at the 12,2,4,6,8 and
10 o'clock positions. The red and blue light blink in a manner
that passes all known tests for being completely random, this is
true regardless of what position the rotary switch is in. Such a
box could be made and still be completely deterministic by just
pre-computing 6 different random sequences and recording them as a
look-up table in the box. Now the box would know which light to flash.
I have another black box. When both boxes have the same setting on
their rotary switch they both produce the same random sequence of
light flashes. This would also be easy to reproduce in a classical
physics world, just record the same 6 random sequences in both boxes.
The set of boxes has another property, if the switches on the 2
boxes are set to opposite positions, 12 and 6 o'clock for example,
there is a total negative correlation; when one flashes red the
other box flashes blue and when one box flashes blue the other
flashes red. This just makes it all the easier to make the boxes
because now you only need to pre-calculate 3 random sequences,
then just change every 1 to 0 and every 0 to 1 to get the other 3
sequences and record all 6 in both boxes.
The boxes have one more feature that makes things very
interesting, if the rotary switch on a box is one notch different
from the setting on the other box then the sequence of light
flashes will on average be different 1 time in 4. How on Earth
could I make the boxes behave like that? Well, I could change on
average one entry in 4 of the 12 o'clock look-up table (hidden
variable) sequence and make that the 2 o'clock table. Then change
1 in 4 of the 2 o'clock and make that the 4 o'clock, and change 1
in 4 of the 4 o'clock and make that the 6 o'clock. So now the
light flashes on the box set at 2 o'clock is different from the
box set at 12 o'clock on average by 1 flash in 4. The box set at 4
o'clock differs from the one set at 12 by 2 flashes in 4, and the
one set at 6 differs from the one set at 12 by 3 flashes in 4.
_BUT_ I said before that boxes with opposite settings should have
a 100% anti-correlation, the flashes on the box set at 12 o'clock
should differ from the box set at 6 o'clock by 4 flashes in 4 NOT
3 flashes in 4. Thus if the boxes work by hidden variables then
when one is set to 12 o'clock and the other to 2 there MUST be a
2/3 correlation, at 4 a 1/3 correlation, and of course at 6 no
correlation at all. A correlation greater than 2/3, such as 3/4,
for adjacent settings produces paradoxes, at least it would if you
expected everything to work mechanistically because of some local
hidden variable involved.
Does this mean it's impossible to make two boxes that have those
specifications? Nope, but it does mean hidden variables can not be
involved and that means something very weird is going on. Actually
it would be quite easy to make a couple of boxes that behave like
that, it's just not easy to understand how that could be.
*
*Photons behave in just this spooky manner, so to make the boxes
all you need it 4 things:
1) A glorified light bulb, something that will make two photons of
unspecified but identical polarizations moving in opposite
directions so you can send one to each box. An excited calcium
atom would do the trick, or you could turn a green photon into two
identical lower energy red photons with a crystal of potassium
dihydrogen phosphate.
2) A light detector sensitive enough to observe just one photon.
Incidentally the human eye is not quite good enough to do that but
frogs can, for frogs when light gets very weak it must stop
getting dimmer and appears to flash instead.
3) A polarizing filter, we've had these for well over a century.
4) Some gears and pulleys so that each time the rotary switch is
advanced one position the filter is advanced by 30 degrees. This
is because it's been known for many years that the amount of light
polarized at 0 degrees that will make it through a polarizing
filter set at X is [COS (x)]^2; and if X = 30 DEGREES (π/6
radians) then the value is .75; if the light is so dim that only
one photon is sent at a time then that translates to the
probability that any individual photon will make it through the
filter is 75%.
The bottom line of all this is that there can not be something
special about a specific photon, some internal difference, some
hidden local variable that determines if it makes it through a
filter or not. Thus if we ignore a superdeterministic conspiracy,
as we should, then one of two things MUST be true:
1) The universe is not realistic, that is, things do NOT exist in
one and only one state both before and after they are observed.
_In the case of Many Worlds it means the very look up table as
described in the above cannot be printed in indelible ink_ but,
because Many Worlds assumes that Schrodinger's Equation means what
it says, _the look up table itself not only can but must exist in
many different versions both before and after a measurement is made._
*
*
2) The universe is non-local, that is, everything influences
everything else and does so without regard for the distances
involved or amount of time involved or even if the events happen
in the past or the future; the future could influence the past.
But _because Many Worlds is non-realistic, and thus doesn't have a
static lookup table, it has no need to resort to any of these
non-local influences to explain experimental results._*
*
*
*Einstein liked non-locality even less than nondeterminism, I'm
not sure how he'd feel about non-realistic theories like Many
Worlds, the idea wasn't discovered until about 10 years after his
death.*
*John K Clark See what's on my new list at Extropolis
<https://groups.google.com/g/extropolis>*
7hn
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