pi can't be irrational in the real world, consequences of that?

Raycyst

As the title says, the number pi can't be irrational in the real world, due to quantitisation of distance. All physical distances must be multiples of plank lengths.

An irrational number can't be expressed as a fraction.

Given that all diameters of circles must be expressed in integer multiples of plank lengths, and all circumferences must also be expressed in integer multiples of plank lengths that means that we can always form a fraction, of one integer over the second integer.

Therefore, pi can never be irrational in a world which has quantitasitation.


Does this have real effects?

It only affects real physical distances.
Does it affect the orbit of electrons in atoms?
Do I win a Nobley prize?


I have more issues. A thought experiment if I may be so bold.
If we have a circle with a diameter of x plank lengths, and we increase the diameter to x+1 plank lengths how will the area and circumference change?

The circumference can only move from one integer multiple of plank lengths to another, it cannot change by 3.14 plank lengths.

Pi must actually be three after all.


This problem is perhaps similar to there being no real pythagorous triangles in our world, because our space is not euclidian.

mikhail

Pi is pi. Real circles aren't Platonic ideals.

Raycyst

Surely the fact that no real circles are actually circles must mean something?

I'm aware that plank lengths are very small. I couldn't draw a circle with line thickness of one plank length.

I couldn't measure a real world diameter to within a million plank lengths I suspect, and so the point is pretty moot.


I'm asking from a theory point of view.
For example, if we calculate an orbit and it isn't an exact multiple of plank lengths what then?
Maybe the orbit will 'snap' to a circumference which does contain an integer number of plank lengths.

But no matter what the circle does the circumference is always exactly divisible by the diameter, when measured in plank lengths.


Does this prove our universe is simulated?
I genuinely suspect quantisation of distance is a big clue that our universe is simulated and exact precision is not possible.

We will never be able to produce simulations which extend to infinite numbers of decimal points either. We use rounding, and so does the universe.

Fourier

The existence of something like the Plank length is consequence of unproven theories such as Loop Quantum Gravity or String Theory.

Currently accepted physics (General Relativity + Quantum Field Theory) has no notion of a Planck length and in fact space is infinitely divisible in them.

Raycyst

thanks Fourier, I had posted the following before I saw your reply. I had no idea that modern theories had continuous space and no quantitisation.


I have new thoughts on this but they're very technical.

If all real physical circles are merely approximations of actual circles with a value of pi that's rational and not irrational as its supposed to be then it definitely follows that larger circles exhibit a value of pi which is closer to the real value.

In other words, pi can always be expressed as a fraction (as both numerator and denominator must be expressed in integer multiples of plank lengths), but larger circles have huge numerators and denominators in the fraction, and therefore the value of pi is much closer to the real value.



In other words, if the diameter is 1 plank length, then the circumference must be either 3 or 4 plank lengths
The value of pi is very inaccurate for small circles.


If your circles diameter is 100 plank lengths then the circumference can be 314 plank lengths.
The value of pi is much closer to the real value here but it is still rational as it will always be if lengths are integer multiples of plank lengths.



Here's the new insight. Probably pretty pointless.
If a circle is hundreds of light years across then the numerator is huge when expressed as an integer multiple of plank lengths. So is the deomninator.

Therefore, the vallue of pi would be very accurate, but still perfectly rational.

Therefore, pi would be accurate to a huge number of decimal places.
But how many decimal places?
More than the few billion humans have calculated?
or less?


I don't know the figures but if a circles diameter was 1x10^30 meters, then its diamter in plank lengths would probably be something like 1x10^60, and it's circumference would be something like 3.1416...x10^60


How many significant and non-recurring decimal places would the calculated value of pi contain in that circumstance?

More or less than the billion digits or so humans have calculated?

Fourier

Raycyst wrote: »

thanks Fourier, I had posted the following before I saw your reply. I had no idea that modern theories had continuous space and no quantitisation.

They do, in fact they require it. There is currently no evidence of quantisation of space or time and quite a bit against it.

Raycyst

Well, if Nick Bostrum is correct when he discusses the possibility of a simulated universe what then?

If the universe is a simulation on a computer system that would require quantitisation of both time and all other quantities wouldn't it?

If humans are simulating something in a computer, even something relatively simple like motion under gravity with three bodies, we cannot do so exactly.

We can get arbitrarily close to the real values by making the time period smaller and smaller. If the time period is very small the bodies don't move very far between each iteration and so accuracy is quite high.

I know calculus can be used when two variables depend on each other and they are constantly varying, but I don't think there's an equivilent for three variables, all depending on one another, and all constantly changing.


If there is no exact mathematical solution to motion of three bodies under gravity then how does the universe get the answer?
It may seem like a strange question but humans cannot get the real answer, we can only get as arbitraily close as we like, but we cannot solve the problem exactly.


If space is not quantitised then our universe is solving the problem exactly, to infinite numbers of decimal places, and not just for three bodies, but for millions of bodies.
I suspect our universe does use rounding at some level. Perhaps this could also be described as a form of error correction.

How does our universe calculate positions to infinite numbers of decimal places?

Fourier

Raycyst wrote: »

Well, if Nick Bostrum is correct when he discusses the possibility of a simulated universe what then?

If the universe is a simulation on a computer system that would require quantitisation of both time and all other quantities wouldn't it?

Who knows, you would have no idea what the theory of computation would be like in the "higher" universe in which ours is embedded, it could allow for continuous computation.

The main point however is that our universe shows no evidence of having a minimum distance.

I know calculus can be used when two variables depend on each other and they are constantly varying, but I don't think there's an equivilent for three variables, all depending on one another, and all constantly changing.

If there is no exact mathematical solution to motion of three bodies under gravity...

You can use calculus for three variables (you can use calculus for an infinite number of variables) and there is an exact mathematical solution to the three body problem, it's been proven to exist and a formula for it has been derived. It just takes a long time to compute.

I suspect our universe does use rounding at some level. Perhaps this could also be described as a form of error correction.

There is no reason to suspect this, as there is no evidence for it.

How does our universe calculate positions to infinite numbers of decimal places?

"Infinite number of decimal" places just means that the number is difficult to express in terms of powers of ten. However pi is just another real number, like 0.5 or 1, from an abstract perspective there is no difference.

Raycyst

Fourier wrote: »

...
...
"Infinite number of decimal" places just means that the number is difficult to express in terms of powers of ten. However pi is just another real number, like 0.5 or 1, from an abstract perspective there is no difference.

I like your entire reply but I'll focus on this for the moment.

I know exactly what you're saying. I was thinking of this very thing just today. If we choose pi as our number base then pi would simply be the number 1.

But surely the whole complexity of the number pi can't just be hand waved away by saying that pi is the number 1 if expressed in the number base pi.

The number base itself using pi would be very complex. What would 10 be in that base?

I know logs can be used to change number bases but I've forgotten how.

The decimal number 10 is probably around 3.1 by coincidence if expressed in base pi.
Well, 10/pi = approx 3.18.

Would the decimal number 10 be 3.18 if expressed in base pi?


Anyway, the reason I was thinking of this earlier was I was thinking what happens if you use 1/3 as your base. It's recurring in base 10.
Can it be used as a base?
It could be changed to 10/3 if needs be.


Anyway, the main point is that pi must be more complex than the number 1.

Morbert

Raycyst wrote: »

An irrational number can't be expressed as a fraction.

Yes it can. E.g pi = tau / 2

Given that all diameters of circles must be expressed in integer multiples of plank lengths

What makes you say this?

[edit] - I see that in a later post you accept that space doesn't have to be quantised.

Fourier

I'm not really speaking of using a certain base or not, but rather that ultimately they are all just members of the real numbers. Pi might be more complex than the number "1", but it depends on what you mean by complex.

Raycyst

Well, in base 10, pi has an infinite decimal expansion which never repeats and which is random. Although it contains an infinite number of digits they can't be predicted.

That's more complex than the number 1. If mathematicans tell me that isn't significant I'd accept that although I'd be a bit confused.

I accept that both pi and 1 are numbers.

I did know that pi is a 'transcendental' number but I have no hope of remembering what that means, or of properly understanding the description.

In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are π and e.

I know pi is merely the circumference of a circle divided by its diameter but how and why does a circle, which seems so simple, contain such complexity?



I am very interested in this stuff. I really like that maths transcends physical universes and is above physical universes.
Is there a Platonic universe containing maths objects?
Would maths exist even if no physical universes did?
Is maths so powerful that it causes the universe?


Earlier, Fourier said that theories of computation may be different in different universes. I don't think that's the case. Maths transcends physical universes, and computer science, (and theories of computation), are all branches of maths and so they also transcend physical universes.
Maths proofs never contain any physical facts and so they don't depend on physical universes. A proof can never be disproven simply because physical facts change.


I accept that in the case of a simulation that the simulation can be paused and arbitrarily high precision can be achieved but not infinite precision, unless the simulation runs for a very long time.

Incredibly, I think it was David Deutsch who wrote in his book that in a contracting universe it'd be possible to build computers which allow for simulations to be run which could have a infinite amount of subjective experience in them. You could implement a simulation of heaven in such a computer.

He does admit that you'd have to constantly research and rebuild your computer so it sounds pretty hard. If it was anyone other than David I'd be a bit suspicious but I reckon it's correct to take him fairly seriously.

It could be that it was Ray Kurzweil and not David Deutsch who wrote that.

David Deutsch is well known for quantum computing, he created one of the first working quantum algorithms (in 1992 with Richard Jozsa says google), so he might have been talking about quantum computers and not classical computation.

Fourier

Raycyst wrote: »

Well, in base 10, pi has an infinite decimal expansion which never repeats and which is random. Although it contains an infinite number of digits they can't be predicted.

That's more complex than the number 1. If mathematicans tell me that isn't significant I'd accept that although I'd be a bit confused.

Is it though? It's more complex than the integer "1", but not the real number "1", which are different objects. The latter is more complex than you think. Again it depends on exactly what you are using to define complex.

Although being simpler, yes I'd agree "1" is simpler, however most numbers are like pi and not like "1". "1" is unusually simple compared to most numbers.

I did know that pi is a 'transcendental' number but I have no hope of remembering what that means, or of properly understanding the description.

Transcendental means that that it's not a power of a rational number (e.g. a square root or a cube root or an nth root, e.t.c.) or a finite sum of powers of rational numbers.

The vast majority of numbers are transcendental, e.g. for every non-transcendental number there are an infinite number of transcendentals. So pi is among the most common type of number.

Earlier, Fourier said that theories of computation may be different in different universes. I don't think that's the case. Maths transcends physical universes, and computer science, (and theories of computation), are all branches of maths and so they also transcend physical universes.

The "theory of computation" refers to the type of computation possible. There are theories of computation where real numbers are used, rather than discrete/finite states. The computers you can build in our universe aren't like this, but there is no reason to think that the "higher" universe wouldn't have such computers.

All such theories of computation are subsets of "Complexity Theory" a branch of mathematics.

It itself has no physical restrictions, but you don't know which individual theory of computation is valid in a given universe.

The Blum–Shub–Smale machine is such a computer, obeying a theory of computation different from that in our own universe and hence cannot be constructed here.

However if we were being simulated from a universe where you could build a Blum–Shub–Smale machine, we wouldn't need to have fundamentally discrete time and space as the Blum–Shub–Smale machine would be capable of infinite precision.

I accept that in the case of a simulation that the simulation can be paused and arbitrarily high precision can be achieved but not infinite precision, unless the simulation runs for a very long time.

Or unless it is running in a universe where you can build a Blum–Shub–Smale machine.

David Deutsch is well known for quantum computing, he created one of the first working quantum algorithms (in 1992 with Richard Jozsa says google), so he might have been talking about quantum computers and not classical computation.

Note quantum computers are so different from classical computers, that they are essentially a different object entirely. Again though, they are both subsets of Complexity theory.

Raycyst

hmmm.

I don't understand it but that Blum–Shub–Smale wikipedia page is very short and they don't describe how those registers work, which can store real numbers of arbitrarily high precision. Also no description how these registers are read to arbitrarily high precision.
It sounds nearly like analogue sensors or something but there is the measurement problem. I suppose if you work directly on the data in the registers you don't need to measure it but no idea how that works.


It could be that quantum registers do allow for numbers of infinite precision.

David Deutsch argues convincingly in his book that if quamtum algoritms are proven to work that that means the many worlds theory of quantum stuff would be proven to be true.
Some algorithms have already worked but I suppose the proof will only be true if quantum computers easily factor numbers large enough to break security crypto systems. If they do that the only explanation as to how the quantum algorithms work is to imagine that the calculation was shared out among huge numbers of universes, all of which were similar enough so that a quantum computer was built in that universe and programmed with the problem.
How else would they work?


Your description of the transcendtal number is very good. It makes sense. Every action taken with rational numbers is knowable. You can square it or cube it but you can't produce an irrational number.

I was actually reading a good page today which described transcendental numbers but I didn't get it as I do now.
http://sprott.physics.wisc.edu/pickover/trans.html

Actually, now that I see that page again he says this.

Transcendental numbers cannot be expressed as the root of any algebraic equation with rational coefficients. This means that pi could not exactly satisfy equations of the type: pi^2 = 10, or 9pi^4 - 240pi^2 + 1492 = 0

I'd got from your description was that there are no equations of the form.
9x^4 - 240x^2 + 1492 = pi
if x is rational, and I can totally see why that'd be the case.

Fourier

Raycyst wrote: »

I don't understand it but that Blum–Shub–Smale wikipedia page is very short

Full details here:
https://projecteuclid.org/euclid.bams/1183555121

But you'll need a strong mathematics background.

and they don't describe how those registers work, which can store real numbers of arbitrarily high precision. Also no description how these registers are read to arbitrarily high precision.

Remember your own post, the mathematics is detached from the physics. The BSS machine is a mathematically possible computational device. How it would be built would depend on the particular laws of physics of a given universe.

It is not possible to build it in ours, hence you are not going to find an example of how the registers are built in terms of matter you find in our reality.

I suppose if you work directly on the data in the registers you don't need to measure it but no idea how that works.

Again, how it "works" would depend on the mechanics of the alternate world in which it can be built. It cannot be built in ours.

It could be that quantum registers do allow for numbers of infinite precision.

They don't allow for infinite precision, that is a proven fact. A review article containing the proof is here:
Bernstein and Vazirani, Quantum complexity theory, SIAM Journal on Computing, 26(5):1411–1473, 1997

However it is highly technical and you would need to know Quantum Mechanics first.

David Deutsch argues convincingly in his book that if quamtum algoritms are proven to work that that means the many worlds theory of quantum stuff would be proven to be true.

All interpretations of quantum mechanics can explain quantum computers, hence this is not true. Deutsch said that when the field of quantum computation was still nascent and believed a quantum computer was a good thought experiment for demonstrating the correctness of the many-worlds interpretation. Further research has since shown all the interpretations validly reproduce quantum computation.

Some algorithms have already worked but I suppose the proof will only be true if quantum computers easily factor numbers large enough to break security crypto systems. If they do that the only explanation as to how the quantum algorithms work is to imagine that the calculation was shared out among huge numbers of universes, all of which were similar enough so that a quantum computer was built in that universe and programmed with the problem.
How else would they work?

Depends on the interpretation. However each interpretation has its own explanation. I'd have to break down the interpretations for you to explain.

Raycyst

ok, good stuff.

Well, I don't see how the Copenhagen intepretation could explain the success of a quantum computer when factorising a huge number with 512 digits.

It's proven that to factor a number of that size is very hard. In the many worlds intrepretation you can at least say that the calculation is farmed out to a near infinite number of universes, which is where the magic comes from. The magic of course being the ability to factor a huge number so quickly. Instead of using a near infinite amount of time we use a near infinite amount of universes.

I know it could be completely false, but it does sound plausible.


In the Copenhagen Interpretation, where the wave function collaspes at some indeterminate point, how does the calculation work?
Why is the universe being so helpful to us, if the universe assists in selecting the correct answer?

(edit: I said 'collaspes at some indeterminate point' above. That obviously means that it's something about when conciousness intervenes and forces the collaspe. Sorry, but that's not very specific and impossible to codify formally I think. So not very reassuring.
At least the Many Worlds just says 'all of the possibilities are all happening' which is great, even if does lead to ludicrous amounts of worlds.)


I will admit that I've never understood how the correct answer is selected from a quantum computer. In other words, all the bits in registers can take values from 0 to 1 and values in between at the same time.
How are numbers encoded and how is the answer selected?





On the issue of computers using real numbers in registers I think it won't happen. I think faster than light travel using something like a Alcubierre drive is more likely, even if it is basically practically impossible, needing galaxy sized amounts of negative energy etc.

https://en.wikipedia.org/wiki/Alcubierre_drive

I have heard of supertasks and stuff like that before though.
It's interesting that you say that a computer using real numbers isn't possible in our universe but it is possible mathematically, and you're not sure if it's possible in other physical universes.

Why can't its action be simulated in this universe?
(edit: I reckon it could easily be simulated but you wouldn't get any of the gains, because your simulation wouldn't have sufficient computing power.)


But the much more important question is, what's the difference between a mathematical universe, where a real computer is possible, and a physical universe, where it isn't possible in ours for example but it may be in others?

What's the fundamental restriction of physical universes that prevents a computer using real registers from working?
I suspect if we can't do it in ours it can't be done in any physical universe.





I was also going on before about how complex pi was and how the universe was great. Perhaps the huge complexity we see in pi is only a result of our own work in uncovering that complexity. If we look for it the complexity is there but you can ignore pi and live your life just fine. I haven't really made my point very well there, but it's our mathematical effort that is producing the complexity in pi, pi just somehow contains it effortlessly.

Fourier

Raycyst wrote: »

Well, I don't see how the Copenhagen intepretation could explain the success of a quantum computer when factorising a huge number with 512 digits.

In the Copenhagen interpretation the computation is carried out by the interactions of different components of the wave function. It would be hard to explain without you knowing QM although I could try.

In the Copenhagen Interpretation, where the wave function collaspes at some indeterminate point, how does the calculation work?
Why is the universe being so helpful to us, if the universe assists in selecting the correct answer?

The different components of the wave-function carry out the computation via their interaction. The circuit is set up in such a way that the wavefunctions amplitude becomes concentrated around the bit values representing the correct answer. The "universe" isn't helping us, the circuits have to be specifically designed to concentrate the amplitude over time. That evolving concentration is the computation.

edit: I said 'collaspes at some indeterminate point' above. That obviously means that it's something about when conciousness intervenes and forces the collaspe. Sorry, but that's not very specific and impossible to codify formally I think.

No interpretation involves consciousness, that's a misunderstanding common from popular books which present it that way. The Copenhagen interpretation does not involve or make reference to consciousness or about "when consciousness intervenes".

I will admit that I've never understood how the correct answer is selected from a quantum computer. In other words, all the bits in registers can take values from 0 to 1 and values in between at the same time.

Well that's down to the complexities of how the circuit is designed. Also it doesn't take values between 0 and 1. It can only take values 0 or 1, or be in a state where it has non-zero probabilities for returning 0 and 1. e.g. a state with 30% chance of 0 and 70% chance of 1, or any other values for the two percentages.

How are numbers encoded and how is the answer selected?

Numbers are encoded in binary like a conventional computer. The answer is selected by concentrating the probability amplitudes on it.

On the issue of computers using real numbers in registers I think it won't happen.

I have already explicitly stated it will not happen, as it cannot be constructed in our world.

I think faster than light travel using something like a Alcubierre drive is more likely

The Alcubierre drive is completely impossible. Quantum Field Theory (particle physics) has already demonstrated that matter cannot be arranged in such a manner.

It's interesting that you say that a computer using real numbers isn't possible in our universe but it is possible mathematically, and you're not sure if it's possible in other physical universes.

Why can't its action be simulated in this universe?

Because computers in our world cannot access real values, only some subset of the rationals. Hence it cannot in anyway be simulated.

What's the fundamental restriction of physical universes that prevents a computer using real registers from working?

Quantum Mechanics prevents any physical quantity from being capable of interacting with another system with a continuous range of values.

I suspect if we can't do it in ours it can't be done in any physical universe.

There is no rational reason to suspect that. People have build examples of laws of physics, different from our own, where it is possible.

Raycyst

Ok, good stuff again.

I'm obviously lost a fair bit here.


On the Copenhagen Interpretation it isn't exactly conciousness that collaspes the wavefunction but the act of measuring. And measuring is a conscious choice. although I accept the choice could be made randonly or automatically, so consciousness isn't important but measurement is.

If we conspire to determine which of the two slits a photon or a electron went through we then lose the interference pattern associated with the two slit experiment.

In other words, the act of measuring at the slits, which allowed us to determine which of the slits a photon or an electron went through, destroys the wave function and collaspes it, and by so doing interference is not possible and interference does not occur and so we don't see an interference pattern on the screen.


What I've said above is all very basic and obvious but what is it about measuring that destroys wave functions?
I've read that it's not always clear when the wave function collaspes in the Copenhagen Intrep.

Humans also become entangled with their own experiments. How do you know that your brain isn't becoming entangled with your experiments? and how do you know that your own conciousness isn't splitting into two?

Those questions above may not make sense due to my misunderstandings of the whole thing.


Is it not possible to take sneaky measurements at the slits to avoid collasping the wave function?
You'd have to be very sneaky.



I'm aware of the quantum bomb testing program and counter factuals.

Counter factual questions seem to give huge capabilities that are impossible in classical physics, such as allowing quantum bombs to be tested.
Are counter factuals used in quantum computing at all or are they something completely different?
In effect, my understanding of counter factuals is that they are 'what if' questions, as in 'what would happen if we shone a photon down that path?', and incredibly we can get answers without actually doing the experiment.



As far as I can remember David Deutsch also said that the two slit experiment also proved the Many Worlds theory but maybe he didn't. And if he did, maybe he was wrong, or maybe he has since changed his mind!

Morbert

If we conspire to determine which of the two slits a photon or a electron went through we then lose the interference pattern associated with the two slit experiment.

Terminology is important here, as people can mean two different processes when they say "collapse".

A) The ascription of classical probabilities to different possible events.

The 'updating' of classical probabilities such that all probabilities go to zero, with the exception of the probability associated with the observed outcome, which goes to one. E.g. If you observe schrodinger's cat and see that it is alive, then the probability that it is alive goes to one, and the probability that it is dead goes to zero.

The disappearance of the interference pattern in the double slit experiment is an example of A, and has nothing to do with consciousness (Neither does B, but A is more relevant to your discussion). The reason the interference pattern disappears is because the quantum state of the system with detectors is

|Ψ> = a|particle passes through slit A>|detector A triggers>
+ b|particle passes through slit B>|detector B triggers>

whereas if there are no detectors, the quantum state is

|Ψ> = a|particle passes through slit A>
+ b|particle passes through slit B>

These different quantum states report different classical probability distributions for where the particle will possibly land, and only the former lets you ascribe classical probabilities to which slits the particles went through and where on the screen they land. The detectors can be conscious beings "watching" the particle, or unconscious detectors. In either case, the interference pattern disappears. If this experiment were somehow carried out after all conscious life had disappeared, there would still be no interference pattern when detectors are present.

Note that there is no destruction of the wavefunction. The wavefunction is just different for the different experimental setups.

Counter factual questions seem to give huge capabilities that are impossible in classical physics, such as allowing quantum bombs to be tested.
Are counter factuals used in quantum computing at all or are they something completely different?
In effect, my understanding of counter factuals is that they are 'what if' questions, as in 'what would happen if we shone a photon down that path?', and incredibly we can get answers without actually doing the experiment.

Different interpretations treat counterfactuals differently, but all valid interpretations make the same predictions. It's just that counterfactual reasoning is counter-intuitive in quantum mechanics, as you have to accommodate the strange relations different observables can have with each other (called commutation relations).

It is these relations, rather than counterfactuals per se, that are exploited in quantum computation.

I'm not sure what you mean by quantum bombs. Are you referring to the Vaidman thought experiment? The thought experiment is for illustrating the strangeness of quantum mechanics rather than for devising a procedure to test bombs.

I'm also confused as to what you mean by getting answers before we do experiments. If you mean we can devise experiments such that quantum mechanics will predict a definitive outcome for a particular measurement, then yes we can do that if we are careful. But we can also do this with classical mechanics.

Fourier

Morbert wrote: »

A) The ascription of classical probabilities to different possible events.

The disappearance of the interference pattern in the double slit experiment is an example of A, and has nothing to do with consciousness (Neither does B, but A is more relevant to your discussion). The reason the interference pattern disappears is because the quantum state of the system with detectors is

The strange thing is, even though decoherence explains this (as well as Galilean or Poincarnvariance selecting the basis), it still fries my mind. That quantum probability leaks off into the environment, leaving behind classical "ignorance based" probability for those observables.
Of course each interpretation has an explanation of this (for this alone, I find the Many-Worlds explanation easier to picture), but still pretty weird!

In another post I broke interpretations into the five groups based on the following assumptions of Bell's theorem:

(a) The world isn't fundamentally random and objects have real objective properties (i.e. an electron definitely is located in one place and has that position as an objective fact, independent of another entity/object observing it.)
(b) There aren't multiple timelines/futures following a single event
(c) The world isn't contextual (i.e. particles don't possess knowledge of the states of all other particles in the universe, the current "context" of the universe)
(d) The future can't send signals to the past
(e) The world isn't superdeterministic (Superdeterminism is that Quantum Mechanics is wrong, but the world is predetermined in such a way that we never do the experiments that show it is wrong)

Hence the five classes would be cases where one of these is rejected. I've never heard how your (A) class event works in theories rejecting (d) (i.e. retrocausal theories, e.g. Cramer's transactional interpretation). Would you happen to know?

Morbert

Hmm, I haven't heard of the transactional interpretation, but it looks similar to the two-state vector formalism, which I have come across.

In case you're not familiar: In the two-state vector formalism, you have your usual quantum state Ψ evolving forwards in time from some initial condition, but you also have a second quantum state Φ evolving backwards in time from some final condition. You use both these states to describe your quantum system + its environment.

I've never heard how your (A) class event works in theories rejecting (d)

The analogue to decoherence in the two-state vector formalism is "two-time decoherence". Like ordinary decoherence, you trace over environmental degrees of freedom. But while in the ordinary formalism, your complete density matrix looks like

rho = |Ψ><Ψ|

under the two-state vector formalism, it looks like

rho = |Ψ><Φ|

And while decoherence normally just reports classical probabilities in the form of a diagonal reduced density matrix, this two-time decoherence will report a single classical outcome. Ψ will correlate with the environment, selecting a preferred basis, and Φ will eliminate all but one of the preferred basis vectors. (This is a time-symmetric process, so you could also say Φ correlates with the environment and Ψ picks a single basis vector).

This paper discusses it in greater detail (particularly section III)