Schrodinger’s cat and the Many Worlds interpretation

[MichaelAlexDavM]

TL;DR Summary

Schrodinger’s cat and the many worlds interpretation presents us with a binary (discrete) outcome but what if there is an infinite number of outcomes i.e. a continuum (uncountable set) of outcomes?

Schrodinger’s Cat and the many worlds interpretation states that the wave function collapse doesn’t happen at all; every possible outcome of an observation actually comes to pass in its own separate universe. We are presented with a binary (discrete) outcome (dead or alive) but what if there are an infinite number of outcomes i.e. a continuum (unaccountably many) number of outcomes?

Instead of possibly killing the cat, let it be anesthetized for a length of time between say 1 and 2 hours, the exact length of time depends on the angle Ѳ, (0 < Ѳ ≤ 2π) of an emitted particle from a radioactive source.

According to the many world interpretation,

• how many universes will be created?
• will there be uncountability many?

 

[PeterDonis]

Moderator's note: Moved thread to the quantum interpretations and foundations forum.

 

[PeroK]

Summary:: Schrodinger’s cat and the many worlds interpretation presents us with a binary (discrete) outcome but what if there is an infinite number of outcomes i.e. a continuum (uncountable set) of outcomes?

Schrodinger’s Cat and the many worlds interpretation states that the wave function collapse doesn’t happen at all; every possible outcome of an observation actually comes to pass in its own separate universe. We are presented with a binary (discrete) outcome (dead or alive) but what if there are an infinite number of outcomes i.e. a continuum (unaccountably many) number of outcomes?

Instead of possibly killing the cat, let it be anesthetized for a length of time between say 1 and 2 hours, the exact length of time depends on the angle Ѳ, (0 < Ѳ ≤ 2π) of an emitted particle from a radioactive source.

According to the many world interpretation,

• how many universes will be created?
• will there be uncountability many?

Everett, who introduced the Many Worlds Interpretation, supposed there were indeed in uncountable infinity of distinct branches in the wavefunction.

Note that the branching of the wavefuction does not create universes; rather, it splits the one universe into multiple "worlds" with different observed experimental results. There remains one universal wavefunction, but it describes a multi-world-like reality.

 

[MichaelAlexDavM]

Everett, who introduced the Many Worlds Interpretation, supposed there were indeed in uncountable infinity of distinct branches in the wavefunction.

Note that the branching of the wavefuction does not create universes; rather, it splits the one universe into multiple "worlds" with different observed experimental results. There remains one universal wavefunction, but it describes a multi-world-like reality.

Perok thanks for your reply. Your point that ' the wavefuction does not create universes; rather, it splits the one universe into multiple "worlds" is well taken. I am guessing this is to do with energy.

My question arose most likely because I do not understand enough about uncountable sets and QM.

• can an uncountable infinite set be split into distinct parts? I picture this set has a continuum with no gaps between the numbers.
• with regards to my example (which I hope is logical), Is it possible for the cat be in a universe where it wakes up after the $\sqrt{61 }$ minutes?

 

[PeroK]

Perok thanks for your reply. Your point that ' the wavefuction does not create universes; rather, it splits the one universe into multiple "worlds" is well taken. I am guessing this is to do with energy.

My question arose most likely because I do not understand enough about uncountable sets and QM.

• can an uncountable infinite set be split into distinct parts? I picture this set has a continuum with no gaps between the numbers.
• with regards to my example (which I hope is logical), Is it possible for the cat be in a universe where it wakes up after the $\sqrt{61 }$ minutes?

The wavefunction is an infinite dimensional vector in Hilbert space. Let's leave the uncountable case out and let's look at an interaction with a large finite or a countable set of results: $\psi_1, \psi_2 \dots$. These are eigenfunctions associated with the measurement interaction. The idea is that the wavefunction is a superposition of these eigenfunctions (or branches in MWI):
$$\Psi = \sum_{n = 1}^{\infty} \psi_n$$
The other important point is that each $\psi_n$ subsquently evolves independently. Not in the strictest sense, but in a practical sense there is "decoherence" between the branches which prevents any significant subsequent mixing of any two. The question for MWI, of course, is how we perceive only one outcome - or think we do.

In fact, in orthodox QM this same idea goes on before a measurement. A system is in a superposition of many eigenstates, all evolving independently. But, in orthodox QM, when a measurement is made this superposition is resolved into one outcome: the one we measure. In MWI, the superposition continues to evolve in its entirety. With each outcome somehow being part of "reality".

In that sense, MWI doesn't put any further mathematical demands on the wavefunction. It's really that the superposition persists after a measurement.

You could try:

http://www.preposterousuniverse.com...ion-of-quantum-mechanics-is-probably-correct/

What's special about $\sqrt{61}$?

Although, you can get your kicks on $\sqrt{66}$!

 

[MichaelAlexDavM]

The wavefunction is an infinite dimensional vector in Hilbert space. Let's leave the uncountable case out and let's look at an interaction with a large finite or a countable set of results: $\psi_1, \psi_2 \dots$. These are eigenfunctions associated with the measurement interaction. The idea is that the wavefunction becomes a superposition of these eigenfunctions (or braches in MWI):
$$\Psi = \sum_{n = 1}^{\infty} \psi_n$$
The other important point is that each $\psi_n$ subsquently evolves independently. Not in the strictest sense, but in a practical sense there is "decoherence" between the branches which prevents any significant subsequent mixing of any two. The question for MWI, of course, is how we perceive only one outcome - or think we do.

In fact, in orthodox QM this same idea goes on before a measurement. A system is in a superposition of many eigenstates, all evolving independently. But, in orthodox QM, when a measurement is made this superposition is resolved into one outcome: the one we measure. In MWI, the superposition continues in its entirety.

In that sense, MWI doesn't put any further mathematical demands on the wavefunction. It's really that the superposition persists after a measurement.

You could try:

http://www.preposterousuniverse.com...ion-of-quantum-mechanics-is-probably-correct/

What's special about $\sqrt{61}$?

I was just using $\sqrt{61}$ as an example of an irrational number and a member of an uncountable set. I thought irrational numbers might cause problems in MW, specifically when the universe branches. I need to do a lot more work not just in QM but also on the underlying maths.

I have just started a QM module, The lecturer is really good but I am sure he is in the "shut up & calculate" camp which I think is understandable given the subject matter. I am already a fan of Sean Carroll but until now I never checked out his website. Thanks for the link and for pointing out the difference between MW & MWI.

 

[PeterDonis]

I was just using $\sqrt{61}$ as an example of an irrational number and a member of an uncountable set.

It's also a member of a countable set: the set of algebraic numbers (numbers that can be obtained from integers by a finite number of algebraic operations, of which taking a square root is one) is countable.

 

[MichaelAlexDavM]

It's also a member of a countable set: the set of algebraic numbers (numbers that can be obtained from integers by a finite number of algebraic operations, of which taking a square root is one) is countable.

Yes, I see that now after researching online ( I read almost all real numbers are transcendental-that's news to me). I should probably have used a transcendental number in my argument.

Are transcendental numbers the reason why the set of irrational numbers is uncountable?

 

[PeterDonis]

Are transcendental numbers the reason why the set of irrational numbers is uncountable?

If you mean, is the set of real numbers that are not transcendental countable, yes, it is--those are just the algebraic numbers.

I'm not sure I would say transcendental numbers are the "reason" the reals are uncountable, since the definition of "transcendental number" is just "not an algebraic number". I would say the reason the reals are uncountable is Cantor's diagonal argument.

 

[Vaxjo]

Dear all,
I want just remind that Schrödinger invented the example with the cat and poison solely to demonstrate the absurdness of the Copenhagen interpretation of QM, this example is just a modification of Einstein's example with dog and gun, from the letter of Einstein to Schrödinger. Nowadays Schrödinger example is really misused...

 

[EPR]

Dear all,
I want just remind that Schrödinger invented the example with the cat and poison solely to demonstrate the absurdness of the Copenhagen interpretation of QM, this example is just a modification of Einstein's example with dog and gun, from the letter of Einstein to Schrödinger. Nowadays Schrödinger example is really misused...

Schrodinger's cat highlights the difficulty in interpreting quantum mechanics(and it's not a conundrum just for the CI). Back when the thought experiment was devised, only the CI existed(1935). The paradox hasn't been resolved fully to this day(even within decoherence) - i don't think it's fair to claim it's been 'misused'.

 

[Nugatory]

i don't think it's fair to claim it's been 'misused'.

“Misuse” is a quite accurate characterization. For every thread starter here who has read what Schrodinger said, we get hundreds who been misled by the popular mischaracterizations of what he said.

 

[Vaxjo]

Schrödinger cat: I strongly recommend to read the correspondence between Einstein and Schrödinger, they discussed CI vs. statistical interpretation; they were in complete agreements that CI is nonsense, and the examples, first Einstein: dog and gun and then its modification by Schrödinger, cat and poison, were elaborated to stress the meaninless of CI...

 

[bahamagreen]

Even before considering Many Worlds, Schrodinger’s cat has always presented the discrete yet potentially confounding not necessarily binary outcome. To measure unconfounded binary outcome, a cat must be prepared so as to be in the ultimate of its nine lives.

 

[EPR]

In QFT as a multi particle field theory, can we say that we end up with a cat, because we always start out with a cat and apply conservation laws which obviously always hold at our scale?
We probe('intrude') quantum fields with crude 'classical' devices and end up with classical results. Field operators create 'cats'. Is this valid? Is there more to the cat story from the point of the best tested physics theory?

 

[Lord Jestocost]

Schrödinger cat: I strongly recommend to read the correspondence between Einstein and Schrödinger, they discussed CI vs. statistical interpretation; they were in complete agreements that CI is nonsense, and the examples, first Einstein: dog and gun and then its modification by Schrödinger, cat and poison, were elaborated to stress the meaninless of CI...

Is Schrödinger's and Einstein's wishful thinking regarding this point really relevant. I don't think so.
As Anthony Leggett puts in “ELEGANCE AND ENIGMA The Quantum Interviews” (Editor: Maximilian Schlosshauer):

"Now, following Schrödinger, let us consider a thought experiment in which the quantum-mechanical description of the final state, as obtained by appropriate solution of the time-dependent Schrödinger equation, contains simultaneously nonzero probability amplitudes for two or more states of the universe that are, by some reasonable criterion, macroscopically distinct (in Schrödinger’s example, this would be “cat alive” and “cat dead”). Of course, just about everyone, including me, would accept that because of, inter alia, the effects of decoherence, it is likely to be impossible, at least for the foreseeable future, to experimentally demonstrate the interference of such states. (On the other hand, as the late John Bell was fond of pointing out, the “foreseeable future” is not a very well-defined concept. In fact, as late as 1999, not a few people were confidently arguing that because of the inevitable effects of decoherence, the projected experiments to demonstrate interference at the level of flux qubits would never work. In this case, the “foreseeable” future lasted approximately one year. As Bell used to emphasize, the answers to fundamental interpretive questions should not depend on the accident of what is or is not currently technologically feasible.) But the crucial point is that the formalism of quantum mechanics itself has changed not one whit between the microscopic and macroscopic levels. Are we then entitled to embrace, at the macrolevel, an interpretation that was forbidden at the microlevel, simply because the evidence against it is no longer available?
I would argue very strongly that we are not, and would therefore draw the conclusion: also at the macrolevel, when the quantum-mechanical description assigns simultaneously nonzero amplitudes to two or more macroscopically distinct possibilities, then it is not the case that each system of the relevant ensemble realizes either one possibility or the other." [Emphasis by LJ]

 

[Vaxjo]

Well, I had a few conversations with Tony Leggett and similar viewpoint is typically presented by Anton Zeilinger; I my impression is that they are very naive in they dream to lift quantum mechanics to macro-level; but,of course, this is their right for dreams... Now, But the crucial point is that the formalism of quantum mechanics itself has changed not one whit between the microscopic and macroscopic levels.
Well, you totally forgot about the indivisible quantum of action given by the Planck constant h. And you position is very natural for people doing QI..., see my recent paper where I elevated the role of h: https://arxiv.org/abs/2003.05718

 

[PeterDonis]

the formalism of quantum mechanics itself has changed not one whit between the microscopic and macroscopic levels

It depends on what you mean by "microscopic" and "macroscopic". The formalism of QM, as it is actually used, treats two cases very differently, but the terms "microscopic" and "macroscopic" don't precisely pick out the two types of cases. A better division would be:

(1) Cases where we can model everything that is happening with unitary, reversible time evolution. This is things like photons passing through beam splitters, silver atoms passing through Stern-Gerlach magnets, buckyballs passing through screens with double slits in them, etc.

(2) Cases where irreversible changes happen that we cannot model with unitary, reversible time evolution. This is things like photons hitting photomultiplier detectors placed downstream of some arrangement of beam splitters, silver atoms hitting detection screens placed downstream of Stern-Gerlach magnets, buckyballs hitting detection screens placed downstream of screens with double slits in them, etc.

The formalism used in (1) is very different from the formalism used in (2).