In other words, in the MWI picture, when does the world split into two, one with the atom aligned and one with the atom anti-aligned?
This one I’m going to do un-rot13ed because this misconception bothers me: The world was split all along according to the modern understanding of the MWI.
Well, once you want to solve real problems, talking about “worlds” becomes less helpful than talking about entanglement. Quantum mechanics is continuous along a dimension that thinking using the word “worlds” tends to be discrete. In fact, the name “many-worlds” was proposed about 13 years after Everett’s seminal paper.
i think you still have to talk about worlds to make actual predictions. You need to specify “a world where observer X sees Y” and a probability weight. of some sort for that world (or that observer, however you want to phrase it).
Nay. In fact, being able to make predictions just from a quantum state, no labels on “worlds,” is an important part of making a quantum-mechanics-only interpretation a reductionist success.
How can you predict ‘observer A sees the electron aligned with the magnetic field’ without being able to point to the section of the wavefunction that represents ‘observer A sees the electron aligned with the magnetic field’?
All decoherence can tell me is that my density matrices evolve towards being diagonal. I still need a framework to extract actual meaning from it.
I’ve never seen a paper make a prediction without some implicit definition of world (or sometimes equivalently of ‘observer’)- if you have a reference, please point me to one.
Yes, definitely, one needs to “point to the section of the wavefunction that represents ‘observer A sees the electron aligned with the magnetic field’.” So to the extent doing that is an implicit definition of “world,” then I agree on that part too.
The reason why I replied, er, emphatically, to “you need to specify [...]” is because any theory treating worlds as basic should not be the simplest way to understand what’s going on—that should be plain ol’ quantum mechanics if the relative state interpretation is to be believed, even though that interpretation is sometimes called “many-worlds.” So you shouldn’t have to talk about worlds.
And sometimes it’s a good idea not to talk about worlds, like when shminux asks you “when does the atom get polarized when it’s in a magnetic field?” One definitely has to use math for that one.
The only way I know to answer shminux’s question rigorously (with the decoherence time scale) in MWI is to make the assumption that a diagonal density of states of represents a classical ensemble of worlds (with weights corresponding to probabilities)- which means explicitly talking about worlds.
Then you can define a decoherence time-scale of some sort. To the best of my knowledge, without some rule-of-thumb for what a world is, all you can say is that “the system evolves to a a density matrix that has 1/sqrt(2) Aligned on one diagonal element, and 1/sqrt(2) anti-aligned on the other.”
The only way I know to answer shminux’s question rigorously (with the decoherence time scale) in MWI is to make the assumption that a diagonal density of states of represents a classical ensemble of worlds (with weights corresponding to probabilities)- which means explicitly talking about worlds.
Yeah, that’s a cool realization: “Hey, this all works if we assume that a density matrix is what an entangled state looks like from inside!” And this happens to be pretty true, though if it’s your definition of “world” then there will be different “worlds” from different perspectives.
But anyhow, to answer shminux I wouldn’t use worlds, I’d just give the decay rate of the top state coupled to a photon mode, to the bottom state plus a photon. The assumption there is not about worlds, but simply that the amplitude-squared measure should be used to describe the properties of the atom.
And this happens to be pretty true, though if it’s your definition of “world” then there will be different “worlds” from different perspectives.
I can’t parse what you mean by perspectives here- do you mean different non-relativistic observers (no relativity, we are using Schroedinger quantum), or do you mean putting a different basis on the Hilbert space?
But anyhow, to answer shminux I wouldn’t use worlds, I’d just give the decay rate of the top state coupled to a photon mode, to the bottom state plus a photon.
Shminux’s question involves an unpolarized beam entering a magnetic field and aligning, not polarized atoms flipping state. These are very different problems.
In the shminux question, you have an entirely off-diagonal density matrix that evolves toward diagonal very quickly when the system becomes entangled with the screen. To extract information, you implicitly or explicitly assume that a diagonal density of states represents an ensemble of classical worlds.
In the question you are answering, you start with only one diagonal element in the density matrix non-zero, and over time the amplitude of that element shrinks while the element of the other diagonal element grows. This is a totally different problem. You still are implicitly thinking about worlds, you’ve just created a different problem where there is no entanglement so it dodges the messy question.
I can’t parse what you mean by perspectives here- do you mean different non-relativistic observers (no relativity, we are using Schroedinger quantum), or do you mean putting a different basis on the Hilbert space?
Hm. So what I mean is that if you have several particles entangled with each other, and you want to know what that “looks like” to one subsystem (or, experimentally, if you’re going to produce a beam of these particles and do a bunch of single-particle measurements), then you have to trace over the other particles in the entangled state. This gives you a reduced density matrix, which is then interpreted as a mixed state. A mixed state between what? Well, “worlds,” of course.
This the definition of “world” I meant when I said “if that’s your definition of world...”
But anyhow, why did I say that different observers will see different lists of worlds? Well, because when you take the partial trace, what you trace over depends on which perspective you want (experimentally, what partial measurement you’re making). If you’re an electron, your worlds are boring—we traced all the complicated externals away and now your perspective just looks like a distribution over single-electron states. If you’re a person, your perspective is much more interesting, your density matrix is much bigger. Or to put it another way, you have more “worlds” to have a distribution over.
You still are implicitly thinking about worlds,
And you’re implicitly thinking about waterfalls, because every cognitive algorithm is isomorphic to a thoght about a waterfall :P
Whose modern understanding of MWI? Give me a mathematical formalism for “world” that allows world count to be conserved as interactions go forward (if the worlds ‘were split all along’, then something like ‘world count’ is constant).
Your understanding also seems to have implications for (“the modern understanding of”) MWI and Bell’s theorem- since you are applying if I know enough about the degrees of freedom “I don’t care about” I can track which world I am in. This should reduce to some sort of hidden variable approach, I would think.
Give me a mathematical formalism for “world” that allows world count to be conserved as interactions go forward (if the worlds ‘were split all along’, then something like ‘world count’ is constant).
As Manfred says, thinking of worlds as discrete isn’t helpful. Anyway, the time evolution dictated by Schrödinger’s equation is unitary, so if it always applies (i.e. there’s no such thing as objective collapse), the measure stays constant.
I know enough about the degrees of freedom “I don’t care about” I can track which world I am in.
If I understand decoherence well enough (probably I don’t), the answer to that is “in you could, yes, but you can’t, because thermodynamics”. The differential equations of quantum field theory are more-or-less time-symmetric (i.e. reversing time is equivalent to boring stuff such as conjugating complex phases and swapping particles with anti-particles and left with right), so the reason stuff appears to be irreversible is the boundary condition that the past had very little entropy. (And I think I’ve seen the argument that given that T-symmetry is equivalent to CP-symmetry, the fact that the past had that little entropy may (or may not) have something to do with the fact that there are so many more particles than antiparticles.)
You have to have a mechanism to separate world as discrete or you have a theory that can’t make predictions. If you want to talk about the aligned/anti-aligned beam in the Stern-Gerlach experiment you have to be able to point and say “this represents the world where observers measure aligned, and this bit over here represents the world where observers measure anti-aligned.” If you can’t do that, you have no theory.
If I understand decoherence well enough (probably I don’t), the answer to that is “in you could, yes, but you can’t, because thermodynamics”.
This has to be wrong, otherwise MWI would predict violations of Bell inequalities.
I think your ‘the world is already split’ interpretation is actually the fundamental misunderstanding- I can’t make any sense of it other than as a hidden variable theory of the type already experimentally ruled out by Aspect-like experiments.
Edit: Unrelated, but to clarify- you can show that (assuming energy is bounded below) a Lorentz invariant Hamiltonian has a combined CPT symmetry, which can mean a lot of things, depending on dimension. T has to be related to CP, but not necessarily the-same-as, unless you have a state where CP^2 = 1.
Generally, you pick up a phase factor after CP^2. The story is exactly like parity (P) if you can embed P^2 in a continuous symmetry, you can define away the phase factor, but if you can’t you are just stuck with it.
This one I’m going to do un-rot13ed because this misconception bothers me: The world was split all along according to the modern understanding of the MWI.
Well, once you want to solve real problems, talking about “worlds” becomes less helpful than talking about entanglement. Quantum mechanics is continuous along a dimension that thinking using the word “worlds” tends to be discrete. In fact, the name “many-worlds” was proposed about 13 years after Everett’s seminal paper.
i think you still have to talk about worlds to make actual predictions. You need to specify “a world where observer X sees Y” and a probability weight. of some sort for that world (or that observer, however you want to phrase it).
Nay. In fact, being able to make predictions just from a quantum state, no labels on “worlds,” is an important part of making a quantum-mechanics-only interpretation a reductionist success.
How can you predict ‘observer A sees the electron aligned with the magnetic field’ without being able to point to the section of the wavefunction that represents ‘observer A sees the electron aligned with the magnetic field’?
All decoherence can tell me is that my density matrices evolve towards being diagonal. I still need a framework to extract actual meaning from it.
I’ve never seen a paper make a prediction without some implicit definition of world (or sometimes equivalently of ‘observer’)- if you have a reference, please point me to one.
Yes, definitely, one needs to “point to the section of the wavefunction that represents ‘observer A sees the electron aligned with the magnetic field’.” So to the extent doing that is an implicit definition of “world,” then I agree on that part too.
The reason why I replied, er, emphatically, to “you need to specify [...]” is because any theory treating worlds as basic should not be the simplest way to understand what’s going on—that should be plain ol’ quantum mechanics if the relative state interpretation is to be believed, even though that interpretation is sometimes called “many-worlds.” So you shouldn’t have to talk about worlds.
And sometimes it’s a good idea not to talk about worlds, like when shminux asks you “when does the atom get polarized when it’s in a magnetic field?” One definitely has to use math for that one.
The only way I know to answer shminux’s question rigorously (with the decoherence time scale) in MWI is to make the assumption that a diagonal density of states of represents a classical ensemble of worlds (with weights corresponding to probabilities)- which means explicitly talking about worlds.
Then you can define a decoherence time-scale of some sort. To the best of my knowledge, without some rule-of-thumb for what a world is, all you can say is that “the system evolves to a a density matrix that has 1/sqrt(2) Aligned on one diagonal element, and 1/sqrt(2) anti-aligned on the other.”
Yeah, that’s a cool realization: “Hey, this all works if we assume that a density matrix is what an entangled state looks like from inside!” And this happens to be pretty true, though if it’s your definition of “world” then there will be different “worlds” from different perspectives.
But anyhow, to answer shminux I wouldn’t use worlds, I’d just give the decay rate of the top state coupled to a photon mode, to the bottom state plus a photon. The assumption there is not about worlds, but simply that the amplitude-squared measure should be used to describe the properties of the atom.
I can’t parse what you mean by perspectives here- do you mean different non-relativistic observers (no relativity, we are using Schroedinger quantum), or do you mean putting a different basis on the Hilbert space?
Shminux’s question involves an unpolarized beam entering a magnetic field and aligning, not polarized atoms flipping state. These are very different problems.
In the shminux question, you have an entirely off-diagonal density matrix that evolves toward diagonal very quickly when the system becomes entangled with the screen. To extract information, you implicitly or explicitly assume that a diagonal density of states represents an ensemble of classical worlds.
In the question you are answering, you start with only one diagonal element in the density matrix non-zero, and over time the amplitude of that element shrinks while the element of the other diagonal element grows. This is a totally different problem. You still are implicitly thinking about worlds, you’ve just created a different problem where there is no entanglement so it dodges the messy question.
Hm. So what I mean is that if you have several particles entangled with each other, and you want to know what that “looks like” to one subsystem (or, experimentally, if you’re going to produce a beam of these particles and do a bunch of single-particle measurements), then you have to trace over the other particles in the entangled state. This gives you a reduced density matrix, which is then interpreted as a mixed state. A mixed state between what? Well, “worlds,” of course.
This the definition of “world” I meant when I said “if that’s your definition of world...”
But anyhow, why did I say that different observers will see different lists of worlds? Well, because when you take the partial trace, what you trace over depends on which perspective you want (experimentally, what partial measurement you’re making). If you’re an electron, your worlds are boring—we traced all the complicated externals away and now your perspective just looks like a distribution over single-electron states. If you’re a person, your perspective is much more interesting, your density matrix is much bigger. Or to put it another way, you have more “worlds” to have a distribution over.
And you’re implicitly thinking about waterfalls, because every cognitive algorithm is isomorphic to a thoght about a waterfall :P
Whose modern understanding of MWI? Give me a mathematical formalism for “world” that allows world count to be conserved as interactions go forward (if the worlds ‘were split all along’, then something like ‘world count’ is constant).
Your understanding also seems to have implications for (“the modern understanding of”) MWI and Bell’s theorem- since you are applying if I know enough about the degrees of freedom “I don’t care about” I can track which world I am in. This should reduce to some sort of hidden variable approach, I would think.
As Manfred says, thinking of worlds as discrete isn’t helpful. Anyway, the time evolution dictated by Schrödinger’s equation is unitary, so if it always applies (i.e. there’s no such thing as objective collapse), the measure stays constant.
If I understand decoherence well enough (probably I don’t), the answer to that is “in you could, yes, but you can’t, because thermodynamics”. The differential equations of quantum field theory are more-or-less time-symmetric (i.e. reversing time is equivalent to boring stuff such as conjugating complex phases and swapping particles with anti-particles and left with right), so the reason stuff appears to be irreversible is the boundary condition that the past had very little entropy. (And I think I’ve seen the argument that given that T-symmetry is equivalent to CP-symmetry, the fact that the past had that little entropy may (or may not) have something to do with the fact that there are so many more particles than antiparticles.)
You have to have a mechanism to separate world as discrete or you have a theory that can’t make predictions. If you want to talk about the aligned/anti-aligned beam in the Stern-Gerlach experiment you have to be able to point and say “this represents the world where observers measure aligned, and this bit over here represents the world where observers measure anti-aligned.” If you can’t do that, you have no theory.
This has to be wrong, otherwise MWI would predict violations of Bell inequalities.
I think your ‘the world is already split’ interpretation is actually the fundamental misunderstanding- I can’t make any sense of it other than as a hidden variable theory of the type already experimentally ruled out by Aspect-like experiments.
Edit: Unrelated, but to clarify- you can show that (assuming energy is bounded below) a Lorentz invariant Hamiltonian has a combined CPT symmetry, which can mean a lot of things, depending on dimension. T has to be related to CP, but not necessarily the-same-as, unless you have a state where CP^2 = 1.
How can it be anything else? Even then, T would equal (CP)^-1.
Generally, you pick up a phase factor after CP^2. The story is exactly like parity (P) if you can embed P^2 in a continuous symmetry, you can define away the phase factor, but if you can’t you are just stuck with it.
That’s still in the reference class I called “boring stuff”, though.