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I was reading decoherence from this site, where I found the following bold words which states that the out-of-phase components do not really get dissappear after decoherence:

What happens to a quantum particle in the real world is that each of its component states gets entangled (separately) with different aspects of its environment. As seen in the page on Quantum Entanglement, when particles become entangled you have to consider them as one single, entangled state (you use the tensor product to calculate the resultant state). So each component of our quantum particle forms separate entangled states. The phases of these states will be altered. This destroys the coherent phase relationships between the components. The components are said to decohere.

If a particle interacts with just a single photon, for example, then the two particles will enter an entangled state and that will be enough to trigger the onset of decoherence (for example a single photon entering the double-slit experiment will be enough to destroy the interference pattern). However, for all interference effects to disappear, the particle must have a macroscopic (rather than a microscopic) effect by forming entanglements with billions of particles in, say, a Geiger counter. This is described in the book Quantum Enigma: "Whenever any property of a microscopic object affects a macroscopic object, that property is 'observed' and becomes a physical reality". In that case, if there are no longer any interference terms then to all intents and purposes the particle is now in a single, quantum state - one of the component eigenstates.

Note that the interference components do not actually disappear - because they are out of phase we just don't notice them at the macroscopic level. In fact, they just get dissipated out into the wider environment. I always imagine them as little ripples in the ocean - we only ever notice the big (macroscopic) waves in the ocean. The little ripples get entangled with other little ripples until it is impossible to tell from which big wave each little ripple came.

Imagine you throw a rock in the sea off the coast of the United Kingdom. After the initial big splash, the ripples dissipate and apparently disappear. But of course, they haven't really disappeared. The ripples have decreased in size, and they have mixed and interfered with other waves, but they have not disappeared. Two weeks later, on the rocky shore of Tierra del Fuego off the Argentinian coast, one of the small waves washing to shore is maybe an imperceptible fraction of one micron higher because of that rock you threw.

So the ripples (interference terms) do not actually disappear. They dissipate into the environment and become effectively undetectable. And it's certainly not possible to associate the microscopic change in the height of the wave in Tierra del Fuego with the rock you threw - there have been so many interactions with other waves along the way. In this sense, the process of decoherence is irreversible - and that's a key feature of decoherence: we can't reverse the process (to regenerate the initial interference components) - they're gone for good. And even the "little ripple" echoes of the interference effects have become imperceptible due to interactions with the environment. Then, for all intents and purposes, the interference effects (ripples) have completely disappeared.

What an extended metaphor (ok, analogy) . But I am fearing I have not understood what he is saying between these lines of metaphor. How do the out-of-phase states get "dissipated to wider environment"? Can anyone help me understand the three bold para of metaphor?

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    $\begingroup$ I would suggest you leave that page and read "Decoherence and the Transition from Quantum to Classical—Revisited" by Wojciech H. Zurek, instead. It's MUCH better and it can be expected to be actually correct. $\endgroup$ – CuriousOne Jun 28 '15 at 6:57
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Right before the part you boldfaced and right after the part you didn't boldface there was a parenthetical part. And the parenthetical included a link. In the link the author is pretty clear that the phases are in the reduced density matrix.

A reduced density matrix is about reproducing the statistics and usually about adding some classical statistics on top and also often about averaging some uncared about degrees of freedoms.

Weinberg is the only person I've seen that takes the density matrix seriously as a physical description of reality as opposed to a convenient way to summarize a partial description of reality.

If you read the parenthetical before the link in the part before the part you boldfaced in your link ... you'll see that it is the size of the Hilbert space that matters. It is the size that comes from interacting with many particles that makes it so hard to ever have these different eigenspaces ever influence each other again.

They become irreversibly orthogonal in a practical sense. Which feels a bit dishonest to say since obviously they were orthogonal all along as time evolution is unitary. Its just the initial state was a combination of all the eigenstates and now you have separated them into eigenstates tensored with environmental degrees of freedom that will not ever interact again with anything.

And in particular each can act like it is the only one. In particular quantum mechanics doesn't actually require the wave functions be unit length (in the L2 sense). When they act like the only one, they don't anymore have to care about how big the other ones are. So that's what really goes on.

As for the phases, whether the phases line up or not only matters if you are summing lots, because then the tend to cancel. That can matter if you have a density matrix and you sum over degrees of freedom, but few people (Weinberg might be one of the few) think a density matrix is a true description of reality the way it actually is.

You can describe the wavefunction as what it is actually us for the actual setup and actual evolution with the actual Hamiltonian. And then you see these initially orthogonal states evolve into also orthogonal states that have their orthogonality in some particular operator's eigenspace decomposition become orthogonal in a tensored way where it coupled to other particles, many many many other particles.

And then because of the sheer size of the Hilbert space they can be forced to no longer interfere with each not just in the L2 scalar product sense (for averages of ensembles and expectation values) but in every possible sense because the so called probability current is also unaffected at all when they do not overlap (the probability current has a subtraction in it, so a small but asymmetric variation on the wave can have a large impact on the so called probability current).

When someone brings up the phase, the phase doesn't matter if they don't overlap. And if they do overlap, then at least some of the analysis of the waves is affected by their being two or more waves instead of one.

If someone wants to make a big deal about the phases not lining up, they want to sum lots of waves. Often that is because they aren't tracking the one actual solution given by the Schrödinger equation's evolution of the one actual initial wavefunction. Why? Maybe they started with a density matrix so don't have a wavefunction to start with. Maybe they really care about thermodynamics and imagine a bunch of unknown possible wavefunctions, which is fine as long as you admit you are doing Statistical Physics on top of an actual theory about the evolution of the actual universe.

Do not confuse the two. And frankly, reading analogies is not a way to avoid confusing the two. Few people actually write down and solve the actual Schrödinger equation for the actual experimental setup. Which is reasonable. But they might not tell you that they aren't doing that. They might find it so reasonable to say that they don't know the actual wavefunction that they start out with a probability of having a bunch of wave functions and then make a theory about the predictions of that ensemble of different wave functions.

And they might make a theory of decoherence about that. Which is totally fine if they tell you that is what they are doing. But phases from a mixed density matrix where you sum over degrees of freedom you don't care about is not a fundamental theory.

It is a very practical theory, designed to agree with realistic things you do and designed to talk about the frequencies of various possibilities.

And the whole point of decoherence theory for some people is to merely figure out when they can start to use regular classical probability theory, statistical analysis and classical frequencies. Nothing more, and not really about nature, more about when nature allows you to use particular math to describe your averages and lumping together that you decided to do. Allows you to do that and get the same answers.

Do not expect to understand the universe by analogy unless it is to help you learn and remember the actual calculations, the actual mathematical models, and the actual situations you use the models for.

It seems like all three are omitted, so it is a pure sham of an understanding. If you study the actual decoherence theory you can learn what they are modeling, how they do the calculations, and what models they are using to model what situations. And then it will be clear when you can use them and what you get out of it.

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    $\begingroup$ Now I am more confused than I ever was about quantum mechanics. $\endgroup$ – CuriousOne Jun 28 '15 at 6:44
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It is not the out of phase states that get dissipated to wider environment. Rather, it is only the interference components with those out of phase states that get dissipated into the wider environment.

I know you got to this question from a question about the Everett ("many worlds") interpretation. If you find Timaeus' more technical description difficult to understand, you can think of it this way. The interference components are the only interactions between the "worlds" (universal eigenstates). During the decoherence process, these interference components are diluted into the wider environment of each world. As a result, each world becomes less able to see the interactions with the other worlds. That doesn't mean the other worlds no longer exist, but it does mean that you no longer have to worry about them, because they no longer have any visible effect on your world.

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A density matrix quantum mechanically is a way to easily model the many body wave function of a many particle system.

It is a matrix whose off diagonal elements keep the phases between the disparate wavefunctions and the amplitude of projecting particle A to particle N. This is the "information" that particle A has for the existence of particle N. Thus coherence describes the system with one very complicated wavefunction.

In principle the analogy of your paragraphs would not be wrong, the off diagonal elements become effectively zero when they are zero within the measurement errors , particularly within the thermal motion of water molecules . In classical physics in principle a deterministic path might be found molecule by molecule but the measurement error would make it irrelevant.

If the quantum density matrix is effectively diagonal it means that the wave function of particle A has no projection on particle N and can be treated independently within measurement uncertainties and of course the Heisenberg uncertainty principle.

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The article isn't bad, but the author rather presents hypothesis as fact. For example look at this bit:

If a particle interacts with just a single photon, for example, then the two particles will enter an entangled state and that will be enough to trigger the onset of decoherence (for example a single photon entering the double-slit experiment will be enough to destroy the interference pattern).

This is a hypothesis about what occurs. An "attempt to explain". And one could attempt to explain things in a totally different fashion, such as this:

The photon is a many-paths wave which interferes with itself. But when you detect it at one slit you perform something like an optical Fourier transform on it and convert it into something pointlike that passes through that slit only. Then when you detect it at the screen you perform something like an optical Fourier transform on it and convert it into something pointlike that appears as a dot on the screen.

enter image description here See Steven Lehar's Intuitive Explanation of Fourier Theory.

He does the same sort of thing here:

This is described in the book Quantum Enigma: "Whenever any property of a microscopic object affects a macroscopic object, that property is 'observed' and becomes a physical reality".

What exists exists. Things don't pop into existence and become a physical reality just because you look at them. And I'm afraid this is wrong:

Imagine you throw a rock in the sea off the coast of the United Kingdom. After the initial big splash, the ripples dissipate and apparently disappear. But of course, they haven't really disappeared. The ripples have decreased in size, and they have mixed and interfered with other waves, but they have not disappeared. Two weeks later, on the rocky shore of Tierra del Fuego off the Argentinian coast, one of the small waves washing to shore is maybe an imperceptible fraction of one micron higher because of that rock you threw.

If it was right, you would never see a flat calm ocean. And you do.

But IMHO this flaw with the analogy isn't really the issue here. I think the real issue is that decoherence is just another attempt to explain, and you shouldn't presume that it's fact.

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