Is the principal objection to decoherence thermodynamical?

[canoe]

I have read any number of books regarding the various interpretations of QM. Personally I find certain of the interpretations (i.e many worlds, consciousness causing the collapse of the wave function, etc.) somewhat of a stretch. It would now seem that the theory of decoherence has elegantly answered the measurement problem and has aided us in sorting through the great quantum muddle. (This is not to say that decoherence is the final word on the subject.) But assuming that decoherence explains the measurement problem (i.e. that measurements are taking place continuously everywhere)...is the primarly weakness in decoherence the fact that any state, no matter how improbable, must eventually recur (Poincare recurrence). I believe that this is essentially the argument in a book I recently read by Alistair Rae entitled "Quantum Theory: Illusion or Reality".

If so, it would seem that all we need do is prove that Poincare Recurrences do not recur (which I admit is a tall order). However, If it could somehow be proven, it would also likely mean that the 2nd Law of Thermodynamics is a hard and fast law and that would mean that all processes are irreversible and that it is reversible processes that are merely an approximation.

 

[Count Iblis]

Microscopic reversibility actually implies the usual results of thermodynamics, including the second law (under reasonable assumptions). If you introduce a fundamental irreversibility in the laws, you better make sure it remains hidden and does not become accessible to observers, otherwise you could exploit it to make an effective Maxwell's demon.


If you imagine Maxwell's demon trying to move the fast molecules to one side of a container, then the reason why this cannot lead to a net decrease of the entropy of the universe is because the Demon uses information about the molecules to take action and that information will have to be ersed at some point. But you cannot really erase information if the laws of physics are exactly unitary. So, you end up dumping the information in the environment and increasing the entropy oif the environment because the environement can be in a larger number of possible states after the dumping of the memory.


But, if you postulate that the laws of physics are fundamentally irreversible, then that means that under a time evolution you can map two different initial states to the same final state. This then makes it impossible even in principle to look at a final state and deduce what the initial state was, so you cannot even in principle go back to the final state.


But if this is the case, then one cannot rule out a process in which the memory of the Demon is cleared without it having to dump the content of the memory in the environment. So, such a process would map any intitial state corresponding to any possible mempory state to a state in which the memory is empty.

 

[canoe]

Microscopic reversibility actually implies the usual results of thermodynamics, including the second law (under reasonable assumptions). If you introduce a fundamental irreversibility in the laws, you better make sure it remains hidden and does not become accessible to observers, otherwise you could exploit it to make an effective Maxwell's demon.


If you imagine Maxwell's demon trying to move the fast molecules to one side of a container, then the reason why this cannot lead to a net decrease of the entropy of the universe is because the Demon uses information about the molecules to take action and that information will have to be ersed at some point. But you cannot really erase information if the laws of physics are exactly unitary. So, you end up dumping the information in the environment and increasing the entropy oif the environment because the environement can be in a larger number of possible states after the dumping of the memory.


But, if you postulate that the laws of physics are fundamentally irreversible, then that means that under a time evolution you can map two different initial states to the same final state. This then makes it impossible even in principle to look at a final state and deduce what the initial state was, so you cannot even in principle go back to the final state.


But if this is the case, then one cannot rule out a process in which the memory of the Demon is cleared without it having to dump the content of the memory in the environment. So, such a process would map any intitial state corresponding to any possible mempory state to a state in which the memory is empty.

Thanks for the post. I am not acquainted with the concept of Maxwell's Demons but I think I get the drift. I don't believe that I said that the laws of physics are irreversible, but I did conjecture that if the 2nd Law of TD was absolute (Entropy always increases - an unmixed state does not recur - a decohered object will never recohere - the moon will never phase out so-to-speak) then that would seem to make our lives easier (QM interpretation wise).

I like the way you stated it in your third paragraph. But rather than postulate that "the laws of physics are irreversible" I might postulate that something might happen to the fundamental quantities of space, time and mass to render a past state irreversible. Just a crazy thought.

 

[Civilized]

But assuming that decoherence explains the measurement problem (i.e. that measurements are taking place continuously everywhere)...is the primarly weakness in decoherence the fact that any state, no matter how improbable, must eventually recur (Poincare recurrence).

The poincare recurrence holds for classical, finite dimensional systems. In particular, any system whose phase space volume is bounded and constant can only wander around for so long before it goes to a point which is arbitrarily near to a given point. But in quantum mechanics even a single electron in space is an infinite-dimensional system, and the recurrence theorem does not apply.

By the way, I'm glad you rejected the more fanciful explanations of the measurement postulate in favor of the decoherence effect. If nothing else this seems like a clear application of Ockam: "it is vain to do with more that which can be done with less."

 

[Count Iblis]

But in quantum mechanics even a single electron in space is an infinite-dimensional system, and the recurrence theorem does not apply.

This is not true. See e.g. section 7.1 on page 22 of this article:

http://arxiv.org/abs/hep-th/0208013

If you look at the proof, you see that it is essentially a consequence of the fact that time evolution in quantum mechanics is trivial, all that happens is that the amplititudes of the energy eigenstates move along circles in the complex plane at certain rates.

 

[Civilized]

Even as I wrote that I was thinking about the almost-periodicity of unitary evolution, it's good to know that this can be rigorously shown to cause a recurrence.

In the quantitative treatments of decoherence which I have seen, the 'environment' is treated as a quantum field, in which case the proof in the paper you provided does not apply (quantum fields evolve in a nonlinear way, and so cannot be decomposed into non-interacting Fourier modes the way that we do in first quantized QM).

I hope this does not seem like a cop out; any fans of the standard model agree that the quantum fields are the primary objects, and that particles are emergent from these fields. Look for Zurek's classic paper "Decoherence and the transition from quantum to classical" and in the last section (page 12 in my copy) titled "Decoherence: How long does it take?" Zurek says:

A tractable model of the environment is afforded by a collection of harmonic oscillators (Feynman and Vernon 1963, Dekker 1981, Caldeira and Leggett 1983a, 1983b,1985, Joos and Zeh 1985, Paz et al. 1993) or, equivalently, by a quantum field (Unruh
and Zurek 1989). If a particle is present, excitations of the field will scatter off the particle. The resulting “ripples” will constitute a record of its position, shape, orientation, and so on, and most important, its instantaneous location and hence its trajectory.

Therefore in a scientific sense there is no issue with talking about a particle in a background field, these terms are well defined in any experiment. In a philosophical sense, the only things which are 'real' are the quantum fields, and these obey nonlinear time evolution and hence do not recurr.

 

[canoe]

The poincare recurrence holds for classical, finite dimensional systems. In particular, any system whose phase space volume is bounded and constant can only wander around for so long before it goes to a point which is arbitrarily near to a given point. But in quantum mechanics even a single electron in space is an infinite-dimensional system, and the recurrence theorem does not apply.

By the way, I'm glad you rejected the more fanciful explanations of the measurement postulate in favor of the decoherence effect. If nothing else this seems like a clear application of Ockam: "it is vain to do with more that which can be done with less."

Civilized:

Thanks for the reply. You have clarified my topic perfectly.

Firstoff, let me preface by stating that I like decoherence. Its explanative...something you can sink your teeth into. The only problem would seem to be the possibility that a macro object could recohere. That would be as sight wouldn't it - to see the moon (or a tablespoon) simply dissappear?

When I first read about this (I believe it was first postulated by Nobel Laureate - Ilya Prigogine) I had the same thought as you - namely why should irreversibility necessarily be mandated for classical systems? I suspect that the simple explanation might be that QT is the more general theory. Therefore due to the correspondence principle whatever holds for the more restrictive theory must hold for the general theory (i.e. QT). Again, one could possibly make the argument that finite dimensional systems differ from entangled wavefunctions. But that would open up a whole new can of worms (and it seems that the Count may have already addressed this in his follow-up post). So let us just presuppose for a moment that (whether we agree with it or not) that the recurrence theorem does apply and that the Count is correct.

So it would seem, if we favor decoherence (and we don't like the notion of an object recohering) we must find an explanation that dictates that all processes are fundamentally irreversible (i.e. that the 2nd law of TD is a real law and not just a statistical one) and that reversible processs are estimations.

Here is a thought on which I would invite comment: If I had a box containing yellow peas (50 count) and green peas (1000 count) that were originally separated (i.e. all yellow peas on one side and all green peas on the other) and shook the box, the recurrence theorem would dictate that the unmixed state is allowed and therefore in the fullness of time must recur. However, what if I caused the box of peas to move within a gravitational field to say a point at which the gravitational field was weaker before the unmixed state recurred, would the recurrence theorem hold (i.e. should that be considered an equivalent state to the initial state).

I invite your comments and thanks for the thoughtful post.

 

[Count Iblis]

quantum fields evolve in a nonlinear way

Only in an semi-classical effective field treatment can you get a nonlinear evolution. In an exact treatment, you have to deal with the functional Schrödinger equation, which is linear.

E.g., you could say that the Euler-Heisenberg equation for the electromagnetic fields contain small nonlinear terms. While that's true, the wavefunctional which assigns an amplitude to any field configuration still evolves in a linear way.

 

[Count Iblis]

Also, note that any mention of the Poincaré recurrence theorem in the paper:

http://arxiv.org/abs/hep-th/0208013

would have been completely irrelevant if it were not valid in QFT.

 

[Civilized]

Only in an semi-classical effective field treatment can you get a nonlinear evolution. In an exact treatment, you have to deal with the functional Schrödinger equation, which is linear.

That's true for Fermion fields, but parts of the the standard model e.g. QCD have nonlinearity built into the Lagrangian itself.

Also, note that any mention of the Poincaré recurrence theorem in the paper...would have been completely irrelevant if it were not valid in QFT.

OK, but the proof in the paper only addresses QM, not QFT. I don't know if the recurrence theorem holds in a QFT framework, so I am not arguing for or against this. All I did was read the proof in the paper you linked and tried to fit that in the quantitative treatments of decoherence I have seen, all of which involve some QFT, so maybe the recurrence theorem does not apply.

Here is a thought on which I would invite comment: If I had a box containing yellow peas (50 count) and green peas (1000 count) that were originally separated (i.e. all yellow peas on one side and all green peas on the other) and shook the box, the recurrence theorem would dictate that the unmixed state is allowed and therefore in the fullness of time must recur. However, what if I caused the box of peas to move within a gravitational field to say a point at which the gravitational field was weaker before the unmixed state recurred, would the recurrence theorem hold (i.e. should that be considered an equivalent state to the initial state).

If you consider yourself external to the system, recurrence will not necessarily ocurr, since the classical recurrence theorem depends on a closed system with no forces that depend on time.

I hadn't thought about it too much, since obviously the recurrence theorem in any form is no threat to any experimental predictions, but it seems to me that the irreversibility is happening on the level of quantum fields.

 

[cesiumfrog]

is the primarly weakness in decoherence [...]Poincare recurrence[?]

No. Firstly, you are mistaken in thinking that there is any inconsistency between thermodynamics and Poincare recurrence, even in ideal classical systems. To see this you need to understand where the second law is derived from (it is just an approximation of the Fluctuation theorem, not a fundamental law). It may help to consider the entropy of toy 1D Brownian motion models, for which the dynamics are solved exactly.

The principle problem with decoherence would seem to be the preferred basis problem. To wit, why does a sugar molecule stay in a chirality (i.e., position) eigenstate, rather than an energy eigenstate? Other interpretations implicitly give special status to position (which is rarely noticed because such interpretations have logical problems more glaring than this), but MWI needs to justify it explicitly.

 

[canoe]

If you consider yourself external to the system, recurrence will not necessarily ocurr, since the classical recurrence theorem depends on a closed system with no forces that depend on time.

I hadn't thought about it too much, since obviously the recurrence theorem in any form is no threat to any experimental predictions, but it seems to me that the irreversibility is happening on the level of quantum fields.

Thanks! We basically agree.

Incidentally, if you were wondering about my box of gravitationally challenged peas...I was thinking of a bare hypothesis once put forth by P. M. Dirac. He conjectured that the Gravitational Constant might, somehow, weaken with time. I don't believe it survived as anything other than a historic endnote (although I don't believe that he ever conceded the possibility). Anyway, if we dare couple your response with Dirac's conjecture we would end up with a solid 2nd law of TD (and no Poincare recurrences).

BTW: Perhaps you could point me to a source Re: the recurrence theorem particularly its dependence on the system being entirely closed. Thanks in advance. You guys have been helpful.

 

[canoe]

No. Firstly, you are mistaken in thinking that there is any inconsistency between thermodynamics and Poincare recurrence, even in ideal classical systems. To see this you need to understand where the second law is derived from (it is just an approximation of the Fluctuation theorem, not a fundamental law). It may help to consider the entropy of toy 1D Brownian motion models, for which the dynamics are solved exactly.

The principle problem with decoherence would seem to be the preferred basis problem. To wit, why does a sugar molecule stay in a chirality (i.e., position) eigenstate, rather than an energy eigenstate? Other interpretations implicitly give special status to position (which is rarely noticed because such interpretations have logical problems more glaring than this), but MWI needs to justify it explicitly.

Thank you for the post.

I do understand the statistical basis of the second law, and to some extent, what Boltzman postulated regarding fluctuations. I also understand that there is no inconsistency between TD and Poincare Recurrences. The original topic was framed to aid me in understanding why a number of semi-technical (and non-technical) books on QT had drawn parallels between the reversibility/irreversibility of TD microstates and decoherence. Apparently decoherence somewhat originated from the work of Ilya Prigogine who was director of the Center for Statistical Mechanics and Thermodynamics in Belgium.

http://en.wikipedia.org/wiki/Ilya_Prigogine

Again thanks for your comments.

 

[Count Iblis]

That's true for Fermion fields, but parts of the the standard model e.g. QCD have nonlinearity built into the Lagrangian itself.

QFT is ordinary quantum mechanics applied to fields. Nonlinearity in the Lagrangian does not mean that you don't have a linear and unitary time evolution for the wave functional. The proof in the paper applies to a general (closed) system described by quantum mechanics and is thus valid in general.

In practice, of course, we don't formulate QFT as ordinary quantum mechanics, because working with a wavefunctional is awkward. But that doesn't mean that deep down, QFT is ordinary quantum mechanics that is now applied to a system with an infinite number of degrees of freedom, instead of a system with only a few degrees of freedom.

Actually, you can interpret the path integral as giving the wave functional. If you write down the path integral with an initial and final boundary condition on the fields, it gives you exactly the amplitude for an initial field configuration to evolve to a final field configuration. This is the exact analogue of the Schrödinger time evolution in ordinary quantum mechanics. This is what is linear, no matter how nonlinear the Lagrangian is.

I think what is misleading here is that when doing second quantization one takes a one particle wavefunction and promotes that to a quantum field. But the interpretation is that this field is the analoge of what a classical variable is in ordinary quantum mechanics. This then becomes an operator just like position and momentum are operators in ordinary quantum mechanics. But then the analogue of the wavefunction in ordinary QM is the wavefunctonal, but this is usually not introduced in QFT texts, as it can be avoided.

Then while you can get an effective one particle (or many particle) nonlinear Schödinger equation back from the Euler-Lagrange equations (e.g. after you have computed the effective action to a few loop), that is then an equation analogous to e.g. the Maxwell equaions, i.e. a classical equation of motion (which can contain quantum corrections).

It is classical because in an exact quantum description you cannot have a classical evolution of fields (commutator between a field and its time derivative does not vanish, so you have an uncertainty relation).

 

[Civilized]

Thanks for the explanation Count Iblis, you have made it clear the sense in which the time evolution in QFT is linear is the same sense that is required in the paper whose proof you linked. It looks like the explanation of decoherence that I had built for myself was flawed, and because of this thread I will have to re-open that investigation .

 

[intervoxel]

A fascinating thread indeed...

I've been playing with the possibility that Maxwel's demon doesn't necessarily increases the entropy of the system. This sort of "unconcious" demon, as a first approximation, could be modeled as

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A door that continuosly changes the opening angle between 0 and pi/2. This scheme can be shown to decrease entropy if you forget for a moment that the door doesn't cost anything and the walls are rigid.

In a more realistic model, this randomness could be obtained for free by letting the door move at will, subject to brownian motion. But now everything - door and wall - suffer brownian bombardment.

A computer simulation of this (classical) model reveals that, contrary to our intuition, nothing happens: Both sides of the box keep the same thermodynamic properties as time goes on (The wall and door were modeled by carbon (12) atoms and the gas was made of xenon (131) atoms)

I wonder if it could somehow be modeled using QM and an identical result confirmed?

 

[Civilized]

This sort of "unconcious" demon, as a first approximation, could be modeled as

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A door that continuosly changes the opening angle between 0 and pi/2. This scheme can be shown to decrease entropy if you forget for a moment that the door doesn't cost anything and the walls are rigid.

I've thought about it but I don't see why this unconscious demon would decrease the entropy.

A computer simulation of this (classical) model reveals that, contrary to our intuition, nothing happens: Both sides of the box keep the same thermodynamic properties as time goes on (The wall and door were modeled by carbon (12) atoms and the gas was made of xenon (131) atoms)

Not to sound smug, but this is what it looks to me like will happen. What is the argument that convinces some people that it will lower the entropy?

I wonder if it could somehow be modeled using QM and an identical result confirmed?

In QM the entropy is constant in between measurements, so we could consider the demon-door as a classical measuring device and treat the particles in the box quantum mechanically, that could be interesting.