Effective Dynamics of Open Quantum Systems: Stochastic vs Unitary Models

[atyy]

There is no sharp cut but a smooth fuzzy boundary, of the same kind as the boundary between the Earth's atmosphere and interplanetary space. The bigger one makes the detector the more classical it becomes as the more accurate become the pointer positions. There is no difference between a classical expectation and a quantum expectation, except by a factor of $\sqrt{\hbar/N}$, and this factor is expected because of the differences between quantum predictions and classical predictions. The difference vanishes in the classical limit $\sqrt{\hbar/N}\to 0$, as it should.

But classical particles have positions. Quantum particles do not. So quantum averaging is producing reality from non-reality.

Another way to see the problem is: why should coarse graining a wave function result in a position? It should simply result in a coarse-grained wave function.

 

[A. Neumaier]

But classical particles have positions. Quantum particles do not. So quantum averaging is producing reality from non-reality.

Only if you assume that the state is unreal. If the state is taken as real, quantum averaging produces position reality from state reality. There is nothing obscure about this.

In fact, single massive particles must have position, too. There can be no doubt that the electrons produced by a small source are in the lab where the source is. This is a position statement, though not a very accurate one. But the uncertainty is consistent with the Heisenberg uncertainty relation. Thus particles have an uncertain position, given by the same formula as the pointer position in statistical mechanics - just applied to the case N=1. In this way, the statistical mechanics interpretation of measurement given in a post in another thread generalizes and becomes my thermal interpretation of quantum mechanics.

 

[atyy]

Only if you assume that the state is unreal. If the state is taken as real, quantum averaging produces position reality from state reality. There is nothing obscure about this.

In fact, single massive particles must have position, too. There can be no doubt that the electrons produced by a small source are in the lab where the source is. This is a position statement, though not a very accurate one. But the uncertainty is consistent with the Heisenberg uncertainty relation. Thus particles have an uncertain position, given by the same formula as the pointer position in statistical mechanics - just applied to the case N=1. In this way, the statistical mechanics interpretation of measurement given in a post in another thread generalizes and becomes my thermal interpretation of quantum mechanics.

Yes, from what you say, if quantum averaging produces position reality from state reality, then single massive particles must have position too.

But then what is special about position - it seems that single massive particles must have momentum too!

As far as I can tell, if you really work this out, then you will get either Bohmian Mechanics or Continuous Spontaneous Localization interpretations. My guess is you are really doing something like CSL, since CSL derives the equations derived under Copenhagen and continuous measurement similar to what B&P do.

http://arxiv.org/abs/math-ph/0512069 p3
"As extended to nondemolition observations continual in time [9]–[15], this approach consists in using the quantum filtering method for the derivation of nonunitary stochastic wave equations describing the quantum dynamics under the observation. Since a particular type of such equations has been taken as a postulate in the phenomenological theory of continuous reduction and spontaneous localization [16]–[20], the question arises whether it is possible to obtain this equation from an appropriate Schroedinger equation."

 

[A. Neumaier]

what is special about position - it seems that single massive particles must have momentum too!

Nothing is special about position; an electron has momentum, too. in fact, the momentum of an electron in a beam is quite well-defined.

you are really doing something like CSL, since CSL derives the equations derived under Copenhagen and continuous measurement similar to what B&P do.

B&P effectively show that the additional dynamical assumptions in CSL are in fact unnecessary. Note by the way that the Markov assumption is used also in the usual decoherence arguments, once (as in realistic models) the dynamics is no longer exactly solvable. In particular, it is also needed in the Bohmian derivation of the Born rule, according to the discussion here. Thus B & P effectively show that also the Bohmian hidden variables can be dispensed with.

My thermal interpretation is slightly different from B&P, and I believe more appropriate since I don't give a special status to the wave function but give reality to the density operator. This avoids the problems you had mentioned with the arbitrariness in the choice of the basis. I haven't yet worked out the corresponding modifications needed in the argument by B & P but I expect no additional difficulties. The equations resulting for the piecewise deterministic stochastic process for the reduced density operator should be identical with those discovered (using collapse arguments) by Wiseman and Milburn.

In contrast to classical mechanics and Bohmian mechanics, the thermal interpretation has (in agreement with experiment) never infinitely precise positions and momenta - these are always inherently uncertain, but with a computable uncertainty.

This is the reason why no dynamical laws are needed in addition to the standard shut-up-and-calculate formulas. Thus the thermal interpretation is an interpretation of QM and QFT without any additional baggage beyond what is used anyway informally in the applications. In particular, unlike in Bohmian mechanics and CSL, there is no need to give position a distinguished role - unless it is selected by the measurement setup as a relevant variable.

It also means that the difficulties of classical field theory with charged point particles, and the difficulties with classical relativistic multiparticle theories are absent since there are no point objects. Uncertain position naturally goes hand in hand with extendedness with a somewhat fuzzy boundary - in the same way as we can locate the position of a city like Vienna on an atlas, but not very accurately due to its extendedness.

 

[vanhees71]

Just one remark. I lost a bit track of the discussion, but one thing I understood, namely that you bring in another approximation, namely the Markov approximation. In the usual physicist's approach to derive the Boltzmann transport equation from the full quantum Kadanoff-Baym equation you have to (a) do a gradient expansion. To "forget" memory, i.e., to make the dynamics Markov, is somewhat subtle. See, e.g.,

Knoll, Jörn, Ivanov, Yu. B., Voskresensky, D .N.: Exact conservation laws of the gradient expanded Kadanoff-Baym equations, Ann. Phys. 293, 126–146, 2001
http://arxiv.org/abs/nucl-th/0102044

 

[A. Neumaier]

you bring in another approximation, namely the Markov approximation. In the usual physicist's approach to derive the Boltzmann transport equation from the full quantum Kadanoff-Baym equation you have to (a) do a gradient expansion. To "forget" memory, i.e., to make the dynamics Markov, is somewhat subtle. See, e.g., [...] Exact conservation laws

There are different ways to make the Markov approximation and one may obtain different results depending on what one neglects. The gradient expansion is just one of them. How difficult the Markov approximation is depends on the particular system modeled. In the context of measurements and decoherence one usually discusses a simplified situation where the detector (including environment) is treated as a quantum system with few degrees of freedom (two level atom, or one scalar particle) coupled to a harmonic heat bath. This is much simpler than deriving the Boltzmann equation from a QFT, where one must take care to ensure that the conservation laws remain valid. In the Boltzmann equation, from the microscopic conservation laws, only entropy conservation is sacrificed; in measurement, energy conservation fails anyway for the measured subsystem, hence one has more freedom.

 

[A. Neumaier]

I lost a bit track of the discussion

You can get again on track by starting at post #80.

 

[A. Neumaier]

the density operator. This avoids the problems you had mentioned with the arbitrariness in the choice of the basis. I haven't yet worked out the corresponding modifications needed in the argument by B & P but I expect no additional difficulties. The equations resulting for the piecewise deterministic stochastic process for the reduced density operator should be identical with those discovered (using collapse arguments) by Wiseman and Milburn.

I just saw that the argument for the density matrix case is indicated in the 2002 book by B &P on p.348-350.

 

[atyy]

B&P effectively show that the additional dynamical assumptions in CSL are in fact unnecessary. Note by the way that the Markov assumption is used also in the usual decoherence arguments, once (as in realistic models) the dynamics is no longer exactly solvable. In particular, it is also needed in the Bohmian derivation of the Born rule, according to the discussion here. Thus B & P effectively show that also the Bohmian hidden variables can be dispensed with.

I read the rest of your comments above too. As a side note, my feeling is that you are not representing B&P's interpretation correctly - I had read other bits of their work before this thread, and my impression was that they were never addressing foundations - they were working within a Copenhagen-style interpretation, just as all conventional "continuous measurement" work does.

However, let me address your interpretation of B&P - if one removes the observer of Copenhagen, and assigns a massive particle a continuous trajectory that exists even without the observer, that is a hidden variable interpretation.

 

[A. Neumaier]

assigns a massive particle a continuous trajectory that exists even without the observer, that is a hidden variable interpretation.

Where are the hidden variables?

I don't assign a continuous trajectory but a tube defining the location. The uncertain pointer position at time $t$ is $\langle X(t)\rangle\pm\sigma_{X(t)}$, where $X(t)$ is the Heisenberg position operator for the center of mass of the pointer at time $t$ and the expectation is taken in the Heisenberg state of the universe (or any sufficiently isolated piece of it). Thus the uncertain position is fully determined by the state - but it is an uncertain position rather than one exact to infinite precision, as for a point. Point trajectories are unphysical, even in classical relativistic mechanics. Thus one shouldn't expect them to exist in quantum mechanics either. They are appropriate only as an approximate description.

 

[Demystifier]

A paper by Weinberg that appeared today may be relevant here:
http://lanl.arxiv.org/abs/1603.06008

 

[atyy]

A paper by Weinberg that appeared today may be relevant here:
http://lanl.arxiv.org/abs/1603.06008

Do you think the infinite time limit taken by Weinberg is similar to that taken by Hepp in http://retro.seals.ch/digbib/view?pid=hpa-001:1972:45::1204 ? Hepp's beautiful result is consistent with the existence of a measurement problem, because a measurement only occurs in infinite time if there is no collapse.

 

[A. Neumaier]

we only get a definite result in infinite time, contrary to observation.

In QM, we also discuss scattering in terms of infinite time, although it is observable already at very short (but not too short) time. In classical statistical mechanics we also get phase transitions only for infinite volume, although they are observed at finite and small (but not too small) volume.

The point is that infinity may already be a very good approximation to a small number when the true dynamics happens at even shorter time or volume scale. Taking the infinite limit just serves to make the mathematics simpler and the effects more definite. Just as in finite volume, observed phase transitions would have smooth and not the observed, essentially discontinuous response functions. The same holds for collapse - at finite times it would be less than perfect, which means very awkward to use.

 

[Demystifier]

Do you think the infinite time limit taken by Weinberg is similar to that taken by Hepp in http://retro.seals.ch/digbib/view?pid=hpa-001:1972:45::1204 ? Hepp's beautiful result is consistent with the existence of a measurement problem, because a measurement only occurs in infinite time if there is no collapse.

In the Weinberg's paper we have exponentially decreasing terms which are negligible at large but finite times, so the infinite time is not essential. But note that "collapse" in Weinberg's paper is not the same thing as "collapse" in most of the literature. For Weinberg, the "collapse" is merely a transition to a density matrix without non-diagonal terms.

 

[stevendaryl]

Yes, but then one still has the classical/quantum cut or macroscopic/microscopic cut - the macroscopic centre of mass is not the classical expectation, since the macroscopic pointer is made of microscopic particles that do not have positions.

Right. Expectation values by themselves are not sufficient for something to have an approximately definite position. To give an extreme example: If there is a 50% probability of my being in Seattle and a 50% chance of being in New York City, then it is not very meaningful to say that, approximately, my location is somewhere in South Dakota. Or to use another example: If my left foot is in boiling water and my right foot is in ice, it's not really meaningful to say that my feet are in water that is approximately 122 degrees F.

Coarse-graining is only going to give you approximately classical objects (with approximately definite positions) if the probability distribution is strongly peaked around the expectation value. That's what I don't understand about environmentally induced collapse. Why should the distribution become strongly peaked? Is there really an argument that it should be? I don't see how there could be such an argument, using just the minimal interpretation of quantum mechanics (just unitary evolution). My feeling is that the mathematics that shows such an effect must, in some nonobvious way, be incorporating a collapse assumption.

 

[Mentz114]

Right. Expectation values by themselves are not sufficient for something to have an approximately definite position. To give an extreme example: If there is a 50% probability of my being in Seattle and a 50% chance of being in New York City, then it is not very meaningful to say that, approximately, my location is somewhere in South Dakota. Or to use another example: If my left foot is in boiling water and my right foot is in ice, it's not really meaningful to say that my feet are in water that is approximately 122 degrees F.

Coarse-graining is only going to give you approximately classical objects (with approximately definite positions) if the probability distribution is strongly peaked around the expectation value. That's what I don't understand about environmentally induced collapse. Why should the distribution become strongly peaked? Is there really an argument that it should be? I don't see how there could be such an argument, using just the minimal interpretation of quantum mechanics (just unitary evolution). My feeling is that the mathematics that shows such an effect must, in some nonobvious way, be incorporating a collapse assumption.

I broadly agree with what you say but it is even more complicated because "Coarse-graining is only going to give you approximately classical objects (with approximately definite positions) if the probability distribution is strongly peaked around the expectation value" is not always the case. For instance a transition from microscopic to macroscopic can be caused by diffusion in which a probability peak is smoothed and spread. Sometimes it takes coherence not decoherence to get to the classical regime.

 

[A. Neumaier]

Expectation values by themselves are not sufficient for something to have an approximately definite position. To give an extreme example: If there is a 50% probability of my being in Seattle and a 50% chance of being in New York City, then it is not very meaningful to say that, approximately, my location is somewhere in South Dakota. Or to use another example: If my left foot is in boiling water and my right foot is in ice, it's not really meaningful to say that my feet are in water that is approximately 122 degrees F.

Note that expectations never come alone. Expectations together with the standard deviations are sufficient for something to have an approximately definite position. Even in your extreme examples, both together (with the usual $3\sigma$ rule in statistics) give an under the given circumstances quite appropriate description of your uncertain location (namely ''somewhere in North America'') in the first case and the assumed description by a single temperature in the second case.

the mathematics that shows such an effect must, in some nonobvious way, be incorporating a collapse assumption.

The mathematics incorporates, in some nonobvious way (through the Markov approximation), an assumption of chaotic behavior, which together with the multimodality coming from the nonlinear evolution of the variables kept in the reduced description of a metastable system, produce dissipation and settlement in one of the local minimizers corresponding to definite measurements. It is not so different from what you get when you bend a classical, rotationally symmetric rod using a force in the direction of the axis of the rod: if the force exceedss the threshold where the straight rod becomes metastable only, the rod will bend into a random, but definite direction. Randomness from Hamiltonian dynamics plus the tiniest amount of uncertainty about the deviations from perfect symmetry.

 

[vanhees71]

In the Weinberg's paper we have exponentially decreasing terms which are negligible at large but finite times, so the infinite time is not essential. But note that "collapse" in Weinberg's paper is not the same thing as "collapse" in most of the literature. For Weinberg, the "collapse" is merely a transition to a density matrix without non-diagonal terms.

This is the ONLY meaning "collapse" can have in physical terms. That's what I fight against all the time in our discussions: There is no instantaneous collapse, but it's all quantum dynamics of relevant (coarse-grained) observables of the macroscopic system (and measurement devices are macroscopic systems). I'll have a careful look at Weinberg's paper this evening. He's THE no-nonsense physicist, and I hope the usual clear statement against any "esoterics" will be given (as in his marvelous textbook on quantum mechanics).

 

[vanhees71]

I broadly agree with what you say but it is even more complicated because "Coarse-graining is only going to give you approximately classical objects (with approximately definite positions) if the probability distribution is strongly peaked around the expectation value" is not always the case. For instance a transition from microscopic to macroscopic can be caused by diffusion in which a probability peak is smoothed and spread. Sometimes it takes coherence not decoherence to get to the classical regime.

Classical behavior is, according to quantum theory, always only "approximate", but it's a so damn good approximation for very many macroscopic observables that it took at least 300 years (from Newton to Heisenberg) to figure out that classical physics is not the full story!

 

[Demystifier]

This is the ONLY meaning "collapse" can have in physical terms.

Perhaps, but we have another, more common expression for that: decoherence.

 

[atyy]

This is the ONLY meaning "collapse" can have in physical terms. That's what I fight against all the time in our discussions: There is no instantaneous collapse, but it's all quantum dynamics of relevant (coarse-grained) observables of the macroscopic system (and measurement devices are macroscopic systems). I'll have a careful look at Weinberg's paper this evening. He's THE no-nonsense physicist, and I hope the usual clear statement against any "esoterics" will be given (as in his marvelous textbook on quantum mechanics).

Unfortunately you are quite wrong (probably due to being mislead by Ballentine and Peres). Weinberg abandoned you already in his textbook, stating collapse explicitly with an equation. Rubi has also abandoned you by using consistent histories, which does not have deterministic unitary evolution as fundamental.

 

[vanhees71]

Yes, I never use the workd "collapse" in context of quantum theory. The only collapse in physics that's really interesting is the gravitational collapse of a star at the end of its life, and that's really happening, while the collapse of the quantum state is just fiction!

 

[vanhees71]

Unfortunately you are quite wrong (probably due to being mislead by Ballentine and Peres). Weinberg abandoned you already in his textbook, stating collapse explicitly with an equation. Rubi has also abandoned you by using consistent histories, which does not have deterministic unitary evolution as fundamental.

Weinberg leaves the answer about which interpretation is correct (or even whether there is a completely satisfactory interpretation) open in his book. He spends quite some time in his book to show that the Born rule cannot be satisfactorily derived from the other postulates (concerning kinematics and dynamics of standard quantum theory):

[Weinberg: Lectures on Quantum mechanics, p. 96]
There is nothing absurd or inconsistent about the decoherent histories
approach in particular, or about the general idea that the state vector serves only
as a predictor of probabilities, not as a complete description of a physical system.
Nevertheless, it would be disappointing if we had to give up the “realist” goal of
finding complete descriptions of physical systems, and of using this description
to derive the Born rule, rather than just assuming it. We can live with the idea
that the state of a physical system is described by a vector in Hilbert space rather
than by numerical values of the positions and momenta of all the particles in the
system, but it is hard to live with no description of physical states at all, only
an algorithm for calculating probabilities. My own conclusion (not universally
shared) is that today there is no interpretation of quantum mechanics that does
not have serious flaws, and that we ought to take seriously the possibility of find-
ing some more satisfactory other theory, to which quantum mechanics is merely
a good approximation.

I would have stopped after the 1st sentence, but of course you can have the view that physics should be more than a mere accurate quantitative description of the world. I don't think so. That's the only purpose of physics (or more generally the natural sciences). Everything else (call it metaphysics or perhaps even religion) is another level of human experience and does not belong to science but to the subjective side of our worldview.

 

[atyy]

Weinberg leaves the answer about which interpretation is correct (or even whether there is a completely satisfactory interpretation) open in his book. He spends quite some time in his book to show that the Born rule cannot be satisfactorily derived from the other postulates (concerning kinematics and dynamics of standard quantum theory):

Sure, but the interpretation you are advocating is not among those he considers having a chance to be correct. Weinberg allows minimal interpretation with collapse, MWI, BM, consistent histories etc. He does not allow Ballentine's and Peres's erroneous handwaving.

 

[vanhees71]

Where does he say so? If I interpret Weinberg correctly, he does not advocate the collapse flavor of Copenhagen Interpretations as satisfactory, because it claims that there's something outside of the validity of quantum dynamics. If I understand him right, there's no solution to this problem yet, i.e., either one should be able to derive the collapse within quantum dynamics, and then it's superfluous or it may be that quantum theory is incomplete and thus must be substituted by some more comprehensive theory that makes the collapse superfluous in some other way.

I think Ballentine's and Peres's view are far from erroneous. They just take QT in the minimal interpretation literally and consider it as a formalism to predict prbabilities. They don't claim "completeness" in the one or the other sense for quantum theory either. I think that's a very valid and pragmatic view, reflecting how QT is used in the labs around the world to invent new experiments, analyze them and describe their outcome. So far the result is that minimally interpreted QT without any collapse explains the outcome of the experiments. The experiments are silent on what may be behind the probabilities predicted by QT, and even whether there is something else behind it. So what?

 

[atyy]

Where does he say so? If I interpret Weinberg correctly, he does not advocate the collapse flavor of Copenhagen Interpretations as satisfactory, because it claims that there's something outside of the validity of quantum dynamics. If I understand him right, there's no solution to this problem yet, i.e., either one should be able to derive the collapse within quantum dynamics, and then it's superfluous or it may be that quantum theory is incomplete and thus must be substituted by some more comprehensive theory that makes the collapse superfluous in some other way.

I think Ballentine's and Peres's view are far from erroneous. They just take QT in the minimal interpretation literally and consider it as a formalism to predict prbabilities. They don't claim "completeness" in the one or the other sense for quantum theory either. I think that's a very valid and pragmatic view, reflecting how QT is used in the labs around the world to invent new experiments, analyze them and describe their outcome. So far the result is that minimally interpreted QT without any collapse explains the outcome of the experiments. The experiments are silent on what may be behind the probabilities predicted by QT, and even whether there is something else behind it. So what?

The collapse in standard terminology is just a way of calculating probabilities, and not necessarily physical. You make the error of asserting that it cannot be physical. Ballentine and Peres mislead people into believing that the minimal interpretation can do without a classical/quantum cut, without state reduction, and has no measurement problem.

BTW, you have understood Weinberg correctly. What he says counters the claim you, Ballentine and Peres make that collapse can be derived from deterministic unitary quantum dynamics in a minimal interpretation.

Also, the idea that if collapse cannot be derived from deterministic unitary quantum dynamics, then a more comprehensive theory is needed is one way of stating that the minimal interpretation does have a measurement problem.

 

[vanhees71]

But there is no measurement problem! We use QT to analyze our experiments and find it to be correct to high accuracy. So where is a problem? It's for sure not in physics. Maybe you (and obviously also Weinberg) are dissatisfied from a philosophical point of view. If you then can come up with a more comprehensive theory, it's great, but it is very unlikely to find a new theory just pondering philosophical quibbles, at least I'm not aware of any great theory being found without a sound and solid foundation in true observations in nature.

 

[atyy]

But there is no measurement problem! We use QT to analyze our experiments and find it to be correct to high accuracy. So where is a problem? It's for sure not in physics. Maybe you (and obviously also Weinberg) are dissatisfied from a philosophical point of view. If you then can come up with a more comprehensive theory, it's great, but it is very unlikely to find a new theory just pondering philosophical quibbles, at least I'm not aware of any great theory being found without a sound and solid foundation in true observations in nature.

It is fine to assert that there is no measurement problem, in the sense that the classical/quantum cut is a feature. However, in any minimal interpretation there is a classical/quantum cut. One can say this is not a problem, as Bohr and Heisenberg did, and I respect that view (actually I quite like it). One can also say there is a problem, as Dirac did. Either way, no one is disagreeing with the success of the theory.

However, what is not acceptable is to say there is no classical/quantum cut and no collapse in a minimal interpretation.

 

[vanhees71]

Where is a classical/quantum cut by applying standard quantum mechanics to predict probabilities and then measure them by looking at large ensembles of preparations?

 

[A. Neumaier]

no collapse in a minimal interpretation.

There is no collapse in a minimal interpretation since nothing at all is asserted about a single system. The statistical assertions about multiple measurements are equivalent to the assertion that the diagonal entries of the density matrices are the observed probabilities. No cut is needed either since the observations are made by quantum systems in the environment, reading macroscopic pointer variables, i.e., expectation values.

Weinberg's view that the collapse in an open system is equivalent to the decay of the off-diagonal entries of the density matrix in the pointer basis is also in agreement with the minimal interpretation - where only the density matrix matters, though not with the Copenhagen interpretation (which makes assertions about the state of single systems).

 

[A. Neumaier]

A paper by Weinberg that appeared today may be relevant here:
http://lanl.arxiv.org/abs/1603.06008

I am quite disappointed with his paper. He just assumes the reduced dynamics of an open system (rather than deriving it from unitarity, with the associated approximate nature this entails), and simply investigates conditions on the coefficients of the Lindblad equation that make it an acceptable description of a measurement process.

 

[atyy]

Where is a classical/quantum cut by applying standard quantum mechanics to predict probabilities and then measure them by looking at large ensembles of preparations?

If the wave function and collapse are not taken to be necessarily real, but the experimental apparatus and results are real, then we have to have a cut somewhere.

 

[Demystifier]

I am quite disappointed with his paper. He just assumes the reduced dynamics of an open system (rather than deriving it from unitarity, with the associated approximate nature this entails), and simply investigates conditions on the coefficients of the Lindblad euation that make it an acceptable description of a measurement process.

If you are disappointed that he didn't derive the Lindblad equation from first principles, well, he cited some references were it was done.
You might also want to see a derivation of the Lindblad equation in the Appendix of my
http://arxiv.org/abs/1502.04324 [JCAP 04 (2015) 002]

 

[A. Neumaier]

If you are disappointed that he didn't derive the Lindblad equation from first principles, well, he cited some references were it was done.
You might also want to see a derivation of the Lindblad equation in the Appendix of my
http://arxiv.org/abs/1502.04324 [JCAP 04 (2015) 002]

I was disappointed that Weinberg didn't really do anything new in the paper. I know many derivations of Lindblad equations under many different assumptions.

 

[Demystifier]

I was disappointed that Weinberg didn't really do anything new in the paper.

That was my impression too.