A Effective Dynamics of Open Quantum Systems: Stochastic vs Unitary Models

[atyy]

Finally, in the paragraph containing (22), the system is described on the third level as a classical stochastic piecewise determinstic (drift and jump) process for the wave function in which the jumps depend stochastically on the measurement results. This is the quantum jump process discussed in post #1. The arguments in this section serve to demonstate that the three descriptions are in some sense equivalent, though the higher the level the more precise the description. In paticular, on the third level, the complete (reduced) quantum measurement process is fully described by the classical PDP, and hence has a fully classical ontology.

Thanks for the replies above, I read those too. I'm going back here to your comment on their other paper, the overview http://arxiv.org/abs/quant-ph/0302047. In their discussion around Eq 22, they do say:

"Physically, $\psi(t)$ represents the state of the reduced system which is conditioned on a specific readout of the measurement carried out on the environment. Consequently, the stochastic evolution depends on the measurement scheme used to monitor the environment."

So if that section applies to their derivation of the Chapman-Kolmogorov equation in http://omnibus.uni-freiburg.de/~breuer/paper/p4041.pdf, then I would expect the measurement of the environment somehow enters one of the assumptions they make, though at this point I am not sure where.

 

[A. Neumaier]

Thanks for the replies above, I read those too. I'm going back here to your comment on their other paper, the overview http://arxiv.org/abs/quant-ph/0302047. In their discussion around Eq 22, they do say:

"Physically, $\psi(t)$ represents the state of the reduced system which is conditioned on a specific readout of the measurement carried out on the environment. Consequently, the stochastic evolution depends on the measurement scheme used to monitor the environment."

So if that section applies to their derivation of the Chapman-Kolmogorov equation in http://omnibus.uni-freiburg.de/~breuer/paper/p4041.pdf, then I would expect the measurement of the environment somehow enters one of the assumptions they make, though at this point I am not sure where.

It is in the dynamics of the detector, which must include enough of the environment to produce irreversible results (and hence determines what is read out). B & P model the latter by assuming separated time scales and the validity of the Markov approximation - which hold only if the detector is big enough to be dissipative. (The latter is typically achieved by including in the detector a heat bath consisting of an infinite number of harmonic oscillators.) Since B & P make these assumptions without deriving them, their analysis holds for general dissipative detectors. But of course for any concrete application one must check (as always in statistical mechanics) that these assumptions are plausible.

In sufficiently idealized settings, these assumptions can actually proved rigorously, but this is beyond the scope of the treatment by B & P. Rigorous results (without the discussion of selecive measurement but probably sufficient to establish the assumptions used by B & P) were first derived by Davies 1974 and later papers with the same title. See also the detailed survey:
H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits. Reviews of Modern Physics, 52 (1980), 569.

In the cases treated by B & P, the discrete PDP process corresponds to photodetection, which measures particle number, which has a discrete spectrum; the diffusion processes correspond to homodyne or heterodyne detection, which measure quadratures, which have a continuous spectrum. B & P obtain the latter from the PDP by a limiting process in the spirit of the traditional approach treating a continuous spectrum as a limit of a discrete spectrum.

 

[atyy]

It is in the dynamics of the detector, which must include enough of the environment to produce irreversible results (and hence determines what is read out). B & P model the latter by assuming separated time scales and the validity of the Markov approximation - which hold only if the detector is big enough to be dissipative. (The latter is typically achieved by including in the detector a heat bath consisting of an infinite number of harmonic oscillators.) Since B & P make these assumptions without deriving them, their analysis holds for general dissipative detectors. But of course for any concrete application one must check (as always in statistical mechanics) that these assumptions are plausible.

In sufficiently idealized settings, these assumptions can actually proved rigorously, but this is beyond the scope of the treatment by B & P. Rigorous results (without the discussion of selecive measurement but probably sufficient to establish the assumptions used by B & P) were first derived by Davies 1974 and later papers with the same title. See also the detailed survey:
H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits. Reviews of Modern Physics, 52 (1980), 569.

In the cases treated by B & P, the discrete PDP process corresponds to photodetection, which measures particle number, which has a discrete spectrum; the diffusion processes correspond to homodyne or heterodyne detection, which measure quadratures, which have a continuous spectrum. B & P obtain the latter from the PDP by a limiting process in the spirit of the traditional approach treating a continuous spectrum as a limit of a discrete spectrum.

So it seems the collapse assumption comes with the Markovian assumption.

In these treatments, the measurement problem is not solved, because unitary evolution alone has no observable outcome (such as a particle position). If we are using the collapse to say when the particle acquires a position, then it is the Markov approximation which causes collapse which determines when a detection is made - which is not satisfactory since it doesn't seem reasonable for an approximation to cause reality.

 

[A. Neumaier]

So it seems the collapse assumption comes with the Markovian assumption.

In these treatments, the measurement problem is not solved, because unitary evolution alone has no observable outcome (such as a particle position). If we are using the collapse to say when the particle acquires a position, then it is the Markov approximation which causes collapse which determines when a detection is made - which is not satisfactory since it doesn't seem reasonable for an approximation to cause reality.

The Markov assumption is used also in classical statistical mechanics to derive hydromechanics or the Boltzmann equation. Thus you seem to propose that classical statistical mechanics is not satisfactory, too. This is a defendable position. But at least the arguments show that to go from unitarity to definite (i.e., irreversible) outcomes in Hamiltonian quantum mechanics one doesn't need to assume more than to go from reversibility to irreversibility in Hamiltonian classical mechanics.

Moreover, I had given references that prove the Markov assumption in the low coupling infinite volume limit. Thus it is sometimes derivable and not an assumption. Your criticism that it is an approximation only is moot since for pointer readings it suffices to have approximately definite outcomes, and these are guaranteed by statistical mechanics for macroscopic observables (with an accuracy of $N^{-1/2}$ where $N$ is of the order of $10^{23}$ or more).

 

[atyy]

The Markov assumption is used also in classical statistical mechanics to derive hydromechanics or the Boltzmann equation. Thus you seem to propose that classical statistical mechanics is not satisfactory, too. This is a defendable position. But at least the arguments show that to go from unitarity to definite (i.e., irreversible) outcomes in Hamiltonian quantum mechanics one doesn't need to assume more than to go from reversibility to irreversibility in Hamiltonian classical mechanics.

Moreover, I had given references that prove the Markov assumption in the low coupling infinite volume limit. Thus it is sometimes derivable and not an assumption. Your criticism that it is an approximation only is moot since for pointer readings it suffices to have approximately definite outcomes, and these are guaranteed by statistical mechanics for macroscopic observables (with an accuracy of $N^{-1/2}$ where $N$ is of the order of $10^{23}$ or more).

It isn't the same. In classical statistical mechanics, a particle has a definite outcome (eg. position) at all times. This is not true in quantum mechanics. It is not sufficient to have approximately definite outcomes.

 

[A. Neumaier]

it is the Markov approximation which causes collapse which determines when a detection is made - which is not satisfactory since it doesn't seem reasonable for an approximation to cause reality.

This is not a cause as in causality but only a cause in the sense of explanation. Thus your claim amounts to: ''it is the Markov approximation which explains collapse which determines when a detection is made - which is not satisfactory since it doesn't seem reasonable for an approximation to explain reality", and here the second half of the sentence is no longer reasonable. Everywhere in physics we explain reality by making approximations. This is the only way we can explain anything at all!

 

[A. Neumaier]

It is not sufficient to have approximately definite outcomes.

Why not? One cannot read a pointer very accurately.

 

[atyy]

Why not? One cannot read a pointer very accurately.

In classical mechanics there is an underlying sharp reality (eg. Newtonian mechanics). Then our inability to read the reality accurately is taken care of by coarse graning and probability. The coarse graning does not cause reality to appear. Reality exists before the coarse graning is done.

In contrast, in quantum mechanics, the sharp reality of a unitarily evolving quantum state is not enough, because it does not specify eg. position or whatever definite measurement outcome is seen. The measurement outcome is part of reality, so it seems that the wave function does not specify all of reality. Consequently, if collapse appears by coarse graning, then the coarse graning is causing reality to appear, which is quite different from classical mechanics.

 

[A. Neumaier]

In classical mechanics there is an underlying sharp reality (eg. Newtonian mechanics). Then our inability to read the reality accurately is taken care of by coarse graning and probability. The coarse graning does not cause reality to appear. Reality exists before the coarse graning is done.

In contrast, in quantum mechanics, the sharp reality of a unitarily evolving quantum state is not enough, because it does not specify eg. position or whatever definite measurement outcome is seen. The measurement outcome is part of reality, so it seems that the wave function does not specify all of reality. Consequently, if collapse appears by coarse graining, then the coarse graning is causing reality to appear, which is quite different from classical mechanics.

Just as in classical mechanics, only the Markov property is assumed. The jump process follows - hence collapse.

Nothing causes reality to appear - reality is, and was before anyone dreamt of quantum mechanics. Whatever is done in the paper is done on paper only - therefore explaining things, not causing anything! Coarse graining explains collapse, and hence explains why QM matches observed reality.

Similarly: In classical mechanics the underlying reality is strictly conservative. There is no dissipation of energy, though the latter characterizes reality. To have dissipation, one must postulate an additional friction axiom that is the classical analogue of the collapse. However, friction is found to arise from the Markov approximation. Thus in your words, classical coarse graining is causing friction to appear - which is not satisfactory since it doesn't seem reasonable for an approximation to cause the reality of friction. In my words, understanding that friction comes from coarse graining is as big an insight as that collapse comes from coarse graining. In both cases, it bridges the difference in the dynamics of an isolated system and that on an open system. The explanation by coarse graining is in both cases fully quantitative and consistent with experiment, hence has all the features a good scientific explanation should have.

 

[atyy]

Just as in classical mechanics, only the Markov property is assumed. The jump process follows - hence collapse.

Nothing causes reality to appear - reality is, and was before anyone dreamt of quantum mechanics. Whatever is done in the paper is done on paper only - therefore explaining things, not causing anything! Coarse graining explains collapse, and hence explains why QM matches observed reality.

Similarly: In classical mechanics the underlying reality is strictly conservative. There is no dissipation of energy, though the latter characterizes reality. To have dissipation, one must postulate an additional friction axiom that is the classical analogue of the collapse. However, friction is found to arise from the Markov approximation. Thus in your words, classical coarse graining is causing friction to appear - which is not satisfactory since it doesn't seem reasonable for an approximation to cause the reality of friction. In my words, understanding that friction comes from coarse graining is as big an insight as that collapse comes from coarse graining. In both cases, it bridges the difference in the dynamics of an isolated system and that on an open system. The explanation by coarse graining is in both cases fully quantitative and consistent with experiment, hence has all the features a good scientific explanation should have.

Don't focus on collapse. Focus on the measurement outcome, which needs no collapse. If one has a unitarily evolving wave function, at what point in time does the particle acquire a position?

It is different from classical physics where the particle has a position, before any coarse graining that makes friction appear.

 

[A. Neumaier]

Don't focus on collapse. Focus on the measurement outcome, which needs no collapse. If one has a unitarily evolving wave function, at what point in time does the particle acquire a position?

It is different from classical physics where the particle has a position, before any coarse graining that makes friction appear.

Both in classical mechanics and in quantum mechanics, the system has a state, which is its only reality. Measurements reveal part of this reality to a certain accuracy. It is a matter of modeling how the measurement results are related to the true reality - the state. In the statistical mechanics of $N$-particle systems, what is measured (both in classical and in quantum mechanics) is the expectation of a macroscopic operator, to an accuracy of order $O(N^{-1/2})$. This is enough to give well-defined pointer readings. Thus no collapse is needed to make the pointer acquire a well-defined position. As a consequence of having definite macroscopic outcomes (plus the Markov approximation) one finds that the dynamics of the subsystem is described by a PDP.

But although the pointer reading is a position measurement of the pointer, what is measured about the particle is not its position but the variable correlated with the pointer reading - which is the photon number or the quadrature. Particle position is as indeterminate as before. Indeed, investigation of the PDP process shows that the collapsed states created by the PDP are approximate eigenstates of the number operator or the quadrature. Thus the PDP can be interpreted in Copenhagen terms as constituting the repeated measurement of particle number or quadrature.

 

[atyy]

Both in classical mechanics and in quantum mechanics, the system has a state, which is its only reality. Measurements reveal part of this reality to a certain accuracy. It is a matter of modeling how the measurement results are related to the true reality - the state. In the statistical mechanics of $N$-particle systems, what is measured (both in classical and in quantum mechanics) is the expectation of a macroscopic operator, to an accuracy of order $O(N^{-1/2})$. This is enough to give well-defined pointer readings. Thus no collapse is needed to make the pointer acquire a well-defined position. As a consequence of having definite macroscopic outcomes (plus the Markov approximation) one finds that the dynamics of the subsystem is described by a PDP.

But although the pointer reading is a position measurement of the pointer, what is measured about the particle is not its position but the variable correlated with the pointer reading - which is the photon number or the quadrature. Particle position is as indeterminate as before. Indeed, investigation of the PDP process shows that the collapsed states created by the PDP are approximate eigenstates of the number operator or the quadrature. Thus the PDP can be interpreted in Copenhagen terms as constituting the repeated measurement of particle number or quadrature.

Referring to the position of the pointer makes no difference - when does the pointer acquire a position?

 

[A. Neumaier]

Referring to the position of the pointer makes no difference - when does the pointer acquire a position?

A macroscopic pointer always has a position, given according to statistical mechanics by the expectation of the operator $\bar X$ corresponding to the center of mass of its $N\gg 1$ particles, to an accuracy of order $N^{-1/2}$ by the law of large numbers. So nothing needs to be acquired - for the pointers I know, this accuracy is much better than the actual reading possible.

 

[atyy]

A macroscopic pointer always has a position, given according to statistical mechanics by the expectation of the operator $\bar X$ corresponding to the center of mass of its $N\gg 1$ particles, to an accuracy of order $N^{-1/2}$ by the law of large numbers. So nothing needs to be acquired - for the pointers I know, this accuracy is much better than the actual reading possible.

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.

 

[A. Neumaier]

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.

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.

 

[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.