What is really that density matrix in QM?

[DarMM]

if entropy represents an intrinsic property of a physical system, the ambiguity associated with the representation carried by a density matrix does not make it an appropriate tool for talking about entropy.

I don't understand, has anybody been discussing basing things on this decomposition?

 

[atyy]

The distinction between proper and improper mixed states breaks down in quantum field theory. See Section 4 of this review of entanglement and open systems in QFT:
https://arxiv.org/abs/quant-ph/0001107
Time evolution of any finite volume system will be non-unitary as everything is an open system in QFT.

Is it be possible to understand that as a case in which the state of every subsystem is an improper mixture?

If so, wouldn't the state of the total system still be pure, so that unitary evolution still applies to the total system, and governs the evolution of the subsystems?

 

[DarMM]

Is it be possible to understand that as a case in which the state of every subsystem is an improper mixture?

If so, wouldn't the state of the total system still be pure, so that unitary evolution still applies to the total system, and governs the evolution of the subsystems?

The total system would be the entire universe. This remains an open question in QFT whether there are global pure states. Complications involved are:

1. QED coloumb fields might always be mixed. This is the mathematically most intractable problem.
2. Such a pure state has no operational meaning. The theorized global purification might be a state over an algebra of self-adjoint operators that includes Wigner's friend type "observing macroscopic systems to the atomic scale" type observables which almost certainly lie outside the observable algebra. Thus over the true algebra of observables the state will still be mixed.
3. Poorly understood complications from QFT in curved backgrounds, e.g. the information loss problem.

 

[vanhees71]

It is time invariant hence stationary but leads to completely wrong predictions for thermal q-expectations such as the internal energy.

How do you come to this conclusion? It depends on the system!

Of course the MEM doesn't tell you what's the relevant information for a given system. That you have to determine yourself. The MEM is a very general principle. It's almost inevitable to make "objective" guesses based on the available information about the system.

Now 2 issues are usually raised against it, also in this thread.

(1) Choice of the "relevant observables" to be constraint in maximizing the entropy

Your criticism is quite common: Given only the constraint that you want a stationary state, constrains the possible choice of "relevant information" to be either averages of conserved quantities (like for energy in the canonical and grand-canonical ensemble; in the latter case you also give the average on one or seveal conserved charge-like quantities) or strict constraints of conservation laws (energy is strictly in an on macroscopic scales very small interval, as for the microcanonical ensemble).

That usually the Gibbs ensembles are preferred, i.e., giving at least constraints on $U=\langle \hat{H} \rangle$ and not some on any other function of $f(\hat{H})$, is due to the fact that one considers the thermodynamical state as a small system coupled (weakly) to some reservoir, defining the canonical and grand-canonical ensembles (depending on whether you allow for exchange of conserved charges or not). Treating than the closed system large (reservoir+system)-system as a microcanonical ensemble (where it doesn't matter, which function $f(\hat{H})$ you look at), you are inevitably let to the usual canonical or grand-canonical ensemble with $\langle \hat{H}_{\text{sys}} \rangle$ is the relevant constraint. In the thermodynamic limit higher cumulants don't play a role. For small systems it can be necessary to take such higher-order constraints into account. The same argument holds if you have other relevant conserved quantities to take into account: In the typical "reservoir situation" for macroscopic thermodynamical systems the additive conserved quantities are the relevant observables and usually not more general functions thereof. For details, see

https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.85.1115

and references therein.

This view is also solidified by the usual dynamical arguments using non-equlibrium descriptions of open quantum systems (master equations, transport equations, etc.): For short-range interactions in the collision term and truncating the BBGKY hierarchy (or the corresponding analogs for more detailed quantum descriptions; as the general Schwinger-Dyson hierarchy of QFT) at the lowest order leads to the standard equilibrium distributions (Bose-Einstein, Fermi-Dirac, Maxwell-Boltzmann), corresponding to the choice of additive conserved quantities in the MEM as discussed above.

(2) Choice of the information measure

This is the question, which type of entropy to use. The Shannon-Jaynes one in the physics context refers to the classical Boltzmann-Gibbs entropy. It's well-known that this doesn't work with lang-range forces present (electromagnetic, but that's pretty harmless, because in many-body systems you usually have Debye screening which comes to the rescue, but there's no such thing for gravity, and that's important for structure formation in the universe and our very existence). Here other (non-additive) entropy forms like Renyi or Tsallis entropies may be the better choice. As far as I know, there's however no generally valid dynamical argument as in the Boltzmann-Gibbs case.

 

[vanhees71]

OK

The same density matrice can represent different mixed states, i.e the spectral decomposition of a density matrix is not unique (unless the state is pure) and there are several ways to achieve the same density matrix by mixing pure states.

/Patrick

No! The statistical operator uniquely determines the state of the system, and its spectral decomposition is unique, if you use a complete set of compatible observables, including the statistical operator. It's a self-adjoint operator!

It's of course right that mixing pure states is not unique, but why should it be?

 

[microsansfil]

No! The statistical operator uniquely determines the state of the system, and its spectral decomposition is unique, if you use a complete set of compatible observables, including the statistical operator. It's a self-adjoint operator!

It's of course right that mixing pure states is not unique, but why should it be?

here



/Patrick

 

[atyy]

The total system would be the entire universe. This remains an open question in QFT whether there are global pure states. Complications involved are:

1. QED coloumb fields might always be mixed. This is the mathematically most intractable problem.
2. Such a pure state has no operational meaning. The theorized global purification might be a state over an algebra of slef-adjoint operators that includes Wigner's friend type "observing macroscopic systems to the atomic scale" type observables which almost certainly lie outside the observable algebra. Thus over the true algebra of observables the state will still be mixed.
3. Poorly understood complications from QFT in curved backgrounds, e.g. the information loss problem.

Is this related: https://arxiv.org/abs/1406.7304 ?

Is it really true then that there is no unitary time evolution in QFT? For measurement, I can buy that there are no mixed states. But I find it hard to buy that there is no unitary evolution. Really? Then there will be complications with the information loss problem, as you say.

 

[DarMM]

Is this related: https://arxiv.org/abs/1406.7304 ?

Is it really true then that there is no unitary time evolution in QFT? For measurement, I can buy that there are no mixed states. But I find it hard to buy that there is no unitary evolution. Really? Then there will be complications with the information loss problem, as you say.

It's probably true, see remark 15 (p.31) in this paper:
https://arxiv.org/abs/1412.5945

 

[atyy]

It's probably true, see remark 15 (p.31) in this paper:
https://arxiv.org/abs/1412.5945

But how about for flat spacetime? In the case of no no pure states in flat spacetime, can we have unitary time evolution?

 

[DarMM]

That's an open question due to issues with infrared representations in QED.

 

[atyy]

That's an open question due to issues with infrared representations in QED.

Well, maybe QED doesn't exist - is this also expected to be a problem in say Yang Mills? Do we have pure states and unitary time evolution in Yang Mills?

 

[vanhees71]

here
View attachment 248039

/Patrick

Sure, that's why I said, if you use a complete set of compatible observables including $\hat{\rho}$. As any operator, a given $\hat{\rho}$ is of course unique. It doesn't depend on it's representation in terms of a complete orthnormal system (CONS), i.e.,
$$\hat{\rho} = \sum_{ij} |i \rangle \langle i|\hat{\rho}|j \rangle \langle j|.$$
If now you have a $\hat{\rho}$ with degnerate eigenvalues $p_j$, there are of course arbitrarily many CONS. Let $|i,\alpha \rangle$ be one such CONS, but in all of them you get
$$\hat{\rho}=\sum_{i} p_i \sum_{\alpha} |i,\alpha \rangle \langle i,\alpha |.$$
The inner sum is just the projector to the uniquely defined "degenerate" eigenspace to the degenerate eigenvalue $p_i$. These projectors are independent of the chosen degenerate CONS since obviously for another CONS $\widetilde{|i,\alpha \rangle}$
$$\hat{P}_i=\sum_{\alpha} |i,\alpha\rangle \langle i,\alpha| = \widetilde{|i,\alpha \rangle}\widetilde{\langle i,\alpha|}.$$

 

[DarMM]

Well, maybe QED doesn't exist - is this also expected to be a problem in say Yang Mills? Do we have pure states and unitary time evolution in Yang Mills?

Yes for Yang-Mills in flat spacetime we should have global pure states.

As for QED not existing I've always found the arguments for this very weak. It's based on the existence of a perturbative Landau pole, but the Gross-Neveu model has a Landau pole perturbatively, but still exists as a well-defined QFT in the non-perturbative sense.

However all of this would still only be in the idealisation of flat space. In curved space for any theory there simply isn't unitary time evolution simply because that's not how time evolution can be modeled for field theories in general, it is to be replaced with the notion of algebraic automorphisms.

 

[DarMM]

I should say I'm slowly in the process of gathering all this info in a coherent form. What nonperturbative QFT in curved spacetime is actually like renders many debates about QM pointless or seriously recasts the issues and I think it would be useful for others to know.

 

[atyy]

I should say I'm slowly in the process of gathering all this info in a coherent form. What nonperturbative QFT in curved spacetime is actually like renders many debates about QM pointless or seriously recasts the issues and I think it would be useful for others to know.

Well, at least it won't affect Bohmian Mechanics :)

 

[DarMM]

Well, at least it won't affect Bohmian Mechanics :)

Well that's because the Bohmians are objectively correct as we all secretly know in our heart but deny in public.

 

[A. Neumaier]

Well, maybe QED doesn't exist - is this also expected to be a problem in say Yang Mills? Do we have pure states and unitary time evolution in Yang Mills?

YangMills has a much worse infrared problem than QED.

 

[A. Neumaier]

In the case of no no pure states in flat spacetime, can we have unitary time evolution?

In principle, yes: Mixed states do not by themselves force a nonunitary evolution.

 

[charters]

I should say I'm slowly in the process of gathering all this info in a coherent form. What nonperturbative QFT in curved spacetime is actually like renders many debates about QM pointless or seriously recasts the issues and I think it would be useful for others to know.

This would be very interesting to read, but I would still be cautious about drawing foundational conclusions (conclusions universal to all quantum theory) from curved QFT, where curvature is purely classical. It isn't clear to me that curved QFT is a fully consistent idea or tells us more truth than flat QFT or isn't a red herring for foundational purposes.

For example Wald says the observables in curved QFT are just local fields and particles are not really valid ideas. But in quantum gravity, local fields are not gauge invariant and not observable. And in string theory, the background curvature itself is created/defined as a coherent state in the Fock space of graviton strings. So, if this facial contradiction stands, I think QG has to trump curved QFT.

 

[DarMM]

Well in a sense then anything could be a red herring when it comes to drawing conclusions. QFT in flat spacetime doesn't take curvature into account, String and Loop QG are partially worked out ideas that could be completely wrong.

The point is more so that most foundational work uses a very idealised form of NRQM and thus misses many subtleties from QFT or even just a realistic application of NRQM. Most of the results I will discuss will be generic to QFT regardless of background.

 

[charters]

The point is more so that most foundational work uses a very idealised form of NRQM and thus misses many subtleties from QFT or even just a realistic application of NRQM. Most of the results I will discuss will be generic to QFT regardless of background.

Yes that's very fair. Look forward to it.

 

[Auto-Didact]

I should say I'm slowly in the process of gathering all this info in a coherent form. What nonperturbative QFT in curved spacetime is actually like renders many debates about QM pointless or seriously recasts the issues and I think it would be useful for others to know.

This issue was already addressed in Cao, 1999. Because QFT (based in SR) is explicitly an EFT and nothing more, it turns out automatically to actually be less interesting from a foundational perspective, i.e. the conceptual issues facing QFT seem at every twist and turn to be completely contingent on the idealized structures which have no clear relation to physical structures in general, nor any actually known strong mathematical basis of consensus coming directly from the practice of pure mathematics; this is the heavy price one pays for advocating operationalism, which w.r.t. QT of course has the minimal interpretation as its poster boy.

 

[DarMM]

explicitly an EFT and nothing more

What is his argument for this point?

less interesting

Less interesting than NRQM?

 

[atyy]

In principle, yes: Mixed states do not by themselves force a nonunitary evolution.

But from DarMM's post #70, it seems that even in flat spacetime for the Type III algebras with no pure states, the question of unitary evolution is open?

 

[Auto-Didact]

Well in a sense then anything could be a red herring when it comes to drawing conclusions. QFT in flat spacetime doesn't take curvature into account, String and Loop QG are partially worked out ideas that could be completely wrong.

The point is more so that most foundational work uses a very idealised form of NRQM and thus misses many subtleties from QFT or even just a realistic application of NRQM. Most of the results I will discuss will be generic to QFT regardless of background.

This is exactly why foundational physics methodology has - as Smolin and Hardy have argued - to be attempted as a deferential team effort which is per capita not too strongly focussed on overt specialization in possibly irrelevant technicalities, definitely not with any single dominant research programme dominating a field but instead with many possibly intercompatible frameworks, where the researchers should have no lasting ties to any single programme but instead rotate periodically. This type of research methodology has proven itself very effective in many other academic fields and human endeavors, but it is practically as far as one can imagine from typical physics training today.

 

[DarMM]

But from DarMM's post #70, it seems that even in flat spacetime for the Type III algebras with no pure states, the question of unitary evolution is open?

It is open. I know that Buchholz thinks there are global pure states and unitary time evolution.

 

[Auto-Didact]

What is his argument for this point?

The book is a collection of arguments made by Weinberg, Nelson, Shankar, Redhead, Teller et al.

From what I remember, there are three responses in the mathematical physics literature to QFT failing as a foundation of physics: 1) the EFT attitude, 2) string theory and 3) mathematics, i.e. the axiomatic, constructive and algebraic approaches. (NB: string theory is generally not even seen as a possible solution for it is conceptually very close to QFT, so close in fact that there is doubt that they actually differ conceptually in a non-trivial sense.)

The argument that QFT is an EFT and nothing more is simply the Wilsonian point of view, which itself is embracing a possibly infinite amount of stable limited domains of validity each with their own separate dynamics and ontology; logically this actually does away with reductionism itself.

Less interesting than NRQM?

Yes, less interesting than the standard problems of NRQM precisely because the conceptual issue in the QFT case is so muddied by foundationally irrelevant contingencies such that we end up with all these meta-problems, whereas in NRQM the issues in contrast are (or have become) quite clear and are therefore explicitly logically and mathematically solvable as demonstrable by the existence of BM and spontaneous collapse models.

 

[atyy]

It is open. I know that Buchholz thinks there are global pure states and unitary time evolution.

What is the status of energy conservation in these subsets of spacetime and the global spacetime? If energy conservation holds, could one have unitary evolution just writing (by analogy) $\dot{\rho} = -i[H,\rho]$ ?

 

[Auto-Didact]

The constructive approach which tries to find a nonpertubative formulation of renormalization is actually from the historical perspective, the most canonical mathematical approach; if quantum gravity exists this will be the mathematics needed to form the conceptual framework of the physical theory.

The problem however is again that the mathematical experience required for achieving the constructive goal is not the kind of mathematics that physicists tend to be familiar with, at least not how they conceptualize it; in stark contrast, most physicists seem to only have a very meager and very weakly generalizae grasp of the branches of mathematics required. Those more in the know tend to be applied or pure mathematicians, e.g. Tao, Villani and Klainerman; see here.

This is partly because the mathematics is actually relatively new, conceptually scattered across multiple fields and that the literature lacks consensus on terminology due to different applied specialists across the sciences reinventing the same mathematical objects without communicating the results to each other; all of this messiness had prevented direct standardization and hence has very much delayed the push into the curriculums even until this day (NB: I poll physics graduates yearly to see if things have changed).

 

[Auto-Didact]

What is the status of energy conservation in these subsets of spacetime and the global spacetime? If energy conservation holds, could one have unitary evolution just writing (by analogy) $\dot{\rho} = -i[H,\rho]$ ?

In general, there typically aren't any conservation laws of particular interest in open systems, with instead other kinds of equations being more interesting to characterize and understand the open system e.g. relative entropy. John Baez and his (former) students have written a lot on these topics.