Is the Quantum/Classical Boundary the most important question in Physics?

[A. Neumaier]

decoherence combined with relativistic constraints generates a superselection rule among macroscopic properties.

I'd be interested in references substantiating this. Or is it just your impression?

 

[atyy]

So originally I quoted Bohr saying that anything can be treated in a quantum way. Are you still disagreeing with this being Bohr's opinion or something else?

No.

I don't understand what you mean by "everything requires a classical treatment". In Quantum Theory the treatment of microscopic systems require an external system to select the Boolean algebra so as to have well defined events/a well defined statistical model. Some degrees of freedom, for example charge, total angular momentum and macroscopic properties, do not seem to require such an external system since their probability theory is Kolmogorovian. However all of this is predicted/given by quantum theory itself, what do you mean by they "require a classical treatment"?

Quantum theory has a classical-quantum cut. Hence the use of quantum theory itself means the assumption of a classical "something".

No, although some relation. First of all it is not state reduction, since you'd still have each outcome with a certain probability, up until you update the state yourself after an observation.

But are you talking about the reduced density matrix?

Secondly it's more that decoherence combined with relativistic constraints generates a superselection rule among macroscopic properties.

Could you give a reference?

 

[EPR]

The pure states form a vector space so of course superposition's are real - its in the very definition of a vector space. Charge is a scaler observable so is always the same when observed.

Thank
Bill

When observed... But what happens to electron's charge when for example an electron is in a state of definite momentum and so is in a superposition of states of many many different positions?

I am not sure if understand this correctly - how do we know electrical charge can induce superselection(and say resolve the measurement problem) if we don't know what happens to charge between observations/measurements?

 

[A. Neumaier]

But what happens to electron's charge when for example an electron is in a state of definite momentum and so is in a superposition of states of many many different positions?

An electron always has charge $-e$, independent of the state it is in.

 

[DarMM]

I'd be interested in references substantiating this. Or is it just your impression?

Even charge superselection is not "proved" as such. Nor lepton number superselection. Do you consider the usual arguments where by the appropriate interference observables can be shown to be unphysical acceptable? If so I can provide references, if not I'll ask more about what you would consider acceptable.

 

[DarMM]

Quantum theory has a classical-quantum cut. Hence the use of quantum theory itself means the assumption of a classical "something".

Describe this cut explicitly though. To my mind in modern quantum theory we require a system to select the Boolean frame to constitute events, however this is not an invocation of classical mechanics. It's that QM doesn't have a single sample space, thus something must select the sample space to give a well-define set of outcomes. That something is the "frame-selector" in one application doesn't mean quantum theory can't be applied to it in another application.
Thus anything can be given a quantum treatment.

But are you talking about the reduced density matrix?

Could you give a reference?

Not exactly with reference to the first part. For the second I shall after the discussion with @A. Neumaier above to see what he is looking for.

 

[EPR]

An electron always has charge $-e$, independent of the state it is in.

Indeed, charge has only 1 possible value so cannot be in a supperposition of different states.

But... since charge also behaves non-classically and obeys the Rules of Quantum Mechanics, how do we know it may lead to superselection? We must assume, right?
In the end, John von Neumann and Niels Bohr had the best clarity of these concepts and this is why their ideas found their way into the textbooks. And Everett's didn't.
Is decoherence taught in textbooks?

 

[A. Neumaier]

Even charge superselection is not "proved" as such. Nor lepton number superselection. Do you consider the usual arguments where by the appropriate interference observables can be shown to be unphysical acceptable? If so I can provide references, if not I'll ask more about what you would consider acceptable.

I just wanted to know what you consider as evidence for your previous assertion since I haven't seen decoherence discussed in a relativistic context.

Isn't the charge superselection rule a theorem?

 

[DarMM]

Isn't the charge superselection rule a theorem?

Yes basically. I should say that wasn't intended to imply doubt about charge superselection. There are remaining issues with characterizing plausible or permitted states, for which there are no stringent criterion. Issues about if QED exists do the appropriately constructed local field operators in a rigorous analogue of the Gupta-Beuler gauge obey enough of the expected properties of that formalism.

Under such "assumptions" as such it is proved here:
https://aip.scitation.org/doi/10.1063/1.1666601
However these arguments are so plausible that this is in essence a proof.

I just wanted to know what you consider as evidence for your previous assertion since I haven't seen decoherence discussed in a relativistic context.

It's the proof given in Chapter 7 of Omnès's text as I mentioned before. Whereby the interference observables for a real device would require a second device with at least $\mathcal{O}\left(10^{10^{18}}\right)$ atoms. Thus such observables do not belong to the observable algebra due to both the actual limit of material in the universe and relativistic reasons, e.g. under General Relativity are so large as to collapse in on themselves and under Special Relativity are too large to operate on the time scales required.

Again Chapter 7 of Omnès's book or Chapter 4 of Streater's Lost Causes (only noticed there that he cites you in that book!)

 

[atyy]

Describe this cut explicitly though. To my mind in modern quantum theory we require a system to select the Boolean frame to constitute events, however this is not an invocation of classical mechanics. It's that QM doesn't have a single sample space, thus something must select the sample space to give a well-define set of outcomes. That something is the "frame-selector" in one application doesn't mean quantum theory can't be applied to it in another application.
Thus anything can be given a quantum treatment.

As far as I can tell, that is the same as saying that every application of quantum mechanics has a classical-quantum cut. Frame selector = classical apparatus.

 

[atyy]

It's the proof given in Chapter 7 of Omnès's text as I mentioned before. Whereby the interference observables for a real device would require a second device with at least $\mathcal{O}\left(10^{10^{18}}\right)$ atoms. Thus such observables do not belong to the observable algebra due to both the actual limit of material in the universe and relativistic reasons, e.g. under General Relativity are so large as to collapse in on themselves and under Special Relativity are too large to operate on the time scales required.

Again Chapter 7 of Omnès's book or Chapter 4 of Streater's Lost Causes (only noticed there that he cites you in that book!)

Which part of Chapter 7? Does it require that one adopt the Consistent Histories interpretation?

 

[DarMM]

As far as I can tell, that is the same as saying that every application of quantum mechanics has a classical-quantum cut. Frame selector = classical apparatus.

The thing is I'm not really sure what is "classical" about it. In an application of QM you need more information for a well posed statistical problem than in classical probability theory. However I'm not sure how this prevents anything from being described by QM.

That's what I don't understand, what prevents a quantum treatment. Give me a system to which QM cannot be applied.

Which part of Chapter 7? Does it require that one adopt the Consistent Histories interpretation?

All of Chapter 7 really. It also requires some lemmas and results from Chapter 6.
Omnès describes a form of Copenhagen in the book, but the argument doesn't require it.

 

[atyy]

The thing is I'm not really sure what is "classical" about it. In an application of QM you need more information for a well posed statistical problem than in classical probability theory. However I'm not sure how this prevents anything from being described by QM.

The term "classical" is just traditional language (Landau & Lifshitz). If you don't like it, you can call it the observer or the measurement apparatus. Whatever it is, the Born rule applies when there is a measurement, but QM itself does not say when a measurement occurs. This is just the traditional measurement problem. The topic of this thread really only makes sense as asking "Is there a measurement problem?"

So my take is that both of these mean the same thing:
Q: Is there a measurement problem. A: Yes
Q: Is there a classical-quantum cut. A: Yes

That's what I don't understand, what prevents a quantum treatment. Give me a system to which QM cannot be applied.

Well, the quantum treatment itself must leave the observer out. So QM cannot apply to the observer. For example, can you apply QM to yourself (your whole self)?

 

[A. Neumaier]

the proof given in Chapter 7 of Omnès's text as I mentioned before. Whereby the interference observables for a real device would require a second device with at least $\mathcal{O}\left(10^{10^{18}}\right)$ atoms. Thus such observables do not belong to the observable algebra due to both the actual limit of material in the universe and relativistic reasons, e.g. under General Relativity are so large as to collapse in on themselves and under Special Relativity are too large to operate on the time scales required.

This is of little weight in the present context since limitations of the existing universe affect just one particular state, whereas a superselection rule is a property of states in general.

 

[DennisN]

Is there any more important question in Physics than this one? [Quantum/Classical Boundary]

Very easy. The most important question in physics is why we don't have hoverboards yet. I'm tired of carrying supplies from stores to my home, and a hoverboard would make my life so much easier.

Seriously, though...

Personally, I think the most important question is why QM and GR don't play well together. Clearly one or the other or both need modification and so far no one has been able to do it.

I also think that is one of the most important questions. And thus, how QM works in (or together with) a dynamic GR spacetime. And then also what the implications of this are for the interior of black holes.

I would like to mention another problem which I personally find deeply interesting, and that is the cosmological constant problem (see e.g. http://aapt.scitation.org/doi/10.1119/1.17850).
And connected to this is also the question of dark energy and the accelerated expansion of the Universe.

Edit: I'd also like to add that I personally understand the cosmological constant problem only on a basic level; I don't have deep enough knowledge about QFT and GR. But I still find it absolutely fascinating.

 

[DarMM]

This is of little weight in the present context since limitations of the existing universe affect just one particular state, whereas a superselection rule is a property of states in general.

What do you mean?

Superselection rules are properties of the observable algebra not states. In fact isn't that the point, that there is more to the physics than just the looking at the states alone can tell you due to restrictions on the observable algebra. If you see Streater's book he calls this a superselection rule.

I also don't understand what you mean by "limitations...affecting one particular state".

 

[DarMM]

The term "classical" is just traditional language (Landau & Lifshitz). If you don't like it, you can call it the observer or the measurement apparatus. Whatever it is, the Born rule applies when there is a measurement, but QM itself does not say when a measurement occurs. This is just the traditional measurement problem.

I said this back in my original post where I said QM tells us neither which event occurs nor the set from it is drawn and that in an experiment we select the Boolean frame with that choice lying outside the theory.

That's why I'm confused, what's the point you're disagreeing with.

Well, the quantum treatment itself must leave the observer out. So QM cannot apply to the observer.

In one application the system that constitutes the selection of the Boolean frame is not given a quantum states. It is mathematically represented by a selection of a Boolean subalgebra, not a statistical operator/density matrix.

However in another application it can be given a quantum state. Thus there is nothing which cannot be treated by QM.

 

[atyy]

I said this back in my original post where I said QM tells us neither which event occurs nor the set from it is drawn and that in an experiment we select the Boolean frame with that choice lying outside the theory.

That's why I'm confused, what's the point you're disagreeing with.

None that I'm aware of. I thought this was clarified many posts back. Basically, there is no disagreement, but for the purposes of this thread it is important that while in one sense, perhaps everything is quantum, in another sense, not everything is quantum. So just trying to use language that is clear.

In one application the system that constitutes the selection of the Boolean frame is not given a quantum states. It is mathematically represented by a selection of a Boolean subalgebra, not a statistical operator/density matrix.

However in another application it can be given a quantum state. Thus there is nothing which cannot be treated by QM.

And can you describe yourself with quantum mechanics?

 

[DarMM]

So just trying to use language that is clear.

From my perspective using Landau and Lifshitz style language is more unclear, even though it has historical precedent. People will take the "cut" to represent some division in systems to which QM applies and it's hard to point to aspects of the formalism itself that represent this "cut".

Where as quantum probability lacking a selection mechanism for both the event and the sample space from which it is drawn is directly related to the formalism and reflects Kochen-Specker contextuality and doesn't leave the idea that (the macro dofs of) a stone is outside quantum theory.

And can you describe yourself with quantum mechanics?

As a matter of practice no, just like any complex macroscopic system such as our devices.
However there is nothing in the formalism forbidding describing me in principal.

Unless this is some kind of question about consciousness.

 

[atyy]

From my perspective using Landau and Lifshitz style language is more unclear, even though it has historical precedent. People will take the "cut" to represent some division in systems to which QM applies and it's hard to point to aspects of the formalism itself that represent this "cut".

Where as quantum probability lacking a selection mechanism for both the event and the sample space from which it is drawn is directly related to the formalism and reflects Kochen-Specker contextuality and doesn't leave the idea that (the macro dofs of) a stone is outside quantum theory.

As a matter of practice no, just like any complex macroscopic system such as our devices.
However there is nothing in the formalism forbidding describing me in principal.

Unless this is some kind of question about consciousness.

Do you believe that there is a measurement problem in quantum mechanics?

 

[DarMM]

Do you believe that there is a measurement problem in quantum mechanics?

People mean many things by that phrase, some split it into two or three separate problems that they then call "small" or "big" and sometimes different groups can mean the same thing but one group views it as a problem and the other views it simply as the way things are, i.e. a property of the theory.

So it depends on what you mean.

 

[A. Neumaier]

What do you mean?

Superselection rules are properties of the observable algebra not states. In fact isn't that the point, that there is more to the physics than just the looking at the states alone can tell you due to restrictions on the observable algebra. If you see Streater's book he calls this a superselection rule.

I also don't understand what you mean by "limitations...affecting one particular state".

Statements about ''the actual limit of material in the universe'' are properties of the state, not the observable algebra. The empty universe and our universe are different states with vastly different amount of material.

On the level of the observable algebra, superselection rules are related to the center of the algebra, and the center does not seem to have any relation to decoherence, as far as I know. That's why I queried you for more information.

The environment induced superselection rules are heavily state-dependent and strictly speaking do not deserve their name.

 

[romsofia]

The "cut" doesn't even need to be brought up when talking about Copenhagen, because Copenhagen doesn't deal with it. Yes, it's mystical that apparently we could make the "cut" appear anywhere in the measurement process, and we get the same results. That's because no where in the formalism does it tell us where to break off, where does the measurement *actually* occur. Apparently, at some point, a break from this quantum system occurs, and a classical outcome becomes reality. So, Copenhagen isn't the interpretation to look at if you care about that. Or in the words of Robert Georch...

But, in any case, no difficulty arises from the failure of the Copenhagen interpretation, in the first experiment, to specify whether the geiger counter was “classical” or “quantum”. The claim of the interpretation is just that:
if you specify to it with sufficient precision what you will do in making a measurement, it will tell you the probability distribution for the results of the measurement.

It does precisely this in every case. It does not have to deal with what “really happens”. It is, in some sense, “consistent”

 

[DarMM]

The environment induced superselection rules are heavily state-dependent and strictly speaking do not deserve their name.

Superselection rules are often defined by the statement that for two sets of states $S_A$ and $S_B$ say, then there is no element of the observable algebra $\mathcal{A}$ that connects these two sets. That's what is going on here. Why is this not a superselection rule then?

 

[A. Neumaier]

Superselection rules are often defined by the statement that for two sets of states $S_A$ and $S_B$ say, then there is no element of the observable algebra $\mathcal{A}$ that connects these two sets. That's what is going on here. Why is this not a superselection rule then?

Because decoherence does not prove this for environment induced superselection rules. At least I haven't seen such a proof, did you?

 

[DarMM]

Because decoherence does not prove this for environment induced superselection rules. At least I haven't seen such a proof, did you?

It shows that the required self-adjoint operator involves systems of $\mathcal{O}\left(10^{10^{18}}\right)$ particles that would have to violate relativity. These don't exist clearly, so this self-adjoint operator isn't an element of the observable algebra.

 

[A. Neumaier]

It shows that the required self-adjoint operator involves systems of $\mathcal{O}\left(10^{10^{18}}\right)$ particles. These don't exist clearly, so this self-adjoint operator isn't an element of the observable algebra.

These don't exist only in a particular state - namely the Heisenberg state specifying our universe. But the observable algebra allows many states in which these exist!

 

[DarMM]

These don't exist only in a particular state - namely the Heisenberg state specifying our universe. But the observable algebra allows many states in which these exist!

It does? The states also require a violation of relativity. How could the observable algebra allow them?

 

[A. Neumaier]

It does?

Yes. A free QFT allows states with arbitrarily many particles.

The states also require a violation of relativity.

This is a different statement, which I do not believe since decoherence is usually discussed in a strictly nonrelativistic framework.

Where precisely is this proved? Page numbers please.

 

[vanhees71]

But in every application there is a cut, so one could just as well say that everything requires a classical treatment. Basically, there is a classical-quantum cut, the topic of this thread.

Where in the description of the Stern-Gerlach experiment and its outcome is the necessity for a cut? You can describe it pretty easily completely by solving the Schrödinger equation for a particle in an inhomogeneous magnetic field of the appropriate kind.

 

[A. Neumaier]

Where in the description of the Stern-Gerlach experiment and its outcome is the necessity for a cut? You can describe it pretty easily completely by solving the Schrödinger equation for a particle in an inhomogeneous magnetic field of the appropriate kind.

The magnetic field and the detector are described in classical terms.

 

[DarMM]

Just to be clear, this is definition of superselection rules given by Wightman in "Superselection Rules: Old and New":

We shall say that a superselection rule operates between subspaces if there are neither spontaneous transitions between their state vectors (i.e. if a selection rule operates between them) and if, in addition to this, there are no measurable quantities with finite matrix elements between their state sectors

In the same paper Wightman also refers to "Superselection rules induced by the environment". So it seems to be considered valid by both Streater and Wightman that there can be superselection rules induced by the environment. Can you perhaps say where they are wrong with this?

 

[DarMM]

Yes. A free QFT allows states with arbitrarily many particles

I don't understand how this is relevant.

This is a different statement, which I do not believe since decoherence is usually discussed in a strictly nonrelativistic framework.

Where precisely is this proved? Page numbers please.

Omnes discusses it on p.269 of his book. He doesn't give it in full detail, but it is easy enough to spell out.

An object of $10^{10^{18}}$ particles would need to be at least several quadrillion light years to not collapse in on itself. At this size it cannot operate fast enough (due to the speed of light) to have full reversible control of the first device, whose thermal fluctuations and so on operate on much smaller time scales.

 

[Morbert]

The magnetic field and the detector are described in classical terms.

But does this imply a classical treatment is required? Omnes states that empirical data are 'best expressed in classical terms' but he still associates them with a quantum mechanical description of the detector.

 

[kith]

However sometimes a state of the form:

$$|\psi\rangle = \frac{1}{\sqrt{2}}\left(|a\rangle + |b\rangle\right)$$​

simply represents classical "ignorance" based probability. Such as when |a⟩|a⟩ and |b⟩|b⟩ belong to different superselection sectors.

If |a \rangle and |b \rangle belong to different superselection sectors, doesn't your definition in post #82 imply that |\psi \rangle can't be prepared in the first place?

 

[DarMM]

If |a \rangle and |b \rangle belong to different superselection sectors, doesn't your definition in post #84 imply that |\psi \rangle can't be prepared in the first place?

Mixed states can be prepared. Post #84 isn't mine, obviously this is just a minor typo, which one of my posts are you referring to?

 

[kith]

Mixed states can be prepared.

Yes but then why did you write down a ket?

I meant post #82 (Thanks, I edited my last post) where you gave the definition of superselection rules.

 

[DarMM]

Yes but then why did you write down a ket?

It's still a vector state, i.e. it is a sum in the Hilbert space of two pure states. It's just that sometimes kets are actually mixed states. That's the "interesting" part about superselection rules.

 

[vanhees71]

Usually a pure state is defined as being represented by a projector, i.e., obeying $\hat{\rho}^2=\hat{\rho}$, i.e., there's a normalized vector, defined up to a phase factor, with $\hat{\rho}=|\psi \rangle \langle \psi|$. There's a one-to-one mapping between Hilbert-space vectors modulo a phase (rays) and the pure states.

In which sense can kets represent mixed states and what has it to do with superselection rules? Don't superselection rules simply say that certain superpositions cannot describe perparable (pure) states? E.g., usually one argues that one cannot have the superposition of states with half-integer and integer spin, because rotations by angles $2 \pi$ should be a symmetry?

 

[kith]

Where in the description of the Stern-Gerlach experiment and its outcome is the necessity for a cut? You can describe it pretty easily completely by solving the Schrödinger equation for a particle in an inhomogeneous magnetic field of the appropriate kind.

The cut is between what you as the user of quantum mechanics consider to be the quantum system and everything else (including the measurement apparatus and yourself).

In a simple description of the Stern-Gerlach experiment, you cut between the spin degree of freedom and everything else. So your quantum mechanical system is a simple two-level system.

In the more sophisticated version you refer to above, you move the cut behind the magnets. So your quantum mechanical system consists of a spin-1/2 particle and the magnet^1. Note that the measurement result gets recorded only when part the quantum system interacts with the screen. This is why in this description, the screen alone constitutes the external measurement apparatus which isn't included in the quantum description.

Can the cut be shifted further? In principle, sure. If you start your experiment, isolate and leave the lab, you could take the spin-1/2 particle, the SG apparatus and the whole lab to be the quantum system which evolves unitarily until you break the isolation by opening the lab door.

Is there a limit how far you can shift the cut even in principle? Yes. If you are doing experiments, you can't include yourself in the description of these experiments.

______
^1 Actually, a semiclassical approximation is used, so the magnet isn't treated completely quantum mechanically.

 

[DarMM]

In which sense can kets represent mixed states and what has it to do with superselection rules?

I'd have to go into the representation theory of the observable algebra. Basically in more advanced contexts pure states and kets aren't the same thing, superselection being one aspect of this.
However I might cover this in a separate thread or Insight.

 

[kith]

It's still a vector state, i.e. it is a sum in the Hilbert space of two pure states.

Yes, mathematically, this state may be possible. But if we consider what states can be actually prepared, this state seems unphysical to me. If we prepare an arbitrary initial state by measuring a complete set of commuting observables of our system, there neither exists a Hamiltonian which leads to your state by unitary time evolution nor do open system dynamics lead to it.

 

[DarMM]

Yes, mathematically, this state may be possible. But if we consider what states can be actually prepared, this state seems unphysical to me. If we prepare an arbitrary initial state by measuring a complete set of commuting observables of our system, there neither exists a Hamiltonian which leads to your state by unitary time evolution nor do open system dynamics lead to it.

Do you mean evolution from an initial pure state under unitary dynamics does not lead to this state? Of course pure states evolve into pure states under unitary dynamics and this state is not pure.

That's the point of superselection rules, that not every sum of two pure states is another pure state.

All of this is immediate from the claim that the state is mixed despite it being a ket. What aspect are disagreeing with or pointing out?

 

[A. Neumaier]

Just to be clear, this is definition of superselection rules given by Wightman in "Superselection Rules: Old and New":

In the same paper Wightman also refers to "Superselection rules induced by the environment". So it seems to be considered valid by both Streater and Wightman that there can be superselection rules induced by the environment. Can you perhaps say where they are wrong with this?

I need to reread the paper to see what they claim with which justification.

It's still a vector state, i.e. it is a sum in the Hilbert space of two pure states. It's just that sometimes kets are actually mixed states. That's the "interesting" part about superselection rules.

Can you explain how a vector state is a mixed state? I understand how a mixed state can be pure in a bigger Hilbert space constructed by the GNS construction, but what you say is the opposite...

I'd have to go into the representation theory of the observable algebra. Basically in more advanced contexts pure states and kets aren't the same thing, superselection being one aspect of this.
However I might cover this in a separate thread or Insight.

I am looking forward to this. For now, can you just give a reference?

 

[DarMM]

Can you explain how a vector state is a mixed state? I understand how a mixed state can be pure in a bigger Hilbert space constructed by the GNS construction, but what you say is the opposite...
...
For now, can you just give a reference?

I think you may have terminology confused. A mixed state is any state which, as an algebraic state, is the sum of another two states, i.e. for all observables:
$$\omega\left(A\right) = c_{1}\omega_{1}\left(A\right) + c_{2}\omega_{2}\left(A\right)$$
A pure being one for which this is not true.
This is equivalent to having vanishing (pure) or nonvanishing (mixed) Von Neumann Entropy.

The GNS construction then allows one to express any state, pure or mixed, as a ket/vector state.
See Robert M. Wald "Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics" p.83:

Note that the GNS construction always expresses any state (pure or mixed) as a vector state

Thus a vector state can be pure or mixed. However a pure state (entropy is zero) is never mixed (entropy non-zero).

 

[vanhees71]

In the SGE you cannot locate the cut between the spin of the atom and everything else. The measurement of the spin component is based on a full entanglement between the atom's spin and its position by its dynamical evolution in presence of the magnetic field. Of course, here we use the semiclassical approximation in treating the magnetic field as classical. I'm not sure, whether a full QED description is feasible.

Even in the usual S-matrix formalism you'd describe the magnetic field as classical background field (it's an interesting question, whether there's a fully relativistic description of the SGE, which I'm not so sure about since the properties of spin are more complicated in the relativistic theory than in the non-relativistic one to begin with).

The other thing with the preparability is also easy: If there's by some principle (like a SSR) no Hamiltonian exists that can prepare the state, then this state is indeed simply unphysical, because it cannot be observed. Isn't this the deeper reason for an SSR (like the spin or charge superselection rule) to begin with?

Another example that comes to my mind are neutrinos. You can't prepare mass eigenstates because there's no Hamiltonian available to do this. Consequently there are no neutrinos that can be interpreted as free particles (asymptotic free single-particle Fock states).

 

[kith]

All of this is immediate from the claim that the state is mixed despite it being a ket. What aspect are disagreeing with or pointing out?

I agree that your ket formally corresponds to a completely mixed state and that we can use it to calculate the correct probabilities.

In order to consider it physical, I'd like to see a description where this state ket emerges from a well-prepared pure state (which necessarily belongs to only one of the superselection sectors) by a time evolution law. I don't see how this could be possible except for the ad hoc way where the density matrix is used in the calculation and only certain special mixed state density matrices are identified with kets by hand later on.

 

[DarMM]

In order to consider it physical, I'd like to see this state emerge from a well-prepared pure state (which necessarily belongs to only one of the superselection sectors) by a time evolution law

Why are you requiring this? All mixed states don't result from unitary time evolution of a pure state.

 

[kith]

Why are you requiring this? All mixed states don't result from unitary time evolution of a pure state.

I don't require unitary time evolution.

/edit: I slightly edited the wording of my last post.

 

[A. Neumaier]

I think you may have terminology confused. A mixed state is any state which, as an algebraic state, is the sum of another two states, i.e. for all observables:
$$\omega\left(A\right) = c_{1}\omega_{1}\left(A\right) + c_{2}\omega_{2}\left(A\right)$$
A pure being one for which this is not true.
This is equivalent to having vanishing (pure) or nonvanishing (mixed) Von Neumann Entropy.

The GNS construction then allows one to express any state, pure or mixed, as a ket/vector state.
See Robert M. Wald "Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics" p.83:Thus a vector state can be pure or mixed. However a pure state (entropy is zero) is never mixed (entropy non-zero).

I know all this. But you were referring to the addition of kets, not of states (linear functionals). A sum of kets is always a pure state (in the Hilbert space of the ket). My question was how it is possible that you refer to it as a mixed state. Or was this only a slip of the pen?