Exploring the Connection Between Quantum Mechanics and Quantum Field Theory

[A. Neumaier]

If you block the partial beam with spin-z down, you are left with a beam with spin-z up. It may be a philosophical problem in how you come to sort out one beam. It's like choosing a red marble rather than a blue just because you like to choose the red one. What's the problem?

I am just trying to explain why stevendaryl is not satisfied with your answers.

The problem is not in that there is a choice in blocking one of the beams. That the beam is blocked may be taken as part of the experimental set-up. The problem is that if one treats this problem quantum mechanically including the blocker and the detector, one apparently ends up (and does so definitely in the oversimplified version used by stevendaryl) in the superposition I wrote down, rather than in one of the two separable states (as observed). So something needs to be explained!

 

[A. Neumaier]

You cannot block the beam in real space, if the beam is only in Hilbert space.

Every experimenter can. The location of the beam is determined by the experimental setting. Thus it is known beforehand whether the experimental setting blocks the left beam.The only question is when and whether a spinning particle actually travels in it.

 

[vanhees71]

If you block the beam, you are introducing hidden variables. You cannot block the beam in real space, if the beam is only in Hilbert space.

I don't introduce hidden variables, but a very visible "beam dump". That can be a big rock or some lead shield or whatever. It's all but hidden! SCNR.

Also a beam is not in Hilbert space but in the lab, e.g., at the LHC in Geneva, hitting one of the detectors in the big experiments. I think a lot of these discussions just come about because many people take the abstract description of quantum theory for the real thing. That's not the case. Already in classical "point mechanics" a stone thrown by a kid is not the (approximate) parabola described as a function $\mathbb{R} \rightarrow \mathbb{R}^3$. The stone is not a set of three real numbers or a vector in an abstract real three-dimensional vector space but it's a rock in the real world. We only describe it more or less accurately with these abstract mathematical tools!

 

[vanhees71]

I am just trying to explain why stevendaryl is not satisfied with your answers.

The problem is not in that there is a choice in blocking one of the beams. That the beam is blocked may be taken as part of the experimental set-up. The problem is that if one treats this problem quantum mechanically including the blocker and the detector, one apparently ends up (and does so definitely in the oversimplified version used by stevendaryl) in the superposition I wrote down, rather than in one of the two separable states (as observed). So something needs to be explained!

But that's my point! You overcomplicate a simple thing like putting a "beam dump" somewhere. Of course, it's not in the superposition if one partial beam is just absorbed by the beam dump. That's just a wrong description of the state after the partial beam hit the beam dump. That's it, but no complicated problems.

 

[A. Neumaier]

But that's my point! You overcomplicate a simple thing like putting a "beam dump" somewhere. Of course, it's not in the superposition if one partial beam is just absorbed by the beam dump. That's just a wrong description of the state after the partial beam hit the beam dump. That's it, but no complicated problems.

But if one uses the complicated description one should still be able to obtain the same final result. That's the whole point of deriving few-particle quantum mechanics from a more comprehensive view in which the equipment is also treated by quantum mechanics. Consistency of QM requires that the final results are the same (within approximation errors), but (according to stevendaryl's arguments) one seemingly gets something essentially different when using the more detailed description.

 

[vanhees71]

Then it's the wrong description, because it contradicts experience with many experiments. To understand energy loss of particles in matter and their absorption, look for Bethe-Block formula. It should be described in many textbooks. I think there's even a section in the Review of Particle Physics on it.

 

[A. Neumaier]

Then it's the wrong description

The question then is, what is the correct many-particle description of the system consisting of particle + blocking screen in a Stern-Gerlach experiment that prepares a collection of (time-separated) spinning particles and in which one of the two beams generated is observed by a photosensitve (or energy-sensitive) blocking screen, while the other beam is used later for further experiments? If QM is valid universally in the lab, there must be a description of this system that develops unitarily from the time a particle is generated well beyond the time the particle is or isn't observed on the screen.

What is your proposed model for this system in QM or QFT, such that precisely one of the two observed outcomes is predicted? I believe that only such a model (including the analysis that it predicts as required) would constitute a solution of what stevendaryl called the selection problem.

 

[vanhees71]

I don't think that such a model is possible nor that it is necessary. The one beam gets blocked by energy loss in the material used to block it. That's described by an effective theory leading to the Bethe-Bloch formula. I don't think that you can describe it in full microscopic detail, but that's not necessary. The effective description is sufficient. That's the case with almost all applications of QT and also classical physics to real-world problems.

 

[A. Neumaier]

I don't think that such a model is possible nor that it is necessary.

I agree that for practical applications such a model is not necessary, and indeed is unduly complicated.

But if such a model were not possible it would mean that QM is no longer applicable to at least one system whose size fits on a lab desk. Do you really want to claim that?

 

[vanhees71]

No, of course not, but if you'd demand to have exact solutions of QT to any real-world observation down to the observation process, then you'd claim that QT is not applicable to any real-world system, but that's a contradiction to over 90 years of successful application of QT, I'd say.

 

[A. Neumaier]

No, of course not, but if you'd demand to have exact solutions of QT to any real-world observation down to the observation process, then you'd claim that QT is not applicable to any real-world system, but that's a contradiction to over 90 years of successful application of QT, I'd say.

For real applications one can use any simplification that leads to results agreeing with experiments.

But for matters of principle (such as the claim that QM is universally valid) one needs to to do more, at least in model situations.
One doesn't want an exact solution - one can allow the standard approximations available for many-particle problems on the formal level. But not classical gross simplifications such as simply throwing away the particle if it has been detected. It is only forbidden to introduce the features one would want to derive!

For example, if one wants to use a simplified picture of a particle disappearing one has to model the particle in Fock space where particle number is variable. However, no matter how the model is chosen, in QM, the modeled degrees of freedom are not allowed to change during the unitary evolution - and whatever is deduced must be deduced formally from the unitary evolution by making appropriate approximations.

The problem boils down to finding a suitable quantum many-particle description covering sufficiently many degrees of freedom such that it describes both the particle and the active part of the screen, together with an approximation procedure that is not based on classical handwaving (we now dump the particle!) but on the usual principles used in quantum mechanics, such as perturbation theory in the interaction picture, diagram expansions, mean field arguments and corrections to it.

 

[stevendaryl]

Having had a long look at the Nieuwenhuizen et al.(1014) treatment I find support for the idea that nature has no cut off/transition between quantum and classical. Quantum mechanics is always in operation - there is only one set of laws. So why do we not see 'cat' states ? At what point can we use classical approximations instead of QM ?

With continuous properties like position there is no problem because a baseball can be in a superposition of 2 position states if the difference between the positions is very small compared to the baseball. How could one ever detect such a thing ?

The point is that the difference between possible positions of a baseball can actually become arbitrarily large. But still, you can ask the same question: How would you detect such a thing? The many-worlds answer is that you can't. There is one "you" that sees a baseball here, and another "you" that sees a baseball there, and there is no "you" that is capable of seeing both. Many-worlds has its own conceptual problems (such as: what does the Born probability rule mean if everything is deterministic), but at least it doesn't have the problem of one set of rules for macroscopic objects and a different set of rules for microscopic objects.

With discrete states the picture is different. If we have a property (operator) with a few possible outcomes we can reduce this to (say) to a binary state by averaging over a few degress of freedom. But defining live and dead states for a cat requires averaging over millions of dof. Adding random phases reduces and eventually destroys interference and the quantum effects, mathematically the equations of motion become trivial when the commutator $\left[\hat{\mathcal{D}},\hat{\mathcal{H}}\right]$ approaches zero. At this point there is no change which predicts that the cat remains forever in its initial state. Since we can only prepare a cat in either state, that is all we can ever see.

I'm sure this is oversimplified and naive but it works for me.

I think you're absolutely right that there is no way to observe a half-live-cat/half-dead-cat.

 

[Mentz114]

... but at least it doesn't have the problem of one set of rules for macroscopic objects and a different set of rules for microscopic objects.

There is only one set of rules. Nature does not make a switch based on some magic criterion you seem to be looking for.

What you call classical mechanics emerges naturally from QM as you average out dof. There are plenty of cases where semi-classical computations give the correct predictions even though one part of the system is 'classical' and another quantum. We decide when calculating where to make the cut.

There is no 'cut' in nature !

 

[stevendaryl]

There is only one set of rules. Nature does not make a switch based on some magic criterion you seem to be looking for.

I'm not saying that nature has different rules for microscopic and macroscopic objects. I'm saying that quantum mechanics as practiced does. If you look at a single electron, or a single atom, or any small system and you describe it quantum-mechanically, where does the Born rule come into play? It doesn't. The Born rule only comes into play when we divide the universe into (the system being measured) + (the device doing the measuring). So WE are the ones who use different rules for microscopic and macroscopic. I'm asking for the justification for this distinction.

Many-worlds doesn't make the distinction. The Bohm interpretation doesn't make the distinction. But it seems to me that any interpretation of quantum mechanics that considers the Born rule to be a primitive postulate of physics has to make such a distinction.

 

[Mentz114]

I'm not saying that nature has different rules for microscopic and macroscopic objects. I'm saying that quantum mechanics as practiced does. If you look at a single electron, or a single atom, or any small system and you describe it quantum-mechanically, where does the Born rule come into play? It doesn't. The Born rule only comes into play when we divide the universe into (the system being measured) + (the device doing the measuring). So WE are the ones who use different rules for microscopic and macroscopic. I'm asking for the justification for this distinction.

Many-worlds doesn't make the distinction. The Bohm interpretation doesn't make the distinction. But it seems to me that any interpretation of quantum mechanics that considers the Born rule to be a primitive postulate of physics has to make such a distinction.

OK. I see that is a bit subtle. I did not know that the status of the Born rule was a problem. I would hope it emerges from the quantum statistical treament (thesis ?) rather than be a postulate. Maybe Born was thinking of statistical mechanics when he added his footnote ?

 

[stevendaryl]

OK. I see that is a bit subtle. I did not know that the status of the Born rule was a problem.\/QUOTE]

Well, it is for me. The Born rule says that if you measure an observable $\hat{O}$, you will get an eigenvalue, with probabilities given by the square of the projection of the wave function onto the subspace corresponding to that eigenvalue. That rule as I wrote it necessarily involves measurement. What is a measurement? To me, a measurement means an interaction between the system being studied and a second system, the measuring device, such that the interaction produces a persistent, macroscopic change in the device corresponding to the value measured. So applying the Born rule seems to me to involve a macroscopic/microscopic distinction.

The decoherence approach is a little subtler: You form the density matrix. Then you "trace out" the degrees of freedom that are unobservable (or uninteresting?). The result looks like a mixed state. Then you can give a statistical or probabilistic interpretation of that mixed state. But that involves two steps that are questionable to me. First, the separation of the degrees of freedom into an unobservable environment plus the system of interest seems very subjective. Second, treating a mixed state that arose from performing a mathematical trace as if it were a mixed state resulting from nondeterminism seems like pretense.

I would hope it emerges from the quantum statistical treament (thesis ?) rather than be a postulate. Maybe Born was thinking of statistical mechanics when he added his footnote ?

There is good reason to believe that IF you are going to interpret quantum mechanics as a probabilistic theory, then the Born rule is pretty much the only sensible choice. But the part that I don't understand is how probabilities arise in the first place.

 

[A. Neumaier]

There is good reason to believe that IF you are going to interpret quantum mechanics as a probabilistic theory, then the Born rule is pretty much the only sensible choice. But the part that I don't understand is how probabilities arise in the first place.

What about my explanation in Chapter 10.5 of my online book? There the Born rule and its probabilities are derived, not postulated; so it might add an element of understanding.

 

[Mentz114]

First, the separation of the degrees of freedom into an unobservable environment plus the system of interest seems very subjective.

Sujective but I would say governed by practicality.

Second, treating a mixed state that arose from performing a mathematical trace as if it were a mixed state resulting from nondeterminism seems like pretense.

Calling the trace a mixed state is pushing it. I see it as a probabilistic statement that gives support to more than one outcome. There's no implication that the outcomes are not exclusive.

I'm not sure what you mean by 'indeterminism'. Is this a special thing reserved for quantum systems ?

 

[stevendaryl]

Calling the trace a mixed state is pushing it.

That's just a definition. A density matrix is a mixed state if it is not of the form [itex]|\psi\rangle\langle \psi|[itex]

I see it as a probabilistic statement that gives support to more than one outcome.

If you start with a wave function evolving deterministically, and then you perform a mathematical operation to make it look like a mixed state, how can it become suddenly probabilistic?

I'm not sure what you mean by 'indeterminism'.

It means not deterministic--there is more than one possible future for a given starting state. I don't see how you can have probability without nondeterminism.

 

[stevendaryl]

What about my explanation in Chapter 10.5 of http://arxiv.org/abs/0810.1019. ? There the Born rule and its probabilities are derived, not postulated; so it might add an element of understanding.

That link is broken.

 

[A. Neumaier]

I don't see how you can have probability without nondeterminism.

Probability theory is even used to analyze the distribution of prime numbers - although this is fully determined by the Peano axioms. To have probability one only needs a concept of expectations with the appropriate properties - not something ''truly random''. (This is why deterministic random number generation is possible.)

 

[A. Neumaier]

That link is broken.

Thanks. I corrected post #262 and checked that the link now works.

 

[reiabretti]

QFT is based on an integral instead of a differential approach to the formulation of problems. This means you need a Green ( propagator) function inside the integral. Moreover instead of working in space -time coordinates, Feynman showed the convenience of working with momentum-energy variables , see the Theory of Positrons , Physical Review 1949.