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Please excuse if some of my terminology is vague, the whole point of this question is to clarify terminology.

In quantum physics, one frequently encounters situations where there are some kind of two distinct physical realms:

  • A hidden realm which cannot be directly observed. More or less, this is the Hilbert space in the quantum mechanics axioms. Often this involves complex numbers and other non-classical formalisms. Examples:
    • The wavefunction in the Schrödinger picture
    • The complex state of a spinor, which undergoes a parity change when the coordinates are rotated by $2\pi$. We cannot measure this parity change as far as I know, but it is there.
    • If the hidden realm could be observed, the EPR paradox would describe a violation of causality, because of the spooky action at a distance.
  • A measurable realm, where we can measure the particles' states. Examples:
    • In the Schrödinger picture, this is the measured quantity, for example the position or momentum of a particle.
    • In the case of the spinor it is the measured spin along a specific axis, for example the Up state along the z axis.
    • The EPR paradox is often resolved by explaining that quantum mechanics carries some non-locality which does not violate causality because it cannot be observed. In my terminology, this means that the non-locality is restricted to the hidden realm. The measurable realm does not have any spooky action at a distance.

In classical theories, the measurable realm is all there is. The whole point of quantum mechanics is that we cannot describe the measurable realm without the formalism of the hidden realm.

Of course these two realms are closely connected:

  • The Copenhagen interpretation says that all events in the measurable realm are determined by the hidden realm, but only in the form of probabilities.
  • Events in the measurable realm ("measurements") do have an effect on the hidden realm through wavefunction collapse.

My question is, is there any (more or less) established terminology for this distinction? I'd prefer to avoid to say hidden realm = Hilbert space. Maybe there are (or will be) theories that describe quantum behavior without Hilbert spaces, and I'm looking for a general terminology that does not involve the technicalities of certain formulations of quantum mechanics.

I don't think that my terminology is suitable, because the term hidden realm carries the connotation of hidden variable theories, which is something else.

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I don't think that my terminology is suitable, because the term hidden realm carries the connotation of hidden variable theories, which is something else.

I don't see them as very different. A hidden variable theory could take as the hidden variable the spinor or the wavefunction as a hidden variable. Many do.

  • Events in the measurable realm ("measurements") do have an effect on the hidden realm through wavefunction collapse.

No. There is no experimental evidence that a collapse dynamical process ever supersede the regular dynamical evolution in the Hilbert Space. The wavefunction is predicted to split, it does split. And eventually the two split parts act independently. But you never see evidence that anything happens other than the the regular evolution in the Hilbert Space. And the Hilbert Space state vector evolves according to the Hilbert Space state vector (and the Hamiltonian). Not according to something else.

  • The Copenhagen interpretation says that all events in the measurable realm are determined by the hidden realm, but only in the form of probabilities.

In order to agree with observations, any interpretation, even Copenhagen, must predict that the Hilbert Space state vector always evolves according to the quantum prescribed evolution, because any deviation would be detectable. What you call the measurable realm isn't a different thing, is is a kind coarse graining. For instance, a Stern-Gerlach device starts by taking a wave in configuration space that has support where in some region it is a beam in the coordinates of one particle and that beam widens and splits and the parts separate and the spin state changes so that the two split beam parts have a well defined spin state for that particle that is an eigenstate to an operator aligned with the device.

And the Schrödinger equation predicts this. And it predicts the sizes of the two split beams, which is relevant when you want later to also have a setup that tracks what proportion of an ensemble split into particular directions.

Every theory predicts this, the Schrödinger equation itself predicts this when you apt it to the actual experimental setup of object and device. And there isn't too. To predict anything else since if you stick detectors inside various portions of the Stern-Gerlach device you'll see it doing this continuous evolution exactly as the Schrödinger equation predicts.

In classical theories, the measurable realm is all there is. The whole point of quantum mechanics is that we cannot describe the measurable realm without the formalism of the hidden realm.

The measurable realm is not distinct. When the regions in configuration space with nonzero support split into two (which starts for instance inside a Stern-Gerlach device) and get to the point where each region can evolve as if it was the only one this is a group of configurations with classical information. It is merely a coarse graining of the actual wave over the actual range of configurations.

  • A hidden realm which cannot be directly observed. More or less, this is the Hilbert space in the quantum mechanics axioms. Often this involves complex numbers and other non-classical formalisms. Examples:

Observations are merely particular kinds of interactions between parts of the whole system. In a Stern-Gerlach device the spin becomes entangled with the position of the same particle. This allows other particles that can become entangled with the position of that particle to allow the split to propagate to a divergence in more than just that one dimension for that one particle (the wave, as always is in configuration space).

Whether you use complex numbers is entirely up to you. You can have two real fields on configuration space and it works just as well.

And you don't need a bunch of axioms since the evolution equation and the initial conditions give you everything, the rest is redundant (at best).

The wavefunction in the Schrödinger picture is the actual mathematical analog of the physical system. And it can be spinor valued, rather than complex valued. And you can do it in a coordinate invariant way, literally have a space of spin states and a confirmation space and have some functions and do it so there is no origin, no basis, and no coordinates. These are all mathematical tricks that may or may not make you life easier.

It's just like with a vector, you can choose a basis or not. As long as you can scale and add and take scalar products you are good.

Spinors are not weird, and a spinor field is simply the first and easiest example of an operator valued field. And operators aren't weird they are perfectly natural geometric objects. A force in relatively is a related to a rotation of the tangent t the worldline in spacetime so it naturally given in terms of rotations in spacetime, it isn't weird to have an electromagnetic field and it isn't weird to have an operator valued field.

  • If the hidden realm could be observed, the EPR paradox would describe a violation of causality, because of the spooky action at a distance.

How is the above even meaningful? The wavefunction is defined as a field on configuration space. It is nonlocal. The object itself is nonlocal. And it evolves according to a PDE, so its evolution (which determined everything) only depends on what is nearby, how can you ask for any more local of an evolution, the causality is clear. The time evolution at a point in configuration space is determined by derivatives evaluated at that point in configuration space. This isn't spooky in the slightest.

  • A measurable realm, where we can measure the particles' states.

This is horrible terminology. When you put something through a Stern-Gerlach device oriented in the z direction the output has the property that no matter how many times you put it through another Stern-Gerlach device oriented in the z direction the beam isn't split. Awesome. That's the kind of thing you can call classical information.

Put it through a Stern-Gerlach device oriented in the x direction and the particle no longer has the property that if you put it through another Stern-Gerlach device oriented in the z direction the beam isn't split. You lost that property. You changed the particle.

Its is disingenuous to call the act of changing something a measurement. It just makes things seem spooky and mysterious. And it is just a name, so if all it does is serve to confuse you, why do it. You can call it polarizing if you want to call it something.

  • In the Schrödinger picture, this is the measured quantity, for example the position or momentum of a particle.

If you go back to the spin example of the Stern-Gerlach device. The position of the particle becomes entangled with the spin of the same particle as the spin of the particle becomes polarized. And the positions associated with the different spins diverges making two different regions of configuration space that have nonzero waves. Then the regions separate even mor win configuration space as the different locations of that one particle lead to divergingly different positions of other particles (really I am just describing the separation in configuration space of two wave packets but since they are defined on the domain of configurations of particles it sounds like particles are moving). This allows the different wave packets to evolve as if they were the only one. But each wave packet has a spin state for that one particle, so the spin state is now associated with that entire wave packet which evolves as if it were the only wave packet. So it dynamically judges itself as the whole world so associates that information with the world. It's a wave over a range of configurations but it associates some fact with the entire world. Remember that every particle of every person and every planet and star is part of the configuration space and so the wave is defined on them all so if the wave in one region acts like it is the only wave then the whole world that is that region thinks it is the whole world.

The randomness and probability is like cell division. If you are a single celled entity and you are going to split and them one of the split versions is going to get killed in 5 hours there is no meaningful way the single cell can answer the question "am I going to get killed in five hours" it's a category mistake to ask that question. What you can ask is that if this happens regularly you can say if something has lots of cells like me in lots of situations like mine and waits until something splits and then asks that split thing to notify an aggregator whether it is alive and then do a follow up message to the aggregator five hours later if still alive, then the aggregator after getting many messages from many cells will be able to report that only half the messages were followed up with a report that the cell was still alive.

And that is where probability comes in. For ensembles of identically prepared systems and about the results of the state of the aggregator that aggregates the reports of the many splits.

  • In the case of the spinor it is the measured spin along a specific axis, for example the Up state along the z axis.

In the case of the spinor, the wavefunction contains a spinor valued field and the field gets polarized in each branch of the split. And the branch later acts like it is the only wave in the world. So the polarized spin state seems like a (potentially new) fact of the world.

  • The EPR paradox is often resolved by explaining that quantum mechanics carries some non-locality which does not violate causality because it cannot be observed.

The object itself is a wave defined on configuration space. Causality is fixed by a PDE defined on the wave. The classical information happens when the regions with non zero wave splits in such a way as to stay split.

Since the classical information appears when it stays split you can't tell exactly when it happens. And there isn't a separate thing trying to tell whether it split. The thing itself splits.

In my terminology, this means that the non-locality is restricted to the hidden realm.

There isn't any nonlocality beyond the fact that the wave is in configuration space to begin with.

The measurable realm does not have any spooky action at a distance.

The classical worlds that focus on results of so called measurements have global properties that are the results of the polarization. They are not facts associated with some region and of course any change in the world is dynamic and involves a continuous change of the wave which means you move from a region with these configurations to regions with ever so slightly different configurations in a eve so slightly different time. But it isn't nonlocal or spooky to anything. Something can look spooky if a region splits and doesn't stay split but that is why it isn't classical information yet if the split parts can meet.

It isn't that hard to look at a solution to a PDE and relate it physical observations. Fancy names aren't required and nothing strange whatsoever is going on. It's just a PDE.

It's clear that an EPR pair evolves strictly local,

I can't tell what you think is clear. The quantum mechanical description of two entangled spin 1/2 particle includes a function $\Psi$ from $\mathbb R^6$ into $\mathbb C^2\otimes \mathbb C^2$ and that part about $\mathbb R^6\neq\mathbb R^3$ means the PDE of the Schrödinger equation is evolving a wave defined in a six dimensional configuration space, not a wave in 3 dimensional Newtonian space.

but a measurement on particle A forces particle B into a specific state.

A measurement will require even more particles than just the two. So then you would have a function from something like $\mathbb R^{3\times 10^{18}}.$ And it forces the region (of configuration space) where the wave is nonzero to split into two.

This is propagated instantaneously.

That sounds like a fairy tale. Among other things, the wave is already in configuration space, so there is nothing to propagates and there is no here or there, there is just some complex numbers at some points in configuration space. And each point in configuration space changes the values of there complex numbers based solely on the potential and spatial variation in that neighborhood of configuration space. It literally doesn't care about distant regions of configuration space.

My point is, if it was possible to measure the whole quantum state of particle B (the hidden state, meaning the whole superposition), it would be possible to propagate information from particle A to B instantaneously, which contradicts causality.

I can't tell what you are doing or trying to say, but either you are discussing a hidden variable theory designed to be wrong, in which case who cares, wrong theories of all types are wrong. Or else, you are not correctly learning how the hidden variable theories that agree with observations actually work. That could happen because you don't want to learn them properly or are trying to learn them from people that didn't learn them properly. And the numbers of both groups are legion.

If you want to correctly learn what a hidden variable theory designed to agree with observations predicts, then you have to learn to not bring assumptions and baggage to the table, just like when you learn any modern physics.

In the simplest example that shows the features involved you can have that $\Psi$ from $\mathbb R^6$ into $\mathbb C^2\otimes \mathbb C^2$ and we can ignore four of those dimensions to look at the function that is roughy $$\Psi(x_1,y_1,z_1,x_2,y_2,z_2)=e^{(x_1^2+z_1^2)/2\sigma ^2}e^{(x_2^2+z_2^2)/2\sigma^2}e^{iky_1}e^{iky_2}\left(\left[\begin{matrix}1\\ 0\end{matrix}\right]\otimes\left[\begin{matrix}1\\ 0\end{matrix}\right]+\left[\begin{matrix}0\\ 1\end{matrix}\right]\otimes\left[\begin{matrix}0\\ 1\end{matrix}\right]\right)$$ in the region where both the $y $s are negative and then over time it evolves to have it widen in the $x$ directions for larger $y$s.

Much like the capita letter Y. As you look at horizontal slices farther up the x width starts to widen and then in the Y you see it split into two branches. The same thing happens to the wave in configuration space. The region were it isn't small splits into two. And the joint spin state changes at the same time and it becomes $\left[\begin{matrix}1\\ 0\end{matrix}\right]\otimes\left[\begin{matrix}1\\ 0\end{matrix}\right]$ in one branch and $\left[\begin{matrix}0\\ 1\end{matrix}\right]\otimes\left[\begin{matrix}0\\ 1\end{matrix}\right]$ in the other branch and it does the changes in spin as it widens and is done with the change by the time it splits.

But the spin changes continuously over time, as does the whole wave. And each part I'm configuration space changes just because of the parts next to it.

I'm describing what the Schrödinger equation predicts and it is the thing that makes the predictions we actually see. The equation doesn't have the word local or nonlocal in it. But it is a Partial Differential Equation (PDE) and as such the values change based solely on what the values are like right nearby, but it is in configuration space so right nearby means configurations that are nearby configurations.

A theory makes models of the world which describe states of the world. Everything is in the model, there isn't some outside thing that sees the model and them whispers to one part of it or another. The model just relates initial conditions to later conditions. In this case the initial conditions are a wavefunction, which is already defined on configuration space. It evolves according to a PDE and so is as local as you could want for something that is always defined on configuration space. Nothing weird happens. It is not the case that some distant region of configuration space is every influenced by something happening in this region of configuration space, not unless the waves at those two places contiguously evolve to affect each other by actually B coming together towards the same common region of configuration space.

And the irony is that measurements are explicitly designed to avoid allowing that, so it is almost be definition the case that a measurement does not allow distant regions of configuration space to affect each other. The state that covers all the configurations in between won't allow them to cross their streams, so they never affect each other.

So I can't figure out why the things you say seem like meaningless non sequiturs or else non science or else the exact opposite of what goes on.

See for example Wikipedia on EPR: ...unless measuring one particle instantaneously affects the other to prevent it, which would involve information being transmitted faster than light as forbidden by the theory of relativity ("spooky action at a distance").

There is a joint spin state, just like the spatial state is a function defined on configuration space. If you want to understand quantum mechanics learn that a function such as $$\Psi(x_1,y_1,z_1,x_2,y_2,z_2)= e^{((x_1-A)^2+(z_1-A)^2)/2\sigma^2}e^{(x_2-B)^2+(z_2-B)^2)/2\sigma^2} \iint dk_1 dk_2e^{k_1^2/2\sigma_k^2}e^{ik_1(y_1-a)}e^{k_2^2/2\sigma_k^2}e^{ik_2(y_2-b)} \left(\left[\begin{matrix}1\\ 0\end{matrix}\right]\otimes\left[\begin{matrix}1\\ 0\end{matrix}\right]+\left[\begin{matrix}0\\ 1\end{matrix}\right]\otimes\left[\begin{matrix}0\\ 1\end{matrix}\right]\right)$$ is a perfectly fine wavefunction with a domain defined on configuration space and with a range in a joint spin state.

And so is $$e^{((x_1-y_1-C)^2/D^2+(z_1-A)^2)/2\sigma^2}e^{(x_2-B)^2+(z_2-B)^2)/2\sigma^2} \iint dk_1 dk_2e^{k_1^2/2\sigma_k^2}e^{ik_1(x_1+y_1-a)}e^{k_2^2/2\sigma_k^2}e^{ik_2(y_2-b)} \left(\left[\begin{matrix}1\\ 0\end{matrix}\right]\otimes\left[\begin{matrix}1\\ 0\end{matrix}\right]\right)$$

(A beam deflected to the left with a factorizable spin state.) And so is $$e^{((x_1+y_1-C)^2/D^2+(z_1-A)^2)/2\sigma^2}e^{(x_2-B)^2+(z_2-B)^2)/2\sigma^2} \iint dk_1 dk_2e^{k_1^2/2\sigma_k^2}e^{ik_1(x_1-y_1-a)}e^{k_2^2/2\sigma_k^2}e^{ik_2(y_2-b)} \left(\left[\begin{matrix}0\\ 1\end{matrix}\right]\otimes\left[\begin{matrix}0\\ 1\end{matrix}\right]\right)$$

(A beam deflected to the right with a factorizable spin state.)

In fact you can have a state like the first one evolve into one like the sum of the last two. And in fact the Schrödinger equation predicts something like that. And the Schrödinger equation is a PDE so each moment the wave changes just a little bit based on what is going on nearby. The joint spin changes a little bit at each point in configuration space. The regions where the wave is large changes a little bit. So it starts with the first particle's wave having the bulk of its current moving along a large letter Y and it forks and flows down two paths as the spin state becomes factorizable in the two regions.

Wikipedia can try to mash together language from different theories in an attempt to oversimplify. But there wasn't a single particle spin state originally, that developed later, and developed for the two different branches in configuration space. So it changed the joint spin state and it created the individual particle spin states (the spin state started out not factorizable and ended up factorizable ad the entanglement passed from the spin of the particles to an entanglement between a single particle's spin and its own position). It did not change the particle's single particle spin state, because it didn't have one prior to the interaction and if someone says otherwise they are either lying, oversimplifying, or dealing with some special case. Usually the middle case.

Which is why I thought you were asking people that respond to your own words what is going on. I thought you didn't want an oversimplification, which is by definition too much and distorts the required essential features so much as to be wrong in the critical aspects.

And that's key. Because when you do the math the function comes from a configuration space and goes into a joint spin state. And it evolves in a continuous and local manner, for functions that are defined on configuration space and go into a joint spin state.

The particle didn't have a state of spin up prior to the interaction, it didn't even have a single particle spin state since the joint spin state wasn't factorizable. It developed one. And it did so over time by the wave splitting in configuration space which is a local process in configuration space. And everything depends on the spin right there in configuration space and the split in configuration space happens right there in configuration space, right where that Y branches.

It isn't a hidden spin, the single particle didn't have it's own spin prior to the interaction. And things aren't changing in a magical way, a PDE is as normal as you can get. The only thing weird is the language some people like to use.

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  • $\begingroup$ Well, thanks for the answer, there are many things I'd like to ask or comment. The most important one: It's clear that an EPR pair evolves strictly local, but a measurement on particle A forces particle B into a specific state. This is propagated instantaneously. My point is, if it was possible to measure the whole quantum state of particle B (the hidden state, meaning the whole superposition), it would be possible to propagate information from particle A to B instantaneously, which contradicts causality. $\endgroup$ – Bass Sep 21 '15 at 14:03
  • $\begingroup$ @BastianTreichler It's all continuous but I don't know why you say the things you say. So while I edited, I don't know if its what you are looking for. Hopefully if you want to understand hidden variables (which you don't need to) then you learn them from a qualified source which includes someone that tries to agree with observations. $\endgroup$ – Timaeus Sep 21 '15 at 15:51
  • $\begingroup$ See for example Wikipedia on EPR: ...unless measuring one particle instantaneously affects the other to prevent it, which would involve information being transmitted faster than light as forbidden by the theory of relativity ("spooky action at a distance"). That's what I'm talking about, sorry if it was unclear. BTW, I'm not sure where I said I wanted to learn hidden variables. I'm just trying to understand QM and its interpretations. $\endgroup$ – Bass Sep 22 '15 at 9:52
  • $\begingroup$ @BastianTreichler Edited $\endgroup$ – Timaeus Sep 22 '15 at 14:57
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There are classical and quantum descriptions of the world. One of the differences of quantum description is paying attention to the process of measurement and how it affects the measured system.

Description of measurements is an integral part of quantum description. Splitting this is into "realms" doesn't make much sense.

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