B Meaning of Wave Function Collapse

[rubi]

Let's treat the first polarizer as state preparation so that we have pure H polarized state after the first polarizer. After the second polarizer the state is say in superposition of being +45deg and absorbed by the second polarizer + being -45deg and passed through the second polarizer. Is it ok so far? Now how do you model interaction with the third polarizer. Only the part that passed the second polarizer interacts with the third polarizer. Or no?

Every polarizer $i$ has an associated unitary operator $U_i$. You start with a state $\psi$. After the first polarizer, the state it $U_1\psi$. After the second one, it is $U_2 U_1\psi$ and after the third polarizer, it is $U_3 U_2 U_1\psi$. No projections are involved. You have to insert projections only if you perform measurements in between.

 

[zonde]

Every polarizer $i$ has an associated unitary operator $U_i$. You start with a state $\psi$. After the first polarizer, the state it $U_1\psi$. After the second one, it is $U_2 U_1\psi$ and after the third polarizer, it is $U_3 U_2 U_1\psi$. No projections are involved. You have to insert projections only if you perform measurements in between.

And what are the components of the state $U_3 U_2 U_1\psi$? One component has to have amplitude who's square is 1/8 given common polarizer angles for this example (H, 45deg, V)

 

[rubi]

And what are the components of the state $U_3 U_2 U_1\psi$? One component has to have amplitude who's square is 1/8 given common polarizer angles for this example (H, 45deg, V)

Well, I'm too lazy right now to write down all the matrices and perform the calculation, but of course, if you choose $U_i$ and $\psi$ correctly and then expand the final state in the basis of your choice, you will get the correct experimental predictions.

 

[Lord Jestocost]

As defined in the standard textbook - Decoherence and the Quantum-to-Classical Transition by Maximilian A. Schopenhauer:

I think you surely meant “Schlosshauer” and not the German philosopher Arthur Schopenhauer (1788 – 1860) (https://en.wikipedia.org/wiki/Arthur_Schopenhauer).

Nevertheless, Schopenhauer’s reasoning and transcendental idealism might be of interest with respect to discussions about quantum mechanics. Schopenhauer described transcendental idealism as a "distinction between the phenomenon and the thing in itself", and a recognition that only the phenomenon is accessible to us because "we know neither ourselves nor things as they are in themselves, but merely as they appear.”

 

[Carpe Physicum]

One way to ground everything in reality is to think purely about the records of experiments that are stored in computer memory. Very often, that's a list of times at which events happened. If you think about APDs, for such devices we might run a wire (or we use a fiber optic cable, or wi-fi, ...) from the APD to the computer that records the data. On that wire, there will be a voltage that most of the time will be near zero voltage, but occasionally an "avalanche" happens, the voltage goes to non-zero (1 volt, 20 volts, whatever), then the hardware checks a clock for the time and records it in memory and then to hard disk for later analysis. The hardware also resets the APD as soon as possible so another avalanche can happen. From a computing and signal analysis point of view, what's just happened was a compression: we could have recorded the voltage picosecond by picosecond to 14-bit accuracy, but we just recorded the time when there was a signal transition from zero to not-zero.
There are certainly experiments that record continuous signals (at finite accuracy and resolution, with a fixed schedule, because it's all going into digital memory), but the analyses that you'll find detailed in physics papers are often on the hunt for discrete structure of some kind, and very often a discrete structure is there to be found.

Everything so far is classical electronics (except the last sentence, which presaged what comes next here) about events and signals. There is no mention of particles or of particle properties whatsoever. Now comes the analysis, where we will introduce the idea that particles (or, more generally, "systems", a field, thing or things that are kinda classical) explain why we see the events and signals that we see. The Correspondence Principle gives us a way, called quantization, to convert a classical dynamics for some kind of classical system (mechanics or electromagnetism) into a differential equation that describes the evolution over time of a "statevector", the Schrödinger equation. The statevector models/predicts the statistics of many different kinds of measurement results (anything that can come out of a mathematical analysis of the raw data of the previous paragraph), and, crucially, how they change over time. Some of those measurement results are "incompatible" with each other, so that properly speaking we can't talk about correlations between incompatible measurements.
The Correspondence Principle is quite tricky because it cannot be a perfect map from a classical dynamics to a quantum dynamics, but it's been a fairly decent guide for the last 90 years, so we're not going to give it up until we have something better. If we find that the quantization of a classical mechanics works well as a model for the signal analysis we do for the raw data, which has to work nicely as the statistics change over time, we pretty much say that the quantized classical system explains the raw data, except of course that we don't as much understand what we're doing when we quantize as we'd like to.
So, @Carpe Physicum, it looks as if you might have left this conversation. If you're still here, I hope you find this a little useful even though it's definitely my idiosyncratic way of thinking about the question. I've tuned the above a little to the computing world because that seems to be your sort of thing, which wasn't hard to do, however, because that's also pretty close to my sort of thing. If you reply, I can point you to the first video on my YouTube channel. I'm trying to figure out whether you really mean Carpe, or perhaps there's a little Carping in your question?

Sorry I did drop for a bit. Very interesting and informative. As a layman it always is amazing how theory and brute reality, as in where the data actually comes from and how it's used, how those two relate. As for the name, yes Carpe, Seize The Physics! (in broken latin of course :) )

 

[stevendaryl]

Every polarizer $i$ has an associated unitary operator $U_i$. You start with a state $\psi$. After the first polarizer, the state it $U_1\psi$. After the second one, it is $U_2 U_1\psi$ and after the third polarizer, it is $U_3 U_2 U_1\psi$. No projections are involved. You have to insert projections only if you perform measurements in between.

Are you talking about a special kind of polarizer, or the usual "polarizing filter" (such as is used in some sunglasses)? In the usual kind of polarizing filter, light that is polarized one way passes through unchanged, while light that is polarized perpendicular to that is absorbed by the filter. Absorption of a photon is irreversible, which usually means it is not described by a unitary transformation. Maybe I'm misunderstanding what you're saying?

I think there are devices that simply rotate the polarization of incoming light, without absorption.

 

[member 648030]

I try to explain what "collapse of the wave function" means without entering "semantic" discussions.
Take a coin and throw it. While the coin is launched, it has no value, it is, so to speak, both head and cross. Since the head and the cross are 1/2 chance to exit, we can formalize the "state" of the coin as a linear combination of "heads" and "cross": indicating with $\psi$ the wave function of the coin we can write:
$\psi = 1/2H+1/2C$
where C means cross, H, head, and 1/2 is the probability factor that exits H (or C).
Suppose the coin lands on a table and you say, "Oh, head out!"
Then the wave function, which was $\psi$ before became H.
In other words there has been a reduction (or collapse) from $\psi$ to H
Or, mathematically formalizing:
$\psi \rightarrow H$
This is the "collapse" of the wave function, neither more nor less

 

[rubi]

Are you talking about a special kind of polarizer, or the usual "polarizing filter" (such as is used in some sunglasses)? In the usual kind of polarizing filter, light that is polarized one way passes through unchanged, while light that is polarized perpendicular to that is absorbed by the filter. Absorption of a photon is irreversible, which usually means it is not described by a unitary transformation. Maybe I'm misunderstanding what you're saying?

I think there are devices that simply rotate the polarization of incoming light, without absorption.

Well, polarizers are understood quite well quantum mechanically in solid-state physics. What happens during an absorption is that photons are scattered into phonons, i.e. lattice vibrations, in the polarizer and the photon-phonon cross section is higher if the photon is polarized appropriately with respect to the lattice structure of the polarizer material. Effectively, this results in a low probability for photons with the wrong polarization to pass. Nevertheless, the whole process is unitary.

 

[Nugatory]

This is the "collapse" of the wave function, neither more nor less

That is not right.
The quantum mechanical state "superposition of A and B" is different from the quantum mechanical state "It is A or B and we don't know which yet", and there is no classical analogy for the former.

 

[stevendaryl]

Well, polarizers are understood quite well quantum mechanically in solid-state physics. What happens during an absorption is that photons are scattered into phonons, i.e. lattice vibrations, in the polarizer and the photon-phonon cross section is higher if the photon is polarized appropriately with respect to the lattice structure of the polarizer material. Effectively, this results in a low probability for photons with the wrong polarization to pass. Nevertheless, the whole process is unitary.

I'm willing to believe that the whole process is unitary if you include photons + phonons + the whole rest of the universe. But it's not a unitary transformation on photon states.

 

[rubi]

I'm willing to believe that the whole process is unitary if you include photons + phonons + the whole rest of the universe. But it's not a unitary transformation on photon states.

You don't need to include all that into the model (especially not the rest of the universe, which is not relevant anyway). For an effective (black box) model, it suffices to associate with the photon a property "absorbed/not absorbed", which allows you to include the transmission probability into the description, and ignore the physical details completely. A polarizer doesn't rotate a vertical photon into a horizontal photon, but rather a transmissible photon into an absorbed photon. The projection happens when we get to know whether the photon was absorbed or not. After all, the detector doesn't measure the polarization but rather just the presence of the photon and thus doesn't project onto the polarization states. If we don't measure that in between, then the whole process is completely unitary and I don't think that's controversial.

 

[stevendaryl]

You don't need to include all that into the model (especially not the rest of the universe, which is not relevant anyway). For an effective (black box) model, it suffices to associate with the photon a property "absorbed/not absorbed", which allows you to include the transmission probability into the description, and ignore the physical details completely. A polarizer doesn't rotate a vertical photon into a horizontal photon, but rather a transmissible photon into an absorbed photon. The projection happens when we get to know whether the photon was absorbed or not. After all, the detector doesn't measure the polarization but rather just the presence of the photon and thus doesn't project onto the polarization states. If we don't measure that in between, then the whole process is completely unitary and I don't think that's controversial.

I think you might be mixing up two different things. As described in the Wikipedia article on photon polarization:

A linear filter transmits one component of a plane wave and absorbs the perpendicular component...

An ideal birefringent crystal transforms the polarization state of an electromagnetic wave without loss of wave energy...A birefringent crystal is a material that has an optic axis with the property that the light has a different index of refraction for light polarized parallel to the axis than it has for light polarized perpendicular to the axis...

It's possible that a polarizing filter can be understood in terms of a birefringent crystal, but absorption is not a unitary transformation on the photon state. Passage through a birefringent crystal leaves the energy of the beam unchanged, while passage through a filter attenuates the energy.

 

[Carpe Physicum]

That is not right.
The quantum mechanical state "superposition of A and B" is different from the quantum mechanical state "It is A or B and we don't know which yet", and there is no classical analogy for the former.

Can you explain how they're different without resorting to math (I pre-appreciate your patience)? Seems like you're just mincing words. He could have said while it's in the air the coin IS heads and tails at the same time. And once it hits the table it collapses to Heads (for ex). Are you maybe saying there is no A and B really, and just some single A/B amalgamated state so to speak? (Which if this is the case then I can see the point that just talking about states really misrepresents the discussion.)

 

[rubi]

I think you might be mixing up two different things. As described in the Wikipedia article on photon polarization:
It's possible that a polarizing filter can be understood in terms of a birefringent crystal, but absorption is not a unitary transformation on the photon state. Passage through a birefringent crystal leaves the energy of the beam unchanged, while passage through a filter attenuates the energy.

No, I'm talking about a linear polarizer. You can model the absorption in the following way (ignoring all the physical details):
$\mathcal H_1 = \mathrm{span}\{h,v\},\, \mathcal H_2 = \mathrm{span}\{n,a\},\, \mathcal H = \mathcal H_1\otimes \mathcal H_2$
$U h\otimes n = h\otimes n,\, U v\otimes n = v \otimes a,\, U h \otimes a = h \otimes a,\, U v \otimes a = v \otimes n$
The Hilbert space $\mathcal H_2$ marks photons as absorbed ($a$) or not absorbed ($n$) and the unitary matrix $U$ models a linear polarizer aligned along the horizontal axis. In a more realistic model, $\mathcal H_2$ would be the Hilbert space of the phonons and the unitary matrix would be determined by some Hamiltonian that describes the photon-phonon interaction. But like I said, a black box description suffices and it's very far from a many-worlds model.

 

[DarMM]

Can you explain how they're different without resorting to math (I pre-appreciate your patience)? Seems like you're just mincing words.

No it's an important difference not word mincing. I'll try to be nontechnical.

Superposed states of spin up and down have different statistics for certain observables compared to just "Spin up or Spin down".

For example "Spin up or Spin down" has an average value when measuring Spin left/right of 0, i.e. comes up Left and Right 50:50.
"Spin up superposed with Spin down" will always come out Spin right, not 50:50 left/right.

 

[zonde]

A polarizer isn't modeled by a projection, but by unitary evolution.

Well, I'm too lazy right now to write down all the matrices and perform the calculation, but of course, if you choose $U_i$ and $\psi$ correctly and then expand the final state in the basis of your choice, you will get the correct experimental predictions.

No, I'm talking about a linear polarizer. You can model the absorption in the following way (ignoring all the physical details):
$\mathcal H_1 = \mathrm{span}\{h,v\},\, \mathcal H_2 = \mathrm{span}\{n,a\},\, \mathcal H = \mathcal H_1\otimes \mathcal H_2$
$U h\otimes n = h\otimes n,\, U v\otimes n = v \otimes a,\, U x \otimes a = x \otimes n$
The Hilbert space $\mathcal H_2$ marks photons as absorbed ($a$) or not absorbed ($n$) and the unitary matrix $U$ models a linear polarizer aligned along the horizontal axis. In a more realistic model, $\mathcal H_2$ would be the Hilbert space of the phonons and the unitary matrix would be determined by some Hamiltonian that describes the photon-phonon interaction. But like I said, a black box description suffices and it's very far from a many-worlds model.

You gave a non-mainstream idea. You refuse to show how it reproduces experimental predictions of mainstream approach. Please provide a reference otherwise it's your personal theory (not interpretation).

 

[atyy]

To give an example of something similar to what rubi is talking about in posts #49, #51 and #64, a unitary description of a polarizing beam splitter is given in Eq 1.22 of http://copilot.caltech.edu/documents/278-weihs_zeillinger_photon_statistics_at_beamsplitters_qip.pdf.

It is true that a polarizer can be modeled by a projection, but it is equally true that it can be modeled by unitary evolution in a larger Hilbert space as long as no measurement is performed.

 

[rubi]

You gave a non-mainstream idea. You refuse to show how it reproduces experimental predictions of mainstream approach. Please provide a reference otherwise it's your personal theory (not interpretation).

Physical models of polarizers are absolutely mainstream and standard solid-state physics. Modeling them by projections may suffice for some simple applications, but is not physically realistic. There is no doubt in the physics community, that the realistic situation is governed by purely unitary evolution. You even need to do that in order to get correct predictions for transmission efficiencies for example, because they follow from concrete calculations of photon-phonon scattering amplitudes. The inner workings of polarizers are part of introductory courses in optics, so it's clearly mainstream physics. If you want a reference, you can look at @atyy's article. The matrix (1.22) is exactly the matrix $U$ of my post #64.

Anyway, my point is that quantum systems are always governed by unitary evolution. Projections are only inserted when we acquire information through measurements. This is not the case in your example in post #48.

 

[stevendaryl]

To give an example of something similar to what rubi is talking about in posts #49, #51 and #64, a unitary description of a polarizing beam splitter is given in Eq 1.22 of http://copilot.caltech.edu/documents/278-weihs_zeillinger_photon_statistics_at_beamsplitters_qip.pdf.

It is true that a polarizer can be modeled by a projection, but it is equally true that it can be modeled by unitary evolution in a larger Hilbert space as long as no measurement is performed.

As I said in my post, I certainly see that some types of polarizers can be modeled by a unitary transformation. But in the case of a standard polarizing filter that absorbs light of one polarization and passes light of the orthogonal polarization, I don't see how that can be a unitary transformation on the state of the light. Maybe you can think of such a filter as a polarizing beam splitter together with an absorber: one of the two beams is directed into the absorber?

 

[zonde]

It is true that a polarizer can be modeled by a projection, but it is equally true that it can be modeled by unitary evolution in a larger Hilbert space as long as no measurement is performed.

It is easy to convince me. Calculate correct prediction for three polarizers experiment's outcome with usual angles (0, 45, 90 deg.,) using unitary evolution.

 

[zonde]

If you want a reference, you can look at @atyy's article. The matrix (1.22) is exactly the matrix $U$ of my post #64.

atyy's reference describes beam splitters. As stevendaryl already said you can turn polarization beam splitter into polarizer by attaching beam dumper to one of the outputs. The part that can't be described by unitary evolution is when you dump part of the beam (one component of the state).

Anyway, my point is that quantum systems are always governed by unitary evolution. Projections are only inserted when we acquire information through measurements. This is not the case in your example in post #48.

You can postpone the update till the beam encounters the next device. If the next device is detector your statement that "projections are only inserted when we acquire information through measurements" will be true. But that is only special case. If the next device is another polarizer at an angle rather than detector your approach breaks down as you have to update the state anyways (or you won't get correct prediction).

 

[rubi]

As I said in my post, I certainly see that some types of polarizers can be modeled by a unitary transformation. But in the case of a standard polarizing filter that absorbs light of one polarization and passes light of the orthogonal polarization, I don't see how that can be a unitary transformation on the state of the light. Maybe you can think of such a filter as a polarizing beam splitter together with an absorber: one of the two beams is directed into the absorber?

The situation is completely analogous. In the case of the polarizing beam splitter, the horizontal mode passes and the vertical mode is converted into a different spatial mode. In the case of an absorbing polarizer, the vertical mode is converted into a phonon mode. The matrix describing the effective dynamics is the same (see my post #64).

atyy's reference describes beam splitters. As stevendaryl already said you can turn polarization beam splitter into polarizer by attaching beam dumper to one of the outputs. The part that can't be described by unitary evolution is when you dump part of the beam (one component of the state).

The description is exactly the same except that the Hilbert space $\mathcal H_2$ now refers to the phonon mode rather than the different spatial mode. You don't need to dump any modes.

You can postpone the update till the beam encounters the next device. If the next device is detector your statement that "projections are only inserted when we acquire information through measurements" will be true. But that is only special case. If the next device is another polarizer at an angle rather than detector your approach breaks down as you have to update the state anyways (or you won't get correct prediction).

No, that's false. There is no projection in this situation. It's a multiplication of unitary matrices, just as I have explained in post #51. There may be a projection at the very end, if you decide to perform a measurement. It's really just standard quantum mechanics.

 

[zonde]

The description is exactly the same except that the Hilbert space $\mathcal H_2$ now refers to the phonon mode rather than the different spatial mode. You don't need to dump any modes.

Phonon modes do not interact with the second device. Only the photon modes that pass the filter interact with second device. Therefore you have to take out phonon modes from consideration. In other words there is no interference (superposition is gone) between phonon modes and passed photon modes at the second polarizer.

No, that's false. There is no projection in this situation. It's a multiplication of unitary matrices, just as I have explained in post #51. There may be a projection at the very end, if you decide to perform a measurement. It's really just standard quantum mechanics.

You know the rules - show that your model can reproduce predictions of standard approach only then it's valid interpretation. Three polarizer experiment is very simple experiment.
Otherwise it's just handwaving.

 

[rubi]

Phonon modes do not interact with the second device. Only the photon modes that pass the filter interact with second device. Therefore you have to take out phonon modes from consideration. In other words there is no interference (superposition is gone) between phonon modes and passed photon modes at the second polarizer.

Neither the phonon modes in the absorbing polarizer nor the redirected beam in the polarizing beam splitter interact with the second polarizer. The situation is completely identical. The phonons are stuck in the first polarizer and end up as heat. You don't have to remove any of these modes from the description. Obviously, they are still physically there, so they must also remain in the model.

You know the rules - show that your model can reproduce predictions of standard approach only then it's valid interpretation. Three polarizer experiment is very simple experiment.
Otherwise it's just handwaving.

This is absolutely standard. And you really have all the information to perform the calculation yourself if it still isn't obvious to you. If you want to learn quantum mechanics, you should do these kind of exercises on your own. It's you who's doing the handwaving. I think none of your 2677 posts ever contained a calculation.

We perform the calculation for the state
$$\psi = h\otimes n\otimes n\otimes n$$
The interesting matrix entries of the $U_i$ are given by:
$$U_1 h\otimes n\otimes n\otimes n = h\otimes n\otimes n\otimes n$$
$$U_2 \frac 1 {\sqrt 2} (h+v)\otimes n\otimes n\otimes n = \frac 1 {\sqrt 2} (h+v)\otimes n\otimes n\otimes n \\ U_2 \frac 1 {\sqrt 2} (h-v)\otimes n\otimes n\otimes n = \frac 1 {\sqrt 2} (h-v)\otimes n\otimes a\otimes n$$
$$U_3 h\otimes n\otimes n\otimes n = h\otimes n\otimes n\otimes a \\ U_3 v\otimes n\otimes n\otimes n = v\otimes n\otimes n\otimes n \\ U_3 x\otimes n\otimes a\otimes n = x\otimes n\otimes a\otimes n$$
The remaining matrix entries are left as an exercise to the reader.
$$U_1 \psi = h\otimes n\otimes n\otimes n \\ U_2 U_1 \psi = \frac 1 2 (h+v)\otimes n\otimes n\otimes n + \frac 1 2 (h-v)\otimes n\otimes a\otimes n \\ U_3 U_2 U_1 \psi = \frac 1 2 h\otimes n\otimes n\otimes a + \frac 1 2 v\otimes n\otimes n\otimes n + \frac 1 2 (h-v)\otimes n\otimes a\otimes n$$
So $P(v\otimes n\otimes n\otimes n) = (\frac 1 2)^2 = \frac 1 4$ as expected and in accordance with Malus law $I=I_0\cos(0^\circ)^2\cos(45^\circ)^2\cos(45^\circ)^2=\frac 1 4 I_0$.

 

[PeterDonis]

You gave a non-mainstream idea. You refuse to show how it reproduces experimental predictions of mainstream approach. Please provide a reference otherwise it's your personal theory (not interpretation).

You know the rules - show that your model can reproduce predictions of standard approach only then it's valid interpretation. Three polarizer experiment is very simple experiment.
Otherwise it's just handwaving.

If you think someone's posts are violating PF rules, you should report it, not try to admonish them in the thread.

 

[physika]

.

You are now talking not about "collapse", but rather superposition. Do you really want to do this?
Zz.

Broken Superposition = Collapse.

 

[bhobba]

.Broken Superposition = Collapse
.

All states are in superposition all the time. The state changes in collapse - not that it is in a superposition. The precise statement is as follows. Take the state its in, expand it in eigenvalues of the observable used to observe it and the outcome of the observation is one of the eigenvalues of that particular expansion - which is a specific superposition - but the state is in tons of other superposition's as well.

Before in the thread I gave the two axioms on which QM is built - that's all you need. Again they are:
1. Associated with every observation on a system is a linear operator, O, whose eigenvalues give the possible outcomes of the observation.
2. The average of the possible outcomes is given by the formula Trace (OS) where S is a positive operator of unit trace, by definition called the state of the system.

If you want to argue what collapse is - that's fine - but understand what the formalism itself says which doesn't even mention collapse.

You can do a google search on what collapse is and get all sorts of views. Comparing them to the formalism given above is an interesting exercise that will deepen your understanding of the basics of QM. Personally I will not advocate any particular view - make up your own mind. Of course I have one - but putting such views forward I do not think is productive, as parts of this thread have shown.

Thanks
Bill

 

[bhobba]

I think you surely meant “Schopenhauer” and not the German philosopher Arthur Schopenhauer (1788 – 1860) (https://en.wikipedia.org/wiki/Arthur_Schopenhauer).

Just so nobody is confused I did mean Schlosshauer and have update the post accordingly

Thanks
Bill

 

[Carpe Physicum]

No. It is more of laymen getting too enamored by the "name" that has been given to an aspect of physics. Maybe for you, you should replace the word "collapse" with a phrase such as "acquire immediately a specific value". After all, you had no problems when the coin that you tossed and landed to attain a particular state of either heads or tails.

Will this make it simpler?

Zz.

Well the thread wandered off into areas where my only response is "my cat's breath smells like catfood". ;) I probably should have left out the part about the formalism, so it's "Seems like at some point the formalism needs to be grounded back to reality". This thread was really a question about how physicists prevent themselves from talking merely about the math, and not the things the math was originally about. A dumb example might be this: a + b = c. a is the number of apples you're given, b is the number of oranges, and c is the total fruit count. So let's say we're getting all theoretical and through various mathematical gyrations we end up with someone working their way to something like the quadratic formula (or any complex formula). And now we're reasoning mathematically about this special formula, etc. I would say at that point we've lost the tie in back to fruit, i.e. reality. You can't square fruit, you can't give fruit coefficients, etc. those mathematical ideas make perfect sense in the context of a math discussion, but don't make sense when you consider the underlying physical things that started the discussion - apples and oranges. How do physicists prevent themselves from reasoning and making conclusions on the math itself, and not the reality to which the math is supposed to correspond?

 

[Mentz114]

How do physicists prevent themselves from reasoning and making conclusions on the math itself, and not the reality to which the math is supposed to correspond?

Short answer - by doing experiments (tests) and by including the concept of 'observable' in the formalism.

 

[Nugatory]

How do physicists prevent themselves from reasoning and making conclusions on the math itself, and not the reality to which the math is supposed to correspond?

You use the math to make predictions, then do experiments and make observations to find out if the predictions are any good. The mathematical formalism of quantum mechanics makes really good predictions - better than any other theory anyone has been able to come up.

 

[Carpe Physicum]

Short answer - by doing experiments (tests) and by including the concept of 'observable' in the formalism.

Would you mind giving a little more detail on how the concept of observable gets included in the formalism, just a high level for a non-math person (though I get the basic terminology to a degree). Mucho appreciated.

 

[Mentz114]

Would you mind giving a little more detail on how the concept of observable gets included in the formalism, just a high level for a non-math person (though I get the basic terminology to a degree). Mucho appreciated.

The mechanism requires graduate level mathematics so it can't be done in a Basic thread. Observables are modeled by mathematical objects called 'operators' which span classical and quantum theory.

But check out the Wiki article to get a taste.

 

[bhobba]

how physicists prevent themselves from talking merely about the math, and not the things the math was originally about.

I don't get this and never have - but it gets bought up a lot. Physicists are not pure mathematicians. We have presentations of QM that are pure math eg:
https://www.amazon.com/dp/0387493859/?tag=pfamazon01-20

They define precisely and mathematically things like observation etc etc. But it is in axiomatic mathematical language. However as Einstein said - 'As far as laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality'. Leaving aside the question of what reality is, in pure math you define your terms as axioms that are accepted as true in your mathematical system (for want of a better word) - there is no question of them being true or false - they are, by the very process used - ie if such and such is true then such and such follows. Physics deals with things like the lines and points of the usual presentation of geometry and the diagrams you draw - these things are quite real - and the theories physicists use deal with other things that are quite real and testable.

How pure math goes about things is not what physicists use. Euclidean geometry you learned at school has many applications eg surveying. Yet terms like line and point can only be given meaning in applying it - they are not precise. To fix this up a very great mathematician, Hilbert, came up with a version in the language of pure math which is purely axiomatic - but not that useful for everyday practical application:
https://en.wikipedia.org/wiki/Hilbert's_axioms

Physical Theories as used by physicists are like the Euclidean Geometry you were taught at school - not the above which is very abstract - its in the form you were taught at school so you can easily apply it in a particular situation like surveying eg what a line and a point are in the diagrams you use to prove its theorems is easy to see. Hilbert's version - quire a bit more difficult - maybe even impossible.

Its the difference between pure and applied math - physics is applied math.

Thanks
Bill

 

[Grinkle]

don't make sense when you consider the underlying physical things that started the discussion - apples and oranges.

I studied control theory in school and I definitely could do math for which I did not have any grounding in reality. As a simple example, I could calculate phase margin for a system well before I had any idea what phase margin meant or implied or why it even mattered. Some of what I learned to calculate I never did catch up with conceptually.

I don't get this and never have

I believe you, Bill, but I definitely get it. At least the way my brain is wired, learning math is a process of repeatedly working through bounded well defined processes (albeit very complex ones sometimes). Keeping that math tied to the real world application is not a bounded / well defined thing - it sometimes requires what I can only think to describe as non-linear intuition or maybe just brilliance / intelligence / insight / genius.

What does it mean to square a fruit? It might mean counting how many fruits can fit in the bottom of a box. To my anecdotal recollection, one of the places students really started to struggle with math was when the dreaded "word problems" were introduced for the first time.

 

[bhobba]

To my anecdotal recollection, one of the places students really started to struggle with math was when the dreaded "word problems" were introduced for the first time.

That's actually true - many have that experience. Me - it was the other way around - I never got math until I tackled algebra and associated word problems plus geometry - that happened in grade 8 when I was 12 - we started school at age 5 where I live - we are gradually moving to 6 like most other places. Then I understood the process of abstraction where you abstract the inessentials away leaving techniques that can be used all over the place. Then when I did my math degree I started to understand pure math where its abstracted away so much it's simply if such and such is true then such and such follows. And later about the interplay between pure and applied math.

I think there is something funny about an applied area like you mention that produces answers you can't physically interpret and IMHO points to a deeper issue - eg the runaway solutions of the Lorentz-Dirac Equation:
https://arxiv.org/pdf/gr-qc/9912045.pdf

The answer to that one is interesting, and alluded to some extent in the above, but needs its own thread.

So if you are getting answers in control theory you can't interpret, I would suggest that requires deeper investigation and may be trying to tell us something important like the Lorentz-Dirac equation.

There is another example in physics - good old re-normalization. When I read about it I thought what a load of bollocks. I was not the only one - the great physicist Ken Wilson thought the same. But he was determined to learn the techniques, pass the exams etc then figure out what was really going on. Why he was the only one to take that attitude - Gell-Mann, Feynman and Dyson all knew it was bollocks - Gel-Mann even discovering the answer but didn't recognize it at the time - really leaves me scratching my head. It took Wilson, whose adviser was Gell-Mann himself, 10 years to work it out. Fortunately for me there are now tons of articles on the answer and I found the solution quite quickly - here is a beginning article on the solution:
https://arxiv.org/pdf/hep-th/0212049.pdf

Interesting story about Wilson. Either Feynman or Gell-Mann could have taken him for his PhD. He knocked of Feynman's door - who gruffly said - what do you want. He asked - what are you doing right now? Feynman said nothing - get lost. He knocked of Murray's door and said the same thing - what are you working on - he opened the door and saw all these equations - Murray said this - Wilson recognized it and said a few things about it, they started discussing it and the next thing you know Murray is his adviser for a PhD that started his work towards solving it.

The fact that in applied areas when you get answers you can't interpret indicates something is wrong with your understanding that needs correcting - it may even be wrong. Only direct contradiction indicates something is amiss in pure math. That's one big difference between the two and depends crucially on one saying something about the world out there and the other just being an axiomatic/logic exercise. I won't use reality because IMHO it's far too loaded a term - but other physicists like Weinberg have no trouble with it - he has a much more direct view on it than me:
http://www.physics.utah.edu/~detar/phys4910/readings/fundamentals/weinberg.html

Personally I side with Dirac for what its worth:
http://philsci-archive.pitt.edu/1614/1/Open_or_Closed-preprint.pdf

We have like your example from control theory things not understood, out of whack, or can be made more elegant all the time - we slowly just keep slugging away at it.

Thanks
Bill

 

[bhobba]

Would you mind giving a little more detail on how the concept of observable gets included in the formalism, just a high level for a non-math person (though I get the basic terminology to a degree). Mucho appreciated.

Sorry - despite what you may have read not all concepts can be explained in English.

For this you need linear algebra:
http://quantum.phys.cmu.edu/CQT/chaps/cqt03.pdf

Linear Algebra is a standard course in virtually all mathematically related areas - Physics, Mathematics, Actuarial science, Finance, Economics, Econometrics, Statistics, Weather Forecasting and probably many others I forgot to mention. It really should, like calculus, be taught at HS, but due to our math phobic age educators think other things more important. But again that is the topic of another thread.

I will explain it using that.

Suppose you have an observation that can have yi (i = 1 to n) outcomes where each yi is a real number.

Pick any orthonormal basis in an n dimensional complex vector space |bi>. Then one can form an operator O = ∑ yi |bi><bi|. This encodes the possible outcomes of the observation

The yi are known as eigenvalues and the |bi> as their corresponding eigenvectors. Thus for any observation can find an operator that has eigenvalues the same as the observation.

I will not go into it because it involves a mathematical theorem (considered difficult) called Gleason's theorem. Gleason was one of those mathematicians not well known but who was in fact a quiet giant. This theorem was tough - but he was fired with the desire to solve it and he famously did. Just out of interest I will post a biography of Gleason:
https://www.ams.org/notices/200910/rtx091001236p.pdf

The theorem says just based on the definition of O I gave (and something called non-contextuality I will not go into - but its a very reasonable assumption considering that we are using vector spaces whose properties should not depend on the basis chosen) that you can calculate the average of the possible outcomes of the observation. It is Average (O) = Trace (OS) where S is something that pops out of the theorem and technically is known as a positive operater of unit trace.

By definition S is called the state of the system - but as presented here it, like probabilities is just something we assign to the thing being observed to aid in calculating that average of the observation.

This is what makes it more than just pure math - we are talking about things we observe. This is also the theories weakness. Presumably this theory can explain the macro world around us. But it is a theory about observations in that world. How can a theory that assumes such in the first place explain it? It seems hopeless, but believe it or not, without going into the details, a lot of progress has been made in doing that. Some issues remain but research is ongoing. We have various interpretations all having a different view. These may or may not be true - but all illuminate what the theory implies. I feel confident we will eventually arrive at a complete solution to this problem - we are almost there.

Thanks
Bill

 

[Carpe Physicum]

Sorry - despite what you may have read not all concepts can be explained in English.

For this you need linear algebra:
http://quantum.phys.cmu.edu/CQT/chaps/cqt03.pdf

Linear Algebra is a standard course in virtually all mathematically related areas - Physics, Mathematics, Actuarial science, Finance, Economics, Econometrics, Statistics, Weather Forecasting and probably many others I forgot to mention. It really should, like calculus, be taught at HS, but due to our math phobic age educators think other things more important. But again that is the topic of another thread.

...
Thanks
Bill

Very helpful. I vaguely remember Linear Algebra in college. But then here's my next question. If the same type of applied math can be applied to Physics, Finance!, etc. then how in the world can we say that it's anything other than a blunt tool that really has no relation to the actual physical nature of things? And then more to my original point, when you guys are doing all this abstract thinking about Hilbert Spaces, Lie Algebras, etc. (I'm faking it of course, just thinking of terms I've seen), are you saying that that thinking also applies to finance problems (which sounds completely absurd to me...almost dirty ;) ). I know I'm missing something. (Again, thanks for your patience.)

 

[Nugatory]

If the same type of applied math can be applied to Physics, Finance!, etc. then how in the world can we say that it's anything other than a blunt tool that really has no relation to the actual physical nature of things?

It feels wrong to describe as having "no real relationship to the actual physical nature of things" a tool that, in the hands of an expert user, describes the physical nature of things with exquisite accuracy. It also feels wrong to use the term "blunt instrument" to describe something so subtle and precise that its error bars are like the wingspan of a good-sized beetle set alongside the continent of north america (roughly one part in $10^8$).

Some of the problem here may be that it is quite an untenable leap from "very generally applicable" to "blunt tool that has no real relationship to the nature of things". Would you make the same argument about the theory of natural numbers? It can be applied to just about everything that can counted - three countable somethings plus two countable somethings make five countable somethings, whether they're zebras or galactic superclusters or bacteria. It's generally a bad habit to impute motives to other people, but I can't help thinking that the distinction you're making in this thread is actually between math that you're comfortable with and math that you're not comfortable with.

And then more to my original point, when you guys are doing all this abstract thinking about Hilbert Spaces, Lie Algebras, etc. (I'm faking it of course, just thinking of terms I've seen), are you saying that that thinking also applies to finance problems

I don't know if those specific topics do, but linear algebra certainly does. It's not quite as ubiquitous as the theory of countable numbers, but there are a lot of linear systems out there.

 

[bhobba]

then how in the world can we say that it's anything other than a blunt tool that really has no relation to the actual physical nature of things?

I think if you want to make your query one that can be answered you need to define precisely 'actual physical nature of things' AND have everybody, including philosophers agree. Good luck with that. In physics we take a simple view - we do not know or care about such things - the actual nature of things is what our theories describe but we do not worry about precisely defining that - we leave that to other disciplines - philosophy is usually what worries about that sort of thing but recently with Kuhn and his like others such as sociologists have got into the act.. And just like a map is not the territory, a mathematical description is not the actual nature of things - without saying what is even meant by that because you will likely get a lot of argument on it - but you will not get an argument on if the description (ie the theory) makes predictions in accord with experiment, everyday experience, observation etc.

And then more to my original point, when you guys are doing all this abstract thinking about Hilbert Spaces, Lie Algebras, etc. (I'm faking it of course, just thinking of terms I've seen), are you saying that that thinking also applies to finance problems (which sounds completely absurd to me...almost dirty ;) ). I know I'm missing something. (Again, thanks for your patience.)

Basically that's why pure math like linear algebra exists - for reasons we do not know the same mathematical ideas occur over and over again. We do not know why:
https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

And that it even crops up in non scientific areas such as Actuarial Science or Finance makes it even weirder.

But just my view - I am with Murray Gell-Mann:


But really nobody knows.

Thanks
Bill

 

[bhobba]

It feels wrong to describe as having "no real relationship to the actual physical nature of things" a tool that, in the hands of an expert user, describes the physical nature of things with exquisite accuracy. It also feels wrong to use the term "blunt instrument" to describe something so subtle and precise that its error bars are like the wingspan of a good-sized beetle set alongside the continent of north america (roughly one part in $10^8$).

It feels to anyone that has actually studied the theory that way (including me), but if you haven't then its very hard to get across what's going on. We simply do not know why, as Wigner says, its like that. In talking to people who have not experienced it I think the best way for them to get a grip on it is its a description of reality - but a description is not the same as the reality it describes. If someone asked me what reality is, its what your theories say it is - but that is just an opinion - I can't prove it. But what I do know is when you study the theory you feel more and more - this is the reality. You may even get sucked into the rabbit hole of Penrose - I did for a while.

To the OP to see the issues in detail read Penrose - The Emperors New Mind and some of his other books.

Thanks
Bill

 

[Carpe Physicum]

Very interesting, especially the dartmouth article. I really like the analogy of map versus territory mentioned above. It's definitely a philosophical puzzle, to use the analogy, that the same map appears to be able to describe vastly different territories in some cases. In the article the author mentions that laws of heredity and physics are two territories that as of yet cannot be described with the same map. (Well he doesn't use the analogy but you get the point.)

So the mystery remains, why in the world would the same map be useful in describing physics and economics? Could it be that the maps are just blunt tools we clever humans force upon the data so to speak? On some other alien planet, their maps could be entirely different, maybe not even what we would call math. I'm reminded of Einstein saying he wanted to know god's thoughts. I kinda don't like that what I think I'm learning is that we (well the theoretical guys) are not figuring out god's thoughts, but only coming up with clever human tricks to handle human obtained data points, whether it's physics or economics. Nothing godlike about it. I truly don't like that idea.

 

[bhobba]

t's definitely a philosophical puzzle

And that is where its should be discussed - not here where by forum rules we do not discuss philosophy.

So please let's get back to the the threads purpose - wave function collapse. Quite a few answers have been given. Wave-function collapse, using the axioms of QM is a simple consequence of those axioms. We have had a discussion on precisely defining it - but regardless of the exact wording of it it's the change in a systems state from observing it. QM is about observations. The state allows us to calculate probabilities of the results of observations - and it naturally changes after the observation. Just like when you flip a coin you will get a head or tail - while flipping it's 50/50 what you will get - when it lands its one or the other. QM - same thing - but more sophisticated because of complex numbers are allowed.

Thanks
Bill

 

[Carpe Physicum]

And that is where its should be discussed - not here where by forum rules we do not discuss philosophy.

So please let's get back to the the threads purpose - wave function collapse. Quite a few answers have been given. Wave-function collapse, using the axioms of QM is a simple consequence of those axioms. We have had a discussion on precisely defining it - but regardless of the exact wording of it it's the change in a systems state from observing it. QM is about observations. The state allows us to calculate probabilities of the results of observations - and it naturally changes after the observation. Just like when you flip a coin you will get a head or tale - while flipping it's 50/50 what you will get - when it lands its one or the other. QM - same thing - but more sophisticated because of complex numbers are allowed.

Thanks
Bill

Understood. Would Physics Forums consider adding a dedicated Philosophy of Science sub forum? The problem I've run into is that general philosophy forums that include philosophy of science, at least the ones I've looked at, tend to devolve into silliness because there are few to none physics practitioners participating. Or if anyone knows of a good forum for it, please post a link.

 

[weirdoguy]

Would Physics Forums consider adding a dedicated Philosophy of Science sub forum?

There's not enough philosophy specialists here to maintain quality on such subforum.

 

[bhobba]

Understood. Would Physics Forums consider adding a dedicated Philosophy of Science sub forum? The problem I've run into is that general philosophy forums that include philosophy of science, at least the ones I've looked at, tend to devolve into silliness because there are few to none physics practitioners participating. Or if anyone knows of a good forum for it, please post a link.

We used to have one but it simply got out of hand. We deal with mainstream science here so it was shut down.

If you want to pursue these issues further a good place to start is Penrose and his books:
https://en.wikipedia.org/wiki/Roger_Penrose

I think, as far as it can be dealt with at the B level, collapse has been pretty much mined, but the thread will remain open for at least a while longer to see what other issues arise. As a group the mentors will make a decision on when it is appropriate to close it, but even after its closed you can always ask for it to be opened by contacting any mentor and it will most definitely be looked at.

Thanks
Bill

 

[bhobba]

There's not enough philosophy specialists here to maintain quality on such subforum.

That was the exact issue and why it got out of hand. I have just done a year long postgraduate philosophy 101, 102 course and enrolled in a graduate certificate on philosophy. I had to pull out because of issues of getting to the library to do research at the time - the certificate was via assignments you needed to research. It also had, unknown to me when I enrolled, a historical philosophical bias ie it was more towards discussing philosophy in a historical context. I was interested more in discussing it in a scientific context - history is not really my thing. I was offered admission to a Masters In Philosophy on what interested me, but that was 3 years part time and I still had the problem of getting to the library. That's solved now - but I am 63 and research is getting a bit beyond me these days although I do what I can.

Thanks
Bill

 

[jeremyfiennes]

When a layman like myself hears the term 'Wave function collapse' is brings to mind physical things. A wave of some sort physically getting smaller or shrinking. Obviously that's not what it is but it does sound like it. In reality, if I have it right it's just a fancy way of saying a measurement has been taken and whatever it was that was being measured has been found to have a value (or range of values). But it might as well be called 'measurement function resolution' or even 'monkeyguts'. And by using "loaded" terms (loaded with physical sounding meaning) confusion might accidently arise. This is similar to web programming with the awful term 'cookies'. We all know it's just a file. But you can imagine a discussion that takes the analogy too far, and wanders into things like, if I mix enough dough, and then add chocalate chips, I can create numerous cookies. And someone replies, well it depends on how you bake the cookies and the type of oven you use. Pretty soon you're talking about cooking itself, instead of file operations and data storage. And if you're not careful you come to conclusions about baking, i.e. about the analogy, and not file storage. Is there a possibility of something like that happening in discussing QM and wave function collapse? Discussions and conclusions are stated having to do with the math (the baking as it were) instead of the thing itself, the files or thing being measured.

I agree entirely. Another good one is "things existing in more than one place at a time", which in terms of everyday language is nonsensical. If a physical 'thing' is here, it cannot be there, and vice versa. What QM-ites mean is that if you make a measurement, you might find it here or you might find it there. But that is not the same. Why can't they say what they mean, rather than trying to confuse us with all this 'wierdness' gobbledigook which (according to them) only they are competent to understand?

 

[stevendaryl]

I agree entirely. Another good one is "things existing in more than one place at a time", which in terms of everyday language is nonsensical. If a physical 'thing' is here, it cannot be there, and vice versa. What QM-ites mean is that if you make a measurement, you might find it here or you might find it there. But that is not the same. Why can't they say what they mean, rather than trying to confuse us with all this 'wierdness' gobbledigook which (according to them) only they are competent to understand?

Nobody is trying to confuse anybody. There are aspects of what's going on that are just not very well understood.

 

[Carpe Physicum]

I think if you want to make your query one that can be answered you need to define precisely 'actual physical nature of things' AND have everybody, including philosophers agree. Good luck with that. In physics we take a simple view - we do not know or care about such things - the actual nature of things is what our theories describe but we do not worry about precisely defining that - we leave that to other disciplines - philosophy is usually what worries about that sort of thing but recently with Kuhn and his like others such as sociologists have got into the act.. And just like a map is not the territory, a mathematical description is not the actual nature of things - without saying what is even meant by that because you will likely get a lot of argument on it - but you will not get an argument on if the description (ie the theory) makes predictions in accord with experiment, everyday experience, observation etc.
Basically that's why pure math like linear algebra exists - for reasons we do not know the same mathematical ideas occur over and over again. We do not know why:
https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

And that it even crops up in non scientific areas such as Actuarial Science or Finance makes it even weirder.

But just my view - I am with Murray Gell-Mann:


But really nobody knows.

Thanks
Bill

Sorry just one more semi-philosophical post that's more directed to the mathies here. If the same math can be used in physics and other "human" areas, in what way could it be true that the math is elegant? I've always thought it was considered elegant because it only applied to the physical universe. Hence the universe is governed by elegant and permanent, emphasis on permanent (not temporary human areas) math. I remember being utterly disappointed to hear a friend who was heavy into advanced economics using some of the same complex math. But economics is merely temporary and human. So what's going on? Are economists just "faking it" so to speak?