Meaning of Wave Function Collapse

[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