Where Can I Find Comprehensive Notes on Quantum Mechanics Interpretations?

Lapidus
Messages
344
Reaction score
12
Does anyone know good notes or a book on quantum mechanics that covers well the interpretational issues? Especially, which deals also with the last fifty or sixty years, i.e. that has Bell, decoherence, GHZ, Aspect experiment, Mach-Zehnder interferometer, delayed-choice, mesocopic Schrödinger cats, Bohm wave mechanics, many-world interpretation, etc. in it.

And a text/ notes that perhaps covers these topics in a more or less down-to-earth pedagogical manner.

I already looked hard, but for strange reasons I could not come up with any findings.

thanks
 
Last edited:
Physics news on Phys.org
I think Ballentines book is a good no-nonsense source for studying such questions:

L. Ballentine, Quantum Mechanics - A modern approach

Another one is

A. Peres, Quantum Theory: Concepts and Methods
 
Lapidus said:
Does anyone know good notes or a book on quantum mechanics that covers well the interpretational issues? Especially, which deals also with the last fifty or sixty years, i.e. that has Bell, decoherence, GHZ, Aspect experiment, Mach-Zehnder interferometer, delayed-choice, mesocopic Schrödinger cats, Bohm wave mechanics, many-world interpretation, etc. in it.

And a text/ notes that perhaps covers these topics in a more or less down-to-earth pedagogical manner.

I already looked hard, but for strange reasons I could not come up with any findings.

thanks


http://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics

references herein.

------

http://plato.stanford.edu/
 
vanhees71 said:
I think Ballentines book is a good no-nonsense source for studying such questions:

L. Ballentine, Quantum Mechanics - A modern approach

Another one is

A. Peres, Quantum Theory: Concepts and Methods

What was it Meatloaf said - you took the words right out of my mouth.

Ballentine also develops QM axiomatically from just two axioms. Interestingly the second is more or less implied by the first by Gleason's Theorem:
http://kof.physto.se/theses/helena-master.pdf

Strange, but true - it really involves just one axiom - the rest follows from rather innocuous observations such as Schrodingers equation etc comes from probabilities should be coordinate independent ie symmetry.

Intrigued - get the book - it had a BIG effect on me - its basically the finest book on QM I have ever read.

After that check out Decoherence and the Quantum-to-Classical Transition by Schlosshauer:
https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20

It examines interpretations in light of the recent advances in decoherence which Ballentine doesn't explore.

Thanks
Bill
 
Last edited by a moderator:
Lapidus said:
Does anyone know good notes or a book on quantum mechanics that covers well the interpretational issues? Especially, which deals also with the last fifty or sixty years, i.e. that has Bell, decoherence, GHZ, Aspect experiment, Mach-Zehnder interferometer, delayed-choice, mesocopic Schrödinger cats, Bohm wave mechanics, many-world interpretation, etc. in it.

And a text/ notes that perhaps covers these topics in a more or less down-to-earth pedagogical manner.

I already looked hard, but for strange reasons I could not come up with any findings.

thanks
My recommendations:
https://www.amazon.com/dp/0199589135/?tag=pfamazon01-20
https://www.amazon.com/dp/110702501X/?tag=pfamazon01-20

If you want something free, then:
http://lanl.arxiv.org/abs/quant-ph/0209123
 
Last edited by a moderator:
  • Like
Likes 1 person
Just a warning: If people simply post links to articles about the interpretation of quantum mechanics, that's fine. But if you actually discuss the interpretation of quantum mechanics, the thread will be closed by the moderators. At least, that's my observation.
 
stevendaryl said:
Just a warning: If people simply post links to articles about the interpretation of quantum mechanics, that's fine. But if you actually discuss the interpretation of quantum mechanics, the thread will be closed by the moderators. At least, that's my observation.
If you discuss some specific interpretation at a thread which is from the beginning opened to be a thread on that specific interpretation, it is usually not closed by the moderators.
 
Thanks everybody!

Demystifier said:

Funny (or not so), after two hours searching the internet, I also felt these two books look best. Unfortunately, the "New quantum age" book has no working kindle format, only "Kindle for PC". I thought about downloading the second book, but now I will look first at the pdf of the same author from the link you gave. Thanks!
 
Last edited by a moderator:
Are "interpretation of QM" threads really closed that quickly here in the forum? I've not that impression. However, the problem is that often such discussions leave the realm of hard sciences (physics) and enter the more philosophical kind ("cargo cult science", as Feynman called it). Then, of course, it's good when the moderators close the thread ;-)).
 
  • #10
vanhees71 said:
Are "interpretation of QM" threads really closed that quickly here in the forum? I've not that impression. However, the problem is that often such discussions leave the realm of hard sciences (physics) and enter the more philosophical kind ("cargo cult science", as Feynman called it). Then, of course, it's good when the moderators close the thread ;-)).

I don't know about soon, but they are all closed eventually--long before they die from lack of new posts.
 
  • #11
If they are going around in circles (which happens almost immediately in many): expect them to be closed more quickly than previously. Ditto for the extended debates about Bell, closed loopholes, etc. that seem to recur all too often and degrade into statements of personal opinion.
 
  • #12
At any rate, i am convinced that the Mentor does not play dice with these threads.
 
  • #13
Determinism usually is a good closure...
 
  • #14
stevendaryl said:
I don't know about soon, but they are all closed eventually--long before they die from lack of new posts.

Usually they're brain-dead from oxygen and new-insight starvation before they run out of new posts.
 
  • #15
:smile:
 
  • #16
DrChinese said:
If they are going around in circles (which happens almost immediately in many): expect them to be closed more quickly than previously. Ditto for the extended debates about Bell, closed loopholes, etc. that seem to recur all too often and degrade into statements of personal opinion.
This is the main reason that interpretation threads are closed. Once everyone has had a chance to advertise for their favorite interpretation the rest of the discussion rapidly devolves into a shouting match about why their favorite is wonderful and why the other person's favorite is stupid.
 
  • #17
vanhees71 said:
Are "interpretation of QM" threads really closed that quickly here in the forum? I've not that impression. However, the problem is that often such discussions leave the realm of hard sciences (physics) and enter the more philosophical kind ("cargo cult science", as Feynman called it). Then, of course, it's good when the moderators close the thread ;-)).

That's my feeling as well.

They aren't closed until everyone has had a chance to put their view, and in the past some have degenerated into philosophy. Personally I think the mods do a really good job monitoring that.

The only other observation I will make is a personal one about decoherence - I seem to go over the same thing again and again. The positive is the issues are really clear in my mind from that practice and I know the references off pat.

Thanks
Bill
 
  • #18
DaleSpam said:
Once everyone has had a chance to advertise for their favorite interpretation the rest of the discussion rapidly devolves into a shouting match about why their favorite is wonderful and why the other person's favorite is stupid.

Yup, I have a very close friend (won't drop names due to discretion o:)) who basically thinks it's all pretty silly, and that the whole business have more in common with "religious war" (experiments is a waste of time since we all get the same results), than fundamental science.

Maybe he's right, I dunno? :rolleyes:
 
  • Like
Likes 1 person
  • #19
DevilsAvocado said:
Maybe he's right, I dunno? :rolleyes:

He is right :wink: :wink: :wink:

But its interesting understanding and comparing them.

Thanks
Bill
 
  • #20
DaleSpam said:
This is the main reason that interpretation threads are closed. Once everyone has had a chance to advertise for their favorite interpretation the rest of the discussion rapidly devolves into a shouting match about why their favorite is wonderful and why the other person's favorite is stupid.
100 % concur.
pompous proselytism, apodictical dogmatism.

i think, various interpretations have possible and real experimental testing, so we have to discourse in that possibilities.
or we have to diferentiate among stand alone models and interpretations.

clear and cut, scientific standing..
 
Last edited:
  • #21
I'd make a clear distinction between "interpretation of quantum theory" and "alternative new theories". Quantum theory in its minimal interpretation (a theory without an interpretation, i.e., without making contact to real-world observations and experiments is not a physical theory at all) defines a clear mathematical scheme and how it is applied to describe the phenomenology they are applicable (sometimes in exact ab-initio solutions of the quantum-theoretical equations, often in terms of approximations like perturbation theory in QFT) to make predictions that can be experimentally tested. Personally, I'm sticking to the minimal interpretation, because it's minimal. In my opinion, a physical theory is complete as soon as I can describe all known reproducible observations with it (that's the status of QT today) or as soon as I have found an experimental fact contradicting it, which doesn't make it obsolete completely but it leads to certain constraints on the applicability range of the theory. That's the case for all of classical physics (Newtonian mechanics was found to be invalid at high velocities and then has to be substituted by (special and general) relativity, electromagnetism needs the extension to relativistic field-theoretical descriptions (first the classical kind a la Maxwell); all of classical physics needs to be substituted by quantum theory after all, etc.). It's pretty likely that also quantum theory one day may be found to be an approximate description of Nature by finding some empirical contradiction to it and it must be substituted by a more comprehensive theory. That's progress of science and thus should not be taken as a "failure" in any way.

Some people are, however, not satisfied by the minimal interpretation because of its probabilistic nature, and they try to find other interpretations of quantum theory. These other interpretations lead to the same predictions about the outcome of measurements and thus from a physics point of view it's still the same theory. There are, however, different categories of "interpretations".

The first category are different mathematical techniques applied to quantum theory. This already started in the very beginning of quantum theory in 1925, when nearly at the same time, Heisenberg, Born, and Jordan came up with what is called "matrix mechanics" and Schrödinger with "wave mechanics". Independently Dirac came up with another formulation which is the most general one and was called "transformation theory" at the time. From the modern point of view, I'd call it representation-independent formulation. The non-relativistic QT was then completely formalized and understood by von Neumann in terms of Hilbert-space theory, which more recently has been reformulated in terms of the socalle "rigged Hilbert-space" formalism, which makes the hand-waving maths we physicists use in our calculations a mathematically strict formulation. Another equivalent (but as far as I know less strict) formulation is then Feynman's path-integral formulation, using functional methods to express the same theory. All this is not "interpretation" in the more narrow sense, but just different mathematical language to formulate the same theory. Which mathematical formalism you use doesn't so much imply which interpretation you follow, although often the path-integral formalism is interpreted somehow in the sense as if it were a new interpretation, a view also Feynman seems to have hold for some time. I think, that's not the case. It's just another way to express one and the same theory, and I (as a follower of the minimal interpretation) can use any mathematical tool without changing my view on interpretation. Which one I use, depends on the problem I like to solve and which one I'm able to use for that problem, but it's totally unimportant concerning my view on the interpretation.

The second category is to add some elements, concerning the philosophical implications of the theory. E.g., in Bohmian mechanics, one adds a kind of "trajectory picture" back into the physics of single particles which has been abandoned before due to the uncertainty relation that states that a particle cannot be prepared in a state such that both position and momentum are both determined to arbitrary precision, which implies that it does not make sense to talk about trajectories of a particle in phase space as is the case in classical mechanics. The Bohmian trajectories are determined by non-local equations with the "wave function" of the particles as "pilot waves". On the other hand it's still an open debate, whether the Bohmian trajectories are observable or not and if so, whether Bohmian mechanics is experimentally disproven already. I think we can say that's pretty undecided today. The same holds for other interpretations as the various flavor of the Copenhagen interpretation, which is a rather vague conglomerate of different ideas like "complementarity" (Bohr), "collapse of the state" (Bohr, Heisenberg, von Neumann), the "cut between quantum and classical behavior" (von Neumann). In my opinion the "collapse idea" is pretty useless if not misleading. At least it brings more problems with it than it solves with regard to Einstein causality in relativistic physics. This was already critizized by Einstein, Poldolsky, and Rosen in their famous paper. Then there is "many-worlds theory", which introduces the quite funny idea of parallel (unobservable!) universes, splitting up at every time one observes some clear fact about a quantum system. I never understood what this bizarre idea might solve in terms of the philosophical quibbles some people have with the (minimally interpreted) quantum theory and its probabilistic nature.

All this are, however, legitimate questions to be asked, and answers to them might lead to a deeper understanding of quantum theory. Of course, anything which doesn't lead to observable consequences that contradict quantum theory in its minimal interpretation, doesn't lead to a new theory in the sense of natural sciences and thus one can well say that such discussions are off topic in a scientific forum like this, and it's as legitimate to close threads on "philsophy" (humanities) without "scientific content", which of course includes also this posting itself ;-)).
 
  • Like
Likes 1 person
  • #22
vanhees71 said:
Some people are, however, not satisfied by the minimal interpretation because of its probabilistic nature

That is not a very good characterization of why people (today) are not satisfied by the minimal interpretation. There are no interpretational difficulties with a stochastic process, where the evolution equations are probabilistic, rather than deterministic. Einstein may not have liked nondeterminism, but that's not anywhere close to the main difficulty in interpreteting quantum mechanics.

The problems that I have with the minimal interpretation is that it seems incoherent. You use the wave function to compute a probability. Fine. But a probabilities of what? Is it a probability of some physical quantity having a certain value? No, that can't be the case. It is not consistent to assume that physical quantities have values before they are measured (because of incompatible observables). So what is it a probability of? You can say that it's the probability, not of something being a certain value, but of an experiment measuring a certain value. But that seems incoherent, as well. Measuring devices are physical objects, themselves. They obey the laws of physics, which presumably includes quantum mechanics. So if it is not consistent to assume that electrons have definite values of properties such as "z component of spin", then how is it consistent to assume that a measurement device has a definite value for something like "the measured value of spin"?

So the minimal interpretation seems completely incoherent to me, as a physical theory. You can make it into a recipe for doing physics by doing as the Copenhagen people suggested, which is to separate reality into macroscopic and microscopic realms, and to assume that in the macroscopic realm, objects have definite macroscopic properties at all times, while in the microscopic realm, there are only wave functions, which are used to compute probabilities for events in the macroscopic realm. I think that works in practice, but it is certainly unsatisfying, because the macroscopic/microscopic distinction seems ad hoc and subjective.
 
  • #23
vanhees71 said:
I'd make a clear distinction between "interpretation of quantum theory" and "alternative new theories". Quantum theory in its minimal interpretation (a theory without an interpretation, i.e., without making contact to real-world observations and experiments is not a physical theory at all) defines a clear mathematical scheme and how it is applied to describe the phenomenology they are applicable (sometimes in exact ab-initio solutions of the quantum-theoretical equations, often in terms of approximations like perturbation theory in QFT) to make predictions that can be experimentally tested. Personally, I'm sticking to the minimal interpretation, because it's minimal. In my opinion, a physical theory is complete as soon as I can describe all known reproducible observations with it (that's the status of QT today) or as soon as I have found an experimental fact contradicting it, which doesn't make it obsolete completely but it leads to certain constraints on the applicability range of the theory. That's the case for all of classical physics (Newtonian mechanics was found to be invalid at high velocities and then has to be substituted by (special and general) relativity, electromagnetism needs the extension to relativistic field-theoretical descriptions (first the classical kind a la Maxwell); all of classical physics needs to be substituted by quantum theory after all, etc.). It's pretty likely that also quantum theory one day may be found to be an approximate description of Nature by finding some empirical contradiction to it and it must be substituted by a more comprehensive theory. That's progress of science and thus should not be taken as a "failure" in any way.

Some people are, however, not satisfied by the minimal interpretation because of its probabilistic nature, and they try to find other interpretations of quantum theory. These other interpretations lead to the same predictions about the outcome of measurements and thus from a physics point of view it's still the same theory. There are, however, different categories of "interpretations".

The first category are different mathematical techniques applied to quantum theory. This already started in the very beginning of quantum theory in 1925, when nearly at the same time, Heisenberg, Born, and Jordan came up with what is called "matrix mechanics" and Schrödinger with "wave mechanics". Independently Dirac came up with another formulation which is the most general one and was called "transformation theory" at the time. From the modern point of view, I'd call it representation-independent formulation. The non-relativistic QT was then completely formalized and understood by von Neumann in terms of Hilbert-space theory, which more recently has been reformulated in terms of the socalle "rigged Hilbert-space" formalism, which makes the hand-waving maths we physicists use in our calculations a mathematically strict formulation. Another equivalent (but as far as I know less strict) formulation is then Feynman's path-integral formulation, using functional methods to express the same theory. All this is not "interpretation" in the more narrow sense, but just different mathematical language to formulate the same theory. Which mathematical formalism you use doesn't so much imply which interpretation you follow, although often the path-integral formalism is interpreted somehow in the sense as if it were a new interpretation, a view also Feynman seems to have hold for some time. I think, that's not the case. It's just another way to express one and the same theory, and I (as a follower of the minimal interpretation) can use any mathematical tool without changing my view on interpretation. Which one I use, depends on the problem I like to solve and which one I'm able to use for that problem, but it's totally unimportant concerning my view on the interpretation.

The second category is to add some elements, concerning the philosophical implications of the theory. E.g., in Bohmian mechanics, one adds a kind of "trajectory picture" back into the physics of single particles which has been abandoned before due to the uncertainty relation that states that a particle cannot be prepared in a state such that both position and momentum are both determined to arbitrary precision, which implies that it does not make sense to talk about trajectories of a particle in phase space as is the case in classical mechanics. The Bohmian trajectories are determined by non-local equations with the "wave function" of the particles as "pilot waves". On the other hand it's still an open debate, whether the Bohmian trajectories are observable or not and if so, whether Bohmian mechanics is experimentally disproven already. I think we can say that's pretty undecided today. The same holds for other interpretations as the various flavor of the Copenhagen interpretation, which is a rather vague conglomerate of different ideas like "complementarity" (Bohr), "collapse of the state" (Bohr, Heisenberg, von Neumann), the "cut between quantum and classical behavior" (von Neumann). In my opinion the "collapse idea" is pretty useless if not misleading. At least it brings more problems with it than it solves with regard to Einstein causality in relativistic physics. This was already critizized by Einstein, Poldolsky, and Rosen in their famous paper. Then there is "many-worlds theory", which introduces the quite funny idea of parallel (unobservable!) universes, splitting up at every time one observes some clear fact about a quantum system. I never understood what this bizarre idea might solve in terms of the philosophical quibbles some people have with the (minimally interpreted) quantum theory and its probabilistic nature.

All this are, however, legitimate questions to be asked, and answers to them might lead to a deeper understanding of quantum theory. Of course, anything which doesn't lead to observable consequences that contradict quantum theory in its minimal interpretation, doesn't lead to a new theory in the sense of natural sciences and thus one can well say that such discussions are off topic in a scientific forum like this, and it's as legitimate to close threads on "philsophy" (humanities) without "scientific content", which of course includes also this posting itself ;-)).

It's unclear to me if the minimal statistical interpretation without the projection postulate is correct. The ability to use a projection operator is a standard part of quantum mechanics, acknowledged eg. by Landau and Lifshitz and Cohen-Tannoudji, and even in modern formulations such as that by Hardy http://arxiv.org/abs/quant-ph/0101012 . So the use of the projection operator must be either derived or postulated. Without projection, the state of a sub-ensemble is undefined, and so filtering measurements used for state preparation cannot be described. If the reduced density matrix is interpreted as a proper mixture, that seems to me an unacknowledged use of the projection postulate, eg:

http://arxiv.org/abs/quant-ph/0312059 (p9) "The reduced density matrix looks like a mixed state density matrix because, if one actually measured an observable of the system, one would expect to get a definite outcome with a certain probability; in terms of measurement statistics, this is equivalent to the situation in which the system is in one of the states from the set of possible outcomes from the beginning, that is, before the measurement. As Pessoa (1998, p. 432) puts it, “taking a partial trace amounts to the statistical version of the projection postulate.”"

http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf (p37) "Ignorance interpretation: The mixed states we find by taking the partial trace over the environment can be interpreted as a proper mixture. Note that this is essentially a collapse postulate."
 
Last edited:
  • #24
stevendaryl said:
That is not a very good characterization of why people (today) are not satisfied by the minimal interpretation. There are no interpretational difficulties with a stochastic process, where the evolution equations are probabilistic, rather than deterministic. Einstein may not have liked nondeterminism, but that's not anywhere close to the main difficulty in interpreteting quantum mechanics.

The problems that I have with the minimal interpretation is that it seems incoherent. You use the wave function to compute a probability. Fine. But a probabilities of what? Is it a probability of some physical quantity having a certain value? No, that can't be the case. It is not consistent to assume that physical quantities have values before they are measured (because of incompatible observables). So what is it a probability of? You can say that it's the probability, not of something being a certain value, but of an experiment measuring a certain value. But that seems incoherent, as well. Measuring devices are physical objects, themselves. They obey the laws of physics, which presumably includes quantum mechanics. So if it is not consistent to assume that electrons have definite values of properties such as "z component of spin", then how is it consistent to assume that a measurement device has a definite value for something like "the measured value of spin"?

So the minimal interpretation seems completely incoherent to me, as a physical theory. You can make it into a recipe for doing physics by doing as the Copenhagen people suggested, which is to separate reality into macroscopic and microscopic realms, and to assume that in the macroscopic realm, objects have definite macroscopic properties at all times, while in the microscopic realm, there are only wave functions, which are used to compute probabilities for events in the macroscopic realm. I think that works in practice, but it is certainly unsatisfying, because the macroscopic/microscopic distinction seems ad hoc and subjective.

The problem here seems to be a misunderstanding of the idea of a "quantum state" within the minimal interpretation. I think it is very important to distinguish between "measurement" and "preoparation". This is very often mixed up and leads to the difficulties you seem to have with the minimal interpretation.

The probabilistic interpretation of the (pure) states in quantum theory, i.e., Born's postulate, implies that you can interpret the states as descriptions of (an equivalence class) of preparations, i.e., clear manipulations on a single quantum system, which prepare them in a certain state in a reproducible way. Then and only then you can prepare ensembles of independent systems in that given state.

This state preparation implies which observables take determined values and which don't. All you can say about the measurement of an observable is "encoded" in the (normalized) state vector (or the corresponding ray), and this is the probability to find a certain value of an observable when you measure it. This prediction can only be tested by preparing a sufficiently large ensemble of equally prepared systems such that you get the probabilities within a statistical limit of accuracy.

There is no necessity to assume a cut between a quantum realm and a classical realm. The classical behavior of macroscopic systems, as are necessarily measurement devices (as indeed already pointed out by Bohr), is understandable from quantum mechanics in terms of standard many-body theory (some coarse graining to average over many "micro states" making up a "macro state", leading to decoherence).
 
  • #25
atyy said:
It's unclear to me if the minimal statistical interpretation without the projection postulate is correct. The ability to use a projection operator is a standard part of quantum mechanics, acknowledged eg. by Landau and Lifshitz and Cohen-Tannoudji, and even in modern formulations such as that by Hardy http://arxiv.org/abs/quant-ph/0101012 . So the use of the projection operator must be either derived or postulated. Without projection, the state of a sub-ensemble is undefined, and so filtering measurements used for state preparation cannot be described. If the reduced density matrix is interpreted as a proper mixture, that seems to me an unacknowledged use of the projection postulate, eg:

http://arxiv.org/abs/quant-ph/0312059 (p9) "The reduced density matrix looks like a mixed state density matrix because, if one actually measured an observable of the system, one would expect to get a definite outcome with a certain probability; in terms of measurement statistics, this is equivalent to the situation in which the system is in one of the states from the set of possible outcomes from the beginning, that is, before the measurement. As Pessoa (1998, p. 432) puts it, “taking a partial trace amounts to the statistical version of the projection postulate.”"

http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf (p37) "Ignorance interpretation: The mixed states we find by taking the partial trace over the environment can be interpreted as a proper mixture. Note that this is essentially a collapse postulate."

What's the problem with the "projection" postulate? I think, it's indeed one of the basic postulates which is part of the foundations of quantum theory. It's closely related to the Born postulate, i.e., how to calculate the probability for the outcome of measurements for a given prepration. There's a very nice discussion on the question, whether the Born postulate can be derived from the other postulates of quantum theory in Weinberg's newest textbook "Quantum Mechanics" with the conclusion that it is an independent postulate.
 
  • #26
vanhees71 said:
There's a very nice discussion on the question, whether the Born postulate can be derived from the other postulates of quantum theory in Weinberg's newest textbook "Quantum Mechanics" with the conclusion that it is an independent postulate.

Weinberg is correct - and it has been known for a long time.

But I find the basis independence assumption of Gleason's Theorem very natural mathematically and personally is what I use use to justify it, which otherwise would seem rather ad-hoc.

Thanks
Bill
 
  • #27
vanhees71 said:
What's the problem with the "projection" postulate? I think, it's indeed one of the basic postulates which is part of the foundations of quantum theory. It's closely related to the Born postulate, i.e., how to calculate the probability for the outcome of measurements for a given prepration. There's a very nice discussion on the question, whether the Born postulate can be derived from the other postulates of quantum theory in Weinberg's newest textbook "Quantum Mechanics" with the conclusion that it is an independent postulate.

Yes, the way Weinberg states the Born rule includes the projection postulate (p26 of http://bks3.books.google.com/books?id=WfTq2W_LBlEC&source=gbs_navlinks_s ). But in your post you indicated that "collapse" was misleading. Isn't "collapse" just another name for the projection postulate?
 
Last edited by a moderator:
  • #28
Born's rule just says what are the probabilities for finding a certain value for an observable when this observable is measured on a system in a given (pure or mixed) state. It does not say that immediately after the measurement the system's state must instantaneously collapse into an eigenstate of this observable. This is most often not the case, because often the system gets destroyed being measured. Of course, there are cases of (almost) von Neumann ideal filter measurements. E.g., one can prepare (almost) pure spin states for an atom using a Stern Gerlach apparatus and just absorbing all partial beams except one, but here I don't need a collapse, but can understand from the quantum dynamics, how the positition of the partial beams gets to (nearly) 100% entangled with the spin state of the particle. Then, I just dump all "unwanted" partial beams into a wall and let all the "wanted" ones through. That's all, leading to the preparation of atoms in a definite spin state. No esoterics like collapses or the like needed.
 
  • #29
vanhees71 said:
Born's rule just says what are the probabilities for finding a certain value for an observable when this observable is measured on a system in a given (pure or mixed) state. It does not say that immediately after the measurement the system's state must instantaneously collapse into an eigenstate of this observable. This is most often not the case, because often the system gets destroyed being measured. Of course, there are cases of (almost) von Neumann ideal filter measurements. E.g., one can prepare (almost) pure spin states for an atom using a Stern Gerlach apparatus and just absorbing all partial beams except one, but here I don't need a collapse, but can understand from the quantum dynamics, how the positition of the partial beams gets to (nearly) 100% entangled with the spin state of the particle. Then, I just dump all "unwanted" partial beams into a wall and let all the "wanted" ones through. That's all, leading to the preparation of atoms in a definite spin state. No esoterics like collapses or the like needed.

Let me see if I understand what you are saying about the Stern Gerlach case. There position and spin are entangled, then the mathematical description of dumping the other beams is the partial trace over all unwanted beams, leaving the reduced density matrix for the wanted beam?
 
  • #30
In a somwhat abstract sense yes. You just choose a subensemble from a larger ensemble. Of course, you ignore the pretty complicated dynamics leading to the absorption of particles in the beam dump ;-).
 
  • #31
vanhees71 said:
In a somwhat abstract sense yes. You just choose a subensemble from a larger ensemble. Of course, you ignore the pretty complicated dynamics leading to the absorption of particles in the beam dump ;-).

The reason I think the partial trace either involves the projection postulate, or must be introduced as a new postulate is that in the naive textbook formalism, the meaning of the partial trace is derived from the projection postulate. I think it is possible to do without the projection postulate in the ensemble interpretation, provided one replaces it with another postulate, such as postulating the interpretation of the partial trace directly. However, I am skeptical that the ensemble interpretation works without the projection postulate or a replacement.

There are similar thoughts that the partial trace has a hidden use of the projection postulate in eg.

http://arxiv.org/abs/quant-ph/0312059 (p9) "The reduced density matrix looks like a mixed state density matrix because, if one actually measured an observable of the system, one would expect to get a definite outcome with a certain probability; in terms of measurement statistics, this is equivalent to the situation in which the system is in one of the states from the set of possible outcomes from the beginning, that is, before the measurement. As Pessoa (1998, p. 432) puts it, “taking a partial trace amounts to the statistical version of the projection postulate.”"

http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf (p37) "Ignorance interpretation: The mixed states we find by taking the partial trace over the environment can be interpreted as a proper mixture. Note that this is essentially a collapse postulate."
 
Last edited:
  • #32
atyy said:
There position and spin are entangled, then the mathematical description of dumping the other beams is the partial trace over all unwanted beams, leaving the reduced density matrix for the wanted beam?
The final state is a superposition. How would you single out a certain term by taking a partial trace?
 
  • #33
kith said:
The final state is a superposition. How would you single out a certain term by taking a partial trace?

It doesn't work the way I initially thought it might. I looked up Ballentine, chapter 9, page 244, and it looks like he includes the environment, has decoherence, traces over the environment and gets a reduced density matrix that has the same form as a proper mixture.
 
  • #34
atyy said:
It doesn't work the way I initially thought it might.
I got a little lost in the last few posts of this discussion. How do you respond to vanhees71's claim about the projection postulate being inequivalent to collapse (which is what he calls extraneous)?

I was with it here:
vanhees71 said:
Born's rule just says what are the probabilities for finding a certain value for an observable when this observable is measured on a system in a given (pure or mixed) state. It does not say that immediately after the measurement the system's state must instantaneously collapse into an eigenstate of this observable. This is most often not the case, because often the system gets destroyed being measured...<snip>... No esoterics like collapses or the like needed.

But then you go back to talking about whether the projection postulate needs to be postulated separately, or can be derived. I thought we're already past that, with vanhees71's granting that it is a separate postulate, but stating that we're debating the merits of collapse, not the postulate of projection (which brings us to the quote above)?
atyy said:
The reason I think the partial trace either involves the projection postulate, or must be introduced as a new postulate is that in the naive textbook formalism, the meaning of the partial trace is derived from the projection postulate.

I only pry because parts of this thread actually appear lucid, which is saying something considering the topic! o:)
 
  • #35
atyy said:
It doesn't work the way I initially thought it might. I looked up Ballentine, chapter 9, page 244, and it looks like he includes the environment, has decoherence, traces over the environment and gets a reduced density matrix that has the same form as a proper mixture.

He actually does it two ways - the partial trace (which he thinks of as the environment changing the system) and a second method where the environment and system are combined and how the apparatus affects that combination.

Of course each view must give the same answer, and he gives a reference where the issue is examined.

I suspect this also has something to say about the factorization problem some worry about concerning decoherence.

Its interesting Ballentine does this because he doesn't really believe decoherence has anything to do with interpretation. I of course respectfully disagree.

Thanks
Bill
 
  • #36
eloheim said:
I got a little lost in the last few posts of this discussion. How do you respond to vanhees71's claim about the projection postulate being inequivalent to collapse (which is what he calls extraneous)?

As usual the correct answer is found in Ballentine - which is just one reason why IMHO it is THE book on QM.

The projection postulate is actually a special type of observation where the system is not destroyed by the observation, and it's probably better viewed as a pure state preparation procedure. For that it is easily seen that the Born rule and physical continuity implies the projection postulate.

Collapse is totally extraneous in the Statistical interpretation, and is arguably extraneous in most versions of Copenhagen.

Thanks
Bill
 
Last edited:
  • Like
Likes 1 person
  • #37
@Eloheim, there are two versions of the Born Rule.

In the first version, stated by Weinberg, if we make a measurement of O, the system that is in state ψ will collapse into an eigenstate |oi> with eigenvalue oi, with probability |<oi|ψ>|2.

In the second version, if we make a measurement of O, the system that is in state ψ will give the result oi with probability |<oi|ψ>|2. So there is no collapse in the second version. If this version is used, most textbooks add collapse as a distinct postulate.

As I understand it, "collapse" and the "projection postulate" are the same, but vanhees71 is using the second version of the Born rule without collapse. Both versions of the Born rule cannot be derived from the Schroedinger equation.

I do accept that there are correct interpretations without collapse, such as Bohmian mechanics and probably also many-worlds. What I am skeptical about is whether the ensemble interpretation without collapse is correct, if one allows filtering type measurements as a means of state preparation.
 
Last edited:
  • Like
Likes 1 person
  • #38
bhobba said:
As usual the correct answer is found in Ballentine - which is just one reason why IMHO it is THE book on QM.

The projection postulate is actually a special type of observation where the system is not destroyed by the observation, and it's probably better viewed as a pure state preparation procedure. For that it is easily seen that the Born rule and physical continuity implies the projection postulate.

Collapse is totally extraneous in the Statistical interpretation, and is arguably extraneous in most versions Copenhagen.

Thanks
Bill

What is physical continuity in the Statistical Interpretation? Is it something like Bohmian trajectories?

BTW, I saw your post before this one too, thanks. I'll take a look at the other way Ballentine does it.
 
  • #39
atyy said:
What is physical continuity in the Statistical Interpretation?

Its simple.

Some measurements are more like filtering in that let's say it has n outcomes, then n different quantum states are also produced associated with each outcome. Let them have numbers 1 to n as the outcomes so the observable is ∑ i |bi><bi|. The Stern-Gerlach experiment is like that - with two outcomes. Suppose k is the outcome. Now let's do the same observation immediately after. Then, from physical continuity, you reasonably expect to get the same outcome, and whatever state is associated with the outcome to have changed by a negligible amount. Again Stern-Gerlach is like that. This assumption, a little math, and the Born rule, shows the state must be the eigenvector associated with the outcome of the observations observable ie |bk><bk| - which, drum-roll, wait for it, is the projection postulate.

Really the projection postulate only applies to 'filtering' type observations like the above. Observations are much more general than that.

Thanks
Bill
 
Last edited:
  • Like
Likes 1 person
  • #40
I don't think it makes sense to talk about "collapse", "state reduction" or the "projection postulate" in the ensemble interpretation because all of them were introduced to avoid macroscopic superpositions and these are not problematic in the ensemble interpretation.

What happens is that after the experiment, the observer redefines what he considers to be part of the system and what constitutes the environment. He doesn't have to do this, it is simply a practical matter.

But thinking further, we don't have something to talk about in the first place, if the observer isn't allowed to define what the system of interest is. You can call this an additional postulate but anytime you apply a scientific theory to a specific experimental situation you implicitly use this postulate. So it is nothing specific to the ensemble interpretation or even QM. In the MWI, you get the factorization problem if you don't use it.
 
  • #41
bhobba said:
Its simple.

Some measurements are more like filtering in that let's say it has n outcomes, then n different quantum states are also produced associated with each outcome. Let them have numbers 1 to n as the outcomes so the observable is ∑ i |bi><bi|. The Stern-Gerlach experiment is like that - with two outcomes. Suppose k is the outcome. Now let's do the same observation immediately after. Then, from physical continuity, you reasonably expect to get the same outcome, and whatever state is associated with the outcome to have changed by a negligible amount. Again Stern-Gerlach is like that. This assumption, a little math, and the Born rule, shows the state must be the eigenvector associated with the outcome of the observations observable ie |bk><bk| - which, drum-roll, wait for it, is the projection postulate.

Really the projection postulate only applies to 'filtering' type observations like the above. Observations are much more general than that.

Thanks
Bill

Yes, that works. But is it bhobba's Statistical Interpretation or Ballentine's? :smile: I don't think Ballentine introduced a "physical continuity" or "immediate repetition of a measurement yields the same result" assumption.
 
  • #42
All things are subject to interpretation whichever interpretation prevails at a given time is a function of power and not truth.
-Friedrich Nietzsche
 
  • #43
atyy said:
Yes, that works. But is it bhobba's Statistical Interpretation or Ballentine's? :smile: I don't think Ballentine introduced a "physical continuity" or "immediate repetition of a measurement yields the same result" assumption.

The continuity assumption is in Ballentine somewhere - although I can't recall exactly where.

Thanks
Bill
 
  • #44
bhobba said:
The continuity assumption is in Ballentine somewhere - although I can't recall exactly where.

Thanks
Bill

Well, if it is it's still hilarious, since in standard texts (like Dirac's) the projection postulate is linked to physical continuity (I believe that's Dirac's term). And if the projection postulate can be derived in Ballentine's framework because he did postulate physical continuity, then his rejection of the projection postulate is wrong, since non-unitary evolution can occur in his framework.

Yet another problem with Ballentine is his comment "The infinities of quantum field theory, of which we have here seen only the first, are somewhat of a hidden scandal. The infinities of quantum field theory, of which we have here seen only the first, are somewhat of a hidden scandal. Most physicists seem content to ignore them because there are procedures (the so-called renormalization theory) which allow us to avoid the infinities in many practical cases. One prominent physicist who was not complacent about the infinities was Dirac." For most physicists, this has been solved at the conceptual level by Wilson in the 1970s http://quantumfrontiers.com/2013/06/18/we-are-all-wilsonians-now/ .
 
Last edited:
  • #45
atyy said:
The infinities of quantum field theory, of which we have here seen only the first, are somewhat of a hidden scandal.

I just love his textbook - overall it's the finest QM book I have ever read and and a BIG effect on me.

But perfect he aren't - the above being a case in point.

It took me while to get it, but once you understand it renormalisation is no more of a scandal than the issues in bog standard EM with point particles which also lead to problems:
http://arxiv.org/pdf/gr-qc/9912045v1.pdf

The issue is simply that its a low energy approximation to a more complete theory so you must have a cutoff. What makes renormalizable theories so nice is you don't need to know the actual cutoff - you simply assume that some such, large cutoff, exists and you can calculate everything. Non renormalizable theories like gravity need an explicit cutoff beyond which the theory is not valid.

Renormalisation is explained VERY well here:
http://arxiv.org/pdf/hep-th/0212049.pdf

I had to go through it a few times, but once I fully understood it I realized renormalisation is not an issue at all.

Thanks
Bill
 
  • #46
Demystifier said:
If you want something free, then:
http://lanl.arxiv.org/abs/quant-ph/0209123

Looking at this, it looks like what Laloë calls the "correlation interpretation" is similar in spirit to what Ballentine was aiming for. In particular, Laloë's Eq 37 (taking into account footnote 41) is the same as Ballentine's Eq 9.30. Laloë says "Equation (37) can be seen as a consequence of the wave packet reduction postulate of quantum mechanics, since we obtained it in this way. But it is also possible to take it as a starting point, as a postulate in itself: it then provides the probability of any sequence of measurements, in a perfectly unambiguous way, without resorting, either to the wave packet reduction, or even to the Schrödinger equation itself. The latter is actually contained in the Heisenberg evolution of projection operators, but it remains true that a direct calculation of the evolution of |ψ> is not really necessary. As for the wave packet reduction, it is also contained in a way in the trace operation of (37), but even less explicitly. If one just uses formula (37), no conflict of postulates takes place, no discontinuous jump of any mathematical quantity; why not then give up entirely the other postulates and just use this single formula for all predictions of results?"

The difference between Laloë's and Ballentine's presentation leading up to the formula is that Laloë acknowledges the projection postulate in its derivation, and that if one gets rid of the projection postulate, then Eq 37 must be postulated directly. Ballentine seems less clear about what is derived, and what is postulated in the steps leading up to his Eq 9.30. Ballentine does say that something new is being defined in this section (on p248 he says "We can still use (9.22) as a definition of the joint probability Prob(A&B|C)"), but I suspect that Ballentine has in this chapter an unacknowledged use of the projection postulate in taking the decohered improper mixture to be a proper mixture on p244, as well as in writing down the density matrix of Eq 9.28 (or he has made use of bhobba's physical continuity, from which the projection postulate can be derived, but if so his rejection of the projection postulate is not justified). ? Comments please!
 
Last edited:
  • #47
atyy said:
but if so his rejection of the projection postulate is not justified).

He doesn't reject the projection postulate. He says it only applies to filtering type measurements and for those it follows from continuity.

Thanks
Bill
 
Last edited:
  • #48
bhobba said:
He doesn't reject the projection postulate. He says it only applies to filtering type measurements and for those it follows from continuity.

Thanks
Bill

I think Ballentine's book would make more sense if that were the case. Ballentine does mention Dirac's use of collapse with what seems to be lack of disapproval right after Ballentine's Eq 9.28, which is where I think Ballentine has used the projection postulate. Then presumably more general measurements (of POVMs) would lead to the generalized collapse rule.

In a philosophy which takes QM to be a limited theory (no wave function of the universe, and classical systems are needed for its application), as I think the Ballentine aims for, there is nothing wrong with collapse. If one really insists on unitary dynamics for the whole universe, I think one is led almost inexorably to many-worlds, if one remains within quantum theory. Bohmian mechanics is inconsistent with quantum theory if it is applied to the whole universe because then it is more natural to have non-equilibrium dynamics, and deviations from quantum theory.

As you have often pointed out, a reason that there may be nothing wrong with collapse is that it seems analogous to statistical updating. I'm not sure how complete the analogy is, but of several different recent constructions, one which seems pretty nice is Leifer and Spekkens's http://arxiv.org/abs/1107.5849 in which they explicitly say "A positive operator valued measure (POVM) is a CPT map from a quantum input to a classical output (the measurement outcome)", and propose that wave function collapse is analogous to a form of belief propagation.

2 cheers for collapse in the ensemble interpretation :-p
 
Last edited:
  • #49
bhobba said:
He doesn't reject the projection postulate. He says it only applies to filtering type measurements and for those it follows from continuity.
He doesn't reject it as a formal tool but he rejects it as a postulate. In the paragraph directly before (9.28) he says "If we consider the result of the subsequent S measurement on only that subensemble for which R ∈ ∆a and ignore the rest, we shall be determining the conditional probability (9.27)." (bolding mine)

So the reason why we use a projected density matrix is that we chose to ignore what happens to the subensemble which is blocked inside the measurement apparatus. We don't have to do this. We could drag it along as long as we want by applying the appropriate time evolution. It just doesn't tell us anything about the subensemble outside the apparatus which is what we use for further experiments. So why should we do this?
 
Last edited:
  • #50
kith said:
He doesn't reject it as a formal tool but he rejects it as a postulate. In the paragraph directly before (9.28) he says "If we consider the result of the subsequent S measurement on only that subensemble for which R ∈ ∆a and ignore the rest, we shall be determining the conditional probability (9.27)." (bolding mine)

So the reason why we use the projection postulate is that we chose to ignore what happens to the subensemble which is blocked inside the measurement apparatus. We don't have to do this. We could drag it along as long as we want by applying the appropriate time evolution. It just doesn't tell us anything about the subensemble outside the apparatus which is what we use for further experiments. So why should we do this?

After 9.27 Ballentine says "Alternatively, we can use (9.22) to define Prob(B|A&ρ) in terms of the other two factors, both of which are known."

Looking at 9.22, it is Prob(A&B|C) = Prob(A|C) Prob(B|A&C).

However, later he says "If the operators R and S do not commute, then (9.26) does not apply. We can still use (9.22) as a definition of the joint probability Prob(A&B|C)"

So yes, in deriving the expression for Prob(B|A&C) in 9.28 he does not need the projection postulate, if he uses 9.22. However, later he uses Prob(B|A&C) to define Prob(A&B|C) in 9.22. So in fact to truly derive 9.30, he must use the projection postulate to get 9.28. Of course, he could just postulate 9.30 directly. The whole discussion is also fine if all steps leading to 9.30 are just hand-wavy, but he doesn't really present it clearly that way.

Laloë's discussion is clearer, because he derives 9.30 explicitly using the projection postulate. Then he says, if we don't want to have a projection postulate, we can treat 9.30 as a postulate instead of considering it derived. (Laloë's Eq 37 is Ballentine's 9.30 - the free paper that Demystifier linked to in post #5. One has to take into account Laloë's footnote 41 to get Laloë's Eq 37 into the same form as Ballentine's 9.30.)
 
Last edited:
Back
Top