A. Neumaier

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For reference purposes and to help focus discussions on Physics Forums in interpretation questions on the real issues, there is a need for fixing the common ground. There is no consensus about the interpretation of quantum mechanics, and – not surprisingly – there is disagreement even among the mentors and science advisors here on Physics Forums. But the following formulation in terms of 7 basic rules of quantum mechanics was agreed upon among the science advisors of Physics Forums in a long and partially heated internal discussion on ”Best Practice to Handle Interpretations in Quantum Physics”, September 24 – October 29, 2017, based on a first draft by @atyy and several improved versions by  @tom.stoer. Other significant contributors to the discussions included @fresh_42, ...
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A. Neumaier

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The article is based on a first draft by @atyy and several improved versions by @tom.stoer. Other significant contributors to the discussions included @fresh_42, @kith, @stevendaryl, and @vanhees71.
I slightly expanded the final version and added headings and links to make it suitable as an insight article. Maybe the participants of the discussion 20 months ago can confirm their continued support or voice disagreements with this public version.
 

Peter Morgan

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I commented on Facebook (in the Facebook group "Quantum Mechanics & Theoretical Physics") as follows:
I invite you to consider the following post, from yesterday, as an alternative to Rule 7: https://www.facebook.com/peter.w.morgan.98/posts/10220443595866417.
One effectively insists that measurements made jointly must be represented by mutually commutative operators:
"4a: Joint observables of a quantum system are represented by mutually commutative self-adjoint operators A, B, ... acting on H."
"Collapse" of the wave function can be shown just to enforce this Rule 4a.
Rule 7 seems to me to be the most contentious of those listed, though I like the "Note that there is no conflict with the unitary evolution in (3) since during a measurement, a system is never isolated".
Obviously this is rather nonstandard, although I believe this is very much more simply put than but very close to the approach of Belavkin [Found.Phys. 24, 685(1994)] that I cite in my https://arxiv.org/abs/1901.00526v3.
 

A. Neumaier

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"4a: Joint observables of a quantum system are represented by mutually commutative self-adjoint operators A, B, ... acting on H."
Although position and momentum do not commute, there are joint position and momentum measurements (e.g., from tracks in bubble chambers or wire chambers), though their accuracy is limited by Heisenberg's uncertainty relation.
Rule 7 seems to me to be the most contentious of those listed
The rules are precise formulations of corresponding statements found more loosely formulated in all textbooks cited (apart from Ballentine). Rule (7) appears there usually in an unqualified (and hence incorrect) form to which your criticism may apply. But I don't understand what you consider contentious in the actual formulation of (7). It surely applies in the cases listed under ''Formal discussion'' of (7):
It is needed to know what is prepared after passing a barrier (e.g., singling out a ray) or polarizer (singling out a polarization state).
 

Peter Morgan

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I agree that the final qualification in the discussion (which I highlighted as very worthwhile, as it seems to me) goes a considerable way towards making Rule (7) uncontentious. If we're talking about what's in the textbooks (which we are), I think everything's just fine, and perhaps I jumped in too fast. I'm happy to jump out again, and I probably should.

On the other hand, one doesn't have to think of a measurement as preparing a new state, by the infamous "collapse" mechanism: one can instead think of a measurement as conditioning what other measurements can be made jointly with it. The equation in my Facebook post is very straightforward: if ##\hat A## has a discrete spectral projection ##\hat A=\sum_i\alpha_i\hat P_i##, then a Lüders operation is given by ##\hat\rho\mapsto\hat\rho_A=\sum_i\hat P_i\hat\rho\hat P_i##, so that, for ##[\hat A,\hat X]\not=0##, $$\mathsf{Tr}[\hat A\hat X\hat\rho_A]=\mathsf{Tr}[\hat A\hat X_A\hat\rho],$$with the Lüders operation enforcing that ##[\hat A,\hat X_A]=0##: even though ##\hat A## and ##\hat X## are not jointly measurable in the state ##\hat\rho##, we can say
  • either that ##\hat A## and ##\hat X## are jointly measurable in the state ##\hat\rho_A##,
  • or that ##\hat A## and ##\hat X_A## are jointly measurable in the state ##\hat\rho##.
In this way of thinking, the state is never changed by measurement, so if we know what the state is before a measurement, we equally know what it is after the measurement. As I say above, this is just what Belavkin suggested, but, I think, put in much simpler terms: everything is just a consequence of the one displayed equation above.
But again, probably best left for the future.
 

A. Neumaier

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one can instead think of a measurement as conditioning what other measurements can be made jointly with it.
The conditioning is always of what is measured afterwards - simultaneous measurement is different and unrelated to the collapse.
a Lüders operation is given by
But this is different from collapse, which says that given the result you can simply work with the projected state - which is what is done in practice. Without projection one must always carry the complete context around (a full ancilla in an extended Hilbert space), which is awkward when making a long sequence of observations.
 

Peter Morgan

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The conditioning is always of what is measured afterwards - simultaneous measurement is different and unrelated to the collapse.
Absolutely. The point of Quantum Non-Demolition (QND) measurements is that one constructs operators that mutually commute even though they represent measurements that are time-like separated. Indeed, it's very easy to check that the Lüders operation ensures that ##[\hat A,\hat X_A]=0## even if ##\hat A## and ##\hat X## represent measurements at time-like separation.

But this is different from collapse, which says that given the result you can simply work with the projected state - which is what is done in practice. Without projection one must always carry the complete context around (a full ancilla in an extended Hilbert space), which is awkward when making a long sequence of observations.
The Lüders operation is what is used to represent "reduction" of the density matrix by measurement in Section II.3.2 and II.3.3 of Busch, Grabowski, and Lahti (which you mention above). On its own, we can think of it as representing a measurement of ##\hat A## without recording the result.
 

A. Neumaier

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The Lüders operation is what is used to represent "reduction" of the density matrix by measurement in Section II.3.2 and II.3.3 of Busch, Grabowski, and Lahti (which you mention above). On its own, we can think of it as representing a measurement of ##\hat A## without recording the result.
Busch et al. nowhere refer to reduction. Your use of the term is nonstandard. What is termed state reduction is a process that turns pure states into pure states. It corresponds to the Lüders operator ##\rho\to P\rho P## discussed at the end of Section II.3.1 and in II.4.
 

Peter Morgan

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Busch et al. nowhere refer to reduction. Your use of the term is nonstandard. What is termed state reduction is a process that turns pure states into pure states. It corresponds to the Lüders operator ##\rho\to P\rho P## discussed at the end of Section II.3.1 and in II.4.
Thanks! ##\hat\rho\mapsto \hat P\hat\rho\hat P## is a Lüders operation (as you say, Section II.3.1), ##\hat\rho\mapsto \sum_i\hat P_i\hat\rho\hat P_i##, is a Lüders transformer (Section II.3.2). I'll have to fix that.
Aiish, important though it is to get names right, even crucial, but names! The Lüders transformer that uses a complete orthogonal set of projection operators to eigenspaces of ##\hat A## transforms ##\hat\rho## to a convex sum of eigenstates of ##\hat A##. The equation$$\mathsf{Tr}[\hat A\hat X\hat\rho_A]=\mathsf{Tr}[\hat A\hat X_A\hat\rho]$$survives this discussion, however, right? What's in question is perhaps how one can use the Lüders transformer applied to a measurement operator ##\hat X\mapsto\hat X_A##, and the equation above, as part of an interpretation of QM. I think I find this helpful, but if someone doesn't want to use it, that's OK.
A Lüders operation ##\hat\rho\mapsto \hat P\hat\rho\hat P## in general only projects pure states to pure states if ##\hat P## projects to a 1-dimensional subspace, right?
 

A. Neumaier

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A Lüders operation ##\hat\rho\mapsto \hat P\hat\rho\hat P## in general only projects pure states to pure states if ##\hat P## projects to a 1-dimensional subspace, right?
It maps an arbitrary pure state with state vector ##\psi## into a pure state with state vector ##\hat P\psi##. That's enough in the present context.

Only if you want to prepare a pure state from an arbitrary mixed state then ##\hat P## must project to a 1-dimensional subspace.
 
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Thank you for the nice article. I would like to ask your opinion on a slightly different aspect of this which is the pedagogical significance of this scheme.

I personally believe that this is the correct way of teaching a theory to the new learners. (That's why I liked the article.) Actually not only quantum mechanics but any physical theory should be thought by giving the postuates in the first encounter and repeating and quoting them all the way throughout the course. Or stating it differenty we have to first teach what a theory is before the theory itself.

(A few years ago I wrote an article on this and published in a Turkish journal easily, but couldn't have the chance to discuss it throughly with any colleagues, so trying my chance here also to get some feedback and opinions.)

Different texts on quantum mechanics have many different approaches like explaining the historical development first, or developing the mathematical framework initally. I even find the method of some great masters like Feynman and Sakurai suspicious when they try to develop the concepts via some thought experiments like double slit or Stern-Gerlach. I have witnessed that the students always seem to get stuck on the "technical" details of the experiment which are irrelevant to the core and that diverges their already fragile attention.

And that's why I believe that a sound scheme of postulates should be emphasized as THE fundemantal thing that matters most. This year I tried this approach on my modern physics course and after exposing them to the postulates I continued by the historical development and I felt that the students were more engaged actually. (It was much easier with special relativity since the postulates can be expressed in daily language; and of course much more challenging with quantum mechanics because of the mathematical language. But I refered to their linear algebra course all the time and said it is nothing but linear algebra, eigenvalues, eigenfunctions, etc...)

So I really would like to hear your opinions (both professors and students) about the pedagocical aspect.
(Sorry if I'm diverting the topic but that is an important part of it I believe...)
 
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A. Neumaier

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So I really would like to hear your opinions (both professors and students) about the pedagocical aspect.
The postulates don't say much without the examples. Thus one has to introduce both in parallel, starting with things that make for an easy bridge, such as optical polarization - see my insight article on the qubit.
 

zonde

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A. Neumaier said:
The only exception I know of is Ballentine 1998, who explicitly rejects rule (7) = his process (9.9).
Bellentine does not reject rule (7) as formulated in your insight article. He just makes distinction between wave function collapse at the moment of detection (which he rejects) and projective measurement (which he calls a filtering-type measurement, see p.246 in his 1998 book).
 

A. Neumaier

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Ballentine does not reject rule (7) as formulated in your insight article. He just makes distinction between wave function collapse at the moment of detection (which he rejects) and projective measurement (which he calls a filtering-type measurement, see p.246 in his 1998 book).
What I say about Ballentine in the Insight article (in the slightly polished formulation of this morning - collapse rejected as fundamental but accepted as effective) was designed to be compatible with what he says in his book. It seems to me also compatible with a suitable interpretation of what you say in this quote.

On p.236-238, Ballentine gives a long argument for his rejection of the conventional formulation of (7) = his (9.9) in the density operator version:
Leslie Ballentine said:
In order to save that interpretation, they postulate a further process that is supposed to lead from the state (9.8) to a so-called “reduced state” (9.9), which is an eigenvector of the indicator variable, with the eigenvalue being the actual observed value of the indicator position. This postulate of reduction of the state vector creates a new problem [...] In all cases in which the initial state is not an eigenstate of the dynamical variable being measured, the final state must involve coherent superpositions of macroscopically distinct indicator eigenvectors. If this situation is unacceptable according to any interpretation, such as A, then that interpretation is untenable.
He accepts it only as an effective view (p.243f)
Leslie Ballentine said:
Thus we see that the so-called “reduced” state is physically significant in certain circumstances. But it is only a phenomenological description of an effect on the system (the neutron and spectrometer) due to its environment (the cause of the noise fluctuations), which has for convenience been left outside of the definition of the system. This “reduction” of the state is not a new fundamental process, and, contrary to the impression given in some of the older literature, it has nothing specifically to do with measurement.
and (rightly, like Landau and Lifshits, but unlike many other textbooks) only under special circumstances (p.247):
Leslie Ballentine said:
This filtering process, which has the effect of removing all values of R except those for which R ∈ Δa, can be regarded as preparing a new state [...] Indeed, the statement by Dirac (1958, p. 36) to the effect that the state immediately after an R measurement must be an eigenstate of R, seems perverse unless its application is restricted to filtering-type measurements.
This is why (7) is formulated in the cautious way given in the Insight article.
 
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zonde

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He accepts it only as an effective view (p.243f)
I don't see it that in p.243 he describes "state reduction" as effective view. He describes how loss of coherence can appear due to experimental imperfections (he contrasts real good experiment with fictional poor experiment). But experimental imperfection have noting to do with measurement. Theory considers idealized experiments, experimentalists report results of good quality experiments and measurements are still present in these idealized theoretical and good quality real experiments. He says:
"This “reduction” of the state is not a new fundamental process, and, contrary to the impression given in some of the older literature, it has nothing specifically to do with measurement."


and (rightly, like Landau and Lifshits, but unlike many other textbooks) only under special circumstances (p.247):
I do not see that in p.246-248 he describes effective view. Yes, projection can be applied only under special circumstances but this is the whole purpose of doing controlled experiments. You isolate some phenomena with the help of controlled environment and make it clearly observable which in other more complicated setups would be hard to pin down and it will depend on correct understanding of other factors.

This is why (7) is formulated in the cautious way given in the Insight article.
Yes, I like formulation of (7) in your Insight article and as I see it is consistent with Ballentine's filtering-type measurement as a fundamental phenomena which needs special circumstances to be clearly observed.
 

atyy

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Thank you for the nice article. I would like to ask your opinion on a slightly different aspect of this which is the pedagogical significance of this scheme.

I personally believe that this is the correct way of teaching a theory to the new learners. (That's why I liked the article.) Actually not only quantum mechanics but any physical theory should be thought by giving the postuates in the first encounter and repeating and quoting them all the way throughout the course. Or stating it differenty we have to first teach what a theory is before the theory itself.

(A few years ago I wrote an article on this and published in a Turkish journal easily, but couldn't have the chance to discuss it throughly with any colleagues, so trying my chance here also to get some feedback and opinions.)

Different texts on quantum mechanics have many different approaches like explaining the historical development first, or developing the mathematical framework initally. I even find the method of some great masters like Feynman and Sakurai suspicious when they try to develop the concepts via some thought experiments like double slit or Stern-Gerlach. I have witnessed that the students always seem to get stuck on the "technical" details of the experiment which are irrelevant to the core and that diverges their already fragile attention.

And that's why I believe that a sound scheme of postulates should be emphasized as THE fundemantal thing that matters most. This year I tried this approach on my modern physics course and after exposing them to the postulates I continued by the historical development and I felt that the students were more engaged actually. (It was much easier with special relativity since the postulates can be expressed in daily language; and of course much more challenging with quantum mechanics because of the mathematical language. But I refered to their linear algebra course all the time and said it is nothing but linear algebra, eigenvalues, eigenfunctions, etc...)

So I really would like to hear your opinions (both professors and students) about the pedagocical aspect.
(Sorry if I'm diverting the topic but that is an important part of it I believe...)
I enjoyed something like that too from my teacher. When I first learned quantum mechanics (Xiao-Gang Wen was the lecturer), the postulates were taught very early, but not in the first lesson. If I recall correctly, the first lecture was about dimensional analysis - to introduce Planck's constant, the lecture 2 was a tour of the ultraviolet problem and old quantum physics, and the postulates were introduced in lecture 3. Then after that wave mechanics was always done in the context of the postulates.

I believe @vanhees71 has advocated something like that in these forums, though I should let him speak for himself.
 

A. Neumaier

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He accepts it only as an effective view (p.243f)
He says: "This “reduction” of the state is not a new fundamental process,
Not fundamental means only effective.
Leslie Ballentine said:
the statement by Dirac (1958, p. 36) to the effect that the state immediately after an R measurement must be an eigenstate of R, seems perverse unless its application is restricted to filtering-type measurements.
projection can be applied only under special circumstances but this is the whole purpose of doing controlled experiments.
Ballentine doesn't restrict to arbitrary controlled experiments but to the much smaller class of ''filtering-type measurements'' by selection, where collapse is equivalent to taking conditional expectations.
I like formulation of (7) in your Insight article and as I see it is consistent with Ballentine's filtering-type measurement as a fundamental phenomena which needs special circumstances to be clearly observed.
whereas Ballentine said explicitly that it is not a fundamental process.
 

zonde

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Not fundamental means only effective.
Yes, and he said that about loss of coherence in poorly performed experiment.

Ballentine doesn't restrict to arbitrary controlled experiments but to the much smaller class of ''filtering-type measurements'' by selection, where collapse is equivalent to taking conditional expectations.
"Arbitrary controlled experiment" is oxymoron. Either "arbitrary experiment" or "controlled experiment".

You can construct more complicated experiments from simpler experiments. Say you use two state preparations that can be described by projective "filtering-type measurements" to produce two beams then you mix the beams together and if you are careful with your setup you can observe interference - this demonstrates that projective "filtering-type measurements" have nothing to do with loss of coherence.

whereas Ballentine said explicitly that it is not a fundamental process.
Ballentine said explicitly that loss of coherence it is not a fundamental process. He was not talking about projective measurement when he said that. You took something he said about one topic and claim that he said that about completely different topic.
Well, for you they maybe are the same topic but not for Ballentine and not for me.
 
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Could there be a different formulation of the axioms of QM if one took the path integral as its basis?

The Schrödinger equation can be derived from the path integral. As a consequence, giving the Schrödinger equation as fundamental suggests the path integral is derived from the Schrödinger equation. But in fact, it is basically the opposite.
 

A. Neumaier

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The Schrödinger equation can be derived from the path integral. As a consequence, giving the Schrödinger equation as fundamental suggests the path integral is derived from the Schrödinger equation.
Yes. The QFT path integral is derived from the QM path integral, which is derived from the Schrödinger equation. Without the latter, one would never know that the path integral formulation is a valid formulation of QM/QFT.
But in fact, it is basically the opposite. Could there be a different formulation of the axioms of QM if one took the path integral as its basis?
No. With the path integral formulation (but without the equivalent traditional formulation), you don't even have a Hilbert space (unless you work in the closed time path setting, which is not common knowledge).

But if you consistently and exclusively do QM in the Heisenberg picture, it looks just like QFT, just with a 1D space-time in place of 4D.
 
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Yes. The QFT path integral is derived from the QM path integral, which is derived from the Schrödinger equation. Without the latter, one would never know that the path integral formulation is a valid formulation of QM/QFT.

No. With the path integral formulation (but without the equivalent traditional formulation), you don't even have a Hilbert space (unless you work in the closed time path setting, which is not common knowledge).

But if you consistently and exclusively do QM in the Heisenberg picture, it looks just like QFT, just with a 1D space-time in place of 4D.
Yes, I understand that with the path integral, there is no Hilbert space. Of course, the first axiom of the Dirac-Von Neumann axioms should be respected.

But if we postulate that the states of that Hilbert space are irreducible representations of the Galilean group (or the Poincaré in the relativistic case), we would have at least the asymptotic states, without needing operators.

I do not know if one could dispense completely of Hilbert space operators though.
 

A. Neumaier

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But if we postulate that the states of that Hilbert space are irreducible representations of the Galilean group (or the Poincaré in the relativistic case).
How do you ensure that in the context of a path integral?
we would have at least the asymptotic states, without needing operators.
Under your assumptions you'd just have a single free particle. Nothing asymptotic here.

Once you have a Hilbert space and a (not necessarily irreducible) unitary representation of such a group, its infinitesimal generators are represented by operators. This gives operators for energy, momentum, angular momentum, and boosts (of the total system).

It is better to study the subject in some more depth than to dabble in unfounded speculations. It takes some time to become familiar with all the relevant relations between the various approaches and to see what which approach offers and misses.
 
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How do you ensure that in the context of a path integral?

Under your assumptions you'd just have a single free particle. Nothing asymptotic here.

Once you have a Hilbert space and a (not necessarily irreducible) unitary representation of such a group, its infinitesimal generators are represented by operators. This gives operators for energy, momentum, angular momentum, and boosts (of the total system).

It is better to study the subject in some more depth than to dabble in unfounded speculations. It takes some time to become familiar with all the relevant relations between the various approaches and to see what which approach offers and misses.
In Chapter 13, and especially 14, Zeidler's Quantum Field Theory vol. 1, it is argued that the Response approach basically dispenses of Hilbert spaces and operators.

Zeidler bases everything in the QA "magic formula" (basically, the definition of the path integral) and the LSZ "magic formula" (which translates correlation functions into the S matrix).

I do not know if his formulation is completely general for all QFT, but it seems clear from this text that he can go directly to empirical numbers, without having to pass through Hilbert spaces or operators.

Edit: Zeidler writes, at the beginning of Chapter 15:

"In Chap. 14, we have described the approach to quantum field theory which
can be traced back to Feynman’s approach in the 1940s based on the Feynman
rules for Feynman diagrams and the representation of propagators by functional
integrals. Typically, this approach does not use operators in Hilbert
spaces, that is, the methods of functional analysis do not play any role
."
 

atyy

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I cannot edit the message above, but the following should read as Edit 2:

It seems that it is possible, in some way, to recover a Hilbert space from the path integral formulation (and in non-relativistic QM, this Hilbert space is the standard Hilbert space):

The usual Hilbert space formulation is primary, and the path integral formulation is secondary. The path integral formulation allows us o do quantum mechanics in the language of statistical mechanics. Not all statistical mechanics path integrals correspond to quantum theories (ie. they make lack unitary evolution etc). The constraints on the path integrals that make them correspond to quantum theories come from the Hilbert space formulation, which is why the Hilbert space formulation is primary.

In the context of relativistic quantum field theory, a set of constraints on path integrals are the Osterwalder-Schrader axioms.
http://www.einstein-online.info/spotlights/path_integrals.html
https://ncatlab.org/nlab/show/Osterwalder-Schrader+theorem
 

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