Exploring the Connection Between Quantum Mechanics and Quantum Field Theory

[vanhees71]

You took my statement out of context. Here I was arguing not about QFT but about the modern foundation of quantum mechanics described in post #128. It is a much more powerful formulation of the Copenhagen interpretation than the usual ones. (Though to save time I didn't make the density matrix version explicit, and that I assumed as Paris a finite-dimensional Hilbert space. For a completely specified set of postulates appropriate for modern quantum mechanics (fully compatible but in detail differing from post #128) see my Postulates for the formal core of quantum mechanics from my theoretical physics FAQ. If you want to discuss these, please do so in a separate thread.)

The postulates 1-6 in your FAQ are indeed standard QT, and I subscribe to them. I've only one question: Why is it sufficient to define the stat. op. as hermitean? I always thought it must be (essentially) self-adjoint.

I also don't think that you have to change anything concerning relativistic QFT. Of course, observables must be defined by (gauge-independent) observables in terms of the quantum fields, and indeed, a particle interpretation is valid only in a very specific limited sense of asymptotic free Fock states.

 

[A. Neumaier]

To me, the comparison with magicians seems more like this:

We see a magician saw a lady in half and then put her back together, unharmed. There are three different reactions possible:

1. Some people say: Wow, that guy really has magical powers.
   
2. Some people (such as "The Amazing Randi") say: There is some trick involved---I want to figure out what it is.
3. Other people say: Why are we focusing on such an extreme, unnatural case? In the vast majority of actual cases, when someone is sawed in half, they don't recover. Let's just worry about these typical cases.

Indeed, the practical implications of quantum mechanics may qualify as magical powers for the uninformed and for people from before 1950, say. People from that time would have viewed as science fiction what is now reality due to transistors, lasers, memory chips and other products of quantum engineering.

But you forgot the Magician's reaction, who says ''I have a sufficient understanding of Nature that I can prepare situations where Nature works predictably in ways that are regarded as magical by the less informed. Since I understand it, for me no magic is involved.''

This is the rational response. We don't need a metaphysical understanding how Nature achieves nonlocality. It is enough to understand Nature enough to utilize its powers. And we do.

 

[stevendaryl]

Indeed, the practical implications of quantum mechanics may qualify as magical powers for the uninformed and for people from before 1950, say. People from that time would have viewed as science fiction what is now reality due to transistors, lasers, memory chips and other products of quantum engineering.

But you forgot the Magician's reaction, who says ''I have a sufficient understanding of Nature that I can prepare situations where Nature works predictably in ways that are regarded as magical by the less informed. Since I understand it, for me no magic is involved.''

It's hard to distinguish between that case and case 3. Especially when the magician, when asked about the sawing the lady in half, changes the topic to those cases where saws work as expected.

 

[A. Neumaier]

It's hard to distinguish between that case and case 3. Especially when the magician, when asked about the sawing the lady in half, changes the topic to those cases where saws work as expected.

Well, my response is for magicians among themselves and their disciples. On stage the magician has to change topic, or he would soon lose his reputation...

 

[Demystifier]

His experiments over the years do not show any hint of "weirdness"

Of course they don't. No experiments do. Weirdness can only be in the interpretation of an experiment, not in the experiment itself.

If I see a flying elephant, there is nothing weird about that. But if I combine it with my expectations that elephants should not fly, then it becomes weird.

 

[atyy]

Everyone seems to make the assumptions that the various forms are equivalent, but few seem prepared to prove it...

It is self-evidently right for position. $P(x) = |\langle x| \psi \rangle|^{2} = |\psi(x)|^{2}$, then the expectation value of x is $\langle x \rangle = \int x P(x) dx$. As long as specifying all moments and cumulants is the same as specifying the probability distribution, then the two forms of the Born rule are equivalent.

The paper by Paris that you cite states on p.2.,

This is an extremely special case of QM, far too special for anything that could claim to be a foundation for all of quantum mechanics. It can serve as a motivation and introduction, but not as a foundation. (And the author doesn't claim to give one.)

If $X$ is a Hermitian operator with a discrete spectrum (which Paris assumes on p.2) then the calculation in Postulate 2 on p.3 is valid and gives a valid derivation of the meaning of the expectation two lines after (1) from the Born rule one line before (1). If the spectrum contains a continuous part, Born's rule as stated in the line before (1) is invalid, as the probability of measuring $x$ inside the continuous spectrum is exactly zero, although a measurement result is always obtained. Instead,
the squared absolute amplitude should give the probability density at $x$. Wikipedia's Born rule has a technical annex for the case a general spectrum that is formally correct but sounds a bit strange for fundamental postulates (that should be reasonably intuitive). But it is not formulated generally enough since the deduction from it,

(which is essentially Born's original interpretation from 1926 - he didn't consider observables other than position coordinates)
doesn't follow but needs even more machinery from functional analysis about the existence of the joint spectrum for a set of commuting self-adjoint operators. it is very strange that the foundations of quantum mechanics should depend on deep results in functional analysis...

Paris goes on to say on p.3,

This is incorrect since according to every interpretation of quantum mechanics, the dynamics of an isolated system is always unitary and it cannot be observed, since observation is possible only when the system interacts with a detector.

The correct formulation (in the finite-dimensional case discussed by Paris) should be:

Postulate 2a. As long as a system is isolated the dynamics of its state is given by the Schroedinger equation. During the interaction with an instrument the state changes in such a way that (in the interaction picture) the state of a system in a pure state $\psi$ before the entering the instrument changes upon leaving the instrument with probability $|P_x\psi|^2$ to a pure state proportional to $P_x\psi$, where $\sum_x P_x^*P_x=1$ (i.e., the $P_x^*P_x$ form a POVM). The $P_x$ are characteristic for the instrument, and can (in principle) be predicted from a quantum treatment of the instrument.

This is the observer-free formulation. It can be complemented by the following assertion involving observation:

Postulate 2b. If the final state is proportional to $P_x\psi$, one can (in principle) deduce the value of $x$ from observations of the instrument and its surrounding. But the change of state happens whether or not the instrument is observed.

There is a corresponding version for mixed states that involves density matrices (which also needs a replacement of Postulate 1 of Paris). The resulting set of postulates is a much better set of postulates for (finite-dimensional) quantum mechanics. In particular, after an appropriate extension to POVMs with infinitely many components, they (unlike the Born rule) fairly faithfully reflect most of what is done in modern QM.

A mutilated, unnecessarily rigid and subjective form of the postulates in the density matrix version was stated by Paris on p.9. Note that in this process he completely changed the postulates! Postulate 1 (pure states) was silently dropped on p.4 where he remarks that ''different ensembles leading to the same density operator are actually the same state, i.e. the density operator provides the natural and most fundamental quantum description of physical systems''. (How can something be more fundamental than the very foundations one starts with? How can obviously different ensembles, if they mean anything physically, ''actually be the same state''? Only by changing the notion of a state.) Postulate 3 (unitarity) is dropped on the same page by observing that ''the action of measuring nothing should be described by the identity operator'', while according to Postulate 3 it should be described by the Hamiltonian dynamics. (He is assuming an interaction picture, without mentioning it anywhere!) Finally, Postulate 2 (the definition of an observable and the Born amplitude squaring rule) is replaced by a new definition of observables in II.1 and a generalized Born rule II.3 that was invented only much later (probably around the time Born died). In II.5 he adds a rule that is in direct conflict with II.3 since a measurement performed in which we find a particular outcome cannot lead to two different states depending on whether or not we record the result.

Thus Paris documents in some detail that modern quantum mechanics is, fundamentally, neither based on state vectors nor on observables being Hermitian operators nor on instantaneous collapse nor on Born's rule for the probability of finding results. Instead, it is based on states described by density matrices, observables described by POVMs, interactions in finite time described by multiplication with a POVM component, and a generalized Born rule for the selection of this component. This generalized setting is necessary and sufficient to describe modern quantum optics experiment at a level where efficiency issues and measuring imperfections can be taken into account.

Apart from the Hilbert space, nothing is kept from the textbook foundations, except that the latter serve as a simplified (but partially misleading) introduction to the whole subject.

Hmmm, probably you are taking the "textbook" formulation too literally, as if the state assigned to the system is really the state assigned to the system. If you take the Copenhagen pure state assigned to a single system as just a convenient fiction, and that the only thing that is real in quantum mechanics is the probability distribution of outcomes, the differences between the final and initial versions are technical, not spiritual.

Also, Paris does have wave function collapse. It's the rule he calls state reduction.

 

[A. Neumaier]

It is self-evidently right for position. $P(x) = |\langle x| \psi \rangle|^{2} = |\psi(x)|^{2}$, then the expectation value of x is $\langle x \rangle = \int x P(x) dx$. As long as specifying all moments and cumulants is the same as specifying the probability distribution, then the two forms of the Born rule are equivalent.

Well, as I said before, this argument can be generalized only to observables that are functions of positions alone. But how do you conclude something about the distribution of $H=p^2/2m+V(x)$?

Hmmm, probably you are taking the "textbook" formulation too literally,

If one cannot take the foundational postulates literally, so that they need another interpretation, what good are they?

as if the state assigned to the system is really the state assigned to the system.

Do you want to suggest that one has to add to the interpretation the additional postulate that the state assigned to the system is not the state assigned to the system. It seems to me that this would solve the problem in a too trivial way since from X equals not X one can deduce arbitrary sense and nonsense.

If you take the Copenhagen pure state assigned to a single system as just a convenient fiction, and that the only thing that is real in quantum mechanics is the probability distribution of outcomes, the differences between the final and initial versions are technical, not spiritual.

This changes the Copenhagen interpretation into the ensemble interpretation.

Also, Paris does have wave function collapse. It's the rule he calls state reduction.

Yes, and so has my version in post #128. This is why I call it a more powerful form of the CI.

 

[atyy]

Well, as I said before, this argument can be generalized only to observables that are functions of positions alone. But how do you conclude something about the distribution of $H=p^2/2m+V(x)##?

Let me keep this as homework that I owe you :P

This changes the Copenhagen interpretation into the ensemble interpretation.

There is no difference between the Copenhagen and the Ensemble interpretation, when both are done correctly. A key point is that Copenhagen = Ensemble has the measurement problem.

 

[vanhees71]

Yes, as stressed several times, roughly speaking the Kopenhagen flavor by Bohr can be described as "minimal interpretation + collaps and quantum-classical cut". The physics part is identical, and thus I prefer the minimal interpretation, because both collapse and the quantum-classical cut are very problematic (to say it mildly ;-)).

 

[vanhees71]

Well, as I said before, this argument can be generalized only to observables that are functions of positions alone. But how do you conclude something about the distribution of $H=p^2/2m+V(x)$?

Of course, we all know, how it is defined,
$$\langle E \rangle=\langle \psi|\hat{H} \langle \psi=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \psi^*(x) \left [-\frac{\Delta}{2m}+V(x) \right ] \psi(x).$$
This is unambiguously defined in the mathematical foundations and has nothing to do with any "interpretation weirdness".

 

[A. Neumaier]

Of course, we all know, how it is defined,
$$\langle E \rangle=\langle \psi|\hat{H} \langle \psi=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \psi^*(x) \left [-\frac{\Delta}{2m}+V(x) \right ] \psi(x).$$
This is unambiguously defined in the mathematical foundations and has nothing to do with any "interpretation weirdness".

The question was not how it is defined (which is part of the QM calculus) but why Born's rule which just says that $|\psi(x)|^2$ is the probability density of $x$ implies that the right hand side is the expected measurement value of $H$. It is used everywhere but is derived nowhere, it seems to me.

 

[vanhees71]

That's the strength of Dirac's formulation compared to the wave-mechanics approach. A pure state is represented by a normalized state vector $|\psi \rangle$ (more precisely the ray, but that's irrelevant for this debate). Then $\hat{H}$ has a complete set of (generalized) eigenvectors $|E \rangle$ (let's for simplicity also forget about the common case that the Hamiltonian is non-degenerate). Then the probability that a system prepared in this state has energy $E$ is according to the Born rule given by
$$P(E)=|\langle E|\psi \rangle|^2,$$
and thus
$$\langle E \rangle = \sum_E P(E) E=\sum_E \langle \psi|E \rangle \langle E|\hat{H} \psi \rangle = \langle \psi|\hat{H} \psi \rangle.$$
The latter expression can now be written in any other representation you like. In the position representation you have, e.g.,
$$\langle E \rangle = \int \mathrm{d}^3 \vec{x}_1 \int \mathrm{d}^3 \vec{x}_2 \langle \psi |\vec{x}_1 \rangle \langle \vec{x}_1|\hat{H}|\vec{x}_2 \rangle \langle \vec{x}_2 |\psi \rangle=\int \mathrm{d}^3 \vec{x}_2 \int \mathrm{d}^3 \vec{x}_2 \psi^*(\vec{x}_1) H(\vec{x}_1,\vec{x}_2) \psi(\vec{x}_2).$$
Now you only have to calculate the matrix element. For the potential it's very simple:
$$V(\vec{x}_1,\vec{x}_2)=\langle \vec{x}_1|V(\hat{x})|\vec{x}_2 \rangle=V(x_2) \delta^{(3)}(\vec{x}_1-\vec{x}_2).$$
For the kinetic part, it's a bit more complicated, but also derivable from the Heisenberg algebra of position and momentum operators.

The first step is to prove
$$\langle \vec{x}|\vec{p} \rangle=\frac{1}{(2 \pi)^{3/2}} \exp(\mathrm{i} \vec{x} \cdot \vec{p}).$$
For simplicity I do this only for the 1-component of position and momentum. That the simultaneous generalized eigenvector of all three momentum components factorizes is clear.

Since $\hat{p}$ is the generator of spatial translations, it's intuitive to look at the operator
$$\hat{X}(\xi)=\exp(\mathrm{i} \xi \hat{p}) \hat{x} \exp(-\mathrm{i} \xi \hat{p}).$$
Taking the derivative wrt. $\xi$ it follows
$$\frac{\mathrm{d}}{\mathrm{d} \xi} \hat{X}(\xi)=-\mathrm{i} \exp(\mathrm{i} \xi \hat{p}) [\hat{x},\hat{p}] \exp(-\mathrm{i} \xi \hat{p}).$$
From the Heisenberg commutation relations this gives
$$\frac{\mathrm{d}}{\mathrm{d} \xi} \hat{X}=1 \; \Rightarrow \; \hat{X}=\hat{x}+\xi \hat{1}.$$
So we have
$$\hat{x} \exp(-\mathrm{i} \xi \hat{p}) |x=0 \rangle=\exp(-\mathrm{i} \xi \hat{p}) \hat{X}(\xi) |x=0 \rangle=\xi \exp \exp(-\mathrm{i} \xi \hat{p}) |x=0 \rangle.$$
Then you have
$$\langle x|p \rangle=\langle \exp(-\mathrm{i} x \hat{p}) x=0|p \rangle=\langle x=0|p \rangle \exp(+\mathrm{i} p x)=N_p \exp(\mathrm{i} p x).$$
The constant $N_p$ is determined by the normalization of the momentum eigenstate as
$$\langle p|p' \rangle=\delta(p-p')=\int \mathrm{d} x \langle p|x \rangle \langle x|p' \rangle=\int \mathrm{d} x N_{p}^* N_p \exp[\mathrm{i}x(p'-p)]=2 \pi |N_{p}|^2 \delta(p-p') \; \Rightarrow \; N_{p}=\frac{1}{\sqrt{2 \pi}}.$$
Of course, the choice of phase is arbitrary.

Now we can also evaluate the expectation value of kinetic energy easily
$$\left \langle \frac{\vec{p}^2}{2m} \right \rangle=\int \mathrm{d}^3 \vec{x} \mathrm{d}^3 \vec{p} \frac{p^2}{2m} \langle \psi|\vec{p} \rangle \langle \vec{p}| \vec{x} \rangle \langle \vec{x} |\psi \rangle=\int \mathrm{d}^3 \vec{x} \mathrm{d}^3 \vec{p} \frac{p^2}{2m} \frac{1}{(2 \pi)^{3/2}} \exp(-\mathrm{i} \vec{p} \cdot \vec{x}) \langle \psi|\vec{p} \rangle \psi(x)= \int \mathrm{d}^3 \vec{x} \mathrm{d}^3 \vec{p} \left [-\frac{\Delta}{2m} \frac{1}{(2 \pi)^{3/2}} \exp(-\mathrm{i} \vec{p} \cdot \vec{x}) \right ] \langle \psi|\vec{p} \rangle \psi(x) = \int \mathrm{d}^3 \vec{x} \int \mathrm{d}^3 \vec{p} \langle \psi|\vec{p} \rangle \langle \vec{p}|\vec{x} \rangle \left (\frac{-\Delta}{2m} \right) \psi(\vec{x}) = \int \mathrm{d}^3 \vec{x} \psi^*(\vec{x}) \left (-\frac{\Delta}{2m} \right) \psi(\vec{x}).$$
So it's not just written down but derived from the fundamental postulates + the specific realization of a quantum theory based on the Heisenberg algebra. To derive the latter from the Galilei group alone is a bit more lengthy. See Ballentine, Quantum Mechanics for that issue (or my QM 2 lecture notes which, however, are in Germany only: http://fias.uni-frankfurt.de/~hees/publ/hqm.pdf ).

 

[A. Neumaier]

That's the strength of Dirac's formulation compared to the wave-mechanics approach.[...] So it's not just written down but derived from the fundamental postulates

From Dirac's postulates (and only if $H$ has no continuous spectrum) but not from Born's. Are Dirac's postulates somewhere available online?

 

[dextercioby]

But there is only one set of postulates in the standard (aka Dirac-von Neumann) formulation. The statistical postulate is:
1. The set of experimentally obtained values of observable A are the spectral values of the self-adjoint operator Â
2. If the state of the system for which one measures A is {p_k, \psi_k}, then the probability to get a_n from disc(Â) is P (a_n) = sum_k p_k <psi_k | P_n| psi_k>, while the probability density of the point alpha from the parametrization space of cont(Â) is P (alpha) = sum_k p_k <psi_k | P_alpha| psi_k>.

The projectors are defined in terms of the Dirac bra/ket spectral decomposition of Â.

 

[A. Neumaier]

But there is only one set of postulates in the standard (aka Dirac-von Neumann) formulation. The statistical postulate is:
1. The set of experimentally obtained values of observable A are the spectral values of the self-adjoint operator Â
2. If the state of the system for which one measures A is {p_k, \psi_k}, then the probability to get a_n from disc(Â) is P (a_n) = sum_k p_k <psi_k | P_n| psi_k>, while the probability density of the point alpha from the parametrization space of cont(A) is P (alpha) = sum_k p_k <psi_k | P_alpha| psi_k>.

The projectors are defined in terms of the Dirac bra/ket spectral decomposition of Â.

This version doesn't cover the argument used by vanhees71 in case H has a continuous spectrum, where the sum must be replaced by an integral.

Which version is in Dirac's book? Or do different editions have different versions? Is one of them applicable to $H=p^2/2m$?

 

[dextercioby]

Of course it does. The continuous spectrum is addressed by an integration in parametrization space. The integral is Riemannian, the parametrization space is a subset of R. The spectral decomposition \Sum_n P_n + \int d alpha P_alpha = Î. This expression makes sense in the rigged Hilbert space formulation of QM, advocated by Arno Böhm and his coworkers.

 

[A. Neumaier]

Of course it does. The continuous spectrum is addressed by an integration in parametrization space.

The question is whether the continuous case (with a Stieltjes integral in place of the sum and the interpretation of matrix elements as probability densities) is in the postulates as formulated by Dirac, or if it is just proceeding by analogy - which would mean that the foundations were not properly formulated.

The rigged Hilbert space is much later than Dirac I think - Gelfand 1964?

 

[dextercioby]

The question is whether the continuous case (with a Stieltjes integral in place of the sum and the interpretation of matrix elements as probability densities) is in the postulates as formulated by Dirac, or if it is just proceeding by analogy - which would mean that the foundations were not properly formulated.

The rigged Hilbert space is much later than Dirac I think - Gelfand 1964?

No, the foundations were properly formulated by von Neumann in 1931, indeed using Stieltjes integrals to define the spectral measures. Dirac's book of 1958 has no precise statement of a set of axioms, yet it has been customary to denote the standard axioms by the name of Dirac and von Neumann (especially for the state reduction/collapse axiom).

The rigged Hilbert spaces were invented by Gel'fand and Kostyuchenko in 1955 and described at large in the 1961 book (4th volume of the famous generalized functions) which was translated to English in 1964. It is not known to me if Arno Böhm knew Russian, it may have been that the book had been first translated to German, or simply someone helped with the translation from Russian. The first use of RHS to QM was made by Arno Böhm in 1964 in a preprint (unfortunately poorly scanned) at the International Center of Theoretical Physics in Trieste.

 

[atyy]

The question is whether the continuous case (with a Stieltjes integral in place of the sum and the interpretation of matrix elements as probability densities) is in the postulates as formulated by Dirac, or if it is just proceeding by analogy - which would mean that the foundations were not properly formulated.

The rigged Hilbert space is much later than Dirac I think - Gelfand 1964?

There are two types of foundations - physical and mathematical. Throughout, I have meant physical while you have often meant mathematical.

The physical foundations were properly formulated by Bohr, Dirac, Heisenberg and von Neumann. Each took a slightly different view, but the key point is that quantum mechanics is a practical operational theory which only makes probabilistic predictions. The wave function, collapse etc are not real. And most importantly, quantum mechanics has a measurement problem.

The mathematical foundations were not complete at the time of von Neumann. POVMs and collapse for continuous variables came later. However, these mathematical tidying up changed no physical concept.

 

[atyy]

The question was not how it is defined (which is part of the QM calculus) but why Born's rule which just says that $|\psi(x)|^2$ is the probability density of $x$ implies that the right hand side is the expected measurement value of $H$. It is used everywhere but is derived nowhere, it seems to me.

No, of course you cannot derive it from the literal Born rule. When we say Born rule nowadays, we mean the generalization, eg. Dirac, von Neumann, and later work, and eg. what vanhees71 did in post #152.

 

[vanhees71]

From Dirac's postulates (and only if $H$ has no continuous spectrum) but not from Born's. Are Dirac's postulates somewhere available online?

I don't know, what you mean by Dirac's vs. Born's postulates. I think the best source for Dirac's point of view is still his famous textbook. What's known as Born's rule is that the modulus squared of the wave function, no matter with respect to which basis, gives the probabilities for the discrete values and probability distributions for the continuous values of the spectrum of the self-adjoint operator. Dirac's handling of distributions (in the sense of generalized functions) was a la physics, i.e., no rigorous. Making it rigorous lead the mathematicians to the development of modern functional analysis. The first mathematically rigorous formulation in form of Hilbert-space theory goes back to John von Neumann, but his physics is a catastrophe, leading to a lot of esoterical debates concerning interpretation. His interpretation is Copenhagen + necessity of a conscious being to take note of the result of a measurement. So it's solipsism in some sense and lead to the famous question by Bell, when the first "collapse" might have happened after the big bang, whether an amoeba is enough to observe something or whether you need some more "conscious" being like a mammal or a human ;-)).

 

[Demystifier]

The first mathematically rigorous formulation in form of Hilbert-space theory goes back to John von Neumann, but his physics is a catastrophe, leading to a lot of esoterical debates concerning interpretation.

Now I am confused. Do you consider his interpretation in terms of consciousness to be a part of physics? That's confusing because at other places you seem to claim the opposite, that such interpretations are not physics.

Or maybe, which I would more naturally expect from you, you would like to divide his work into three aspects: mathematics, physics, and interpretation? But in that case it would not be fair to call his physics a catastrophe. His insight that measurement involves entanglement with wave functions of macroscopic apparatuses is an amazing physical insight widely adopted in modern theory of quantum measurements, irrespective of interpretations.

 

[vanhees71]

Of course, von Neumann's interpretation is no physics but esoterics. I'm totally baffled that somebody of his caliber could come to such an idea. I think his merits concerning QT are completely mathematical, namely to have put it on a solid mathematically strict ground in terms of Hilbert-space theory (mostly in the formulation as "wave mechanics".

 

[Demystifier]

Of course, von Neumann's interpretation is no physics but esoterics. I'm totally baffled that somebody of his caliber could come to such an idea.

I agree on this.

I think his merits concerning QT are completely mathematical,

But disagree on that. I think he had physical merits too.

 

[atyy]

I agree on this.

But the greatness of von Neumann is that he saw clearly, like Bohr and Dirac, that Copenhagen has a measurement problem. The great merit of these physicists is that they are very concerned about physics, unlike Peres (which is a marvellous book), but is completely misleading in not stating the measurement problem clearly, and even hinting that it does not exist in the Ensemble interpretation.

Also I don't think von Neumann's idea of consciousness causing collapse is that different from Bohr or even Landau and Lifshitz's classical/quantum cut, which is a subjective cut. It's the same as Dirac agreeing that there is an observer problem - somehow there has to be an observer/consciousness/classical-quantum cut, which are more or less the same thing.

 

[vanhees71]

That's the great miracle. After all this time people think that there is a measurement problem, but where is it when accepting the minimal interpretation?

Where is the necessity of a classical/quantum cut or a collapse? I just need real-world lab equipment and experimentalists able to handle it to do measurements on whatever system they can prepare in whatever way, make a model within QT and compare my prediction to the oustcome of the measurements. Both my prediction and the mesurements are probabilistic and statistical, respectively. The more than 90 years of application of QT to real-world experimental setups and real-world observations are a great success story So where is the real physics problem? There may be a problem in some metaphysical sense, depending on the believe or world view of the one or the other physicist, but no problem concerning the natural-science side of affairs.

 

[atyy]

That's the great miracle. After all this time people think that there is a measurement problem, but where is it when accepting the minimal interpretation?

Where is the necessity of a classical/quantum cut or a collapse? I just need real-world lab equipment and experimentalists able to handle it to do measurements on whatever system they can prepare in whatever way, make a model within QT and compare my prediction to the oustcome of the measurements. Both my prediction and the mesurements are probabilistic and statistical, respectively. The more than 90 years of application of QT to real-world experimental setups and real-world observations are a great success story So where is the real physics problem? There may be a problem in some metaphysical sense, depending on the believe or world view of the one or the other physicist, but no problem concerning the natural-science side of affairs.

A simple way to see it is that even in the minimal interpretation, one has deterministic unitary evolution and probabilistic evolution due to the Born rule. If one extends deterministic evolution to the whole universe, then there is no room for probability. So the wave function cannot extend to the whole universe. Deciding where it stops, and when the boundary between deterministic evolution and stochastic evolution is is the classical/quantum cut.

 

[vanhees71]

I've never claimed that QT is applicable to a single "event" like the "entire universe" ;-)).

 

[Demystifier]

Minimal ensemble interpretation is not a solution of the measurement problem. It is a clever way of avoiding talk about the measurement problem.

 

[Demystifier]

I've never claimed that QT is applicable to a single "event" like the "entire universe" ;-)).

How about single electron?

 

[vanhees71]

QT makes probabilistic predictions about the behavior of a single electron. You can take a single electron and prepare it very often in the same state and statistically analyse the result to test the probabilistic predictions. A single measurement on a single electron doesn't tell much concerning the validity of the probabilistic predictions.

 

[atyy]

I've never claimed that QT is applicable to a single "event" like the "entire universe" ;-)).

Yes, so one needs an ensemble of subsystems of the universe. The choice of subsystem is the classical/quantum cut.

 

[vanhees71]

This is a bit too short an answer to be convincing. Why is choosing a subsystem of the universe the classical/quantum cut? Matter as we know it cannot be described completely by classical physics at all. So how can just taking a lump of matter as the choice of a subsystem define a classical/quantum cut?

 

[atyy]

This is a bit too short an answer to be convincing. Why is choosing a subsystem of the universe the classical/quantum cut? Matter as we know it cannot be described completely by classical physics at all. So how can just taking a lump of matter as the choice of a subsystem define a classical/quantum cut?

Well, if you agree that quantum mechanics cannot describe the whole universe, but it can describe subsystems of it, then it seems that at some point quantum mechanics stops working.

 

[stevendaryl]

That's the great miracle. After all this time people think that there is a measurement problem, but where is it when accepting the minimal interpretation?

The Born interpretation itself to me seems to require a choice of basis before it can be applied. The rule gives the probability for obtaining various values for the results of measurements. I don't see how you can make sense of the Born rule without talking about measurements. How can you possible compare QM to experiment unless you have a rule saying: If you do such and such, you will get such and such value? (or: if you do such and such many times, the values will be distributed according to such and such probability)