Clarification of the postulates of QM

[stevendaryl]

Why do you need one of the splits to formulate Born's rule?

Well, the formulation used in standard QM is: When you measure an observable, you get an eigenvalue of the corresponding Hermitian operator, with probabilities given by the square of the projection of the wave function onto the subspace with that value of the operator. So that formulation uses the concept of "when you measure..." What makes an interaction into a measurement of an observable? I don't really think that the density matrix formulation is any different, conceptually: "The expectation value of an observable corresponding to an operator $\hat{O}$ is the trace of $\rho \hat{O}$". How can you make sense of "expectation value of an observable" without making "measuring an observable" into something separate from other interactions?

Either "measurement" and "observation" are primitive concepts, which bakes the distinction into the formalism, or else they are derived concepts. As a derived concept, you might say something like "An interaction counts as a measurement of an observable if afterward there is a persistent macroscopic record of the value". But that involves the macroscopic/microscopic split.

 

[stevendaryl]

You just prepare your system and measure the observable you like to observe, and then the quantum state you've prepared tells you the probabilities to find values of this observable.

I consider that sentence to already have the split. "The probabilities to find values of this observable" already makes the distinction between observations and other interactions.

 

[vanhees71]

Well, the formulation used in standard QM is: When you measure an observable, you get an eigenvalue of the corresponding Hermitian operator, with probabilities given by the square of the projection of the wave function onto the subspace with that value of the operator. So that formulation uses the concept of "when you measure..." What makes an interaction into a measurement of an observable? I don't really think that the density matrix formulation is any different, conceptually: "The expectation value of an observable corresponding to an operator $\hat{O}$ is the trace of $\rho \hat{O}$". How can you make sense of "expectation value of an observable" without making "measuring an observable" into something separate from other interactions?

Either "measurement" and "observation" are primitive concepts, which bakes the distinction into the formalism, or else they are derived concepts. As a derived concept, you might say something like "An interaction counts as a measurement of an observable if afterward there is a persistent macroscopic record of the value". But that involves the macroscopic/microscopic split.

You measure an observable by using the adequate measurement apparatus. How else? What you quote are just the postulates of the formalism, and a state of a system is not a self-adjoint trace-class operator in some Hilbert space or a measurement some projection operator to an eigenspace of a self-adjoint operator representing an observable but real-world preparation procedures and real-world measurement apparati, defining the quantities operationally. Of course at a certain point you must assume that the measurement apparatus measures what you want to measure, but that's so with observables within classical physics either.

 

[stevendaryl]

You measure an observable by using the adequate measurement apparatus. How else?

And what makes a measurement apparatus adequate? I think there is no way to formulate that in a non-circular way without making the kind of split I'm talking about.

 

[stevendaryl]

Of course at a certain point you must assume that the measurement apparatus measures what you want to measure, but that's so with observables within classical physics either.

No, that's not true. In classical physics, it's derivable. And also, in classical physics, the laws don't refer to observables at all. The distinction isn't baked into classical physics, so there is no need for the split.

 

[atyy]

The measurement is due to interaction of the measured system with a measurement apparatus. I also never bought the argument that this is something different than the interactions described by quantum theory. This doesn't make sense either! On the one hand we have to use quantum theory, driven by observations that tell us that the classical theory is only an approximation. So to claim a measurement doesn't follow the laws of QT is not very satisfactory, and I don't see, why one should use this assumption nowadays, where we have understood much better the emergence of classical behavior of macroscopic systems from quantum theory than the "founding fathers" of QT could have known in the beginning. The interaction of a particle with a detector follows the rules of quantum theory and thus is described as a local interaction between the measured system.

I don't want to bash the textbook by Cohen-Tanoudji et al, but I think you should get the postulates as precise as possible, because this helps tremendously to study the subject.

But if we follow your reasoning, and quantum theory applies to everything, then there should be a wave function of the universe, and it should make physical sense without many-worlds or Bohmian mechanics.

 

[vanhees71]

Here, I have a problem of course, because a probabilistic interpretation makes it necessary to be able to prepare an ensemble of systems. For the universe as a whole that's impossible. On the other hand, it's anyway fictitious since we can never observe the universe as a whole ;-).

 

[A. Neumaier]

we can never observe the universe as a whole ;-).

?

We can evaluate lots of observables of the universe as a whole by measuring its local fields and currents at particular positions of interest.

 

[vanhees71]

Even with an infinite lifetime we can only observe a part of the universe:

https://en.wikipedia.org/wiki/List_of_cosmological_horizons

 

[A. Neumaier]

But if [...] quantum theory applies to everything, then there should be a wave function of the universe, and it should make physical sense without many-worlds or Bohmian mechanics.

There must be a state of the universe, but not necessarily a wave function -- it could be a mixed state. It does indeed make physical sense:

According to my http://arnold-neumaier.at/physfaq/cei/ , we need a Hilbert space carrying a representation of the standard model plus some (not yet decided) form of gravity, unitary dynamics for operators, density operators for Heisenberg states, the definition of $\langle A\rangle:=\mbox{tr}~\rho A$ as mathematical framework, and for its interpretation a single rule:

Upon measuring a Hermitian operator $A$, the measured result will be approximately $\bar A=\langle A\rangle$, with an uncertainty
at least of the order of $\sigma_A=\sqrt{\langle (A-\bar A)^2\rangle}$. If the measurement can be sufficiently often repeated (on an object with the same or sufficiently similar state) then $\sigma_A$ will be a lower bound on the standard deviation of the measurement results.

Everyone doing quantum mechanics uses these rules (even those adhering to the shut-up-and-calculate mode of working), and they apply universally. No probabilistic interpretation beyond that is needed, so it applies also to the single universe we live in. Everything deduced in quantum field theory about macroscopic properties follows, and one has a completely self-consistent setting. The transition to classicality is automatic and needs no deep investigations - the classical situation is simply the limit of a huge number of particles. Whereas on the microscopic level, uncertainties of single events are large, so that state determination must be based by the statistics of multiple events with a similar preparation.

We cannot expect to measure all the observables of the whole universe, and perhaps never determine its precise state. But measuring all observables or finding its exact state is already out of the question for a small quantum system such as a shaken bottle of water. What matters for a successful physics of the universe is only that we can model (and then predict) the observables that are accessible to measurement.

 

[A. Neumaier]

Even with an infinite lifetime we can only observe a part of the universe:

sure, but this only means that we cannot expect to measure all the observables of the whole universe, and perhaps never determine its precise state.

But measuring all observables or finding its exact state is already out of the question for a small quantum system such as a shaken bottle of water, though nobody deduces from this impossibility that its state is fictitious.

 

[atyy]

Here, I have a problem of course, because a probabilistic interpretation makes it necessary to be able to prepare an ensemble of systems. For the universe as a whole that's impossible. On the other hand, it's anyway fictitious since we can never observe the universe as a whole ;-).

But don't you think Dirac, Landau & Lifshitz, Cohen-Tanoudji etc may have taken this into account when they state collapse as a postulate?

ie. you are not able to really defend quantum mechanics as applying to the whole universe.

 

[vanhees71]

sure, but this only means that we cannot expect to measure all the observables of the whole universe, and perhaps never determine its precise state.

Yes, and that's why "the state of the entire universe" is an empty phrase since, if I claim to know the "state of the entire universe" and give it to you in terms of some Stat. Op. you can never empirically check my claim. So I can claim whatever I like. Some theorists love such ideas since they cannot not disproven by observation, but than that's (perhaps interesting) math but no physics!

 

[vanhees71]

But don't you think Dirac, Landau & Lifshitz, Cohen-Tanoudji etc may have taken this into account when they state collapse as a postulate?

ie. you are not able to really defend quantum mechanics as applying to the whole universe.

Yes, but I don't want to defend any theory as applying to the whole universe since it's not observable.

Of course, it's an assumption made in cosmology that there's no preferred place or direction in the universe (at least in the large-scale coarse grained picture) and that all physical laws are thus the same at any point and time in this universe (cosmological principle), and so far we have not seen any contradiction to this assumption, which mathematically boils down to the statement that the large-scale coarse grained spacetime is described by a Friedmann-Lemaitre-Robertson-Walker metric, by the "local" observations we are able to make today, and you can thus keep the cosmological principle as hypothesis about the "state" of the entire universe, but you'll never be able to finally check it completely, because there are regions in spacetime we can never observe (given the observational fact that we live in an "accelerating" universe there's a "future horizon").

 

[A. Neumaier]

Yes, and that's why "the state of the entire universe" is an empty phrase

By the same reasoning, the state of a piece of metal would also be an empty phrase, since we never know it exactly, and we can probe only a few of its observables. But probing these variables is sufficient; it is the conventional test for adequacy of a proposed state of the metal. Otherwise no solid state physics would be possible.

Exactly the same holds for the universe as a whole.

if I claim to know the "state of the entire universe" and give it to you in terms of some Stat. Op. you can never empirically check my claim. So I can claim whatever I like.

One can claim whatever one likes about any system, but the claim is no physics if it is easily falsified. In case the system is the whole universe, you can claim whatever you like but unless your claim is very informed it can be easily falsified by computing from the state the expectation values of the electromagnetic field at points where we can measure it. It is very difficult to come up with a state that cannot be falsified in this or similar ways. For this would be a state that is compatible with everything we have ever empirically observed in the universe! Thus knowing this state amounts to knowing all physics accessible to us.

Thus, as for a metal, one must be content with describing this state approximately, but this is not impossible.

 

[vanhees71]

Sure, I can make an assumption about the state of my local environment or parts of it (like a piece of metal), and that's well testable by observation. What's also testable in cosmology are the observations in my "space-time neighborhood", including the redshift of far-distant objects to determine the Hubble diagram with better and better accuracy (assuming of course certain laws on the luminosity of the objects to determine the distance). But here I just probe a very coarse grained classical picture of the universe, and that's sufficient to describe the observables. But this is far from having a description of the "state of the entire universe" and in fact not involving any quantum theory at all (it's just GR and a very crude model for the "matter" described as an ideal fluid). The same is true for the piece of metal, I can describe by some very coarse grained macroscopic (thermodynamic) observables like temperature.

 

[A. Neumaier]

But this is far from having a description of the "state of the entire universe" and in fact not involving any quantum theory at all (it's just GR and a very crude model for the "matter" described as an ideal fluid). The same is true for the piece of metal, I can describe it by some very coarse grained macroscopic (thermodynamic) observables like temperature.

Yes, the whole universe and a piece of metal are completely analogous in this respect.

All descriptions in physics are either very coarse-grained or of very small objects. The detailed state can be found with a good approximation only for fairly stationary sources of very small objects. But this doesn't mean that the detailed state (of the metal or the whole universe) doesn't exist or that talking about it is an empty phrase. Even in classical mechanics, it is impossible to know a highly accurate state of a many-particle system (not even of the solar system with sun, planets, planetoids, and comets treated as rigid bodies) but its existence is never questioned.

Thus there is no physical reason to question the existence of the state of the whole universe, even though all its details may be unknown for ever.

 

[martinbn]

@A. Neumaier : Isn't there a difference though? For the universe it is unknowable in principle for the metal only in practice.

 

[A. Neumaier]

The detailed state of a piece of metal is unknowable in principle. To disprove this you'd have to propose a Gedankenexperiment how to find it. This cannot even be done for a classical model of the metal. I haven't seen any idea in the literature that would indicate how to reliably detect a single classical particle position anywhere in the deep interior of a piece of metal. Exact classical positions of multiparticle systems are therefore metaphysical assumptions.

 

[vanhees71]

Yes, the whole universe and a piece of metal are completely analogous in this respect.

All descriptions in physics are either very coarse-grained or of very small objects. The detailed state can be found with a good approximation only for fairly stationary sources of very small objects. But this doesn't mean that the detailed state (of the metal or the whole universe) doesn't exist or that talking about it is an empty phrase. Even in classical mechanics, it is impossible to know a highly accurate state of a many-particle system (not even of the solar system with sun, planets, planetoids, and comets treated as rigid bodies) but its existence is never questioned.

Thus there is no physical reason to question the existence of the state of the whole universe, even though all its details may be unknown for ever.

The only fundamental question is, how you then define the meaning of a probabilistic statement for something that is one single event and cannot be reproduced in terms of an ensemble. Of course, also the single bar of metal is well-described by macroscopic quantities, and there the averaging/coarse-graining is over many microscopic details/sufficiently large space-time "fluid cells", making up an ensemble in some sense.

 

[A. Neumaier]

The only fundamental question is, how you then define the meaning of a probabilistic statement for something that is one single event and cannot be reproduced in terms of an ensemble. Of course, also the single bar of metal is well-described by macroscopic quantities, and there the averaging/coarse-graining is over many microscopic details/sufficiently large space-time "fluid cells", making up an ensemble in some sense.

I had already answered this in post #115. You can easily check that in the quantum field theory of macroscopic objects, the averaging is always done inside the definition of the macroscopic operator to be measured; this is sufficient to guarantee very small uncertainties $\sigma_A$ of macroscopic observables $A$. Thus one does not need an additional averaging in terms of multiple experiments on similarly prepared copies of the system. Since all quantities of interest in a study of the universe as a whole are macroscopic, they are well-determined by the state.

 

[vanhees71]

I had already answered this in http://https//www.physicsforums.com/posts/5496600/ . You can easily check that in the quantum field theory of macroscopic objects, the averaging is always done inside the definition of the macroscopic operator to be measured; this is sufficient to guarantee very small uncertainties $\sigma_A$ of macroscopic observables $A$. Thus one does not need an additional averaging in terms of multiple experiments on similarly prepared copies of the system. Since all quantities of interest in a study of the universe as a whole are macroscopic, they are well-determined by the state.

I agree with all of that, except that I don't know what you mean by "study of the universe as a whole". What we study are pretty local tiny parts of the universe in our neighborhood. That we call this "study of the universe as a whole" is entirely based on the assumption of the Cosmological Principle, which never can be checked by experiment. Although cosmology is nowadays a very successful branch of physics, one should not forget this problem in connection with what we call the "scientific method" in all other branches of physics!

 

[A. Neumaier]

I don't know what you mean by "study of the universe as a whole" [...] which never can be checked by experiment

In 1978, Penzias and Wislon got a Nobel prize for experimentally checking an earlier theoretical prediction about the universe as a whole.

The Nobel Prize in Physics 1978 was divided, one half awarded to Pyotr Leonidovich Kapitsa "for his basic inventions and discoveries in the area of low-temperature physics", the other half jointly to Arno Allan Penzias and Robert Woodrow Wilson "for their discovery of cosmic microwave background radiation".

From the press release:

The discovery of Penzias and Wilson was a fundamental one: it has made it possible to obtain information about cosmic processes that took place a very long time ago, at the time of the creation of the universe.

Studying what took place at the time of the creation of the universe is surely a study of the universe as a whole. And no doubt it is a quantum phenomenon, since everything in the universe is one, though much of it can be described in a classical approximation. It is therefore reassuring to know that there are no fundamental obstacles in quantum field theory that prevent it to be applied even to the largest possible quantum system.

 

[Ilja]

The question is if there is such a thing as "studying the creation of the universe".

The only "evidence" about the creation of the universe is a singularity of the Einstein equations of GR. But such a singularity is not reasonably evidence about a "creation of the universe", but, instead, evidence for a failure of GR, which is plausible given that in the very early universe quantum gravity effects would become relevant, so that it is anyway well-known that the best (according to the mainstream) existing theory of gravity, classical GR, is no longer applicable anyway.

On the other hand, given the homogeneity of the CMBR, it is reasonably plausible that studying this radiation is studying the whole observable part of the universe.

What we know from observation is that the straightforward GR model, without "inflation", fails. That means, we know that there was some time in the very "early" universe where $a''(\tau)>0$. This is something very different from a naive meaning of "inflation" (which would be $a'(\tau)$ very large, see http://ilja-schmelzer.de/gravity/inflation.php ), but this is not the point I want to make. With $a''(\tau)>0$ in the very early universe, it is not even clear if there is a singularity. There can be, as well, some minimal value, with some big crunch before, a big bounce. The only observational evidence is about $a''(\tau)>0$, nothing more.

 

[A. Neumaier]

The question is if there is such a thing as "studying the creation of the universe".

I corrected the phrase I had used to studying ''what took place at the time of the creation of the universe'', which is the exact wording used by the Nobel prize committee. You'll have to argue with them whether their formulation was adequate, not with me.

 

[Ilja]

Ok, if any of the members of the Nobel committee reads this - I think you made an error here.