The Time Symmetry Debate in Quantum Theory

[Jano L.]

Determinism is simply a special case of probabilistic theories - the only allowed probabilities are zero and one.

That works only for discrete space of events. If the space of events is continuous (like in classical mechanics), we cannot restrict the probability values in probabilistic model only to 0 and 1.

 

[bhobba]

That works only for discrete space of events. If the space of events is continuous (like in classical mechanics), we cannot restrict the probability values in probabilistic model only to 0 and 1.

Yes you can - the predicted values have a value of 1 - all the rest zero. You will have to use Dirac Delta functions - but hey welcome to applied math.

What you don't have is a continuous transition which is an issue in QM but not classically.

Thanks
Bill

 

[Jano L.]

Why should the wave function be describing both the position and momentum of a quantum as if it was a classical particle? why an equation that has features of a wave equation but even more clearly of a diffusion equation should be interpreted as fundamentally related to a classical particle behaviour rather than to a field?

The basic reason is that it is the most natural and simplest interpretation in quantum chemistry.

To paraphrase Slater, after the success of de Broglie wave hypothesis for individual electrons, people wondered, what to do for multi-particle system, like the helium atom or hydrogen molecule. Should we introduce different $\psi$ functions for each different particle (field-like idea of the electron) or should the many-particle system be described by one big $\psi$ function?

It turned out that the close analogy to Hamilton-Jacobi theory works well even for many particle systems, i.e. one $\psi$ function describes the whole system.

Why are often electrons thought of as particles and not fields? Well, the particle is much simpler concept and it is quite natural if you look at the equation which people use to calculate properties of the atoms and molecules:

$$
\partial_t \psi(\mathbf r_1, \mathbf r_2, ...) = \frac{1}{i\hbar} \left[\sum_a \frac{\hat{\mathbf p}_a^2}{2m_a} + U(\mathbf r_1, \mathbf r_2, ...)\right] \psi
$$

The function $\psi$ is defined on configuration space of the system of particles. The variables $\mathbf r_1, ...$ are possible positions of those particles.

 

[Jano L.]

Yes you can - the predicted values have a value of 1 - all the rest zero. You will have to use Dirac Delta functions - but hey welcome to applied math.

Probabilistic models on continuous spaces usually work with regular probability distributions. What you suggested would require a probabilistic model which would assign singular probability to one event in continuous space as a result of the calculation. Can you give an example of such model ?

 

[bhobba]

Probabilistic models on continuous spaces usually work with regular probability distributions. What you suggested would require a probabilistic model which would assign singular probability to one event in continuous space as a result of the calculation. Can you give an example of such model ?

The probability distribution of finding something at position x with certainty is delta(x-x') where delta is the Dirac Delta function. x in that equation is the positions predicted by classical mechanics.

Thanks
Bill

 

[Jano L.]

I think the problem is in the following. Probabilities are measures and measures do not capture the essential character of certainty that determinism gives us.

In your example with delta distribution, if the probability distribution on space of x' is $\delta(x'-x)$, the only thing we can infer from it is that the probability that the particle is out of x is 0.

We cannot infer that the particle is at x with certainty. It can still get out of x, as long as it spends infinitely less time there. However, if it can get out of x, it can have some significant influence there.

In a deterministic model, if the particle is at x, it really is there and never gets anywhere else; it cannot have influence at other points than at x.

 

[bhobba]

We cannot infer that the particle is at x with certainty. It can still get out of x, as long as it spends infinitely less time there.

I have zero idea what you are trying to say. It has zero probability of being anywhere other than x because, for any point other than x, you can always find points near x to take the integral around that will give zero. But if it includes x you get one. That has one and only one interpretation - the particle is with a dead cert at x. And the only probabilities that enter into it are zero and one - just like I asserted.

Thanks
Bill

 

[Jano L.]

That has one and only one interpretation - the particle is with a dead cert at x.

There is difference between probability 1 and absolute certainty.

Probability 1 means only that the particle will be found out of x insignificant number of times, in other words, measure of such cases is zero. Still, in a long enough series of measurements, one may find particle out of x million times.

Absolute certainty of x would require that the particle will never be found out of x. Such kind of certainty cannot be described by measure on the event space, but it can be described by statements like "particle will be found at x", used in deterministic models.

 

[DevilsAvocado]

1.) If you interpret the electron position to be described only by probabilities at any instant, then it must be the case that to maintain equal probabilities over individual shells over some period of time, the electron position must be discontinuous in space.

2.) If instead you interpret the electron path to be continuous then the particle must not be more fundamentally described by a probability distribution which could only acts as a statistical analysis. Since we have assumed the particle has a set path whether we can observe that path or not.

Both 1 and 2 assume that one particle is always one particle and that the particle must exist somewhere in space at all times. With these assumptions we can observe that the particle must either follow a continuous path or be discontinuous in space.

I might be missing something here; but isn’t the problem that you treat the electron as a particle instead of a spherical cloud of probability, which we know must be correct since a rotating charge classically orbiting around the nucleus, would constantly lose energy in form of electromagnetic radiation, and finally collapse into the nucleus...

 

[DevilsAvocado]

Why can't the instantaneous determination of spin of entangled particles be based in deterministic law?

A non-local deterministic theory can, but a local deterministic theory cannot.

By the way, isn’t this talk about determinism a bit ‘superfluous’...? I mean, the Schrödinger equation is perfectly deterministic...

Isn’t the real problem that we don’t know exactly (in mathematical terms) what happens at measurement, when QM suddenly ‘transforms’ into a probabilistic theory?

Who cares if it’s deterministic or probabilistic? As long as we fully understand what’s going on.

(no problem to start ‘stochastic’ unpredictable chaos in a deterministic system)

 

[DevilsAvocado]

why an equation that has features of a wave equation but even more clearly of a diffusion equation should be interpreted as fundamentally related to a classical particle behaviour rather than to a field? Do we think in terms of heat particles when dealing with the heat equation?

I think the reason is – you can’t put sunlight in your pocket, but a marble you can.

 

[DevilsAvocado]

I don't have it handy. Can you give the basic points where Ballentine argues that puzzlement about QM not giving deterministic trajectories but amplitude probabilities has nothing to do with insisting about classical particle behaviour?

I don’t think Ballentine do away with the measurement problem in chapter 3:

Kinematics and Dynamics
The results of Ch. 2 constitute what is sometimes called “the formal structure of quantum mechanics”. Although much has been written about its interpretation, derivation from more elementary axioms, and possible generalization, it has by itself very little physical content. It is not possible to solve a single physical problem with that formalism until one obtains correspondence rules that identify particular dynamical variables with particular operators. This will be done in the present chapter. The fundamental physical variables, such as linear and angular momentum, are closely related to space–time symmetry transformations. The study of these transformations serves a dual purpose: a fundamental one by identifying the operators for important dynamical variables, and a practical one by introducing the concepts and techniques of symmetry transformations.

Where’s the “symmetry” in this?

https://www.youtube.com/watch?v=ZJ-0PBRuthc

 

[bhobba]

There is difference between probability 1 and absolute certainty.

No. Kolgmorgrov's axioms are clear on this point:
http://en.wikipedia.org/wiki/Probability_axioms
'This is the assumption of unit measure: that the probability that some elementary event in the entire sample space will occur is 1. More specifically, there are no elementary events outside the sample space.'

If something has probability 1 it must occur.

In my example let's for simplicity consider the 1 dimensional case. If you pick a point other than x you can find an interval that doesn't have x in it but includes that point. The probability of the particle being in that interval is integral delta (x-x') over that interval - which is zero. It can't be in that interval so can't be at the point. Do the same thing but at x and you get 1 - it must be in that interval. This means it must be at x.

The ability to define only 0 and 1 and such being deterministic is one of the key ingredients in the Kochen-Specker theorem:
http://en.wikipedia.org/wiki/Kochen–Specker_theorem

Thanks
Bill

 

[bhobba]

Isn’t the real problem that we don’t know exactly (in mathematical terms) what happens at measurement, when QM suddenly ‘transforms’ into a probabilistic theory?

Its probabilistic from the outset. What is deterministic is the state that allows you calculate probabilities.

Who cares if it’s deterministic or probabilistic? As long as we fully understand what’s going on.

That's the whole problem with QM - what one person thinks is a full understanding to another is an anathema.

Thanks
Bill

 

[T0mr]

I might be missing something here; but isn’t the problem that you treat the electron as a particle instead of a spherical cloud of probability, which we know must be correct since a rotating charge classically orbiting around the nucleus, would constantly lose energy in form of electromagnetic radiation, and finally collapse into the nucleus...

The electron in orbit was given as an example of a possible path. I have read about this argument before. That an classically orbiting electron should emit radiation presumably because it is a charged object and an accelerated charged object (changing direction) will emit electromagnetic radiation. Yet if you were to put opposite charge on two spheres, one light and one heavy, and then set the lighter in orbit (in space) around the heavier would the two spheres not act just as the two body problem for gravitational force. Or take for instance an electromagnet, as electrons move at relatively similar speeds around the coils, no electromagnetic fields are emitted just magnetic fields. Magnets do not die out rapidly. So that argument as being the motivation for adopting a completely probabilistic approach to the atom seems to me to be inadequate.

By the way, isn’t this talk about determinism a bit ‘superfluous’...? I mean, the Schrödinger equation is perfectly deterministic...

A wave function can be deterministic in the sense that it can determine the probabilities of a property of a quantum object. But it really isn't deterministic in the sense that it can determine what those properties actually are. For example, a wave function will not give the position of the object (or position of object's center) even when given the classical initial conditions required. A wave function gives many objects and the concern may be with only one object. If you ask it where the center of an electron is at time t, it will reply "I do not know but here are an infinite number of possible options." That kind of process really is not in the spirit of determinism. A deterministic function in my mind is something that is one to one, not one to infinity(with odds).

 

[bhobba]

I don’t think Ballentine do away with the measurement problem in chapter 3:

That was not my assertion - my assertion was its not based on a particle model.

Where’s the “symmetry” in this?

The symmetry is in the laws of physics. It doesn't matter where you do the experiment, in what direction its done or when you do it the same laws of physics apply and the same outcome will occur. To be specific since this is QM the probabilities will be the same. This implies physical laws and is one of the very deep things modern physics has taught us - to be specific it was one of the great insights of Wigner and part of the reason he got a Nobel prize. For classical physics it was the great mathematician Emily Noether that discovered it. It was Feynman that showed how the two were related.

This is in fact the defining property of an inertial frame - the Earth isn't exactly inertial but for many practical purposes such as this experiment it is.

Thanks
Bill

 

[TrickyDicky]

That was not my assertion - my assertion was its not based on a particle model.

I think you are missing my point for some reason. I was relating the measurement problem precisely to the insistence in analogies with the classical particle model.
Now oddly enough you try to refute this by highlighting the similarities of the classical mechanics model(which is a particle model isn't it?) with quantum mechanics.

The symmetry is in the laws of physics. It doesn't matter where you do the experiment, in what direction its done or when you do it the same laws of physics apply and the same outcome will occur. To be specific since this is QM the probabilities will be the same. This implies physical laws and is one of the very deep things modern physics has taught us - to be specific it was one of the great insights of Wigner and part of the reason he got a Nobel prize. For classical physics it was the great mathematician Emily Noether that discovered it. It was Feynman that showed how the two were related.

This is in fact the defining property of an inertial frame - the Earth isn't exactly inertial but for many practical purposes such as this experiment it is.

I'm not one to be convinced of the mathematical beauty of symmetries, but in this thread we are actually centering on departures of quantum theory from those classical symmetries.
If you are so fascinated by symmetries you surely must feel how awkwardly those symmetries are spoiled in QFT/QM by a lot of things that can be reduced to the measurment problem for brevity and the fact that the nice path integral formulation in order to obtain sane results (to many decimal places) needs to recurr to arbitrary procedures like regularization that are neither physically nor mathematically justified, yeah I know, welcome to applied mathematics but I thought you had some fondness of symmetries and mathematical beauty.
So you make a rather strange mix of demanding the aesthetic value of symmetries one side and practical purposes that are only fulfilled thru rather ugly manouvers (in terms of mathematical rigour that is). Not to mention the incompatibility between GR and QM. Certainly some symmetry is not right here.

To insist in the example I used earlier, the Schrodinger equation is basically a heat equation with a Wick rotation, this introduces time reversibility thru i but again there is nothing fundamental in the equation that makes us use the particle picture other than being used to it from classical mechanics and the practical matters that Jano L. mentioned, that have nothing to do with fundamental issues. That practical use is compatible with considering those local observables simply as excitations of quantum relativistic fields. What I was pointing out was that IMO as soon as one stops thinking about particles with trajectories (that is with simultaneous position and momentum) the probabilistic issues and the measurement problem/collpse of wf or even entanglement and other "quantum weird stuff" gets downgraded just by concentrating on the fields picture.

 

[bhobba]

I think you are missing my point for some reason. I was relating the measurement problem precisely to the insistence in analogies with the classical particle model.

I think you are missing my point. QM does not insist on analogies with a classical particle model. All it assumes is position is an observable - which is a fact. The rest follows from symmetries - no other assumptions at all. It's forced on us - no escaping it. For example momentum exists because the laws of physics are space translation invariant which means it's generator has certain properties and those properties imply the momentum operator. No particle assumption - yet momentum exists. The same with energy. No assumption about it at all yet time translation symmetry, similar to momentum, means it exists and implies the Schrodinger equation. It's a very deep insight.

This is a very important point and I think we need to get it sorted before discussing other stuff.

BTW its true QFT does shed considerable light on QM - see for example:
https://www.amazon.com/dp/9812381767/?tag=pfamazon01-20

But that's not because a particle analogy was chosen - symmetries force it onto us - no escaping.

Thanks
Bill

 

[TrickyDicky]

I think you are missing my point. QM does not insist on analogies with a classical particle model. All it assumes is position is an observable - which is a fact. The rest follows from symmetries - no other assumptions at all. It's forced on us - no escaping it. For example momentum exists because the laws of physics are space translation invariant which means it's generator has certain properties and those properties imply the momentum operator. No particle assumption - yet momentum exists. The same with energy. No assumption about it at all yet time translation symmetry, similar to momentum, means it exists and implies the Schrodinger equation. It's a very deep insight.

This is a very important point and I think we need to get it sorted before discussing other stuff.

BTW its true QFT does shed considerable light on QM - see for example:
https://www.amazon.com/dp/9812381767/?tag=pfamazon01-20

But that's not because a particle analogy was chosen - symmetries force it onto us - no escaping.

I can't argue with someone that while insisting on classical symmetries like time and space translation keeps saying there are no analogies with classical mechanics(even after recommending me the classic by Landau to have a better understanding of QM) or that classical Newtonian mechanics are unrelated to a particle picture.

 

[bhobba]

I can't argue with someone that while insisting on classical symmetries like time and space translation keeps saying there are no analogies with classical mechanics(even after recommending me the classic by Landau to have a better understanding of QM) or that classical Newtonian mechanics are unrelated to a particle picture.

Sigh. It looks like there is a schism here that's difficult to overcome. For example 'classical symmetries' - there is nothing classical about the POR that these symmetries derive from - its true relativistically, QM, classically, all sorts of ways. It is these symmetries that determine the dynamics of QM - not an analogy to anything. Even if you know nothing of classical physics exactly the same equations result.

The reason I recommended Landau is it shows classical mechanics is really about symmetry so its no surprise QM is also about it. In fact the reason classical mechanics is about symmetry is because QM is about symmetry. Its not the other way around - QM is not based on classical analogies - it based on the implications of symmetry and that implies that classical mechanics is also. In QM the symmetries are in the quantum state and observables - in classical mechanics its in the Lagrangian. But as Feynman showed the Lagrangian follows from the rules governing states and observables.

Thanks
Bill

 

[stevendaryl]

I think you are missing my point for some reason. I was relating the measurement problem precisely to the insistence in analogies with the classical particle model.

Maybe you could say a little more about the connection between the two (the measurement problem, and classical particle model). The basic axioms of quantum mechanics don't mention particles or classical mechanics. They are something like:

1. The set of possible states of a system are the normalized vectors of a Hilbert space.
2. Each observable/measurement corresponds to a self-adjoint linear operator.
3. The results of a measurement corresponding to an operator $O$ always produces an eigenvalue of $O$.
4. The probability that a measurement of an observable corresponding to operator $O$ produces outcome $o_i$ is given by:
   $| \langle \psi | P_i |\psi \rangle |^2$, where $P_i$ is the projection operator onto the subspace of the Hilbert space corresponding to the states with eigenvalue $o_i$.
5. Immediately after the measurement, if the result was $o_i$, then the system will be in state $P_i | \psi \rangle$.
6. Between measurements, the system evolves according to equation:
   $\dfrac{d}{dt} | \psi \rangle = -i H | \psi \rangle$ (in units where h-bar = 1), where $H$ is the Hamiltonian operator, the operator associated with the total energy of the system.

I'm not worried too much about whether I've got these exactly right, or whether there is disagreement about them, or whether some axioms can be proved to follow from others, or from more basic considerations. I'm just putting a straw man collection out there so that you can say which one has to do with assuming a classical notion of "particle". I don't see it.

 

[bhobba]

[*]Between measurements, the system evolves according to equation:
$\dfrac{d}{dt} | \psi \rangle = -i H | \psi \rangle$ (in units where h-bar = 1), where $H$ is the Hamiltonian operator, the operator associated with the total energy of the system.

I think at least one of his concerns is the form of the Hamiltonian (ie H = (p^2)/2*m + V(x)) is the same as classical physics. I suspect he believes that is because of classical analogies. My point is that the form follows from symmetry considerations (that's what Ballentine proves in Chapter 3) - not classical analogies. The same is true in classical mechanics which is the point of my reference to Landau's classic (he proves that form from symmetry considerations as well). That Classical Mechanics is like this is because QM is like this - not the other way around.

Thanks
Bill

 

[TrickyDicky]

Sigh. It looks like there is a schism here that's difficult to overcome. For example 'classical symmetries' - there is nothing classical about the POR that these symmetries derive from - its true relativistically, QM, classically, all sorts of ways.
It is these symmetries that determine the dynamics of QM - not an analogy to anything. Even if you know nothing of classical physics exactly the same equations result.

The reason I recommended Landau is it shows classical mechanics is really about symmetry so its no surprise QM is also about it. In fact the reason classical mechanics is about symmetry is because QM is about symmetry. Its not the other way around - QM is not based on classical analogies - it based on the implications of symmetry and that implies that classical mechanics is also. In QM the symmetries are in the quantum state and observables - in classical mechanics its in the Lagrangian. But as Feynman showed the Lagrangian follows from the rules governing states and observables.

Thanks
Bill

It's no use to keep circling around this. I'm not disagreeing with this rather with the way you present it anyway.
Instead and since I think you favor the statistical interpretation, QFT is also related to statistical field theory by a Wick rotation. I sometimes entertain myself relating ensemble properties to field properties.

 

[TrickyDicky]

Maybe you could say a little more about the connection between the two (the measurement problem, and classical particle model). The basic axioms of quantum mechanics don't mention particles or classical mechanics. They are something like:

1. The set of possible states of a system are the normalized vectors of a Hilbert space.
2. Each observable/measurement corresponds to a self-adjoint linear operator.
3. The results of a measurement corresponding to an operator $O$ always produces an eigenvalue of $O$.
4. The probability that a measurement of an observable corresponding to operator $O$ produces outcome $o_i$ is given by:
   $| \langle \psi | P_i |\psi \rangle |^2$, where $P_i$ is the projection operator onto the subspace of the Hilbert space corresponding to the states with eigenvalue $o_i$.
5. Immediately after the measurement, if the result was $o_i$, then the system will be in state $P_i | \psi \rangle$.
6. Between measurements, the system evolves according to equation:
   $\dfrac{d}{dt} | \psi \rangle = -i H | \psi \rangle$ (in units where h-bar = 1), where $H$ is the Hamiltonian operator, the operator associated with the total energy of the system.

I'm not worried too much about whether I've got these exactly right, or whether there is disagreement about them, or whether some axioms can be proved to follow from others, or from more basic considerations. I'm just putting a straw man collection out there so that you can say which one has to do with assuming a classical notion of "particle". I don't see it.

Well if in axioms 4 and 5 you associate the observables with properties of a classical particle (considering a classical particle an object with a trajectory) you hit the measurement problem, otherwise you don't.
Besides if one thinks about the reason one chooses a linear space as axiomatic in the first place one realizes it demands point particles, just like Euclidean space in classical mechanics demands classical point particles.

 

[TrickyDicky]

I think at least one of his concerns is the form of the Hamiltonian (ie H = (p^2)/2*m + V(x)) is the same as classical physics. I suspect he believes that is because of classical analogies. My point is that the form follows from symmetry considerations (that's what Ballentine proves in Chapter 3) - not classical analogies. The same is true in classical mechanics which is the point of my reference to Landau's classic (he proves that form from symmetry considerations as well). That Classical Mechanics is like this is because QM is like this - not the other way around.

Thanks
Bill

Certainly Schrodinger derived the Hamiltonian from classical mechanics whether one considers it a historical accident or not.

 

[bhobba]

Besides if one thinks about the reason one chooses a linear space as axiomatic in the first place one realizes it demands point particles, just like Euclidean space in classical mechanics demands classical point particles.

I don't get that at all.

For example check out the link I posted before:
http://arxiv.org/pdf/0911.0695v1.pdf

Precisely where in that axiomatic development is point particles assumed?

Thanks
Bill

 

[bhobba]

Certainly Schrodinger derived the Hamiltonian from classical mechanics whether one considers it a historical accident or not.

The point though is a lot of water has passed under the bridge since then and its real basis is now known. Like I said Wigner got a Nobel prize in part for figuring it out.

Thanks
Bill

 

[TrickyDicky]

The point though is a lot of water has passed under the bridge since then and its real basis is now known. Like I said Wigner got a Nobel prize in part for figuring it out.

Thanks
Bill

Thus my "whether one considers it a historical accident or not".

 

[stevendaryl]

Well if in axioms 4 and 5 you associate the observables with properties of a classical particle (considering a classical particle an object with a trajectory) you hit the measurement problem, otherwise you don't.
Besides if one thinks about the reason one chooses a linear space as axiomatic in the first place one realizes it demands point particles, just like Euclidean space in classical mechanics demands classical point particles.

I don't see that, at all. To me, the "measurement problem" is the conceptual difficulty that on the one hand, a measurement has an abstract role in the axioms of quantum mechanics, as obtaining an eigenvalue of a self-adjoint linear operator, and it has a physical/pragmatic/empirical role in actual experiments as a procedure performed using equipment. What is the relationship between these two notions of measurement? The axioms of quantum mechanics don't make it clear.

I don't see that it has anything particularly to do with particles.

There is a pragmatic issue, which is that we don't really know how to measure arbitrary observables. There are only a few that we know how to measure: position, momentum, energy, angular momentum. It might be correct to say that we only know how to measure observables with classical analogs. But you seem to be saying something different, that the measurement problem only comes up because we're insisting on classical analogues. I don't see that, at all. We can certainly pick an observable with no classical analog. The measurement problem doesn't go away, it becomes worse, because we don't know how to measure it.

 

[TrickyDicky]

I don't see that, at all. To me, the "measurement problem" is the conceptual difficulty that on the one hand, a measurement has an abstract role in the axioms of quantum mechanics, as obtaining an eigenvalue of a self-adjoint linear operator, and it has a physical/pragmatic/empirical role in actual experiments as a procedure performed using equipment. What is the relationship between these two notions of measurement? The axioms of quantum mechanics don't make it clear.

I don't see that it has anything particularly to do with particles.

There is a pragmatic issue, which is that we don't really know how to measure arbitrary observables. There are only a few that we know how to measure: position, momentum, energy, angular momentum. It might be correct to say that we only know how to measure observables with classical analogs. But you seem to be saying something different, that the measurement problem only comes up because we're insisting on classical analogues. I don't see that, at all. We can certainly pick an observable with no classical analog. The measurement problem doesn't go away, it becomes worse, because we don't know how to measure it.

Well if you reduce the discussion to abstract observables without attributing them to any particular object be it a particle, a field or whatever, you don't have a way to connect it with the physical/pragmatic side so no measurement problem for you, but as Ballentine warned in the quote posted by devil then you don't really have a physical theory but just a set of abstract axioms without connection with experiment.
Quote by Quantum Mechanics - A Modern Development, Leslie E. Ballentine (1998)
Kinematics and Dynamics:
"The results of Ch. 2 constitute what is sometimes called “the formal structure of quantum mechanics”. Although much has been written about its interpretation, derivation from more elementary axioms, and possible generalization, it has by itself very little physical content. It is not possible to solve a single physical problem with that formalism until one obtains correspondence rules that identify particular dynamical variables with particular operators."
It is in those correspondence rules that the problem arises, and depending on how one interprets the Born rule, for instance, you might have a smaller or bigger problem.
I would say the Born rule was devised having the particle picture of classical mechanics in mind. Don't you?
When you were talking about observables were you thinking about them in terms of properties of particles, fields...?

 

[bhobba]

It is in those correspondence rules that the problem arises, and depending on how one interprets the Born rule, for instance, you might have a smaller or bigger problem.

I can't follow you there. I know Ballentine pretty well and what he shows is based on the invarience of probabilities the dynamics follows.

I would say the Born rule was devised having the particle picture of classical mechanics in mind. Don't you?

What do you mean by devised? Historically - probably - but so? We now know it follows from much more general considerations having nothing to do with particles eg Gleason's theorem.

Thanks
Bill

 

[stevendaryl]

Well if you reduce the discussion to abstract observables without attributing them to any particular object be it a particle, a field or whatever, you don't have a way to connect it with the physical/pragmatic side so no measurement problem for you, but as Ballentine warned in the quote posted by devil then you don't really have a physical theory but just a set of abstract axioms without connection with experiment.

I think that's what I intended to say: that the "measurement problem" is about the connection between the notion of "observable" that is a primitive in the quantum theory, and the "observable" that is something that requires a measurement apparatus and a measurement procedure. But I don't see how that supports the claim that the measurement problem has anything to do, intrinsically, with classical properties of particles.

I would say the Born rule was devised having the particle picture of classical mechanics in mind. Don't you?

No, I don't see much of a connection between the two. The Born rule about probabilities, or something like it, is forced on us by the assumption, or empirical fact, that an observation always produces an eigenvalue of the corresponding operator, and that operators don't commute (so it's not possible for all observables to have definite values simultaneously). I don't see that there is anything particularly particle-like about any of this.

When you were talking about observables were you thinking about them in terms of properties of particles, fields...?

They are properties of a system, as a whole. The electric and magnetic field at a point in space is an observable. The mass, position, momentum, magnetic moment of a lump of iron are all observables. Yes, those observables are all macroscopic sums of observables associated with individual atoms of the iron, but the observables don't require particles to make sense of them. So I really don't understand the point you are making about the relationship between observables and particles.

 

[Jano L.]

No. Kolgmorgrov's axioms are clear on this point:
http://en.wikipedia.org/wiki/Probability_axioms
'This is the assumption of unit measure: that the probability that some elementary event in the entire sample space will occur is 1. More specifically, there are no elementary events outside the sample space.'

If something has probability 1 it must occur.

I think it will be easier to explain this by example. Consider random process that produces series of red points somewhere in a unit disk with uniform probability density. The probability of the event that the next point will concide with any point A of the disk is equal to 0.

However, after the event occurs, some point of the disk will be red. At that instant, an event with probability 0 has happened.

Actually, all events that happen in such random process are events that have probability 0.

So "event has probability 0" does not mean "impossible event".

Similarly, "probability 1" does not mean "certain event". Consider probability that the red point will land at point with both coordinates irrational. This can be shown to be equal to 1 in standard measure theory. However, there is still infinity of points that have rational coordinates, and these can happen - they are part of the disk.

In the language of abstract theory, all this is just a manifestation of the fact that equal measures do not imply that the sets are equal.

 

[Jano L.]

I would say the Born rule was devised having the particle picture of classical mechanics in mind. Don't you?

Very good point. As far as know, there are actually two Born rules, although people tend to think they are the same.

The first rule, well working in scattering and quantum chemistry, is the assumption that

$$
|\psi(\mathbf r)|^2 ~\Delta V,
$$

gives probability that the particle is in the small volume element $\Delta V$ around $\mathbf r$.

This really refers to particles and their configuration.

The second rule, I think proposed after the first one, is that

$$
p_k = |\langle \phi_k|\psi \rangle|^2
$$

gives the probability that the system in state $\psi$ will manifest energy $E_k$ (or get into state $\phi_k$ in other versions) when "measurement of energy" is performed (or even spontaneously, due to interaction with environment in other versions). This is more abstract and does not require particles.

We should really distinguish these two rules. The first one is easy and does not depend on the measurement problem, and is gauge-invariant.

The second is difficult to understand, because it is connected to measurements and is gauge-dependent - if we choose different gauge to calculate $\psi$, we get different $p_k$.

 

[lugita15]

No. Kolgmorgrov's axioms are clear on this point:
http://en.wikipedia.org/wiki/Probability_axioms
'This is the assumption of unit measure: that the probability that some elementary event in the entire sample space will occur is 1. More specifically, there are no elementary events outside the sample space.'

If something has probability 1 it must occur.

It's true that if an event will definitely occur, then it must have probability 1. But it's not the case that if an event has probability 1, it will definitely occur. See this wikipedia page.