Exploring the Fundamental Postulates of QM: Are They Truly Ad-Hoc and Strange?

[stevendaryl]

The idea that I used as the starting point is that a theory must be able to assign probabilities to statements of the form

"If you use the measuring device \delta on the object \pi, the result will be in the set E".​

So your approach is to let "measuring device" be an abstract term. But what is supposed to be the interpretation? Suppose we have as a simple case a universe consisting of nothing but a single spin-1/2 particle fixed in place (so the only degrees of freedom are from spin). I assume that the "observables" in this case are associated with the set of 2x2 hermitian matrices. Which means, in terms of Pauli spin matrices \sigma_i, that they are of the form:
A + B_i \sigma_i, where A, B_x, B_y, B_z are 4 real numbers. So for this toy theory, each such matrix is a measuring device?

 

[Fredrik]

So your approach is to let "measuring device" be an abstract term. But what is supposed to be the interpretation? Suppose we have as a simple case a universe consisting of nothing but a single spin-1/2 particle fixed in place (so the only degrees of freedom are from spin). I assume that the "observables" in this case are associated with the set of 2x2 hermitian matrices. Which means, in terms of Pauli spin matrices \sigma_i, that they are of the form:
A + B_i \sigma_i, where A, B_x, B_y, B_z are 4 real numbers. So for this toy theory, each such matrix is a measuring device?

The set Δ whose members I call "measuring devices" contains elements that correspond to the actual measuring devices mentioned by the theory's correspondence rules. But I do not require that every element of Δ corresponds to an actual measuring device. We can take Δ to be a larger set, if that's convenient.

In a quantum theory defined by a 2-dimensional Hilbert space, the set of self-adjoint operators is our Δ/~ (i.e. the set of all equivalence classes of measuring devices). This is a 4-dimensional vector space over ℝ, that's spanned by $\{\sigma_1,\sigma_2,\sigma_3,I\}$. The sigmas correspond to measuring devices that measure spin in one of three orthogonal directions. The identity matrix corresponds to a measuring device that always gives us the result 1, no matter how the system was prepared before the measurement. Since every self-adjoint operator is a linear combination of these four, every self-adjoint operator corresponds to an actual measuring device (assuming that linear combinations of observables make sense).

Regarding the meaning of linear combinations... I defined scalar multiplication earlier. I haven't really thought addition through. Strocchi appears to be doing something like this: If we denote the expectation value of an observable X by E(X|s) when the system is in state s, then addition can be defined by saying that A+B is the observable such that E(A+B|s)=E(A|s)+E(B|s) for all states s. (I haven't verified that this definition works).

 

[kith]

To me, the strangeness is already put in at the very beginning, when a "measurement" is given fundamental status in the axioms. As I said, a "measurement" is not a fundamental, atomic entity, but is a special kind of interaction whereby the state of one system (the observer, or measuring device) becomes correlated, in a persistent way, with the state of another system (the thing being observed or measured).

As you know, physics obtains its knowledge by measurements. So first of all, the resulting theories are theories about measurements. In QM, it is not straightforward how to extrapolate the theory about measurements to a theory about what "really happens". From the viewpoint of common forms of realism, this is a problem of course. But why should we expect such a straightforward extrapolation in the first place?

Also conceptually, the Many Worlds interpretation is quite straightforward and tells us what really happens. However, it's still hard to accept from the viewpoint of naive realism.

 

[stevendaryl]

As you know, physics obtains its knowledge by measurements.

Sure.

So first of all, the resulting theories are theories about measurements.

I don't think that follows at all. That's like saying: "Nowadays, many people learn about physics over the internet. So for them, a theory of physics is a theory of web browsers."

We learn about physics by measurements, but measurements are not the subject of physics. (Well, there can certainly be a subfield of physics, the theory of measurement, but that's not all of physics.) We use measurements to figure out how the world works, and then we apply that knowledge in situations where there are no measurements around--such as the Earth prior to the formation of life, or inside a star, or whatever.

I absolutely reject the assumption that a theory of physics is a theory of measurement.

 

[bhobba]

To me, the strangeness is already put in at the very beginning, when a "measurement" is given fundamental status in the axioms. As I said, a "measurement" is not a fundamental, atomic entity, but is a special kind of interaction whereby the state of one system (the observer, or measuring device) becomes correlated, in a persistent way, with the state of another system (the thing being observed or measured).

Then, I suspect, to you, that entanglement basically leads to QM would be a very pertinent aspect.

My view is a few approaches with reasonable foundations lead to QM but unfortunately, like the paper I linked, require a certain amount of mathematical sophistication such as the Schur-Auerbach lemma from group theory. Unfortunately there are some people who don't like this mathematical aspect of physical theories and in some quarters there is a resistance to it with for example claims SR is simply math and can't represent physical reality. I had long discussions (if that's what you would call them) with people of that bent when I posted a lot on sci.physics.relativity - they just simply can't get the idea that physics is not about easily visualizeable pictures they carry around in their head.

Thanks
Bill

 

[kith]

That's like saying: "Nowadays, many people learn about physics over the internet. So for them, a theory of physics is a theory of web browsers."

This analogy is valid if you make the assumption that measurements uncover an independent reality. But this is exactly the assumption I questioned in my previous post.

 

[bhobba]

As you know, physics obtains its knowledge by measurements. So first of all, the resulting theories are theories about measurements.

Yes of course. But that in itself raises a fundamental issue - measurement apparatus are classical objects and QM is a fundamental theory about the constituents of those classical objects so we have a 'cut' in how we view nature right at the foundations of QM. This leads to stuff like the Von Newmann regress and the introduction of consciousness causes collapse most would think a bit too far out to be taken seriously. My view is a fully quantum theory of measurement is required and indeed much progress in that area has been made but a few issues still remain such as proving the basis singled out by decoherence does not depend on how the system is decomposed. I believe we are not far away from a full resolution but until all the i's are dotted at t's crossed I for one still think some mystery remains. And who knows - dotting the i's and crossing the t's may show up something truly surprising.

Thanks
Bill

 

[bhobba]

This analogy is valid if you make the assumption that measurements uncover an independent reality. But this is exactly the assumption I questioned in my previous post.

Bingo - we have a winner. That is the rock bottom foundational issue with QM IMHO.

Thanks
Bill

 

[kith]

Bill, similar to your thoughts, I think that the universal wavefunction and decoherence give a quite complete realistic picture. It is just that I also see the appeal in the C* approach which is much closer to the scientific practise than unobservable entities like the universal wavefunction.

 

[Fredrik]

Yes of course. But that in itself raises a fundamental issue - measurement apparatus are classical objects and QM is a fundamental theory about the constituents of those classical objects so we have a 'cut' in how we view nature right at the foundations of QM. This leads to stuff like the Von Newmann regress and the introduction of consciousness causes collapse most would think a bit too far out to be taken seriously. My view is a fully quantum theory of measurement is required and indeed much progress in that area has been made but a few issues still remain such as proving the basis singled out by decoherence does not depend on how the system is decomposed. I believe we are not far away from a full resolution but until all the i's are dotted at t's crossed I for one still think some mystery remains. And who knows - dotting the i's and crossing the t's may show up something truly surprising.

What do you mean by a fully quantum theory of measurement, if QM isn't one already? (Keep in mind that QM includes decoherence). And what is it required for?

The von Neumann regress, if you mean what I think you mean, doesn't have anything to do with the consciousness causes collapse idea. The former is just the observation about what a theory is, and the latter is at best a wild speculation about reality.

I think the idea that the basis is independent of the decomposition is as likely to be true as the idea that 2x is independent of x.

 

[bhobba]

What do you mean by a fully quantum theory of measurement, if QM isn't one already? (Keep in mind that QM includes decoherence). And what is it required for?

A theory of measurement that does not include the a priori existence of classical measurement devices like Copenhagen does. It is only of recent times such theories have emerged - but as yet have not been full worked out eg my understanding is that the emergence of a classical domain from QM is not quite 100% complete - we are almost there - but not quite - at least that's what I have read.

The von Neumann regress, if you mean what I think you mean, doesn't have anything to do with the consciousness causes collapse idea. The former is just the observation about what a theory is, and the latter is at best a wild speculation about reality.

Von Neumann was one of the first, or maybe even the first, to examine the measurement process fully quantum mechanically. What that showed is where the wavefunction collapse occurs could be placed anywhere and if we keep following it all the way back to the observer it is only at consciousness something different comes into it. To some such as Wigner this is where they placed the collapse:
http://en.wikipedia.org/wiki/Interp...rpretation:_consciousness_causes_the_collapse

'In his treatise The Mathematical Foundations of Quantum Mechanics, John von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the Schrödinger equation (the universal wave function). He also described how measurement could cause a collapse of the wave function. This point of view was prominently expanded on by Eugene Wigner, who argued that human experimenter consciousness (or maybe even dog consciousness) was critical for the collapse, but he later abandoned this interpretation'

I have an aged copy of Von Newmann's text, and while it been years since I have read it (its actually one of the first books I learned QM from because it was mathematically more in line with the Hilbert spaces I studied in my math degree - other texts were not quite as transparent to me until I learned a bit about Rigged Hilbert Spaces - but that is another issue) I seem to recall that's pretty much what he did.

Wigner abandoned it when he heard of some early work on decoherence by Zurek.

I think the idea that the basis is independent of the decomposition is as likely to be true as the idea that 2x is independent of x.

And you may be right - however I prefer not to be so sure about it until the theorems demonstrating it unequivocally one way or another are forthcoming and they have had time to be checked. It is interesting standard texts on dechorenece such as the one I have make no mention of it - there may be something already known about it we are missing - I am adopting a wait and see attitude to it..

Thanks
Bill