I Measurement results and two conflicting interpretations

[edmund cavendish]

Sir Roger Penrose posits a new physics in the area between classical and quantum. The penrose diosi theorum suggests observer independent gravitationally induced collapse of the wave function. Might this have some experimental validity?

 

[vanhees71]

No; this was never true. You were misreading the concept of a beable. Maybe this is the source of our continuing misunderstandings.

According to Bell, a beable of a system is a property of the system that exists and is predictable independently of whether one measures anything. A measurement is the reading of a measurement value from a measurement device interacting with the system that is guaranteed to approximate the value of a beable of that system within the claimed accuracy of the measurement device.

In classical mechanics, the exact positions and momenta of all particles in the Laplacian universe are the beables. and a measurement device for a particular prepared particle is a macroscopic body coupled to this particle, with a pointer such that the pointer reading approximates some position coordinate in a suitable coordinate system. Clearly, any given measurement never yields the exact position but only an approximation of it.

In a Stern-Gerlach experiment with a single particle, the beables are the three real-valued components of the q-expectation $\bar S$ of the spin vector $S$, and the location of the spot produced on the screen is the pointer. Because of the semiclassical analysis of the experimental arrangement, the initial beam carrying the particle splits in the magnetic field into two beams, hence only two spots carry enough intensity to produce a response. Thus the pointer can measure only a single bit of $\bar S$. This is very little information, whence the error is predictably large.

The thermal interpretation predicts everything: the spin vector, the two beams, the two spots, and the (very low) accuracy with which these spots measure the beable $S_3$.

Then "beable" is an empty phrase, because we cannot know about any property of a system without measuring (or observing) it. Physics deals with what's observable and measuarable and not philosophical fictitions that cannot be observed and measured.

In classical mechanics as well as in quantum mechanics the positions and momenta of all particles are observables that can (in principle) always be measured as precisely as you wish. There's no difference between classical and quantum mechanics here (of course with the caveat that there are no point particles existing and thus to measure the "position" must sometimes be specified more precisely, but let's keep it at the simple level).

Spin is in some sense special, because it's an observable without a classical analogon. Within non-relativistic quantum theory spin is an observable axial-vector quantity with the properties of an angular momentum. The three components are thus observables (within non-relativistic QT). I don't see, why you emphasize the semiclassical description (i.e., keeping the description of the external magnetic field classical) so much. That's unimportant for the very general issue we are discussing here. I've never thought about a quantum-field theoretical description of the SGE. It's perhaps even an interesting formal question, but it's totally irrelevant for the issues we are discussing here.

Now, in the SGE what's measured is one of the spin components (or to be very precise the component of the magnetic moment related to this spin component) wrt. the direction determined by the magnetic field. The pointer variable is the position of the particle, the inhomogeneous magnetic field designed such that it leads to a (nearly) 100% entanglement between position and this spin component. In this way it's even a preparation procedure for this spin component (or in other words the polarization state of the particle). The error is practically negligible, if the experiment is well designed.

I think, already the interpretation of the SGE in terms of the thermal interpretation is orthogonal to what's really achieved in the lab already in 1922 by Stern and Gerlach, while the minimal standard interpretation precisely describes these findings very well.

 

[vanhees71]

If I find the result $+1/2$ or $-1/2$ with certainty, I can be sure that the measurement error according to the thermal interpretation is exactly $1/2$, since this is the absolute value of the difference between the measured value and the true value (defined in the thermal interpretation to be the q-expectation $0$). As a consequence, I can be sure that the standard deviation of the measurement results is also exactly $1/2$.

This contradicts the empirical outcome of the SGE. Even in the lab at university with its rather limited budget you can get very well separated beams of Ag atoms!

 

[A. Neumaier]

Even in the lab at university with its rather limited budget you can get very well separated beams of Ag atoms!

Yes, but the spots are supposed to measure a very tiny spin of the order of $\hbar$. On a scale where the positions of the two silver spots represent the numbers $\pm \hbar/2$, these positions are approximations of any number of the order of $\hbar$ with an error of the order $\hbar$, in particular, one of the q-expectation of the spin, as the thermal interpretations claims.

This contradicts the empirical outcome of the SGE.

There cannot be a contradiction in cases where no significant relative accuracy is claimed!