atyy said:
But Landau and Lifshitz do have the collapse postulate. This is where Ballentine deviates from LL. So if Ballentine's interpretation is "minimal", then LL (and Weinberg, Nielsen and Chuang, etc) is certainly not among them. As for EPR, if the measurement is simultaneous in one frame, it cannot be simultaneous in another frame. The collapse postulate deals with non-simultaneous measurements, and it is correct to use collapse to calculate the probabilities for such results.
http://arxiv.org/abs/1007.3977
Incidentally Ballentine also got the "uncertainty principle" wrong in his 1970 review. There are several "uncertainty principles" coming from the commutation relations. The most common one is, as you say, about what states can be prepared, and is not about simultaneous measurement. However, this does not mean that simultaneous measurement of canonically conjugate observables is possible. Ballentine makes a misleading or wrong remark in his 1970 review, which again the textbooks state - canonically conjugate observables cannot be simultaneously and accurately measured on an arbitrary unknown state. In his 1970 review, Ballentine deliberately questions this standard principle, but his counterexample fails. It is true that in this case the standard theory was not fully worked out until recently, eg
http://arxiv.org/abs/1306.1565. Still the standard heuristic was based on the fact that conjugate observables require different setups to measure. There were exceptions known at Ballentine's time, for special states as given by Park and Margenau. However, where Park and Margenau's exception is valid, Ballentine's is not. It's fine to make mistakes accidentally, but regarding the limitation on simultaneous measurement by non-commutation and in the questioning of the collapse postulate, Ballentine is deliberately contrary to well-known standard theory stated in many undergraduate and graduate textbooks.
Although we discuss this in the wrong subforum, I think it's a very interesting discussion. Perhaps we should open a new thread in the quantum physics forum.
Ok, I always thought from the classical textbooks on QT LL, Dirac, and even good old Sommerfeld's "Atombau und Spektrallinien" are among the books, who don't introduce the collapse argument, but I can be mistaken. I've to look it through more carefully again. But it's very clear that the collapse postulate is unnecessary and leads to the well-known contradictions which were criticized by Einstein, Podolsky, and Rosen in their famous paper.
I'm pretty sure that LL get the uncertainty relation, as it is derived in the standard way in textbooks, right. It's a statement about states. In the minimal interpretation it's clear what this means: Each state encodes the probabilities for any possible measurement on the system. This implies that I need an ensemble of equally but independent prepared systems to represent this state. The usual uncertainty relation (the Heisenberg-Robertson uncertainty relation) uses Born's rule and the positive definiteness of the scalar product of the Hilbert space to derive the following relatin about the standard deviations of observables,
\Delta A \Delta B \geq \frac{1}{2} \left \langle \frac{1}{\mathrm{i}} [\hat{A},\hat{B}] \right \rangle.
The expectation values to calculate the standard deviations and the right-hand side of the inequality have to be taken wrt. to the same (pure or mixed) state. This means you have to use one ensemble to measure observable A and another ensemble to measure B and determine its standard deviations. Then these standard deviations obey the above given uncertainty relation (supposed QT is right, and that's the case as far as we know). In this interpretation, and there is no other interpretation possible given the QT formalism, you never make a measurement of both incompatible observables on the very same system.
More recently, other kinds of measurements than these complete measurements of one observable (or a compatible set of observables) have been considered, socalled "weak or incomplete measurements", and the description of such procedures has been developed in terms of positive operator valued measures. I'm far from being an expert in this topic. As far as I understand, for such measurements, where you measure in some sense two incompatible observables both not precisely but with a certain a-priori inaccuracy. Then one can derive uncertainty-disturbance relations which are different in both their definition and their physical meaning than the Heisenberg-Robertson uncertainty relation. A while ago, I tried to discuss an easily understandable example about spin measurements on neutrons. Nobody ever responded. So it seem not to be very interesting for people in this forum. In the community it's also a quite heated debate, how to interpreat the measurement-disturbance relations; some authors contradict others claiming to the contrary that the usual Heisenberg-Robertson uncertainty relation also holds for certain types of measurement in the sense of a measurement-distrubance relation. I think, however, one has to consider each concrete experimental setup to interpret what's measured and what meaning uncertainty relations of whatever kind may have.
The original meaning of the Heisenberg-Robertson uncertainty relation is, talking about precise measurements of the two observables on then necessarily distinct ensembles, however, is very clear. Ironically Heisenberg's very first paper was wrong in misinterpreting the relation as measurement-disturbance relation. It was made clear on the example of position-momentum uncertainty and the observation of an electron by scattering of light, the "Heisenberg microscope" gedanken experiment. Very shortly after the first paper, Bohr pointed out the interpretation mistake on the very same example very clearly, coming to the conclusion that it's a property of states and has nothing to do with perturbation of the system by measurements.
In fact, if you want to experimentally check the Heisenberg-Robertson uncertainty relation with significance you have to measure both observables (on each separate ensemble) with much higher accuracy than the quantum mechanical standard deviations due to the state preparation! This underlines the more the fact that you must do the measurements of each observable on distinct ensembles.
