Sorry, late reply.
PeterDonis said:
It's not "the general view". It's math.
PeterDonis said:
Again, you're getting it backwards. The uncertainty relations are derived, mathematically, from the fact that the observables do not commute.
Yes, but does deriving something necessarily mean it is directly caused by it? You could argue it doesn't imply much more than an inferential relationship where if you assume something, you can demonstrate that something else holds. Similar to how correlation doesn't imply causation. I am not disagreeing about uncertainty relations being derived from non-commuting nature of observables. I think my line on this, to clarify the original post of mine you replied to, will be something like the non-commutativity and uncertainty relations are more or less tapping into the same phenomena, pwrhaps from just a different kind of angle. But the angle from uncertainty relations personally has always seemed more explanatorily interesting to me.
I think I will have to re-emphasize the point of Busch's work. He is saying that if you loosen the constraints on measurements by allowing them to be noisy or unsharp, you can have "measurements" of position and momentum which minimally disturb each other, and the reason for this is that by injecting noise into the measurements, you are making them respect uncertainty relations.
Again, I am not saying that uncertainty relations are not derived from non-commutativity. But I am arguing that the relationship is maybe more nuanced and goes both ways if, by altering uncertainty, you can minimize disturbance and produce commuting functions which are "smeared" versions of position and momentum observables. What has been identified by Busch is also essentially the same as what is being said in my point about the Husimi distribution.
PeterDonis said:
Again, none of this contradicts anything I've said. The non-commutation of the observables is still behind all of it, because that is what requires there to be a "squeezing parameter" which can take a range of values, specifying how much of the joint uncertainty due to non-commutation of the observables you want to allocate to each one. If two observables commute, there is no "squeezing parameter" because there is no tradeoff to be made; the observables have joint eigenstates in which both can have sharp values.
I understand what you're saying but my concern is this: asked what it would take to get a distribution for both eigenstates, I don't see anything stopping me from just saying "take away the minimum uncertainty constraint". That seems to be the essence of the (in)ability for observables to share common eigenstates in the Husimi picture. There are precisely two limits of the parameter where we have sharp observables of one but not the other. If there was no minimum uncertainty limit, it seems there would be nothing stopping a joint probability distribution of both sharp observables simultaneously sharing common eigenstates. When you reintroduce the uncertainty constraint, it then becomes clear that there must be disturbances purely for the reason that the eigenstates can no longer exist in the same joint probability distributions. Instead, it is only logically possible to represent eigenstate statistics in two separate contexts, and you then can kind of see why the product rule and law of total probability breaks down in quantum mechanics. Valid marginalizations for an observable can only be made in one context, not the other, whereas classical probability allows you to find marginal probabilities from a single joint probability distribution.
Injecting noise into the measurements will then minimize the disturbance because the uncertainty isn't redundant, it has an active role in determining what is allowable. With enough uncertainty you can have genuine joint distributions for position and momentum without requiring disturbance, ones that exist among allowable Husimi distributions, but which just happen to not be the distributions physicists want.
So, from what I can see, it looks like uncertainty relations have significant explanatory value in regard to non-commutativity and incompatibility. I am obviously not making rigorous statements but it just
looks like to me that the minimum uncertainty constraint is having an active role in making the difference for measurability, and so I am inclined to say that the uncertainty relations and non-commutativity are just different ways of tapping into what is fundamentally the same phenomena.
PeterDonis said:
Which chapter number is that?
15.
PeterDonis said:
The paper you reference here is not limited to QM, but is about a generalized formulation that covers other models--models which are less fundamental than QM (and which ultimately have to be approximations to QM). I don't think you can conclude anything about fundamentals like the "origin" of something from this.
Well, the quantum mechanical case exists within the generalized case so it would make it seem valid to me. This generalized formulation may be applied to descriptions of systems less fundamental
physically but if quantum mechanical relations are also a special case then I think arguably it
could be talking about something which is
mathematically more fundamental.