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Suppose you have a resolution of the identity $\hat{\mathbb{1}}=\sum_i\hat{p_i}$ (pairwise othogonal), and construct two (non-degenerate) pvm observables, $\hat{B}=\sum_ib_i\hat{p_i}$ and $\hat{C}=\sum_ic_i\hat{p_i}$, both using that same $\hat{p_i}$ resolution, but with $b_i\not=c_i$ (at least not all of them). However, all $b_i,c_i$ have the same kind of units, e.g., length or mass, etc.

Now, if there's some (dimensionless) constant, $k$, such that $c_i=kb_i$, then I think we can clearly interpret $\hat B$ as an observable measured in, say, meters, and interpret $\hat C$ as an observable measuring exactly the same physical quantity in, say, cm.

But now suppose there's no relation between the $b_i,c_i$'s. Mathematically, they're both still canonical observables. But what are they measuring that can be interpreted as physically different? As I understand (or maybe misunderstand?) it, it's the $\hat{p_i}$ resolution of the identity that's the physically significant/interpretable part of the mathematical model. The $b_i,c_i$ eigenvalues are pretty much just labels that tell you which $\hat{p_i}$ subspace contains the state of the system after measurement. In other words, if you have some measuring apparatus for your observable, with a needle that points to the measurement outcome, you just label each possible needle position with a corresponding eigenvalue. So in what way are our $\hat B, \hat C$ observables physically different?

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What you are correctly pointing out is that the calibrated scale attached to a measuring arrangement is arbitrary insomuch as it doesn't change the nature of the thing being measured.

The thing is, conventional Quantum Mechanics identifies observables with self-adjoint operators (or equivalently their associated resolutions of the identity) and this object has the measurement scale $\langle\mathbb{R},\mathscr{B}(\mathbb{R})\rangle$ built into it.

This is convenient (since physics usually deals with measured quantities) but you don't have to live with it.

To remove the measurement scale, the mathematical object you want to think about is a maximal Boolean algebra of projection operators. Basically, you divorce the projection operators in your spectral decomposition from the numerical values attached to them and then flesh that family of projection operators out into a maximal family of yes/no questions.

These questions are not measured against some arbitrary scale, and can represent empirical questions like "what is the probability that the detector will fire" directly (as opposed to asking for the probability of an outcome having a numerical value in some interval).

Physics Footnotes
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  • Thanks, Derek. Yeah, I'm familiar with what you're saying from Chapter 5 of "Foundations of QM" by Jauch, which, given the flavor of your answer, I'm betting you're familiar with, too (do I win?:). So I think, but am not quite sure, you're saying $\hat B,\hat C$ both measure the same physical quantity. Is that right? In which case there's an equivalence class of all observables constructed using the same $\hat{p_i}$ resolution. Right again? Or wrong both times? Or what, exactly? Thanks. –  Jun 20 '16 at 08:10
  • Hey @JohnForkosh. Yes, Jauch, Piron, Beltrametti & Cassinelli, etc., are all familiar to me from asking precisely these sorts of questions when I first met QM from a mathematical point of view. I guess you could use the term equivalence class here, by saying that any spectral resolutions over the same Boolean algebra of projectors are equivalent in the sense that they are measuring (or partially measuring) the same quantity. So basically I would call you right both times ;-) – Physics Footnotes Jun 20 '16 at 08:28
  • Thanks again, Derek. Yeah, Beltrametti&Cassinelli's a really terrific book, too. I'd been struggling for a long time with a different problem, about the lattice of propositions, when I finally came across the covering property on page 98 of B&C. And, poof, problem solved in 10 seconds flat. The only other place I've come across it is Section 4 of http://arxiv.org/abs/1211.5627 which is really quite nice (and the price is right), in case you haven't seen it. And they're both also among the very few places I've ever seen any discussion of the coordinatization problem. Thanks again. –  Jun 20 '16 at 08:40
  • B&C is the clearest book on the mathematical foundations of QM I've ever read. It should be a standard textbook even now, 35 years later! And I'll check out the link John - thanks. – Physics Footnotes Jun 20 '16 at 08:45
  • Yeah, B&C's great, but I'd have suggested they significantly expand their teeny-weeny discussion of measurable/Borel sets (along with Lebesgue integration over $\mu$-measurable functions, etc, etc). Without all that prerequisite background, the naturalness-cum-inevitability of subsequent discussions/definitions about probability spaces, observables, etc, isn't as crystal clear as it could be. –  Jun 21 '16 at 01:43